--- /srv/rebuilderd/tmp/rebuilderd4sBUEK/inputs/macaulay2-common_1.26.06+ds-5_all.deb +++ /srv/rebuilderd/tmp/rebuilderd4sBUEK/out/macaulay2-common_1.26.06+ds-5_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2026-10-24 01:47:31.000000 debian-binary │ --rw-r--r-- 0 0 0 567932 2026-10-24 01:47:31.000000 control.tar.xz │ --rw-r--r-- 0 0 0 33668792 2026-10-24 01:47:31.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 567944 2026-10-24 01:47:31.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 33670328 2026-10-24 01:47:31.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.26.06+ds-5 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 324507 │ │ │ +Installed-Size: 324508 │ │ │ Depends: fonts-katex (>= 0.16.10+~cs6.1.0), libjs-katex (>= 0.16.10+~cs6.1.0) │ │ │ Section: math │ │ │ Priority: optional │ │ │ Multi-Arch: foreign │ │ │ Homepage: http://macaulay2.com │ │ │ Description: Software system for algebraic geometry research (common files) │ │ │ Macaulay 2 is a software system for algebraic geometry research, written by │ │ ├── ./md5sums │ │ │ ├── ./md5sums │ │ │ │┄ Files differ ├── data.tar.xz │ ├── data.tar │ │ ├── file list │ │ │ @@ -3610,28 +3610,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7817 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1918 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1748 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1747 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_expected__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_linear__System__On__Rational__Surface.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1889 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1596 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_slow__Adjunction__Calculation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2944 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_special__Families__Of__Sommese__Vande__Ven.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 9411 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11522 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_expected__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_linear__System__On__Rational__Surface.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9773 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9772 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9934 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9512 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_slow__Adjunction__Calculation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12677 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_special__Families__Of__Sommese__Vande__Ven.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13977 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AlgebraicSplines/ │ │ │ @@ -3867,18 +3867,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 77417 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4425 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3108 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4444 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289778 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -4524,26 +4524,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells.out │ │ │ -rw-r--r-- 0 root (0) root (0) 574 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells_lp__Z__Z_cm__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1427 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_complex_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 539 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_describe_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 266 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_dim_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 291 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_dim_lp__Cell_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_face__Poset_lp__Cell__Complex_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_hull__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 961 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_hull__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 446 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Cycle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 257 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Free_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Simplex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Well__Defined_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_is__Well__Defined_lp__Cell_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_max__Cells.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_net_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_new__Cell.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_new__Simplex__Cell.out │ │ │ --rw-r--r-- 0 root (0) root (0) 741 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 734 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_ring_lp__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 519 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_skeleton_lp__Z__Z_cm__Cell__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_subcomplex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 821 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_taylor__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 524 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_tex__Math_lp__Cell__Complex_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/ │ │ │ @@ -4568,26 +4568,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7989 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells_lp__Z__Z_cm__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13005 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_complex_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_describe_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_dim_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6576 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_dim_lp__Cell_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7340 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_face__Poset_lp__Cell__Complex_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8613 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8611 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6824 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Cycle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6966 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Free_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7587 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Minimal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Simplex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8375 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Well__Defined_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_is__Well__Defined_lp__Cell_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6644 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_max__Cells.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_net_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8648 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_new__Cell.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7546 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_new__Simplex__Cell.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8801 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8794 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_ring_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8618 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_skeleton_lp__Z__Z_cm__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8967 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_subcomplex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6783 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_taylor__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7312 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/_tex__Math_lp__Cell__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 28743 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CellularResolutions/html/index.html │ │ │ @@ -4691,49 +4691,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12305 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4872 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123136 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4376 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4375 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3442 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Toric__Chow__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ --rw-r--r-- 0 root (0) root (0) 341 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ --rw-r--r-- 0 root (0) root (0) 760 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2014 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 759 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 666 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 624 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 623 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1049 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Multi__Proj__Coord__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Output.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3327 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Toric__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_is__Multi__Homogeneous.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_probabilistic_spalgorithm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 55 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 23815 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11361 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Toric__Variety__Valid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chern.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9699 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Toric__Chow__Ring.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10506 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18685 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6076 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6025 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10504 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18683 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6077 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6024 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6134 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6588 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6587 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Multi__Proj__Coord__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16520 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Segre.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Toric__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5032 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_bertini__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_configuring_sp__Bertini.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_is__Multi__Homogeneous.html │ │ │ @@ -4748,15 +4748,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2624 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Chordal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Ring__Element_sp_pc_sp__Chordal__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1273 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/___Ring__Map_sp__Chordal__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 178 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_adjacent__Minors__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2151 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Elim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 969 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1787 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1202 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1196 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Tria.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2559 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal_spnetworks_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 429 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_chromatic__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 237 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_codim__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1454 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_components_lp__Chordal__Net_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_constraint__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/example-output/_digraph_lp__Chordal__Net_rp.out │ │ │ @@ -4789,15 +4789,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4099 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Get__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Ring__Element_sp_pc_sp__Chordal__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7440 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/___Ring__Map_sp__Chordal__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5263 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_adjacent__Minors__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Elim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7480 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8536 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8530 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9053 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Tria.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11044 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chordal_spnetworks_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5626 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_chromatic__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7282 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_codim__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8333 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_components_lp__Chordal__Net_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5913 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_constraint__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7887 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Chordal/html/_digraph_lp__Chordal__Net_rp.html │ │ │ @@ -4990,23 +4990,23 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 36669 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26692 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15442 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CodingTheory/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 13418 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 13259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1045 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 982 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/example-output/_cohom__Calg_lp__Normal__Toric__Variety_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/___Silent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10305 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8623 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/_cohom__Calg_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25757 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5957 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3766 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CohomCalg/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 108157 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/___Coincident__Root__Locus_sp_st_sp__Coincident__Root__Locus.out │ │ │ @@ -5089,15 +5089,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__G__G__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1959 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Complete__Intersection__Resolutions.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4598 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash__Total.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2743 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1023 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Ext__Module__Data.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2102 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___S2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1754 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 761 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Tate__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_complexity.out │ │ │ @@ -5121,24 +5121,24 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1851 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 857 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1336 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10478 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_new__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1284 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_odd__Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_regularity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1314 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_splittings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 377 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ --rw-r--r-- 0 root (0) root (0) 449 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 379 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 450 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___A__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5136 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Augmentation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6508 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__G__G__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9295 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___B__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6167 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Check.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15460 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15457 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash__Total.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10528 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Ext__Module__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5485 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5732 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Hom__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5331 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Layered.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Lift.html │ │ │ @@ -5183,18 +5183,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_module__As__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23243 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_new__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8937 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_odd__Ext__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6089 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_psi__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_regularity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7584 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_splittings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5425 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_stable__Hom.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5774 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_tensor__With__Components.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5070 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_to__Array.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 56230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35572 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Complexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 740220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Complexes/example-output/ │ │ │ @@ -5548,29 +5548,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9600 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 55660 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 36621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Gauss_sq_sphypergeometric_spfunction.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3309 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3310 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_base__Fraction__Field.out │ │ │ -rw-r--r-- 0 root (0) root (0) 448 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_connection__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 905 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_gauge__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_gauge__Transform.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_is__Epsilon__Factorized.out │ │ │ -rw-r--r-- 0 root (0) root (0) 395 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_is__Integrable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 472 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_normal__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 978 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_pfaffian__System.out │ │ │ -rw-r--r-- 0 root (0) root (0) 347 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/_standard__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 46697 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8055 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Gauss_sq_sphypergeometric_spfunction.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9429 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9430 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5833 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_base__Fraction__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_connection__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9078 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8897 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_gauge__Transform.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7864 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_is__Epsilon__Factorized.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_is__Integrable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8418 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConnectionMatrices/html/_normal__Form.html │ │ │ @@ -5792,136 +5792,136 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 38839 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CpMackeyFunctors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 239248 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2310 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Codim__Bs__Inv.out │ │ │ --rw-r--r-- 0 root (0) root (0) 19789 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out │ │ │ --rw-r--r-- 0 root (0) root (0) 527 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 19785 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 526 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1795 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2551 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2314 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp_eq_eq_sp__Rational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2465 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp_st_sp__Rational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2051 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5318 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp_vb_sp__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3568 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp_vb_vb_sp__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6045 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out │ │ │ --rw-r--r-- 0 root (0) root (0) 42548 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6051 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 42549 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1530 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_coefficients_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 33238 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_describe_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 515 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_exceptional__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1047 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_flatten_lp__Rational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 492 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6559 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_graph_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4857 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11552 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 46184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1572 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3447 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Isomorphism_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 927 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Morphism.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6113 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1086 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_map_lp__Rational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 18540 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 18541 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4711 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4713 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_quadro__Quadric__Cremona__Transformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3884 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2826 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5782 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Polynomial__Ring_cm__List_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6394 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_segre.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6867 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cremona__Transformation.out │ │ │ --rw-r--r-- 0 root (0) root (0) 23419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cubic__Transformation.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Quadratic__Transformation.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6868 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cremona__Transformation.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 23418 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cubic__Transformation.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3632 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Quadratic__Transformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1876 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_substitute_lp__Rational__Map_cm__Polynomial__Ring_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1509 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_super_lp__Rational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_to__External__String_lp__Rational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1341 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_to__External__String_lp__Rational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4842 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/example-output/_to__Map.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Blow__Up__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6275 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Certify.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11951 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11950 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5616 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Codim__Bs__Inv.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4254 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Dominant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8724 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8723 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Num__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7897 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5086 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9371 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp_eq_eq_sp__Rational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8642 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp_st_sp__Rational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8516 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5564 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp_us_st.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp_vb_sp__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9873 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp_vb_vb_sp__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17785 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17163 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 52131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17169 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 52132 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5099 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_coefficient__Ring_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7778 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_coefficients_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 41012 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4920 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_degree_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_degrees_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_describe_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5224 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_entries_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_exceptional__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6490 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_flatten_lp__Rational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6761 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6762 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5942 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_force__Inverse__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14385 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_graph.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_graph_lp__Ring__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11059 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_ideal_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6238 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_image_lp__Rational__Map_cm__String_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6296 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_image_lp__Rational__Map_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 54044 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_inverse_lp__Rational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8464 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8463 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5564 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_is__Inverse__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6626 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_is__Inverse__Map_lp__Rational__Map_cm__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7314 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_is__Isomorphism_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_is__Morphism.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13299 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13298 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7739 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_map_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6065 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_matrix_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5138 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_parametrize.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_parametrize_lp__Ideal_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24583 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_parametrize_lp__Ideal_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4806 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7766 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_point_lp__Quotient__Ring_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14919 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14921 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6855 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10515 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_quadro__Quadric__Cremona__Transformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10697 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12834 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Polynomial__Ring_cm__List_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14313 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14315 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8810 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_segre.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_source_lp__Rational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13965 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html │ │ │ --rw-r--r-- 0 root (0) root (0) 30512 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10894 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13966 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 30511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10893 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8931 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_substitute_lp__Rational__Map_cm__Polynomial__Ring_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_super_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5196 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_target_lp__Rational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7139 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7138 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/_to__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 66239 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 66235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37204 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16290 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cremona/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cyclotomic/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cyclotomic/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 6101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cyclotomic/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cyclotomic/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 455 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cyclotomic/example-output/_cyclotomic__Field.out │ │ │ @@ -5939,15 +5939,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3784 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Cyclotomic/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 881585 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2851 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Base_spchange_spand_sptensor_spwith_spnon-__D__G_sptypes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7201 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2706 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Module_sp__Maps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 644 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Building_sp__D__G_spalgebras_spfrom_spexisting_sp__D__G_spalgebras.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1278 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Building_sp__D__G_spmodules_cm_spsubmodules_cm_spand_spquotients.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Computing_spmodule_spdifferentials_spand_spvisualizing_sp__D__G_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1841 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Algebra__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Algebra__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 884 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Algebra_sp_sl_sp__D__G__Ideal.out │ │ │ @@ -5971,31 +5971,31 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Quotient__Module_sp_st_st_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Quotient__Module_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 938 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule_sp_eq_eq_sp__D__G__Submodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1428 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule_sp_pl_sp__D__G__Submodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule_sp_st_st_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___D__G__Submodule_sp_st_st_sp__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 601 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 600 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1463 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1882 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1509 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1849 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Quotient__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_us__Z__Z_sp__D__G__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_us__Z__Z_sp__D__G__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1966 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_us__Z__Z_sp__D__G__Quotient__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1588 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Homology_spof_sp__D__G_spmodules_spand_sp__D__G_spmodule_spmaps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1900 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Image_cm_spkernel_cm_spand_spcokernel_spof_sp__D__G_spmodule_spmaps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1758 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Module-like_spoperations_spon_sp__D__G_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Operations_spon_sp__D__G_sp__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Operations_spon_sp__D__G_sp__Submodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1201 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Pruning_sp__D__G_spmodules_cm_spsubmodules_cm_spquotients_cm_spand_spmaps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Ring__Element_sp_pc_sp__D__G__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1644 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Semifree_spresolutions_spof_sp__D__G_spmodules.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5183 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5179 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1320 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Well-definedness_cm_spacyclicity_cm_spand_spquasi-isomorphism.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_acyclic__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1894 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_acyclic__Closure_lp..._cm__End__Degree_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_acyclic__Closure_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_adjoin__Generators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_adjoin__Variables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_ambient.out │ │ │ @@ -6029,18 +6029,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 830 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_generator__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1151 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_generator__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 683 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Basis.out │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Basis_lp__Z__Z_cm__D__G__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1392 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Boundary__Preimage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1429 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Boundary__Preimage_lp__D__G__Module_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2072 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Boundary__Preimage_lp__D__G__Module_cm__Vector_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Deg__N__Module.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 860 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Deg__N__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1535 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Generators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out │ │ │ --rw-r--r-- 0 root (0) root (0) 854 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 852 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1583 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class_lp__D__G__Module_cm__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3481 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology_lp__D__G__Module__Map_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_identity__D__G__Algebra__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2324 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_identity__D__G__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_image_lp__D__G__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_intersect_lp__D__G__Ideal_cm__D__G__Ideal_rp.out │ │ │ @@ -6132,15 +6132,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1248 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_zeroth__Homology.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 25 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8690 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Accessors_spand_spcache_spmanagement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6021 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Assert__Well__Defined.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14858 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Base_spchange_spand_sptensor_spwith_spnon-__D__G_sptypes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17579 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17580 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Module_sp__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Building_sp__D__G_spalgebras_spfrom_spexisting_sp__D__G_spalgebras.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Building_sp__D__G_spmodules_cm_spsubmodules_cm_spand_spquotients.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11694 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Computing_spmodule_spdifferentials_spand_spvisualizing_sp__D__G_spmodules.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___D__G__Algebra.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14042 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___D__G__Algebra__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5535 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___D__G__Algebra__Map_sp_st_sp__D__G__Algebra__Map.html │ │ │ @@ -6172,17 +6172,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9904 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___D__G__Submodule_sp_eq_eq_sp__D__G__Submodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9469 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___D__G__Submodule_sp_pl_sp__D__G__Submodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9266 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___D__G__Submodule_sp_st_st_sp__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___D__G__Submodule_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10240 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___D__G_spmodule_spprimitives_spand_spchain-complex_spexport.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9385 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___End__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8611 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Gen__Degree__Limit.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7677 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8701 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11688 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7676 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8702 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11689 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Quotient__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5692 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_us__Z__Z_sp__D__G__Algebra.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11962 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_us__Z__Z_sp__D__G__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11073 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_us__Z__Z_sp__D__G__Quotient__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12629 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Homology_spof_sp__D__G_spmodules_spand_sp__D__G_spmodule_spmaps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13680 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Image_cm_spkernel_cm_spand_spcokernel_spof_sp__D__G_spmodule_spmaps.html │ │ │ @@ -6192,15 +6192,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Operations_spon_sp__D__G_sp__Submodules.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11082 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Pruning_sp__D__G_spmodules_cm_spsubmodules_cm_spquotients_cm_spand_spmaps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Rel__Degree__Limit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7440 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Ring__Element_sp_pc_sp__D__G__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14350 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Semifree_spresolutions_spof_sp__D__G_spmodules.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Start__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6388 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___T__M__O__Limit.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13876 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13872 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7794 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Verbosity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13366 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/___Well-definedness_cm_spacyclicity_cm_spand_spquasi-isomorphism.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18319 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_acyclic__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_acyclic__Closure_lp..._cm__End__Degree_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_acyclic__Closure_lp..._cm__Start__Degree_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10049 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_acyclic__Closure_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14348 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_acyclic__Closure_lp__Ring_rp.html │ │ │ @@ -6245,24 +6245,24 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Basis.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6613 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Basis_lp..._cm__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9023 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Basis_lp__Z__Z_cm__D__G__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5783 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Basis_lp__Z__Z_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9447 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Boundary__Preimage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Boundary__Preimage_lp__D__G__Module_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12589 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Boundary__Preimage_lp__D__G__Module_cm__Vector_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9036 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Deg__N__Module.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9037 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Deg__N__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Generators.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10218 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Generators_lp..._cm__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8111 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Generators_lp..._cm__Start__Degree_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7930 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Generators_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13909 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7756 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra_lp..._cm__Gen__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6537 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra_lp..._cm__Rel__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7847 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7738 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7736 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10863 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class_lp__D__G__Module_cm__Vector_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10800 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13061 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology_lp__D__G__Module__Map_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6572 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_identity__D__G__Algebra__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10178 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_identity__D__G__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10402 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_image_lp__D__G__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/DGAlgebras/html/_intersect_lp__D__G__Ideal_cm__D__G__Ideal_rp.html │ │ │ @@ -6791,24 +6791,24 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5144 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3271 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EigenSolver/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 14843 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 891 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 892 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 892 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 890 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 9078 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 9079 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/example-output/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 7465 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7965 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16623 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16009 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16010 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Elimination/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 98399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EliminationMatrices/example-output/ │ │ │ @@ -7025,37 +7025,37 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EngineTests/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8600 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EngineTests/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7192 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EngineTests/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 11703 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 366 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 367 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_multiple__Cover.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2168 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2170 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 48 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5876 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5877 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_multiple__Cover.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12190 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12192 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5731 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3654 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EnumerationCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52732 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Equivariant__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/___Priority__Queue.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Monomial__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 576 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 346 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_build__E__Ring_lp__Ring_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_delete__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 348 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_exponent__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 449 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_inc__Orbit.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_insert_lp__Priority__Queue_cm__Thing_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 248 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_length_lp__Priority__Queue_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_merge__P__Q.out │ │ │ -rw-r--r-- 0 root (0) root (0) 161 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_min_lp__Priority__Queue_rp.out │ │ │ @@ -7070,15 +7070,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/___Shift.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4045 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/___Symmetrize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9278 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_build__E__Monomial__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_build__E__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6808 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_build__E__Ring_lp__Ring_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6096 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_delete__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6673 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9197 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9199 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6816 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6327 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_exponent__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_inc__Orbit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_insert_lp__Priority__Queue_cm__Thing_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_length_lp__Priority__Queue_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6679 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_merge__P__Q.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/EquivariantGB/html/_min_lp__Priority__Queue_rp.html │ │ │ @@ -7353,53 +7353,53 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FGLM/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4746 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FGLM/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3159 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FGLM/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 145597 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 26064 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 26068 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Point__Options.out │ │ │ --rw-r--r-- 0 root (0) root (0) 19478 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 19477 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1053 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out │ │ │ -rw-r--r-- 0 root (0) root (0) 337 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Good__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Random__Submatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Largest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_choose__Submatrix__Smallest__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 533 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_get__Submatrix__Of__Rank.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1787 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1786 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out │ │ │ -rw-r--r-- 0 root (0) root (0) 275 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Rank__At__Least.out │ │ │ --rw-r--r-- 0 root (0) root (0) 437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 436 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out │ │ │ --rw-r--r-- 0 root (0) root (0) 26534 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 26535 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 273 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/example-output/_reorder__Polynomial__Ring.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6309 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Det__Strategy.html │ │ │ --rw-r--r-- 0 root (0) root (0) 47081 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 47085 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5548 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Max__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Min__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4622 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Modulus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6983 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Point__Options.html │ │ │ --rw-r--r-- 0 root (0) root (0) 33597 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 33596 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14937 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4042 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/___Verify__Non__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9365 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Good__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5860 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Random__Submatrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6494 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Submatrix__Largest__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6410 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_choose__Submatrix__Smallest__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10290 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_get__Submatrix__Of__Rank.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11450 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11449 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Dim__At__Most.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9681 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5539 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_is__Rank__At__Least_lp..._cm__Threads_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10381 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html │ │ │ --rw-r--r-- 0 root (0) root (0) 46337 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 46338 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/_reorder__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25040 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7776 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FastMinors/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 26090 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ @@ -7642,15 +7642,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 105269 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Bounds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Frobenius__Thresholds.out │ │ │ -rw-r--r-- 0 root (0) root (0) 793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/___Guess__Strategy.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_compare__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4034 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2460 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 760 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__Jumping__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__F__P__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 828 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_is__Simple__Normal__Crossing.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 617 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 12 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5732 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Bounds.html │ │ │ @@ -7662,15 +7662,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10699 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Guess__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4815 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Return__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4919 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Search.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4885 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Standard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Use__Special__Algorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14844 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_compare__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ --rw-r--r-- 0 root (0) root (0) 24663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__Jumping__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10037 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__Simple__Normal__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20074 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7138 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/ │ │ │ @@ -7722,15 +7722,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 732 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 170 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_normal__Toric__Variety_lp__G__K__M__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8063 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pullback_lp__Equivariant__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pushforward.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_set__Indicator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_trivial__K__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_underlying__Graph_lp__Moment__Graph_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/ │ │ │ @@ -7773,15 +7773,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9369 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7658 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7819 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_normal__Toric__Variety_lp__G__K__M__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 20015 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pullback_lp__Equivariant__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7948 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pushforward.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7166 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_set__Indicator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5918 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_trivial__K__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5739 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -8841,42 +8841,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 71515 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Graphs/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 57702 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Graphs/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27654 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Graphs/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56750 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 33163 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 33567 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Constraints.out │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Vector.out │ │ │ -rw-r--r-- 0 root (0) root (0) 30368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2709 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Stratum.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_linear__Part.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13295 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ --rw-r--r-- 0 root (0) root (0) 14741 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13418 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 15152 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6751 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 675 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_smaller__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 960 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_standard__Monomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1344 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_tail__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___All__Standard.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5493 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___Minimalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9986 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Constraints.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8639 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 42308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Stratum.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_linear__Part.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15086 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23626 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24140 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15170 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7464 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_smaller__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_standard__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8669 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_tail__Monomials.html │ │ │ --rw-r--r-- 0 root (0) root (0) 50311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 50715 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11356 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5776 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18071 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 705 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -9020,15 +9020,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50644 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 920 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/___Appell__F1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2967 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/___Canonical_sp__Series_sp__Tutorial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 630 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Expts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1007 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Expts__Mult.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4890 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4891 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5004 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_diff__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_distraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 768 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_euler__Operators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_gkz.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_is__Torus__Fixed.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_put__Weyl__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2252 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out │ │ │ @@ -9038,15 +9038,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6851 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Appell__F1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Canonical_sp__Series_sp__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Theta__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Wto__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_create__Theta__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Expts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8063 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Expts__Mult.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11336 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11337 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_diff__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_distraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7893 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_euler__Operators.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_gkz.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_indicial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_is__Torus__Fixed.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4094 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_nilsson__Support.html │ │ │ @@ -9060,21 +9060,21 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 21857 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 5770 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/___Homotopy__Lie__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1937 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_ad.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_allgens.out │ │ │ --rw-r--r-- 0 root (0) root (0) 11142 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 11141 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1542 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket__Matrix.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 9218 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_ad.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9121 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_allgens.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22563 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22562 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8702 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15674 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6094 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HyperplaneArrangements/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HyperplaneArrangements/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 233569 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HyperplaneArrangements/dump/rawdocumentation.dump │ │ │ @@ -9247,17 +9247,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Frac__P_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 707 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Fractions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1444 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__P__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_idealizer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Keep_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Limit_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 28072 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 28068 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 370 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Variable_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2397 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3674 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ring_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_is__Normal_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2065 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_make__S2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 496 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ring__From__Fractions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2043 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_test__Huneke__Question.out │ │ │ @@ -9282,17 +9282,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6584 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_ic__P__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_idealizer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5167 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_idealizer_lp..._cm__Index_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6731 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_idealizer_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7446 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Keep_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7883 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Limit_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 57929 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 57925 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Variable_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10218 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14448 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14878 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ring_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12894 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_is__Normal_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9808 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_make__S2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_make__S2_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7958 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_ring__From__Fractions.html │ │ │ @@ -9308,15 +9308,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 690 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_action__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 415 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_ambient_lp__Ring__Of__Invariants_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_cyclic__Factors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 557 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_defining__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 617 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_degrees__Ring_lp__Diagonal__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1384 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_diagonal__Action.out │ │ │ -rw-r--r-- 0 root (0) root (0) 300 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_dim_lp__Group__Action_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1736 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1734 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3560 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert__Series.out │ │ │ -rw-r--r-- 0 root (0) root (0) 668 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_finite__Action.out │ │ │ -rw-r--r-- 0 root (0) root (0) 677 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_generators_lp__Finite__Group__Action_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 555 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_generators_lp__Ring__Of__Invariants_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 636 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_group__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hilbert__Ideal.out │ │ │ @@ -9363,15 +9363,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_action__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6098 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_ambient_lp__Ring__Of__Invariants_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_cyclic__Factors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7058 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_defining__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6766 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_degrees__Ring_lp__Diagonal__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11550 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_diagonal__Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_dim_lp__Group__Action_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7048 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7046 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10826 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert__Series.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7237 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_finite__Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6659 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_generators_lp__Finite__Group__Action_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_generators_lp__Ring__Of__Invariants_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7031 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_group.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6767 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_group__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10725 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/html/_hilbert__Ideal.html │ │ │ @@ -9530,41 +9530,41 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/JSONRPC/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8344 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/JSONRPC/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4365 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/JSONRPC/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 78221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2737 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2738 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2643 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2465 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp4.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___J__J.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5996 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5994 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 634 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets__Projection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets__Radical.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Affine__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1209 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2492 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1463 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Quotient__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_jets_lp__Z__Z_cm__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_lifting__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_lifting__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4670 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/example-output/_principal__Component.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 574 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 70 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 9302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18072 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10914 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Example_sp4.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___J__J.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Saturate.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16417 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5304 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6457 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4821 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Base.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4736 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Info.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5086 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4864 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Max__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7373 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Projection.html │ │ │ @@ -9585,62 +9585,62 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101880 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1949 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2785 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2786 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1036 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3425 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1006 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ --rw-r--r-- 0 root (0) root (0) 267 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2260 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1005 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 269 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6823 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_correspondence__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 804 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_cox__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6871 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6874 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2293 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_homotopy__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 720 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_product__Of__Projective__Spaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1031 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1075 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1638 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_relative__Resolution__Twists.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out │ │ │ -rw-r--r-- 0 root (0) root (0) 129 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 758 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_scheme__In__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3037 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_schreyer__Name.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_small__Diagonal.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Fine__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/___Scrolls.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_all__Gradings.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9710 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9711 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6739 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Carpet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Homotopies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12167 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9965 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10795 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6909 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10794 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6911 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_correspondence__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_cox__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9602 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15071 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15074 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_gorenstein__Double.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_hankel__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8285 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_homotopy__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8194 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_product__Of__Projective__Spaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7292 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution__Twists.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8496 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6025 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7697 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_scheme__In__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_schreyer__Name.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_small__Diagonal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 32029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/toc.html │ │ │ @@ -9764,27 +9764,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1849 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LAYOUT.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33504 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2858 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2411 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1436 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_gcd__L__L__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/example-output/_is__L__L__L.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 28 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4270 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___B__K__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Cohen__Engine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4330 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Cohen__Top__Level.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4133 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Givens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3626 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Hermite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11003 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10007 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Change__Matrix_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 22718 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 22717 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4178 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___N__T__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3979 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__F__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3999 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4181 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4015 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__R__R.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4048 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__X__D.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Threshold.html │ │ │ @@ -10098,15 +10098,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 231 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_comp__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_diagonal__Multidegrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Mins.out │ │ │ -rw-r--r-- 0 root (0) root (0) 701 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ -rw-r--r-- 0 root (0) root (0) 754 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_irrelevant__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 994 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Linear__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 904 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_is__Quasi__Linear.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 976 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_multigraded__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_partial__Regularities.out │ │ │ -rw-r--r-- 0 root (0) root (0) 577 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_regularity__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_support__Of__Tor.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Headline │ │ │ @@ -10114,29 +10114,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7100 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_diagonal__Multidegrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6096 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Mins.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9598 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7370 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8254 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Linear__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9218 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_is__Quasi__Linear.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8398 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8580 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8581 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10649 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_multigraded__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9034 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_partial__Regularities.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8046 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_regularity__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_support__Of__Tor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25981 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15960 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68685 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 746 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/___Local__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/___Local__Rings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1024 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_is__Well__Defined_lp__Local__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_lift__Up.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 412 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Mingens.out │ │ │ -rw-r--r-- 0 root (0) root (0) 574 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Modulo.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Prune.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2639 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/example-output/_local__Resolution.out │ │ │ @@ -10151,15 +10151,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_char_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_coefficient__Ring_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4299 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_degree__Length_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4169 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_degrees_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4159 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_dim_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_frac_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4284 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_generators_lp__Local__Ring_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10319 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_is__Commutative_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5576 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_is__Well__Defined_lp__Local__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10614 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_lift__Up.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7356 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_local__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7226 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_local__Mingens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8032 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_local__Modulo.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7324 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LocalRings/html/_local__Prune.html │ │ │ @@ -10356,15 +10356,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1999 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Expression.out │ │ │ -rw-r--r-- 0 root (0) root (0) 788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Matrix_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1449 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 548 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Flat__Monoid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_at_at_sp__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 836 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_us_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 415 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__F.out │ │ │ -rw-r--r-- 0 root (0) root (0) 194 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6869 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Grassmannian.out │ │ │ @@ -10431,15 +10431,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2458 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 587 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal_sp-_sp__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Multigraded__Betti__Tally.out │ │ │ -rw-r--r-- 0 root (0) root (0) 218 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__Hash__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 795 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 298 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__Matrix_sp_us_sp__Sequence_sp_eq_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1770 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 699 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net__File.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net_sp_vb_sp__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net_sp_vb_vb_sp__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Numbered__Vertical__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 430 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Option.out │ │ │ -rw-r--r-- 0 root (0) root (0) 571 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Option__Table_sp_gt_gt_sp__Function.out │ │ │ @@ -10471,15 +10471,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp_sl_sp__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1854 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5127 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schreyer_sporders.out │ │ │ -rw-r--r-- 0 root (0) root (0) 998 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Schubert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Self__Initializing__Type.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Sequence.out │ │ │ --rw-r--r-- 0 root (0) root (0) 940 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 950 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp-_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_sh_qu_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_st_st_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.10.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1721 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.8.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.9.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.2.13.out │ │ │ @@ -10611,15 +10611,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apply_lp__Z__Z_cm__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apropos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ascii.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asinh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_assigning_spvalues.out │ │ │ --rw-r--r-- 0 root (0) root (0) 235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 234 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atanh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 773 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ @@ -10634,25 +10634,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_borel_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 725 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_break.out │ │ │ -rw-r--r-- 0 root (0) root (0) 996 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_breakpoint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cache.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1307 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out │ │ │ -rw-r--r-- 0 root (0) root (0) 16980 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_can__Use__Hilbert__Hint.out │ │ │ --rw-r--r-- 0 root (0) root (0) 588 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 591 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8939 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_characters.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1523 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_clean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Monomial__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Quotient__Ring_rp.out │ │ │ @@ -10668,15 +10668,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1539 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1550 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4287 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ @@ -10694,15 +10694,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 740 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 113 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cosh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cot.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_coth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cover__Map_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cover_lp__Module_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 314 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 318 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_create__Task.out │ │ │ -rw-r--r-- 0 root (0) root (0) 483 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_creating_span_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_creating_spand_spwriting_spfiles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_csc.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_csch.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1090 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Column__Number.out │ │ │ @@ -10764,15 +10764,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1720 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21213 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21211 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 782 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3782 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erfc.out │ │ │ @@ -10914,15 +10914,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1039 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inputting_spa_spmatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 592 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_insert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spassignment_spmethods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 936 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__R__Ri_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ @@ -10957,15 +10957,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Mutable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Open.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Output__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Prime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Primitive.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ready_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Real.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 460 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Skew__Commutative.out │ │ │ @@ -11062,15 +11062,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ --rw-r--r-- 0 root (0) root (0) 82 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 83 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ -rw-r--r-- 0 root (0) root (0) 83 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1637 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1730 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_merge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2673 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 928 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method__Options_lp__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6785 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ @@ -11162,16 +11162,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pack.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packages.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packing_spmonomials_spfor_spefficiency.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pad.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1059 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pairs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel__Apply.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8673 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8672 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parse.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parsing_spprecedence_cm_spin_spdetail.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partitions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_path.out │ │ │ @@ -11224,15 +11224,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 796 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3975 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_lp__Matrix_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1400 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 94 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 669 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Mutable__Matrix_lp__Z__Z_cm__Z__Z_cm__R__R_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Subset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 843 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__List_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1109 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Q__Q_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Type_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Z__Z_cm__Ideal_rp.out │ │ │ @@ -11249,15 +11249,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_real__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_recursion__Depth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reduce__Hilbert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reduced__Row__Echelon__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regex__Quote.out │ │ │ --rw-r--r-- 0 root (0) root (0) 712 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 659 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 238 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularized__Beta.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularized__Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_relativize__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 713 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remainder_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove.out │ │ │ @@ -11320,15 +11320,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_shuffle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sign.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2452 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_simple_sp__Groebner_spbasis_spcomputations_spover_spvarious_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_singular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_size2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_smith__Normal__Form_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4508 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 891 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_some__Terms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort__Columns.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_span.out │ │ │ @@ -11396,15 +11396,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 687 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_terms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 730 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_tests.out │ │ │ -rw-r--r-- 0 root (0) root (0) 214 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_tex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_tex__Math.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3455 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_the_spdebugger.out │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_thread__Local.out │ │ │ -rw-r--r-- 0 root (0) root (0) 134 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_throw.out │ │ │ --rw-r--r-- 0 root (0) root (0) 143 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 142 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 186 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 141 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Absolute__Path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 296 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__C__C.out │ │ │ -rw-r--r-- 0 root (0) root (0) 428 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__External__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Field_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 94 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Lower.out │ │ │ @@ -11449,15 +11449,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_value_lp__Symbol_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 201 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_values.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_variables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 770 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__Monoid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1063 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vector.out │ │ │ --rw-r--r-- 0 root (0) root (0) 12384 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 12390 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1121 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_viewing_spthe_spsymbols_spdefined_spso_spfar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 774 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_wedge__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_weight__Range.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_what_spa_spclass_spis.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_what_spis_spa_sp__Groebner_spbasis_qu.out │ │ │ -rw-r--r-- 0 root (0) root (0) 754 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_while.out │ │ │ -rw-r--r-- 0 root (0) root (0) 133 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_width_lp__Net_rp.out │ │ │ @@ -11628,15 +11628,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4930 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Function__Body.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Function__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4208 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Function_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5711 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Function_sp_at_at_sp__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7991 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Function_sp_us_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4654 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__B__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__C_spgarbage_spcollector.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5839 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5840 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__F.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4653 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Lex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3764 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__N__U_sp__M__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6957 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Rev__Lex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Galois__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6177 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Gamma.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5167 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___General__Ordered__Monoid.html │ │ │ @@ -11784,15 +11784,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Monomial__Size.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4748 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13865 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Multigraded__Betti__Tally.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7583 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutable__Hash__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutable__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19757 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutable__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6139 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutable__Matrix_sp_us_sp__Sequence_sp_eq_sp__Thing.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10853 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3922 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___N__C__Lex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3745 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___N__T__L_splibrary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10931 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6731 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Net__File.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6153 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Net_sp^_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Net_sp_vb_sp__Net.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5139 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Net_sp_vb_vb_sp__Net.html │ │ │ @@ -11889,15 +11889,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8049 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Schubert.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Schur_springs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Scripted__Functor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5455 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___See__Also.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8656 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Self__Initializing__Type.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Separate__Exec.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21206 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Sequence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Set.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5993 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Set_sp-_sp__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5471 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Set_sp_sh_qu_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Set_sp_st_st_sp__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4634 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Singular-__Factory.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Singular_sp__Book_sp1.1.10.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Singular_sp__Book_sp1.1.8.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6262 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Singular_sp__Book_sp1.1.9.html │ │ │ @@ -12191,15 +12191,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_argument.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6102 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_ascii.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5806 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_asin.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4794 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_asinh.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_assert.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8161 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_assigning_spvalues.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_associative_spalgebras.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5666 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_atan.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6507 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_atan2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5074 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_atanh.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_augmented_spassignment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7020 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_autoload.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4315 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_backtrace.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5961 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_base__Filename.html │ │ │ @@ -12218,17 +12218,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7027 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_binary_spmethods.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_binomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5900 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_block__Matrix__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4967 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_borel_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8096 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_break.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_breakpoint.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cache.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7256 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7260 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27037 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_can__Use__Hilbert__Hint.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7355 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17375 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_capture.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3727 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cdd_pl.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_ceiling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5049 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_ceiling_lp__Number_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4937 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_center__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9347 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Base.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5958 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html │ │ │ @@ -12266,15 +12266,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12939 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_cm_sp1.8.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7067 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_cm_sp1.9.1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4047 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_cm_sp1.9.2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10394 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_cm_sp1.9.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6630 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_changes_spto_sp__Macaulay2_cm_spby_spversion.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5340 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_char.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_characters.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13571 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4278 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_chi.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6379 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8805 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_clean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_clear__All.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3950 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_clear__Echo.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_clear__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6795 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_close.html │ │ │ @@ -12305,15 +12305,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4886 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_combinatorics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10851 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_combine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6036 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_command__Interpreter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_command__Line.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4236 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_comments.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5843 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_common__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8034 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_commonest.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10632 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9658 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_commutative_spalgebra.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6077 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_comodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_compact__Matrix__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_compare__Exchange.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4510 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3986 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_complete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_components.html │ │ │ @@ -12343,15 +12343,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5804 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cosh.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cot.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5894 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_coth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5014 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cover.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5690 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cover__Map_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5059 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cover_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cover_lp__Module_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5661 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_create__Task.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8195 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_creating_spa_sppackage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_creating_span_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6063 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_creating_spand_spwriting_spfiles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5805 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_csc.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_csch.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3554 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_csdp.html │ │ │ @@ -12450,15 +12450,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_eigenvectors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_eigenvectors_lp..._cm__Hermitian_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_eint.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5452 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5585 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Timing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4901 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elementary_sparithmetic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5930 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elements.html │ │ │ --rw-r--r-- 0 root (0) root (0) 31087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elimination_spof_spvariables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 31085 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elimination_spof_spvariables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10538 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_endl.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3927 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_engine__Debug__Level.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6047 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_entries.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5465 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_entries_lp__Vector_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3915 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_environment.html │ │ │ @@ -12695,15 +12695,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6381 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_install__Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_install__Package.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4829 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_installed__Packages.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8528 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_installing_spassignment_spmethods.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7838 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_installing_spaugmented_spassignment_spmethods.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7510 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_installing_spmethods.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4658 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instance.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5641 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5769 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_integers_spmodulo_spa_spprime.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6736 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_integrate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_interpreter__Depth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9911 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_intersect.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10244 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_intersect_lp__Ideal_cm__Ideal_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_intersect_lp__R__Ri_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5415 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_intersect_lp__Set_cm__Set_rp.html │ │ │ @@ -12748,15 +12748,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7649 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Mutable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Normal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6969 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Open.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Output__File_lp__File_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5958 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Prime.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4855 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Primitive.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11253 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11255 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Quotient__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Quotient__Of.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4305 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Quotient__Of_lp__Ring_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4390 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Quotient__Of_lp__Type_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6758 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Quotient__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5533 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Ready_lp__File_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4979 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Ready_lp__Task_rp.html │ │ │ @@ -12891,15 +12891,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8021 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_matrix_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7056 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_matrix_lp__Mutable__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_matrix_lp__Ring__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6878 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_matrix_lp__Ring_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_matrix_lp__Ring_cm__Ring__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_matrix_lp__Vector_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7680 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5102 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4189 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7583 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Position.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12836 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_merge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6293 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_merge__Pairs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20108 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6336 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_method__Options_lp__Function_rp.html │ │ │ @@ -13041,16 +13041,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11126 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_packages.html │ │ │ -rw-r--r-- 0 root (0) root (0) 80348 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6569 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_packing_spmonomials_spfor_spefficiency.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5576 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_pad.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_pager.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_pairs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7165 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel__Apply.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17837 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html │ │ │ --rw-r--r-- 0 root (0) root (0) 20622 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17838 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6305 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parse.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21117 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parsing_spprecedence_cm_spin_spdetail.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18784 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_partition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9384 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_partitions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8781 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parts.html │ │ │ @@ -13125,15 +13125,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7083 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_quotient__Remainder_sq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19764 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_quotient_lp__Matrix_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11273 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_quotient_springs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11523 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_quotient_sq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8924 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5408 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8653 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7742 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__Mutable__Matrix_lp__Z__Z_cm__Z__Z_cm__R__R_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__Subset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8994 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random_lp__List_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8867 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random_lp__Module_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random_lp__Q__Q_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8374 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random_lp__Type_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10376 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random_lp__Z__Z_cm__Ideal_rp.html │ │ │ @@ -13157,15 +13157,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7438 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5944 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_recursion__Depth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_recursion__Limit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7178 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reduce__Hilbert.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19143 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reduced__Row__Echelon__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15307 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_regex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_regex__Quote.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6808 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24530 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_regular_spexpressions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10538 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_regularity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6089 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_regularized__Beta.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5870 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_regularized__Gamma.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4694 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_relativize__Filename.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7237 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remainder.html │ │ │ @@ -13263,15 +13263,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5155 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sin.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6069 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4998 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sinh.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4288 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_size.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6421 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_size2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sleep.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12916 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_smith__Normal__Form_lp__Matrix_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 19597 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 19592 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_some__Terms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sort.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9329 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sort__Columns.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sort__Columns_lp..._cm__Degree__Order_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sort__Columns_lp..._cm__Monomial__Order_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6432 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sort_lp..._cm__Degree__Order_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8479 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_sort_lp..._cm__Monomial__Order_eq_gt..._rp.html │ │ │ @@ -13368,15 +13368,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_tex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5643 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_tex__Math.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_the_spdebugger.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4157 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_the_spengine_spof_sp__Macaulay2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3690 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_the_spinterpreter_spof_sp__Macaulay2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5527 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_thread__Local.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4973 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_throw.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5190 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_times.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5527 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_timing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_to__Absolute__Path.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_to__C__C.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6011 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_to__C__Ci.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6735 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_to__External__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7298 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_to__Field_lp__Ring_rp.html │ │ │ @@ -13444,15 +13444,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_values.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_variables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4700 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_vars.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_vars_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5196 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_vars_lp__Monoid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_vars_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_vector.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17533 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17539 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8764 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_view__Help.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_viewing_spthe_spsymbols_spdefined_spso_spfar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4557 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_wait.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6935 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_wedge__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_weight__Range.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9460 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_what_spa_spclass_spis.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8286 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_what_spis_spa_sp__Groebner_spbasis_qu.html │ │ │ @@ -13889,15 +13889,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 199964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___A__S__M__Full__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 266 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___A__S__M__Random__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 670 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___A__S__M__To__Monotone__Triangle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1727 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Functions_spfor_spinvestigating_sppermutations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7736 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Initial_spideals_spof_sp__A__S__M_spideals.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_sp__A__S__M_spvarieties.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6282 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_sp__A__S__M_spvarieties.out │ │ │ -rw-r--r-- 0 root (0) root (0) 44253 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_spmatrix_sp__Schubert_spvarieties.out │ │ │ -rw-r--r-- 0 root (0) root (0) 680 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___K__Polynomial__A__S__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Pipe__Dream.out │ │ │ -rw-r--r-- 0 root (0) root (0) 535 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_anti__Diag__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 410 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_augmented__Essential__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_augmented__Rothe__Diagram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 257 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_avoids__All__Patterns.out │ │ │ @@ -13910,15 +13910,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_double__Schubert__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_entrywise__Max__Rank__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_entrywise__Min__Rank__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 219 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_essential__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 258 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_first__Descent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1732 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_fulton__Gens.out │ │ │ -rw-r--r-- 0 root (0) root (0) 445 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_get__A__S__M.out │ │ │ --rw-r--r-- 0 root (0) root (0) 676 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_grothendieck__Polynomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 677 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_grothendieck__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_initial__Ideals__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_inverse__Of.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_is__A__S__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_is__A__S__M__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 178 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_is__A__S__M__Union.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_is__C__D__G.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_is__Cartwright__Sturmfels.out │ │ │ @@ -13959,15 +13959,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 526 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___A__S__M__Full__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5458 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___A__S__M__Random__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6645 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___A__S__M__To__Monotone__Triangle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19509 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Functions_spfor_spinvestigating_sppermutations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Initial_spideals_spof_sp__A__S__M_spideals.html │ │ │ --rw-r--r-- 0 root (0) root (0) 31079 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 31080 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 58223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_spmatrix_sp__Schubert_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5931 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___K__Polynomial__A__S__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5966 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Pipe__Dream.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8374 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_anti__Diag__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6611 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_augmented__Essential__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6710 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_augmented__Rothe__Diagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6075 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_avoids__All__Patterns.html │ │ │ @@ -13980,15 +13980,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12035 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_double__Schubert__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5783 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_entrywise__Max__Rank__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5770 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_entrywise__Min__Rank__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6132 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_essential__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5741 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_first__Descent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8314 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_fulton__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6105 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_get__A__S__M.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6670 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6671 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5759 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_initial__Ideals__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5324 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_inverse__Of.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_is__A__S__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6486 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_is__A__S__M__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_is__A__S__M__Union.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6321 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_is__C__D__G.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6455 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/_is__Cartwright__Sturmfels.html │ │ │ @@ -14028,15 +14028,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 48709 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 41318 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18178 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MatrixSchubert/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 349743 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1789 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1035 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid_sp_eq_eq_sp__Matroid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 766 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid_sp_pl_pl_sp__Matroid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2370 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid_sp_pl_sp__Matroid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1025 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroids.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/___Working_spwith_sp__Chow_springs_spof_spmatroids.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_affine__Geometry.out │ │ │ @@ -14065,15 +14065,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4033 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_flats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 840 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_fundamental__Circuit.out │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Cycles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Isos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 838 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Representation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 516 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Separation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_ground__Set.out │ │ │ --rw-r--r-- 0 root (0) root (0) 464 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 465 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 218 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_hyperplanes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5177 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_ideal__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1002 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_ideal__Orlik__Solomon__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 801 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_ideal_lp__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 276 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_independence__Complex_lp__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_independent__Sets_lp__Matroid_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 432 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_indices__Of.out │ │ │ @@ -14087,15 +14087,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_is__Non__Crossing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_is__Positively__Orientable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_is__Positively__Oriented.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2333 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_is__Quotient.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_is__Simple_lp__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_is__Well__Defined_lp__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_iso__Types.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1659 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_isomorphism_lp__Matroid_cm__Matroid_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1660 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_isomorphism_lp__Matroid_cm__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 325 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_lattice__Of__Flats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_linear__Subclass.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_linear__Subclass_lp..._cm__Check__Well__Defined_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_loops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3381 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_matroid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 772 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_matroid_lp..._cm__Entry__Mode_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 914 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_max__Weight__Basis.out │ │ │ @@ -14128,15 +14128,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_tutte__Polynomial_lp__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_uniform__Matroid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 293 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/example-output/_wheel.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 26 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5258 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/___Check__Well__Defined.html │ │ │ --rw-r--r-- 0 root (0) root (0) 36585 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 36586 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/___Matroid_sp_eq_eq_sp__Matroid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7121 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/___Matroid_sp_pl_pl_sp__Matroid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12986 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/___Matroid_sp_pl_sp__Matroid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8065 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/___Matroid_sp_us_sp__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/___Working_spwith_sp__Chow_springs_spof_spmatroids.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_affine__Geometry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_all__Matroids.html │ │ │ @@ -14165,15 +14165,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_flats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_fundamental__Circuit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_get__Cycles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8643 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_get__Isos.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7483 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_get__Representation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_get__Separation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_ground__Set.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7481 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7482 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_hyperplanes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13203 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_ideal__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8840 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_ideal__Orlik__Solomon__Algebra.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8469 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_ideal_lp__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6564 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_independence__Complex_lp__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10306 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_independent__Sets_lp__Matroid_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7934 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_indices__Of.html │ │ │ @@ -14187,15 +14187,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5810 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_is__Non__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6073 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_is__Positively__Orientable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6092 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_is__Positively__Oriented.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8768 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_is__Quotient.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7484 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_is__Simple_lp__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8820 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_is__Well__Defined_lp__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6837 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_iso__Types.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10522 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_lattice__Of__Flats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_linear__Subclass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8286 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_linear__Subclass_lp..._cm__Check__Well__Defined_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_loops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18357 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_matroid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_matroid_lp..._cm__Entry__Mode_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7659 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Matroids/html/_max__Weight__Basis.html │ │ │ @@ -14241,29 +14241,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6919 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MergeTeX/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5117 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MergeTeX/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3155 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MergeTeX/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 34985 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 772 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_is__Prime_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2748 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_minimal__Primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 906 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_minimal_spprimes_spof_span_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 727 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical.out │ │ │ --rw-r--r-- 0 root (0) root (0) 870 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical__Containment.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 871 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical__Containment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical_spof_span_spideal.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 46 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 6046 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6045 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6716 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_is__Prime_lp__Ideal_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_minimal__Primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6098 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_minimal_spprimes_spof_span_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11471 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8780 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8781 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical_spof_span_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10088 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8701 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Miura/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Miura/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 15856 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Miura/dump/rawdocumentation.dump │ │ │ @@ -14299,43 +14299,43 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 31134 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 260 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_hom__Ideal__Polytope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_hom__Ideal__Polytope_lp..._cm__Coefficient__Ring_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 535 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_m__Mixed__Volume.out │ │ │ -rw-r--r-- 0 root (0) root (0) 896 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_mixed__Multiplicity.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2838 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2837 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal_lp..._cm__Variable__Base__Name_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_sec__Milnor__Numbers.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 648 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 30 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7769 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_hom__Ideal__Polytope.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7252 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_hom__Ideal__Polytope_lp..._cm__Coefficient__Ring_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_m__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10444 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_mixed__Multiplicity.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12547 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12546 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal_lp..._cm__Variable__Base__Name_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7822 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_sec__Milnor__Numbers.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13418 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4779 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MixedMultiplicity/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20012 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 709 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_cm__Ring__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2987 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2986 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 53 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6629 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp..._cm__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9390 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_cm__Ring__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12129 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/_xi.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9767 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5454 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4111 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ModuleDeformations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 132316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/dump/rawdocumentation.dump │ │ │ @@ -14343,18 +14343,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Monodromy__Solver__Options.out │ │ │ -rw-r--r-- 0 root (0) root (0) 492 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/___Point__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_complete__Graph__Init.out │ │ │ -rw-r--r-- 0 root (0) root (0) 418 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_compute__Mixed__Volume.out │ │ │ -rw-r--r-- 0 root (0) root (0) 407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_create__Seed__Pair.out │ │ │ --rw-r--r-- 0 root (0) root (0) 943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 942 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Augment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_flower__Graph__Init.out │ │ │ --rw-r--r-- 0 root (0) root (0) 530 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_get__Track__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 520 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_get__Track__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out │ │ │ -rw-r--r-- 0 root (0) root (0) 427 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_cm__Abstract__Point_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1395 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Solve_lp__System_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 956 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__E.out │ │ │ -rw-r--r-- 0 root (0) root (0) 442 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_potential__Lower__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 300 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_select__Best__Edge__And__Direction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 482 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_solve__Family.out │ │ │ @@ -14366,18 +14366,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4658 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/___Homotopy__Node.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8762 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/___Monodromy__Solver__Options.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7120 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/___Point__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5033 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4875 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_complete__Graph__Init.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_compute__Mixed__Volume.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7992 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_create__Seed__Pair.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Augment.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4848 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_flower__Graph__Init.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6923 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6913 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6377 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_homotopy__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_make__Batch__Potential.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17402 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_monodromy__Group.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5111 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_monodromy__Solve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11860 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_monodromy__Solve_lp__System_cm__Abstract__Point_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_monodromy__Solve_lp__System_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7508 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MonodromySolver/html/_potential__E.html │ │ │ @@ -14608,28 +14608,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 19293 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13033 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7613 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedBGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 41382 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2242 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2240 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 719 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_compute__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 723 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_interpolate__Component.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_max__Grading.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_trim__Basis__In__Degree.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5655 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Coefficient__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4675 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Previous__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4934 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Return__Target__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5196 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Interpolation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4831 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/___Use__Matroid.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12040 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12038 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6786 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9408 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_compute__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10282 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_interpolate__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8192 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_max__Grading.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9654 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_trim__Basis__In__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14199 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/master.html │ │ │ @@ -14638,28 +14638,28 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 369 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/___N__P.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_get__Gen__Elts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_gr__Gr.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_hilbert__Sequence.out │ │ │ --rw-r--r-- 0 root (0) root (0) 413 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 414 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 262 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out │ │ │ -rw-r--r-- 0 root (0) root (0) 573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 596 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_multiplicity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1239 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_print__Hilbert__Sequence.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 47 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6408 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/___N__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7564 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_get__Gen__Elts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6944 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_gr__Gr.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_hilbert__Sequence.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6258 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5924 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7049 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Reduction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6696 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_multiplicity__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7139 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_print__Hilbert__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16013 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10683 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/master.html │ │ │ @@ -14713,31 +14713,31 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5421 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 380295 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1954 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp!.out │ │ │ --rw-r--r-- 0 root (0) root (0) 20221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 20222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 670 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_pl_pl_sp__Embedded__Projective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1106 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Fano_lp__Z__Z_cm__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 374 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___G__G.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5148 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___G__G_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1057 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Hom_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 329 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_bs_bs_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_bs_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_eq_eq_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 390 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 447 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 905 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_st_st_sp__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1341 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1274 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1340 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1275 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_lt_lt_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1848 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 772 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_sp__Multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 763 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 744 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp_vb_vb_sp__Multirational__Map.out │ │ │ @@ -14756,42 +14756,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_coefficient__Ring_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cone__Of__Lines.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_conormal__Variety_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_cycle__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 671 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_decompose_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 158 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1004 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 427 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1006 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 428 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degrees_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 722 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2266 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 155 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dim_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 482 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_dual_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 809 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_entries_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_euler_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_factor_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1459 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_fiber__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 673 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_force__Image_lp__Multirational__Map_cm__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1856 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1857 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 324 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_hilbert__Polynomial_lp__Embedded__Projective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 384 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_ideal_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1018 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1406 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 934 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1017 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 801 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1405 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 931 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Member_lp__Multirational__Map_cm__R__A__T_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 754 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Subset_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Well__Defined_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linear__Span.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1090 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linearly__Normal__Embedding.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1088 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linearly__Normal__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 279 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 534 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 664 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1186 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multirational__Map_lp__Multirational__Map_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2768 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_parametrize_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 490 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_permute_lp__Multiprojective__Variety_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1237 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out │ │ │ @@ -14806,15 +14806,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_random_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_rational__Map_lp__Multiprojective__Variety_cm__Tally_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 440 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_ring_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_schubert__Cycle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_sectional__Genus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 468 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6206 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multiprojective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1051 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_shape_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_shortcuts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3242 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 512 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_singular__Locus_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_super_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 352 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_support_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_tangent__Cone_lp__Embedded__Projective__Variety_cm__Embedded__Projective__Variety_rp.out │ │ │ @@ -14828,15 +14828,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_⋂.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_⋃.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 641 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 14292 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8321 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp!.html │ │ │ --rw-r--r-- 0 root (0) root (0) 29238 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 29239 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_pl_pl_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10104 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Fano_lp__Z__Z_cm__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___G__G.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11626 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___G__G_lp__Z__Z_cm__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6741 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Grassmannian__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Hom_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25182 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety.html │ │ │ @@ -14846,16 +14846,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6585 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_eq_eq_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6690 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_pc_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7362 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_pl_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7082 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7152 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_st_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7958 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multiprojective__Variety_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8387 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8485 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8486 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5983 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_eq_eq_sp__Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5864 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_lt_eq_eq_gt_sp__Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8487 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_lt_lt_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7012 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_st_sp__Multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9148 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8005 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_vb_sp__Multiprojective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8973 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp_vb_sp__Multirational__Map.html │ │ │ @@ -14879,42 +14879,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_coefficient__Ring_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9838 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_cone__Of__Lines.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7060 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_conormal__Variety_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_cycle__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7199 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_decompose_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree__Sequence.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multiprojective__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_cm__Option_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6664 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_cm__Option_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5774 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degrees_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multiprojective__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8529 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5440 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_dim_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6678 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_dual_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6762 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_entries_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6959 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_euler_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8914 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_factor_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9367 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_fiber__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7546 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_force__Image_lp__Multirational__Map_cm__Multiprojective__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9887 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6874 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_hilbert__Polynomial_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5750 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_ideal_lp__Multiprojective__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10440 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7325 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8021 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6852 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Member_lp__Multirational__Map_cm__R__A__T_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7026 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6108 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Subset_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7091 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Well__Defined_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6731 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linear__Span.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7054 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7052 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5627 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7442 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multirational__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14039 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multirational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7824 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multirational__Map_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6159 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multirational__Map_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6985 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multirational__Map_lp__Multirational__Map_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10482 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_parametrize_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7036 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_permute_lp__Multiprojective__Variety_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7961 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_point_lp__Multiprojective__Variety_rp.html │ │ │ @@ -14931,15 +14931,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_rational__Map_lp__List_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8357 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_rational__Map_lp__Multiprojective__Variety_cm__Tally_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5983 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_ring_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7010 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_schubert__Cycle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5586 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_sectional__Genus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6287 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre__Embedding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multiprojective__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7510 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_shape_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9285 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_shortcuts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8654 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5942 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_singular__Locus_lp__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5751 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_source_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6910 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_super_lp__Multirational__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6862 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_support_lp__Multiprojective__Variety_rp.html │ │ │ @@ -15314,26 +15314,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 29101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NCAlgebra/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 139526 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 266 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 496 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1408 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1405 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 165 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_count__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_filter__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2806 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Bipartite__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3579 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 146 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_graph6__To__Sparse6.out │ │ │ --rw-r--r-- 0 root (0) root (0) 500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__To__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 152 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_is__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_neighborhood__Complements.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_new__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_only__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_relabel__Bipartite.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_relabel__Graph.out │ │ │ @@ -15343,26 +15343,26 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_string__To__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 575 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 18 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6190 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5916 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8723 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7193 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14397 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8007 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8305 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13917 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14290 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9860 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8645 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6704 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8897 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8895 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9145 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7341 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6575 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9142 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/_relabel__Graph.html │ │ │ @@ -15376,15 +15376,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8306 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 131317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 493 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1405 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1406 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 826 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 124 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ @@ -15398,15 +15398,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Isomorphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6397 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5908 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8749 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8750 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13355 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8002 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8282 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Graphs.html │ │ │ @@ -15688,20 +15688,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1011 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 912 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_matrix_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_max_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1557 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1559 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1002 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__List_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1965 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1599 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1337 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_polytope_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2219 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_projective_spspace.out │ │ │ @@ -15803,20 +15803,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10776 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Simplicial_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13861 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Smooth_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9792 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_making_spnormal_sptoric_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12713 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_matrix_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9319 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_max_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10617 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10619 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_nef__Generators_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10534 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19370 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Matrix_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10007 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12299 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10951 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10778 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Toric__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11019 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_polytope_lp__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11296 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_projective_spspace.html │ │ │ @@ -16123,40 +16123,40 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40622 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6381 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 146200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1058 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1060 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Numerical__Interpolation__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Pseudo__Witness__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_is__On__Image.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1224 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Degree.out │ │ │ --rw-r--r-- 0 root (0) root (0) 573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 576 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 712 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Sample.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Nullity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1569 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Source__Sample.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1201 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_pseudo__Witness__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1181 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_real__Point.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 594 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 43 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 7347 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7349 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Max__Threads.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9039 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Numerical__Interpolation__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11485 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Pseudo__Witness__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9731 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_is__On__Image.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13013 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10876 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8818 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Degree_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9530 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9533 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10027 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Sample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Nullity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Source__Sample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18968 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11048 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set_lp..._cm__Software_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8997 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34829 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/index.html │ │ │ @@ -16341,59 +16341,59 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 691 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Degree__Shifts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1610 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1610 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map_sp__Vector__In__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 395 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 285 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Module__In__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Number_sp_st_sp__Vector__In__Width.out │ │ │ --rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 494 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 260 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Polynomial__O__I__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Polynomial__O__I__Algebra_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1049 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Ring__Element_sp_st_sp__Vector__In__Width.out │ │ │ --rw-r--r-- 0 root (0) root (0) 474 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Top__Nonminimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Top__Nonminimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 446 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Vector__In__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 998 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Vector__In__Width_sp-_sp__Vector__In__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 900 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Vector__In__Width_sp_pl_sp__Vector__In__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_degree_lp__Vector__In__Width_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1193 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe__Full.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe__Full.out │ │ │ -rw-r--r-- 0 root (0) root (0) 986 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__Free__O__I__Module__Map_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 919 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__O__I__Resolution_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 918 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__O__I__Resolution_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 233 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Basis__Elements.out │ │ │ -rw-r--r-- 0 root (0) root (0) 716 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Free__O__I__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 167 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Rank.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Schreyer__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 533 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_get__Width.out │ │ │ -rw-r--r-- 0 root (0) root (0) 890 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_image_lp__Free__O__I__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1124 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_install__Generators__In__Width.out │ │ │ --rw-r--r-- 0 root (0) root (0) 507 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 506 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 410 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Homogeneous_lp__Free__O__I__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 831 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Homogeneous_lp__Vector__In__Width_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__O__I__G__B.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 901 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__O__I__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Zero.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Zero_lp__Free__O__I__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Zero_lp__Vector__In__Width_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 548 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_lead__Coefficient_lp__Vector__In__Width_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_lead__Monomial_lp__Vector__In__Width_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_lead__Term_lp__Vector__In__Width_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 750 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_make__Free__O__I__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 445 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_make__Polynomial__O__I__Algebra.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2211 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Free__O__I__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 312 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Free__O__I__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Module__In__Width_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 397 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Polynomial__O__I__Algebra_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Vector__In__Width_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 792 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 709 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Orbit.out │ │ │ -rw-r--r-- 0 root (0) root (0) 474 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Res.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1068 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Syz.out │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_ranks.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1811 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 509 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_restricted__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 655 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_terms_lp__Vector__In__Width_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_to__String_lp__Free__O__I__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 145 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_to__String_lp__Polynomial__O__I__Algebra_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 687 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_use_lp__Module__In__Width_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/.Headline │ │ │ @@ -16406,64 +16406,64 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9413 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8023 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map_sp__Vector__In__Width.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5966 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5908 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Module__In__Width.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6635 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Number_sp_st_sp__Vector__In__Width.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5376 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Monomial__Order.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7829 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6739 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6538 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Polynomial__O__I__Algebra.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5823 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Polynomial__O__I__Algebra_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7120 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Ring__Element_sp_st_sp__Vector__In__Width.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Row__Down__Col__Down.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4009 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Row__Down__Col__Up.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4009 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Row__Up__Col__Down.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Row__Up__Col__Up.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5901 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5900 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6489 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Variable__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9720 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Vector__In__Width.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7042 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Vector__In__Width_sp-_sp__Vector__In__Width.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6884 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Vector__In__Width_sp_pl_sp__Vector__In__Width.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_degree_lp__Vector__In__Width_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6877 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__Free__O__I__Module__Map_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6472 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6471 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_get__Basis__Elements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6333 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_get__Free__O__I__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5415 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_get__Rank.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8161 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_get__Schreyer__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6017 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_get__Width.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6756 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_image_lp__Free__O__I__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7536 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_install__Generators__In__Width.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6731 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6730 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6402 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Homogeneous_lp__Free__O__I__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6791 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Homogeneous_lp__Vector__In__Width_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7707 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7706 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Zero.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6552 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Zero_lp__Free__O__I__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Zero_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_lead__Coefficient_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6695 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_lead__Monomial_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_lead__Term_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8446 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_make__Free__O__I__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6668 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_make__Polynomial__O__I__Algebra.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9163 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6178 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5829 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Module__In__Width_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5989 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5988 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5534 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Polynomial__O__I__Algebra_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5969 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6547 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Orbit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8699 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10079 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Syz.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_ranks.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_restricted__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6530 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_terms_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5535 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_to__String_lp__Free__O__I__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5625 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_to__String_lp__Polynomial__O__I__Algebra_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_use_lp__Module__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 40245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24456 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/master.html │ │ │ @@ -16480,31 +16480,31 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 528 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Chain__Complex_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 420 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Chain__Complex_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 346 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Chain__Complex_sp_pl_pl_sp__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 520 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Chain__Complex_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Chain__Complex_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 976 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Chain__Complex_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 377 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Chain__Complex_sp_us_sp__Z__Z_sp_eq_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5598 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Fast__Nonminimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5597 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Fast__Nonminimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Graded__Module__Map_sp_vb_sp__Graded__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 514 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Graded__Module__Map_sp_vb_vb_sp__Graded__Module__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Graded__Module_sp_st_st_sp__Graded__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 554 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Graded__Module_sp_st_st_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 971 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___H__H^__Z__Z_sp__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___H__H_sp__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___H__H_us__Z__Z_sp__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Hom_lp__Module_cm__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 703 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Ring_sp^_sp__Betti__Tally.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp__Graded__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_complete_lp__Chain__Complex_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1392 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_cone_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dd.out │ │ │ -rw-r--r-- 0 root (0) root (0) 536 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dual_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1335 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1130 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extracting_spinformation_spfrom_spchain_spcomplexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_free_spresolutions_spof_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_graded__Module__Map.out │ │ │ @@ -16551,15 +16551,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_st_st_sp__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4754 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_st_st_sp__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4369 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_st_st_sp__Graded__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6364 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_st_st_sp__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_us_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6568 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Chain__Complex_sp_us_sp__Z__Z_sp_eq_sp__Thing.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18353 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module__Map_sp_vb_sp__Graded__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6242 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module__Map_sp_vb_vb_sp__Graded__Module__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4309 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4353 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module_sp_st_st_sp__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6286 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Graded__Module_sp_st_st_sp__Graded__Module.html │ │ │ @@ -16569,25 +16569,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5804 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___H__H_sp__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___H__H_sp__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___H__H_us__Z__Z_sp__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5159 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___H__H_us__Z__Z_sp__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Hom_lp__Module_cm__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5592 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Ring_sp^_sp__Betti__Tally.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10626 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10627 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8721 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5197 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6785 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6065 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6429 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_chain__Complex_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5792 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_complete_lp__Graded__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_components_lp__Chain__Complex_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7787 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7783 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7537 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_cone_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dd.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5986 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4042 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_dual_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8623 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7937 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extracting_spinformation_spfrom_spchain_spcomplexes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_free_spresolutions_spof_spmodules.html │ │ │ @@ -16657,22 +16657,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6580 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OpenMath/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5219 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OpenMath/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OpenMath/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84707 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 17052 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 17047 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1080 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2081 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 16187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Oscillators.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1682 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_all__Unique__Principal__Minors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10260 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_find__Real__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Angles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2330 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_identify__Stability.out │ │ │ -rw-r--r-- 0 root (0) root (0) 459 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_is__Stable__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10680 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Jacobian.out │ │ │ @@ -16680,22 +16680,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 673 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 23276 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_osc__System.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1924 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_standard__Sols.out │ │ │ -rw-r--r-- 0 root (0) root (0) 252 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/_vertex__Spanning__Polynomial.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 33 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 25578 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7903 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7811 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6858 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3730 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___Harrington-__Schenck-__Stillman.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8974 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_all__Unique__Principal__Minors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19877 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_find__Real__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7031 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Angles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10740 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_identify__Stability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7660 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_is__Stable__Solution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20186 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Jacobian.html │ │ │ @@ -17484,15 +17484,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49935 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 313 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/___All__Random.out │ │ │ -rw-r--r-- 0 root (0) root (0) 476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/___Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 735 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/___Verify__Points.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1730 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 619 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Make__Ring__Maps.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1456 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1010 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 463 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_affine__Points__Mat.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_expected__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 234 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_min__Max__Resolution.out │ │ │ @@ -17505,15 +17505,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 307 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_projective__Points__By__Intersection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 333 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_random__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/example-output/_random__Points__Mat.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 14 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/___All__Random.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6314 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/___Verify__Points.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10310 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points__By__Intersection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Make__Ring__Maps.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8320 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7153 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points__By__Intersection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_affine__Points__Mat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6214 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_expected__Betti.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Points/html/_min__Max__Resolution.html │ │ │ @@ -17932,15 +17932,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2469 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Hibi_spideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__Intersection_splattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Example_co_sp__L__C__M-lattices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Poset_sp_us_st.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2468 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 306 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_adjoin__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 668 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_all__Relations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_antichains.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_atoms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 253 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_boolean__Lattice.out │ │ │ @@ -18059,15 +18059,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9670 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Hibi_spideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__Intersection_splattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Example_co_sp__L__C__M-lattices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 38979 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5891 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6014 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5677 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Poset_sp_us_st.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8850 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8852 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6226 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6257 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_adjoin__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_all__Relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7035 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_antichains.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7208 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5816 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_atoms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5957 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_augment__Poset.html │ │ │ @@ -18222,30 +18222,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 83569 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2696 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/___Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3736 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated__Primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated_spprimes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 411 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_irreducible__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_is__Primary.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 627 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_localize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1837 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary__Decomposition_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_primary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1390 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 423 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_remove__Lowest__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_strategies_spfor_spcomputing_spprimary_spdecomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 242 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_top__Components.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 17869 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4849 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated_spprimes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5941 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_irreducible__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8411 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_is__Primary.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8912 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8911 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_localize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8542 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Component.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14541 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21275 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary__Decomposition_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5914 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_primary_spdecomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10065 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_remove__Lowest__Dimension.html │ │ │ @@ -18650,15 +18650,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 13630 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3258 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1713 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21924 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21923 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2741 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4091 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1645 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2176 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -18692,15 +18692,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9263 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 10050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 7030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16374 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20058 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8895 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 47772 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 47771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11186 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4325 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12642 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Pfaffians_spon_spquadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Singularities_spof_splifting_spof_sptype_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 14894 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12432 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9486 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.html │ │ │ @@ -18960,27 +18960,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/___Zero__Mean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2643 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_disturb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1300 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1502 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5291 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1143 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1159 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7432 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7072 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14710 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13046 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4959 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52982 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/ │ │ │ @@ -18994,15 +18994,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 427 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_expected__Betti_lp__Z__Z_cm__Z__Z_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1436 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_hartshorne__Rao__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_hilbert__Numerator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 807 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_image__Under__Rational__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1305 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_known__Unirational__Component__Of__Space__Curves.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_nodal__Plane__Curve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2925 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_random__Canonical__Curve__Genus8with8__Points.out │ │ │ --rw-r--r-- 0 root (0) root (0) 485 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 484 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_random__Curve__Genus14__Degree18in__P6.out │ │ │ -rw-r--r-- 0 root (0) root (0) 393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_random__Curve__Genus8__Degree14in__P6.out │ │ │ -rw-r--r-- 0 root (0) root (0) 21956 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/_space__Curve.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 35 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/___Random__Space__Curves.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_canonical__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5627 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_canonical__Curve__Genus14.html │ │ │ @@ -19014,15 +19014,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6112 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_expected__Betti_lp__Z__Z_cm__Z__Z_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9851 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_hartshorne__Rao__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5858 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_hilbert__Numerator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7074 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_image__Under__Rational__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8515 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_known__Unirational__Component__Of__Space__Curves.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7937 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_nodal__Plane__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9493 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_random__Canonical__Curve__Genus8with8__Points.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7705 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7704 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_random__Curve__Genus8__Degree14in__P6.html │ │ │ -rw-r--r-- 0 root (0) root (0) 31204 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/_space__Curve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16066 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7325 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/ │ │ │ @@ -19045,25 +19045,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5165 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86199 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Alexander__Probability.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 447 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 430 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 906 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 287 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 384 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ @@ -19080,28 +19080,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5977 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_is__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7962 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Addition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7486 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6753 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7981 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Elements__From__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6456 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Pure__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7555 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7614 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal__Chain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9884 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Sparse__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7646 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18182 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Toric__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5925 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_reg__Seq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6209 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5930 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free_lp__Z__Z_cm__Ring_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25798 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25781 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17675 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 151393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/___C__M__Stats.out │ │ │ @@ -19219,15 +19219,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 292 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1455 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 379 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_find__A__Non__Zero__Minor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1379 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_generic__Projection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 841 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_get__Random__Linear__Forms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_projection__To__Hypersurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Coordinate__Change.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1226 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 51 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4968 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Decomposition__Strategy.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6079 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Dimension__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Extend__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Max__Coordinates__To__Replace.html │ │ │ @@ -19238,15 +19238,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 13620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_find__A__Non__Zero__Minor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11624 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_generic__Projection.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8119 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_generic__Projection_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10362 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_get__Random__Linear__Forms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_projection__To__Hypersurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9256 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Coordinate__Change.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12644 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12643 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14120 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 26365 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7040 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RationalMaps/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RationalMaps/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 117469 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RationalMaps/dump/rawdocumentation.dump │ │ │ @@ -19492,15 +19492,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_is__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3453 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_jacobian__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_minimal__Reduction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_minimal__Reduction_lp..._cm__Tries_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 255 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 972 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_reduction__Number.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5811 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Algebra.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3968 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3965 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 417 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_special__Fiber.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_special__Fiber__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1090 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_symmetric__Kernel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2858 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_symmetric__Kernel_lp..._cm__Variable_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_versal__Embedding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_which__Gm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/ │ │ │ @@ -19519,15 +19519,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 19295 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_jacobian__Dual.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12079 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_minimal__Reduction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_minimal__Reduction_lp..._cm__Basis__Element__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6155 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_minimal__Reduction_lp..._cm__Tries_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8847 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_multiplicity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_reduction__Number.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16698 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Algebra.html │ │ │ --rw-r--r-- 0 root (0) root (0) 17752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 17749 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11514 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal_lp..._cm__Degree__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal_lp..._cm__Minimal__Generators_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9738 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal_lp..._cm__Pair__Limit_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10553 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_special__Fiber.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11873 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_special__Fiber__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6125 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReesAlgebra/html/_symmetric__Algebra__Ideal.html │ │ │ @@ -19570,19 +19570,19 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 20000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReflexivePolytopesDB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10180 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReflexivePolytopesDB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5744 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ReflexivePolytopesDB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 6654 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3690 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/example-output/_m__Regularity.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3689 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/example-output/_m__Regularity.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 53 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/html/___C__M.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12240 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6816 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5042 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3369 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Regularity/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 36656 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/example-output/ │ │ │ @@ -19723,80 +19723,80 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 114689 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1231 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/___Grass.out │ │ │ -rw-r--r-- 0 root (0) root (0) 454 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_affine__Discriminant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2692 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_affine__Resultant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2420 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6589 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out │ │ │ --rw-r--r-- 0 root (0) root (0) 17439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 17437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1436 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_conormal__Variety.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4382 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1757 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1756 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7159 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_dualize.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7649 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7647 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2051 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_generic__Polynomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5201 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3139 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3134 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6760 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6759 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7136 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 122874 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5787 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_veronese.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 41 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6151 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5450 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Affine__Chart__Proj.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5388 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Assume__Ordinary.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5168 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Duality.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9362 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6713 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Grass_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4939 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/___Singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_affine__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8873 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_affine__Resultant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10851 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10853 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14555 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25300 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25298 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12271 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_conormal__Variety_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12275 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9436 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12273 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_dualize.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15055 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15053 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_generic__Polynomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9390 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9385 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8421 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12774 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12773 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10211 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14575 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14574 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 130702 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16477 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7970 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/_veronese.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19795 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20915 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7538 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37681 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 415 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5992 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5995 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5453 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Keep__Statistics__Command.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5714 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/___Pre__Run__Script.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5211 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Child.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Parent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7798 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23723 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5849 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2__Return__Answer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6510 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2_lp..._cm__Keep__Files_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7746 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_suggestions_spfor_spusing_sp__Run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9870 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7677 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5065 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SCMAlgebras/ │ │ │ @@ -19839,15 +19839,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6849 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SCSCP/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4011 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SCSCP/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 106534 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 436 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___Gate.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2516 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2515 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 525 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_arithmetic_spwith_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 389 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_c__Code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 830 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_compressing_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 623 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_creating_spgates.out │ │ │ -rw-r--r-- 0 root (0) root (0) 142 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_declare__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 635 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_differentiating_spcircuits.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1597 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/example-output/_evaluate_lp__S__L__Program_cm__Mutable__Matrix_cm__Mutable__Matrix_rp.out │ │ │ @@ -19888,15 +19888,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5039 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_make__S__L__Program.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8509 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_measuring_spthe_spsize_spof_spcircuits.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6142 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_print__As__S__L__P.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6784 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_support.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_undeclare__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6744 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_value__Hash__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9003 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_working_spwith_spgate_spmatrices.html │ │ │ --rw-r--r-- 0 root (0) root (0) 39799 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 39798 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23485 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7224 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42792 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3218 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/_sl2__Equivariant__Constant__Rank__Matrix.out │ │ │ @@ -20131,33 +20131,33 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 65194 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 46080 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22244 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SRdeformations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44195 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2583 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11853 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_are__Pseudo__Inverses.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6809 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_laplacians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_numeric__Rank.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9340 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8680 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_pseudo__Inverse.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3925 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Laplacian.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3939 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___Projection.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11844 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11843 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12984 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15064 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_are__Pseudo__Inverses.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_check__S__V__D__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10109 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10108 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_laplacians.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3690 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_new__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9288 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_numeric__Rank.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18201 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18248 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_pseudo__Inverse.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4139 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_pseudo__Inverse1.html │ │ │ @@ -20181,24 +20181,24 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 54413 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_annihilator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_ideal_spquotients_spand_spsaturation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_is__Supported__In__Zero__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1834 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/example-output/_saturate.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 9102 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_annihilator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_ideal_spquotients_spand_spsaturation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7056 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_is__Supported__In__Zero__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8786 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_module_spquotients_cm_spsaturation_cm_spand_spannihilator.html │ │ │ --rw-r--r-- 0 root (0) root (0) 18629 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 18627 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16015 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp__Module_cm__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12905 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/_saturate_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16273 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4484 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Saturation/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/ │ │ │ @@ -20540,15 +20540,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 327215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 910 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Basis.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2350 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Class__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 910 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___E__H__P__Variables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Eor__H.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1608 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Group__Acting.out │ │ │ --rw-r--r-- 0 root (0) root (0) 565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Memoize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 564 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 701 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Odd__Or__Even.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___S__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Schur__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 954 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Schur__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 719 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Schur__Ring__Indexed__Variable__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5676 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Schur__Rings.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_branch.out │ │ │ @@ -20561,15 +20561,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1749 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_dim_lp__List_cm__Schur__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_e__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_h__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_internal__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 978 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_internal__Product_lp__Class__Function_cm__Class__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 853 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_internal__Product_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi.out │ │ │ --rw-r--r-- 0 root (0) root (0) 446 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 447 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 638 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_kostka__Number.out │ │ │ -rw-r--r-- 0 root (0) root (0) 674 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_modification__Rule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_numgens_lp__Schur__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 840 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_p__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3004 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_partitions_lp__Set_cm__Basic__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_plethysm.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_plethysm_lp__Basic__List_cm__Ring__Element_rp.out │ │ │ @@ -20601,15 +20601,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 10650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Basis.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Class__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8459 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___E__H__P__Variables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8366 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Eor__H.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Group__Acting.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7052 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Memoize.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7051 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Memoize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Odd__Or__Even.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7628 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___S__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18186 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Schur__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9588 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Schur__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Schur__Ring__Indexed__Variable__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15493 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_branch.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8489 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_centralizer__Size.html │ │ │ @@ -20621,15 +20621,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 13928 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_dim_lp__List_cm__Schur__Ring__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7389 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_e__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7744 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_h__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8482 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_internal__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8482 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_internal__Product_lp__Class__Function_cm__Class__Function_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9278 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_internal__Product_lp__Ring__Element_cm__Ring__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6506 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6507 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9589 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_kostka__Number.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11827 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_modification__Rule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_numgens_lp__Schur__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8469 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_p__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10372 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_partitions_lp__Set_cm__Basic__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12769 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_plethysm.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/_plethysm_lp__Basic__List_cm__Ring__Element_rp.html │ │ │ @@ -20717,29 +20717,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Multi__Hom.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_make__Product__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 453 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_projective__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_projective__Degrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 876 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre.out │ │ │ --rw-r--r-- 0 root (0) root (0) 720 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 719 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 97 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 7825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_chow__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8540 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_contained__In__Singular__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9961 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_intersection__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9843 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6705 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Multi__Hom.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6078 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_make__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6616 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_make__Product__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7935 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_multiplicity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9492 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_projective__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_projective__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9006 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8876 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8875 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17740 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13728 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SegreClasses/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SemidefiniteProgramming/example-output/ │ │ │ @@ -21065,15 +21065,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10013 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_coimage_lp__Simplicial__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 27943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_cokernel_lp__Simplicial__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 28194 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_components_lp__Simplicial__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 17297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_components_lp__Simplicial__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_degree_lp__Simplicial__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 24923 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_direct__Sum_lp__Simplicial__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8737 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_ext__Power.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6778 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_exterior__Inclusion.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6779 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_exterior__Inclusion.out │ │ │ -rw-r--r-- 0 root (0) root (0) 45108 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_face_sldegeneracy_spmaps_spof_spa_spsimplicial_spmodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7472 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_forget__Complex_lp__Simplicial__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1587 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_forget__Degeneracy_lp__Simplicial__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_id_sp_us_sp__Simplicial__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 36794 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_image_lp__Simplicial__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 60726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_induced__Map_lp__Simplicial__Module_cm__Simplicial__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 12255 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_is__Commutative_lp__Simplicial__Module__Map_rp.out │ │ │ @@ -21136,15 +21136,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 36316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_components_lp__Simplicial__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_components_lp__Simplicial__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11527 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_degree_lp__Simplicial__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35440 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_direct__Sum_lp__Simplicial__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20083 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_ext__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6089 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_ext__Power_lp..._cm__Degeneracy_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5837 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_ext__Power_lp..._cm__Top__Degree_eq_gt..._rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16431 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_exterior__Inclusion.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16432 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_exterior__Inclusion.html │ │ │ -rw-r--r-- 0 root (0) root (0) 55086 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_face_sldegeneracy_spmaps_spof_spa_spsimplicial_spmodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5827 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Complex_lp..._cm__Remember__Summands_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16703 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Complex_lp__Simplicial__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Degeneracy_lp__Simplicial__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7153 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_id_sp_us_sp__Simplicial__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 45212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_image_lp__Simplicial__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_induced__Map_lp..._cm__Degree_eq_gt..._rp.html │ │ │ @@ -21415,18 +21415,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1122 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 646 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Multidimensional__Matrix_sp_us_sp__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 597 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Ring__Element_sp_st_sp__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Discriminant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 930 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Resultant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 385 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Resultant_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1023 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 930 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1364 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 927 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1362 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 973 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dim_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_entries_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 394 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 471 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_exponents_lp__Sparse__Resultant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2043 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_flattening.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_generic__Laurent__Polynomials.out │ │ │ @@ -21440,15 +21440,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 634 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_random__Multidimensional__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2126 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_rank_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1772 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_reverse__Shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_ring_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 457 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2104 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sort__Shape.out │ │ │ -rw-r--r-- 0 root (0) root (0) 12400 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Discriminant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 55254 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 55251 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ -rw-r--r-- 0 root (0) root (0) 911 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 35 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 10344 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix_sp-_sp__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6547 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Multidimensional__Matrix_sp_eq_eq_sp__Multidimensional__Matrix.html │ │ │ @@ -21458,18 +21458,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6397 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Ring__Element_sp_st_sp__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5642 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Discriminant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6199 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Discriminant_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5670 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6728 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/___Sparse__Resultant_sp__Thing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5400 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_char_lp__Sparse__Discriminant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_char_lp__Sparse__Resultant_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8391 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8469 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8909 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7102 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8392 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8907 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6324 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_dim_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5913 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_entries_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5753 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents_lp__Sparse__Discriminant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_exponents_lp__Sparse__Resultant_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_flattening.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7962 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_generic__Laurent__Polynomials.html │ │ │ @@ -21483,15 +21483,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6781 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_random__Multidimensional__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8496 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_rank_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_reverse__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5830 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_ring_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5791 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8089 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sort__Shape.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20355 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ --rw-r--r-- 0 root (0) root (0) 65737 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 65734 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/_sylvester__Matrix_lp__Multidimensional__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22135 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21170 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11236 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 178042 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump │ │ │ @@ -21538,15 +21538,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_permute__Polynomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_power__Sum__Symmetric__Polynomials_lp__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_reading__Word_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1936 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 410 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_row__Permutation__Tableaux_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 335 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_row__Stabilizer_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 563 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_schur__Polynomial_lp__List_cm__Partition_cm__Polynomial__Ring_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 45653 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 45652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1508 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_semistandard__Tableaux_lp__Partition_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 206 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_size_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 603 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_sort__Columns__Tableau_lp__Specht__Module__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_sort__Columns__Tableau_lp__Young__Tableau_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_specht__Module__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 873 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_specht__Polynomial_lp__Young__Tableau_cm__Polynomial__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2337 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_specht__Polynomials_lp__Partition_cm__Polynomial__Ring_rp.out │ │ │ @@ -21602,15 +21602,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8870 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_permute__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6031 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_power__Sum__Symmetric__Polynomials_lp__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5786 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_reading__Word_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11257 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6291 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_row__Permutation__Tableaux_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5977 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_row__Stabilizer_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_schur__Polynomial_lp__List_cm__Partition_cm__Polynomial__Ring_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 53740 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 53739 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7430 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_semistandard__Tableaux_lp__Partition_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5377 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_size_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6697 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_sort__Columns__Tableau_lp__Specht__Module__Element_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_sort__Columns__Tableau_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7627 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Module__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7563 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Polynomial_lp__Young__Tableau_cm__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8925 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_specht__Polynomials_lp__Partition_cm__Polynomial__Ring_rp.html │ │ │ @@ -21656,27 +21656,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_is__Admissible__G__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_map_lp__Congruence__Of__Curves_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1356 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_mirror__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_normal__Sheaf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Cubic__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 967 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 654 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1966 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13809 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_polarized__K3surface.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13811 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_polarized__K3surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1488 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 438 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surfaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 744 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__External__String_lp__Hodge__Special__Fourfold_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5170 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 5171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 525 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 524 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 610 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6551 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7158 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Congruence__Of__Curves_sp__Embedded__Projective__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10032 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Cubic__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11408 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/___Doubly__Special__Cubic__Fourfold.html │ │ │ @@ -21715,29 +21715,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5784 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_mirror__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_normal__Sheaf.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8548 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8257 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8384 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25217 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6716 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7886 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9392 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5826 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surfaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_swap.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6169 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__External__String_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10948 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5579 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_trisecant__Flop.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7138 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 40566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35967 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpectralSequences/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpectralSequences/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 230760 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpectralSequences/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpectralSequences/example-output/ │ │ │ @@ -22749,21 +22749,21 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 806 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_compatible__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_decompose__Fraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 488 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1817 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1492 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ @@ -22800,21 +22800,21 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_compatible__Ideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_decompose__Fraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10782 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12579 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5408 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16914 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16916 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8725 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8387 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15975 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15983 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Pure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Rational.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16563 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16562 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5995 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8061 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34923 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ @@ -23022,28 +23022,28 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24955 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Lineage__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1467 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7067 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ --rw-r--r-- 0 root (0) root (0) 653 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1034 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1155 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 535 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 826 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2876 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8702 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Lineage__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6781 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6895 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7172 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13648 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21706 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6872 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 67031 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ @@ -23147,25 +23147,25 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 20320 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 944 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_cm__Volumes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_dual__Deg__Codim.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1537 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1536 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1032 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_polar__Degrees.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Force__Amat.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4331 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/___Text__Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8972 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7635 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_cm__Volumes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7596 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_dual__Deg__Codim.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8942 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8974 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/_polar__Degrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9945 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4638 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricInvariants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricTopology/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 54964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump │ │ │ @@ -23382,15 +23382,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 166062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Chirotope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 815 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulation.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2275 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2276 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 18048 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_all__Triangulations_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_bistellar__Flip.out │ │ │ -rw-r--r-- 0 root (0) root (0) 440 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_chirotope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_degree__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_delaunay__Subdivision.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_delaunay__Weights.out │ │ │ -rw-r--r-- 0 root (0) root (0) 383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_fine__Star__Triangulation.out │ │ │ @@ -23450,15 +23450,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11148 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Triangulation__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_secondary__Cone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_some__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_vectors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_volume__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_wall__Circuits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 39758 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 39759 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49559 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/___Betti1_lp__Triplet_rp.out │ │ │ @@ -24243,15 +24243,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9346 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 226354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Def__Param.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Comparison__Theorem.out │ │ │ -rw-r--r-- 0 root (0) root (0) 406 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Tangent__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 862 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_correct__Deformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 774 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 562 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_ext__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_families.out │ │ │ @@ -24280,15 +24280,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5903 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Correction__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7898 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Def__Param.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Degree__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5123 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Highest__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9653 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Nested__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5248 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Polynomial__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6257 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Sanity__Check.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8262 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7148 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Verbose.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4635 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Versal__Deformation__Results.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_check__Comparison__Theorem.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7831 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_check__Tangent__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10134 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_correct__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5625 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_correction__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10071 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_cotangent__Cohomology1.html │ │ │ @@ -24412,15 +24412,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_flattening__Relations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Flat__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 110 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__From__Disk.out │ │ │ -rw-r--r-- 0 root (0) root (0) 990 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Range__Of__One__Parameter__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1616 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Smoothing__Family__With__Versal__Deformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 842 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_give1683__Format.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_hilbert__Burch__Matrices.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1293 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1292 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3279 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Unfolding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1617 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_prune__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 15838 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_restricted__Unfolding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_satisfies__Degree__Condition1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_smoothness__With__Reductions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 968 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_to__Do__List.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/ │ │ │ @@ -24449,15 +24449,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Range__Of__One__Parameter__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Smoothing__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Smoothing__Family__With__Versal__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_give1683__Format.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_hilbert__Burch__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_improve__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14619 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_is__Smoothing__Family.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Range.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8148 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Range.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12736 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Unfolding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_prune__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_restricted__Unfolding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9758 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_satisfies__Degree__Condition1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_smoothness__With__Reductions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_solving__Flattening__Relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_test__Bound.html │ │ │ @@ -24480,15 +24480,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ceiling_lp__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clean__Support.out │ │ │ -rw-r--r-- 0 root (0) root (0) 470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clear__Cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 254 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Basic__List_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 335 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficients_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2714 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1719 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1715 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1174 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_embed__As__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_find__Element__Of__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_gbs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 831 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Linear__Diophantine__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Divisors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 336 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ideal__Power.out │ │ │ @@ -24512,16 +24512,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 441 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Zero__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_map__To__Projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_non__Cartier__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 780 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_positive__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ramification__Divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4359 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4357 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ring_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__Q__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 458 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__R__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 576 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_torsion__Submodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 364 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_trim_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 174 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_zero__Divisor.out │ │ │ @@ -24551,15 +24551,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6646 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_ceiling_lp__R__Weil__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_clean__Support.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6255 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_clear__Cache.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6610 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_coefficient_lp__Basic__List_cm__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6757 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_coefficients_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19944 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_divisor.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12509 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_embed__As__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_find__Element__Of__Degree.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7522 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_gbs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8578 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_get__Linear__Diophantine__Solution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7449 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_get__Prime__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5944 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_get__Prime__Divisors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6547 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_ideal__Power.html │ │ │ @@ -24584,16 +24584,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6166 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Zero__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8335 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_map__To__Projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8549 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_non__Cartier__Locus.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6896 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_positive__Part.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7362 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_primes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9323 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_ramification__Divisor.html │ │ │ --rw-r--r-- 0 root (0) root (0) 20586 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9763 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20584 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5344 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_ring_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6343 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__Q__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7093 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__R__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7414 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__Weil__Divisor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7033 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_torsion__Submodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_trim_lp__Basic__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/html/_zero__Divisor.html │ │ │ @@ -25221,312 +25221,312 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/macaulay2-common/ │ │ │ -rw-r--r-- 0 root (0) root (0) 8541 2026-10-24 01:47:31.000000 ./usr/share/doc/macaulay2-common/changelog.Debian.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 86835 2026-10-24 01:37:48.000000 ./usr/share/doc/macaulay2-common/copyright │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc-base/ │ │ │ -rw-r--r-- 0 root (0) root (0) 577 2026-10-23 21:15:50.000000 ./usr/share/doc-base/macaulay2-common.macaulay2 │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/info/ │ │ │ -rw-r--r-- 0 root (0) root (0) 35705 2026-10-24 01:47:31.000000 ./usr/share/info/A1BrouwerDegrees.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14112 2026-10-24 01:47:31.000000 ./usr/share/info/AInfinity.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14113 2026-10-24 01:47:31.000000 ./usr/share/info/AInfinity.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10507 2026-10-24 01:47:31.000000 ./usr/share/info/AbstractSimplicialComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1490 2026-10-24 01:47:31.000000 ./usr/share/info/AbstractToricVarieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7476 2026-10-24 01:47:31.000000 ./usr/share/info/AdjointIdeal.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8780 2026-10-24 01:47:31.000000 ./usr/share/info/AdjunctionForSurfaces.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 23008 2026-10-24 01:47:31.000000 ./usr/share/info/AlgebraicSplines.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8653 2026-10-24 01:47:31.000000 ./usr/share/info/AllMarkovBases.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3084 2026-10-24 01:47:31.000000 ./usr/share/info/AnalyzeSheafOnP1.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 38025 2026-10-24 01:47:31.000000 ./usr/share/info/AssociativeAlgebras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13736 2026-10-24 01:47:31.000000 ./usr/share/info/BGG.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1733 2026-10-24 01:47:31.000000 ./usr/share/info/BIBasis.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 16570 2026-10-24 01:47:31.000000 ./usr/share/info/BeginningMacaulay2.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 1535 2026-10-24 01:47:31.000000 ./usr/share/info/Benchmark.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 1545 2026-10-24 01:47:31.000000 ./usr/share/info/Benchmark.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 48030 2026-10-24 01:47:31.000000 ./usr/share/info/BernsteinSato.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 21319 2026-10-24 01:47:31.000000 ./usr/share/info/Bertini.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 41855 2026-10-24 01:47:31.000000 ./usr/share/info/BettiCharacters.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 21318 2026-10-24 01:47:31.000000 ./usr/share/info/Bertini.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 41858 2026-10-24 01:47:31.000000 ./usr/share/info/BettiCharacters.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3384 2026-10-24 01:47:31.000000 ./usr/share/info/BinomialEdgeIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10986 2026-10-24 01:47:31.000000 ./usr/share/info/Binomials.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 20088 2026-10-24 01:47:31.000000 ./usr/share/info/BoijSoederberg.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 16451 2026-10-24 01:47:31.000000 ./usr/share/info/Book3264Examples.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1823 2026-10-24 01:47:31.000000 ./usr/share/info/BooleanGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8487 2026-10-24 01:47:31.000000 ./usr/share/info/Brackets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1070 2026-10-24 01:47:31.000000 ./usr/share/info/Browse.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6532 2026-10-24 01:47:31.000000 ./usr/share/info/Bruns.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 22607 2026-10-24 01:47:31.000000 ./usr/share/info/CellularResolutions.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 18878 2026-10-24 01:47:31.000000 ./usr/share/info/ChainComplexExtras.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 22601 2026-10-24 01:47:31.000000 ./usr/share/info/CellularResolutions.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 18875 2026-10-24 01:47:31.000000 ./usr/share/info/ChainComplexExtras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4414 2026-10-24 01:47:31.000000 ./usr/share/info/ChainComplexOperations.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 26092 2026-10-24 01:47:31.000000 ./usr/share/info/CharacteristicClasses.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 22867 2026-10-24 01:47:31.000000 ./usr/share/info/Chordal.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 26071 2026-10-24 01:47:31.000000 ./usr/share/info/CharacteristicClasses.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 22852 2026-10-24 01:47:31.000000 ./usr/share/info/Chordal.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3732 2026-10-24 01:47:31.000000 ./usr/share/info/Classic.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 40902 2026-10-24 01:47:31.000000 ./usr/share/info/CodingTheory.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 5344 2026-10-24 01:47:31.000000 ./usr/share/info/CohomCalg.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 5337 2026-10-24 01:47:31.000000 ./usr/share/info/CohomCalg.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 24995 2026-10-24 01:47:31.000000 ./usr/share/info/CoincidentRootLoci.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 59610 2026-10-24 01:47:31.000000 ./usr/share/info/CompleteIntersectionResolutions.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 59613 2026-10-24 01:47:31.000000 ./usr/share/info/CompleteIntersectionResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 233501 2026-10-24 01:47:31.000000 ./usr/share/info/Complexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12241 2026-10-24 01:47:31.000000 ./usr/share/info/ConformalBlocks.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14544 2026-10-24 01:47:31.000000 ./usr/share/info/ConnectionMatrices.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14541 2026-10-24 01:47:31.000000 ./usr/share/info/ConnectionMatrices.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7172 2026-10-24 01:47:31.000000 ./usr/share/info/ConvexInterface.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2611 2026-10-24 01:47:31.000000 ./usr/share/info/ConwayPolynomials.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8622 2026-10-24 01:47:31.000000 ./usr/share/info/CorrespondenceScrolls.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7449 2026-10-24 01:47:31.000000 ./usr/share/info/CotangentSchubert.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 29813 2026-10-24 01:47:31.000000 ./usr/share/info/CpMackeyFunctors.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 96527 2026-10-24 01:47:31.000000 ./usr/share/info/Cremona.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 96510 2026-10-24 01:47:31.000000 ./usr/share/info/Cremona.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1982 2026-10-24 01:47:31.000000 ./usr/share/info/Cyclotomic.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 187220 2026-10-24 01:47:31.000000 ./usr/share/info/DGAlgebras.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 187239 2026-10-24 01:47:31.000000 ./usr/share/info/DGAlgebras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4885 2026-10-24 01:47:31.000000 ./usr/share/info/DecomposableSparseSystems.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6062 2026-10-24 01:47:31.000000 ./usr/share/info/Depth.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 18058 2026-10-24 01:47:31.000000 ./usr/share/info/DeterminantalRepresentations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19250 2026-10-24 01:47:31.000000 ./usr/share/info/DiffAlg.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 12524 2026-10-24 01:47:31.000000 ./usr/share/info/DirectSummands.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 12523 2026-10-24 01:47:31.000000 ./usr/share/info/DirectSummands.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3452 2026-10-24 01:47:31.000000 ./usr/share/info/Dmodules.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15778 2026-10-24 01:47:31.000000 ./usr/share/info/EagonResolution.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 45554 2026-10-24 01:47:31.000000 ./usr/share/info/EdgeIdeals.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 45556 2026-10-24 01:47:31.000000 ./usr/share/info/EdgeIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2730 2026-10-24 01:47:31.000000 ./usr/share/info/EigenSolver.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4475 2026-10-24 01:47:31.000000 ./usr/share/info/Elimination.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4468 2026-10-24 01:47:31.000000 ./usr/share/info/Elimination.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 21429 2026-10-24 01:47:31.000000 ./usr/share/info/EliminationMatrices.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6396 2026-10-24 01:47:31.000000 ./usr/share/info/EliminationTemplates.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7235 2026-10-24 01:47:31.000000 ./usr/share/info/EllipticCurves.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1691 2026-10-24 01:47:31.000000 ./usr/share/info/EllipticIntegrals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2459 2026-10-24 01:47:31.000000 ./usr/share/info/EngineTests.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 2941 2026-10-24 01:47:31.000000 ./usr/share/info/EnumerationCurves.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 9712 2026-10-24 01:47:31.000000 ./usr/share/info/EquivariantGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 2942 2026-10-24 01:47:31.000000 ./usr/share/info/EnumerationCurves.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 9710 2026-10-24 01:47:31.000000 ./usr/share/info/EquivariantGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7474 2026-10-24 01:47:31.000000 ./usr/share/info/EuclideanDistanceDegree.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 20413 2026-10-24 01:47:31.000000 ./usr/share/info/ExampleSystems.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19434 2026-10-24 01:47:31.000000 ./usr/share/info/ExteriorExtensions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5181 2026-10-24 01:47:31.000000 ./usr/share/info/ExteriorIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10414 2026-10-24 01:47:31.000000 ./usr/share/info/ExteriorModules.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2583 2026-10-24 01:47:31.000000 ./usr/share/info/FGLM.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 31638 2026-10-24 01:47:31.000000 ./usr/share/info/FastMinors.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 5253 2026-10-24 01:47:31.000000 ./usr/share/info/FiniteFittingIdeals.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 31632 2026-10-24 01:47:31.000000 ./usr/share/info/FastMinors.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 5252 2026-10-24 01:47:31.000000 ./usr/share/info/FiniteFittingIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1043 2026-10-24 01:47:31.000000 ./usr/share/info/FirstPackage.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 22601 2026-10-24 01:47:31.000000 ./usr/share/info/ForeignFunctions.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 22599 2026-10-24 01:47:31.000000 ./usr/share/info/ForeignFunctions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7467 2026-10-24 01:47:31.000000 ./usr/share/info/FormalGroupLaws.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7408 2026-10-24 01:47:31.000000 ./usr/share/info/FourTiTwo.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7670 2026-10-24 01:47:31.000000 ./usr/share/info/FourierMotzkin.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 17499 2026-10-24 01:47:31.000000 ./usr/share/info/FrobeniusThresholds.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 17489 2026-10-24 01:47:31.000000 ./usr/share/info/FrobeniusThresholds.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2002 2026-10-24 01:47:31.000000 ./usr/share/info/FunctionFieldDesingularization.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 36098 2026-10-24 01:47:31.000000 ./usr/share/info/GKMVarieties.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 36100 2026-10-24 01:47:31.000000 ./usr/share/info/GKMVarieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 23605 2026-10-24 01:47:31.000000 ./usr/share/info/GameTheory.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2724 2026-10-24 01:47:31.000000 ./usr/share/info/GenericInitialIdeal.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15376 2026-10-24 01:47:31.000000 ./usr/share/info/GeometricDecomposability.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 86361 2026-10-24 01:47:31.000000 ./usr/share/info/GradedLieAlgebras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 38557 2026-10-24 01:47:31.000000 ./usr/share/info/GraphicalModels.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 22852 2026-10-24 01:47:31.000000 ./usr/share/info/GraphicalModelsMLE.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14394 2026-10-24 01:47:31.000000 ./usr/share/info/Graphics.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 51498 2026-10-24 01:47:31.000000 ./usr/share/info/Graphs.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 22330 2026-10-24 01:47:31.000000 ./usr/share/info/GroebnerStrata.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4747 2026-10-24 01:47:31.000000 ./usr/share/info/GroebnerWalk.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4463 2026-10-24 01:47:31.000000 ./usr/share/info/Hadamard.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 22384 2026-10-24 01:47:31.000000 ./usr/share/info/GroebnerStrata.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4745 2026-10-24 01:47:31.000000 ./usr/share/info/GroebnerWalk.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4466 2026-10-24 01:47:31.000000 ./usr/share/info/Hadamard.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4157 2026-10-24 01:47:31.000000 ./usr/share/info/HigherCIOperators.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 24610 2026-10-24 01:47:31.000000 ./usr/share/info/HighestWeights.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5264 2026-10-24 01:47:31.000000 ./usr/share/info/HodgeIntegrals.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 12512 2026-10-24 01:47:31.000000 ./usr/share/info/HolonomicSystems.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 6739 2026-10-24 01:47:31.000000 ./usr/share/info/HomotopyLieAlgebra.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 41857 2026-10-24 01:47:31.000000 ./usr/share/info/HyperplaneArrangements.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 12517 2026-10-24 01:47:31.000000 ./usr/share/info/HolonomicSystems.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 6729 2026-10-24 01:47:31.000000 ./usr/share/info/HomotopyLieAlgebra.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 41850 2026-10-24 01:47:31.000000 ./usr/share/info/HyperplaneArrangements.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9253 2026-10-24 01:47:31.000000 ./usr/share/info/IncidenceCorrespondenceCohomology.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6096 2026-10-24 01:47:31.000000 ./usr/share/info/IntegerProgramming.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 26754 2026-10-24 01:47:31.000000 ./usr/share/info/IntegralClosure.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 42595 2026-10-24 01:47:31.000000 ./usr/share/info/InvariantRing.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 26763 2026-10-24 01:47:31.000000 ./usr/share/info/IntegralClosure.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 42596 2026-10-24 01:47:31.000000 ./usr/share/info/InvariantRing.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12127 2026-10-24 01:47:31.000000 ./usr/share/info/InverseSystems.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9403 2026-10-24 01:47:31.000000 ./usr/share/info/InvolutiveBases.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 9407 2026-10-24 01:47:31.000000 ./usr/share/info/Isomorphism.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 9406 2026-10-24 01:47:31.000000 ./usr/share/info/Isomorphism.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3263 2026-10-24 01:47:31.000000 ./usr/share/info/JSON.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4999 2026-10-24 01:47:31.000000 ./usr/share/info/JSONRPC.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 20222 2026-10-24 01:47:31.000000 ./usr/share/info/Jets.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 24266 2026-10-24 01:47:31.000000 ./usr/share/info/K3Carpets.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 20220 2026-10-24 01:47:31.000000 ./usr/share/info/Jets.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 24275 2026-10-24 01:47:31.000000 ./usr/share/info/K3Carpets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7219 2026-10-24 01:47:31.000000 ./usr/share/info/K3Surfaces.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4839 2026-10-24 01:47:31.000000 ./usr/share/info/Kronecker.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19956 2026-10-24 01:47:31.000000 ./usr/share/info/KustinMiller.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 10675 2026-10-24 01:47:31.000000 ./usr/share/info/LLLBases.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 10679 2026-10-24 01:47:31.000000 ./usr/share/info/LLLBases.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1924 2026-10-24 01:47:31.000000 ./usr/share/info/LanguageServer.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 10242 2026-10-24 01:47:31.000000 ./usr/share/info/LatticePolytopes.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 10243 2026-10-24 01:47:31.000000 ./usr/share/info/LatticePolytopes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 11907 2026-10-24 01:47:31.000000 ./usr/share/info/LexIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 54647 2026-10-24 01:47:31.000000 ./usr/share/info/LieAlgebraRepresentations.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 11180 2026-10-24 01:47:31.000000 ./usr/share/info/LinearTruncations.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 11574 2026-10-24 01:47:31.000000 ./usr/share/info/LocalRings.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 11184 2026-10-24 01:47:31.000000 ./usr/share/info/LinearTruncations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 11572 2026-10-24 01:47:31.000000 ./usr/share/info/LocalRings.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14659 2026-10-24 01:47:31.000000 ./usr/share/info/M0nbar.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8784 2026-10-24 01:47:31.000000 ./usr/share/info/MCMApproximations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7822 2026-10-24 01:47:31.000000 ./usr/share/info/MRDI.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 1077880 2026-10-24 01:47:31.000000 ./usr/share/info/Macaulay2Doc.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 1077866 2026-10-24 01:47:31.000000 ./usr/share/info/Macaulay2Doc.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14655 2026-10-24 01:47:31.000000 ./usr/share/info/MacaulayPosets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4315 2026-10-24 01:47:31.000000 ./usr/share/info/MapleInterface.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6011 2026-10-24 01:47:31.000000 ./usr/share/info/Markov.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 30709 2026-10-24 01:47:31.000000 ./usr/share/info/MatchingFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 164097 2026-10-24 01:47:31.000000 ./usr/share/info/MatrixFactorizations.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 37754 2026-10-24 01:47:31.000000 ./usr/share/info/MatrixSchubert.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 37757 2026-10-24 01:47:31.000000 ./usr/share/info/MatrixSchubert.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 67589 2026-10-24 01:47:31.000000 ./usr/share/info/Matroids.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1333 2026-10-24 01:47:31.000000 ./usr/share/info/MergeTeX.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 7086 2026-10-24 01:47:31.000000 ./usr/share/info/MinimalPrimes.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 7089 2026-10-24 01:47:31.000000 ./usr/share/info/MinimalPrimes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3760 2026-10-24 01:47:31.000000 ./usr/share/info/Miura.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 7730 2026-10-24 01:47:31.000000 ./usr/share/info/MixedMultiplicity.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 6317 2026-10-24 01:47:31.000000 ./usr/share/info/ModuleDeformations.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14641 2026-10-24 01:47:31.000000 ./usr/share/info/MonodromySolver.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 7731 2026-10-24 01:47:31.000000 ./usr/share/info/MixedMultiplicity.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 6320 2026-10-24 01:47:31.000000 ./usr/share/info/ModuleDeformations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14639 2026-10-24 01:47:31.000000 ./usr/share/info/MonodromySolver.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 20461 2026-10-24 01:47:31.000000 ./usr/share/info/MonomialAlgebras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12695 2026-10-24 01:47:31.000000 ./usr/share/info/MonomialIntegerPrograms.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5727 2026-10-24 01:47:31.000000 ./usr/share/info/MonomialOrbits.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 10050 2026-10-24 01:47:31.000000 ./usr/share/info/Msolve.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 10048 2026-10-24 01:47:31.000000 ./usr/share/info/Msolve.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10658 2026-10-24 01:47:31.000000 ./usr/share/info/MultiGradedRationalMap.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10203 2026-10-24 01:47:31.000000 ./usr/share/info/MultigradedBGG.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6956 2026-10-24 01:47:31.000000 ./usr/share/info/MultigradedImplicitization.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 7810 2026-10-24 01:47:31.000000 ./usr/share/info/MultiplicitySequence.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 7808 2026-10-24 01:47:31.000000 ./usr/share/info/MultiplicitySequence.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7534 2026-10-24 01:47:31.000000 ./usr/share/info/MultiplierIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4215 2026-10-24 01:47:31.000000 ./usr/share/info/MultiplierIdealsDim2.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 72089 2026-10-24 01:47:31.000000 ./usr/share/info/MultiprojectiveVarieties.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 72106 2026-10-24 01:47:31.000000 ./usr/share/info/MultiprojectiveVarieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 21488 2026-10-24 01:47:31.000000 ./usr/share/info/NAGtypes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 85049 2026-10-24 01:47:31.000000 ./usr/share/info/NCAlgebra.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 15867 2026-10-24 01:47:31.000000 ./usr/share/info/Nauty.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14719 2026-10-24 01:47:31.000000 ./usr/share/info/NautyGraphs.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 15859 2026-10-24 01:47:31.000000 ./usr/share/info/Nauty.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14730 2026-10-24 01:47:31.000000 ./usr/share/info/NautyGraphs.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7533 2026-10-24 01:47:31.000000 ./usr/share/info/NeuralIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4174 2026-10-24 01:47:31.000000 ./usr/share/info/NoetherNormalization.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 34956 2026-10-24 01:47:31.000000 ./usr/share/info/NoetherianOperators.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9612 2026-10-24 01:47:31.000000 ./usr/share/info/NonPrincipalTestIdeals.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 111299 2026-10-24 01:47:31.000000 ./usr/share/info/NormalToricVarieties.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 111308 2026-10-24 01:47:31.000000 ./usr/share/info/NormalToricVarieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 31276 2026-10-24 01:47:31.000000 ./usr/share/info/Normaliz.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3131 2026-10-24 01:47:31.000000 ./usr/share/info/NumericSolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 36449 2026-10-24 01:47:31.000000 ./usr/share/info/NumericalAlgebraicGeometry.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13554 2026-10-24 01:47:31.000000 ./usr/share/info/NumericalCertification.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 17848 2026-10-24 01:47:31.000000 ./usr/share/info/NumericalImplicitization.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 17846 2026-10-24 01:47:31.000000 ./usr/share/info/NumericalImplicitization.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3201 2026-10-24 01:47:31.000000 ./usr/share/info/NumericalLinearAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 31715 2026-10-24 01:47:31.000000 ./usr/share/info/NumericalSchubertCalculus.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 24930 2026-10-24 01:47:31.000000 ./usr/share/info/NumericalSemigroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 20278 2026-10-24 01:47:31.000000 ./usr/share/info/OIGroebnerBases.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 46304 2026-10-24 01:47:31.000000 ./usr/share/info/OldChainComplexes.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 20279 2026-10-24 01:47:31.000000 ./usr/share/info/OIGroebnerBases.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 46290 2026-10-24 01:47:31.000000 ./usr/share/info/OldChainComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2108 2026-10-24 01:47:31.000000 ./usr/share/info/OnlineLookup.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1529 2026-10-24 01:47:31.000000 ./usr/share/info/OpenMath.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 27489 2026-10-24 01:47:31.000000 ./usr/share/info/Oscillators.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 27475 2026-10-24 01:47:31.000000 ./usr/share/info/Oscillators.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 27404 2026-10-24 01:47:31.000000 ./usr/share/info/PHCpack.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2646 2026-10-24 01:47:31.000000 ./usr/share/info/PackageCitations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1714 2026-10-24 01:47:31.000000 ./usr/share/info/PackageTemplate.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8780 2026-10-24 01:47:31.000000 ./usr/share/info/Padic.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9367 2026-10-24 01:47:31.000000 ./usr/share/info/Parametrization.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7247 2026-10-24 01:47:31.000000 ./usr/share/info/Parsing.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 32147 2026-10-24 01:47:31.000000 ./usr/share/info/PathSignatures.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 30041 2026-10-24 01:47:31.000000 ./usr/share/info/PencilsOfQuadrics.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 30040 2026-10-24 01:47:31.000000 ./usr/share/info/PencilsOfQuadrics.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4753 2026-10-24 01:47:31.000000 ./usr/share/info/Permanents.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 18989 2026-10-24 01:47:31.000000 ./usr/share/info/Permutations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19598 2026-10-24 01:47:31.000000 ./usr/share/info/PhylogeneticTrees.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 49993 2026-10-24 01:47:31.000000 ./usr/share/info/PieriMaps.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12261 2026-10-24 01:47:31.000000 ./usr/share/info/PlaneCurveLinearSeries.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 10920 2026-10-24 01:47:31.000000 ./usr/share/info/Points.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 10922 2026-10-24 01:47:31.000000 ./usr/share/info/Points.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 71927 2026-10-24 01:47:31.000000 ./usr/share/info/Polyhedra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1814 2026-10-24 01:47:31.000000 ./usr/share/info/Polymake.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 18845 2026-10-24 01:47:31.000000 ./usr/share/info/PolyominoIdeals.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 54811 2026-10-24 01:47:31.000000 ./usr/share/info/Posets.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 54819 2026-10-24 01:47:31.000000 ./usr/share/info/Posets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9899 2026-10-24 01:47:31.000000 ./usr/share/info/PositivityToricBundles.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 15879 2026-10-24 01:47:31.000000 ./usr/share/info/PrimaryDecomposition.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 15873 2026-10-24 01:47:31.000000 ./usr/share/info/PrimaryDecomposition.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10313 2026-10-24 01:47:31.000000 ./usr/share/info/Probability.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2875 2026-10-24 01:47:31.000000 ./usr/share/info/PseudomonomialPrimaryDecomposition.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2445 2026-10-24 01:47:31.000000 ./usr/share/info/Pullback.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5379 2026-10-24 01:47:31.000000 ./usr/share/info/PushForward.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 38105 2026-10-24 01:47:31.000000 ./usr/share/info/Python.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9607 2026-10-24 01:47:31.000000 ./usr/share/info/QthPower.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6092 2026-10-24 01:47:31.000000 ./usr/share/info/QuadraticIdealExamplesByRoos.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6876 2026-10-24 01:47:31.000000 ./usr/share/info/Quasidegrees.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 56521 2026-10-24 01:47:31.000000 ./usr/share/info/QuaternaryQuartics.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 56519 2026-10-24 01:47:31.000000 ./usr/share/info/QuaternaryQuartics.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14084 2026-10-24 01:47:31.000000 ./usr/share/info/QuillenSuslin.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 24198 2026-10-24 01:47:31.000000 ./usr/share/info/RInterface.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8672 2026-10-24 01:47:31.000000 ./usr/share/info/RandomComplexes.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 13651 2026-10-24 01:47:31.000000 ./usr/share/info/RandomCurves.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4058 2026-10-24 01:47:31.000000 ./usr/share/info/RandomCurvesOverVerySmallFiniteFields.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 15022 2026-10-24 01:47:31.000000 ./usr/share/info/RandomIdeals.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 24197 2026-10-24 01:47:31.000000 ./usr/share/info/RInterface.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8681 2026-10-24 01:47:31.000000 ./usr/share/info/RandomComplexes.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 13652 2026-10-24 01:47:31.000000 ./usr/share/info/RandomCurves.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4057 2026-10-24 01:47:31.000000 ./usr/share/info/RandomCurvesOverVerySmallFiniteFields.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 15010 2026-10-24 01:47:31.000000 ./usr/share/info/RandomIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 30515 2026-10-24 01:47:31.000000 ./usr/share/info/RandomMonomialIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3586 2026-10-24 01:47:31.000000 ./usr/share/info/RandomObjects.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 15034 2026-10-24 01:47:31.000000 ./usr/share/info/RandomPoints.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 18617 2026-10-24 01:47:31.000000 ./usr/share/info/RationalMaps.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 15036 2026-10-24 01:47:31.000000 ./usr/share/info/RandomPoints.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 18616 2026-10-24 01:47:31.000000 ./usr/share/info/RationalMaps.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1703 2026-10-24 01:47:31.000000 ./usr/share/info/RationalPoints.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 10732 2026-10-24 01:47:31.000000 ./usr/share/info/RationalPoints2.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 10733 2026-10-24 01:47:31.000000 ./usr/share/info/RationalPoints2.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 20539 2026-10-24 01:47:31.000000 ./usr/share/info/ReactionNetworks.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10301 2026-10-24 01:47:31.000000 ./usr/share/info/RealRoots.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 37338 2026-10-24 01:47:31.000000 ./usr/share/info/ReesAlgebra.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 37334 2026-10-24 01:47:31.000000 ./usr/share/info/ReesAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12568 2026-10-24 01:47:31.000000 ./usr/share/info/ReflexivePolytopesDB.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 2743 2026-10-24 01:47:31.000000 ./usr/share/info/Regularity.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 2740 2026-10-24 01:47:31.000000 ./usr/share/info/Regularity.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6354 2026-10-24 01:47:31.000000 ./usr/share/info/RelativeCanonicalResolution.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6019 2026-10-24 01:47:31.000000 ./usr/share/info/ResLengthThree.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7484 2026-10-24 01:47:31.000000 ./usr/share/info/ResidualIntersections.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4689 2026-10-24 01:47:31.000000 ./usr/share/info/ResolutionsOfStanleyReisnerRings.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6905 2026-10-24 01:47:31.000000 ./usr/share/info/ResultantComplexes.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 45727 2026-10-24 01:47:31.000000 ./usr/share/info/Resultants.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8747 2026-10-24 01:47:31.000000 ./usr/share/info/RunExternalM2.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 45726 2026-10-24 01:47:31.000000 ./usr/share/info/Resultants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8756 2026-10-24 01:47:31.000000 ./usr/share/info/RunExternalM2.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3682 2026-10-24 01:47:31.000000 ./usr/share/info/SCMAlgebras.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4075 2026-10-24 01:47:31.000000 ./usr/share/info/SCSCP.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 15318 2026-10-24 01:47:31.000000 ./usr/share/info/SLPexpressions.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 15314 2026-10-24 01:47:31.000000 ./usr/share/info/SLPexpressions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8266 2026-10-24 01:47:31.000000 ./usr/share/info/SLnEquivariantMatrices.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 51962 2026-10-24 01:47:31.000000 ./usr/share/info/SRdeformations.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 21940 2026-10-24 01:47:31.000000 ./usr/share/info/SVDComplexes.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 51965 2026-10-24 01:47:31.000000 ./usr/share/info/SRdeformations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 21945 2026-10-24 01:47:31.000000 ./usr/share/info/SVDComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2643 2026-10-24 01:47:31.000000 ./usr/share/info/SagbiGbDetection.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 9618 2026-10-24 01:47:31.000000 ./usr/share/info/Saturation.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 64650 2026-10-24 01:47:31.000000 ./usr/share/info/Schubert2.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 9613 2026-10-24 01:47:31.000000 ./usr/share/info/Saturation.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 64655 2026-10-24 01:47:31.000000 ./usr/share/info/Schubert2.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6088 2026-10-24 01:47:31.000000 ./usr/share/info/SchurComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 25425 2026-10-24 01:47:31.000000 ./usr/share/info/SchurFunctors.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 68707 2026-10-24 01:47:31.000000 ./usr/share/info/SchurRings.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 68712 2026-10-24 01:47:31.000000 ./usr/share/info/SchurRings.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7965 2026-10-24 01:47:31.000000 ./usr/share/info/SchurVeronese.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2652 2026-10-24 01:47:31.000000 ./usr/share/info/SectionRing.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 7699 2026-10-24 01:47:31.000000 ./usr/share/info/SegreClasses.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 7700 2026-10-24 01:47:31.000000 ./usr/share/info/SegreClasses.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7469 2026-10-24 01:47:31.000000 ./usr/share/info/SemidefiniteProgramming.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8394 2026-10-24 01:47:31.000000 ./usr/share/info/Seminormalization.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2627 2026-10-24 01:47:31.000000 ./usr/share/info/Serialization.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8333 2026-10-24 01:47:31.000000 ./usr/share/info/SimpleDoc.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8332 2026-10-24 01:47:31.000000 ./usr/share/info/SimpleDoc.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 76318 2026-10-24 01:47:31.000000 ./usr/share/info/SimplicialComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7232 2026-10-24 01:47:31.000000 ./usr/share/info/SimplicialDecomposability.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 167492 2026-10-24 01:47:31.000000 ./usr/share/info/SimplicialModules.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 167495 2026-10-24 01:47:31.000000 ./usr/share/info/SimplicialModules.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 3044 2026-10-24 01:47:31.000000 ./usr/share/info/SimplicialPosets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 37861 2026-10-24 01:47:31.000000 ./usr/share/info/SlackIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13866 2026-10-24 01:47:31.000000 ./usr/share/info/SpaceCurves.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 27481 2026-10-24 01:47:31.000000 ./usr/share/info/SparseResultants.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 37572 2026-10-24 01:47:31.000000 ./usr/share/info/SpechtModule.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 40448 2026-10-24 01:47:31.000000 ./usr/share/info/SpecialFanoFourfolds.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 27515 2026-10-24 01:47:31.000000 ./usr/share/info/SparseResultants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 37575 2026-10-24 01:47:31.000000 ./usr/share/info/SpechtModule.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 40453 2026-10-24 01:47:31.000000 ./usr/share/info/SpecialFanoFourfolds.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 289560 2026-10-24 01:47:31.000000 ./usr/share/info/SpectralSequences.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 12690 2026-10-24 01:47:31.000000 ./usr/share/info/StatGraphs.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2632 2026-10-24 01:47:31.000000 ./usr/share/info/StatePolytope.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7999 2026-10-24 01:47:31.000000 ./usr/share/info/StronglyStableIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1915 2026-10-24 01:47:31.000000 ./usr/share/info/Style.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 29770 2026-10-24 01:47:31.000000 ./usr/share/info/SubalgebraBases.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15727 2026-10-24 01:47:31.000000 ./usr/share/info/SumsOfSquares.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5366 2026-10-24 01:47:31.000000 ./usr/share/info/SuperLinearAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2753 2026-10-24 01:47:31.000000 ./usr/share/info/SwitchingFields.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14348 2026-10-24 01:47:31.000000 ./usr/share/info/SymbolicPowers.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14350 2026-10-24 01:47:31.000000 ./usr/share/info/SymbolicPowers.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2316 2026-10-24 01:47:31.000000 ./usr/share/info/SymmetricPolynomials.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14494 2026-10-24 01:47:31.000000 ./usr/share/info/TSpreadIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 30234 2026-10-24 01:47:31.000000 ./usr/share/info/Tableaux.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1559 2026-10-24 01:47:31.000000 ./usr/share/info/TangentCone.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 47061 2026-10-24 01:47:31.000000 ./usr/share/info/TateOnProducts.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 47062 2026-10-24 01:47:31.000000 ./usr/share/info/TateOnProducts.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 22580 2026-10-24 01:47:31.000000 ./usr/share/info/TensorComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2548 2026-10-24 01:47:31.000000 ./usr/share/info/TerraciniLoci.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2115 2026-10-24 01:47:31.000000 ./usr/share/info/TestAudit.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 31168 2026-10-24 01:47:31.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 31174 2026-10-24 01:47:31.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15163 2026-10-24 01:47:31.000000 ./usr/share/info/Text.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 23201 2026-10-24 01:47:31.000000 ./usr/share/info/ThinSincereQuivers.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8350 2026-10-24 01:47:31.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8574 2026-10-24 01:47:31.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13976 2026-10-24 01:47:31.000000 ./usr/share/info/Topcom.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7724 2026-10-24 01:47:31.000000 ./usr/share/info/TorAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6967 2026-10-24 01:47:31.000000 ./usr/share/info/ToricHigherDirectImages.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4013 2026-10-24 01:47:31.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4012 2026-10-24 01:47:31.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9922 2026-10-24 01:47:31.000000 ./usr/share/info/ToricTopology.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 31419 2026-10-24 01:47:31.000000 ./usr/share/info/ToricVectorBundles.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2026-10-24 01:47:31.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33244 2026-10-24 01:47:31.000000 ./usr/share/info/Triangulations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33248 2026-10-24 01:47:31.000000 ./usr/share/info/Triangulations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7304 2026-10-24 01:47:31.000000 ./usr/share/info/Triplets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 11010 2026-10-24 01:47:31.000000 ./usr/share/info/Tropical.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10699 2026-10-24 01:47:31.000000 ./usr/share/info/TropicalToric.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5763 2026-10-24 01:47:31.000000 ./usr/share/info/Truncations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2026-10-24 01:47:31.000000 ./usr/share/info/Units.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4688 2026-10-24 01:47:31.000000 ./usr/share/info/VNumber.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8937 2026-10-24 01:47:31.000000 ./usr/share/info/Valuations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 44396 2026-10-24 01:47:31.000000 ./usr/share/info/Varieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19350 2026-10-24 01:47:31.000000 ./usr/share/info/VectorFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 51845 2026-10-24 01:47:31.000000 ./usr/share/info/VectorGraphics.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 41360 2026-10-24 01:47:31.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13242 2026-10-24 01:47:31.000000 ./usr/share/info/VirtualResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10436 2026-10-24 01:47:31.000000 ./usr/share/info/Visualize.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33371 2026-10-24 01:47:31.000000 ./usr/share/info/WeierstrassSemigroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 38095 2026-10-24 01:47:31.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33368 2026-10-24 01:47:31.000000 ./usr/share/info/WeierstrassSemigroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 38084 2026-10-24 01:47:31.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10618 2026-10-24 01:47:31.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33219 2026-10-24 01:47:31.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14718 2026-10-24 01:47:31.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33194 2026-10-24 01:47:31.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14717 2026-10-24 01:47:31.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19544 2026-10-24 01:47:31.000000 ./usr/share/info/WittVectors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8865 2026-10-24 01:47:31.000000 ./usr/share/info/XML.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 49515 2026-10-24 01:47:31.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 49516 2026-10-24 01:47:31.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 10748 2026-10-23 21:15:50.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/Macaulay2/Style/katex/contrib/copy-tex.min.js -> ../../../../javascript/katex/contrib/copy-tex.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/Macaulay2/Style/katex/contrib/render-a11y-string.min.js -> ../../../../javascript/katex/contrib/render-a11y-string.js │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/Macaulay2/Style/katex/fonts/KaTeX_AMS-Regular.ttf -> ../../../../fonts/truetype/katex/KaTeX_AMS-Regular.ttf │ │ ├── ./usr/share/doc/Macaulay2/A1BrouwerDegrees/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ Z2V0VHJhY2U= │ │ │ #:len=2312 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIHRyYWNlIG92ZXIg │ │ │ JGskIGZvciBhbiBlbGVtZW50IGluIGEgZmluaXRlIGRpbWVuc2lvbmFsICRrJCAtYWxnZWJyYSIs │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ aGFzTWluaW1hbE11bHQoUmluZyxJbmZpbml0ZU51bWJlcik= │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU1OCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaGFzTWluaW1hbE11bHQsUmluZyxJbmZpbml0ZU51 │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ @@ -10,25 +10,25 @@ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ - -- 2.44098s elapsed │ │ │ + -- 1.76023s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ - -- 2.39145s elapsed │ │ │ + -- 2.21339s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ @@ -95,28 +95,28 @@ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 2.44098s elapsed
│ │ │ + -- 1.76023s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o3 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o3 : Complex
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i4 : elapsedTime burkeResolution(M, 7, Check => true)
│ │ │ - -- 2.39145s elapsed
│ │ │ + -- 2.21339s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o4 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o4 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -23,24 +23,24 @@ │ │ │ │ i2 : M = coker vars R │ │ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ │ │ 1 │ │ │ │ o2 : R-module, quotient of R │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ │ - -- 2.44098s elapsed │ │ │ │ + -- 1.76023s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o3 : Complex │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ │ - -- 2.39145s elapsed │ │ │ │ + -- 2.21339s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ ZmFjZXRzKEFic3RyYWN0U2ltcGxpY2lhbENvbXBsZXgp │ │ │ #:len=368 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmYWNldHMsQWJzdHJhY3RTaW1wbGljaWFsQ29tcGxl │ │ ├── ./usr/share/doc/Macaulay2/AbstractToricVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ QWJzdHJhY3RUb3JpY1ZhcmlldGllcw== │ │ │ #:len=379 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibGlua3MgYWJzdHJhY3Qgc2ltcGxpY2lh │ │ │ bCAobm9ybWFsKSB0b3JpYyB2YXJpZXRpZXMgdG8gU2NodWJlcnQyIiwgRGVzY3JpcHRpb24gPT4g │ │ ├── ./usr/share/doc/Macaulay2/AdjointIdeal/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=25 │ │ │ dHJhY2VNYXRyaXgoSWRlYWwsTWF0cml4KQ== │ │ │ #:len=268 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE3MCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodHJhY2VNYXRyaXgsSWRlYWwsTWF0cml4KSwidHJh │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ YWRqdW5jdGlvblByb2Nlc3MoSWRlYWwsWlop │ │ │ #:len=310 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjc4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGp1bmN0aW9uUHJvY2VzcyxJZGVhbCxaWiksImFk │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ o8 : BettiTally │ │ │ │ │ │ i9 : c=codim I │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ i10 : elapsedTime fI=res I │ │ │ - -- .0257838s elapsed │ │ │ + -- .0378869s elapsed │ │ │ │ │ │ 1 14 33 28 8 │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o10 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ @@ -87,30 +87,30 @@ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : phi=map(P2,Pn,H); │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ i15 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .609185s elapsed │ │ │ + -- .741131s elapsed │ │ │ │ │ │ 0 1 │ │ │ o15 = total: 1 11 │ │ │ 0: 1 . │ │ │ 1: . 3 │ │ │ 2: . 8 │ │ │ │ │ │ o15 : BettiTally │ │ │ │ │ │ i16 : I'== I │ │ │ │ │ │ o16 = true │ │ │ │ │ │ i17 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 6.98144s elapsed │ │ │ + -- 6.3908s elapsed │ │ │ │ │ │ i18 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o18 = Tally{(1, 1, total: 1 2) => 5} │ │ │ 0: 1 2 │ │ │ 0 1 │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ @@ -79,40 +79,40 @@ │ │ │ 1: . . │ │ │ 2: . . │ │ │ 3: . 8 │ │ │ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : elapsedTime sub(I,H) │ │ │ - -- .0200526s elapsed │ │ │ + -- .0228224s elapsed │ │ │ │ │ │ o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) │ │ │ │ │ │ o14 : Ideal of P2 │ │ │ │ │ │ i15 : phi=map(P2,Pn,H); │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ i16 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .0565566s elapsed │ │ │ + -- .0792646s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 1.96041s elapsed │ │ │ + -- 1.8465s elapsed │ │ │ │ │ │ i19 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o19 = Tally{(0, 34, total: 1 15) => 1} │ │ │ 0: 1 . │ │ │ 1: . . │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ @@ -154,15 +154,15 @@ │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i10 : elapsedTime fI=res I
│ │ │ - -- .0257838s elapsed
│ │ │ + -- .0378869s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn
│ │ │                                            
│ │ │        0       1        2        3        4
│ │ │  
│ │ │  o10 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2: . 12 │ │ │ │ │ │ │ │ o8 : BettiTally │ │ │ │ i9 : c=codim I │ │ │ │ │ │ │ │ o9 = 4 │ │ │ │ i10 : elapsedTime fI=res I │ │ │ │ - -- .0257838s elapsed │ │ │ │ + -- .0378869s elapsed │ │ │ │ │ │ │ │ 1 14 33 28 8 │ │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o10 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ @@ -222,15 +222,15 @@ │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .609185s elapsed
│ │ │ + -- .741131s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -243,15 +243,15 @@
│ │ │  
│ │ │  o16 = true
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 6.98144s elapsed
│ │ │ + -- 6.3908s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                            0 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -110,28 +110,28 @@
│ │ │ │            6: . 7
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o14 : RingMap P2 <-- Pn
│ │ │ │  i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .609185s elapsed
│ │ │ │ + -- .741131s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o15 = total: 1 11
│ │ │ │            0: 1  .
│ │ │ │            1: .  3
│ │ │ │            2: .  8
│ │ │ │  
│ │ │ │  o15 : BettiTally
│ │ │ │  i16 : I'== I
│ │ │ │  
│ │ │ │  o16 = true
│ │ │ │  i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 6.98144s elapsed
│ │ │ │ + -- 6.3908s elapsed
│ │ │ │  i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                            0 1
│ │ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │ │                         0: 1 2
│ │ │ │                            0 1
│ │ │ │              (1, 3, total: 1 3) => 8
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o13 : BettiTally
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0200526s elapsed
│ │ │ + -- .0228224s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │ │ │ │ │ │ │ │ │ │ @@ -215,15 +215,15 @@ │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .0565566s elapsed
│ │ │ + -- .0792646s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │ @@ -235,15 +235,15 @@ │ │ │ │ │ │ o17 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 1.96041s elapsed
│ │ │ + -- 1.8465s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                             0  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,36 +82,36 @@
│ │ │ │            0: 1 .
│ │ │ │            1: . .
│ │ │ │            2: . .
│ │ │ │            3: . 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : elapsedTime sub(I,H)
│ │ │ │ - -- .0200526s elapsed
│ │ │ │ + -- .0228224s elapsed
│ │ │ │  
│ │ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │ │  
│ │ │ │  o14 : Ideal of P2
│ │ │ │  i15 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o15 : RingMap P2 <-- Pn
│ │ │ │  i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .0565566s elapsed
│ │ │ │ + -- .0792646s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 1.96041s elapsed
│ │ │ │ + -- 1.8465s elapsed
│ │ │ │  i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                             0  1
│ │ │ │  o19 = Tally{(0, 34, total: 1 15) => 1}
│ │ │ │                          0: 1  .
│ │ │ │                          1: .  .
│ │ │ │                          2: .  .
│ │ ├── ./usr/share/doc/Macaulay2/AlgebraicSplines/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c3RhbmxleVJlaXNuZXIoTGlzdCxMaXN0KQ==
│ │ │  #:len=286
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjI0NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3RhbmxleVJlaXNuZXIsTGlzdCxMaXN0KSwic3Rh
│ │ ├── ./usr/share/doc/Macaulay2/AllMarkovBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  cmFuZG9tTWFya292KE1hdHJpeCk=
│ │ │  #:len=265
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21NYXJrb3YsTWF0cml4KSwicmFuZG9tTWFy
│ │ ├── ./usr/share/doc/Macaulay2/AnalyzeSheafOnP1/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  YW5hbHl6ZShNb2R1bGUp
│ │ │  #:len=251
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbmFseXplLE1vZHVsZSksImFuYWx5emUoTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/AssociativeAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c2VxdWVuY2VUb1ZhcmlhYmxlU3ltYm9scw==
│ │ │  #:len=228
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgsICJ1bmRvY3VtZW50ZWQiID0+IHRy
│ │ │  dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InNlcXVlbmNl
│ │ ├── ./usr/share/doc/Macaulay2/BGG/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  dGF0ZVJlc29sdXRpb24=
│ │ │  #:len=2005
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluaXRlIHBpZWNlIG9mIHRoZSBUYXRl
│ │ │  IHJlc29sdXRpb24iLCAibGluZW51bSIgPT4gNzUzLCBJbnB1dHMgPT4ge1NQQU57VFR7Im0ifSwi
│ │ ├── ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out
│ │ │ @@ -114,26 +114,26 @@
│ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │  
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │  
│ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ + -- used 0.805454s (cpu); 0.608557s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │  o14 : BettiTally
│ │ │  
│ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ + -- used 0.669115s (cpu); 0.563462s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html
│ │ │ @@ -258,15 +258,15 @@
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ + -- used 0.805454s (cpu); 0.608557s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -277,15 +277,15 @@
│ │ │          

│ │ │            
With the form pureResolution(p,q,D) we can directly create the situation of pureResolution(M,D) where M is generic product(m_i+1) x #D-1+sum(m_i) matrix of linear forms defined over a ring with product(m_i+1) * #D-1+sum(m_i) variables of characteristic p, created by the script. For a given number of variables in A this runs much faster than taking a random matrix M.

│ │ │          

│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,27 +192,27 @@
│ │ │ │  o18 : ActionOnComplex
│ │ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │ │  
│ │ │ │  o19 = Complex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- 1.06712s elapsed
│ │ │ │ + -- .790935s elapsed
│ │ │ │  
│ │ │ │  o20 = Character over kk
│ │ │ │  
│ │ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │ │                   |                4    2         4    2
│ │ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │ │        (2, {13})  |  1   1  1  1            1                  1
│ │ │ │  
│ │ │ │  o20 : Character
│ │ │ │  i21 : elapsedTime a2 = character A2
│ │ │ │ - -- 36.3539s elapsed
│ │ │ │ + -- 27.9166s elapsed
│ │ │ │  
│ │ │ │  o21 = Character over kk
│ │ │ │  
│ │ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │ │                   |                4    2         4    2
│ │ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ │ @@ -319,15 +319,15 @@
│ │ │ │  i30 : M = Is2 / I2;
│ │ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │ │  
│ │ │ │  o31 = Module with 6 actors
│ │ │ │  
│ │ │ │  o31 : ActionOnGradedModule
│ │ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ │ - -- 16.4277s elapsed
│ │ │ │ + -- 12.9409s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over kk
│ │ │ │  
│ │ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │ │  
│ │ │ │  o32 : Character
│ │ │ │  i33 : b/T
│ │ ├── ./usr/share/doc/Macaulay2/BinomialEdgeIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=3
│ │ │  YmVp
│ │ │  #:len=336
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQmlub21pYWwgZWRnZSBpZGVhbCIsIERl
│ │ │  c2NyaXB0aW9uID0+ICgiYmVpIGlzIGEgc3lub255bSBmb3IgIixUT3tuZXcgRG9jdW1lbnRUYWcg
│ │ ├── ./usr/share/doc/Macaulay2/Binomials/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Ymlub21pYWxJc1ByaW1l
│ │ │  #:len=1312
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGVzdCBmb3IgcHJpbWVuZXNzIG9mIGEg
│ │ │  Ymlub21pYWwgaWRlYWwiLCAibGluZW51bSIgPT4gMTU4MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJJ
│ │ ├── ./usr/share/doc/Macaulay2/BoijSoederberg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bWF0cml4KEJldHRpVGFsbHksWlop
│ │ │  #:len=291
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAyMCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWF0cml4LEJldHRpVGFsbHksWlopLCJtYXRyaXgo
│ │ ├── ./usr/share/doc/Macaulay2/Book3264Examples/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  SW50ZXJzZWN0aW9uIFRoZW9yeSBTZWN0aW9uIDUuMg==
│ │ │  #:len=1578
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQmFzaWNzIG9mIHZlY3RvciBidW5kbGVz
│ │ │  IGFuZCBDaGVybiBjbGFzc2VzIiwgRGVzY3JpcHRpb24gPT4gKERJVntQQVJBe1RFWHsiSW4gU2No
│ │ ├── ./usr/share/doc/Macaulay2/BooleanGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  Z2JCb29sZWFuKElkZWFsKQ==
│ │ │  #:len=1781
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZSBHcm9lYm5lciBCYXNpcyBm
│ │ │  b3IgSWRlYWxzIGluIEJvb2xlYW4gUG9seW5vbWlhbCBRdW90aWVudCBSaW5nIiwgImxpbmVudW0i
│ │ ├── ./usr/share/doc/Macaulay2/Brackets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  R0NBbGdlYnJh
│ │ │  #:len=1521
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiAoRElWe1BBUkF7VEVYeyJBbiBvYmpl
│ │ │  Y3Qgb2YgY2xhc3MgR0NBbGdlYnJhIHJlcHJlc2VudHMgYSBHcmFzc21hbm4tQ2F5bGV5IGFsZ2Vi
│ │ ├── ./usr/share/doc/Macaulay2/Browse/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  QnJvd3Nl
│ │ │  #:len=397
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBtZXRob2QgZm9yIGJyb3dzaW5nIGFu
│ │ │  ZCBleGFtaW5pbmcgTWFjYXVsYXkyIGRhdGEgc3RydWN0dXJlcyIsIERlc2NyaXB0aW9uID0+ICgi
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  aXNTeXp5Z3koTW9kdWxlLFpaKQ==
│ │ │  #:len=228
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc1N5enlneSxNb2R1bGUsWlopLCJpc1N5enlneShN
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out
│ │ │ @@ -230,15 +230,15 @@
│ │ │            0: 1 . . . .
│ │ │            1: . 4 2 . .
│ │ │            2: . 1 6 5 1
│ │ │  
│ │ │  o22 : BettiTally
│ │ │  
│ │ │  i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ + -- used 0.562666s (cpu); 0.372499s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
│ │ │  
│ │ │  i24 : betti res j
│ │ │  
│ │ │               0 1 2 3 4
│ │ │  o24 = total: 1 3 6 5 1
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html
│ │ │ @@ -385,15 +385,15 @@
│ │ │  
│ │ │  o22 : BettiTally
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,27 +26,27 @@
│ │ │ │  i3 : P2 = convexHull matrix {{2,-2,0},{1,1,0}};
│ │ │ │  i4 : P3 = convexHull matrix {{-2,-2,0},{1,-1,0}};
│ │ │ │  i5 : P4 = convexHull matrix {{-2,2,0},{-1,-1,0}};
│ │ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │ │  i7 : C = cellComplex(R,F);
│ │ │ │  i8 : facePoset C
│ │ │ │  
│ │ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 |
│ │ │ │ -                      | 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 |
│ │ │ │ -                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 |
│ │ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 |
│ │ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 |
│ │ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ │ +                      | 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ │ +                      | 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ │ +                      | 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │ │  
│ │ │ │  o8 : Poset
│ │ │ │  The labels on the vertices can be controlled via the optional parameter Labels
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.html
│ │ │ @@ -134,16 +134,16 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          

│ │ │                
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ + -- used 0.669115s (cpu); 0.563462s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │ ├── html2text {}
│ │ │ │ @@ -161,30 +161,30 @@
│ │ │ │        | -30a-29b -29a-24b -47a-39b 38a+2b   |
│ │ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o13 : Matrix A  <-- A
│ │ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ │ - -- used 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ │ + -- used 0.805454s (cpu); 0.608557s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o14 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ │ │  
│ │ │ │  o14 : BettiTally
│ │ │ │  With the form pureResolution(p,q,D) we can directly create the situation of
│ │ │ │  pureResolution(M,D) where M is generic product(m_i+1) x #D-1+sum(m_i) matrix of
│ │ │ │  linear forms defined over a ring with product(m_i+1) * #D-1+sum(m_i) variables
│ │ │ │  of characteristic p, created by the script. For a given number of variables in
│ │ │ │  A this runs much faster than taking a random matrix M.
│ │ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ │ - -- used 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ │ + -- used 0.669115s (cpu); 0.563462s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BIBasis/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  YmlCYXNpcyguLi4sdG9Hcm9lYm5lcj0+Li4uKQ==
│ │ │  #:len=240
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODMsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JpQmFzaXMsdG9Hcm9lYm5lcl0sImJpQmFzaXMoLi4u
│ │ ├── ./usr/share/doc/Macaulay2/BeginningMacaulay2/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  QmVnaW5uaW5nTWFjYXVsYXky
│ │ │  #:len=29304
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTWF0aGVtYXRpY2lhbnMnIEludHJvZHVj
│ │ │  dGlvbiB0byAgTWFjYXVsYXkyIiwgRGVzY3JpcHRpb24gPT4gKERJVntQQVJBe1RFWHsiV2UgYXNz
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  QmVuY2htYXJr
│ │ │  #:len=347
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3RhbmRhcmQgTWFjYXVsYXkyIGJlbmNo
│ │ │  bWFya3MiLCBEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMgcGFja2FnZSBwcm92
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sat Oct 24 03:20:06 UTC 2026
│ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Mon Oct 26 10:52:45 UTC 2026
│ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .195653 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -80,19 +80,19 @@
│ │ │          

│ │ │            
The tests available are: 
"deg2generic" -- gb of a generic ideal of codimension 2 and degree 2
"gb4by4comm" -- gb of the ideal of generic commuting 4 by 4 matrices over ZZ/101
"gb3445" -- gb of an ideal with elements of degree 3,4,4,5 in 8 variables
"gbB148" -- gb of Bayesian graph ideal #148
"res39" -- res of a generic 3 by 9 matrix over ZZ/101
"resG25" -- res of the coordinate ring of Grassmannian(2,5)
"yang-gb1" -- an example of Yang-Hui He arising in string theory
"yang-subring" -- an example of Yang-Hui He

│ │ │          

│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          

│ │ │                
i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sat Oct 24 03:20:06 UTC 2026
│ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Mon Oct 26 10:52:45 UTC 2026
│ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998  
│ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .195653 seconds
│ │ │              

│ │ │        
│ │ │        

│ │ │          

│ │ │            
For the programmer

│ │ │ ├── html2text {}
│ │ │ │ @@ -23,18 +23,18 @@
│ │ │ │  "gb3445" -- gb of an ideal with elements of degree 3,4,4,5 in 8 variables
│ │ │ │  "gbB148" -- gb of Bayesian graph ideal #148
│ │ │ │  "res39" -- res of a generic 3 by 9 matrix over ZZ/101
│ │ │ │  "resG25" -- res of the coordinate ring of Grassmannian(2,5)
│ │ │ │  "yang-gb1" -- an example of Yang-Hui He arising in string theory
│ │ │ │  "yang-subring" -- an example of Yang-Hui He
│ │ │ │  i1 : runBenchmarks "res39"
│ │ │ │ --- beginning computation Sat Oct 24 03:20:06 UTC 2026
│ │ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1
│ │ │ │ -(2026-10-15) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250
│ │ │ │ +-- beginning computation Mon Oct 26 10:52:45 UTC 2026
│ │ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2099.998
│ │ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .195653 seconds
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_u_n_B_e_n_c_h_m_a_r_k_s is a _c_o_m_m_a_n_d.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Benchmark.m2:319:0.
│ │ ├── ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  RGxvY2FsaXplQWxsKElkZWFsLFJpbmdFbGVtZW50KQ==
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhEbG9jYWxpemVBbGwsSWRlYWwsUmluZ0VsZW1lbnQp
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  U3RlcHNGb3JJbmNyZWFzZQ==
│ │ │  #:len=268
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzM3Mywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiU3RlcHNGb3JJbmNyZWFzZSIsIlN0ZXBzRm9ySW5j
│ │ │ @@ -515,15 +515,15 @@
│ │ │  Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbUNvbXBsZXgifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57fX0sU1BBTntUTzJ7bmV3
│ │ │  IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pVXNlckhvbW90b3B5LFJhbmRvbVJlYWxdLCJiZXJ0
│ │ │  aW5pVXNlckhvbW90b3B5KC4uLixSYW5kb21SZWFsPT4uLi4pIiwiQmVydGluaSJ9LCJSYW5kb21S
│ │ │  ZWFsIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZhbHVlICIsInt9
│ │ │  In0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHsiVG9wRGlyZWN0
│ │ │  b3J5IiwiVG9wRGlyZWN0b3J5IiwiQmVydGluaSJ9LCJUb3BEaXJlY3RvcnkifSxUVHsiID0+ICJ9
│ │ │ -LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTIzNjY5LTAv
│ │ │ +LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiXCIvdG1wL00yLTMwMTA5LTAv
│ │ │  MFwiIn0sIiwgIixTUEFOeyJPcHRpb24gdG8gY2hhbmdlIGRpcmVjdG9yeSBmb3IgZmlsZSBzdG9y
│ │ │  YWdlLiJ9fSxTUEFOe1RPMntuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JlcnRpbmlVc2VySG9tb3Rv
│ │ │  cHksVmVyYm9zZV0sImJlcnRpbmlVc2VySG9tb3RvcHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0
│ │ │  aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQg
│ │ │  dmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwg
│ │ │  b3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsi
│ │ │  YmVydGluaVVzZXJIb21vdG9weSIsImJlcnRpbmlVc2VySG9tb3RvcHkiLCJCZXJ0aW5pIn0sIEtl
│ │ │ @@ -1100,15 +1100,15 @@
│ │ │  ZXJ0aW5pUGFyYW1ldGVySG9tb3RvcHkoLi4uLFJhbmRvbVJlYWw9Pi4uLikiLCJCZXJ0aW5pIn0s
│ │ │  IlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFs
│ │ │  dWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGljaCBkZXNpZ25hdGVzIHN5bWJvbHMv
│ │ │  c3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0byBiZSBhIHJhbmRvbSByZWFsIG51
│ │ │  bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BBTntUTzJ7bmV3IERvY3VtZW50VGFn
│ │ │  IGZyb20geyJUb3BEaXJlY3RvcnkiLCJUb3BEaXJlY3RvcnkiLCJCZXJ0aW5pIn0sIlRvcERpcmVj
│ │ │  dG9yeSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIsICIsU1BBTnsiZGVmYXVsdCB2YWx1ZSAiLCJc
│ │ │ -Ii90bXAvTTItMjM2NjktMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │ +Ii90bXAvTTItMzAxMDktMC8wXCIifSwiLCAiLFNQQU57Ik9wdGlvbiB0byBjaGFuZ2UgZGlyZWN0
│ │ │  b3J5IGZvciBmaWxlIHN0b3JhZ2UuIn19LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVBhcmFtZXRlckhvbW90b3B5LFZlcmJvc2VdLCJiZXJ0aW5pUGFyYW1ldGVySG9tb3Rv
│ │ │  cHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRU
│ │ │  eyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9w
│ │ │  dGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwgb3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiYmVydGluaVBhcmFtZXRlckhvbW90b3B5IiwiYmVy
│ │ │  dGluaVBhcmFtZXRlckhvbW90b3B5IiwiQmVydGluaSJ9LCBLZXkgPT4gYmVydGluaVBhcmFtZXRl
│ │ │ @@ -2449,15 +2449,15 @@
│ │ │  YWw9Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGlj
│ │ │  aCBkZXNpZ25hdGVzIHN5bWJvbHMvc3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0
│ │ │  byBiZSBhIHJhbmRvbSByZWFsIG51bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BB
│ │ │  TntUTzJ7bmV3IERvY3VtZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFRvcERpcmVj
│ │ │  dG9yeV0sImJlcnRpbmlaZXJvRGltU29sdmUoLi4uLFRvcERpcmVjdG9yeT0+Li4uKSIsIkJlcnRp
│ │ │  bmkifSwiVG9wRGlyZWN0b3J5In0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZh
│ │ │ -dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0yMzY2OS0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │ +dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0zMDEwOS0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │  IGNoYW5nZSBkaXJlY3RvcnkgZm9yIGZpbGUgc3RvcmFnZS4ifX0sU1BBTntUTzJ7bmV3IERvY3Vt
│ │ │  ZW50VGFnIGZyb20ge1tiZXJ0aW5pWmVyb0RpbVNvbHZlLFVzZVJlZ2VuZXJhdGlvbl0sImJlcnRp
│ │ │  bmlaZXJvRGltU29sdmUoLi4uLFVzZVJlZ2VuZXJhdGlvbj0+Li4uKSIsIkJlcnRpbmkifSwiVXNl
│ │ │  UmVnZW5lcmF0aW9uIn0sVFR7IiA9PiAifSxUVHsiLi4uIn0sIiwgIixTUEFOeyJkZWZhdWx0IHZh
│ │ │  bHVlICIsIi0xIn0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVplcm9EaW1Tb2x2ZSxWZXJib3NlXSwiYmVydGluaVplcm9EaW1Tb2x2ZSguLi4sVmVy
│ │ │  Ym9zZT0+Li4uKSIsIkJlcnRpbmkifSwiVmVyYm9zZSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │              
  • HomVariableGroup => ..., default value {}, an option to group variables and use multihomogeneous homotopies
    • 
│ │ │              
  • M2Precision (missing documentation)
│ │ │   => ..., default value 53, 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30109-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list whose entries are lists of solutions for each target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30109-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list whose entries are lists of solutions for each
│ │ │ │              target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │   => ..., default value 53, 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │              
  • RandomReal (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30109-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S0, a list, a list of solutions to the target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │              value {},
│ │ │ │            o HomVariableGroup (missing documentation) => ..., default value {},
│ │ │ │            o M2Precision (missing documentation) => ..., default value 53,
│ │ │ │            o OutputStyle (missing documentation) => ..., default value
│ │ │ │              "OutPoints",
│ │ │ │            o RandomComplex (missing documentation) => ..., default value {},
│ │ │ │            o RandomReal (missing documentation) => ..., default value {},
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30109-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S0, a _l_i_s_t, a list of solutions to the target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method calls Bertini to track a user-defined homotopy. The user needs to
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │   => ..., default value "main_data", 
│ │ │              
  • NameSolutionsFile (missing documentation)
│ │ │   => ..., default value "raw_solutions", 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30109-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • UseRegeneration (missing documentation)
│ │ │   => ..., default value -1, 
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list of points that are contained in the variety of F
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30109-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o UseRegeneration (missing documentation) => ..., default value -1,
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list of points that are contained in the variety of F
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  YWN0aW9uKE1vZHVsZSxMaXN0KQ==
│ │ │  #:len=290
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzgzNywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYWN0aW9uLE1vZHVsZSxMaXN0KSwiYWN0aW9uKE1v
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out
│ │ │ @@ -76,15 +76,15 @@
│ │ │  i8 : A = action(RI,S7)
│ │ │  
│ │ │  o8 = Complex with 15 actors
│ │ │  
│ │ │  o8 : ActionOnComplex
│ │ │  
│ │ │  i9 : elapsedTime c = character A
│ │ │ - -- .449182s elapsed
│ │ │ + -- .405787s elapsed
│ │ │  
│ │ │  o9 = Character over QQ
│ │ │        
│ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out
│ │ │ @@ -100,15 +100,15 @@
│ │ │  i6 : A=action(RI,S6)
│ │ │  
│ │ │  o6 = Complex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
│ │ │  
│ │ │  i7 : elapsedTime c=character A
│ │ │ - -- 1.06265s elapsed
│ │ │ + -- .641243s elapsed
│ │ │  
│ │ │  o7 = Character over QQ
│ │ │        
│ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out
│ │ │ @@ -187,28 +187,28 @@
│ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │  
│ │ │  o19 = Complex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- 1.06712s elapsed
│ │ │ + -- .790935s elapsed
│ │ │  
│ │ │  o20 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │                   |                4    2         4    2
│ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │        (2, {13})  |  1   1  1  1            1                  1
│ │ │  
│ │ │  o20 : Character
│ │ │  
│ │ │  i21 : elapsedTime a2 = character A2
│ │ │ - -- 36.3539s elapsed
│ │ │ + -- 27.9166s elapsed
│ │ │  
│ │ │  o21 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │                   |                4    2         4    2
│ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ @@ -308,15 +308,15 @@
│ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
│ │ │  
│ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ - -- 16.4277s elapsed
│ │ │ + -- 12.9409s elapsed
│ │ │  
│ │ │  o32 = Character over kk
│ │ │         
│ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │  
│ │ │  o32 : Character
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
│ │ │ @@ -167,15 +167,15 @@
│ │ │  
│ │ │  o8 : ActionOnComplex
    • 
│ │ │              
    
│ │ │                
    i9 : elapsedTime c = character A
│ │ │ - -- .449182s elapsed
│ │ │ + -- .405787s elapsed
│ │ │  
│ │ │  o9 = Character over QQ
│ │ │        
│ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ │ ├── html2text {}
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │  o7 : List
│ │ │ │  i8 : A = action(RI,S7)
│ │ │ │  
│ │ │ │  o8 = Complex with 15 actors
│ │ │ │  
│ │ │ │  o8 : ActionOnComplex
│ │ │ │  i9 : elapsedTime c = character A
│ │ │ │ - -- .449182s elapsed
│ │ │ │ + -- .405787s elapsed
│ │ │ │  
│ │ │ │  o9 = Character over QQ
│ │ │ │  
│ │ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime c=character A
│ │ │ - -- 1.06265s elapsed
│ │ │ + -- .641243s elapsed
│ │ │  
│ │ │  o7 = Character over QQ
│ │ │        
│ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ │ ├── html2text {}
│ │ │ │ @@ -113,15 +113,15 @@
│ │ │ │  o5 : List
│ │ │ │  i6 : A=action(RI,S6)
│ │ │ │  
│ │ │ │  o6 = Complex with 11 actors
│ │ │ │  
│ │ │ │  o6 : ActionOnComplex
│ │ │ │  i7 : elapsedTime c=character A
│ │ │ │ - -- 1.06265s elapsed
│ │ │ │ + -- .641243s elapsed
│ │ │ │  
│ │ │ │  o7 = Character over QQ
│ │ │ │  
│ │ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime a1 = character A1
│ │ │ - -- 1.06712s elapsed
│ │ │ + -- .790935s elapsed
│ │ │  
│ │ │  o20 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │                   |                4    2         4    2
│ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │ @@ -331,15 +331,15 @@
│ │ │  
│ │ │  o20 : Character
    
│ │ │              
    
│ │ │                
    i21 : elapsedTime a2 = character A2
│ │ │ - -- 36.3539s elapsed
│ │ │ + -- 27.9166s elapsed
│ │ │  
│ │ │  o21 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │                   |                4    2         4    2
│ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ @@ -483,15 +483,15 @@
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │              
    
│ │ │                
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 16.4277s elapsed
│ │ │ + -- 12.9409s elapsed
│ │ │  
│ │ │  o32 = Character over kk
│ │ │         
│ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ + -- used 0.562666s (cpu); 0.372499s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
    
│ │ │              
    
│ │ │                
    i24 : betti res j
│ │ │ ├── html2text {}
│ │ │ │ @@ -230,15 +230,15 @@
│ │ │ │  o22 = total: 1 5 8 5 1
│ │ │ │            0: 1 . . . .
│ │ │ │            1: . 4 2 . .
│ │ │ │            2: . 1 6 5 1
│ │ │ │  
│ │ │ │  o22 : BettiTally
│ │ │ │  i23 : time j=bruns F.dd_3;
│ │ │ │ - -- used 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ │ + -- used 0.562666s (cpu); 0.372499s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of S
│ │ │ │  i24 : betti res j
│ │ │ │  
│ │ │ │               0 1 2 3 4
│ │ │ │  o24 = total: 1 3 6 5 1
│ │ │ │            0: 1 . . . .
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  c3ViY29tcGxleChDZWxsQ29tcGxleCxMaXN0KQ==
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU2OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ViY29tcGxleCxDZWxsQ29tcGxleCxMaXN0KSwi
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.out
│ │ │ @@ -12,27 +12,27 @@
│ │ │  
│ │ │  i6 : F = polyhedralComplex {P1,P2,P3,P4};
│ │ │  
│ │ │  i7 : C = cellComplex(R,F);
│ │ │  
│ │ │  i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 |
│ │ │ -                      | 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 |
│ │ │ -                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ +                      | 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cell__Complex_lp__Ring_cm__Simplicial__Complex_rp.out
│ │ │ @@ -24,15 +24,15 @@
│ │ │  
│ │ │  o7 : CellComplex
│ │ │  
│ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │                        5   4    3 2   2 3     4   5
│ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2
│ │ │ -               1 => {x y , x y , x y , x y , x y, x y , x y , x y , x y , x y , x y, x y }
│ │ │ +                      4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2
│ │ │ +               1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y }
│ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_cells.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │  
│ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │  
│ │ │  i7 : cells(C)
│ │ │  
│ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with label z, Cell of dimension 0 with label x}}
│ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label x, Cell of dimension 0 with label z, Cell of dimension 0 with label y}}
│ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │  
│ │ │  o7 : HashTable
│ │ │  
│ │ │  i8 : R = QQ;
│ │ │  
│ │ │  i9 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_hull__Complex.out
│ │ │ @@ -19,22 +19,22 @@
│ │ │                             
│ │ │       -1     0      1      2
│ │ │  
│ │ │  o4 : Complex
│ │ │  
│ │ │  i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       5 4   3 5    4 5   2        2    4 4    5 3
│ │ │ -o5 = {x y , x y z, x y , x y*z, x*y z, x y z, x y z}
│ │ │ +         2    4 4    5 3    5 4   3 5    4 5   2
│ │ │ +o5 = {x*y z, x y z, x y z, x y , x y z, x y , x y*z}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : cells(2,H)/cellLabel
│ │ │  
│ │ │ -       5 4    4 5
│ │ │ +       4 5    5 4
│ │ │  o6 = {x y z, x y z}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : isMinimal H
│ │ │  
│ │ │  o7 = true
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out
│ │ │ @@ -22,20 +22,20 @@
│ │ │  
│ │ │  i11 : T = new HashTable from {v0 => a^2*b, v1 => b*c^2, v2 => b^2, v3 => a*c};
│ │ │  
│ │ │  i12 : relabeledC = relabelCellComplex(C,T);
│ │ │  
│ │ │  i13 : for c in cells(0,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -             2      2   2
│ │ │ -o13 = {a*c, a b, b*c , b }
│ │ │ +        2      2   2
│ │ │ +o13 = {a b, b*c , b , a*c}
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -          2    2   2   2      2 2       2
│ │ │ -o14 = {a*b c, a b*c , a b*c, a b , a*b*c }
│ │ │ +        2 2   2   2     2    2 2       2
│ │ │ +o14 = {b c , a b*c , a*b c, a b , a*b*c }
│ │ │  
│ │ │  o14 : List
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/example-output/_taylor__Complex.out
│ │ │ @@ -25,20 +25,20 @@
│ │ │  
│ │ │             1                       3
│ │ │  o5 = -1 : S  <------------------- S  : 0
│ │ │                  | -x2 -y2 -z2 |
│ │ │  
│ │ │            3                           3
│ │ │       0 : S  <----------------------- S  : 1
│ │ │ -               {2} | -y2 0   -z2 |
│ │ │ -               {2} | x2  -z2 0   |
│ │ │ -               {2} | 0   y2  x2  |
│ │ │ +               {2} | -z2 -y2 0   |
│ │ │ +               {2} | 0   x2  -z2 |
│ │ │ +               {2} | x2  0   y2  |
│ │ │  
│ │ │            3                   1
│ │ │       1 : S  <--------------- S  : 2
│ │ │ +               {4} | y2  |
│ │ │                 {4} | -z2 |
│ │ │                 {4} | -x2 |
│ │ │ -               {4} | y2  |
│ │ │  
│ │ │  o5 : ComplexMap
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cell__Complex_lp__Ring_cm__Polyhedral__Complex_rp.html
│ │ │ @@ -117,27 +117,27 @@
│ │ │                
    i7 : C = cellComplex(R,F);
    
│ │ │              
    
│ │ │                
    i8 : facePoset C
│ │ │  
│ │ │ -o8 = Relation Matrix: | 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 |
│ │ │ -                      | 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 |
│ │ │ -                      | 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 |
│ │ │ -                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 |
│ │ │ +o8 = Relation Matrix: | 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 |
│ │ │ +                      | 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 |
│ │ │ +                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
│ │ │ +                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
│ │ │                        | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
│ │ │  
│ │ │  o8 : Poset
    
│ │ │              
    
│ │ │                
    i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │  
│ │ │                        5   4    3 2   2 3     4   5
│ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
│ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2
│ │ │ -               1 => {x y , x y , x y , x y , x y, x y , x y , x y , x y , x y , x y, x y }
│ │ │ +                      4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3   5 4   4 2
│ │ │ +               1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y , x y , x y }
│ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │  
│ │ │  o8 : HashTable
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,18 +41,18 @@
│ │ │ │  
│ │ │ │  o7 : CellComplex
│ │ │ │  i8 : applyValues(cells C, l -> apply(l,cellLabel))
│ │ │ │  
│ │ │ │                        5   4    3 2   2 3     4   5
│ │ │ │  o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }
│ │ │ │  }
│ │ │ │ -                      5 3   5 4   4 2   4 4   5    3 3   5 2   2 4   5 3   5 4
│ │ │ │ -5    5 2
│ │ │ │ -               1 => {x y , x y , x y , x y , x y, x y , x y , x y , x y , x y ,
│ │ │ │ -x y, x y }
│ │ │ │ +                      4 4   5    3 3   5 2   2 4   5 3   5 4   5    5 2   5 3
│ │ │ │ +5 4   4 2
│ │ │ │ +               1 => {x y , x y, x y , x y , x y , x y , x y , x y, x y , x y ,
│ │ │ │ +x y , x y }
│ │ │ │                        5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
│ │ │ │                 2 => {x y , x y , x y , x y , x y , x y , x y , x y }
│ │ │ │  
│ │ │ │  o8 : HashTable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_e_l_l_C_o_m_p_l_e_x -- create a cell complex
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_cells.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │                
    i6 : C = cellComplex(R,{exy,vz});
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : cells(C)
│ │ │  
│ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with label z, Cell of dimension 0 with label x}}
│ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label x, Cell of dimension 0 with label z, Cell of dimension 0 with label y}}
│ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │  
│ │ │  o7 : HashTable
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,16 +19,16 @@
│ │ │ │  i2 : vx = newSimplexCell({},x);
│ │ │ │  i3 : vy = newSimplexCell({},y);
│ │ │ │  i4 : vz = newSimplexCell({},z);
│ │ │ │  i5 : exy = newSimplexCell {vx,vy};
│ │ │ │  i6 : C = cellComplex(R,{exy,vz});
│ │ │ │  i7 : cells(C)
│ │ │ │  
│ │ │ │ -o7 = HashTable{0 => {Cell of dimension 0 with label y, Cell of dimension 0 with
│ │ │ │ -label z, Cell of dimension 0 with label x}}
│ │ │ │ +o7 = HashTable{0 => {Cell of dimension 0 with label x, Cell of dimension 0 with
│ │ │ │ +label z, Cell of dimension 0 with label y}}
│ │ │ │                 1 => {Cell of dimension 1 with label x*y}
│ │ │ │  
│ │ │ │  o7 : HashTable
│ │ │ │  i8 : R = QQ;
│ │ │ │  i9 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}};
│ │ │ │  i10 : C = cellComplex(R,P);
│ │ │ │  i11 : cells C
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_hull__Complex.html
│ │ │ @@ -114,25 +114,25 @@
│ │ │  o4 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : cells(1,H)/cellLabel
│ │ │  
│ │ │ -       5 4   3 5    4 5   2        2    4 4    5 3
│ │ │ -o5 = {x y , x y z, x y , x y*z, x*y z, x y z, x y z}
│ │ │ +         2    4 4    5 3    5 4   3 5    4 5   2
│ │ │ +o5 = {x*y z, x y z, x y z, x y , x y z, x y , x y*z}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : cells(2,H)/cellLabel
│ │ │  
│ │ │ -       5 4    4 5
│ │ │ +       4 5    5 4
│ │ │  o6 = {x y z, x y z}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,21 +39,21 @@
│ │ │ │  o4 = S  <-- S  <-- S  <-- S
│ │ │ │  
│ │ │ │       -1     0      1      2
│ │ │ │  
│ │ │ │  o4 : Complex
│ │ │ │  i5 : cells(1,H)/cellLabel
│ │ │ │  
│ │ │ │ -       5 4   3 5    4 5   2        2    4 4    5 3
│ │ │ │ -o5 = {x y , x y z, x y , x y*z, x*y z, x y z, x y z}
│ │ │ │ +         2    4 4    5 3    5 4   3 5    4 5   2
│ │ │ │ +o5 = {x*y z, x y z, x y z, x y , x y z, x y , x y*z}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : cells(2,H)/cellLabel
│ │ │ │  
│ │ │ │ -       5 4    4 5
│ │ │ │ +       4 5    5 4
│ │ │ │  o6 = {x y z, x y z}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : isMinimal H
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : H2 = hullComplex (3/2,I)
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html
│ │ │ @@ -141,26 +141,26 @@
│ │ │                
    i12 : relabeledC = relabelCellComplex(C,T);
    
│ │ │              
    
│ │ │                
    i13 : for c in cells(0,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -             2      2   2
│ │ │ -o13 = {a*c, a b, b*c , b }
│ │ │ +        2      2   2
│ │ │ +o13 = {a b, b*c , b , a*c}
│ │ │  
│ │ │  o13 : List
    
│ │ │              
    
│ │ │                
    i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │  
│ │ │ -          2    2   2   2      2 2       2
│ │ │ -o14 = {a*b c, a b*c , a b*c, a b , a*b*c }
│ │ │ +        2 2   2   2     2    2 2       2
│ │ │ +o14 = {b c , a b*c , a*b c, a b , a*b*c }
│ │ │  
│ │ │  o14 : List
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,22 +31,22 @@
│ │ │ │  i8 : v1 = verts#1;
│ │ │ │  i9 : v2 = verts#2;
│ │ │ │  i10 : v3 = verts#3;
│ │ │ │  i11 : T = new HashTable from {v0 => a^2*b, v1 => b*c^2, v2 => b^2, v3 => a*c};
│ │ │ │  i12 : relabeledC = relabelCellComplex(C,T);
│ │ │ │  i13 : for c in cells(0,relabeledC) list cellLabel(c)
│ │ │ │  
│ │ │ │ -             2      2   2
│ │ │ │ -o13 = {a*c, a b, b*c , b }
│ │ │ │ +        2      2   2
│ │ │ │ +o13 = {a b, b*c , b , a*c}
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  i14 : for c in cells(1,relabeledC) list cellLabel(c)
│ │ │ │  
│ │ │ │ -          2    2   2   2      2 2       2
│ │ │ │ -o14 = {a*b c, a b*c , a b*c, a b , a*b*c }
│ │ │ │ +        2 2   2   2     2    2 2       2
│ │ │ │ +o14 = {b c , a b*c , a*b c, a b , a*b*c }
│ │ │ │  
│ │ │ │  o14 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_e_l_l_L_a_b_e_l -- return the label of a cell
│ │ │ │      * _R_i_n_g_M_a_p_ _*_*_ _C_e_l_l_C_o_m_p_l_e_x -- tensors labels via a ring map
│ │ │ │  ********** WWaayyss ttoo uussee rreellaabbeellCCeellllCCoommpplleexx:: **********
│ │ │ │      * relabelCellComplex(CellComplex,HashTable)
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/html/_taylor__Complex.html
│ │ │ @@ -115,23 +115,23 @@
│ │ │  
│ │ │             1                       3
│ │ │  o5 = -1 : S  <------------------- S  : 0
│ │ │                  | -x2 -y2 -z2 |
│ │ │  
│ │ │            3                           3
│ │ │       0 : S  <----------------------- S  : 1
│ │ │ -               {2} | -y2 0   -z2 |
│ │ │ -               {2} | x2  -z2 0   |
│ │ │ -               {2} | 0   y2  x2  |
│ │ │ +               {2} | -z2 -y2 0   |
│ │ │ +               {2} | 0   x2  -z2 |
│ │ │ +               {2} | x2  0   y2  |
│ │ │  
│ │ │            3                   1
│ │ │       1 : S  <--------------- S  : 2
│ │ │ +               {4} | y2  |
│ │ │                 {4} | -z2 |
│ │ │                 {4} | -x2 |
│ │ │ -               {4} | y2  |
│ │ │  
│ │ │  o5 : ComplexMap
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -35,23 +35,23 @@ │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o5 = -1 : S <------------------- S : 0 │ │ │ │ | -x2 -y2 -z2 | │ │ │ │ │ │ │ │ 3 3 │ │ │ │ 0 : S <----------------------- S : 1 │ │ │ │ - {2} | -y2 0 -z2 | │ │ │ │ - {2} | x2 -z2 0 | │ │ │ │ - {2} | 0 y2 x2 | │ │ │ │ + {2} | -z2 -y2 0 | │ │ │ │ + {2} | 0 x2 -z2 | │ │ │ │ + {2} | x2 0 y2 | │ │ │ │ │ │ │ │ 3 1 │ │ │ │ 1 : S <--------------- S : 2 │ │ │ │ + {4} | y2 | │ │ │ │ {4} | -z2 | │ │ │ │ {4} | -x2 | │ │ │ │ - {4} | y2 | │ │ │ │ │ │ │ │ o5 : ComplexMap │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _h_u_l_l_C_o_m_p_l_e_x_(_M_o_n_o_m_i_a_l_I_d_e_a_l_) -- gives the hull complex of a monomial ideal │ │ │ │ ********** WWaayyss ttoo uussee ttaayylloorrCCoommpplleexx:: ********** │ │ │ │ * taylorComplex(MonomialIdeal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ RUtSZXNvbHV0aW9uKE1vbm9taWFsSWRlYWwp │ │ │ #:len=292 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3MSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoRUtSZXNvbHV0aW9uLE1vbm9taWFsSWRlYWwpLCJF │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ @@ -63,15 +63,15 @@ │ │ │ o11 : ChainComplex │ │ │ │ │ │ i12 : isMinimalChainComplex E │ │ │ │ │ │ o12 = false │ │ │ │ │ │ i13 : time m = minimize (E[1]); │ │ │ - -- used 0.296573s (cpu); 0.217064s (thread); 0s (gc) │ │ │ + -- used 0.443344s (cpu); 0.321002s (thread); 0s (gc) │ │ │ │ │ │ i14 : isQuasiIsomorphism m │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : E[1] == source m │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ @@ -27,18 +27,18 @@ │ │ │ i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5); │ │ │ │ │ │ i6 : mods = for i from 0 to max C list pushForward(f, C_i); │ │ │ │ │ │ i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S)); │ │ │ │ │ │ i8 : time m = resolutionOfChainComplex C; │ │ │ - -- used 0.0995083s (cpu); 0.0987657s (thread); 0s (gc) │ │ │ + -- used 0.131993s (cpu); 0.129253s (thread); 0s (gc) │ │ │ │ │ │ i9 : time n = cartanEilenbergResolution C; │ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc) │ │ │ + -- used 0.161628s (cpu); 0.162765s (thread); 0s (gc) │ │ │ │ │ │ i10 : betti source m │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ o10 = total: 1 19 80 181 312 484 447 156 │ │ │ 0: 1 3 3 1 . . . . │ │ │ 1: . . 1 3 3 . . . │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html │ │ │ @@ -186,15 +186,15 @@ │ │ │
    │ │ │

    Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -306,15 +306,15 @@ │ │ │ │ │ │ o21 : A │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -160,15 +160,15 @@ │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o14 = ideal (x x - x x x , x x ) │ │ │ │ 0 3 1 2 4 2 5 │ │ │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ i15 : time csmK=CSM(A,K) │ │ │ │ - -- used 0.399939s (cpu); 0.310473s (thread); 0s (gc) │ │ │ │ + -- used 1.07842s (cpu); 0.430348s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o15 = 7h h + 5h h + 4h h + h + 3h h + h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o15 : A │ │ │ │ i16 : csmKHash= CSM(A,K,Output=>HashForm) │ │ │ │ @@ -199,15 +199,15 @@ │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o21 = 9h h + 9h h + 9h h + 3h + 7h h + 3h + 3h + 2h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 1 2 │ │ │ │ │ │ │ │ o21 : A │ │ │ │ i22 : time CSM(A,K,m) │ │ │ │ - -- used 0.182686s (cpu); 0.107681s (thread); 0s (gc) │ │ │ │ + -- used 0.259978s (cpu); 0.121032s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o22 = 7h h + 5h h + 4h h + h + 3h h + h │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o22 : A │ │ │ │ In the case where the ambient space is a toric variety which is not a product │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ @@ -77,15 +77,15 @@ │ │ │ │ │ │ o2 : NormalToricVariety │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.296573s (cpu); 0.217064s (thread); 0s (gc)
    │ │ │ + -- used 0.443344s (cpu); 0.321002s (thread); 0s (gc) │ │ │
    │ │ │ 
    i14 : isQuasiIsomorphism m
│ │ │  
│ │ │  o14 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -81,15 +81,15 @@ │ │ │ │ o11 : ChainComplex │ │ │ │ i12 : isMinimalChainComplex E │ │ │ │ │ │ │ │ o12 = false │ │ │ │ Now we minimize the result. The free summand we added to the end maps to zero, │ │ │ │ and thus is part of the minimization. │ │ │ │ i13 : time m = minimize (E[1]); │ │ │ │ - -- used 0.296573s (cpu); 0.217064s (thread); 0s (gc) │ │ │ │ + -- used 0.443344s (cpu); 0.321002s (thread); 0s (gc) │ │ │ │ i14 : isQuasiIsomorphism m │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : E[1] == source m │ │ │ │ │ │ │ │ o15 = true │ │ │ │ i16 : E' = target m │ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ @@ -134,21 +134,21 @@ │ │ │     		│ │ │ 
    i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S));
    │ │ │
    │ │ │ 
    i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.0995083s (cpu); 0.0987657s (thread); 0s (gc)
    │ │ │ + -- used 0.131993s (cpu); 0.129253s (thread); 0s (gc) │ │ │
    │ │ │ 
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc)
    │ │ │ + -- used 0.161628s (cpu); 0.162765s (thread); 0s (gc) │ │ │
    │ │ │ 
    i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,17 +49,17 @@
│ │ │ │  
│ │ │ │  o4 : RingMap R <-- S
│ │ │ │  i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5);
│ │ │ │  i6 : mods = for i from 0 to max C list pushForward(f, C_i);
│ │ │ │  i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-
│ │ │ │  1),mods_i,substitute(matrix C.dd_i,S));
│ │ │ │  i8 : time m = resolutionOfChainComplex C;
│ │ │ │ - -- used 0.0995083s (cpu); 0.0987657s (thread); 0s (gc)
│ │ │ │ + -- used 0.131993s (cpu); 0.129253s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc)
│ │ │ │ + -- used 0.161628s (cpu); 0.162765s (thread); 0s (gc)
│ │ │ │  i10 : betti source m
│ │ │ │  
│ │ │ │               0  1  2   3   4   5   6   7
│ │ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │ │            0: 1  3  3   1   .   .   .   .
│ │ │ │            1: .  .  1   3   3   .   .   .
│ │ │ │            2: .  1  3   3   2   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexOperations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  c3ltMihDb21wbGV4KQ==
│ │ │  #:len=259
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW0yLENvbXBsZXgpLCJzeW0yKENvbXBsZXgpIiwi
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  TXVsdGlQcm9qQ29vcmRSaW5n
│ │ │  #:len=2206
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQSBxdWljayB3YXkgdG8gYnVpbGQgdGhl
│ │ │  IGNvb3JkaW5hdGUgcmluZyBvZiBhIHByb2R1Y3Qgb2YgcHJvamVjdGl2ZSBzcGFjZXMiLCAibGlu
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out
│ │ │ @@ -83,15 +83,15 @@
│ │ │                2              2
│ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │                0 3    1 2 4   2 5
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.399939s (cpu); 0.310473s (thread); 0s (gc)
│ │ │ + -- used 1.07842s (cpu); 0.430348s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
│ │ │  
│ │ │ @@ -124,15 +124,15 @@
│ │ │          2 2     2         2     2             2
│ │ │  o21 = 9h h  + 9h h  + 9h h  + 3h  + 7h h  + 3h  + 3h  + 2h
│ │ │          1 2     1 2     1 2     1     1 2     2     1     2
│ │ │  
│ │ │  o21 : A
│ │ │  
│ │ │  i22 : time CSM(A,K,m)
│ │ │ - -- used 0.182686s (cpu); 0.107681s (thread); 0s (gc)
│ │ │ + -- used 0.259978s (cpu); 0.121032s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out
│ │ │ @@ -9,28 +9,28 @@
│ │ │  i2 : U = toricProjectiveSpace 7
│ │ │  
│ │ │  o2 = U
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │  
│ │ │  i3 : time CSM U
│ │ │ - -- used 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ + -- used 0.339448s (cpu); 0.234003s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │  o3 : -----------------------------------------------------------------------------------------------
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
│ │ │  
│ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ + -- used 0.4979s (cpu); 0.388648s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │  
│ │ │  i4 : I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ + -- used 1.12446s (cpu); 0.495958s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o5 : ------
│ │ │          6
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ + -- used 2.56033s (cpu); 2.24246s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -53,29 +53,29 @@
│ │ │  i8 : S=QQ[s_0..s_3];
│ │ │  
│ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │  
│ │ │  o9 : Ideal of S
│ │ │  
│ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ + -- used 0.376171s (cpu); 0.271716s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ + -- used 0.127364s (cpu); 0.127372s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out
│ │ │ @@ -21,20 +21,20 @@
│ │ │               2                                                        2
│ │ │       - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
│ │ │               3         0 4        1 4         2 4         3 4         4
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0376966s (cpu); 0.0369951s (thread); 0s (gc)
│ │ │ + -- used 0.0810226s (cpu); 0.0614135s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc)
│ │ │ + -- used 0.377593s (cpu); 0.218185s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
│ │ │  
│ │ │  i6 : EulerIHash=Euler(I,Output=>HashForm);
│ │ │  
│ │ │  i7 : A=ring EulerIHash#"CSM"
│ │ │  
│ │ │ @@ -62,20 +62,20 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x )
│ │ │          0 3
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.0792323s (cpu); 0.0754648s (thread); 0s (gc)
│ │ │ + -- used 0.149429s (cpu); 0.0933068s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc)
│ │ │ + -- used 0.308498s (cpu); 0.13444s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │  
│ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out
│ │ │ @@ -13,12 +13,12 @@
│ │ │              2    2    2
│ │ │  o3 = ideal(x  + x  + x  - 1)
│ │ │              1    2    3
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time EulerAffine I
│ │ │ - -- used 0.047108s (cpu); 0.0467349s (thread); 0s (gc)
│ │ │ + -- used 0.0924451s (cpu); 0.0749482s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ + -- used 5.29899s (cpu); 1.4999s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │  o3 : ----------
│ │ │          3   3
│ │ │        (h , h )
│ │ │          1   2
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ + -- used 5.05853s (cpu); 1.46128s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out
│ │ │ @@ -3,43 +3,43 @@
│ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ + -- used 1.15173s (cpu); 0.608333s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ + -- used 0.0920395s (cpu); 0.0536011s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ + -- used 0.0901452s (cpu); 0.0523829s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ + -- used 3.18952s (cpu); 1.25734s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          7
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ + -- used 0.689722s (cpu); 0.284922s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html
│ │ │ @@ -239,15 +239,15 @@
│ │ │  
│ │ │  o14 : Ideal of R
    │ │ │
    │ │ │ 
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.399939s (cpu); 0.310473s (thread); 0s (gc)
│ │ │ + -- used 1.07842s (cpu); 0.430348s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
    │ │ │
    │ │ │ 
    i22 : time CSM(A,K,m)
│ │ │ - -- used 0.182686s (cpu); 0.107681s (thread); 0s (gc)
│ │ │ + -- used 0.259978s (cpu); 0.121032s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
    │ │ │
    │ │ │ 
    i3 : time CSM U
│ │ │ - -- used 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ + -- used 0.339448s (cpu); 0.234003s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ @@ -93,15 +93,15 @@
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
    │ │ │
    │ │ │ 
    i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ + -- used 0.4979s (cpu); 0.388648s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,30 +16,30 @@
│ │ │ │  o1 : Package
│ │ │ │  i2 : U = toricProjectiveSpace 7
│ │ │ │  
│ │ │ │  o2 = U
│ │ │ │  
│ │ │ │  o2 : NormalToricVariety
│ │ │ │  i3 : time CSM U
│ │ │ │ - -- used 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ │ + -- used 0.339448s (cpu); 0.234003s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ │ │  o3 : --------------------------------------------------------------------------
│ │ │ │  ---------------------
│ │ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x ,
│ │ │ │  - x  + x , - x  + x )
│ │ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5
│ │ │ │  0    6     0    7
│ │ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ │ - -- used 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ │ + -- used 0.4979s (cpu); 0.388648s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    │ │ │
    │ │ │ 
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ + -- used 1.12446s (cpu); 0.495958s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -114,15 +114,15 @@
│ │ │         h
│ │ │          1
    │ │ │
    │ │ │ 
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ + -- used 2.56033s (cpu); 2.24246s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o9 : Ideal of S
    │ │ │
    │ │ │ 
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ + -- used 0.376171s (cpu); 0.271716s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │ @@ -164,15 +164,15 @@
│ │ │          h
│ │ │           1
    │ │ │
    │ │ │ 
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ + -- used 0.127364s (cpu); 0.127372s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,28 +32,28 @@
│ │ │ │  using the regenerative cascade implemented in Bertini. This is done by choosing
│ │ │ │  the option bertini, provided Bertini is _i_n_s_t_a_l_l_e_d_ _a_n_d_ _c_o_n_f_i_g_u_r_e_d.
│ │ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │ │  i4 : I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ │ + -- used 1.12446s (cpu); 0.495958s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │ │         1      1      1      1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o5 : ------
│ │ │ │          6
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ │ - -- used 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ │ + -- used 2.56033s (cpu); 2.24246s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          6
│ │ │ │ @@ -62,28 +62,28 @@
│ │ │ │  
│ │ │ │  o7 = 2
│ │ │ │  i8 : S=QQ[s_0..s_3];
│ │ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │ │  
│ │ │ │  o9 : Ideal of S
│ │ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ │ + -- used 0.376171s (cpu); 0.271716s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ │ + -- used 0.127364s (cpu); 0.127372s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -130,23 +130,23 @@
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │
    │ │ │ 
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0376966s (cpu); 0.0369951s (thread); 0s (gc)
│ │ │ + -- used 0.0810226s (cpu); 0.0614135s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    │ │ │
    │ │ │ 
    i5 : time Euler I
│ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc)
│ │ │ + -- used 0.377593s (cpu); 0.218185s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    │ │ │
    │ │ │ 
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    │ │ │ @@ -194,23 +194,23 @@ │ │ │
    │ │ │

    Note that the ideal J above is a complete intersection, thus we may change the method option which may speed computation in some cases. We may also note that the ideal generated by the first 2 generators of I defines a smooth scheme and input this information into the method. This may also improve computation speed.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.0792323s (cpu); 0.0754648s (thread); 0s (gc)
│ │ │ + -- used 0.149429s (cpu); 0.0933068s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    │ │ │
    │ │ │ 
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc)
│ │ │ + -- used 0.308498s (cpu); 0.13444s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
    │ │ │
    │ │ │
    │ │ │

    Now consider an example in \PP^2 \times \PP^2.

    │ │ │ ├── html2text {} │ │ │ │ @@ -74,19 +74,19 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 │ │ │ │ - 14254x - 11226x x + 2653x x + 12365x x - 10226x x - 12696x ) │ │ │ │ 3 0 4 1 4 2 4 3 4 4 │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : time Euler(I,InputIsSmooth=>true) │ │ │ │ - -- used 0.0376966s (cpu); 0.0369951s (thread); 0s (gc) │ │ │ │ + -- used 0.0810226s (cpu); 0.0614135s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : time Euler I │ │ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc) │ │ │ │ + -- used 0.377593s (cpu); 0.218185s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 4 │ │ │ │ i6 : EulerIHash=Euler(I,Output=>HashForm); │ │ │ │ i7 : A=ring EulerIHash#"CSM" │ │ │ │ │ │ │ │ o7 = A │ │ │ │ │ │ │ │ @@ -114,19 +114,19 @@ │ │ │ │ o9 : Ideal of R │ │ │ │ Note that the ideal J above is a complete intersection, thus we may change the │ │ │ │ method option which may speed computation in some cases. We may also note that │ │ │ │ the ideal generated by the first 2 generators of I defines a smooth scheme and │ │ │ │ input this information into the method. This may also improve computation │ │ │ │ speed. │ │ │ │ i10 : time Euler(J,Method=>DirectCompleteInt) │ │ │ │ - -- used 0.0792323s (cpu); 0.0754648s (thread); 0s (gc) │ │ │ │ + -- used 0.149429s (cpu); 0.0933068s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = 2 │ │ │ │ i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1}) │ │ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc) │ │ │ │ + -- used 0.308498s (cpu); 0.13444s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = 2 │ │ │ │ Now consider an example in \PP^2 \times \PP^2. │ │ │ │ i12 : R=MultiProjCoordRing({2,2}) │ │ │ │ │ │ │ │ o12 = R │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ │ │ │ o3 : Ideal of R │ │ │
    │ │ │ 
    i4 : time EulerAffine I
│ │ │ - -- used 0.047108s (cpu); 0.0467349s (thread); 0s (gc)
│ │ │ + -- used 0.0924451s (cpu); 0.0749482s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    │ │ │
    │ │ │
    │ │ │

    Observe that the algorithm is a probabilistic algorithm and may give a wrong answer with a small but nonzero probability. Read more under probabilistic algorithm.

    │ │ │ ├── html2text {} │ │ │ │ @@ -23,15 +23,15 @@ │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ o3 = ideal(x + x + x - 1) │ │ │ │ 1 2 3 │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : time EulerAffine I │ │ │ │ - -- used 0.047108s (cpu); 0.0467349s (thread); 0s (gc) │ │ │ │ + -- used 0.0924451s (cpu); 0.0749482s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ Observe that the algorithm is a probabilistic algorithm and may give a wrong │ │ │ │ answer with a small but nonzero probability. Read more under _p_r_o_b_a_b_i_l_i_s_t_i_c │ │ │ │ _a_l_g_o_r_i_t_h_m. │ │ │ │ ********** WWaayyss ttoo uussee EEuulleerrAAffffiinnee:: ********** │ │ │ │ * EulerAffine(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ @@ -75,15 +75,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ + -- used 5.29899s (cpu); 1.4999s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ @@ -92,15 +92,15 @@
│ │ │        (h , h )
│ │ │          1   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ + -- used 5.05853s (cpu); 1.46128s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,28 +16,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ │ + -- used 5.29899s (cpu); 1.4999s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ │ │  o3 : ----------
│ │ │ │          3   3
│ │ │ │        (h , h )
│ │ │ │          1   2
│ │ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ │ + -- used 5.05853s (cpu); 1.46128s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time CSM I
│ │ │ - -- used 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ + -- used 1.15173s (cpu); 0.608333s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -88,15 +88,15 @@
│ │ │         h
│ │ │          1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ + -- used 0.0920395s (cpu); 0.0536011s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -110,15 +110,15 @@
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time Chern I
│ │ │ - -- used 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ + -- used 0.0901452s (cpu); 0.0523829s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,42 +9,42 @@
│ │ │ │  input ideal is known to define a smooth subscheme setting this option to true
│ │ │ │  will speed up computations (it is set to false by default).
│ │ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ │ + -- used 1.15173s (cpu); 0.608333s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ │ + -- used 0.0920395s (cpu); 0.0536011s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o4 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o4 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  Note that one could, equivalently, use the command _C_h_e_r_n instead in this case.
│ │ │ │  i5 : time Chern I
│ │ │ │ - -- used 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ │ + -- used 0.0901452s (cpu); 0.0523829s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ + -- used 3.18952s (cpu); 1.25734s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -92,15 +92,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ + -- used 0.689722s (cpu); 0.284922s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,28 +18,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ │ + -- used 3.18952s (cpu); 1.25734s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o3 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          7
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ │ - -- used 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ │ + -- used 0.689722s (cpu); 0.284922s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  UmluZ01hcCBDaG9yZGFsTmV0
│ │ │  #:len=1424
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgcmluZyBtYXAgdG8gYSBjaG9y
│ │ │  ZGFsIG5ldHdvcmsiLCAibGluZW51bSIgPT4gODg5LCBJbnB1dHMgPT4ge1NQQU57VFR7ImYifSwi
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out
│ │ │ @@ -16,32 +16,32 @@
│ │ │  
│ │ │  o2 : Digraph
│ │ │  
│ │ │  i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
│ │ │  
│ │ │  i4 : tree = elimTree G
│ │ │  
│ │ │ -o4 = ElimTree{a => b   }
│ │ │ +o4 = ElimTree{a => c}
│ │ │                b => c
│ │ │                c => d
│ │ │ -              d => null
│ │ │ +              d => b
│ │ │  
│ │ │  o4 : ElimTree
│ │ │  
│ │ │  i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │                       "c0"=>c, "d0"=>d, "c1"=>c, "d1"=>d};
│ │ │  
│ │ │  i6 : eqs = hashTable{"a0" => ({a},{}), "a1" => ({},{}),
│ │ │                       "b0" => ({b},{}), "b1" => ({},{}), "b2" => ({b},{}),
│ │ │                       "c0" => ({c},{}), "c1" => ({},{}),
│ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
│ │ │  
│ │ │  i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │  
│ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ -                 b => {b,  , b}
│ │ │ -                 a => {a,  }
│ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │                   c => { , c}
│ │ │ +                 d => { , d}
│ │ │ +                 b => {b,  , b}
│ │ │  
│ │ │  o7 : ChordalNet
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/example-output/_components_lp__Chordal__Net_cm__Z__Z_rp.out
│ │ │ @@ -27,14 +27,14 @@
│ │ │  o5 : HashTable
│ │ │  
│ │ │  i6 : components(N)
│ │ │  
│ │ │  o6 = HashTable{4 => {ideal (x , x , x , x ), ideal (x , x , x , x )}                                                                                                                                                                                }
│ │ │                               1   3   5   7           2   4   6   8
│ │ │                 5 => {ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x )}
│ │ │ -                             1   3   4   6   7           1   2   4   6   7           1   2   4   5   7           1   3   5   6   8           1   3   4   6   8           2   3   5   6   8           2   4   5   7   8           2   3   5   7   8
│ │ │ +                             1   3   4   6   7           1   2   4   6   7           1   2   4   5   7           2   4   5   7   8           2   3   5   7   8           2   3   5   6   8           1   3   5   6   8           1   3   4   6   8
│ │ │                 6 => {}
│ │ │                 7 => {}
│ │ │  
│ │ │  o6 : HashTable
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html
│ │ │ @@ -107,18 +107,18 @@
│ │ │                
    i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
    
│ │ │              
    
│ │ │                
    i4 : tree = elimTree G
│ │ │  
│ │ │ -o4 = ElimTree{a => b   }
│ │ │ +o4 = ElimTree{a => c}
│ │ │                b => c
│ │ │                c => d
│ │ │ -              d => null
│ │ │ +              d => b
│ │ │  
│ │ │  o4 : ElimTree
    
│ │ │              
    
│ │ │                
    i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │ @@ -133,18 +133,18 @@
│ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
    
│ │ │              
    
│ │ │                
    i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │  
│ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ -                 b => {b,  , b}
│ │ │ -                 a => {a,  }
│ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │                   c => { , c}
│ │ │ +                 d => { , d}
│ │ │ +                 b => {b,  , b}
│ │ │  
│ │ │  o7 : ChordalNet
    
│ │ │              
    
│ │ │          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,32 +32,32 @@
│ │ │ │               d0 => {}
│ │ │ │               d1 => {}
│ │ │ │  
│ │ │ │  o2 : Digraph
│ │ │ │  i3 : G = chordalGraph digraph hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
│ │ │ │  i4 : tree = elimTree G
│ │ │ │  
│ │ │ │ -o4 = ElimTree{a => b   }
│ │ │ │ +o4 = ElimTree{a => c}
│ │ │ │                b => c
│ │ │ │                c => d
│ │ │ │ -              d => null
│ │ │ │ +              d => b
│ │ │ │  
│ │ │ │  o4 : ElimTree
│ │ │ │  i5 : rnk = hashTable{"a0"=>a, "a1"=>a, "b0"=>b, "b1"=>b, "b2"=>b,
│ │ │ │                       "c0"=>c, "d0"=>d, "c1"=>c, "d1"=>d};
│ │ │ │  i6 : eqs = hashTable{"a0" => ({a},{}), "a1" => ({},{}),
│ │ │ │                       "b0" => ({b},{}), "b1" => ({},{}), "b2" => ({b},{}),
│ │ │ │                       "c0" => ({c},{}), "c1" => ({},{}),
│ │ │ │                       "d0" => ({},{}), "d1" => ({d},{}) };
│ │ │ │  i7 : chordalNet(eqs,rnk,tree,DG)
│ │ │ │  
│ │ │ │ -o7 = ChordalNet{ d => { , d}    }
│ │ │ │ -                 b => {b,  , b}
│ │ │ │ -                 a => {a,  }
│ │ │ │ +o7 = ChordalNet{ a => {a,  }    }
│ │ │ │                   c => { , c}
│ │ │ │ +                 d => { , d}
│ │ │ │ +                 b => {b,  , b}
│ │ │ │  
│ │ │ │  o7 : ChordalNet
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_s_p_l_a_y_N_e_t -- displays a chordal network using Graphivz
│ │ │ │      * _d_i_g_r_a_p_h_(_C_h_o_r_d_a_l_N_e_t_) -- digraph associated to a chordal network
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _c_h_o_r_d_a_l_N_e_t_(_H_a_s_h_T_a_b_l_e_,_H_a_s_h_T_a_b_l_e_,_E_l_i_m_T_r_e_e_,_D_i_g_r_a_p_h_) -- construct chordal
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/html/_components_lp__Chordal__Net_cm__Z__Z_rp.html
│ │ │ @@ -123,15 +123,15 @@
│ │ │            
│ │ │              
│ │ │                
    i6 : components(N)
│ │ │  
│ │ │  o6 = HashTable{4 => {ideal (x , x , x , x ), ideal (x , x , x , x )}                                                                                                                                                                                }
│ │ │                               1   3   5   7           2   4   6   8
│ │ │                 5 => {ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x )}
│ │ │ -                             1   3   4   6   7           1   2   4   6   7           1   2   4   5   7           1   3   5   6   8           1   3   4   6   8           2   3   5   6   8           2   4   5   7   8           2   3   5   7   8
│ │ │ +                             1   3   4   6   7           1   2   4   6   7           1   2   4   5   7           2   4   5   7   8           2   3   5   7   8           2   3   5   6   8           1   3   5   6   8           1   3   4   6   8
│ │ │                 6 => {}
│ │ │                 7 => {}
│ │ │  
│ │ │  o6 : HashTable
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,16 +44,16 @@
│ │ │ │  }
│ │ │ │                               1   3   5   7           2   4   6   8
│ │ │ │                 5 => {ideal (x , x , x , x , x ), ideal (x , x , x , x , x ),
│ │ │ │  ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x , x ,
│ │ │ │  x ), ideal (x , x , x , x , x ), ideal (x , x , x , x , x ), ideal (x , x , x ,
│ │ │ │  x , x )}
│ │ │ │                               1   3   4   6   7           1   2   4   6   7
│ │ │ │ -1   2   4   5   7           1   3   5   6   8           1   3   4   6   8
│ │ │ │ -2   3   5   6   8           2   4   5   7   8           2   3   5   7   8
│ │ │ │ +1   2   4   5   7           2   4   5   7   8           2   3   5   7   8
│ │ │ │ +2   3   5   6   8           1   3   5   6   8           1   3   4   6   8
│ │ │ │                 6 => {}
│ │ │ │                 7 => {}
│ │ │ │  
│ │ │ │  o6 : HashTable
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  It is assumed that the _c_h_a_i_n_s of the network define prime ideals.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Classic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bW9ub21pYWxJZGVhbChTdHJpbmcp
│ │ │  #:len=1235
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibWFrZSBhIG1vbm9taWFsIGlkZWFsIHVz
│ │ │  aW5nIGNsYXNzaWMgTWFjYXVsYXkgc3ludGF4IiwgImxpbmVudW0iID0+IDE0NCwgSW5wdXRzID0+
│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  VmFuaXNoaW5nSWRlYWw=
│ │ │  #:len=1237
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmFuaXNoaW5nIGlkZWFsIG9mIGFuIGV2
│ │ │  YWx1YXRpb24gY29kZSIsICJsaW5lbnVtIiA9PiA1MTU3LCBJbnB1dHMgPT4ge1NQQU57VFR7IkMi
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  Y29ob21DYWxnKE5vcm1hbFRvcmljVmFyaWV0eSk=
│ │ │  #:len=2221
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibG9jYWxseSBzdGFzaGVkIGNvaG9tb2xv
│ │ │  Z3kgdmVjdG9ycyBmcm9tIENvaG9tQ2FsZyIsICJsaW5lbnVtIiA9PiAyODIsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out
│ │ │ @@ -184,15 +184,15 @@
│ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, -1, -1}}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.09141s elapsed
│ │ │ + -- 3.56093s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -265,45 +265,45 @@
│ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │  
│ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │  
│ │ │  o22 : List
│ │ │  
│ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .3809s elapsed
│ │ │ + -- .533434s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 11.5175s elapsed
│ │ │ + -- 11.1725s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
│ │ │  
│ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │  
│ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │  
│ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .397928s elapsed
│ │ │ + -- .535095s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .553978s elapsed
│ │ │ - -- .554009s elapsed
│ │ │ + -- .531175s elapsed
│ │ │ + -- .531206s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -314,15 +314,15 @@
│ │ │  
│ │ │  o19 : List    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.09141s elapsed
│ │ │ + -- 3.56093s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -404,25 +404,25 @@
│ │ │  
│ │ │  o22 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .3809s elapsed
│ │ │ + -- .533434s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 11.5175s elapsed
│ │ │ + -- 11.1725s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -438,26 +438,26 @@
│ │ │  
│ │ │  o26 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .397928s elapsed
│ │ │ + -- .535095s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .553978s elapsed
│ │ │ - -- .554009s elapsed
│ │ │ + -- .531175s elapsed
│ │ │ + -- .531206s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -182,15 +182,15 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, -1, -1}}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ │ - -- 3.09141s elapsed
│ │ │ │ + -- 3.56093s elapsed
│ │ │ │  
│ │ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -262,42 +262,42 @@
│ │ │ │                         {2, 2, 3, 1, -4, -6} => {{0, 1, 0, 0, 0}, {{1, 1x1*x2}}}
│ │ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │ │  
│ │ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │ │  
│ │ │ │  o22 : List
│ │ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ │ - -- .3809s elapsed
│ │ │ │ + -- .533434s elapsed
│ │ │ │  
│ │ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X
│ │ │ │  (0,0,1,2,0,-1))
│ │ │ │ - -- 11.5175s elapsed
│ │ │ │ + -- 11.1725s elapsed
│ │ │ │  
│ │ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o24 : List
│ │ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │ │  
│ │ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ │ - -- .397928s elapsed
│ │ │ │ + -- .535095s elapsed
│ │ │ │  
│ │ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j
│ │ │ │  (X, OO_X(0,0,1,2,-2,-1))
│ │ │ │ - -- .553978s elapsed
│ │ │ │ - -- .554009s elapsed
│ │ │ │ + -- .531175s elapsed
│ │ │ │ + -- .531206s elapsed
│ │ │ │  
│ │ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o28 : List
│ │ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ │ │  _c_o_h_o_m_C_a_l_g computes cohomology vectors by calling CohomCalg. It also stashes
│ │ │ │  it's results in the toric variety's cache table, so computations need not be
│ │ ├── ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  Y29tcGxleHJhbmsoUmluZ0VsZW1lbnQp
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV4cmFuayxSaW5nRWxlbWVudCksImNvbXBs
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  RWlzZW5idWRTaGFtYXNoVG90YWwoTW9kdWxlKQ==
│ │ │  #:len=349
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTE2OSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoRWlzZW5idWRTaGFtYXNoVG90YWwsTW9kdWxlKSwi
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : len = 10
│ │ │  
│ │ │  o6 = 10
│ │ │  
│ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ + -- used 9.28624s (cpu); 7.26969s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -140,37 +140,37 @@
│ │ │  i19 : R1 = R/ideal ff
│ │ │  
│ │ │  o19 = R1
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ + -- used 0.0971383s (cpu); 0.0969447s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
│ │ │  
│ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ + -- used 1.51133s (cpu); 1.10893s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │  
│ │ │  o21 : Complex
│ │ │  
│ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ + -- used 1.3973s (cpu); 1.08444s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out
│ │ │ @@ -2,21 +2,21 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ + -- used 0.955934s (cpu); 0.588924s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ + -- used 0.216773s (cpu); 0.164466s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ + -- used 4.524e-06s (cpu); 3.044e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : twoMonomials(2,3)
│ │ │ - -- used 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ + -- used 1.33511s (cpu); 0.922604s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ + -- used 0.839858s (cpu); 0.568997s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ + -- used 0.201821s (cpu); 0.190064s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o6 = 10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ + -- used 9.28624s (cpu); 7.26969s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -300,28 +300,28 @@
│ │ │  
│ │ │  o19 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ + -- used 0.0971383s (cpu); 0.0969447s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ + -- used 1.51133s (cpu); 1.10893s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │ @@ -333,15 +333,15 @@
│ │ │          
    
│ │ │            
    The function also deals correctly with complexes F where min F is not 0:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ + -- used 1.3973s (cpu); 1.08444s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  o5 = R
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : len = 10
│ │ │ │  
│ │ │ │  o6 = 10
│ │ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ │ - -- used 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ │ + -- used 9.28624s (cpu); 7.26969s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S
│ │ │ │  \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /
│ │ │ │  S   \68     /    S   \76
│ │ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |-------
│ │ │ │  -|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |-
│ │ │ │  -------|   <-- |--------|
│ │ │ │ @@ -165,36 +165,36 @@
│ │ │ │  o18 : Matrix R  <-- R
│ │ │ │  i19 : R1 = R/ideal ff
│ │ │ │  
│ │ │ │  o19 = R1
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ │ - -- used 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ │ + -- used 0.0971383s (cpu); 0.0969447s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        0       1       2        3        4
│ │ │ │  
│ │ │ │  o20 : Complex
│ │ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ │ - -- used 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ │ + -- used 1.51133s (cpu); 1.10893s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │ │  
│ │ │ │        0         1         2          3          4
│ │ │ │  
│ │ │ │  o21 : Complex
│ │ │ │  The function also deals correctly with complexes F where min F is not 0:
│ │ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ │ - -- used 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ │ + -- used 1.3973s (cpu); 1.08444s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        -2      -1      0        1        2
│ │ │ │  
│ │ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html
│ │ │ @@ -84,23 +84,23 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
    
│ │ │                
    i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ + -- used 0.955934s (cpu); 0.588924s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ + -- used 0.216773s (cpu); 0.164466s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ + -- used 4.524e-06s (cpu); 3.044e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,23 +18,23 @@
│ │ │ │  appropriate syzygy M of M0 = R/(m1+m2) where m1 and m2 are monomials of the
│ │ │ │  same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ │ - -- used 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ │ + -- used 0.955934s (cpu); 0.588924s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ │ + -- used 0.216773s (cpu); 0.164466s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ │ + -- used 4.524e-06s (cpu); 3.044e-06s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ssuummTTwwooMMoonnoommiiaallss:: **********
│ │ │ │      * sumTwoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html
│ │ │ @@ -88,25 +88,25 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : twoMonomials(2,3)
│ │ │ - -- used 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ + -- used 1.33511s (cpu); 0.922604s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ + -- used 0.839858s (cpu); 0.568997s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ + -- used 0.201821s (cpu); 0.190064s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,25 +20,25 @@
│ │ │ │  that is, for an appropriate syzygy M of M0 = R/(m1, m2) where m1 and m2 are
│ │ │ │  monomials of the same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : twoMonomials(2,3)
│ │ │ │ - -- used 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ │ + -- used 1.33511s (cpu); 0.922604s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ │ + -- used 0.839858s (cpu); 0.568997s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │ │        {{3, 3}, {2, 3}} => 1
│ │ │ │  
│ │ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ │ + -- used 0.201821s (cpu); 0.190064s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ttwwooMMoonnoommiiaallss:: **********
│ │ │ │      * twoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  cHJ1bmVDb21wbGV4KC4uLixQcnVuaW5nTWFwPT4uLi4p
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzQ4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcnVuZUNvbXBsZXgsUHJ1bmluZ01hcF0sInBydW5l
│ │ ├── ./usr/share/doc/Macaulay2/ConformalBlocks/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsRGl2aXNvck0wbmJhcg==
│ │ │  #:len=1172
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY2xhc3Mgb2YgdGhl
│ │ │  IGNhbm9uaWNhbCBkaXZpc29yIG9uIHRoZSBtb2R1bGkgc3BhY2Ugb2Ygc3RhYmxlIG4tcG9pbnRl
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=39
│ │ │  aXNFcHNpbG9uRmFjdG9yaXplZChNYXRyaXgsUmluZ0VsZW1lbnQp
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0Vwc2lsb25GYWN0b3JpemVkLE1hdHJpeCxSaW5n
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out
│ │ │ @@ -27,18 +27,18 @@
│ │ │       - ϵ*z*dy + 2ϵ  - ϵ, x*dx + y*dy + z*dz - 2ϵ)
│ │ │  
│ │ │  o7 : Ideal of D
│ │ │  
│ │ │  i8 : assert(holonomicRank I == 4)
│ │ │  
│ │ │  i9 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- 4.39641s elapsed
│ │ │ + -- 3.38421s elapsed
│ │ │  
│ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.46882s elapsed
│ │ │ + -- 4.43708s elapsed
│ │ │  
│ │ │  i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o11 = || 2ϵ/x                                                                                                                                                                                                                                                                                                                                                                                                                                    -y/x                                                                                                                                                                                                                                                                                                                                                                                                                                                                   -z/x                                                                                                                                                                                                                                                                                                                                                                                                                                                  0                                                                                                                             |                                                                                                                                                                                                                                                                                                                                     |
│ │ │        || (4x2y2ϵ^2+4xy2zϵ^2-2x2z2ϵ^2-2y2z2ϵ^2-4xz3ϵ^2+x3zϵ-3xy2zϵ+2xz3ϵ)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                               (2x3y2ϵ-2x2y3ϵ+2x3yzϵ-2xy3zϵ-x3z2ϵ+x2yz2ϵ-xy2z2ϵ+y3z2ϵ-2x3yz+2xy3z)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                          (-2x2y2zϵ-x3z2ϵ-3xy2z2ϵ+x2z3ϵ+y2z3ϵ+4xz4ϵ+2xy2z2-2xz4)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                      (-xyz+xz2+yz2-z3)/(2x2y+2xy2-x2z-2xyz-y2z)                                                                                    |                                                                                                                                                                                                                                                                                                                                     |
│ │ │        || (-2xyz2ϵ^2-2y2z2ϵ^2-4yz3ϵ^2+2x2y2ϵ+x2yzϵ+xy2zϵ+2y2z2ϵ+2yz3ϵ)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                 (x2yz2ϵ+2xy2z2ϵ+y3z2ϵ+2xyz3ϵ+2y2z3ϵ-2x2y3-x2y2z-xy3z-x2yz2-y3z2-xyz3-y2z3)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                                  (2x2y2ϵ+x2yzϵ+xy2zϵ-2x2z2ϵ+xyz2ϵ+y2z2ϵ-2xz3ϵ+2yz3ϵ-2x2y2-x2yz-xy2z+x2z2-y2z2+xz3-yz3)/(2x3y2+x3yz+x2y2z-x3z2-xy2z2-x2z3-xyz3)                                                                                                                                                                                                                                                                                                                         (-yz+z2)/(2xy-xz-yz)                                                                                                          |                                                                                                                                                                                                                                                                                                                                     |
│ │ │ @@ -56,24 +56,24 @@
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i12 : F = baseFractionField D;
│ │ │  
│ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │  
│ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .606974s elapsed
│ │ │ + -- .636657s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
│ │ │  
│ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.30033s elapsed
│ │ │ + -- 1.25306s elapsed
│ │ │  
│ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .857943s elapsed
│ │ │ + -- .969889s elapsed
│ │ │  
│ │ │  i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o17 = || 0                            1                      0                                         0                                                      |                                                                         |
│ │ │        || (-2ϵ^2+ϵ)/(x2-z2)            (3xϵ+zϵ-2x)/(x2-z2)    (yϵ+zϵ)/(x2-z2)                           (-y-z)/(x-z)                                           |                                                                         |
│ │ │        || 0                            0                      0                                         1                                                      |                                                                         |
│ │ │ @@ -96,18 +96,18 @@
│ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │  
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
│ │ │  
│ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .393587s elapsed
│ │ │ + -- .389688s elapsed
│ │ │  
│ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- 1.06124s elapsed
│ │ │ + -- .850595s elapsed
│ │ │  
│ │ │  i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |                                     |
│ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |                                     |
│ │ │        || 0       0           ϵ/(x+z) (-yϵ+zϵ)/(x2+xy+xz+yz) |                                     |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out
│ │ │ @@ -16,18 +16,18 @@
│ │ │  
│ │ │                     2
│ │ │  o6 = {1, dx, dy, dy }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- .32932s elapsed
│ │ │ + -- .326403s elapsed
│ │ │  
│ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .168768s elapsed
│ │ │ + -- .228216s elapsed
│ │ │  
│ │ │  i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o9 = || 0                                                       1                                               0                                                                     0                                                                      ||
│ │ │       || 0                                                       -1/x                                            1/x                                                                   y/x                                                                    ||
│ │ │       || -1/2xy                                                  -1/y                                            (-x-3y+1)/2xy                                                         (-x-y+1)/2x                                                            ||
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html
│ │ │ @@ -123,21 +123,21 @@
│ │ │          
    
│ │ │            
    Then, we compute the system in connection form and verify that it meets the integrability conditions.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,15 +82,15 @@
│ │ │ │                2                 2
│ │ │ │       - a*c + e         - b*c + f
│ │ │ │       ----------*v, x + ----------*v)
│ │ │ │        d*e - a*f         d*e - a*f
│ │ │ │  
│ │ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i6 : time phi^** q
│ │ │ │ - -- used 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ │ + -- used 0.479291s (cpu); 0.274562s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                  e        d        c        b        a
│ │ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │ │                  f        f        f        f        f
│ │ │ │  
│ │ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i7 : oo == p
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html
│ │ │ @@ -139,59 +139,59 @@
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
    
│ │ │                
    i9 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- 4.39641s elapsed
    
│ │ │ + -- 3.38421s elapsed
│ │ │              
    
│ │ │                
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.46882s elapsed
    
│ │ │ + -- 4.43708s elapsed
│ │ │              
    
│ │ │                
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -172,30 +172,30 @@
│ │ │              
    		
│ │ │                
    i13 : B = {1_D,dx,dy,dx*dy};
    
│ │ │              
    
│ │ │                
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .606974s elapsed
│ │ │ + -- .636657s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.30033s elapsed
    
│ │ │ + -- 1.25306s elapsed
│ │ │              
    
│ │ │                
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .857943s elapsed
    
│ │ │ + -- .969889s elapsed
│ │ │              
    
│ │ │                
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -232,21 +232,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .393587s elapsed
    
│ │ │ + -- .389688s elapsed
│ │ │              
    
│ │ │                
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- 1.06124s elapsed
    
│ │ │ + -- .850595s elapsed
│ │ │              
    
│ │ │                
    i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o7 : Ideal of D
│ │ │ │  First, we check that the system has finite holonomic rank using _h_o_l_o_n_o_m_i_c_R_a_n_k.
│ │ │ │  i8 : assert(holonomicRank I == 4)
│ │ │ │  Then, we compute the system in connection form and verify that it meets the
│ │ │ │  integrability conditions.
│ │ │ │  i9 : elapsedTime A = pfaffianSystem I;
│ │ │ │ - -- 4.39641s elapsed
│ │ │ │ + -- 3.38421s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.46882s elapsed
│ │ │ │ + -- 4.43708s elapsed
│ │ │ │  i11 : netList A
│ │ │ │  
│ │ │ │        +----------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │ @@ -227,22 +227,22 @@
│ │ │ │  -----------------------------------------------------------------------------------+
│ │ │ │  Next, we use _g_a_u_g_e_ _m_a_t_r_i_x for changing base to a base given by suitable set of
│ │ │ │  standard monomials, and compute the _g_a_u_g_e_ _t_r_a_n_s_f_o_r_m with respect to this gauge
│ │ │ │  matrix.
│ │ │ │  i12 : F = baseFractionField D;
│ │ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ │ - -- .606974s elapsed
│ │ │ │ + -- .636657s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.30033s elapsed
│ │ │ │ + -- 1.25306s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- .857943s elapsed
│ │ │ │ + -- .969889s elapsed
│ │ │ │  i17 : netList A1
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  --------------------------------------------------------------------------+
│ │ │ │  o17 = || 0                            1                      0
│ │ │ │  0                                                      |
│ │ │ │ @@ -300,17 +300,17 @@
│ │ │ │                {0, ϵ*(x^2-z^2), 0, ϵ*(x+y)*(x+z)},
│ │ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o18 : Matrix F  <-- F
│ │ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ │ - -- .393587s elapsed
│ │ │ │ + -- .389688s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- 1.06124s elapsed
│ │ │ │ + -- .850595s elapsed
│ │ │ │  i21 : netList A2
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------+
│ │ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |
│ │ │ │  |
│ │ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html
│ │ │ @@ -105,21 +105,21 @@
│ │ │          
    
│ │ │            
    Finally, we can compute the connection matrices.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- .32932s elapsed
    
│ │ │ + -- .326403s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .168768s elapsed
    
│ │ │ + -- .228216s elapsed
│ │ │              
    
│ │ │                
    i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,17 +20,17 @@
│ │ │ │  
│ │ │ │                     2
│ │ │ │  o6 = {1, dx, dy, dy }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Finally, we can compute the connection matrices.
│ │ │ │  i7 : elapsedTime A = pfaffianSystem I;
│ │ │ │ - -- .32932s elapsed
│ │ │ │ + -- .326403s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .168768s elapsed
│ │ │ │ + -- .228216s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./usr/share/doc/Macaulay2/ConvexInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  bUNvbnZleEh1bGxGYWNlcyguLi4sdG9GaWxlPT4uLi4p
│ │ │  #:len=273
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzU3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttQ29udmV4SHVsbEZhY2VzLHRvRmlsZV0sIm1Db252
│ │ ├── ./usr/share/doc/Macaulay2/ConwayPolynomials/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  Y29ud2F5UG9seW5vbWlhbA==
│ │ │  #:len=1659
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJvdmlkZSBhIENvbndheSBwb2x5bm9t
│ │ │  aWFsIiwgRGVzY3JpcHRpb24gPT4gKERJVntIRUFERVIyeyJTeW5vcHNpcyJ9LFVMe0xJe0RMeyJj
│ │ ├── ./usr/share/doc/Macaulay2/CorrespondenceScrolls/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=52
│ │ │  cHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyguLi4sQ29lZmZpY2llbnRGaWVsZD0+Li4uKQ==
│ │ │  #:len=367
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzcwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcm9kdWN0T2ZQcm9qZWN0aXZlU3BhY2VzLENvZWZm
│ │ ├── ./usr/share/doc/Macaulay2/CotangentSchubert/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  TGFiZWxz
│ │ │  #:len=183
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDA5LCAidW5kb2N1bWVudGVkIiA9PiB0
│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJMYWJlbHMi
│ │ ├── ./usr/share/doc/Macaulay2/CpMackeyFunctors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  bWFrZUJ1cm5zaWRlTWFja2V5RnVuY3RvcihaWik=
│ │ │  #:len=345
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlQnVybnNpZGVNYWNrZXlGdW5jdG9yLFpaKSwi
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZGVzY3JpYmUoUmF0aW9uYWxNYXAp
│ │ │  #:len=1096
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGVzY3JpYmUgYSByYXRpb25hbCBtYXAi
│ │ │  LCAibGluZW51bSIgPT4gOTgzLCBJbnB1dHMgPT4ge1NQQU57VFR7InBoaSJ9LCIsICIsU1BBTnsi
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,27 +13,27 @@
│ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
│ │ │  
│ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ + -- used 1.55264s (cpu); 1.23061s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
│ │ │  
│ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ + -- used 1.65484s (cpu); 1.16581s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -62,27 +62,27 @@
│ │ │          0,2 1,3    0,1 2,3
│ │ │  
│ │ │                  ZZ
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i9 : time ChernClass G
│ │ │ - -- used 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ + -- used 0.152207s (cpu); 0.152177s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
│ │ │  
│ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ + -- used 0.0421118s (cpu); 0.0181553s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out
│ │ │ @@ -1,56 +1,56 @@
│ │ │  -- -*- M2-comint -*- hash: 10433409267944421825
│ │ │  
│ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │  
│ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ + -- used 0.00566984s (cpu); 0.0056673s (thread); 0s (gc)
│ │ │  
│ │ │              ZZ              ZZ                3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o2 = map (------[t ..t ], ------[x ..x ], {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   300007  0   9       2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
│ │ │  
│ │ │  i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ + -- used 0.0567073s (cpu); 0.0567084s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │                  ZZ
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
│ │ │  
│ │ │  i4 : time degreeMap phi
│ │ │ - -- used 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ + -- used 0.0372848s (cpu); 0.037157s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ + -- used 0.633196s (cpu); 0.513987s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ + -- used 0.0914524s (cpu); 0.0914625s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ + -- used 0.0041893s (cpu); 0.00419242s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                                                             ------[x ..x ]
│ │ │                 ZZ                                                          300007  0   9
│ │ │  o7 : RingMap ------[t ..t ] <-- ----------------------------------------------------------------------------------------------------
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i8 : time psi = inverseMap phi
│ │ │ - -- used 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ + -- used 0.657052s (cpu); 0.541959s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -76,32 +76,32 @@
│ │ │                                                          ------[x ..x ]
│ │ │                                                          300007  0   9                                                   ZZ
│ │ │  o8 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[t ..t ]
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ + -- used 0.0119239s (cpu); 0.0119286s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ + -- used 0.199603s (cpu); 0.199611s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ + -- used 7.52838s (cpu); 6.8613s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ + -- used 0.00275207s (cpu); 0.0027582s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -147,15 +147,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │  
│ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ + -- used 0.0757898s (cpu); 0.075791s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -217,15 +217,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i14 : time phi^(-1)
│ │ │ - -- used 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ + -- used 0.699254s (cpu); 0.591838s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -275,71 +275,71 @@
│ │ │                         x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x
│ │ │                          6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │                        }
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
│ │ │  
│ │ │  i15 : time degrees phi^(-1)
│ │ │ - -- used 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ + -- used 0.364033s (cpu); 0.312703s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ + -- used 0.085042s (cpu); 0.0268094s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ + -- used 0.00453372s (cpu); 0.00454109s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i18 : time describe phi^(-1)
│ │ │ - -- used 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ + -- used 0.0130698s (cpu); 0.0130788s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │        number of minimal representatives: 1
│ │ │        dimension base locus: 4
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ + -- used 0.0132473s (cpu); 0.0132547s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i21 : time degrees f
│ │ │ - -- used 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ + -- used 1.51799s (cpu); 1.14509s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ + -- used 1.7761e-05s (cpu); 1.7288e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ + -- used 0.00177047s (cpu); 0.00177482s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out
│ │ │ @@ -3,18 +3,18 @@
│ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │  
│ │ │                  ZZ
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i2 : time EulerCharacteristic I
│ │ │ - -- used 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ + -- used 0.692097s (cpu); 0.288497s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ + -- used 0.051737s (cpu); 0.0212687s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  o3 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i4 : time phi! ;
│ │ │ - -- used 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ + -- used 0.0865409s (cpu); 0.0705998s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │  
│ │ │  i5 : describe phi
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │ @@ -37,15 +37,15 @@
│ │ │  
│ │ │  o8 = rational map defined by forms of degree 2
│ │ │       source variety: PP^4
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i9 : time phi! ;
│ │ │ - -- used 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ + -- used 0.0712487s (cpu); 0.0588081s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i10 : describe phi
│ │ │  
│ │ │  o10 = rational map defined by forms of degree 2
│ │ │        source variety: PP^4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │       - a*c + e         - b*c + f
│ │ │       ----------*v, x + ----------*v)
│ │ │        d*e - a*f         d*e - a*f
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │  
│ │ │  i6 : time phi^** q
│ │ │ - -- used 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ + -- used 0.479291s (cpu); 0.274562s (thread); 0s (gc)
│ │ │  
│ │ │                  e        d        c        b        a
│ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │                  f        f        f        f        f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out
│ │ │ @@ -47,50 +47,50 @@
│ │ │                                                                            P7
│ │ │  o3 : Ideal of -------------------------------------------------------------------------------------------------------------------------
│ │ │                 2 2                2 2                                        2 2                                                    2 2
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │  
│ │ │  i4 : time SegreClass X
│ │ │ - -- used 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ + -- used 1.18697s (cpu); 0.686282s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ + -- used 0.578796s (cpu); 0.399267s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ + -- used 0.0404911s (cpu); 0.0269575s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ + -- used 0.368464s (cpu); 0.181236s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -98,22 +98,22 @@
│ │ │  
│ │ │  i8 : o4 == o6 and o5 == o7
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ -= ------[x ..x ]
│ │ │ -  100003  0   6
│ │ │ +       ZZ
│ │ │ +o9 = ------[x ..x ]
│ │ │ +     100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
│ │ │  
│ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ + -- used 0.0752016s (cpu); 0.0752054s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -122,15 +122,15 @@
│ │ │                                                           ------[y ..y ]
│ │ │                                                           100003  0   9                                                   ZZ
│ │ │  o10 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[x ..x ]
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │  
│ │ │  i11 : time SegreClass phi
│ │ │ - -- used 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ + -- used 0.449255s (cpu); 0.293058s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -150,27 +150,27 @@
│ │ │                                                            100003  0   9
│ │ │  o12 : Ideal of ----------------------------------------------------------------------------------------------------
│ │ │                 (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │  
│ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ + -- used 0.512456s (cpu); 0.340242s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
│ │ │  
│ │ │  i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ + -- used 1.99806s (cpu); 1.20752s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out
│ │ │ @@ -17,32 +17,32 @@
│ │ │  
│ │ │  o3 = QQ[u ..u ]
│ │ │           0   5
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ + -- used 0.000518171s (cpu); 0.000511273s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │  
│ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ + -- used 0.150227s (cpu); 0.150234s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ + -- used 0.595499s (cpu); 0.475293s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -113,48 +113,48 @@
│ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │                   ZZ
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
│ │ │  
│ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ + -- used 0.0506466s (cpu); 0.0506457s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ + -- used 3.07714s (cpu); 2.20173s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ + -- used 3.4806e-05s (cpu); 3.2523e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ + -- used 4.83467s (cpu); 3.52399s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
│ │ │  
│ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │  
│ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │ @@ -169,15 +169,15 @@
│ │ │  i20 : T q
│ │ │  
│ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │  o20 : List
│ │ │  
│ │ │  i21 : time f = rationalMap T
│ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ + -- used 4.01634s (cpu); 2.92522s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out
│ │ │ @@ -54,15 +54,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ + -- used 0.596918s (cpu); 0.407878s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │  
│ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ + -- used 0.343778s (cpu); 0.236792s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i6 : assert(psi == psi')
│ │ │  
│ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-3*x_6*x_7+2*x_0*x_8+11*x_1*x_8-37*x_3*x_8-23*x_4*x_8-33*x_6*x_8+8*x_0*x_9+10*x_1*x_9-25*x_2*x_9-9*x_3*x_9+3*x_4*x_9+24*x_5*x_9-27*x_6*x_9-5*x_0*x_10+28*x_1*x_10+37*x_2*x_10+9*x_4*x_10+27*x_6*x_10-25*x_0*x_11+9*x_2*x_11+27*x_4*x_11-27*x_5*x_11,x_2^2+17*x_2*x_3-14*x_3^2-13*x_2*x_4+34*x_3*x_4+44*x_0*x_5-30*x_2*x_5+27*x_3*x_5+31*x_2*x_6-36*x_3*x_6-x_0*x_7+13*x_1*x_7+8*x_3*x_7+9*x_5*x_7+46*x_6*x_7+41*x_0*x_8-7*x_1*x_8-34*x_3*x_8-9*x_4*x_8-46*x_6*x_8-17*x_0*x_9+32*x_1*x_9-8*x_2*x_9-35*x_3*x_9-46*x_4*x_9+26*x_5*x_9+17*x_6*x_9+15*x_0*x_10+35*x_1*x_10+34*x_2*x_10+20*x_4*x_10+14*x_0*x_11+36*x_1*x_11+35*x_2*x_11-17*x_4*x_11,x_1*x_2-40*x_2*x_3+28*x_3^2-x_0*x_4+5*x_2*x_4-16*x_3*x_4+5*x_0*x_5-36*x_2*x_5+37*x_3*x_5+48*x_2*x_6-5*x_1*x_7-5*x_3*x_7+x_5*x_7+20*x_6*x_7+10*x_0*x_8+34*x_1*x_8+41*x_3*x_8-x_4*x_8+x_6*x_8+40*x_0*x_9-32*x_1*x_9+5*x_2*x_9-11*x_3*x_9-20*x_4*x_9+45*x_5*x_9-14*x_6*x_9-25*x_0*x_10+45*x_1*x_10-41*x_2*x_10-46*x_4*x_10+8*x_6*x_10-28*x_0*x_11+11*x_2*x_11+14*x_4*x_11-8*x_5*x_11),{t_4^2+t_0*t_5+t_1*t_5+35*t_2*t_5+10*t_3*t_5+25*t_4*t_5-5*t_5^2-14*t_0*t_6-14*t_1*t_6-5*t_2*t_6-13*t_4*t_6+37*t_5*t_6+22*t_6^2-31*t_3*t_7+26*t_4*t_7+12*t_5*t_7-45*t_6*t_7-46*t_3*t_8+37*t_4*t_8+28*t_5*t_8+33*t_6*t_8,t_3*t_4+4*t_0*t_5+39*t_1*t_5-40*t_2*t_5+40*t_3*t_5+26*t_4*t_5-20*t_5^2+41*t_0*t_6+36*t_1*t_6-22*t_2*t_6+36*t_4*t_6-30*t_5*t_6-13*t_6^2-25*t_3*t_7+5*t_4*t_7-35*t_5*t_7+10*t_6*t_7+11*t_3*t_8+46*t_4*t_8+29*t_5*t_8+28*t_6*t_8,t_2*t_4-5*t_0*t_5-40*t_1*t_5+12*t_2*t_5+47*t_3*t_5+37*t_4*t_5+25*t_5^2-27*t_0*t_6-22*t_1*t_6+27*t_2*t_6-23*t_4*t_6+5*t_5*t_6-13*t_6^2-39*t_3*t_7-29*t_4*t_7+9*t_5*t_7+39*t_6*t_7+36*t_3*t_8+13*t_4*t_8+26*t_5*t_8+37*t_6*t_8,t_0*t_4-t_0*t_5-8*t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+15*t_0*t_6+15*t_1*t_6+5*t_2*t_6+15*t_4*t_6-38*t_5*t_6-22*t_6^2+31*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_2*t_3-t_0*t_5-t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+14*t_0*t_6+14*t_1*t_6+5*t_2*t_6+14*t_4*t_6-31*t_5*t_6-24*t_6^2+32*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_1*t_3-7*t_1*t_5+t_1*t_6+t_4*t_6-7*t_5*t_6+2*t_6^2-t_3*t_7,t_0*t_3-46*t_0*t_5-39*t_1*t_5-43*t_2*t_5-41*t_3*t_5-26*t_4*t_5-28*t_5^2-35*t_0*t_6-36*t_1*t_6+20*t_2*t_6-36*t_4*t_6+9*t_5*t_6+15*t_6^2+26*t_3*t_7-5*t_4*t_7+35*t_5*t_7-10*t_6*t_7-10*t_3*t_8-46*t_4*t_8+47*t_5*t_8-25*t_6*t_8,t_2^2-46*t_1*t_4-33*t_0*t_5-45*t_1*t_5-39*t_2*t_5-39*t_3*t_5-46*t_4*t_5-29*t_5^2-48*t_0*t_6-38*t_1*t_6-30*t_2*t_6+19*t_4*t_6-44*t_5*t_6-47*t_6^2-36*t_0*t_7-46*t_1*t_7+t_2*t_7-44*t_3*t_7+48*t_4*t_7-14*t_5*t_7+4*t_6*t_7-36*t_0*t_8-46*t_1*t_8+47*t_2*t_8-34*t_3*t_8-24*t_4*t_8-12*t_5*t_8-47*t_6*t_8+47*t_7*t_8,t_1*t_2+6*t_1*t_5+5*t_0*t_6-2*t_1*t_6-t_4*t_6-t_5*t_6+5*t_0*t_7+t_1*t_7-2*t_2*t_7-7*t_5*t_7+2*t_6*t_7-2*t_1*t_8+3*t_7*t_8,t_0*t_2+t_1*t_4+5*t_0*t_5+32*t_1*t_5-20*t_2*t_5-47*t_3*t_5-37*t_4*t_5-25*t_5^2+19*t_0*t_6+22*t_1*t_6-25*t_2*t_6+25*t_4*t_6-5*t_5*t_6+13*t_6^2+5*t_0*t_7+t_1*t_7+39*t_3*t_7+28*t_4*t_7-9*t_5*t_7-39*t_6*t_7+4*t_0*t_8+t_1*t_8-36*t_3*t_8-14*t_4*t_8-26*t_5*t_8-37*t_6*t_8,t_0*t_1-39*t_1*t_4+40*t_1*t_5-37*t_0*t_6-39*t_1*t_6+19*t_4*t_6-39*t_5*t_6-38*t_0*t_7+39*t_1*t_7+19*t_2*t_7+18*t_5*t_7-19*t_6*t_7+19*t_1*t_8+20*t_7*t_8,t_0^2+12*t_1*t_4+20*t_0*t_5+27*t_1*t_5-8*t_2*t_5+37*t_3*t_5+28*t_4*t_5+30*t_5^2-46*t_0*t_6+24*t_1*t_6-40*t_2*t_6+25*t_4*t_6+16*t_5*t_6-35*t_6^2+29*t_0*t_7+12*t_1*t_7-35*t_2*t_7-8*t_3*t_7-18*t_4*t_7+42*t_5*t_7-12*t_6*t_7-6*t_0*t_8+12*t_1*t_8-15*t_3*t_8+9*t_4*t_8+20*t_5*t_8-30*t_6*t_8+4*t_7*t_8})
│ │ │  
│ │ │ @@ -192,15 +192,15 @@
│ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │  
│ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ + -- used 2.86736s (cpu); 2.31325s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ + -- used 3.59065s (cpu); 2.94829s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out
│ │ │ @@ -9,27 +9,27 @@
│ │ │                                   2                  2                             2                                       2                                                2                                                           2                                                                       2                                                                              2                                                                                            2         2                 2                             2                                       2                                              2                                                           2                                                                   2                                                                               2                                                                                          2        2                   2                            2                                      2                                                  2                                                          2                                                                      2                                                                               2                                                                                            2        2                   2                             2                                      2                                                 2                                                          2                                                                    2                                                                               2                                                                                          2         2                2                          2                                      2                                                 2                                                           2                                                                       2                                                                            2                                                                                          2        2                  2                         2                                       2                                                2                                                           2                                                                      2                                                                               2                                                                                        2       2                  2                             2                                    2                                                2                                                          2                                                                    2                                                                               2                                                                                       2       2                 2                           2                                      2                                                 2                                                            2                                                                       2                                                                                2                                                                                           2        2                   2                             2                                       2                                                  2                                                            2                                                                   2                                                                            2                                                                                          2      2                 2                            2                                       2                                                2                                                         2                                                                        2                                                                               2                                                                                          2     2                   2                             2                                      2                                                   2                                                          2                                                                     2                                                                               2                                                                                          2         2                2                            2                                       2                                                 2                                                           2                                                                      2                                                                                  2                                                                                              2      2                  2                            2                                    2                                                2                                                            2                                                                    2                                                                                2                                                                                          2       2                  2                            2                                    2                                                   2                                                        2                                                                         2                                                                               2                                                                                           2       2                  2                             2                                       2                                                 2                                                          2                                                                       2                                                                               2                                                                                       2
│ │ │  o4 = map (ringP8, ringP14, {- 95x  + 181x x  + 1028x  - 1384x x  - 1455x x  + 559x  - 502x x  + 1264x x  - 162x x  + 1209x  - 180x x  - 504x x  - 1168x x  - 676x x  + 501x  + 73x x  + 1263x x  + 1035x x  + 844x x  + 1593x x  + 785x  + 982x x  - 412x x  + 1335x x  + 1136x x  + 826x x  + 1078x x  + 1158x  + 335x x  - 982x x  - 1479x x  - 15x x  + 1363x x  + 1397x x  - 575x x  - 71x  + 1255x x  - 1138x x  - 1590x x  + 604x x  + 1182x x  - 63x x  - 1382x x  - 1255x x  - 613x , - 1444x  + 575x x  + 767x  - 1495x x  + 1631x x  - 217x  - 294x x  - 1511x x  - 504x x  - 1284x  - 1459x x  + 152x x  + 141x x  - 10x x  - 95x  + 1056x x  + 654x x  + 1397x x  - 930x x  + 578x x  - 696x  + 759x x  + 733x x  + 505x x  - 609x x  + 526x x  - 659x x  + 846x  + 1253x x  - 1519x x  + 635x x  + 576x x  + 54x x  - 1261x x  - 822x x  - 257x  - 986x x  + 356x x  - 1488x x  - 1561x x  - 850x x  - 85x x  - 1350x x  - 783x x  - 1335x , - 871x  + 1006x x  - 1399x  - 1636x x  - 699x x  - 769x  - 307x x  - 1645x x  - 502x x  - 719x  + 1405x x  + 870x x  - 1133x x  + 425x x  - 1203x  - 1601x x  + 117x x  - 382x x  + 318x x  - 117x x  - 560x  + 1135x x  + 1468x x  + 869x x  - 943x x  - 335x x  - 1218x x  + 201x  - 11x x  + 540x x  - 710x x  - 489x x  + 1605x x  + 1663x x  - 423x x  + 1246x  + 97x x  - 644x x  + 1655x x  + 1219x x  + 1476x x  + 1355x x  + 1594x x  + 893x x  + 1150x , - 143x  + 1240x x  - 1042x  + 1649x x  + 1024x x  + 794x  + 1442x x  - 1263x x  + 537x x  - 82x  - 734x x  - 1569x x  - 798x x  - 366x x  + 1289x  - 569x x  - 254x x  + 237x x  - 1234x x  - 807x x  + 264x  - 202x x  - 616x x  + 44x x  + 1465x x  + 685x x  + 1630x x  - 406x  - 123x x  - 4x x  + 1583x x  + 1235x x  + 162x x  + 1034x x  - 1035x x  + 737x  + 660x x  + 1459x x  - 359x x  - 1291x x  + 1638x x  - 325x x  - 631x x  + 73x x  - 1471x , - 1340x  + 31x x  - 994x  - 880x x  - 89x x  + 574x  + 760x x  - 1054x x  + 772x x  - 239x  - 443x x  + 1240x x  + 637x x  - 1423x x  + 320x  - 1363x x  - 1139x x  - 158x x  - 325x x  - 1578x x  + 32x  + 695x x  + 305x x  + 1012x x  + 1492x x  + 1290x x  + 1579x x  - 342x  - 83x x  - 104x x  + 998x x  - 92x x  + 1554x x  + 201x x  - 237x x  + 160x  - 228x x  - 543x x  - 1147x x  - 376x x  + 1313x x  + 603x x  + 106x x  - 1361x x  + 699x , - 228x  - 1510x x  + 277x  - 4x x  - 22x x  - 1526x  + 234x x  + 969x x  + 1253x x  - 1426x  - 1474x x  + 947x x  + 194x x  - 316x x  - 988x  - 1211x x  + 1087x x  + 536x x  - 491x x  + 870x x  - 659x  + 1490x x  - 469x x  + 1190x x  + 807x x  + 650x x  + 448x x  - 1353x  - 218x x  + 759x x  - 253x x  + 830x x  - 1080x x  - 143x x  - 1313x x  - 374x  - 180x x  + 741x x  + 742x x  - 1254x x  + 458x x  - 345x x  + 597x x  + 1567x x  - 31x , 1120x  + 709x x  - 1538x  - 1048x x  - 162x x  - 1518x  - 73x x  + 380x x  + 533x x  - 286x  + 1374x x  - 74x x  - 22x x  + 1535x x  - 1071x  - 839x x  - 560x x  + 928x x  + 335x x  - 1008x x  + 810x  - 448x x  - 357x x  - 107x x  + 40x x  + 784x x  - 1423x x  + 1276x  + 147x x  + 443x x  - 598x x  - 1077x x  - 1214x x  + 322x x  - 1408x x  + 72x  - 63x x  - 1513x x  - 791x x  + 11x x  + 77x x  + 836x x  - 1100x x  + 1637x x  - 788x , 1331x  + 318x x  - 704x  + 51x x  + 275x x  + 1149x  + 1526x x  + 768x x  + 414x x  - 782x  - 262x x  + 686x x  - 380x x  + 1377x x  + 1077x  + 1650x x  - 1129x x  - 508x x  + 846x x  + 1513x x  + 460x  - 1626x x  - 1024x x  + 862x x  + 1352x x  - 188x x  - 1382x x  - 650x  + 55x x  - 326x x  + 1037x x  + 705x x  - 667x x  + 1483x x  + 1661x x  - 1652x  - 1052x x  - 692x x  - 542x x  + 162x x  + 582x x  - 1369x x  + 934x x  + 1392x x  + 1227x , - 346x  + 1408x x  - 1225x  - 1536x x  - 1028x x  - 985x  - 210x x  - 1312x x  + 915x x  + 1633x  - 202x x  - 1636x x  - 1653x x  - 480x x  - 1260x  - 813x x  - 1623x x  - 1429x x  + 1094x x  - 747x x  + 955x  + 898x x  - 795x x  - 35x x  - 566x x  + 1631x x  - 324x x  + 926x  - 132x x  - 9x x  - 1290x x  - 543x x  + 902x x  + 735x x  - 342x x  - 400x  + 900x x  - 463x x  + 694x x  - 1262x x  - 1449x x  - 448x x  - 1402x x  - 731x x  - 996x , 301x  + 166x x  - 955x  - 739x x  - 1199x x  - 319x  + 1047x x  - 532x x  + 902x x  + 1195x  - 663x x  + 1215x x  - 534x x  - 332x x  - 973x  + 772x x  - 308x x  + 315x x  - 454x x  - 483x x  - 239x  - 1313x x  - 419x x  - 1340x x  - 1388x x  - 1340x x  - 1665x x  - 333x  - 465x x  - 1084x x  + 676x x  - 1612x x  - 288x x  + 11x x  - 1170x x  - 189x  + 498x x  - 889x x  + 693x x  + 1460x x  - 473x x  - 414x x  - 122x x  - 1659x x  - 1421x , 14x  - 1049x x  + 1506x  + 1235x x  + 642x x  - 1034x  + 460x x  + 150x x  + 760x x  - 1246x  - 1407x x  + 1570x x  + 1403x x  - 1610x x  - 431x  + 574x x  + 893x x  - 657x x  + 417x x  + 1362x x  + 224x  + 268x x  + 1097x x  + 1132x x  + 148x x  + 1331x x  - 77x x  - 756x  + 228x x  + 136x x  - 1484x x  - 1478x x  - 13x x  + 1620x x  - 701x x  - 769x  - 760x x  - 492x x  - 1077x x  - 1249x x  - 834x x  - 395x x  - 1358x x  - 988x x  + 113x , - 1634x  - 13x x  + 805x  - 21x x  - 1655x x  + 1479x  - 1510x x  - 646x x  + 225x x  - 1411x  + 1227x x  - 1108x x  + 1291x x  - 59x x  - 142x  + 586x x  - 676x x  + 655x x  - 1476x x  + 453x x  - 1076x  - 1152x x  + 1373x x  - 1191x x  - 416x x  + 699x x  + 317x x  + 825x  - 1560x x  - 488x x  - 1035x x  - 1561x x  - 644x x  - 1178x x  - 1320x x  + 158x  + 889x x  + 1444x x  - 1486x x  - 1211x x  + 1269x x  - 1228x x  + 568x x  + 1591x x  + 1207x , 105x  - 538x x  - 1222x  - 277x x  + 716x x  - 1067x  - 428x x  + 154x x  - 469x x  + 77x  + 538x x  - 179x x  + 921x x  - 223x x  + 1093x  - 262x x  + 1299x x  + 631x x  + 1486x x  - 1280x x  - 121x  - 50x x  - 978x x  - 694x x  - 531x x  + 505x x  + 1412x x  - 1061x  + 1202x x  + 448x x  - 187x x  + 1276x x  - 121x x  + 1361x x  + 697x x  + 682x  + 1592x x  + 705x x  - 227x x  - 7x x  - 1423x x  - 1446x x  - 1578x x  + 1511x x  + 917x , 1270x  - 391x x  - 1116x  - 287x x  + 653x x  + 1643x  + 1623x x  + 514x x  - 14x x  - 90x  + 1232x x  - 1434x x  + 1296x x  + 1522x x  + 136x  - 623x x  - 607x x  + 18x x  + 896x x  - 29x x  + 1059x  - 1053x x  + 1643x x  + 1652x x  - 1190x x  - 1073x x  + 1470x x  - 944x  - 93x x  - 187x x  - 994x x  - 1415x x  - 229x x  - 796x x  + 1642x x  + 1600x  - 344x x  + 905x x  + 1032x x  - 538x x  - 891x x  + 1243x x  + 1290x x  + 490x x  - 1148x , 1613x  + 175x x  - 1346x  - 1000x x  - 1217x x  - 729x  - 1296x x  + 1456x x  + 745x x  + 539x  + 525x x  - 811x x  + 753x x  + 1362x x  + 1629x  - 840x x  + 513x x  + 429x x  + 842x x  + 1414x x  - 308x  + 1415x x  - 1461x x  - 1135x x  + 701x x  + 766x x  + 785x x  + 1503x  + 147x x  + 929x x  - 1220x x  - 853x x  + 493x x  + 226x x  + 1416x x  + 280x  - 7x x  + 1632x x  + 520x x  + 1259x x  + 157x x  + 1596x x  + 655x x  - 42x x  - 586x })
│ │ │                                   0       0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4       1 4        2 4       3 4       4      0 5        1 5        2 5       3 5        4 5       5       0 6       1 6        2 6        3 6       4 6        5 6        6       0 7       1 7        2 7      3 7        4 7        5 7       6 7      7        0 8        1 8        2 8       3 8        4 8      5 8        6 8        7 8       8         0       0 1       1        0 2        1 2       2       0 3        1 3       2 3        3        0 4       1 4       2 4      3 4      4        0 5       1 5        2 5       3 5       4 5       5       0 6       1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8      5 8        6 8       7 8        8        0        0 1        1        0 2       1 2       2       0 3        1 3       2 3       3        0 4       1 4        2 4       3 4        4        0 5       1 5       2 5       3 5       4 5       5        0 6        1 6       2 6       3 6       4 6        5 6       6      0 7       1 7       2 7       3 7        4 7        5 7       6 7        7      0 8       1 8        2 8        3 8        4 8        5 8        6 8       7 8        8        0        0 1        1        0 2        1 2       2        0 3        1 3       2 3      3       0 4        1 4       2 4       3 4        4       0 5       1 5       2 5        3 5       4 5       5       0 6       1 6      2 6        3 6       4 6        5 6       6       0 7     1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8       2 8        3 8        4 8       5 8       6 8      7 8        8         0      0 1       1       0 2      1 2       2       0 3        1 3       2 3       3       0 4        1 4       2 4        3 4       4        0 5        1 5       2 5       3 5        4 5      5       0 6       1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7      3 7        4 7       5 7       6 7       7       0 8       1 8        2 8       3 8        4 8       5 8       6 8        7 8       8        0        0 1       1     0 2      1 2        2       0 3       1 3        2 3        3        0 4       1 4       2 4       3 4       4        0 5        1 5       2 5       3 5       4 5       5        0 6       1 6        2 6       3 6       4 6       5 6        6       0 7       1 7       2 7       3 7        4 7       5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8      8       0       0 1        1        0 2       1 2        2      0 3       1 3       2 3       3        0 4      1 4      2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5       0 6       1 6       2 6      3 6       4 6        5 6        6       0 7       1 7       2 7        3 7        4 7       5 7        6 7      7      0 8        1 8       2 8      3 8      4 8       5 8        6 8        7 8       8       0       0 1       1      0 2       1 2        2        0 3       1 3       2 3       3       0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5        4 5       5        0 6        1 6       2 6        3 6       4 6        5 6       6      0 7       1 7        2 7       3 7       4 7        5 7        6 7        7        0 8       1 8       2 8       3 8       4 8        5 8       6 8        7 8        8        0        0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4        1 4        2 4       3 4        4       0 5        1 5        2 5        3 5       4 5       5       0 6       1 6      2 6       3 6        4 6       5 6       6       0 7     1 7        2 7       3 7       4 7       5 7       6 7       7       0 8       1 8       2 8        3 8        4 8       5 8        6 8       7 8       8      0       0 1       1       0 2        1 2       2        0 3       1 3       2 3        3       0 4        1 4       2 4       3 4       4       0 5       1 5       2 5       3 5       4 5       5        0 6       1 6        2 6        3 6        4 6        5 6       6       0 7        1 7       2 7        3 7       4 7      5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8        8     0        0 1        1        0 2       1 2        2       0 3       1 3       2 3        3        0 4        1 4        2 4        3 4       4       0 5       1 5       2 5       3 5        4 5       5       0 6        1 6        2 6       3 6        4 6      5 6       6       0 7       1 7        2 7        3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8       5 8        6 8       7 8       8         0      0 1       1      0 2        1 2        2        0 3       1 3       2 3        3        0 4        1 4        2 4      3 4       4       0 5       1 5       2 5        3 5       4 5        5        0 6        1 6        2 6       3 6       4 6       5 6       6        0 7       1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8        2 8        3 8        4 8        5 8       6 8        7 8        8      0       0 1        1       0 2       1 2        2       0 3       1 3       2 3      3       0 4       1 4       2 4       3 4        4       0 5        1 5       2 5        3 5        4 5       5      0 6       1 6       2 6       3 6       4 6        5 6        6        0 7       1 7       2 7        3 7       4 7        5 7       6 7       7        0 8       1 8       2 8     3 8        4 8        5 8        6 8        7 8       8       0       0 1        1       0 2       1 2        2        0 3       1 3      2 3      3        0 4        1 4        2 4        3 4       4       0 5       1 5      2 5       3 5      4 5        5        0 6        1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7        3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8       4 8        5 8        6 8       7 8        8       0       0 1        1        0 2        1 2       2        0 3        1 3       2 3       3       0 4       1 4       2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5        0 6        1 6        2 6       3 6       4 6       5 6        6       0 7       1 7        2 7       3 7       4 7       5 7        6 7       7     0 8        1 8       2 8        3 8       4 8        5 8       6 8      7 8       8
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │  
│ │ │  i5 : time degreeMap phi
│ │ │ - -- used 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ + -- used 0.0584314s (cpu); 0.0584316s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14)
│ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │  
│ │ │                                   2                  2                           2                                      2                                                 2                                                           2                                                                   2                                                                              2                                                                                          2        2                  2                              2                                       2                                                2                                                             2                                                                  2                                                                              2                                                                                            2        2                  2                             2                                       2                                                2                                                           2                                                                      2                                                                              2                                                                                         2         2                 2                            2                                       2                                                  2                                                             2                                                                    2                                                                                2                                                                                             2       2                   2                            2                                     2                                                2                                                          2                                                                  2                                                                                   2                                                                                            2        2                2                           2                                      2                                                  2                                                            2                                                                      2                                                                                 2                                                                                          2   2                   2                           2                                     2                                                  2                                                           2                                                                    2                                                                              2                                                                                         2      2                  2                           2                                      2                                                  2                                                             2                                                                       2                                                                              2                                                                                          2         2                  2                            2                                     2                                                 2                                                              2                                                                    2                                                                               2                                                                                        2
│ │ │  o6 = map (ringP8, ringP8, {- 780x  - 506x x  + 1537x  - 132x x  - 928x x  + 386x  - 102x x  + 422x x  + 725x x  - 1073x  - 905x x  - 830x x  + 1500x x  + 276x x  + 1533x  - 653x x  + 1558x x  + 939x x  - 1432x x  + 462x x  - 329x  - 92x x  + 661x x  - 1298x x  - 684x x  + 70x x  - 715x x  + 1093x  + 581x x  + 329x x  + 454x x  - 911x x  - 84x x  - 1452x x  - 809x x  + 1202x  + 1353x x  + 1503x x  + 482x x  + 893x x  - 643x x  + 598x x  + 110x x  + 1064x x  - 472x , - 522x  - 583x x  + 1339x  + 1535x x  - 1317x x  + 1113x  - 169x x  + 1440x x  - 1657x x  + 721x  + 40x x  - 1576x x  - 367x x  + 257x x  - 1454x  + 1612x x  + 1529x x  - 1068x x  + 560x x  - 1441x x  + 608x  - 92x x  - 1006x x  + 285x x  + 102x x  - 397x x  + 66x x  - 643x  - 38x x  + 1380x x  + 1069x x  - 426x x  + 1147x x  + 982x x  + 10x x  - 662x  + 16x x  + 1561x x  + 1597x x  + 512x x  + 1288x x  - 1253x x  + 1317x x  + 1481x x  - 354x , - 640x  - 1551x x  + 469x  + 1482x x  - 1593x x  - 986x  + 471x x  + 612x x  + 1228x x  + 1156x  - 731x x  + 1503x x  - 628x x  + 674x x  - 799x  + 1137x x  + 844x x  + 589x x  - 666x x  + 829x x  - 1024x  - 170x x  + 450x x  + 1497x x  + 1204x x  - 907x x  + 1621x x  - 417x  + 1297x x  + 1444x x  + 4x x  + 398x x  + 996x x  - 1031x x  + 239x x  + 303x  + 1215x x  - 83x x  + 1571x x  - 1543x x  - 925x x  - 694x x  + 151x x  - 520x x  + 880x , - 1210x  - 222x x  + 185x  + 245x x  + 1059x x  - 322x  + 238x x  + 962x x  + 1260x x  - 1581x  + 50x x  + 1352x x  - 1465x x  + 1555x x  + 1333x  + 1362x x  + 1365x x  + 1168x x  - 1401x x  + 149x x  - 652x  + 1378x x  - 557x x  - 112x x  + 26x x  + 315x x  + 111x x  + 1592x  - 283x x  - 1454x x  + 907x x  + 212x x  + 400x x  + 1049x x  - 882x x  - 1429x  - 183x x  + 1571x x  - 1286x x  - 1179x x  + 1319x x  + 240x x  - 1100x x  + 1500x x  - 348x , 1051x  - 1325x x  + 1354x  - 346x x  - 1532x x  - 466x  + 163x x  - 659x x  - 291x x  + 966x  + 789x x  + 393x x  + 403x x  - 1199x x  - 570x  - 93x x  - 492x x  - 418x x  + 713x x  - 1323x x  - 1384x  - 830x x  - 54x x  - 306x x  + 709x x  + 421x x  - 954x x  - 299x  + 1053x x  - 1080x x  + 686x x  + 170x x  - 1272x x  - 1661x x  + 1235x x  + 1553x  - 1454x x  - 1411x x  - 1195x x  - 962x x  + 737x x  - 390x x  + 957x x  + 1538x x  + 1234x , - 509x  + 9x x  - 1563x  - 710x x  - 642x x  + 541x  + 220x x  - 1214x x  - 16x x  + 1008x  - 1088x x  + 755x x  - 886x x  - 1433x x  + 1154x  + 1627x x  - 1547x x  - 951x x  + 866x x  + 163x x  - 1142x  - 668x x  + 1361x x  + 1324x x  - 490x x  + 282x x  - 1133x x  - 612x  + 805x x  - 126x x  + 1296x x  - 973x x  + 1271x x  - 1646x x  + 844x x  + 1073x  - 1452x x  - 1112x x  - 141x x  + 176x x  - 1579x x  - 78x x  + 848x x  - 1365x x  + 711x , x  + 1543x x  - 1076x  + 493x x  - 526x x  + 868x  - 582x x  - 996x x  + 206x x  - 419x  + 1258x x  - 391x x  + 1002x x  - 1539x x  + 931x  - 1504x x  + 810x x  + 324x x  + 1356x x  + 313x x  + 772x  + 299x x  + 1186x x  + 718x x  + 407x x  - 64x x  - 828x x  - 1393x  + 94x x  - 290x x  - 766x x  + 950x x  - 640x x  + 265x x  - 1640x x  - 1403x  - 126x x  + 891x x  - 1519x x  - 927x x  - 1335x x  - 1448x x  - x x  - 1103x x  - 1152x , 821x  + 558x x  - 1174x  - 168x x  + 986x x  + 790x  + 549x x  + 817x x  + 1396x x  + 695x  + 1211x x  + 878x x  - 1061x x  - 1244x x  - 880x  + 1409x x  - 567x x  + 1240x x  + 1126x x  - 1262x x  + 490x  + 1553x x  + 1276x x  + 805x x  + 576x x  - 1076x x  + 1617x x  - 495x  - 750x x  - 277x x  + 544x x  + 1479x x  - 784x x  - 64x x  - 1203x x  + 405x  + 1013x x  + 604x x  + 1301x x  + 1003x x  + 235x x  + 696x x  + 939x x  - 714x x  - 879x , - 1452x  + 727x x  - 1159x  + 449x x  - 1169x x  + 732x  + 575x x  - 600x x  + 924x x  - 837x  + 1298x x  - 860x x  + 1010x x  + 774x x  + 319x  + 1087x x  - 1120x x  + 1439x x  + 1175x x  - 1648x x  + 985x  - 1317x x  - 878x x  + 399x x  - 1339x x  + 70x x  - 463x x  + 470x  - 628x x  - 907x x  + 748x x  + 98x x  + 1150x x  + 1140x x  + 1308x x  + 621x  + 369x x  - 991x x  - 1186x x  + 61x x  - 907x x  - 681x x  - 1528x x  + 717x x  + 854x })
│ │ │                                   0       0 1        1       0 2       1 2       2       0 3       1 3       2 3        3       0 4       1 4        2 4       3 4        4       0 5        1 5       2 5        3 5       4 5       5      0 6       1 6        2 6       3 6      4 6       5 6        6       0 7       1 7       2 7       3 7      4 7        5 7       6 7        7        0 8        1 8       2 8       3 8       4 8       5 8       6 8        7 8       8        0       0 1        1        0 2        1 2        2       0 3        1 3        2 3       3      0 4        1 4       2 4       3 4        4        0 5        1 5        2 5       3 5        4 5       5      0 6        1 6       2 6       3 6       4 6      5 6       6      0 7        1 7        2 7       3 7        4 7       5 7      6 7       7      0 8        1 8        2 8       3 8        4 8        5 8        6 8        7 8       8        0        0 1       1        0 2        1 2       2       0 3       1 3        2 3        3       0 4        1 4       2 4       3 4       4        0 5       1 5       2 5       3 5       4 5        5       0 6       1 6        2 6        3 6       4 6        5 6       6        0 7        1 7     2 7       3 7       4 7        5 7       6 7       7        0 8      1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1       1       0 2        1 2       2       0 3       1 3        2 3        3      0 4        1 4        2 4        3 4        4        0 5        1 5        2 5        3 5       4 5       5        0 6       1 6       2 6      3 6       4 6       5 6        6       0 7        1 7       2 7       3 7       4 7        5 7       6 7        7       0 8        1 8        2 8        3 8        4 8       5 8        6 8        7 8       8       0        0 1        1       0 2        1 2       2       0 3       1 3       2 3       3       0 4       1 4       2 4        3 4       4      0 5       1 5       2 5       3 5        4 5        5       0 6      1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7        4 7        5 7        6 7        7        0 8        1 8        2 8       3 8       4 8       5 8       6 8        7 8        8        0     0 1        1       0 2       1 2       2       0 3        1 3      2 3        3        0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5       4 5        5       0 6        1 6        2 6       3 6       4 6        5 6       6       0 7       1 7        2 7       3 7        4 7        5 7       6 7        7        0 8        1 8       2 8       3 8        4 8      5 8       6 8        7 8       8   0        0 1        1       0 2       1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5       2 5        3 5       4 5       5       0 6        1 6       2 6       3 6      4 6       5 6        6      0 7       1 7       2 7       3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8        4 8        5 8    6 8        7 8        8      0       0 1        1       0 2       1 2       2       0 3       1 3        2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5        2 5        3 5        4 5       5        0 6        1 6       2 6       3 6        4 6        5 6       6       0 7       1 7       2 7        3 7       4 7      5 7        6 7       7        0 8       1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1        1       0 2        1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4       3 4       4        0 5        1 5        2 5        3 5        4 5       5        0 6       1 6       2 6        3 6      4 6       5 6       6       0 7       1 7       2 7      3 7        4 7        5 7        6 7       7       0 8       1 8        2 8      3 8       4 8       5 8        6 8       7 8       8
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │  
│ │ │  i7 : time degreeMap phi'
│ │ │ - -- used 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ + -- used 1.72901s (cpu); 0.937432s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out
│ │ │ @@ -5,14 +5,14 @@
│ │ │  o2 : Ideal of P6
│ │ │  
│ │ │  i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
│ │ │ + -- used 0.000871528s (cpu); 0.000864713s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │                        - x  + x x
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
│ │ │ + -- used 0.089818s (cpu); 0.029284s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │                                    ZZ                                 ZZ
│ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
│ │ │                                  190181  0   1   2   3   4          190181  0   1   2   3   4   5
│ │ │ @@ -173,15 +173,15 @@
│ │ │  i8 : projectiveDegrees p2
│ │ │  
│ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : time g = graph p2;
│ │ │ - -- used 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
│ │ │ + -- used 0.0620512s (cpu); 0.0416268s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │  
│ │ │  i11 : g_1;
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │                        x  - x x
│ │ │                         1    0 3
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │  
│ │ │  i3 : time ideal phi
│ │ │ - -- used 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ + -- used 0.00461882s (cpu); 0.00459893s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -108,15 +108,15 @@
│ │ │                        y
│ │ │                         4
│ │ │                       }
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
│ │ │  
│ │ │  i6 : time ideal phi'
│ │ │ - -- used 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ + -- used 0.373298s (cpu); 0.203632s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │                        w w  - w w  + w w
│ │ │                         2 4    1 5    0 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │  
│ │ │  i2 : time psi = inverseMap phi
│ │ │ - -- used 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ + -- used 0.0920367s (cpu); 0.0920367s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -158,15 +158,15 @@
│ │ │  o4 = map (QQ[w ..w  ], QQ[w ..w  ], {w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   + w  w   - w w  , w w   - w w   + w w   + w w   + w w  , w  w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   + w  w   - w w  , w w   - w w   + w w   + w w   + w w  , w  w   - w  w   - w  w   + w  w   - w w  , w w   - w w   - w w   + w w   + w w  , w  w   - w  w   - w  w   + w  w   - w w  , w  w   - w  w   - w  w   + w  w   - w w  , w w   - w w   - w w   + w w   + w w  , w w   - w w   - w w   + w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   + w w   - w w  , w w   - w w   + w w   + w w   - w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   + w w   - w w  , w w   - w w   - w w   + w w   - w w  , w w  - w w  - w w  + w w  - w w })
│ │ │                0   26       0   26     21 22    20 23    15 24    10 25    0 26   19 22    18 23    16 24    11 25    1 26   19 20    18 21    17 24    12 25    2 26   15 19    16 21    17 23    13 25    3 26   10 19    11 21    12 23    13 24    4 26   0 19    1 21    2 23    3 24    4 25   15 18    16 20    17 22    14 25    5 26   10 18    11 20    12 22    14 24    6 26   0 18    1 20    2 22    5 24    6 25   12 16    11 17    13 18    14 19    7 26   2 16    1 17    3 18    5 19    7 25   12 15    10 17    13 20    14 21    8 26   11 15    10 16    13 22    14 23    9 26   2 15    0 17    3 20    5 21    8 25   1 15    0 16    3 22    5 23    9 25   5 13    3 14    7 15    8 16    9 17   5 12    2 14    6 17    8 18    7 20   3 12    2 13    4 17    8 19    7 21   5 11    1 14    6 16    9 18    7 22   3 11    1 13    4 16    9 19    7 23   2 11    1 12    4 18    6 19    7 24   7 10    8 11    9 12    6 13    4 14   5 10    0 14    6 15    9 20    8 22   3 10    0 13    4 15    9 21    8 23   2 10    0 12    4 20    6 21    8 24   1 10    0 11    4 22    6 23    9 24   4 5    3 6    0 7    1 8    2 9
│ │ │  
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ │  
│ │ │  i5 : time psi = inverseMap phi
│ │ │ - -- used 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ + -- used 0.348558s (cpu); 0.241573s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   + w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w  , w w   - w w   + w w   + w  w   - w  w  , - w w   + w w   + w  w   + w w   - w w  , - w w   + w w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , w  w   - w  w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w  - w w  - w w   + w w   - w w  , - w w  + w w  + w w   - w w   + w w  , w w  - w w  - w w  + w w   - w w  })
│ │ │                0   26       0   26       5 22    8 23    14 24    13 25    0 26     5 18    8 19    14 20    10 25    1 26     5 16    8 17    13 20    10 24    2 26     5 15    14 17    13 19    10 23    3 26     5 21    20 23    19 24    17 25    4 26     8 15    14 16    13 18    10 22    6 26     8 21    20 22    18 24    16 25    7 26   17 18    16 19    15 20    10 21    9 26     13 21    17 22    16 23    15 24    11 26     14 21    19 22    18 23    15 25    12 26   0 21    4 22    7 23    12 24    11 25     4 18    7 19    12 20    1 21    9 25     4 16    7 17    11 20    2 21    9 24     4 15    12 17    11 19    3 21    9 23     7 15    12 16    11 18    6 21    9 22   12 13    11 14    0 15    3 22    6 23   10 12    9 14    1 15    3 18    6 19   10 11    9 13    2 15    3 16    6 17   8 9    7 10    1 16    2 18    6 20   5 9    4 10    1 17    2 19    3 20   8 11    7 13    0 16    2 22    6 24   5 11    4 13    0 17    2 23    3 24   8 12    7 14    0 18    1 22    6 25   5 12    4 14    0 19    1 23    3 25   5 7    4 8    0 20    1 24    2 25     5 6    3 8    0 10    1 13    2 14   4 6    3 7    0 9    1 11    2 12
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │                        - -------x  + ---------x x  + ------------x x  - ----------x x  - -----x  - -----------x x  + -------------x x x  + -------------x x x  - --------x x  - ----------x x  + -------------x x x  - ----------x x  - -----------x x  + ----------x x  + ------x  + -----------x x  + ----------x x x  - -----------x x x  - -------x x  + -------------x x x  + ------------x x x x  - -----------x x x  + -----------x x x  - ------------x x x  + ----------x x  - -----------x x  - ------------x x x  - ---------x x  - ------------x x x  - -----------x x x  + -----------x x  - ----------x x  + -------x x  + --------x x  + ------x  + ---------x x  - ------------x x x  - -------------x x x  - ----------x x  + --------------x x x  + -------------x x x x  - ------------x x x  + -------------x x x  + ------------x x x  + ----------x x  + -----------x x x  - -------------x x x x  - ----------x x x  + --------------x x x x  - -------------x x x x  + -------------x x x  - ------------x x x  + ---------x x x  - ------------x x x  + ---------x x  - ---------x x  - -----------x x x  - ----------x x  + -----------x x x  + -----------x x x  + ----------x x  - -----------x x x  - -----------x x x  - ------------x x x  - ----------x x  + ---------x x  - ------x x  - --------x x  - ----------x x  - -----x
│ │ │                           290304 0    3888000  0 1    2939328000  0 1    163296000 0 1   20250 1    228614400  0 2    41150592000  0 1 2    41150592000  0 1 2    3888000 1 2     3572100  0 2    10287648000  0 1 2    342921600 1 2    114307200  0 2    63504000  1 2    25200 2     76204800  0 3    42336000  0 1 3    428652000  0 1 3    212625 1 3     5334336000  0 2 3    9601804800  0 1 2 3    489888000  1 2 3    222264000  0 2 3    12002256000 1 2 3    66679200  2 3    666792000  0 3     666792000  0 1 3    47628000 1 3    1333584000  0 2 3    444528000  1 2 3    777924000  2 3    55566000  0 3    105840 1 3    3472875 2 3    11025 3    4665600  0 4    2939328000  0 1 4     4898880000  0 1 4    29160000  1 4     41150592000  0 2 4    20575296000  0 1 2 4    4898880000  1 2 4    20575296000  0 2 4    1371686400  1 2 4    95256000  2 4     40824000  0 3 4     8573040000  0 1 3 4    11664000  1 3 4     24004512000  0 2 3 4    34292160000  1 2 3 4    12002256000  2 3 4     333396000  0 3 4    5292000  1 3 4    1333584000  2 3 4    3969000  3 4    6804000  0 4    272160000  0 1 4    58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4    476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4    30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i3 : time inverse phi
│ │ │ - -- used 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ + -- used 0.0784975s (cpu); 0.0784981s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out
│ │ │ @@ -40,18 +40,18 @@
│ │ │                        - t  + t t
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time isBirational phi
│ │ │ - -- used 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ + -- used 0.02786s (cpu); 0.0278608s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ + -- used 0.0317464s (cpu); 0.0191403s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │  
│ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.33576s (cpu); 2.59272s (thread); 0s (gc)
│ │ │ + -- used 2.80761s (cpu); 2.41858s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │  
│ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │       C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-23*x_1*x_7+8*x_2*x_7-22*x_3*x_7+20*x_4*x_7-15*x_5*x_7,x_2*x_5+47*x_5^2-40*x_0*x_6+37*x_1*x_6-25*x_2*x_6-22*x_3*x_6-8*x_4*x_6+27*x_5*x_6+15*x_6^2-23*x_0*x_7-42*x_1*x_7+27*x_2*x_7+35*x_3*x_7+39*x_4*x_7+24*x_5*x_7,x_1*x_5+15*x_5^2+49*x_0*x_6+8*x_1*x_6-31*x_2*x_6+9*x_3*x_6+38*x_4*x_6-36*x_5*x_6-30*x_6^2-33*x_0*x_7+26*x_1*x_7+32*x_2*x_7+27*x_3*x_7+6*x_4*x_7+36*x_5*x_7,x_0*x_5+30*x_5^2-11*x_0*x_6-38*x_1*x_6+13*x_2*x_6-32*x_3*x_6-30*x_4*x_6+4*x_5*x_6-28*x_6^2-30*x_0*x_7-6*x_1*x_7-45*x_2*x_7+34*x_3*x_7+20*x_4*x_7+48*x_5*x_7,x_3*x_4+46*x_5^2-37*x_0*x_6+27*x_1*x_6+33*x_2*x_6+8*x_3*x_6-32*x_4*x_6+42*x_5*x_6-34*x_6^2-37*x_0*x_7-28*x_1*x_7+10*x_2*x_7-27*x_3*x_7-42*x_4*x_7-8*x_5*x_7,x_2*x_4-25*x_5^2-4*x_0*x_6+2*x_1*x_6-31*x_2*x_6-5*x_3*x_6+16*x_4*x_6-24*x_5*x_6+31*x_6^2-30*x_0*x_7+32*x_1*x_7+12*x_2*x_7-40*x_3*x_7+3*x_4*x_7-28*x_5*x_7,x_0*x_4+15*x_5^2+48*x_0*x_6-50*x_1*x_6+46*x_2*x_6-48*x_3*x_6-23*x_4*x_6-28*x_5*x_6+39*x_6^2+38*x_1*x_7-5*x_3*x_7+5*x_4*x_7-34*x_5*x_7,x_3^2-31*x_5^2+41*x_0*x_6-30*x_1*x_6-4*x_2*x_6+43*x_3*x_6+23*x_4*x_6+7*x_5*x_6+31*x_6^2-19*x_0*x_7+25*x_1*x_7-49*x_2*x_7-16*x_3*x_7-45*x_4*x_7+25*x_5*x_7,x_2*x_3+13*x_5^2-45*x_0*x_6-22*x_1*x_6+33*x_2*x_6-26*x_3*x_6-21*x_4*x_6+34*x_5*x_6-21*x_6^2-47*x_0*x_7-10*x_1*x_7+29*x_2*x_7-46*x_3*x_7-x_4*x_7+20*x_5*x_7,x_1*x_3+22*x_5^2+4*x_0*x_6+3*x_1*x_6+45*x_2*x_6+37*x_3*x_6+17*x_4*x_6+36*x_5*x_6-2*x_6^2-31*x_0*x_7+3*x_1*x_7-12*x_2*x_7+19*x_3*x_7+28*x_4*x_7+30*x_5*x_7,x_0*x_3-47*x_5^2-43*x_0*x_6+6*x_1*x_6-40*x_2*x_6+21*x_3*x_6+26*x_4*x_6-5*x_5*x_6-5*x_6^2+4*x_0*x_7-15*x_1*x_7+18*x_2*x_7-31*x_3*x_7+50*x_4*x_7-46*x_5*x_7,x_2^2+4*x_5^2+31*x_0*x_6+41*x_1*x_6+31*x_2*x_6+28*x_3*x_6+42*x_4*x_6-28*x_5*x_6-4*x_6^2-7*x_0*x_7+15*x_1*x_7-9*x_2*x_7+31*x_3*x_7+3*x_4*x_7+7*x_5*x_7,x_1*x_2-46*x_5^2-6*x_0*x_6-50*x_1*x_6+32*x_2*x_6-10*x_3*x_6+42*x_4*x_6+33*x_5*x_6+18*x_6^2-9*x_0*x_7-20*x_1*x_7+45*x_2*x_7-9*x_3*x_7+10*x_4*x_7-8*x_5*x_7,x_0*x_2-9*x_5^2+34*x_0*x_6-45*x_1*x_6+19*x_2*x_6+24*x_3*x_6+23*x_4*x_6-37*x_5*x_6-44*x_6^2+24*x_0*x_7-33*x_2*x_7+41*x_3*x_7-40*x_4*x_7+4*x_5*x_7,x_1^2+x_1*x_4+x_4^2-28*x_5^2-33*x_0*x_6-17*x_1*x_6+11*x_3*x_6+20*x_4*x_6+25*x_5*x_6-21*x_6^2-22*x_0*x_7+24*x_1*x_7-14*x_2*x_7+5*x_3*x_7-39*x_4*x_7-18*x_5*x_7,x_0*x_1-47*x_5^2-5*x_0*x_6-9*x_1*x_6-45*x_2*x_6+48*x_3*x_6+45*x_4*x_6-29*x_5*x_6+3*x_6^2+29*x_0*x_7+40*x_1*x_7+46*x_2*x_7+27*x_3*x_7-36*x_4*x_7-39*x_5*x_7,x_0^2-31*x_5^2+36*x_0*x_6-30*x_1*x_6-10*x_2*x_6+42*x_3*x_6+9*x_4*x_6+34*x_5*x_6-6*x_6^2+48*x_0*x_7-47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);
│ │ │ @@ -21,12 +21,12 @@
│ │ │  
│ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │  
│ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 4.05735s (cpu); 2.55688s (thread); 0s (gc)
│ │ │ + -- used 4.66494s (cpu); 3.20213s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out
│ │ │ @@ -6,23 +6,23 @@
│ │ │  o1 = map (QQ[x ..x ], QQ[y ..y  ], {- 5x x  + x x  + x x  + 35x x  - 7x x  + x x  - x x  - 49x  - 5x x  + 2x x  - x x  + 27x x  - 4x  + x x  - 7x x  + 2x x  - 2x x  + 14x x  - 4x x , - x x  - 6x x  - 5x x  + 2x x  + x x  + x x  - 5x x  - x x  + 2x x  + 7x x  - 2x x  + 2x x  - 3x x , - 25x  + 9x x  + 10x x  - 2x x  - x  + 29x x  - x x  - 7x x  - 13x x  + 3x x  + x x  - x x  + 2x x  - x x  + 7x x  - 2x x  - 8x x  + 2x x  - 3x x , x x  + x x  + x  + 7x x  - 9x x  + 12x x  - 4x  + 2x x  + 2x x  - 14x x  + 4x x  + x x  - x x  - 14x x  + x x , - 5x x  + x x  - 7x x  + 8x x  - 5x x  + 2x x  - x x  + x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , x x  + x  - 7x x  - 8x x  + x x  + x x  + 2x x  - x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  + x  - 8x x  + x x  + 6x x  - 2x  + x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  - 7x x  + x x  + x x  - 7x x  + 2x  - x x , - 4x x  + x x  - x  - 7x x  + 8x x  + x x  - x x  - 6x x  + 2x  - x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , - 5x x  + 2x  + x x  - x  - x x  + 8x x  - 10x x  + 2x x  + 2x x  - 2x x  + 14x x  - 4x x  + 5x x  - 3x x  - 2x x  + 7x x  - 2x x  - 3x x , - 5x x  + x x  + x x  - 4x x  - x x  + x x  + x x , x x  - x x  + 5x x  + x x  - 14x x  - x x  - 8x x  - 8x x  + 2x x  + 4x x  + 2x x  + 4x x  + 3x x  - 7x x  + 2x x  - 3x x })
│ │ │                0   8       0   11        0 3    2 4    3 4      0 5     2 5    3 5    4 5      5     0 6     2 6    4 6      5 6     6    4 7     5 7     6 7     4 8      5 8     6 8     1 2     1 5     0 6     1 6    4 6    5 6     0 7    1 7     2 7     5 7     6 7     1 8     7 8       0     0 2      0 4     2 4    4      0 5    2 5     4 5      0 6     4 6    5 6    0 7     2 7    4 7     5 7     6 7     0 8     4 8     7 8   2 4    3 4    4     2 5     4 5      5 6     6     3 7     4 7      5 7     6 7    3 8    4 8      5 8    6 8      0 4    2 4     2 5     4 5     0 6     2 6    4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   0 4    4     1 5     4 5    0 6    1 6     4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   2 3    4     4 5    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8   1 3     1 5    1 6    4 6     5 6     6    3 7      0 3    3 4    4     0 5     4 5    0 6    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8      0 2     2    2 4    4    2 5     4 5      0 6     5 6     2 7     4 7      5 7     6 7     0 8     2 8     4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7   0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2 7     0 8     1 8     5 8     6 8     7 8
│ │ │  
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │  
│ │ │  i2 : time kernel(phi,1)
│ │ │ - -- used 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ + -- used 0.028148s (cpu); 0.0281438s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ + -- used 1.43024s (cpu); 0.669271s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out
│ │ │ @@ -26,15 +26,15 @@
│ │ │                8           9
│ │ │  
│ │ │                   ZZ
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i3 : time parametrize L
│ │ │ - -- used 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ + -- used 0.00587398s (cpu); 0.00587336s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -116,15 +116,15 @@
│ │ │               5 9           6 9           7 9           8 9           9
│ │ │  
│ │ │                   ZZ
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i5 : time parametrize Q
│ │ │ - -- used 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ + -- used 0.647252s (cpu); 0.447437s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 3560583829489988690
│ │ │  
│ │ │  i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
│ │ │  
│ │ │  i2 : time p = point source f
│ │ │ - -- used 0.298623s (cpu); 0.198159s (thread); 0s (gc)
│ │ │ + -- used 0.316484s (cpu); 0.193877s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -20,12 +20,12 @@
│ │ │                                                             -----[y ..y  ]
│ │ │                                                             33331  0   11
│ │ │  o2 : Ideal of -------------------------------------------------------------------------------------------------------
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i3 : time p == f^* f p
│ │ │ - -- used 0.176944s (cpu); 0.176946s (thread); 0s (gc)
│ │ │ + -- used 0.130669s (cpu); 0.130669s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │                       0   4              0   5       1    0 2     1 2    0 3     2    1 3     1 3    0 4     2 3    1 4     3    2 4
│ │ │  
│ │ │  o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ]
│ │ │                          0   4                 0   5
│ │ │  
│ │ │  i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0278401s (cpu); 0.0272526s (thread); 0s (gc)
│ │ │ + -- used 0.0376683s (cpu); 0.0251078s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │                           0   5
│ │ │  o4 : RingMap ------------------ <-- GF 109561[t ..t ]
│ │ │               x x  - x x  + x x                 0   4
│ │ │                2 3    1 4    0 5
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.019353s (cpu); 0.0187436s (thread); 0s (gc)
│ │ │ + -- used 0.0279112s (cpu); 0.0156626s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a rational octic surface
│ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ @@ -48,21 +48,21 @@
│ │ │            300007  0   6   300007  0   6     2 4    1 5          0 4          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6   2 3    0 5          1 3          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6        0 3         1 4         3 4         4          0 5         1 5         2 5          3 5          4 5         5         3 6          4 6         5 6          0 1          1         0 2          1 2         2          1 4          1 5         2 5          0 6         1 6         2 6         0          1         0 2         1 2         2         1 4          4         0 5         1 5          2 5          4 5         5         0 6         1 6          2 6          3 6         4 6         5 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │  
│ │ │  i7 : time projectiveDegrees phi
│ │ │ - -- used 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc)
│ │ │ + -- used 6.7461e-05s (cpu); 5.7891e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc)
│ │ │ + -- used 2.7546e-05s (cpu); 2.7512e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
│ │ │  
│ │ │                  ZZ
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │  
│ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ + -- used 0.136262s (cpu); 0.136263s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │                     0         1         2         3        4         5
│ │ │  
│ │ │  o4 : Ideal of X
│ │ │  
│ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │  
│ │ │  i6 : time phi = rationalMap D
│ │ │ - -- used 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ + -- used 0.0373871s (cpu); 0.037389s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -123,13 +123,13 @@
│ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │                       }
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │  
│ │ │  i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ + -- used 1.72776s (cpu); 0.747668s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cremona__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330846641081
│ │ │  
│ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ + -- used 1.28106s (cpu); 1.15351s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cubic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1730018912715498288
│ │ │  
│ │ │  i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ + -- used 0.113796s (cpu); 0.113797s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -62,15 +62,15 @@
│ │ │                        8x x  - 12x x  + 24x  - 11x x  + 17x x x  - 24x x  - 10x x  + 11x x  - 3x  - 6x x  + 28x x x  - 70x x  - 21x x x  + 47x x x  - 13x x  - 14x x  + 66x x  - 22x x  - 20x  + 2x x  - 2x x x  - 10x x  - 11x x x  + 8x x x  - 5x x  + 3x x x  + 23x x x  - 11x x x  - 12x x  + 3x x  - 3x x  - 2x x  + 3x x  + x  - 11x x  + 14x x x  + 34x x  - 6x x x  - 16x x x  + 3x x  - 15x x x  - 66x x x  + 12x x x  + 30x x  - 19x x x  + 2x x x  - 5x x x  - 2x x x  - 7x x  + 6x x  + 21x x  - 3x x  - 21x x  + x x  + 5x  - 8x x  + 7x x x  - 32x x  - 13x x x  + 28x x x  - 9x x  + 70x x x  - 27x x x  - 36x x  + x x x  + 4x x x  - 7x x x  - 2x x x  + 3x x  - 25x x x  - 23x x x  + 4x x x  + 27x x x  - 14x x x  - 9x x  - 2x x  + 10x x  - 6x x  - 10x x  + 3x x  - 2x x
│ │ │                          0 1      0 1      1      0 2      0 1 2      1 2      0 2      1 2     2     0 3      0 1 3      1 3      0 2 3      1 2 3      2 3      0 3      1 3      2 3      3     0 4     0 1 4      1 4      0 2 4     1 2 4     2 4     0 3 4      1 3 4      2 3 4      3 4     0 4     1 4     2 4     3 4    4      0 5      0 1 5      1 5     0 2 5      1 2 5     2 5      0 3 5      1 3 5      2 3 5      3 5      0 4 5     1 4 5     2 4 5     3 4 5     4 5     0 5      1 5     2 5      3 5    4 5     5     0 6     0 1 6      1 6      0 2 6      1 2 6     2 6      1 3 6      2 3 6      3 6    0 4 6     1 4 6     2 4 6     3 4 6     4 6      0 5 6      1 5 6     2 5 6      3 5 6      4 5 6     5 6     0 6      1 6     2 6      3 6     4 6     5 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ + -- used 0.0346402s (cpu); 0.034641s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Quadratic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1729200582376678705
│ │ │  
│ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ + -- used 0.095649s (cpu); 0.0956524s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -50,15 +50,15 @@
│ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ + -- used 0.00933303s (cpu); 0.00933396s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_to__External__String_lp__Rational__Map_rp.out
│ │ │ @@ -7,34 +7,34 @@
│ │ │  i2 : str = toExternalString phi;
│ │ │  
│ │ │  i3 : #str
│ │ │  
│ │ │  o3 = 6927
│ │ │  
│ │ │  i4 : time phi' = value str;
│ │ │ - -- used 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ + -- used 0.0314253s (cpu); 0.0312877s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ + -- used 0.00765564s (cpu); 0.00760529s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │       number of minimal representatives: 1
│ │ │       dimension base locus: 1
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
│ │ │  
│ │ │  i6 : time describe inverse phi'
│ │ │ - -- used 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ + -- used 0.006149s (cpu); 0.00613904s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html
│ │ │ @@ -102,30 +102,30 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ + -- used 1.55264s (cpu); 1.23061s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
    
│ │ │              
    
│ │ │                
    i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ + -- used 1.65484s (cpu); 1.16581s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -172,30 +172,30 @@
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i9 : time ChernClass G
│ │ │ - -- used 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ + -- used 0.152207s (cpu); 0.152177s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
    
│ │ │              
    
│ │ │                
    i10 : time ChernClass(G,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ + -- used 0.0421118s (cpu); 0.0181553s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,26 +39,26 @@
│ │ │ │                 2                           2
│ │ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │ │                          0   4
│ │ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ │ + -- used 1.55264s (cpu); 1.23061s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o3 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o3 : -----
│ │ │ │          5
│ │ │ │         H
│ │ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ │ + -- used 1.65484s (cpu); 1.16581s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o4 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          5
│ │ │ │ @@ -88,26 +88,26 @@
│ │ │ │          0,2 1,3    0,1 2,3
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i9 : time ChernClass G
│ │ │ │ - -- used 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ │ + -- used 0.152207s (cpu); 0.152177s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          9      8      7      6      5      4     3
│ │ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o9 : -----
│ │ │ │         10
│ │ │ │        H
│ │ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ │ + -- used 0.0421118s (cpu); 0.0181553s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -90,24 +90,24 @@
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i2 : time EulerCharacteristic I
│ │ │ - -- used 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ + -- used 0.692097s (cpu); 0.288497s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ + -- used 0.051737s (cpu); 0.0212687s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,20 +31,20 @@
│ │ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i2 : time EulerCharacteristic I
│ │ │ │ - -- used 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ │ + -- used 0.692097s (cpu); 0.288497s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ │ + -- used 0.051737s (cpu); 0.0212687s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = 10
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  No test is made to see if the projective variety is smooth.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_u_l_e_r_(_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- topological Euler characteristic of a
│ │ │ │        (smooth) projective variety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i4 : time phi! ;
│ │ │ - -- used 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ + -- used 0.0865409s (cpu); 0.0705998s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    
│ │ │              
    
│ │ │                
    i5 : describe phi
│ │ │ @@ -132,15 +132,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i9 : time phi! ;
│ │ │ - -- used 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ + -- used 0.0712487s (cpu); 0.0588081s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
    
│ │ │              
    
│ │ │                
    i10 : describe phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i3 : describe phi
│ │ │ │  
│ │ │ │  o3 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i4 : time phi! ;
│ │ │ │ - -- used 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ │ + -- used 0.0865409s (cpu); 0.0705998s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │ │  i5 : describe phi
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  i8 : describe phi
│ │ │ │  
│ │ │ │  o8 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i9 : time phi! ;
│ │ │ │ - -- used 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ │ + -- used 0.0712487s (cpu); 0.0588081s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │ │  i10 : describe phi
│ │ │ │  
│ │ │ │  o10 = rational map defined by forms of degree 2
│ │ │ │        source variety: PP^4
│ │ │ │        target variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i6 : time phi^** q
│ │ │ - -- used 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ + -- used 0.479291s (cpu); 0.274562s (thread); 0s (gc)
│ │ │  
│ │ │                  e        d        c        b        a
│ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │                  f        f        f        f        f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i4 : time SegreClass X
│ │ │ - -- used 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ + -- used 1.18697s (cpu); 0.686282s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ + -- used 0.578796s (cpu); 0.399267s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i6 : time SegreClass(X,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ + -- used 0.0404911s (cpu); 0.0269575s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ + -- used 0.368464s (cpu); 0.181236s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -208,25 +208,25 @@
│ │ │          
    
│ │ │          
    The method also accepts as input a ring map phi representing a rational map $\Phi:X\dashrightarrow Y$ between projective varieties. In this case, the method returns the push-forward to the Chow ring of the ambient projective space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it basically computes SegreClass ideal matrix phi. In the next example, we compute the Segre class of the base locus of a birational map $\mathbb{G}(1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ -= ------[x ..x ]
│ │ │ -  100003  0   6
│ │ │ +       ZZ
│ │ │ +o9 = ------[x ..x ]
│ │ │ +     100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ + -- used 0.0752016s (cpu); 0.0752054s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -238,15 +238,15 @@
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i11 : time SegreClass phi
│ │ │ - -- used 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ + -- used 0.449255s (cpu); 0.293058s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -272,30 +272,30 @@
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ + -- used 0.512456s (cpu); 0.340242s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
    
│ │ │              
    
│ │ │                
    i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ + -- used 1.99806s (cpu); 1.20752s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,47 +81,47 @@
│ │ │ │                 2 2                2 2                                        2
│ │ │ │  2                                                    2 2
│ │ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x
│ │ │ │  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1
│ │ │ │  6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │ │  i4 : time SegreClass X
│ │ │ │ - -- used 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ │ + -- used 1.18697s (cpu); 0.686282s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ │ - -- used 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ │ + -- used 0.578796s (cpu); 0.399267s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o5 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ │ + -- used 0.0404911s (cpu); 0.0269575s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ │ + -- used 0.368464s (cpu); 0.181236s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │          8
│ │ │ │ @@ -134,22 +134,22 @@
│ │ │ │  method returns the push-forward to the Chow ring of the ambient projective
│ │ │ │  space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it
│ │ │ │  basically computes SegreClass ideal matrix phi. In the next example, we compute
│ │ │ │  the Segre class of the base locus of a birational map $\mathbb{G}
│ │ │ │  (1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
│ │ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │ │  
│ │ │ │ -o9     ZZ
│ │ │ │ -= ------[x ..x ]
│ │ │ │ -  100003  0   6
│ │ │ │ +       ZZ
│ │ │ │ +o9 = ------[x ..x ]
│ │ │ │ +     100003  0   6
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},
│ │ │ │  {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ │ + -- used 0.0752016s (cpu); 0.0752054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                          ZZ
│ │ │ │                                                        ------[y ..y ]
│ │ │ │                                                        100003  0   9
│ │ │ │  ZZ              2                              2
│ │ │ │  o10 = map (--------------------------------------------------------------------
│ │ │ │  --------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │  o10 : RingMap -----------------------------------------------------------------
│ │ │ │  ----------------------------------- <-- ------[x ..x ]
│ │ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y
│ │ │ │  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6
│ │ │ │  1 7    0 8   2 3    1 4    0 5
│ │ │ │  i11 : time SegreClass phi
│ │ │ │ - -- used 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ │ + -- used 0.449255s (cpu); 0.293058s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │          10
│ │ │ │ @@ -198,26 +198,26 @@
│ │ │ │  ------------------------------------
│ │ │ │                 (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y
│ │ │ │  - y y  + y y , y y  - y y  + y y )
│ │ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2
│ │ │ │  6    1 7    0 8   2 3    1 4    0 5
│ │ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │ │        time SegreClass B
│ │ │ │ - -- used 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ │ + -- used 0.512456s (cpu); 0.340242s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o13 : -----
│ │ │ │          10
│ │ │ │         H
│ │ │ │  i14 : -- Segre class of B in P^9
│ │ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ │ - -- used 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ │ + -- used 1.99806s (cpu); 1.20752s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             9       8       7      6     5
│ │ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o14 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ + -- used 0.000518171s (cpu); 0.000511273s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │ @@ -124,23 +124,23 @@
│ │ │            
    
│ │ │          
    Now we compute first the degree of the forms defining the abstract map psi and then the corresponding concrete rational map.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ + -- used 0.150227s (cpu); 0.150234s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time rationalMap psi
│ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ + -- used 0.595499s (cpu); 0.475293s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -238,15 +238,15 @@
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    
│ │ │              
    
│ │ │                
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ + -- used 0.0506466s (cpu); 0.0506457s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -258,26 +258,26 @@
│ │ │            
    
│ │ │          
    The degree of the forms defining the abstract map T can be obtained by the following command:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ + -- used 3.07714s (cpu); 2.20173s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself is defined by linear forms:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : time T2 = T * T
│ │ │ - -- used 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ + -- used 3.4806e-05s (cpu); 3.2523e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -286,15 +286,15 @@
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    
│ │ │              
    
│ │ │                
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ + -- used 4.83467s (cpu); 3.52399s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself leaves a random point fixed:
    
│ │ │          
│ │ │ @@ -327,15 +327,15 @@
│ │ │            
│ │ │          
    
│ │ │          
    We now compute the concrete rational map corresponding to T:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i21 : time f = rationalMap T
│ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ + -- used 4.01634s (cpu); 2.92522s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,32 +35,32 @@
│ │ │ │  i3 : P5 := QQ[u_0..u_5]
│ │ │ │  
│ │ │ │  o3 = QQ[u ..u ]
│ │ │ │           0   5
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ │ - -- used 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ │ + -- used 0.000518171s (cpu); 0.000511273s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: given by a function
│ │ │ │  
│ │ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │ │  Now we compute first the degree of the forms defining the abstract map psi and
│ │ │ │  then the corresponding concrete rational map.
│ │ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ │ - -- used 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ │ + -- used 0.150227s (cpu); 0.150234s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ │ + -- used 0.595499s (cpu); 0.475293s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: {
│ │ │ │ @@ -139,48 +139,48 @@
│ │ │ │  o13 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o13 : Ideal of -----[x ..x ]
│ │ │ │                 65521  0   3
│ │ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ │ - -- used 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ │ + -- used 0.0506466s (cpu); 0.0506457s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  The degree of the forms defining the abstract map T can be obtained by the
│ │ │ │  following command:
│ │ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ │ + -- used 3.07714s (cpu); 2.20173s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3
│ │ │ │  We verify that the composition of T with itself is defined by linear forms:
│ │ │ │  i16 : time T2 = T * T
│ │ │ │ - -- used 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ │ + -- used 3.4806e-05s (cpu); 3.2523e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ │ + -- used 4.83467s (cpu); 3.52399s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 1
│ │ │ │  We verify that the composition of T with itself leaves a random point fixed:
│ │ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │ │  
│ │ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │ @@ -193,15 +193,15 @@
│ │ │ │  i20 : T q
│ │ │ │  
│ │ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │  o20 : List
│ │ │ │  We now compute the concrete rational map corresponding to T:
│ │ │ │  i21 : time f = rationalMap T
│ │ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ │ + -- used 4.01634s (cpu); 2.92522s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ + -- used 0.596918s (cpu); 0.407878s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -205,15 +205,15 @@
│ │ │            
    
│ │ │                
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ + -- used 0.343778s (cpu); 0.236792s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i6 : assert(psi == psi')
    
│ │ │ @@ -300,15 +300,15 @@
│ │ │            
    
│ │ │                
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ + -- used 2.86736s (cpu); 2.31325s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -372,15 +372,15 @@
│ │ │              
    		
│ │ │                
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ + -- used 3.59065s (cpu); 2.94829s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  -- approximateInverseMap: step 4 of 10
│ │ │ │  -- approximateInverseMap: step 5 of 10
│ │ │ │  -- approximateInverseMap: step 6 of 10
│ │ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ │  -- approximateInverseMap: step 8 of 10
│ │ │ │  -- approximateInverseMap: step 9 of 10
│ │ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ │ - -- used 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ │ + -- used 0.596918s (cpu); 0.407878s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                    ZZ
│ │ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │ │                                  ZZ
│ │ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -252,15 +252,15 @@
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ │ + -- used 0.343778s (cpu); 0.236792s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i6 : assert(psi == psi')
│ │ │ │  A more complicated example is the following (here _i_n_v_e_r_s_e_M_a_p takes a lot of
│ │ │ │  time!).
│ │ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-
│ │ │ │  22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  
│ │ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time
│ │ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ │ + -- used 2.86736s (cpu); 2.31325s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = -- rational map --
│ │ │ │                                  ZZ
│ │ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │               {
│ │ │ │                                  2
│ │ │ │ @@ -526,15 +526,15 @@
│ │ │ │  o9 = false
│ │ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ │ + -- used 3.59065s (cpu); 2.94829s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │                {
│ │ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    
│ │ │              
    
│ │ │                
    i5 : time degreeMap phi
│ │ │ - -- used 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ + -- used 0.0584314s (cpu); 0.0584316s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │              
    
│ │ │                
    i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    
│ │ │              
    
│ │ │                
    i7 : time degreeMap phi'
│ │ │ - -- used 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ + -- used 1.72901s (cpu); 0.937432s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -266,15 +266,15 @@
│ │ │ │  4       0 5       1 5       2 5       3 5        4 5       5        0 6
│ │ │ │  1 6        2 6       3 6       4 6       5 6        6       0 7       1 7
│ │ │ │  2 7       3 7       4 7       5 7        6 7       7     0 8        1 8       2
│ │ │ │  8        3 8       4 8        5 8       6 8      7 8       8
│ │ │ │  
│ │ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │ │  i5 : time degreeMap phi
│ │ │ │ - -- used 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ │ + -- used 0.0584314s (cpu); 0.0584316s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 1
│ │ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a
│ │ │ │  general subspace of P^14
│ │ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have
│ │ │ │  degree equal to deg(G(1,5))=14)
│ │ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  0 5        1 5        2 5        3 5        4 5       5        0 6       1 6
│ │ │ │  2 6        3 6      4 6       5 6       6       0 7       1 7       2 7      3
│ │ │ │  7        4 7        5 7        6 7       7       0 8       1 8        2 8
│ │ │ │  3 8       4 8       5 8        6 8       7 8       8
│ │ │ │  
│ │ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │ │  i7 : time degreeMap phi'
│ │ │ │ - -- used 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ │ + -- used 1.72901s (cpu); 0.937432s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = 14
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ │ │      * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s -- projective degrees of a rational map between
│ │ │ │        projective varieties
│ │ │ │  ********** WWaayyss ttoo uussee ddeeggrreeeeMMaapp:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
    
│ │ │ + -- used 0.000871528s (cpu); 0.000864713s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  
│ │ │ │  o2 : Ideal of P6
│ │ │ │  i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │ │  
│ │ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ │ - -- used 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
│ │ │ │ + -- used 0.000871528s (cpu); 0.000864713s (thread); 0s (gc)
│ │ │ │  i5 : Phi;
│ │ │ │  
│ │ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional
│ │ │ │  subvariety of PP^9)
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  If the declaration is false, nonsensical answers may result.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_graph.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
    
│ │ │ + -- used 0.089818s (cpu); 0.029284s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │ @@ -277,15 +277,15 @@
│ │ │            
│ │ │          
│ │ │          
    When the source of the rational map is a multi-projective variety, the method returns all the projections.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │  
│ │ │ │                        w w  - w w  + w w
│ │ │ │                         2 4    1 5    0 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │ │  i2 : time psi = inverseMap phi
│ │ │ │ - -- used 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ │ + -- used 0.0920367s (cpu); 0.0920367s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = -- rational map --
│ │ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w
│ │ │ │  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13
│ │ │ │  14   15   16   17   18   19   20
│ │ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w
│ │ │ │ @@ -216,15 +216,15 @@
│ │ │ │  15    9 20    8 22   3 10    0 13    4 15    9 21    8 23   2 10    0 12    4
│ │ │ │  20    6 21    8 24   1 10    0 11    4 22    6 23    9 24   4 5    3 6    0 7
│ │ │ │  1 8    2 9
│ │ │ │  
│ │ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │ │                   0   26          0   26
│ │ │ │  i5 : time psi = inverseMap phi
│ │ │ │ - -- used 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ │ + -- used 0.348558s (cpu); 0.241573s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  ,
│ │ │ │  - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   -
│ │ │ │  w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   -
│ │ │ │  w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   +
│ │ │ │  w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   +
│ │ │ │  w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse_lp__Rational__Map_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time g = graph p2;
│ │ │ - -- used 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
    
│ │ │ + -- used 0.0620512s (cpu); 0.0416268s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │                        - x  + x x
│ │ │ │                           3    2 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in
│ │ │ │  PP^5)
│ │ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ │ - -- used 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
│ │ │ │ + -- used 0.089818s (cpu); 0.029284s (thread); 0s (gc)
│ │ │ │  i4 : p1
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │                                    ZZ                                 ZZ
│ │ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y
│ │ │ │  , y , y , y , y ]) defined by
│ │ │ │                                  190181  0   1   2   3   4          190181  0
│ │ │ │ @@ -192,15 +192,15 @@
│ │ │ │  
│ │ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  When the source of the rational map is a multi-projective variety, the method
│ │ │ │  returns all the projections.
│ │ │ │  i9 : time g = graph p2;
│ │ │ │ - -- used 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
│ │ │ │ + -- used 0.0620512s (cpu); 0.0416268s (thread); 0s (gc)
│ │ │ │  i10 : g_0;
│ │ │ │  
│ │ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ │ │  of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │ │  i11 : g_1;
│ │ │ │  
│ │ │ │  o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_ideal_lp__Rational__Map_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │              
    
│ │ │                
    i3 : time ideal phi
│ │ │ - -- used 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ + -- used 0.00461882s (cpu); 0.00459893s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -200,15 +200,15 @@
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    
│ │ │              
    
│ │ │                
    i6 : time ideal phi'
│ │ │ - -- used 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ + -- used 0.373298s (cpu); 0.203632s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                         2
│ │ │ │                        x  - x x
│ │ │ │                         1    0 3
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i3 : time ideal phi
│ │ │ │ - -- used 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ │ + -- used 0.00461882s (cpu); 0.00459893s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2                                     2
│ │ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2
│ │ │ │       x x , x  - x x )
│ │ │ │ @@ -121,15 +121,15 @@
│ │ │ │                        y
│ │ │ │                         4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of
│ │ │ │  PP^5 x PP^4 to PP^4)
│ │ │ │  i6 : time ideal phi'
│ │ │ │ - -- used 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ │ + -- used 0.373298s (cpu); 0.203632s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -157,15 +157,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    
│ │ │              
    
│ │ │                
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ + -- used 0.0920367s (cpu); 0.0920367s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -255,15 +255,15 @@
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ + -- used 0.348558s (cpu); 0.241573s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   + w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w  , w w   - w w   + w w   + w  w   - w  w  , - w w   + w w   + w  w   + w w   - w w  , - w w   + w w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , w  w   - w  w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w  - w w  - w w   + w w   - w w  , - w w  + w w  + w w   - w w   + w w  , w w  - w w  - w w  + w w   - w w  })
│ │ │                0   26       0   26       5 22    8 23    14 24    13 25    0 26     5 18    8 19    14 20    10 25    1 26     5 16    8 17    13 20    10 24    2 26     5 15    14 17    13 19    10 23    3 26     5 21    20 23    19 24    17 25    4 26     8 15    14 16    13 18    10 22    6 26     8 21    20 22    18 24    16 25    7 26   17 18    16 19    15 20    10 21    9 26     13 21    17 22    16 23    15 24    11 26     14 21    19 22    18 23    15 25    12 26   0 21    4 22    7 23    12 24    11 25     4 18    7 19    12 20    1 21    9 25     4 16    7 17    11 20    2 21    9 24     4 15    12 17    11 19    3 21    9 23     7 15    12 16    11 18    6 21    9 22   12 13    11 14    0 15    3 22    6 23   10 12    9 14    1 15    3 18    6 19   10 11    9 13    2 15    3 16    6 17   8 9    7 10    1 16    2 18    6 20   5 9    4 10    1 17    2 19    3 20   8 11    7 13    0 16    2 22    6 24   5 11    4 13    0 17    2 23    3 24   8 12    7 14    0 18    1 22    6 25   5 12    4 14    0 19    1 23    3 25   5 7    4 8    0 20    1 24    2 25     5 6    3 8    0 10    1 13    2 14   4 6    3 7    0 9    1 11    2 12
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i3 : time inverse phi
│ │ │ - -- used 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ + -- used 0.0784975s (cpu); 0.0784981s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ │ ├── html2text {}
│ │ │ │ @@ -290,15 +290,15 @@
│ │ │ │  58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4
│ │ │ │  476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4
│ │ │ │  30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │ │  i3 : time inverse phi
│ │ │ │ - -- used 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ │ + -- used 0.0784975s (cpu); 0.0784981s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html
│ │ │ @@ -128,24 +128,24 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ + -- used 0.02786s (cpu); 0.0278608s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ + -- used 0.0317464s (cpu); 0.0191403s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,20 +58,20 @@
│ │ │ │                        - t  + t t
│ │ │ │                           3    2 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in
│ │ │ │  PP^5)
│ │ │ │  i3 : time isBirational phi
│ │ │ │ - -- used 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ │ + -- used 0.02786s (cpu); 0.0278608s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ │ + -- used 0.0317464s (cpu); 0.0191403s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_D_o_m_i_n_a_n_t -- whether a rational map is dominant
│ │ │ │  ********** WWaayyss ttoo uussee iissBBiirraattiioonnaall:: **********
│ │ │ │      * isBirational(RationalMap)
│ │ │ │      * isBirational(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.33576s (cpu); 2.59272s (thread); 0s (gc)
│ │ │ + -- used 2.80761s (cpu); 2.41858s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : P7 = ZZ/101[x_0..x_7];
    
│ │ │ @@ -120,15 +120,15 @@
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 4.05735s (cpu); 2.55688s (thread); 0s (gc)
│ │ │ + -- used 4.66494s (cpu); 3.20213s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : P8 = ZZ/101[x_0..x_8];
│ │ │ │  i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},
│ │ │ │  {x_4..x_8}};
│ │ │ │  
│ │ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 3.33576s (cpu); 2.59272s (thread); 0s (gc)
│ │ │ │ + -- used 2.80761s (cpu); 2.41858s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │ │       C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-
│ │ │ │  26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-
│ │ │ │  24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  
│ │ │ │  o5 : Ideal of P7
│ │ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │ │  
│ │ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 4.05735s (cpu); 2.55688s (thread); 0s (gc)
│ │ │ │ + -- used 4.66494s (cpu); 3.20213s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_B_i_r_a_t_i_o_n_a_l -- whether a rational map is birational
│ │ │ │  ********** WWaayyss ttoo uussee iissDDoommiinnaanntt:: **********
│ │ │ │      * isDominant(RationalMap)
│ │ │ │      * isDominant(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html
│ │ │ @@ -93,26 +93,26 @@
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ + -- used 0.028148s (cpu); 0.0281438s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time kernel(phi,2)
│ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ + -- used 1.43024s (cpu); 0.669271s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │  4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7
│ │ │ │  0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2
│ │ │ │  7     0 8     1 8     5 8     6 8     7 8
│ │ │ │  
│ │ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │ │                   0   8          0   11
│ │ │ │  i2 : time kernel(phi,1)
│ │ │ │ - -- used 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ │ + -- used 0.028148s (cpu); 0.0281438s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ │ + -- used 1.43024s (cpu); 0.669271s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                             2
│ │ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │  
│ │ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_parametrize_lp__Ideal_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time parametrize L
│ │ │ - -- used 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ + -- used 0.00587398s (cpu); 0.00587336s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -205,15 +205,15 @@
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time parametrize Q
│ │ │ - -- used 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ + -- used 0.647252s (cpu); 0.447437s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │       - 849671x  + 3034137x )
│ │ │ │                8           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o2 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i3 : time parametrize L
│ │ │ │ - -- used 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ │ + -- used 0.00587398s (cpu); 0.00587336s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │       1211601x x  - 2168594x x  - 1801762x x  + 3022242x x  + 3618789x )
│ │ │ │               5 9           6 9           7 9           8 9           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o4 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i5 : time parametrize Q
│ │ │ │ - -- used 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ │ + -- used 0.647252s (cpu); 0.447437s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_point_lp__Quotient__Ring_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time p = point source f
│ │ │ - -- used 0.298623s (cpu); 0.198159s (thread); 0s (gc)
│ │ │ + -- used 0.316484s (cpu); 0.193877s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -104,15 +104,15 @@
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time p == f^* f p
│ │ │ - -- used 0.176944s (cpu); 0.176946s (thread); 0s (gc)
│ │ │ + -- used 0.130669s (cpu); 0.130669s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  _p_o_i_n_t_(_M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_).
│ │ │ │  Below we verify the birationality of a rational map.
│ │ │ │  i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to
│ │ │ │  PP^8)
│ │ │ │  i2 : time p = point source f
│ │ │ │ - -- used 0.298623s (cpu); 0.198159s (thread); 0s (gc)
│ │ │ │ + -- used 0.316484s (cpu); 0.193877s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │ │               10        11   9         11   8        11   7        11   6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │ │             11   5        11   4        11   3         11   2        11   1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o2 : Ideal of -----------------------------------------------------------------
│ │ │ │  --------------------------------------
│ │ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y
│ │ │ │  y  - y y  + y y , y y  - y y  + y y )
│ │ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11
│ │ │ │  3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ │  i3 : time p == f^* f p
│ │ │ │ - -- used 0.176944s (cpu); 0.176946s (thread); 0s (gc)
│ │ │ │ + -- used 0.130669s (cpu); 0.130669s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t -- pick a random K rational point on the scheme X
│ │ │ │        defined by I
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * point(PolynomialRing)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │                          0   4                 0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0278401s (cpu); 0.0272526s (thread); 0s (gc)
│ │ │ + -- used 0.0376683s (cpu); 0.0251078s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -122,15 +122,15 @@
│ │ │                2 3    1 4    0 5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.019353s (cpu); 0.0187436s (thread); 0s (gc)
│ │ │ + -- used 0.0279112s (cpu); 0.0156626s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ @@ -148,25 +148,25 @@
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time projectiveDegrees phi
│ │ │ - -- used 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc)
│ │ │ + -- used 6.7461e-05s (cpu); 5.7891e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc)
│ │ │ + -- used 2.7546e-05s (cpu); 2.7512e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │                       0   4              0   5       1    0 2     1 2    0 3
│ │ │ │  2    1 3     1 3    0 4     2 3    1 4     3    2 4
│ │ │ │  
│ │ │ │  o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ]
│ │ │ │                          0   4                 0   5
│ │ │ │  i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0278401s (cpu); 0.0272526s (thread); 0s (gc)
│ │ │ │ + -- used 0.0376683s (cpu); 0.0251078s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │ │  
│ │ │ │             GF 109561[x ..x ]
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │                GF 109561[x ..x ]
│ │ │ │                           0   5
│ │ │ │  o4 : RingMap ------------------ <-- GF 109561[t ..t ]
│ │ │ │               x x  - x x  + x x                 0   4
│ │ │ │                2 3    1 4    0 5
│ │ │ │  i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.019353s (cpu); 0.0187436s (thread); 0s (gc)
│ │ │ │ + -- used 0.0279112s (cpu); 0.0156626s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a
│ │ │ │  rational octic surface
│ │ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ │ @@ -119,21 +119,21 @@
│ │ │ │  4 5         5         0 6         1 6          2 6          3 6         4 6
│ │ │ │  5 6
│ │ │ │  
│ │ │ │                 ZZ                 ZZ
│ │ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │ │               300007  0   6      300007  0   6
│ │ │ │  i7 : time projectiveDegrees phi
│ │ │ │ - -- used 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc)
│ │ │ │ + -- used 6.7461e-05s (cpu); 5.7891e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.7546e-05s (cpu); 2.7512e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = {4, 1}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  Another way to use this method is by passing an integer i as second argument.
│ │ │ │  However, this is equivalent to first projectiveDegrees(phi,NumDegrees=>i) and
│ │ │ │  generally it is not faster.
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ + -- used 0.136262s (cpu); 0.136263s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-
│ │ │ │  x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o2 : Ideal of -----[x ..x ]
│ │ │ │                33331  0   6
│ │ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ │ - -- used 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ │ + -- used 0.136262s (cpu); 0.136263s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                      ZZ
│ │ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │ │                    33331  0   1   2   3   4   5   6
│ │ │ │                      ZZ
│ │ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  ,
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │              
│ │ │                
    i5 : D = new Tally from {H => 2,C => 1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ + -- used 0.0373871s (cpu); 0.037389s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -224,15 +224,15 @@
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ + -- used 1.72776s (cpu); 0.747668s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    See also the package WeilDivisors, which provides general tools for working with divisors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  o4 = ideal(- 32646x  - 28377x  + 26433x  - 29566x  + 3783x  + 26696x )
│ │ │ │                     0         1         2         3        4         5
│ │ │ │  
│ │ │ │  o4 : Ideal of X
│ │ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │ │  i6 : time phi = rationalMap D
│ │ │ │ - -- used 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ │ + -- used 0.0373871s (cpu); 0.037389s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │                                    ZZ
│ │ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │ │                                  65521  0   1   2   3   4   5
│ │ │ │               {
│ │ │ │                   2                  2
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │                                                    2                         2
│ │ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │ │  i7 : time ? image(phi,"F4")
│ │ │ │ - -- used 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ │ + -- used 1.72776s (cpu); 0.747668s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │ │       hypersurfaces of degree 2
│ │ │ │  See also the package _W_e_i_l_D_i_v_i_s_o_r_s, which provides general tools for working
│ │ │ │  with divisors.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_t_i_o_n_a_l_M_a_p -- makes a rational map
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ + -- used 1.28106s (cpu); 1.15351s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │              K, according to the classification given in Table 1 of _S_p_e_c_i_a_l
│ │ │ │              _c_u_b_i_c_ _C_r_e_m_o_n_a_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _P_6_ _a_n_d_ _P_7.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A Cremona transformation is said to be special if the base locus scheme is
│ │ │ │  smooth and irreducible. To ensure this condition, the field K must be large
│ │ │ │  enough but no check is made.
│ │ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ │ - -- used 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ │ + -- used 1.28106s (cpu); 1.15351s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │ │        source variety: PP^3
│ │ │ │        target variety: PP^3
│ │ │ │        dominance: true
│ │ │ │        birationality: true
│ │ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ + -- used 0.113796s (cpu); 0.113797s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ + -- used 0.0346402s (cpu); 0.034641s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special cubic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 2 of _S_p_e_c_i_a_l_ _c_u_b_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _p_r_o_j_e_c_t_i_v_e
│ │ │ │              _s_p_a_c_e_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialCubicTransformation 9
│ │ │ │ - -- used 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ │ + -- used 0.113796s (cpu); 0.113797s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6
│ │ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -323,15 +323,15 @@
│ │ │ │  6     4 6      0 5 6      1 5 6     2 5 6      3 5 6      4 5 6     5 6     0 6
│ │ │ │  1 6     2 6      3 6     4 6     5 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ │ + -- used 0.0346402s (cpu); 0.034641s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^6
│ │ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ + -- used 0.095649s (cpu); 0.0956524s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ + -- used 0.00933303s (cpu); 0.00933396s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special quadratic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 1 of _E_x_a_m_p_l_e_s_ _o_f_ _s_p_e_c_i_a_l_ _q_u_a_d_r_a_t_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s
│ │ │ │              _i_n_t_o_ _c_o_m_p_l_e_t_e_ _i_n_t_e_r_s_e_c_t_i_o_n_s_ _o_f_ _q_u_a_d_r_i_c_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ │ - -- used 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ │ + -- used 0.095649s (cpu); 0.0956524s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6   7   8
│ │ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │                                                     2
│ │ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ │ + -- used 0.00933303s (cpu); 0.00933396s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^8
│ │ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html
│ │ │ @@ -93,23 +93,23 @@
│ │ │  
│ │ │  o3 = 6927
    
│ │ │              
    
│ │ │                
    i4 : time phi' = value str;
│ │ │ - -- used 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ + -- used 0.0314253s (cpu); 0.0312877s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ + -- used 0.00765564s (cpu); 0.00760529s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ @@ -118,15 +118,15 @@
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │              
    
│ │ │                
    i6 : time describe inverse phi'
│ │ │ - -- used 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ + -- used 0.006149s (cpu); 0.00613904s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,32 +19,32 @@
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i2 : str = toExternalString phi;
│ │ │ │  i3 : #str
│ │ │ │  
│ │ │ │  o3 = 6927
│ │ │ │  i4 : time phi' = value str;
│ │ │ │ - -- used 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ │ + -- used 0.0314253s (cpu); 0.0312877s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ │ + -- used 0.00765564s (cpu); 0.00760529s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^3
│ │ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ │       number of minimal representatives: 1
│ │ │ │       dimension base locus: 1
│ │ │ │       degree base locus: 5
│ │ │ │       coefficient ring: ZZ/33331
│ │ │ │  i6 : time describe inverse phi'
│ │ │ │ - -- used 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ │ + -- used 0.006149s (cpu); 0.00613904s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = rational map defined by forms of degree 2
│ │ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │ │       target variety: PP^3
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/index.html
│ │ │ @@ -63,29 +63,29 @@
│ │ │              
    		
│ │ │                
    i1 : ZZ/300007[t_0..t_6];
    
│ │ │              
    
│ │ │                
    i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ + -- used 0.00566984s (cpu); 0.0056673s (thread); 0s (gc)
│ │ │  
│ │ │              ZZ              ZZ                3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o2 = map (------[t ..t ], ------[x ..x ], {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   300007  0   9       2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
    
│ │ │              
    
│ │ │                
    i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ + -- used 0.0567073s (cpu); 0.0567084s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │ @@ -93,43 +93,43 @@
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │              
    
│ │ │                
    i4 : time degreeMap phi
│ │ │ - -- used 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ + -- used 0.0372848s (cpu); 0.037157s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ + -- used 0.633196s (cpu); 0.513987s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ + -- used 0.0914524s (cpu); 0.0914625s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ + -- used 0.0041893s (cpu); 0.00419242s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -141,15 +141,15 @@
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i8 : time psi = inverseMap phi
│ │ │ - -- used 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ + -- used 0.657052s (cpu); 0.541959s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -161,44 +161,44 @@
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ + -- used 0.0119239s (cpu); 0.0119286s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ + -- used 0.199603s (cpu); 0.199611s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ + -- used 7.52838s (cpu); 6.8613s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │          
    We repeat the example using the type RationalMap and using deterministic methods.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │        2
│ │ │ │  o6 = y T
│ │ │ │          1,2
│ │ │ │  
│ │ │ │  o6 : R[T   ..T   ]
│ │ │ │          1,1   1,3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s
│ │ │ │ +Finding easy relations           :  -- used 0.19201s (cpu); 0.0407282s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -134,15 +134,15 @@
│ │ │  
│ │ │  o5 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ + -- used 0.00275207s (cpu); 0.0027582s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -247,15 +247,15 @@
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │              
    
│ │ │                
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ + -- used 0.0757898s (cpu); 0.075791s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -320,15 +320,15 @@
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i14 : time phi^(-1)
│ │ │ - -- used 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ + -- used 0.699254s (cpu); 0.591838s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -381,49 +381,49 @@
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │              
    
│ │ │                
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ + -- used 0.364033s (cpu); 0.312703s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ + -- used 0.085042s (cpu); 0.0268094s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ + -- used 0.00453372s (cpu); 0.00454109s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i18 : time describe phi^(-1)
│ │ │ - -- used 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ + -- used 0.0130698s (cpu); 0.0130788s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ @@ -432,41 +432,41 @@
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ + -- used 0.0132473s (cpu); 0.0132547s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i21 : time degrees f
│ │ │ - -- used 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ + -- used 1.51799s (cpu); 1.14509s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ + -- used 1.7761e-05s (cpu); 1.7288e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ + -- used 0.00177047s (cpu); 0.00177482s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  map) from a list of $m+1$ homogeneous elements of the same degree in $K
│ │ │ │  [x_0,...,x_n]/I$.
│ │ │ │  Below is an example using the methods provided by this package, dealing with a
│ │ │ │  birational transformation $\Phi:\mathbb{P}^6 \dashrightarrow \mathbb{G}
│ │ │ │  (2,4)\subset\mathbb{P}^9$ of bidegree $(3,3)$.
│ │ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ │ + -- used 0.00566984s (cpu); 0.0056673s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              ZZ              ZZ                3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │  3                2    2                2                                 2
│ │ │ │  2    2                                  2        2                      2
│ │ │ │  2                        2                         2    2                 2
│ │ │ │  3                2    2
│ │ │ │ @@ -52,43 +52,43 @@
│ │ │ │  0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4
│ │ │ │  3 4 5    2 5    3 6    2 4 6
│ │ │ │  
│ │ │ │                 ZZ                 ZZ
│ │ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │ │               300007  0   6      300007  0   9
│ │ │ │  i3 : time J = kernel(phi,2)
│ │ │ │ - -- used 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ │ + -- used 0.0567073s (cpu); 0.0567084s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o3 : Ideal of ------[x ..x ]
│ │ │ │                300007  0   9
│ │ │ │  i4 : time degreeMap phi
│ │ │ │ - -- used 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ │ + -- used 0.0372848s (cpu); 0.037157s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ │ + -- used 0.633196s (cpu); 0.513987s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ │ + -- used 0.0914524s (cpu); 0.0914625s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ │ + -- used 0.0041893s (cpu); 0.00419242s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                                         ZZ
│ │ │ │                                                                       ------[x
│ │ │ │  ..x ]
│ │ │ │              ZZ                                                       300007  0
│ │ │ │  9                                                  3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │ @@ -123,15 +123,15 @@
│ │ │ │  o7 : RingMap ------[t ..t ] <-- -----------------------------------------------
│ │ │ │  -----------------------------------------------------
│ │ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  -
│ │ │ │  x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5
│ │ │ │  2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ │  i8 : time psi = inverseMap phi
│ │ │ │ - -- used 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ │ + -- used 0.657052s (cpu); 0.541959s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                         ZZ
│ │ │ │                                                       ------[x ..x ]
│ │ │ │                                                       300007  0   9
│ │ │ │  ZZ              3                2               2    2
│ │ │ │  2                          2     2        2                               2
│ │ │ │  2               2             2                       3
│ │ │ │ @@ -164,31 +164,31 @@
│ │ │ │  o8 : RingMap ------------------------------------------------------------------
│ │ │ │  ---------------------------------- <-- ------[t ..t ]
│ │ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │  1 6    0 8   2 4    1 5    0 7
│ │ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ │ - -- used 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ │ + -- used 0.0119239s (cpu); 0.0119286s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ │ + -- used 0.199603s (cpu); 0.199611s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ │ + -- used 7.52838s (cpu); 6.8613s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  We repeat the example using the type _R_a_t_i_o_n_a_l_M_a_p and using deterministic
│ │ │ │  methods.
│ │ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ │ + -- used 0.00275207s (cpu); 0.0027582s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -233,15 +233,15 @@
│ │ │ │                            3                2    2
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ │ - -- used 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ │ + -- used 0.0757898s (cpu); 0.075791s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                                     ZZ
│ │ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │ @@ -304,15 +304,15 @@
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i14 : time phi^(-1)
│ │ │ │ - -- used 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ │ + -- used 0.699254s (cpu); 0.591838s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                                     ZZ
│ │ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │  ]) defined by
│ │ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │ │                {
│ │ │ │ @@ -373,67 +373,67 @@
│ │ │ │                          6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6
│ │ │ │  9     4 6 9    2 7 9    4 7 9     0 9
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9
│ │ │ │  to PP^6)
│ │ │ │  i15 : time degrees phi^(-1)
│ │ │ │ - -- used 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ │ + -- used 0.364033s (cpu); 0.312703s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ │ + -- used 0.085042s (cpu); 0.0268094s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ │ + -- used 0.00453372s (cpu); 0.00454109s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = rational map defined by forms of degree 3
│ │ │ │        source variety: PP^6
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i18 : time describe phi^(-1)
│ │ │ │ - -- used 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ │ + -- used 0.0130698s (cpu); 0.0130788s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 = rational map defined by forms of degree 3
│ │ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        target variety: PP^6
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ │        number of minimal representatives: 1
│ │ │ │        dimension base locus: 4
│ │ │ │        degree base locus: 24
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ │ - -- used 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ │ + -- used 0.0132473s (cpu); 0.0132547s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety
│ │ │ │  of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │ │  i21 : time degrees f
│ │ │ │ - -- used 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ │ + -- used 1.51799s (cpu); 1.14509s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.7761e-05s (cpu); 1.7288e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ │ + -- used 0.00177047s (cpu); 0.00177482s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20
│ │ │ │  hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1,
│ │ │ │  1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2,
│ │ │ │  0},{2, 0})
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/Cyclotomic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  Q3ljbG90b21pYw==
│ │ │  #:len=478
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY3ljbG90b21pYyBmaWVsZHMiLCBEZXNj
│ │ │  cmlwdGlvbiA9PiAoRU17IkN5Y2xvdG9taWMifSwiIGlzIGEgcGFja2FnZSBmb3IgY3ljbG90b21p
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  c291cmNlKERHQWxnZWJyYU1hcCk=
│ │ │  #:len=1484
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIHNvdXJjZSBvZiBhIERHIGFsZ2Vi
│ │ │  cmEgbWFwIiwgImxpbmVudW0iID0+IDkyNTUsIElucHV0cyA9PiB7U1BBTntUVHsicGhpIn0sIiwg
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out
│ │ │ @@ -155,15 +155,15 @@
│ │ │                                    2       2       2         2 2       2 2        2 2        2 2       2 2            2 2           2 2            2           2           2
│ │ │         Differential => {a, b, c, a T   , b T   , c T   , a*b c T   , b c T   , -a b T   , -a c T   , b c T   T   , -a c T   T   , b c T   T   , -a T   T   , c T   T   , b T   T   }
│ │ │                                      1,1     1,2     1,3         1,1       2,1        2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3 2,4     1,2 2,4
│ │ │  
│ │ │  o16 : DGAlgebra
│ │ │  
│ │ │  i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0495233s (cpu); 0.0274118s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │        Underlying algebra => R[S   ..S   ]
│ │ │                                 1,1   1,4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0418334s (cpu); 0.0290018s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i6 : describe HB
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │                                      2
│ │ │        Differential => {a, b, c, d, a S   }
│ │ │                                        1,1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0384504s (cpu); 0.0250352s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │        Underlying algebra => R[T   ..T   ]
│ │ │                                 1,1   1,3
│ │ │        Differential => {a, b, c}
│ │ │  
│ │ │  o2 : DGAlgebra
│ │ │  
│ │ │  i3 : HA = HH A
│ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.211384s (cpu); 0.0536302s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = HA
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │  
│ │ │  o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
│ │ │               1,1   1,3      1,1   1,2     1,2   1,3
│ │ │  
│ │ │  o4 : DGAlgebraMap
│ │ │  
│ │ │  i5 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0397315s (cpu); 0.0262886s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │        Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
│ │ │                                     |     0    |  |     0    |
│ │ │                                     |     0    |  |     0    |
│ │ │  
│ │ │  o4 : DGModule
│ │ │  
│ │ │  i5 : HM = homology M
│ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.240287s (cpu); 0.0630181s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ │  
│ │ │                                                  4
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module__Map.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │        Natural => {0, 0} | 1 0 0 |
│ │ │                   {2, 2} | 0 1 0 |
│ │ │                   {2, 2} | 0 0 1 |
│ │ │  
│ │ │  o4 : DGModuleMap
│ │ │  
│ │ │  i5 : h = homology idM
│ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.212795s (cpu); 0.0606195s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │       {3, 4} | 0 1 0 0 |
│ │ │       {3, 4} | 0 0 1 0 |
│ │ │       {3, 4} | 0 0 0 1 |
│ │ │  
│ │ │  o5 : Matrix
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │                                1                                                             {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |     3
│ │ │                                                                                      
│ │ │                                                                                     2
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.103198s (cpu); 0.0378152s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i8 : ideal HKR
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │  i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-c^2*d^2}
│ │ │  
│ │ │  o9 = R'
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.691391s (cpu); 0.592007s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 1.1027s (cpu); 0.84719s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_cycles.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra A
│ │ │ -Finding easy relations           :  -- used 0.0210175s (cpu); 0.0187759s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0359181s (cpu); 0.0228538s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Deg__N__Module.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │        Underlying algebra => R[T   ..T   ]
│ │ │                                 1,1   1,3
│ │ │        Differential => {x, y, z}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HA = HH KR;
│ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.088231s (cpu); 0.0295435s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : H0 = zerothHomology KR
│ │ │  
│ │ │  o6 = H0
│ │ │  
│ │ │  o6 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0223745s (cpu); 0.0212429s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0651693s (cpu); 0.029287s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
│ │ │  
│ │ │ @@ -46,15 +46,15 @@
│ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0919266s (cpu); 0.08901s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.196636s (cpu); 0.122994s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -122,15 +122,15 @@
│ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0836547s (cpu); 0.0683607s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -159,14 +159,14 @@
│ │ │         Underlying algebra => S[T   ..T   ]
│ │ │                                  1,1   1,4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0421419s (cpu); 0.0298108s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
│ │ │ @@ -43,15 +43,15 @@
│ │ │  o6 = y T
│ │ │          1,2
│ │ │  
│ │ │  o6 : R[T   ..T   ]
│ │ │          1,1   1,3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.19201s (cpu); 0.0407282s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │  
│ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │                                    
│ │ │       0      1      2      3      4
│ │ │  
│ │ │  o5 : Complex
│ │ │  
│ │ │  i6 : HKR = HH(KR)
│ │ │ - -- used 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ + -- used 0.147735s (cpu); 0.125634s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
│ │ │  
│ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
│ │ │ @@ -68,15 +68,15 @@
│ │ │                 2
│ │ │  o9 = (true, x y T   T   T    - x x y T   T   T   )
│ │ │               2 2 1,1 1,2 1,3    1 2 2 1,2 1,3 1,4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.596064s (cpu); 0.508805s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.865078s (cpu); 0.748554s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T   T   T   T
│ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │  
│ │ │  o10 : R[T   ..T   ]
│ │ │           1,1   1,5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -27,15 +27,15 @@
│ │ │                                 1,1   1,4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.302961s (cpu); 0.267678s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.742914s (cpu); 0.606823s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
│ │ │ @@ -294,15 +294,15 @@
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0495233s (cpu); 0.0274118s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -218,15 +218,15 @@
│ │ │ │                                      1,1     1,2     1,3         1,1       2,1
│ │ │ │  2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3
│ │ │ │  2,4     1,2 2,4
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s
│ │ │ │ +Finding easy relations           :  -- used 0.0495233s (cpu); 0.0274118s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -179,15 +179,15 @@
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │                                 1,1   1,4
│ │ │ │        Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  One can compute the homology algebra of a DGAlgebra using the homology (or HH)
│ │ │ │  command.
│ │ │ │  i5 : HB = HH B
│ │ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s
│ │ │ │ +Finding easy relations           :  -- used 0.0418334s (cpu); 0.0290018s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │ @@ -87,15 +87,15 @@
│ │ │ │                                 1,1   1,4   2,1
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a S   }
│ │ │ │                                        1,1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s
│ │ │ │ +Finding easy relations           :  -- used 0.0384504s (cpu); 0.0250352s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o2 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o2 = {Ring => R                          }
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,3
│ │ │ │        Differential => {a, b, c}
│ │ │ │  
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : HA = HH A
│ │ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s
│ │ │ │ +Finding easy relations           :  -- used 0.211384s (cpu); 0.0536302s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = HA
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  
│ │ │  o4 : DGAlgebraMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -255,15 +255,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0418334s (cpu); 0.0290018s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0384504s (cpu); 0.0250352s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i3 : HA = HH A
│ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.211384s (cpu); 0.0536302s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = HA
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
    
│ │ │                
    i5 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0397315s (cpu); 0.0262886s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  i4 : g = dgAlgebraMap(K1, K2, matrix{{Y_(1,2), Y_(1,3)}})
│ │ │ │  
│ │ │ │  o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
│ │ │ │               1,1   1,3      1,1   1,2     1,2   1,3
│ │ │ │  
│ │ │ │  o4 : DGAlgebraMap
│ │ │ │  i5 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s
│ │ │ │ +Finding easy relations           :  -- used 0.0397315s (cpu); 0.0262886s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o4 : DGModule
    
│ │ │              
    
│ │ │                
    i5 : HM = homology M
│ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.240287s (cpu); 0.0630181s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ │  
│ │ │                                                  4
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,15 +57,15 @@
│ │ │ │        Generator degrees => {{0, 0}, {2, 2}, {2, 2}}
│ │ │ │        Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
│ │ │ │                                     |     0    |  |     0    |
│ │ │ │                                     |     0    |  |     0    |
│ │ │ │  
│ │ │ │  o4 : DGModule
│ │ │ │  i5 : HM = homology M
│ │ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s
│ │ │ │ +Finding easy relations           :  -- used 0.240287s (cpu); 0.0630181s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module__Map.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o4 : DGModuleMap
    
│ │ │              
    
│ │ │                
    i5 : h = homology idM
│ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.212795s (cpu); 0.0606195s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │       {3, 4} | 0 1 0 0 |
│ │ │       {3, 4} | 0 0 1 0 |
│ │ │       {3, 4} | 0 0 0 1 |
│ │ │  
│ │ │  o5 : Matrix
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,15 +59,15 @@
│ │ │ │                      1,1   1,2
│ │ │ │        Natural => {0, 0} | 1 0 0 |
│ │ │ │                   {2, 2} | 0 1 0 |
│ │ │ │                   {2, 2} | 0 0 1 |
│ │ │ │  
│ │ │ │  o4 : DGModuleMap
│ │ │ │  i5 : h = homology idM
│ │ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s
│ │ │ │ +Finding easy relations           :  -- used 0.212795s (cpu); 0.0606195s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │ │       {3, 4} | 0 1 0 0 |
│ │ │ │       {3, 4} | 0 0 1 0 |
│ │ │ │       {3, 4} | 0 0 0 1 |
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -143,15 +143,15 @@
│ │ │          
    
│ │ │            
    Since the Koszul complex is a DG algebra, its homology is itself an algebra.  One can obtain this algebra using the command homology, homologyAlgebra, or HH (all commands work).  This algebra structure can detect whether or not the ring is a complete intersection or Gorenstein.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  Since the Koszul complex is a DG algebra, its homology is itself an algebra.
│ │ │ │  One can obtain this algebra using the command homology, homologyAlgebra, or HH
│ │ │ │  (all commands work). This algebra structure can detect whether or not the ring
│ │ │ │  is a complete intersection or Gorenstein.
│ │ │ │  i7 : HKR = HH KR
│ │ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s
│ │ │ │ +Finding easy relations           :  -- used 0.103198s (cpu); 0.0378152s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = HKR
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i8 : ideal HKR
│ │ │ │  
│ │ │ │ @@ -94,16 +94,16 @@
│ │ │ │  i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-
│ │ │ │  c^2*d^2}
│ │ │ │  
│ │ │ │  o9 = R'
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ │ -Finding easy relations           :  -- used 0.691391s (cpu); 0.592007s
│ │ │ │ -(thread); 0s (gc)
│ │ │ │ +Finding easy relations           :  -- used 1.1027s (cpu); 0.84719s (thread);
│ │ │ │ +0s (gc)
│ │ │ │  
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ │ │  
│ │ │ │  o11 = 34
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_cycles.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra A
│ │ │ │ -Finding easy relations           :  -- used 0.0210175s (cpu); 0.0187759s
│ │ │ │ +Finding easy relations           :  -- used 0.0359181s (cpu); 0.0228538s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Deg__N__Module.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.103198s (cpu); 0.0378152s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.691391s (cpu); 0.592007s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 1.1027s (cpu); 0.84719s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra A
│ │ │ -Finding easy relations           :  -- used 0.0210175s (cpu); 0.0187759s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0359181s (cpu); 0.0228538s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i5 : HA = HH KR;
│ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s (thread); 0s (gc)
    
│ │ │ +Finding easy relations           :  -- used 0.088231s (cpu); 0.0295435s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : H0 = zerothHomology KR
│ │ │  
│ │ │  o6 = H0
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o4 = {Ring => R                          }
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,3
│ │ │ │        Differential => {x, y, z}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : HA = HH KR;
│ │ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s
│ │ │ │ +Finding easy relations           :  -- used 0.088231s (cpu); 0.0295435s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  i6 : H0 = zerothHomology KR
│ │ │ │  
│ │ │ │  o6 = H0
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  i7 : M1 = getDegNModule(1, H0, HA)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0223745s (cpu); 0.0212429s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0651693s (cpu); 0.029287s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o7 : List
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0919266s (cpu); 0.08901s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.196636s (cpu); 0.122994s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0836547s (cpu); 0.0683607s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0421419s (cpu); 0.0298108s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0223745s (cpu); 0.0212429s
│ │ │ │ +Finding easy relations           :  -- used 0.0651693s (cpu); 0.029287s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra)
│ │ │ │  since R is a complete intersection.
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  o6 : DGAlgebra
│ │ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0919266s (cpu); 0.08901s
│ │ │ │ +Finding easy relations           :  -- used 0.196636s (cpu); 0.122994s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │ @@ -130,15 +130,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s
│ │ │ │ +Finding easy relations           :  -- used 0.0836547s (cpu); 0.0683607s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = HA
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  One can check that HA has Poincare duality since R is Gorenstein.
│ │ │ │  If your DGAlgebra has generators in even degrees, then one must specify the
│ │ │ │ @@ -166,15 +166,15 @@
│ │ │ │  o20 = {Ring => S                          }
│ │ │ │         Underlying algebra => S[T   ..T   ]
│ │ │ │                                  1,1   1,4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s
│ │ │ │ +Finding easy relations           :  -- used 0.0421419s (cpu); 0.0298108s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │  o6 : R[T   ..T   ]
│ │ │          1,1   1,3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.19201s (cpu); 0.0407282s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ + -- used 0.147735s (cpu); 0.125634s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │        1      4      6      4      1
│ │ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │ │  
│ │ │ │       0      1      2      3      4
│ │ │ │  
│ │ │ │  o5 : Complex
│ │ │ │  i6 : HKR = HH(KR)
│ │ │ │ - -- used 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ │ + -- used 0.147735s (cpu); 0.125634s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o6 = HKR
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  The following is the graded canonical module of R:
│ │ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product.html
│ │ │ @@ -197,15 +197,15 @@
│ │ │          
    
│ │ │            
    Given cycles z1,z2,z3 such that z1*z2 and z2*z3 are boundaries, the Massey triple product of the homology classes represented by z1,z2 and z3 is the homology class of lift12*z3 + z1*lift23.  To see this, we compute and check:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s
│ │ │ │ +Finding easy relations           :  -- used 0.302961s (cpu); 0.267678s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = H
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_tor__Algebra_lp__Ring_cm__Ring_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  i2 : M = coker matrix {{a^3*b^3*c^3*d^3}};
│ │ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s
│ │ │ │ +Finding easy relations           :  -- used 0.742914s (cpu); 0.606823s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DecomposableSparseSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  c29sdmVEZWNvbXBvc2FibGVTeXN0ZW0oTGlzdCk=
│ │ │  #:len=338
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzb2x2ZURlY29tcG9zYWJsZVN5c3RlbSxMaXN0KSwi
│ │ ├── ./usr/share/doc/Macaulay2/Depth/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  U2VlZA==
│ │ │  #:len=219
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTZWVkIiwiU2VlZCIsIkRlcHRoIn0sIFByaW1hcnlU
│ │ ├── ./usr/share/doc/Macaulay2/DeterminantalRepresentations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  Z2VuZXJhbGl6ZWRNaXhlZERpc2NyaW1pbmFudChMaXN0KQ==
│ │ │  #:len=367
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmFsaXplZE1peGVkRGlzY3JpbWluYW50LExp
│ │ ├── ./usr/share/doc/Macaulay2/DiffAlg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  bGluZWFyQ29tYg==
│ │ │  #:len=2390
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2VuZXJpYyBsaW5lYXIgY29tYmluYXRp
│ │ │  b24gb2YgZWxlbWVudHMiLCAibGluZW51bSIgPT4gODYzLCBJbnB1dHMgPT4ge1NQQU57VFR7Ikwi
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  dGFsbHlTdW1tYW5kcw==
│ │ │  #:len=2086
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFsbHkgYSBsaXN0IG9mIG1vZHVsZXMg
│ │ │  dXAgdG8gaXNvbW9ycGhpc20iLCAibGluZW51bSIgPT4gNjgwLCBJbnB1dHMgPT4ge1NQQU57VFR7
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/example-output/___Direct__Summands.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : F = frobeniusPushforward(1, OO_X);
│ │ │  
│ │ │  i4 : elapsedTime tallySummands summands F
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │ - -- 3.35396s elapsed
│ │ │ + -- 3.43524s elapsed
│ │ │  
│ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ @@ -32,15 +32,15 @@
│ │ │                      {1} | 0  0   -2z z   3y  x  |
│ │ │              1
│ │ │             R  => 1
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ - -- 11.5598s elapsed
│ │ │ + -- 11.6307s elapsed
│ │ │  
│ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │                      {1} | x        az (-a-3)y |
│ │ │                      {1} | (2a-2)z  3y x       |
│ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │                      {1} | 3az      -y       x       |
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/example-output/_is__Indecomposable.out
│ │ │ @@ -54,14 +54,14 @@
│ │ │  
│ │ │                              19
│ │ │  o8 : S-module, quotient of S
│ │ │  
│ │ │  i9 : assert(2 == rank FHM)
│ │ │  
│ │ │  i10 : assert elapsedTime isIndecomposable FHM
│ │ │ - -- 1.00223s elapsed
│ │ │ + -- 1.14854s elapsed
│ │ │  
│ │ │  i11 : assert({FHM} == summands FHM)
│ │ │  
│ │ │  i12 : assert FHM.cache.isIndecomposable
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/html/_is__Indecomposable.html
│ │ │ @@ -170,15 +170,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -124,15 +124,15 @@
│ │ │ │  0     0      0    0      0      0      0      -x_4   -x_3    x_3  x_2    x_1
│ │ │ │  -x_0   |
│ │ │ │  
│ │ │ │                              19
│ │ │ │  o8 : S-module, quotient of S
│ │ │ │  i9 : assert(2 == rank FHM)
│ │ │ │  i10 : assert elapsedTime isIndecomposable FHM
│ │ │ │ - -- 1.00223s elapsed
│ │ │ │ + -- 1.14854s elapsed
│ │ │ │  i11 : assert({FHM} == summands FHM)
│ │ │ │  i12 : assert FHM.cache.isIndecomposable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_r_e_c_t_S_u_m_m_a_n_d_s -- compute the direct summands of a module or coherent
│ │ │ │        sheaf
│ │ │ │      * _f_i_n_d_I_d_e_m_p_o_t_e_n_t_s -- construct idempotent endomorphisms
│ │ │ │  ********** WWaayyss ttoo uussee iissIInnddeeccoommppoossaabbllee:: **********
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/html/index.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,24 +25,24 @@
│ │ │ │  _T_i_m_e_L_i_m_i_t). The method will return null if it cannot find a hypergraph within
│ │ │ │  the branch and time limits.
│ │ │ │  i1 : R = QQ[x_1..x_5];
│ │ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ │ +                              1   2   3     3   4     1   2   4   5
│ │ │ │                  "ring" => R
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o3 : HyperGraph
│ │ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ │ +                              1   4   5     1   3     2   3   4   5
│ │ │ │                  "ring" => R
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o4 : HyperGraph
│ │ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch
│ │ │ │  limit reached
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_spanning__Tree.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns a breadth first spanning tree of a graph.
│ │ │ │  i1 : R = QQ[x_1..x_6];
│ │ │ │  i2 : C = cycle R; -- a 6-cycle
│ │ │ │  i3 : spanningTree C
│ │ │ │  
│ │ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │ │                             1   2   3   4   5   6
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : T = graph {x_1*x_2,x_2*x_3, x_1*x_4,x_1*x_5,x_5*x_6}; -- a tree (no
│ │ │ │  cycles)
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : G = graph {x_1*x_2,x_2*x_3,x_3*x_1,x_4*x_5,x_5*x_6,x_6*x_4}; -- two three
│ │ │ │  cycles
│ │ │ │  i7 : spanningTree G
│ │ │ │  
│ │ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ │ +                         1   3     2   3     4   6     5   6
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │ │                             1   2   3   4   5   6
│ │ │ │  
│ │ │ │  o7 : Graph
│ │ │ │  ********** WWaayyss ttoo uussee ssppaannnniinnggTTrreeee:: **********
│ │ │ │      * spanningTree(Graph)
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  emVyb0RpbVNvbHZlKElkZWFsKQ==
│ │ │  #:len=254
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh6ZXJvRGltU29sdmUsSWRlYWwpLCJ6ZXJvRGltU29s
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/example-output/___Eigen__Solver.out
│ │ │ @@ -15,14 +15,14 @@
│ │ │       a*b*e*f + a*d*e*f + c*d*e*f, a*b*c*d*e + a*b*c*d*f + a*b*c*e*f +
│ │ │       ------------------------------------------------------------------------
│ │ │       a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1)
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │  
│ │ │  i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .217098s elapsed
│ │ │ + -- .259372s elapsed
│ │ │  
│ │ │  i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,22 +29,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ │ + -- used 0.00324358s (cpu); 0.00324157s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                        2    2             2           2
│ │ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ │ + -- used 0.00182395s (cpu); 0.00182622s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html
│ │ │ @@ -102,26 +102,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,22 +30,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ │ + -- used 0.0033255s (cpu); 0.00332237s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                        2    2             2           2
│ │ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ │ + -- used 0.0018705s (cpu); 0.00187313s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.596064s (cpu); 0.508805s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.865078s (cpu); 0.748554s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T   T   T   T
│ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │  
│ │ │  o10 : R[T   ..T   ]
│ │ │           1,1   1,5
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │  Note that the first return value of _g_e_t_B_o_u_n_d_a_r_y_P_r_e_i_m_a_g_e indicates that the
│ │ │ │  inputs are indeed boundaries, and the second value is the lift of the boundary
│ │ │ │  along the differential.
│ │ │ │  Given cycles z1,z2,z3 such that z1*z2 and z2*z3 are boundaries, the Massey
│ │ │ │  triple product of the homology classes represented by z1,z2 and z3 is the
│ │ │ │  homology class of lift12*z3 + z1*lift23. To see this, we compute and check:
│ │ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ │ -Finding easy relations           :  -- used 0.596064s (cpu); 0.508805s
│ │ │ │ +Finding easy relations           :  -- used 0.865078s (cpu); 0.748554s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2
│ │ │ │  o10 = x x y z T   T   T   T
│ │ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │ │  
│ │ │ │  o10 : R[T   ..T   ]
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html
│ │ │ @@ -124,15 +124,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
    
│ │ │                
    i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.302961s (cpu); 0.267678s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.742914s (cpu); 0.606823s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    i9 : assert(2 == rank FHM)
    
│ │ │              
    
│ │ │                
    i10 : assert elapsedTime isIndecomposable FHM
│ │ │ - -- 1.00223s elapsed
    
│ │ │ + -- 1.14854s elapsed
│ │ │              
    
│ │ │                
    i11 : assert({FHM} == summands FHM)
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime tallySummands summands F
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │ - -- 3.35396s elapsed
│ │ │ + -- 3.43524s elapsed
│ │ │  
│ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ - -- 11.5598s elapsed
│ │ │ + -- 11.6307s elapsed
│ │ │  
│ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │                      {1} | x        az (-a-3)y |
│ │ │                      {1} | (2a-2)z  3y x       |
│ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │                      {1} | 3az      -y       x       |
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │  i1 : R = ZZ/7[x,y,z]/(x^3+y^3+z^3);
│ │ │ │  i2 : X = Proj R;
│ │ │ │  i3 : F = frobeniusPushforward(1, OO_X);
│ │ │ │  i4 : elapsedTime tallySummands summands F
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │ - -- 3.35396s elapsed
│ │ │ │ + -- 3.43524s elapsed
│ │ │ │  
│ │ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │                      {1} | 3z -3y x   0   -2z -y |
│ │ │ │                      {1} | 0  0   -2z z   3y  x  |
│ │ │ │              1
│ │ │ │             R  => 1
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ │ - -- 11.5598s elapsed
│ │ │ │ + -- 11.6307s elapsed
│ │ │ │  
│ │ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │ │                      {1} | x        az (-a-3)y |
│ │ │ │                      {1} | (2a-2)z  3y x       |
│ │ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │ │                      {1} | 3az      -y       x       |
│ │ ├── ./usr/share/doc/Macaulay2/Dmodules/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  RG1vZHVsZXM=
│ │ │  #:len=758
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRC1tb2R1bGVzIHBhY2thZ2UgY29sbGVj
│ │ │  dGlvbiIsICJsaW5lbnVtIiA9PiAxNDIsICJmaWxlbmFtZSIgPT4gIkRtb2R1bGVzLm0yIiwgRGVz
│ │ ├── ./usr/share/doc/Macaulay2/EagonResolution/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  VHJhbnNwb3Nl
│ │ │  #:len=1380
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVHJhbnNwb3NlID0+IGZhbHNlLCBkZWZh
│ │ │  dWx0IG9wdGlvbiBmb3IgcGljdHVyZSIsICJsaW5lbnVtIiA9PiAxMzUwLCBJbnB1dHMgPT4ge1NQ
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Y29tcGxlbWVudEdyYXBo
│ │ │  #:len=2011
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY29tcGxlbWVudCBv
│ │ │  ZiBhIGdyYXBoIG9yIGh5cGVyZ3JhcGgiLCAibGluZW51bSIgPT4gMjA3OSwgSW5wdXRzID0+IHtT
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : R = QQ[x_1..x_5];
│ │ │  
│ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ +                              1   2   3     3   4     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
│ │ │  
│ │ │  i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ +                              1   4   5     1   3     2   3   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
│ │ │  
│ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch limit reached
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_spanning__Tree.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : R = QQ[x_1..x_6];
│ │ │  
│ │ │  i2 : C = cycle R; -- a 6-cycle
│ │ │  
│ │ │  i3 : spanningTree C
│ │ │  
│ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : T = graph {x_1*x_2,x_2*x_3, x_1*x_4,x_1*x_5,x_5*x_6}; -- a tree (no cycles)
│ │ │ @@ -21,15 +21,15 @@
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : G = graph {x_1*x_2,x_2*x_3,x_3*x_1,x_4*x_5,x_5*x_6,x_6*x_4}; -- two three cycles
│ │ │  
│ │ │  i7 : spanningTree G
│ │ │  
│ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ +                         1   3     2   3     4   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o7 : Graph
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -93,28 +93,28 @@
│ │ │              
    
│ │ │                
    i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ +                              1   2   3     3   4     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o3 : HyperGraph
    
│ │ │              
    
│ │ │                
    i4 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o4 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ +                              1   4   5     1   3     2   3   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : spanningTree C
│ │ │  
│ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
    
│ │ │                
    i7 : spanningTree G
│ │ │  
│ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ +                         1   3     2   3     4   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o7 : Graph
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .217098s elapsed
    
│ │ │ + -- .259372s elapsed
│ │ │              
    
│ │ │                
    i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a*b*e*f + a*d*e*f + c*d*e*f, a*b*c*d*e + a*b*c*d*f + a*b*c*e*f +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1)
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[a..f]
│ │ │ │  i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ │ - -- .217098s elapsed
│ │ │ │ + -- .259372s elapsed
│ │ │ │  i4 : #sols -- 156 solutions
│ │ │ │  
│ │ │ │  o4 = 156
│ │ │ │  The authors would like to acknowledge the June 2020 Macaulay2 workshop held
│ │ │ │  virtually at Warwick, where this package was first developed.
│ │ │ │  RReeffeerreenncceess:
│ │ │ │      * [1] Sturmfels, Bernd. Solving systems of polynomial equations. No. 97.
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZWxpbWluYXRlKElkZWFsLExpc3Qp
│ │ │  #:len=200
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgxLCAidW5kb2N1bWVudGVkIiA9PiB0
│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhlbGltaW5h
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ + -- used 0.00324358s (cpu); 0.00324157s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ + -- used 0.00182395s (cpu); 0.00182622s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ + -- used 0.0033255s (cpu); 0.00332237s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ + -- used 0.0018705s (cpu); 0.00187313s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │  
│ │ │        8    5
│ │ │  o3 = x  + x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ + -- used 1.79118s (cpu); 1.51139s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -73,15 +73,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ + -- used 0.0181234s (cpu); 0.0181234s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  
│ │ │        8    5
│ │ │  o4 = x  + x  + x*c + d
│ │ │  
│ │ │  o4 : R
│ │ │  
│ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ + -- used 1.99035s (cpu); 1.67582s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -75,15 +75,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ + -- used 0.0165426s (cpu); 0.0165405s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -108,26 +108,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ + -- used 0.00324358s (cpu); 0.00324157s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ + -- used 0.00182395s (cpu); 0.00182622s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ + -- used 0.0033255s (cpu); 0.00332237s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ + -- used 0.0018705s (cpu); 0.00187313s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ + -- used 1.79118s (cpu); 1.51139s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -169,15 +169,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ + -- used 0.0181234s (cpu); 0.0181234s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  i3 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o3 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ │ + -- used 1.79118s (cpu); 1.51139s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ │ + -- used 0.0181234s (cpu); 0.0181234s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : R
    
│ │ │              
    
│ │ │                
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ + -- used 1.99035s (cpu); 1.67582s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,15 +168,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ + -- used 0.0165426s (cpu); 0.0165405s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i4 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o4 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o4 : R
│ │ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ │ + -- used 1.99035s (cpu); 1.67582s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -85,15 +85,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ │ + -- used 0.0165426s (cpu); 0.0165405s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/EliminationMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  bWF4Q29sKE1hdHJpeCk=
│ │ │  #:len=257
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExNSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWF4Q29sLE1hdHJpeCksIm1heENvbChNYXRyaXgp
│ │ ├── ./usr/share/doc/Macaulay2/EliminationTemplates/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  Y29weVRlbXBsYXRl
│ │ │  #:len=2172
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29waWVzIEVsaW1pbmF0aW9uVGVtcGxh
│ │ │  dGUgb2JqZWN0IiwgImxpbmVudW0iID0+IDk4NywgSW5wdXRzID0+IHtTUEFOe1RUeyJFIn0sIiwg
│ │ ├── ./usr/share/doc/Macaulay2/EllipticCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  RWxsUG9pbnRXID09IEVsbFBvaW50Vw==
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzQ3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgPT0sRWxsUG9pbnRXLEVsbFBvaW50Vyks
│ │ ├── ./usr/share/doc/Macaulay2/EllipticIntegrals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RWxsaXB0aWNJbnRlZ3JhbHM=
│ │ │  #:len=584
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMg
│ │ │  cGFja2FnZSBwcm92aWRlcyBzb21lIGZ1bmN0aW9ucyBmb3IgY29tcHV0aW5nIGVsbGlwdGljIGlu
│ │ ├── ./usr/share/doc/Macaulay2/EngineTests/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  RW5naW5lVGVzdHM=
│ │ │  #:len=311
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSB0ZXN0IHN1aXRlIGZvciB0aGUgTWFj
│ │ │  YXVsYXkyIGVuZ2luZSIsICJsaW5lbnVtIiA9PiA0NzksICJmaWxlbmFtZSIgPT4gIkVuZ2luZVRl
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RW51bWVyYXRpb25DdXJ2ZXM=
│ │ │  #:len=747
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRW51bWVyYXRpb24gb2YgcmF0aW9uYWwg
│ │ │  Y3VydmVzIHZpYSB0b3J1cyBhY3Rpb25zIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7IiJ9
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1331975673177
│ │ │  
│ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ + -- used 0.0318941s (cpu); 0.0318926s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out
│ │ │ @@ -37,83 +37,83 @@
│ │ │  i6 : rationalCurve(2) - rationalCurve(1)/8
│ │ │  
│ │ │  o6 = 609250
│ │ │  
│ │ │  o6 : QQ
│ │ │  
│ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ + -- used 0.429037s (cpu); 0.355463s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ + -- used 0.157938s (cpu); 0.157946s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ + -- used 5.77906s (cpu); 4.99077s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ + -- used 0.257513s (cpu); 0.180642s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ + -- used 5.93841s (cpu); 5.17564s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ + -- used 1.80961s (cpu); 1.53876s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ + -- used 8.23861s (cpu); 6.5053s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ + -- used 1.85724s (cpu); 1.58465s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ + -- used 8.1847s (cpu); 6.48293s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ + -- used 7.99333s (cpu); 6.35304s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
    Computes the number of lines on a general hypersurface of degree 2n - 3 in \mathbb P^n.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the number of lines on a general hypersurface of degree
│ │ │ │              2n - 3 in \mathbb P^n
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Computes the number of lines on a general hypersurface of degree 2n - 3 in
│ │ │ │  \mathbb P^n.
│ │ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ │ - -- used 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ │ + -- used 0.0318941s (cpu); 0.0318926s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html
│ │ │ @@ -157,15 +157,15 @@
│ │ │            
    The numbers of conics on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ + -- used 0.0318941s (cpu); 0.0318926s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ + -- used 0.429037s (cpu); 0.355463s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ @@ -173,27 +173,27 @@
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -203,15 +203,15 @@
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ + -- used 0.157938s (cpu); 0.157946s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │              
    
│ │ │                
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ + -- used 5.77906s (cpu); 4.99077s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ + -- used 0.257513s (cpu); 0.180642s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │              
    
│ │ │ @@ -219,15 +219,15 @@
│ │ │            
    The numbers of rational curves of degree 3 on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ + -- used 5.93841s (cpu); 5.17564s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │ @@ -235,27 +235,27 @@
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ + -- used 1.80961s (cpu); 1.53876s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │              
    
│ │ │                
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ + -- used 8.23861s (cpu); 6.5053s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │              
    
│ │ │ @@ -263,15 +263,15 @@
│ │ │            
    The number of rational curves of degree 4 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ + -- used 1.85724s (cpu); 1.58465s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │              
    
│ │ │ @@ -279,25 +279,25 @@
│ │ │            
    The numbers of rational curves of degree 4 on general complete intersections of types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ + -- used 8.1847s (cpu); 6.48293s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │              
    
│ │ │                
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ + -- used 7.99333s (cpu); 6.35304s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,85 +59,85 @@
│ │ │ │  
│ │ │ │  o6 = 609250
│ │ │ │  
│ │ │ │  o6 : QQ
│ │ │ │  The numbers of conics on general complete intersection Calabi-Yau threefolds
│ │ │ │  can be computed as follows:
│ │ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ │ - -- used 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ │ + -- used 0.429037s (cpu); 0.355463s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ │ + -- used 0.157938s (cpu); 0.157946s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ │ + -- used 5.77906s (cpu); 4.99077s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        8564575000  422690816           4834592  11239424
│ │ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │ │            27          27                 3        27
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  The number of rational curves of degree 3 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ │ - -- used 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ │ + -- used 0.257513s (cpu); 0.180642s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 317206375
│ │ │ │  
│ │ │ │  o10 : QQ
│ │ │ │  The numbers of rational curves of degree 3 on general complete intersection
│ │ │ │  Calabi-Yau threefolds can be computed as follows:
│ │ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ │ - -- used 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ │ + -- used 5.93841s (cpu); 5.17564s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ │ + -- used 1.80961s (cpu); 1.53876s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ │ + -- used 8.23861s (cpu); 6.5053s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = 3883914084
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  The number of rational curves of degree 4 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ │ - -- used 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ │ + -- used 1.85724s (cpu); 1.58465s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 242467530000
│ │ │ │  
│ │ │ │  o14 : QQ
│ │ │ │  The numbers of rational curves of degree 4 on general complete intersections of
│ │ │ │  types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
│ │ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ │ + -- used 8.1847s (cpu); 6.48293s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ │ + -- used 7.99333s (cpu); 6.35304s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 1139448384
│ │ │ │  
│ │ │ │  o16 : QQ
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_e_C_o_v_e_r -- Multiple coverings of rational curves on Calabi-Yau
│ │ │ │        threefolds
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  YnVpbGRFTW9ub21pYWxNYXAoUmluZyxSaW5nLExpc3Qp
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIzNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYnVpbGRFTW9ub21pYWxNYXAsUmluZyxSaW5nLExp
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out
│ │ │ @@ -10,34 +10,34 @@
│ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │                    1   1 0   1 0   0
│ │ │  
│ │ │  o3 : RingMap R <-- S
│ │ │  
│ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .0022445 seconds
│ │ │ -     -- used .0006884 seconds
│ │ │ +     -- used .00233466 seconds
│ │ │ +     -- used .00080397 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00370187 seconds
│ │ │ -     -- used .00494948 seconds
│ │ │ +     -- used .00446286 seconds
│ │ │ +     -- used .00576887 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00769783 seconds
│ │ │ -     -- used .118616 seconds
│ │ │ +     -- used .0144329 seconds
│ │ │ +     -- used .133835 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0180006 seconds
│ │ │ -     -- used .319378 seconds
│ │ │ +     -- used .0201126 seconds
│ │ │ +     -- used .365356 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .039338 seconds
│ │ │ -     -- used 1.12239 seconds
│ │ │ +     -- used .0512176 seconds
│ │ │ +     -- used 1.19082 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html
│ │ │ @@ -106,34 +106,34 @@
│ │ │  o3 : RingMap R <-- S
    
│ │ │              
    
│ │ │                
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .0022445 seconds
│ │ │ -     -- used .0006884 seconds
│ │ │ +     -- used .00233466 seconds
│ │ │ +     -- used .00080397 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00370187 seconds
│ │ │ -     -- used .00494948 seconds
│ │ │ +     -- used .00446286 seconds
│ │ │ +     -- used .00576887 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00769783 seconds
│ │ │ -     -- used .118616 seconds
│ │ │ +     -- used .0144329 seconds
│ │ │ +     -- used .133835 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0180006 seconds
│ │ │ -     -- used .319378 seconds
│ │ │ +     -- used .0201126 seconds
│ │ │ +     -- used .365356 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .039338 seconds
│ │ │ -     -- used 1.12239 seconds
│ │ │ +     -- used .0512176 seconds
│ │ │ +     -- used 1.19082 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,34 +33,34 @@
│ │ │ │                    2               2
│ │ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │ │                    1   1 0   1 0   0
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- S
│ │ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │ │  3
│ │ │ │ -     -- used .0022445 seconds
│ │ │ │ -     -- used .0006884 seconds
│ │ │ │ +     -- used .00233466 seconds
│ │ │ │ +     -- used .00080397 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00370187 seconds
│ │ │ │ -     -- used .00494948 seconds
│ │ │ │ +     -- used .00446286 seconds
│ │ │ │ +     -- used .00576887 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00769783 seconds
│ │ │ │ -     -- used .118616 seconds
│ │ │ │ +     -- used .0144329 seconds
│ │ │ │ +     -- used .133835 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0180006 seconds
│ │ │ │ -     -- used .319378 seconds
│ │ │ │ +     -- used .0201126 seconds
│ │ │ │ +     -- used .365356 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .039338 seconds
│ │ │ │ -     -- used 1.12239 seconds
│ │ │ │ +     -- used .0512176 seconds
│ │ │ │ +     -- used 1.19082 seconds
│ │ │ │  (49, 217)
│ │ │ │  
│ │ │ │                                     2
│ │ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EuclideanDistanceDegree/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=40
│ │ │  bnVtZXJpY1dlaWdodEVERGVncmVlKC4uLixTdWJtb2RlbD0+Li4uKQ==
│ │ │  #:len=341
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDkxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tudW1lcmljV2VpZ2h0RUREZWdyZWUsU3VibW9kZWxd
│ │ ├── ./usr/share/doc/Macaulay2/ExampleSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  aGVhcnQ=
│ │ │  #:len=1516
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gOC1kaW1lbnNpb25hbCBlY29ub21p
│ │ │  Y3MgcHJvYmxlbSIsICJsaW5lbnVtIiA9PiA0OCwgSW5wdXRzID0+IHtTUEFOe1RUeyJrayJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorExtensions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  YmFzZXM=
│ │ │  #:len=602
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2l2ZXMgYSBsaXN0IG9mIHRoZSBiYXNl
│ │ │  cyBvZiB0aGUgZGlmZmVyZW50IGdyYWRlZCBwaWVjZXMgb2YgdGhlIEV4dGVyaW9yRXh0ZW5zaW9u
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  c29sdmVNYWNhdWxheUV4cGFuc2lvbihMaXN0KQ==
│ │ │  #:len=301
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDMwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzb2x2ZU1hY2F1bGF5RXhwYW5zaW9uLExpc3QpLCJz
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorModules/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  YWxtb3N0U3RhYmxlTW9kdWxlKE1vZHVsZSk=
│ │ │  #:len=292
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbG1vc3RTdGFibGVNb2R1bGUsTW9kdWxlKSwiYWxt
│ │ ├── ./usr/share/doc/Macaulay2/FGLM/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  ZmdsbShJZGVhbCxSaW5nKQ==
│ │ │  #:len=211
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzk2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmZ2xtLElkZWFsLFJpbmcpLCJmZ2xtKElkZWFsLFJp
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  cmVjdXJzaXZlTWlub3JzKFpaLE1hdHJpeCk=
│ │ │  #:len=272
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVjdXJzaXZlTWlub3JzLFpaLE1hdHJpeCksInJl
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out
│ │ │ @@ -414,50 +414,50 @@
│ │ │                 3 2 4     3 6
│ │ │  o27 = ideal(12x x x  - 4x x )
│ │ │                 3 7 9     3 9
│ │ │  
│ │ │  o27 : Ideal of S
│ │ │  
│ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ + -- used 0.0599983s (cpu); 0.0598348s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ + -- used 0.331385s (cpu); 0.243281s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ + -- used 0.500315s (cpu); 0.417961s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ + -- used 0.229284s (cpu); 0.232737s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ + -- used 0.315939s (cpu); 0.235888s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ + -- used 0.391271s (cpu); 0.302793s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ + -- used 0.154093s (cpu); 0.155419s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ + -- used 15.3772s (cpu); 12.169s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
│ │ │  
│ │ │  i36 : peek StrategyDefault
│ │ │  
│ │ │  o36 = OptionTable{GRevLexLargest => 0      }
│ │ │                    GRevLexSmallest => 16
│ │ │ @@ -488,15 +488,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ + -- used 0.517179s (cpu); 0.429275s (thread); 0s (gc)
│ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │  
│ │ │  o37 : Ideal of S
│ │ │  
│ │ │  i38 : peek StrategyDefaultNonRandom
│ │ │  
│ │ │  o38 = OptionTable{GRevLexLargest => 0      }
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i41 : ptsStratGeometric = new OptionTable from (options chooseGoodMinors)#PointOptions;
│ │ │  
│ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │  
│ │ │  o42 = true
│ │ │  
│ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ + -- used 0.925658s (cpu); 0.767166s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
│ │ │  
│ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │                    DimensionFunction => dim
│ │ │ @@ -559,49 +559,49 @@
│ │ │  o44 : OptionTable
│ │ │  
│ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │  
│ │ │  o45 = false
│ │ │  
│ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ + -- used 0.760476s (cpu); 0.589151s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ + -- used 2.70501s (cpu); 2.32634s (thread); 0s (gc)
│ │ │  
│ │ │  o47 = true
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ + -- used 1.55627s (cpu); 1.29166s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
│ │ │ + -- used 3.83936s (cpu); 3.56341s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
│ │ │ + -- used 3.16256s (cpu); 2.45827s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ + -- used 0.338401s (cpu); 0.341445s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
│ │ │ + -- used 3.86797s (cpu); 2.9763s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
│ │ │ + -- used 4.43077s (cpu); 3.58577s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ + -- used 69.0795s (cpu); 58.5017s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ + -- used 3.93541s (cpu); 3.00692s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
│ │ │  
│ │ │  i56 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  o2 : Ideal of S
│ │ │  
│ │ │  i3 : dim (S/J)
│ │ │  
│ │ │  o3 = 4
│ │ │  
│ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ + -- used 4.38621s (cpu); 2.60071s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ + -- used 17.3336s (cpu); 10.6186s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time regularInCodimension(1, S/J, Verbose=>true)
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -127,15 +127,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ + -- used 4.75603s (cpu); 2.82964s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 452.908 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -175,15 +175,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ + -- used 0.350214s (cpu); 0.197276s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -197,15 +197,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ + -- used 0.335923s (cpu); 0.189692s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -219,15 +219,15 @@
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ + -- used 0.54881s (cpu); 0.31502s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 6, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -262,15 +262,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ + -- used 1.6985s (cpu); 1.08017s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 5, and computed = 5
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -313,15 +313,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ + -- used 1.21185s (cpu); 0.751148s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 15, and computed = 14
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5509279875405941999
│ │ │  
│ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │  
│ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 4.1651s elapsed
│ │ │ + -- 3.7471s elapsed
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : peek StrategyDefault
│ │ │  
│ │ │  o3 = OptionTable{GRevLexLargest => 0      }
│ │ │                   GRevLexSmallest => 16
│ │ │ @@ -16,12 +16,12 @@
│ │ │                   LexSmallest => 16
│ │ │                   LexSmallestTerm => 16
│ │ │                   Points => 0
│ │ │                   Random => 16
│ │ │                   RandomNonzero => 16
│ │ │  
│ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.67457s elapsed
│ │ │ + -- 1.54762s elapsed
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out
│ │ │ @@ -16,29 +16,29 @@
│ │ │  i5 : r = rank myDiff;
│ │ │  
│ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.000947066s (cpu); 0.00263955s (thread); 0s (gc)
│ │ │ + -- used 0.00400117s (cpu); 0.00417061s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : I = ideal(x_2^8*x_10^3-3*x_1*x_2^7*x_10^2*x_11+3*x_1^2*x_2^6*x_10*x_11^2-x_1^3*x_2^5*x_11^3,x_5^5*x_6^3*x_11^3-3*x_5^6*x_6^2*x_11^2*x_12+3*x_5^7*x_6*x_11*x_12^2-x_5^8*x_12^3,x_1^5*x_2^3*x_4^3-3*x_1^6*x_2^2*x_4^2*x_5+3*x_1^7*x_2*x_4*x_5^2-x_1^8*x_5^3,x_6^8*x_11^3-3*x_5*x_6^7*x_11^2*x_12+3*x_5^2*x_6^6*x_11*x_12^2-x_5^3*x_6^5*x_12^3,x_8^3*x_10^8-3*x_7*x_8^2*x_10^7*x_11+3*x_7^2*x_8*x_10^6*x_11^2-x_7^3*x_10^5*x_11^3,x_2^8*x_4^3-3*x_1*x_2^7*x_4^2*x_5+3*x_1^2*x_2^6*x_4*x_5^2-x_1^3*x_2^5*x_5^3,-x_6^3*x_11^8+3*x_5*x_6^2*x_11^7*x_12-3*x_5^2*x_6*x_11^6*x_12^2+x_5^3*x_11^5*x_12^3,-x_6^3*x_7^3*x_9^5+3*x_4*x_6^2*x_7^2*x_9^6-3*x_4^2*x_6*x_7*x_9^7+x_4^3*x_9^8,x_8^8*x_10^3-3*x_7*x_8^7*x_10^2*x_11+3*x_7^2*x_8^6*x_10*x_11^2-x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
│ │ │  
│ │ │                 ZZ
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │  
│ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00400074s (cpu); 0.00251447s (thread); 0s (gc)
│ │ │ + -- used 0.00361314s (cpu); 0.0034771s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc)
│ │ │ + -- used 0.00189832s (cpu); 0.00382314s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out
│ │ │ @@ -7,17 +7,17 @@
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : pdim(module I)
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.296971s (cpu); 0.159933s (thread); 0s (gc)
│ │ │ + -- used 0.430958s (cpu); 0.249096s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc)
│ │ │ + -- used 0.0142396s (cpu); 0.0166226s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │  
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ + -- used 0.638836s (cpu); 0.568079s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ + -- used 1.60164s (cpu); 1.44213s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : I1 == I2
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out
│ │ │ @@ -19,44 +19,44 @@
│ │ │  i6 : S = T/I;
│ │ │  
│ │ │  i7 : dim S
│ │ │  
│ │ │  o7 = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc)
│ │ │ + -- used 1.32249s (cpu); 0.841829s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc)
│ │ │ + -- used 11.1853s (cpu); 7.1376s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : R = QQ[c, f, g, h]/ideal(g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
│ │ │  
│ │ │  i11 : dim(R)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0279591s (cpu); 0.0261627s (thread); 0s (gc)
│ │ │ + -- used 0.0270262s (cpu); 0.0243831s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc)
│ │ │ + -- used 0.684727s (cpu); 0.380789s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc)
│ │ │ + -- used 0.425532s (cpu); 0.268548s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc)
│ │ │ + -- used 0.473376s (cpu); 0.254374s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -381,15 +381,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc)
│ │ │ + -- used 11.4598s (cpu); 7.49946s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -462,15 +462,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc)
│ │ │ + -- used 2.15602s (cpu); 1.42355s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -486,68 +486,68 @@
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 30, and computed = 25
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 30, and computed = 25.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i18 : time regularInCodimension(2, S, VerifyNonRegular=>true)
│ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc)
│ │ │ + -- used 2.08497s (cpu); 1.19351s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = false
│ │ │  
│ │ │  i19 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i21 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc)
│ │ │ + -- used 0.733147s (cpu); 0.425419s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc)
│ │ │ + -- used 0.853632s (cpu); 0.416929s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc)
│ │ │ + -- used 0.498479s (cpu); 0.327804s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc)
│ │ │ + -- used 0.593409s (cpu); 0.341934s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = true
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i27 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc)
│ │ │ + -- used 4.08395s (cpu); 2.37479s (thread); 0s (gc)
│ │ │  
│ │ │  i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc)
│ │ │ + -- used 4.07422s (cpu); 2.39475s (thread); 0s (gc)
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc)
│ │ │ + -- used 0.461171s (cpu); 0.300656s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc)
│ │ │ + -- used 0.684033s (cpu); 0.42462s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc)
│ │ │ + -- used 2.37898s (cpu); 1.62521s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc)
│ │ │ + -- used 2.01944s (cpu); 1.34737s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = true
│ │ │  
│ │ │  i34 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -577,71 +577,71 @@
│ │ │          
    
│ │ │            
    Here the $1$ passed to the function says how many minors to compute.  For instance, let's compute 8 minors for each of these strategies and see if that was enough to verify that the ring is regular in codimension 1.  In other words, if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/J$ has dimension 3).
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ + -- used 0.0599983s (cpu); 0.0598348s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ + -- used 0.331385s (cpu); 0.243281s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ + -- used 0.500315s (cpu); 0.417961s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ + -- used 0.229284s (cpu); 0.232737s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ + -- used 0.315939s (cpu); 0.235888s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ + -- used 0.391271s (cpu); 0.302793s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ + -- used 0.154093s (cpu); 0.155419s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ + -- used 15.3772s (cpu); 12.169s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Indeed, in this example, even computing determinants of 1,000 random submatrices is not typically enough to verify that $V(J)$ is regular in codimension 1.  On the other hand, Points is almost always quite effective at finding valuable submatrices, but can be quite slow.  In this particular example, we can see that LexSmallestTerm also performs very well (and does it quickly). Since different strategies work better or worse on different examples, the default strategy actually mixes and matches various strategies.  The default strategy, which we now elucidate,
    
│ │ │ @@ -688,15 +688,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ + -- used 0.517179s (cpu); 0.429275s (thread); 0s (gc)
│ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │  
│ │ │  o37 : Ideal of S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -779,15 +779,15 @@
│ │ │  
│ │ │  o42 = true
    
│ │ │              
    
│ │ │                
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ + -- used 0.925658s (cpu); 0.767166s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    
│ │ │              
    
│ │ │                
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │ @@ -811,15 +811,15 @@
│ │ │  
│ │ │  o45 = false
    
│ │ │              
    
│ │ │                
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ + -- used 0.760476s (cpu); 0.589151s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Other options may also be passed to the RandomPoints package via the PointOptions option.
    
│ │ │ @@ -827,71 +827,71 @@
│ │ │          
    
│ │ │            
    regularInCodimension:  It is reasonable to think that you should find a few minors (with one strategy or another), and see if perhaps the minors you have computed so far are enough to verify our ring is regular in codimension 1.  This is exactly what regularInCodimension does.  One can control at a fine level how frequently new minors are computed, and how frequently the dimension of what we have computed so far is checked, by the option codimCheckFunction.  For more on that, see RegularInCodimensionTutorial and regularInCodimension.  Let us finish running regularInCodimension on our example with several different strategies.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ + -- used 2.70501s (cpu); 2.32634s (thread); 0s (gc)
│ │ │  
│ │ │  o47 = true
    
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ + -- used 1.55627s (cpu); 1.29166s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
    
│ │ │ + -- used 3.83936s (cpu); 3.56341s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
    
│ │ │ + -- used 3.16256s (cpu); 2.45827s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ + -- used 0.338401s (cpu); 0.341445s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
    
│ │ │ + -- used 3.86797s (cpu); 2.9763s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
    
│ │ │ + -- used 4.43077s (cpu); 3.58577s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ + -- used 69.0795s (cpu); 58.5017s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ + -- used 3.93541s (cpu); 3.00692s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If regularInCodimension outputs nothing, then it couldn't verify that the ring was regular in that codimension.  We set MaxMinors => 100 to keep it from running too long with an ineffective strategy.  Again, even though GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular example, in others they perform better than other strategies.  Note similar considerations also apply to projDim.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -438,44 +438,44 @@
│ │ │ │  o27 : Ideal of S
│ │ │ │  Here the $1$ passed to the function says how many minors to compute. For
│ │ │ │  instance, let's compute 8 minors for each of these strategies and see if that
│ │ │ │  was enough to verify that the ring is regular in codimension 1. In other words,
│ │ │ │  if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/
│ │ │ │  J$ has dimension 3).
│ │ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ │ + -- used 0.0599983s (cpu); 0.0598348s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ │ + -- used 0.331385s (cpu); 0.243281s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ │ + -- used 0.500315s (cpu); 0.417961s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ │ + -- used 0.229284s (cpu); 0.232737s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ │ + -- used 0.315939s (cpu); 0.235888s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ │ + -- used 0.391271s (cpu); 0.302793s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ │ + -- used 0.154093s (cpu); 0.155419s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ │ + -- used 15.3772s (cpu); 12.169s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o35 = 1
│ │ │ │  Indeed, in this example, even computing determinants of 1,000 random
│ │ │ │  submatrices is not typically enough to verify that $V(J)$ is regular in
│ │ │ │  codimension 1. On the other hand, Points is almost always quite effective at
│ │ │ │  finding valuable submatrices, but can be quite slow. In this particular
│ │ │ │  example, we can see that LexSmallestTerm also performs very well (and does it
│ │ │ │ @@ -518,15 +518,15 @@
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ │ + -- used 0.517179s (cpu); 0.429275s (thread); 0s (gc)
│ │ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │ │  
│ │ │ │  o37 : Ideal of S
│ │ │ │  we can see different minors being chosen via different strategies.
│ │ │ │  Note, if one asks chooseGoodMinors for more than one minor, then any time a
│ │ │ │  Points strategy is selected, the point is found on $J$ plus the ideal of all
│ │ │ │  minors computed thus far.
│ │ │ │ @@ -587,15 +587,15 @@
│ │ │ │  i41 : ptsStratGeometric = new OptionTable from (options
│ │ │ │  chooseGoodMinors)#PointOptions;
│ │ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │ │  
│ │ │ │  o42 = true
│ │ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratGeometric))
│ │ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ │ + -- used 0.925658s (cpu); 0.767166s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o43 = 2
│ │ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that
│ │ │ │  value
│ │ │ │  
│ │ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │ │                    DimensionFunction => dim
│ │ │ │ @@ -609,60 +609,60 @@
│ │ │ │  
│ │ │ │  o44 : OptionTable
│ │ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │ │  
│ │ │ │  o45 = false
│ │ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratRational))
│ │ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ │ + -- used 0.760476s (cpu); 0.589151s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = 2
│ │ │ │  Other options may also be passed to the _R_a_n_d_o_m_P_o_i_n_t_s package via the
│ │ │ │  _P_o_i_n_t_O_p_t_i_o_n_s option.
│ │ │ │  rreegguullaarrIInnCCooddiimmeennssiioonn:: It is reasonable to think that you should find a few
│ │ │ │  minors (with one strategy or another), and see if perhaps the minors you have
│ │ │ │  computed so far are enough to verify our ring is regular in codimension 1. This
│ │ │ │  is exactly what regularInCodimension does. One can control at a fine level how
│ │ │ │  frequently new minors are computed, and how frequently the dimension of what we
│ │ │ │  have computed so far is checked, by the option codimCheckFunction. For more on
│ │ │ │  that, see _R_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n_T_u_t_o_r_i_a_l and _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n. Let us finish
│ │ │ │  running regularInCodimension on our example with several different strategies.
│ │ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefault)
│ │ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ │ + -- used 2.70501s (cpu); 2.32634s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o47 = true
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ │ + -- used 1.55627s (cpu); 1.29166s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
│ │ │ │ + -- used 3.83936s (cpu); 3.56341s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
│ │ │ │ + -- used 3.16256s (cpu); 2.45827s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ │ + -- used 0.338401s (cpu); 0.341445s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
│ │ │ │ + -- used 3.86797s (cpu); 2.9763s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
│ │ │ │ + -- used 4.43077s (cpu); 3.58577s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ │ + -- used 69.0795s (cpu); 58.5017s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ │ + -- used 3.93541s (cpu); 3.00692s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o55 = true
│ │ │ │  If regularInCodimension outputs nothing, then it couldn't verify that the ring
│ │ │ │  was regular in that codimension. We set MaxMinors => 100 to keep it from
│ │ │ │  running too long with an ineffective strategy. Again, even though
│ │ │ │  GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular
│ │ │ │  example, in others they perform better than other strategies. Note similar
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html
│ │ │ @@ -86,23 +86,23 @@
│ │ │          
    
│ │ │            
    It is the cone over $P^2 \times E$ where $E$ is an elliptic curve.  We have embedded it with a Segre embedding inside $P^8$.  In particular, this example is even regular in codimension 3.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ + -- used 4.38621s (cpu); 2.60071s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ + -- used 17.3336s (cpu); 10.6186s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the ideal made up of a small number of minors of the Jacobian matrix. In this example, instead of computing all relevant 1465128 minors to compute the singular locus, and then trying to compute the dimension of the ideal they generate, we instead compute a few of them.  regularInCodimension returns true if it verified that the ring is regular in codim 1 or 2 (respectively) and null if not.  Because of the randomness that exists in terms of selecting minors, the execution time can actually vary quite a bit.   Let's take a look at what is occurring by using the Verbose option.  We go through the output and explain what each line is telling us.
    
│ │ │ @@ -217,15 +217,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ + -- used 4.75603s (cpu); 2.82964s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 452.908 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -273,15 +273,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ + -- used 0.350214s (cpu); 0.197276s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -306,15 +306,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ + -- used 0.335923s (cpu); 0.189692s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -339,15 +339,15 @@
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ + -- used 0.54881s (cpu); 0.31502s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 6, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -393,15 +393,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ + -- used 1.6985s (cpu); 1.08017s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 5, and computed = 5
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -452,15 +452,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ + -- used 1.21185s (cpu); 0.751148s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 15, and computed = 14
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  i3 : dim (S/J)
│ │ │ │  
│ │ │ │  o3 = 4
│ │ │ │  It is the cone over $P^2 \times E$ where $E$ is an elliptic curve. We have
│ │ │ │  embedded it with a Segre embedding inside $P^8$. In particular, this example is
│ │ │ │  even regular in codimension 3.
│ │ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ │ + -- used 4.38621s (cpu); 2.60071s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ │ + -- used 17.3336s (cpu); 10.6186s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the
│ │ │ │  ideal made up of a small number of minors of the Jacobian matrix. In this
│ │ │ │  example, instead of computing all relevant 1465128 minors to compute the
│ │ │ │  singular locus, and then trying to compute the dimension of the ideal they
│ │ │ │  generate, we instead compute a few of them. regularInCodimension returns true
│ │ │ │ @@ -152,15 +152,15 @@
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ │ + -- used 4.75603s (cpu); 2.82964s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 452.908 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 6
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -213,15 +213,15 @@
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ │ + -- used 0.350214s (cpu); 0.197276s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -244,15 +244,15 @@
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ │ + -- used 0.335923s (cpu); 0.189692s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -282,15 +282,15 @@
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ │ + -- used 0.54881s (cpu); 0.31502s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 3, and computed = 3
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -343,15 +343,15 @@
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ │ + -- used 1.6985s (cpu); 1.08017s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 2, and computed = 2
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ │ @@ -410,15 +410,15 @@
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ │ + -- used 1.21185s (cpu); 0.751148s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │              
    		
│ │ │                
    i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 4.1651s elapsed
│ │ │ + -- 3.7471s elapsed
│ │ │  
│ │ │  o2 = true
    
│ │ │              
    
│ │ │  In this particular example, on one machine, we list average time to completion of each of the above strategies after 100 runs.        
      
│ │ │            
    • StrategyDefault: 1.65 seconds
      • 
│ │ │ @@ -127,15 +127,15 @@
│ │ │            
    • StrategyPoints: choose all submatrices via Points.
      • 
│ │ │            
    • StrategyDefaultWithPoints: like StrategyDefault but replaces the Random and RandomNonZero submatrices as with matrices chosen as in Points.
      • 
│ │ │          
    
│ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported.  It begins as the default strategy, but the user can modify it.

    Using a single heuristic  Alternatively, if the user only wants to use say LexSmallestTerm they can set, Strategy to point to that symbol, instead of a creating a custom strategy HashTable.  For example:         
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.67457s elapsed
│ │ │ + -- 1.54762s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-
│ │ │ │  b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-
│ │ │ │  e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-
│ │ │ │  a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-
│ │ │ │  g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-
│ │ │ │  h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ │ - -- 4.1651s elapsed
│ │ │ │ + -- 3.7471s elapsed
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  In this particular example, on one machine, we list average time to completion
│ │ │ │  of each of the above strategies after 100 runs.
│ │ │ │      * StrategyDefault: 1.65 seconds
│ │ │ │      * StrategyRandom: 8.32 seconds
│ │ │ │      * StrategyDefaultNonRandom: 0.99 seconds
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported. It
│ │ │ │  begins as the default strategy, but the user can modify it.
│ │ │ │  
│ │ │ │  UUssiinngg aa ssiinnggllee hheeuurriissttiicc Alternatively, if the user only wants to use say
│ │ │ │  LexSmallestTerm they can set, Strategy to point to that symbol, instead of a
│ │ │ │  creating a custom strategy HashTable. For example:
│ │ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ │ - -- 1.67457s elapsed
│ │ │ │ + -- 1.54762s elapsed
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _S_t_r_a_t_e_g_y_D_e_f_a_u_l_t is an _o_p_t_i_o_n_ _t_a_b_l_e.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/FastMinors.m2:2037:0.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.000947066s (cpu); 0.00263955s (thread); 0s (gc)
│ │ │ + -- used 0.00400117s (cpu); 0.00417061s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The function works by computing gb(I, PairLimit=>f(i)) for successive values of i.  Here f(i) is a function that takes t, some approximation of the base degree value of the polynomial ring (for example, in a standard graded polynomial ring, this is probably expected to be \{1\}).  And i is a counting variable. You can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>( (i) -> f(i) ), the default function is SPairsFunction=>i->ceiling(1.5^i)   Perhaps more commonly however, the user may want to instead tell the function to compute for larger values of i.  This is done via the option PairLimit.  This is the max value of i to be plugged into SPairsFunction before the function gives up.  In other words, PairLimit=>5 will tell the function to check codimension 5 times.
    
│ │ │ @@ -141,24 +141,24 @@
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00400074s (cpu); 0.00251447s (thread); 0s (gc)
│ │ │ + -- used 0.00361314s (cpu); 0.0034771s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc)
│ │ │ + -- used 0.00189832s (cpu); 0.00382314s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Notice in the first case the function returned null, because the depth of search was not high enough.  It only computed codim 5 times.  The second returned true, but it did so as soon as the answer was found (and before we hit the PairLimit limit).
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,15 +38,15 @@
│ │ │ │               30      12
│ │ │ │  o4 : Matrix R   <-- R
│ │ │ │  i5 : r = rank myDiff;
│ │ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ │ - -- used 0.000947066s (cpu); 0.00263955s (thread); 0s (gc)
│ │ │ │ + -- used 0.00400117s (cpu); 0.00417061s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  The function works by computing gb(I, PairLimit=>f(i)) for successive values of
│ │ │ │  i. Here f(i) is a function that takes t, some approximation of the base degree
│ │ │ │  value of the polynomial ring (for example, in a standard graded polynomial
│ │ │ │  ring, this is probably expected to be \{1\}). And i is a counting variable. You
│ │ │ │  can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>
│ │ │ │ @@ -72,20 +72,20 @@
│ │ │ │  x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-
│ │ │ │  3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ │ - -- used 0.00400074s (cpu); 0.00251447s (thread); 0s (gc)
│ │ │ │ + -- used 0.00361314s (cpu); 0.0034771s (thread); 0s (gc)
│ │ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc)
│ │ │ │ + -- used 0.00189832s (cpu); 0.00382314s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  Notice in the first case the function returned null, because the depth of
│ │ │ │  search was not high enough. It only computed codim 5 times. The second returned
│ │ │ │  true, but it did so as soon as the answer was found (and before we hit the
│ │ │ │  PairLimit limit).
│ │ │ │  ********** WWaayyss ttoo uussee iissCCooddiimmAAttLLeeaasstt:: **********
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html
│ │ │ @@ -104,23 +104,23 @@
│ │ │  
│ │ │  o3 = 2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.296971s (cpu); 0.159933s (thread); 0s (gc)
│ │ │ + -- used 0.430958s (cpu); 0.249096s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc)
│ │ │ + -- used 0.0142396s (cpu); 0.0166226s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The option MaxMinors can be used to control how many minors are computed at each step. If this is not specified, the number of minors is a function of the dimension $d$ of the polynomial ring and the possible minors $c$. Specifically it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors => ZZ to specify that a fixed integer is used for each step.  Alternatively, the user can control the number of minors computed at each step by setting the option MaxMinors => List.  In this case, the list specifies how many minors to be computed at each step, (working backwards). Finally, you can also set MaxMinors to be a custom function of the dimension $d$ of the polynomial ring and the maximum number of minors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,19 +44,19 @@
│ │ │ │  i2 : I = ideal((x^3+y)^2, (x^2+y^2)^2, (x+y^3)^2, (x*y)^2);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : pdim(module I)
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ │ - -- used 0.296971s (cpu); 0.159933s (thread); 0s (gc)
│ │ │ │ + -- used 0.430958s (cpu); 0.249096s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc)
│ │ │ │ + -- used 0.0142396s (cpu); 0.0166226s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 1
│ │ │ │  The option MaxMinors can be used to control how many minors are computed at
│ │ │ │  each step. If this is not specified, the number of minors is a function of the
│ │ │ │  dimension $d$ of the polynomial ring and the possible minors $c$. Specifically
│ │ │ │  it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors
│ │ │ │  => ZZ to specify that a fixed integer is used for each step. Alternatively, the
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html
│ │ │ @@ -97,23 +97,23 @@
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ + -- used 0.638836s (cpu); 0.568079s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ + -- used 1.60164s (cpu); 1.44213s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : I1 == I2
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  strategy for minors
│ │ │ │  i1 : R = QQ[x,y];
│ │ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │ │  
│ │ │ │               6      7
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ │ - -- used 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ │ + -- used 0.638836s (cpu); 0.568079s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ │ + -- used 1.60164s (cpu); 1.44213s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : I1 == I2
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r_s -- ideal generated by minors
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html
│ │ │ @@ -139,23 +139,23 @@
│ │ │  
│ │ │  o7 = 3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc)
│ │ │ + -- used 1.32249s (cpu); 0.841829s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc)
    
│ │ │ + -- used 11.1853s (cpu); 7.1376s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -173,39 +173,39 @@
│ │ │  
│ │ │  o11 = 2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0279591s (cpu); 0.0261627s (thread); 0s (gc)
│ │ │ + -- used 0.0270262s (cpu); 0.0243831s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc)
│ │ │ + -- used 0.684727s (cpu); 0.380789s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc)
│ │ │ + -- used 0.425532s (cpu); 0.268548s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc)
│ │ │ + -- used 0.473376s (cpu); 0.254374s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The function works by choosing interesting looking submatrices, computing their determinants, and periodically (based on a logarithmic growth setting), computing the dimension of a subideal of the Jacobian. The option Verbose can be used to see this in action.
    
│ │ │ @@ -538,15 +538,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc)
│ │ │ + -- used 11.4598s (cpu); 7.49946s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -627,15 +627,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc)
│ │ │ + -- used 2.15602s (cpu); 1.42355s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -659,15 +659,15 @@
│ │ │          
    
│ │ │            
    If you set the option VerifyNonRegular => true, then Macaulay2 will try to verify that the ring is not regular in codimension n.  Turning this on means that when the set where the minors computed so far has codimension n, then it evaluates the matrix at the generic point of a minimal prime of that set.  If that evaluated Jacobian matrix has too low of a rank, then one has verified that variety is not regular in codimemsion n.  We consider the same example as above, but notice now the function returns false instead of true.  This sometimes can be slower and sometimes can be faster.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i18 : time regularInCodimension(2, S, VerifyNonRegular=>true)
│ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc)
│ │ │ + -- used 2.08497s (cpu); 1.19351s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │          
    
│ │ │            
    This function has many options which allow you to fine tune the strategy used to find interesting minors. You can pass it a HashTable specifying the strategy via the option Strategy.  See LexSmallest for how to construct this HashTable. The default strategy is StrategyDefault, which seems to work well on the examples we have explored.  However, caution must be exercised, because, even in the examples above, certain strategies work well while others do not.  In the Abelian surface example, LexSmallest works very well, while LexSmallestTerm does not even typically correctly identify the ring as nonsingular (this is because there are a small number of entries with nonzero constant terms, which are selected repeatedly). However, in our first example, the LexSmallestTerm is much faster, and Random does not perform well at all.
    
│ │ │ @@ -687,39 +687,39 @@
│ │ │              
│ │ │                
    i21 : StrategyCurrent#LexSmallestTerm = 0;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc)
│ │ │ + -- used 0.733147s (cpu); 0.425419s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc)
│ │ │ + -- used 0.853632s (cpu); 0.416929s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc)
│ │ │ + -- used 0.498479s (cpu); 0.327804s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc)
│ │ │ + -- used 0.593409s (cpu); 0.341934s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i26 : StrategyCurrent#LexSmallest = 0;
    
│ │ │ @@ -729,51 +729,51 @@
│ │ │              
│ │ │                
    i27 : StrategyCurrent#LexSmallestTerm = 100;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc)
    
│ │ │ + -- used 4.08395s (cpu); 2.37479s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc)
    
│ │ │ + -- used 4.07422s (cpu); 2.39475s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc)
│ │ │ + -- used 0.461171s (cpu); 0.300656s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc)
│ │ │ + -- used 0.684033s (cpu); 0.42462s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc)
│ │ │ + -- used 2.37898s (cpu); 1.62521s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc)
│ │ │ + -- used 2.01944s (cpu); 1.34737s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The minimum number of minors computed before checking the codimension can also be controlled by an option MinMinorsFunction.  This is should be a function of a single variable, the number of minors computed.  Finally, via the option CodimCheckFunction, you can pass the regularInCodimension a function which controls how frequently the codimension of the partial Jacobian ideal is computed.  By default this is the floor of 1.3^k. Finally, passing the option Modulus => p will do the computation after changing the coefficient ring to ZZ/p.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,19 +77,19 @@
│ │ │ │  
│ │ │ │  o5 : Ideal of T
│ │ │ │  i6 : S = T/I;
│ │ │ │  i7 : dim S
│ │ │ │  
│ │ │ │  o7 = 3
│ │ │ │  i8 : time regularInCodimension(1, S)
│ │ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc)
│ │ │ │ + -- used 1.32249s (cpu); 0.841829s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : time regularInCodimension(2, S)
│ │ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc)
│ │ │ │ + -- used 11.1853s (cpu); 7.1376s (thread); 0s (gc)
│ │ │ │  There are numerous examples where regularInCodimension is several orders of
│ │ │ │  magnitude faster that calls of dim singularLocus.
│ │ │ │  The following is a (pruned) affine chart on an Abelian surface obtained as a
│ │ │ │  product of two elliptic curves. It is nonsingular, as our function verifies. If
│ │ │ │  one does not prune it, then the dim singularLocus call takes an enormous amount
│ │ │ │  of time, otherwise the running times of dim singularLocus and our function are
│ │ │ │  frequently about the same.
│ │ │ │ @@ -97,27 +97,27 @@
│ │ │ │  (g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-
│ │ │ │  f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-
│ │ │ │  c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
│ │ │ │  i11 : dim(R)
│ │ │ │  
│ │ │ │  o11 = 2
│ │ │ │  i12 : time (dim singularLocus (R))
│ │ │ │ - -- used 0.0279591s (cpu); 0.0261627s (thread); 0s (gc)
│ │ │ │ + -- used 0.0270262s (cpu); 0.0243831s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -1
│ │ │ │  i13 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc)
│ │ │ │ + -- used 0.684727s (cpu); 0.380789s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc)
│ │ │ │ + -- used 0.425532s (cpu); 0.268548s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : time regularInCodimension(2, R)
│ │ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc)
│ │ │ │ + -- used 0.473376s (cpu); 0.254374s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  The function works by choosing interesting looking submatrices, computing their
│ │ │ │  determinants, and periodically (based on a logarithmic growth setting),
│ │ │ │  computing the dimension of a subideal of the Jacobian. The option Verbose can
│ │ │ │  be used to see this in action.
│ │ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │ │ @@ -445,15 +445,15 @@
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc)
│ │ │ │ + -- used 11.4598s (cpu); 7.49946s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4
│ │ │ │  minors, we will compute up to 327.599 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 9, and computed = 9
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │ │ @@ -550,15 +550,15 @@
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc)
│ │ │ │ + -- used 2.15602s (cpu); 1.42355s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4
│ │ │ │  minors, we will compute up to 30 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 9, and computed = 9
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │ │ @@ -589,15 +589,15 @@
│ │ │ │  that when the set where the minors computed so far has codimension n, then it
│ │ │ │  evaluates the matrix at the generic point of a minimal prime of that set. If
│ │ │ │  that evaluated Jacobian matrix has too low of a rank, then one has verified
│ │ │ │  that variety is not regular in codimemsion n. We consider the same example as
│ │ │ │  above, but notice now the function returns false instead of true. This
│ │ │ │  sometimes can be slower and sometimes can be faster.
│ │ │ │  i18 : time regularInCodimension(2, S, VerifyNonRegular=>true)
│ │ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc)
│ │ │ │ + -- used 2.08497s (cpu); 1.19351s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  This function has many options which allow you to fine tune the strategy used
│ │ │ │  to find interesting minors. You can pass it a HashTable specifying the strategy
│ │ │ │  via the option Strategy. See _L_e_x_S_m_a_l_l_e_s_t for how to construct this HashTable.
│ │ │ │  The default strategy is StrategyDefault, which seems to work well on the
│ │ │ │  examples we have explored. However, caution must be exercised, because, even in
│ │ │ │ @@ -607,49 +607,49 @@
│ │ │ │  because there are a small number of entries with nonzero constant terms, which
│ │ │ │  are selected repeatedly). However, in our first example, the LexSmallestTerm is
│ │ │ │  much faster, and Random does not perform well at all.
│ │ │ │  i19 : StrategyCurrent#Random = 0;
│ │ │ │  i20 : StrategyCurrent#LexSmallest = 100;
│ │ │ │  i21 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc)
│ │ │ │ + -- used 0.733147s (cpu); 0.425419s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc)
│ │ │ │ + -- used 0.853632s (cpu); 0.416929s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc)
│ │ │ │ + -- used 0.498479s (cpu); 0.327804s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc)
│ │ │ │ + -- used 0.593409s (cpu); 0.341934s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o25 = true
│ │ │ │  i26 : StrategyCurrent#LexSmallest = 0;
│ │ │ │  i27 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc)
│ │ │ │ + -- used 4.08395s (cpu); 2.37479s (thread); 0s (gc)
│ │ │ │  i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc)
│ │ │ │ + -- used 4.07422s (cpu); 2.39475s (thread); 0s (gc)
│ │ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc)
│ │ │ │ + -- used 0.461171s (cpu); 0.300656s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc)
│ │ │ │ + -- used 0.684033s (cpu); 0.42462s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc)
│ │ │ │ + -- used 2.37898s (cpu); 1.62521s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = true
│ │ │ │  i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc)
│ │ │ │ + -- used 2.01944s (cpu); 1.34737s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = true
│ │ │ │  The minimum number of minors computed before checking the codimension can also
│ │ │ │  be controlled by an option MinMinorsFunction. This is should be a function of a
│ │ │ │  single variable, the number of minors computed. Finally, via the option
│ │ │ │  CodimCheckFunction, you can pass the regularInCodimension a function which
│ │ │ │  controls how frequently the codimension of the partial Jacobian ideal is
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  bmV4dERlZ3JlZShNYXRyaXgsWlosUmluZyk=
│ │ │  #:len=288
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzY0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhuZXh0RGVncmVlLE1hdHJpeCxaWixSaW5nKSwibmV4
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out
│ │ │ @@ -81,23 +81,23 @@
│ │ │  
│ │ │  i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │  
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
│ │ │  
│ │ │  i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00253889s (cpu); 0.00253616s (thread); 0s (gc)
│ │ │ + -- used 0.00378621s (cpu); 0.00378545s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │  
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ - -- used 0.00662056s (cpu); 0.00662794s (thread); 0s (gc)
│ │ │ + -- used 0.00810437s (cpu); 0.00810589s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : I==J
│ │ │  
│ │ │  o17 = true
│ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html
│ │ │ @@ -207,26 +207,26 @@
│ │ │                8      8
│ │ │  o14 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00253889s (cpu); 0.00253616s (thread); 0s (gc)
│ │ │ + -- used 0.00378621s (cpu); 0.00378545s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │  
│ │ │  o15 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ - -- used 0.00662056s (cpu); 0.00662794s (thread); 0s (gc)
│ │ │ + -- used 0.00810437s (cpu); 0.00810589s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : I==J
│ │ │ ├── html2text {}
│ │ │ │ @@ -95,22 +95,22 @@
│ │ │ │                2      6
│ │ │ │  o13 : Matrix R  <-- R
│ │ │ │  i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │ │  
│ │ │ │                8      8
│ │ │ │  o14 : Matrix R  <-- R
│ │ │ │  i15 : time I=co1Fitting(K3)
│ │ │ │ - -- used 0.00253889s (cpu); 0.00253616s (thread); 0s (gc)
│ │ │ │ + -- used 0.00378621s (cpu); 0.00378545s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ │ - -- used 0.00662056s (cpu); 0.00662794s (thread); 0s (gc)
│ │ │ │ + -- used 0.00810437s (cpu); 0.00810589s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : I==J
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  Note that our method is a bit faster for this small example, and for rank 2
│ │ │ │  quotients of S^3=\mathbb{Z}[x,y]^3 the time difference is massive.
│ │ ├── ./usr/share/doc/Macaulay2/FirstPackage/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  Rmlyc3RQYWNrYWdl
│ │ │  #:len=509
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZXhhbXBsZSBNYWNhdWxheTIgcGFj
│ │ │  a2FnZSIsICJsaW5lbnVtIiA9PiA1MywgImZpbGVuYW1lIiA9PiAiRmlyc3RQYWNrYWdlLm0yIiwg
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  dmFsdWUodWludDE2KQ==
│ │ │  #:len=273
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTcxOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodmFsdWUsdWludDE2KSwidmFsdWUodWludDE2KSIs
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Object.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9050}
│ │ │ +o2 = int32{Address => 0x7fdd7e29d2a0}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9050
│ │ │ +o3 = 0x7fdd7e29d2a0
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : class x
│ │ │  
│ │ │  o4 = int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  
│ │ │  o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog}
│ │ │  
│ │ │  o2 : ForeignObject of type char**
│ │ │  
│ │ │  i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70}
│ │ │ +o3 = {0x7fdd7e2e99b0, 0x7fdd7e2e99a0, 0x7fdd7e2e9990}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
│ │ │  
│ │ │  i4 : x = charstarstar {"foo", "bar", "baz"}
│ │ │  
│ │ │  o4 = {foo, bar, baz}
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type_sp__Visible__List.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {foo, bar}
│ │ │  
│ │ │  o1 : ForeignObject of type char**
│ │ │  
│ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ +o2 = {0x7fdd7e2e9870, 0x7fdd7e2e9860, 0x7fdd7e2e9850}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
│ │ │  
│ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │  
│ │ │  o3 = int32[2]*
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Type_sp__Pointer.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1730835169888399450
│ │ │  
│ │ │  i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7f40f26a3da0
│ │ │ +o1 = 0x7fdd7e52e860
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f40f26a3da0
│ │ │ +o2 = 0x7fdd7e52e860
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Pointer.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d97f0
│ │ │ +o2 = 0x7fdd7e2c6dc0
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 : int ptr
│ │ │  
│ │ │  o3 = 5
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp_st_spvoidstar.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1731230829183683930
│ │ │  
│ │ │  i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f40f21d91f0
│ │ │ +o1 = 0x7fdd7e2c6990
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : int * ptr
│ │ │  
│ │ │  o2 = 5
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = myunion
│ │ │  
│ │ │  o1 : ForeignUnionType
│ │ │  
│ │ │  i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.91261e-310}
│ │ │ +o2 = HashTable{"bar" => 6.94593e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
│ │ │  
│ │ │  i3 : myunion pi
│ │ │  
│ │ │  o3 = HashTable{"bar" => 3.14159   }
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Pointer.out
│ │ │ @@ -4,28 +4,28 @@
│ │ │  
│ │ │  o1 = 20
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9960}
│ │ │ +o2 = int32{Address => 0x7fdd7e29d130}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9960
│ │ │ +o3 = 0x7fdd7e29d130
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f40f21d9965
│ │ │ +o4 = 0x7fdd7e29d135
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f40f21d995d
│ │ │ +o5 = 0x7fdd7e29d12d
│ │ │  
│ │ │  o5 : Pointer
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Shared__Library.out
│ │ │ @@ -4,10 +4,10 @@
│ │ │  
│ │ │  o1 = mpfr
│ │ │  
│ │ │  o1 : SharedLibrary
│ │ │  
│ │ │  i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f40f9b43ac0, mpfr}
│ │ │ +o2 = SharedLibrary{0x7fdd93161ac0, mpfr}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/__st_spvoidstar_sp_eq_sp__Thing.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d9220
│ │ │ +o2 = 0x7fdd7e2c69c0
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 : *ptr = int 6
│ │ │  
│ │ │  o3 = 6
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_address.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1730181884377373595
│ │ │  
│ │ │  i1 : address int
│ │ │  
│ │ │ -o1 = 0x562fbd063080
│ │ │ +o1 = 0x5561cf7b9080
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f40f21d95f0
│ │ │ +o2 = 0x7fdd7e2c6bd0
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Function.out
│ │ │ @@ -78,14 +78,14 @@
│ │ │  
│ │ │  o16 = free
│ │ │  
│ │ │  o16 : ForeignFunction
│ │ │  
│ │ │  i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f0558070f60
│ │ │ +o17 = 0x7f7a40070f60
│ │ │  
│ │ │  o17 : ForeignObject of type void*
│ │ │  
│ │ │  i18 : registerFinalizer(x, free)
│ │ │  
│ │ │  i19 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_get__Memory.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 10647988412767280310
│ │ │  
│ │ │  i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f40f50ca360
│ │ │ +o1 = 0x7fdd7e8ca880
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f40f21d9ae0
│ │ │ +o2 = 0x7fdd7e2c6a60
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f40f21d9a00
│ │ │ +o3 = 0x7fdd7e2c6960
│ │ │  
│ │ │  o3 : ForeignObject of type void*
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.out
│ │ │ @@ -17,18 +17,18 @@
│ │ │  o3 = finalizer
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
│ │ │  
│ │ │  i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f40dc0848b0
│ │ │ -freeing memory at 0x7f40dc084870
│ │ │ -freeing memory at 0x7f40dc084830
│ │ │ -freeing memory at 0x7f40dc0841d0
│ │ │ -freeing memory at 0x7f40dc0841f0
│ │ │ -freeing memory at 0x7f40dc0848d0
│ │ │ -freeing memory at 0x7f40dc084810
│ │ │ -freeing memory at 0x7f40dc084850
│ │ │ -freeing memory at 0x7f40dc084890
│ │ │ +freeing memory at 0x7fdd680848b0
│ │ │ +freeing memory at 0x7fdd68084890
│ │ │ +freeing memory at 0x7fdd680841f0
│ │ │ +freeing memory at 0x7fdd680841d0
│ │ │ +freeing memory at 0x7fdd68084830
│ │ │ +freeing memory at 0x7fdd68084870
│ │ │ +freeing memory at 0x7fdd68084850
│ │ │ +freeing memory at 0x7fdd68084810
│ │ │ +freeing memory at 0x7fdd680848d0
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_value_lp__Foreign__Object_rp.out
│ │ │ @@ -20,21 +20,21 @@
│ │ │  
│ │ │  o4 = 5
│ │ │  
│ │ │  o4 : RR (of precision 53)
│ │ │  
│ │ │  i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f40f21fd990
│ │ │ +o5 = 0x7fdd7e2e9f90
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f40f21fd990
│ │ │ +o6 = 0x7fdd7e2e9f90
│ │ │  
│ │ │  o6 : Pointer
│ │ │  
│ │ │  i7 : x = charstar "Hello, world!"
│ │ │  
│ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
│ │ │ @@ -69,27 +69,27 @@
│ │ │  o1 : ForeignObject of type int32
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9050}
    
│ │ │ +o2 = int32{Address => 0x7fdd7e29d2a0}
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    To get this, use address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9050
│ │ │ +o3 = 0x7fdd7e29d2a0
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Use class to determine the type of the object.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,19 +10,19 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : peek x
│ │ │ │  
│ │ │ │ -o2 = int32{Address => 0x7f40f21d9050}
│ │ │ │ +o2 = int32{Address => 0x7fdd7e29d2a0}
│ │ │ │  To get this, use _a_d_d_r_e_s_s.
│ │ │ │  i3 : address x
│ │ │ │  
│ │ │ │ -o3 = 0x7f40f21d9050
│ │ │ │ +o3 = 0x7fdd7e29d2a0
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Use _c_l_a_s_s to determine the type of the object.
│ │ │ │  i4 : class x
│ │ │ │  
│ │ │ │  o4 = int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │  o2 : ForeignObject of type char**
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70}
│ │ │ +o3 = {0x7fdd7e2e99b0, 0x7fdd7e2e99a0, 0x7fdd7e2e9990}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Foreign pointer arrays may be subscripted using _.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │           "lazy", "dog"}
│ │ │ │  
│ │ │ │  o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type char**
│ │ │ │  i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o3 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70}
│ │ │ │ +o3 = {0x7fdd7e2e99b0, 0x7fdd7e2e99a0, 0x7fdd7e2e9990}
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type void**
│ │ │ │  Foreign pointer arrays may be subscripted using __.
│ │ │ │  i4 : x = charstarstar {"foo", "bar", "baz"}
│ │ │ │  
│ │ │ │  o4 = {foo, bar, baz}
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type_sp__Visible__List.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  o1 : ForeignObject of type char**
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ +o2 = {0x7fdd7e2e9870, 0x7fdd7e2e9860, 0x7fdd7e2e9850}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : charstarstar {"foo", "bar"}
│ │ │ │  
│ │ │ │  o1 = {foo, bar}
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type char**
│ │ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o2 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ │ +o2 = {0x7fdd7e2e9870, 0x7fdd7e2e9860, 0x7fdd7e2e9850}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void**
│ │ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ │  
│ │ │ │  o3 = int32[2]*
│ │ │ │  
│ │ │ │  o3 : ForeignPointerArrayType
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Type_sp__Pointer.html
│ │ │ @@ -78,24 +78,24 @@
│ │ │            
    To cast a Macaulay2 pointer to a foreign object with a pointer type, give the type followed by the pointer.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7f40f26a3da0
│ │ │ +o1 = 0x7fdd7e52e860
│ │ │  
│ │ │  o1 : Pointer
    
│ │ │              
    
│ │ │                
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f40f26a3da0
│ │ │ +o2 = 0x7fdd7e52e860
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,20 +15,20 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  To cast a Macaulay2 pointer to a foreign object with a pointer type, give the
│ │ │ │  type followed by the pointer.
│ │ │ │  i1 : ptr = address int 0
│ │ │ │  
│ │ │ │ -o1 = 0x7f40f26a3da0
│ │ │ │ +o1 = 0x7fdd7e52e860
│ │ │ │  
│ │ │ │  o1 : Pointer
│ │ │ │  i2 : voidstar ptr
│ │ │ │  
│ │ │ │ -o2 = 0x7f40f26a3da0
│ │ │ │ +o2 = 0x7fdd7e52e860
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void*
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _F_o_r_e_i_g_n_P_o_i_n_t_e_r_T_y_p_e_ _P_o_i_n_t_e_r -- cast a Macaulay2 pointer to a foreign
│ │ │ │        pointer
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d97f0
│ │ │ +o2 = 0x7fdd7e2c6dc0
│ │ │  
│ │ │  o2 : Pointer
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : int ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7f40f21d97f0
│ │ │ │ +o2 = 0x7fdd7e2c6dc0
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f40f21d91f0
│ │ │ +o1 = 0x7fdd7e2c6990
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    
│ │ │              
    
│ │ │                
    i2 : int * ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t, of type T;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This is syntactic sugar for T value ptr (see _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r) for
│ │ │ │  dereferencing pointers.
│ │ │ │  i1 : ptr = voidstar address int 5
│ │ │ │  
│ │ │ │ -o1 = 0x7f40f21d91f0
│ │ │ │ +o1 = 0x7fdd7e2c6990
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  i2 : int * ptr
│ │ │ │  
│ │ │ │  o2 = 5
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  o1 : ForeignUnionType
    
│ │ │              
    
│ │ │                
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.91261e-310}
│ │ │ +o2 = HashTable{"bar" => 6.94593e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : myunion = foreignUnionType("myunion", {"foo" => int, "bar" => double})
│ │ │ │  
│ │ │ │  o1 = myunion
│ │ │ │  
│ │ │ │  o1 : ForeignUnionType
│ │ │ │  i2 : myunion 27
│ │ │ │  
│ │ │ │ -o2 = HashTable{"bar" => 6.91261e-310}
│ │ │ │ +o2 = HashTable{"bar" => 6.94593e-310}
│ │ │ │                 "foo" => 27
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type myunion
│ │ │ │  i3 : myunion pi
│ │ │ │  
│ │ │ │  o3 = HashTable{"bar" => 3.14159   }
│ │ │ │                 "foo" => 1413754136
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html
│ │ │ @@ -69,50 +69,50 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
    
│ │ │                
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9960}
    
│ │ │ +o2 = int32{Address => 0x7fdd7e29d130}
│ │ │              
    
│ │ │          
    
│ │ │            
    These pointers can be accessed using address.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9960
│ │ │ +o3 = 0x7fdd7e29d130
│ │ │  
│ │ │  o3 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Simple arithmetic can be performed on pointers.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f40f21d9965
│ │ │ +o4 = 0x7fdd7e29d135
│ │ │  
│ │ │  o4 : Pointer
    
│ │ │              
    
│ │ │                
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f40f21d995d
│ │ │ +o5 = 0x7fdd7e29d12d
│ │ │  
│ │ │  o5 : Pointer
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,30 +10,30 @@
│ │ │ │  i1 : x = int 20
│ │ │ │  
│ │ │ │  o1 = 20
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : peek x
│ │ │ │  
│ │ │ │ -o2 = int32{Address => 0x7f40f21d9960}
│ │ │ │ +o2 = int32{Address => 0x7fdd7e29d130}
│ │ │ │  These pointers can be accessed using _a_d_d_r_e_s_s.
│ │ │ │  i3 : ptr = address x
│ │ │ │  
│ │ │ │ -o3 = 0x7f40f21d9960
│ │ │ │ +o3 = 0x7fdd7e29d130
│ │ │ │  
│ │ │ │  o3 : Pointer
│ │ │ │  Simple arithmetic can be performed on pointers.
│ │ │ │  i4 : ptr + 5
│ │ │ │  
│ │ │ │ -o4 = 0x7f40f21d9965
│ │ │ │ +o4 = 0x7fdd7e29d135
│ │ │ │  
│ │ │ │  o4 : Pointer
│ │ │ │  i5 : ptr - 3
│ │ │ │  
│ │ │ │ -o5 = 0x7f40f21d995d
│ │ │ │ +o5 = 0x7fdd7e29d12d
│ │ │ │  
│ │ │ │  o5 : Pointer
│ │ │ │  ******** MMeennuu ********
│ │ │ │      * _n_u_l_l_P_o_i_n_t_e_r -- the null pointer
│ │ │ │      * _a_d_d_r_e_s_s -- pointer to type or object
│ │ │ │      * _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r -- dereference a pointer
│ │ │ │  ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa ppooiinntteerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │  o1 : SharedLibrary
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f40f9b43ac0, mpfr}
    
│ │ │ +o2 = SharedLibrary{0x7fdd93161ac0, mpfr}
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    Menu
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │  i1 : mpfr = openSharedLibrary "mpfr"
│ │ │ │  
│ │ │ │  o1 = mpfr
│ │ │ │  
│ │ │ │  o1 : SharedLibrary
│ │ │ │  i2 : peek mpfr
│ │ │ │  
│ │ │ │ -o2 = SharedLibrary{0x7f40f9b43ac0, mpfr}
│ │ │ │ +o2 = SharedLibrary{0x7fdd93161ac0, mpfr}
│ │ │ │  ******** MMeennuu ********
│ │ │ │      * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library
│ │ │ │  ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa sshhaarreedd lliibbrraarryy:: **********
│ │ │ │      * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa sshhaarreedd lliibbrraarryy:: **********
│ │ │ │      * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) -- see
│ │ │ │        _f_o_r_e_i_g_n_F_u_n_c_t_i_o_n -- construct a foreign function
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/__st_spvoidstar_sp_eq_sp__Thing.html
│ │ │ @@ -83,15 +83,15 @@
│ │ │  o1 : ForeignObject of type int32
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d9220
│ │ │ +o2 = 0x7fdd7e2c69c0
│ │ │  
│ │ │  o2 : Pointer
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : *ptr = int 6
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7f40f21d9220
│ │ │ │ +o2 = 0x7fdd7e2c69c0
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : *ptr = int 6
│ │ │ │  
│ │ │ │  o3 = 6
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_address.html
│ │ │ @@ -76,29 +76,29 @@
│ │ │            
      If x is a foreign type, then this returns the address to the ffi_type struct used by libffi to identify the type.
      
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : address int
│ │ │  
│ │ │ -o1 = 0x562fbd063080
│ │ │ +o1 = 0x5561cf7b9080
│ │ │  
│ │ │  o1 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If x is a foreign object, then this returns the address to the object.  It behaves like the & "address-of" operator in C.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f40f21d95f0
│ │ │ +o2 = 0x7fdd7e2c6bd0
│ │ │  
│ │ │  o2 : Pointer
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,22 +11,22 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _p_o_i_n_t_e_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  If x is a foreign type, then this returns the address to the ffi_type struct
│ │ │ │  used by libffi to identify the type.
│ │ │ │  i1 : address int
│ │ │ │  
│ │ │ │ -o1 = 0x562fbd063080
│ │ │ │ +o1 = 0x5561cf7b9080
│ │ │ │  
│ │ │ │  o1 : Pointer
│ │ │ │  If x is a foreign object, then this returns the address to the object. It
│ │ │ │  behaves like the & "address-of" operator in C.
│ │ │ │  i2 : address int 5
│ │ │ │  
│ │ │ │ -o2 = 0x7f40f21d95f0
│ │ │ │ +o2 = 0x7fdd7e2c6bd0
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  ********** WWaayyss ttoo uussee aaddddrreessss:: **********
│ │ │ │      * address(ForeignObject)
│ │ │ │      * address(ForeignType)
│ │ │ │      * address(Nothing) (missing documentation)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Function.html
│ │ │ @@ -237,15 +237,15 @@
│ │ │  o16 : ForeignFunction
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f0558070f60
│ │ │ +o17 = 0x7f7a40070f60
│ │ │  
│ │ │  o17 : ForeignObject of type void*
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : registerFinalizer(x, free)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -95,15 +95,15 @@
│ │ │ │  i16 : free = foreignFunction("free", void, voidstar)
│ │ │ │  
│ │ │ │  o16 = free
│ │ │ │  
│ │ │ │  o16 : ForeignFunction
│ │ │ │  i17 : x = malloc 8
│ │ │ │  
│ │ │ │ -o17 = 0x7f0558070f60
│ │ │ │ +o17 = 0x7f7a40070f60
│ │ │ │  
│ │ │ │  o17 : ForeignObject of type void*
│ │ │ │  i18 : registerFinalizer(x, free)
│ │ │ │  ********** WWaayyss ttoo uussee ffoorreeiiggnnFFuunnccttiioonn:: **********
│ │ │ │      * foreignFunction(Pointer,String,ForeignType,VisibleList)
│ │ │ │      * foreignFunction(SharedLibrary,String,ForeignType,ForeignType)
│ │ │ │      * foreignFunction(SharedLibrary,String,ForeignType,VisibleList)
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_get__Memory.html
│ │ │ @@ -82,43 +82,43 @@
│ │ │            
    Allocate n bytes of memory using the GC garbage collector.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f40f50ca360
│ │ │ +o1 = 0x7fdd7e8ca880
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If the memory will not contain any pointers, then set the Atomic option to true.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f40f21d9ae0
│ │ │ +o2 = 0x7fdd7e2c6a60
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Alternatively, a foreign object type T may be specified.  In this case, the number of bytes and whether the Atomic option should be set will be determined automatically.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f40f21d9a00
│ │ │ +o3 = 0x7fdd7e2c6960
│ │ │  
│ │ │  o3 : ForeignObject of type void*
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,30 +14,30 @@
│ │ │ │            o Atomic => ..., default value false
│ │ │ │      * Outputs:
│ │ │ │            o an instance of the type _v_o_i_d_s_t_a_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Allocate n bytes of memory using the _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r.
│ │ │ │  i1 : ptr = getMemory 8
│ │ │ │  
│ │ │ │ -o1 = 0x7f40f50ca360
│ │ │ │ +o1 = 0x7fdd7e8ca880
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  If the memory will not contain any pointers, then set the Atomic option to
│ │ │ │  _t_r_u_e.
│ │ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │ │  
│ │ │ │ -o2 = 0x7f40f21d9ae0
│ │ │ │ +o2 = 0x7fdd7e2c6a60
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void*
│ │ │ │  Alternatively, a foreign object type T may be specified. In this case, the
│ │ │ │  number of bytes and whether the Atomic option should be set will be determined
│ │ │ │  automatically.
│ │ │ │  i3 : ptr = getMemory int
│ │ │ │  
│ │ │ │ -o3 = 0x7f40f21d9a00
│ │ │ │ +o3 = 0x7fdd7e2c6960
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type void*
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a
│ │ │ │        foreign object
│ │ │ │  ********** WWaayyss ttoo uussee ggeettMMeemmoorryy:: **********
│ │ │ │      * getMemory(ForeignType)
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.html
│ │ │ @@ -105,23 +105,23 @@
│ │ │              
│ │ │                
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f40dc0848b0
│ │ │ -freeing memory at 0x7f40dc084870
│ │ │ -freeing memory at 0x7f40dc084830
│ │ │ -freeing memory at 0x7f40dc0841d0
│ │ │ -freeing memory at 0x7f40dc0841f0
│ │ │ -freeing memory at 0x7f40dc0848d0
│ │ │ -freeing memory at 0x7f40dc084810
│ │ │ -freeing memory at 0x7f40dc084850
│ │ │ -freeing memory at 0x7f40dc084890
    
│ │ │ +freeing memory at 0x7fdd680848b0
│ │ │ +freeing memory at 0x7fdd68084890
│ │ │ +freeing memory at 0x7fdd680841f0
│ │ │ +freeing memory at 0x7fdd680841d0
│ │ │ +freeing memory at 0x7fdd68084830
│ │ │ +freeing memory at 0x7fdd68084870
│ │ │ +freeing memory at 0x7fdd68084850
│ │ │ +freeing memory at 0x7fdd68084810
│ │ │ +freeing memory at 0x7fdd680848d0
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,23 +31,23 @@
│ │ │ │  i3 : finalizer = x -> (print("freeing memory at " | net x); free x)
│ │ │ │  
│ │ │ │  o3 = finalizer
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
│ │ │ │  i5 : collectGarbage()
│ │ │ │ -freeing memory at 0x7f40dc0848b0
│ │ │ │ -freeing memory at 0x7f40dc084870
│ │ │ │ -freeing memory at 0x7f40dc084830
│ │ │ │ -freeing memory at 0x7f40dc0841d0
│ │ │ │ -freeing memory at 0x7f40dc0841f0
│ │ │ │ -freeing memory at 0x7f40dc0848d0
│ │ │ │ -freeing memory at 0x7f40dc084810
│ │ │ │ -freeing memory at 0x7f40dc084850
│ │ │ │ -freeing memory at 0x7f40dc084890
│ │ │ │ +freeing memory at 0x7fdd680848b0
│ │ │ │ +freeing memory at 0x7fdd68084890
│ │ │ │ +freeing memory at 0x7fdd680841f0
│ │ │ │ +freeing memory at 0x7fdd680841d0
│ │ │ │ +freeing memory at 0x7fdd68084830
│ │ │ │ +freeing memory at 0x7fdd68084870
│ │ │ │ +freeing memory at 0x7fdd68084850
│ │ │ │ +freeing memory at 0x7fdd68084810
│ │ │ │ +freeing memory at 0x7fdd680848d0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_e_t_M_e_m_o_r_y -- allocate memory using the garbage collector
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a
│ │ │ │        foreign object
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_value_lp__Foreign__Object_rp.html
│ │ │ @@ -121,24 +121,24 @@
│ │ │            
      Foreign pointer objects are converted to Pointer objects.
      
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f40f21fd990
│ │ │ +o5 = 0x7fdd7e2e9f90
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    
│ │ │              
    
│ │ │                
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f40f21fd990
│ │ │ +o6 = 0x7fdd7e2e9f90
│ │ │  
│ │ │  o6 : Pointer
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Foreign string objects are converted to strings.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,20 +34,20 @@
│ │ │ │  
│ │ │ │  o4 = 5
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  Foreign pointer objects are converted to _P_o_i_n_t_e_r objects.
│ │ │ │  i5 : x = voidstar address int 5
│ │ │ │  
│ │ │ │ -o5 = 0x7f40f21fd990
│ │ │ │ +o5 = 0x7fdd7e2e9f90
│ │ │ │  
│ │ │ │  o5 : ForeignObject of type void*
│ │ │ │  i6 : value x
│ │ │ │  
│ │ │ │ -o6 = 0x7f40f21fd990
│ │ │ │ +o6 = 0x7fdd7e2e9f90
│ │ │ │  
│ │ │ │  o6 : Pointer
│ │ │ │  Foreign string objects are converted to strings.
│ │ │ │  i7 : x = charstar "Hello, world!"
│ │ │ │  
│ │ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/FormalGroupLaws/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  c2VyaWVzKFJpbmdFbGVtZW50LFpaKQ==
│ │ │  #:len=1234
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29uc3RydWN0aW5nIGEgZm9ybWFsIHNl
│ │ │  cmllcyIsICJsaW5lbnVtIiA9PiAzNzIsIElucHV0cyA9PiB7U1BBTntUVHsicyJ9LCIsICIsU1BB
│ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  dG9yaWNHcm9lYm5lcg==
│ │ │  #:len=2522
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2FsY3VsYXRlcyBhIEdyb2VibmVyIGJh
│ │ │  c2lzIG9mIHRoZSB0b3JpYyBpZGVhbCBJX0EsIGdpdmVuIEE7IGludm9rZXMgXCJncm9lYm5lclwi
│ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/example-output/_put__Matrix.out
│ │ │ @@ -6,27 +6,27 @@
│ │ │       | 1 2 3 4 |
│ │ │  
│ │ │                2       4
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15423-0/0
│ │ │ +o2 = /tmp/M2-19303-0/0
│ │ │  
│ │ │  i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15423-0/0
│ │ │ +o3 = /tmp/M2-19303-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : putMatrix(F,A)
│ │ │  
│ │ │  i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15423-0/0
│ │ │ +o5 = /tmp/M2-19303-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : getMatrix(s)
│ │ │  
│ │ │  o6 = | 1 1 1 1 |
│ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html
│ │ │ @@ -84,36 +84,36 @@
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15423-0/0
    
│ │ │ +o2 = /tmp/M2-19303-0/0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15423-0/0
│ │ │ +o3 = /tmp/M2-19303-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : putMatrix(F,A)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15423-0/0
│ │ │ +o5 = /tmp/M2-19303-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : getMatrix(s)
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,24 +16,24 @@
│ │ │ │  o1 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ │ │  
│ │ │ │                2       4
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  i2 : s = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-15423-0/0
│ │ │ │ +o2 = /tmp/M2-19303-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-15423-0/0
│ │ │ │ +o3 = /tmp/M2-19303-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-15423-0/0
│ │ │ │ +o5 = /tmp/M2-19303-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : getMatrix(s)
│ │ │ │  
│ │ │ │  o6 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FourierMotzkin/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  Zm91cmllck1vdHpraW4=
│ │ │  #:len=3387
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW50ZXJjaGFuZ2UgaW5lcXVhbGl0eS9n
│ │ │  ZW5lcmF0b3IgcmVwcmVzZW50YXRpb24gb2YgYSBwb2x5aGVkcmFsIGNvbmUiLCAibGluZW51bSIg
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  aXNGUFQoLi4uLFFHb3JlbnN0ZWluSW5kZXg9Pi4uLik=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0ZQVCxRR29yZW5zdGVpbkluZGV4XSwiaXNGUFQo
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out
│ │ │ @@ -155,31 +155,31 @@
│ │ │  i26 : numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true) -- FinalAttempt improves the estimate slightly
│ │ │  
│ │ │  o26 = {.142067, .144}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ + -- used 2.17893s (cpu); 1.53593s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ + -- used 1.23762s (cpu); 0.970297s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ + -- used 1.08731s (cpu); 0.795003s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out
│ │ │ @@ -43,34 +43,34 @@
│ │ │  o12 = 220
│ │ │  
│ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │  
│ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │  
│ │ │  i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ + -- used 0.00585223s (cpu); 0.00585268s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ + -- used 0.407935s (cpu); 0.322993s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
│ │ │  
│ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │  
│ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ + -- used 0.367506s (cpu); 0.275686s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ + -- used 1.74408s (cpu); 1.36429s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
│ │ │  
│ │ │  i21 : R = ZZ/3[x,y];
│ │ │  
│ │ │  i22 : M = ideal(x, y);
│ │ │  
│ │ │ @@ -85,34 +85,34 @@
│ │ │  o24 = 8
│ │ │  
│ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │  
│ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ + -- used 0.983969s (cpu); 0.793997s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ + -- used 1.51361s (cpu); 1.16685s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
│ │ │  
│ │ │  i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ + -- used 2.02513s (cpu); 1.65145s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ + -- used 0.633252s (cpu); 0.543155s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
│ │ │  
│ │ │  i32 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html
│ │ │ @@ -368,37 +368,37 @@
│ │ │          
    
│ │ │            
    The computations performed when FinalAttempt is set to true are often slow, and often fail to improve the estimate, and for this reason, this option should be used sparingly. It is often more effective to increase the values of Attempts or DepthOfSearch, instead.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -228,29 +228,29 @@
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  The computations performed when FinalAttempt is set to true are often slow, and
│ │ │ │  often fail to improve the estimate, and for this reason, this option should be
│ │ │ │  used sparingly. It is often more effective to increase the values of Attempts
│ │ │ │  or DepthOfSearch, instead.
│ │ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ │ - -- used 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ │ + -- used 2.17893s (cpu); 1.53593s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ │ + -- used 1.23762s (cpu); 0.970297s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ │ + -- used 1.08731s (cpu); 0.795003s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o29 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o29 : QQ
│ │ │ │  As seen in several examples above, when the exact answer is not found, a list
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html
│ │ │ @@ -197,23 +197,23 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ + -- used 2.17893s (cpu); 1.53593s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ + -- used 1.23762s (cpu); 0.970297s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ + -- used 1.08731s (cpu); 0.795003s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
    
│ │ │                  		
│ │ │                
    i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
    
│ │ │              
    
│ │ │                
    i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ + -- used 0.00585223s (cpu); 0.00585268s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ + -- used 0.407935s (cpu); 0.322993s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The valid values for the option ContainmentTest are FrobeniusPower, FrobeniusRoot, and StandardPower. The default value of this option depends on what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to StandardPower if frobeniusNu is passed an ideal $I$. We describe the consequences of setting ContainmentTest to each of these values below.
    
│ │ │ @@ -230,23 +230,23 @@
│ │ │              
│ │ │                
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ + -- used 0.367506s (cpu); 0.275686s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ + -- used 1.74408s (cpu); 1.36429s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Finally, when ContainmentTest is set to FrobeniusPower, then instead of producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as defined in the paper Frobenius Powers by Hernández, Teixeira, and Witt, which can be computed with the function frobeniusPower, from the TestIdeals package. In particular, frobeniusNu(e,I,J) and frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower) need not agree. However, they will agree when $I$ is a principal ideal.
    
│ │ │ @@ -292,46 +292,46 @@
│ │ │              
│ │ │                
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ + -- used 0.983969s (cpu); 0.793997s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ + -- used 1.51361s (cpu); 1.16685s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ + -- used 2.02513s (cpu); 1.65145s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ + -- used 0.633252s (cpu); 0.543155s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The option AtOrigin (default value true) can be turned off to tell frobeniusNu to effectively do the computation over all possible maximal ideals $J$ and take the minimum.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  algorithms, namely diagonal polynomials, binomials, forms in two variables, and
│ │ │ │  polynomials whose factors are in simple normal crossing. This feature can be
│ │ │ │  disabled by setting the option UseSpecialAlgorithms (default value true) to
│ │ │ │  false.
│ │ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │ │  i15 : time frobeniusNu(3, f)
│ │ │ │ - -- used 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ │ + -- used 0.00585223s (cpu); 0.00585268s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ │ + -- used 0.407935s (cpu); 0.322993s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 3756
│ │ │ │  The valid values for the option ContainmentTest are FrobeniusPower,
│ │ │ │  FrobeniusRoot, and StandardPower. The default value of this option depends on
│ │ │ │  what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to
│ │ │ │  FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to
│ │ │ │  StandardPower if frobeniusNu is passed an ideal $I$. We describe the
│ │ │ │ @@ -133,19 +133,19 @@
│ │ │ │  is contained in $J$. The output is unaffected, but this option often speeds up
│ │ │ │  computations, specially when a polynomial or principal ideal is passed as the
│ │ │ │  second argument.
│ │ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by
│ │ │ │  default
│ │ │ │ - -- used 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ │ + -- used 0.367506s (cpu); 0.275686s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ │ + -- used 1.74408s (cpu); 1.36429s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = 499
│ │ │ │  Finally, when ContainmentTest is set to FrobeniusPower, then instead of
│ │ │ │  producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead
│ │ │ │  outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius
│ │ │ │  power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the
│ │ │ │  $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as
│ │ │ │ @@ -167,31 +167,31 @@
│ │ │ │  The function frobeniusNu works by searching through the list of potential
│ │ │ │  integers $n$ and checking containments of $I^n$ in the specified Frobenius
│ │ │ │  power of $J$. The way this search is approached is specified by the option
│ │ │ │  Search, which can be set to Binary (the default value) or Linear.
│ │ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ │ - -- used 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ │ + -- used 0.983969s (cpu); 0.793997s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ │ + -- used 1.51361s (cpu); 1.16685s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 1124
│ │ │ │  i29 : M = ideal(x, y, z);
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ │ - -- used 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ │ + -- used 2.02513s (cpu); 1.65145s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ │ + -- used 0.633252s (cpu); 0.543155s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 97
│ │ │ │  The option AtOrigin (default value true) can be turned off to tell frobeniusNu
│ │ │ │  to effectively do the computation over all possible maximal ideals $J$ and take
│ │ │ │  the minimum.
│ │ │ │  i32 : R = ZZ/7[x,y];
│ │ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  YXJjcw==
│ │ │  #:len=3089
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJpbnRzIG5vZGUgbGFiZWxzIGZvciB0
│ │ │  aGUgZGVzaW5ndWxhcml6YXRpb24gdHJlZSIsICJsaW5lbnVtIiA9PiA2NTIsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  UlJFRk1ldGhvZA==
│ │ │  #:len=209
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjEzMCwgInVuZG9jdW1lbnRlZCIgPT4g
│ │ │  dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiUlJFRk1l
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out
│ │ │ @@ -208,21 +208,21 @@
│ │ │        | 3/7 5/4 3/7  10   |
│ │ │        | 6/7 2/9 5    3/2  |
│ │ │  
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
│ │ │  
│ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.68501s (cpu); 0.426849s (thread); 0s (gc)
│ │ │ + -- used 1.99181s (cpu); 0.527732s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc)
│ │ │ + -- used 2.68323s (cpu); 1.13926s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
│ │ │  
│ │ │  i29 :
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html
│ │ │ @@ -391,25 +391,25 @@
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.68501s (cpu); 0.426849s (thread); 0s (gc)
│ │ │ + -- used 1.99181s (cpu); 0.527732s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc)
│ │ │ + -- used 2.68323s (cpu); 1.13926s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -241,21 +241,21 @@
│ │ │ │  o26 = | 7   6   3/10 10/9 |
│ │ │ │        | 3/7 5/4 3/7  10   |
│ │ │ │        | 6/7 2/9 5    3/2  |
│ │ │ │  
│ │ │ │                 3       4
│ │ │ │  o26 : Matrix QQ  <-- QQ
│ │ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ │ - -- used 0.68501s (cpu); 0.426849s (thread); 0s (gc)
│ │ │ │ + -- used 1.99181s (cpu); 0.527732s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = an "equivariant K-class" on a GKM variety
│ │ │ │  
│ │ │ │  o27 : KClass
│ │ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc)
│ │ │ │ + -- used 2.68323s (cpu); 1.13926s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = an "equivariant K-class" on a GKM variety
│ │ │ │  
│ │ │ │  o28 : KClass
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_e_n_e_r_a_l_i_z_e_d_F_l_a_g_V_a_r_i_e_t_y -- makes a generalized flag variety as a GKM
│ │ │ │        variety
│ │ ├── ./usr/share/doc/Macaulay2/GameTheory/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  cHJvYmFiaWxpdHlSaW5nKExpc3Qp
│ │ │  #:len=262
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU0Niwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocHJvYmFiaWxpdHlSaW5nLExpc3QpLCJwcm9iYWJp
│ │ ├── ./usr/share/doc/Macaulay2/GenericInitialIdeal/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  Z2luKElkZWFsKQ==
│ │ │  #:len=242
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTczLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnaW4sSWRlYWwpLCJnaW4oSWRlYWwpIiwiR2VuZXJp
│ │ ├── ./usr/share/doc/Macaulay2/GeometricDecomposability/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  SXNJZGVhbFVubWl4ZWQ=
│ │ │  #:len=1529
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3BlY2lmeSB3aGV0aGVyIGFuIGlkZWFs
│ │ │  IGlzIHVubWl4ZWQiLCAibGluZW51bSIgPT4gMTkwNCwgU2VlQWxzbyA9PiBESVZ7SEVBREVSMnsi
│ │ ├── ./usr/share/doc/Macaulay2/GradedLieAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  WlogXyBFeHRBbGdlYnJh
│ │ │  #:len=932
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2V0IHRoZSB6ZXJvIGVsZW1lbnQiLCBE
│ │ │  ZXNjcmlwdGlvbiA9PiAoRElWe0hFQURFUjJ7IlN5bm9wc2lzIn0sVUx7TEl7REx7ImNsYXNzIiA9
│ │ ├── ./usr/share/doc/Macaulay2/GraphicalModels/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  VmFyaWFibGVOYW1l
│ │ │  #:len=616
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl
│ │ │  IGluZGV0ZXJtaW5hdGUgbmFtZSBpbiBtYXJrb3ZSaW5nIiwgImxpbmVudW0iID0+IDE4ODgsIFNl
│ │ ├── ./usr/share/doc/Macaulay2/GraphicalModelsMLE/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  c29sdmVyTUxFKExpc3QsR3JhcGgp
│ │ │  #:len=274
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk1OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc29sdmVyTUxFLExpc3QsR3JhcGgpLCJzb2x2ZXJN
│ │ ├── ./usr/share/doc/Macaulay2/Graphics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  cGljdHVyZVpvbmUoU3BoZXJlKQ==
│ │ │  #:len=747
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZCB0aGUgem9uZSB0aGF0IGNvbnRh
│ │ │  aW5zIHRoZSBzcGhlcmUiLCAibGluZW51bSIgPT4gMTExNywgSW5wdXRzID0+IHtTUEFOe1RUeyJz
│ │ ├── ./usr/share/doc/Macaulay2/Graphs/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  cmV2ZXJzZUJyZWFkdGhGaXJzdFNlYXJjaA==
│ │ │  #:len=1565
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicnVucyBhIHJldmVyc2UgYnJlYWR0aCBm
│ │ │  aXJzdCBzZWFyY2ggb24gdGhlIGRpZ3JhcGggc3RhcnRpbmcgYXQgYSBzcGVjaWZpZWQgbm9kZSIs
│ │ ├── ./usr/share/doc/Macaulay2/Graphs/example-output/_new__Digraph.out
│ │ │ @@ -32,12 +32,12 @@
│ │ │                                           5 => {6}
│ │ │                                           6 => {}
│ │ │  
│ │ │  o2 : SortedDigraph
│ │ │  
│ │ │  i3 : keys H
│ │ │  
│ │ │ -o3 = {digraph, map, newDigraph}
│ │ │ +o3 = {map, newDigraph, digraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o2 : SortedDigraph
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : keys H
│ │ │  
│ │ │ -o3 = {digraph, map, newDigraph}
│ │ │ +o3 = {map, newDigraph, digraph}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │                                           4 => {}
│ │ │ │                                           5 => {6}
│ │ │ │                                           6 => {}
│ │ │ │  
│ │ │ │  o2 : SortedDigraph
│ │ │ │  i3 : keys H
│ │ │ │  
│ │ │ │ -o3 = {digraph, map, newDigraph}
│ │ │ │ +o3 = {map, newDigraph, digraph}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_o_p_S_o_r_t -- topologically sort the vertices of a digraph
│ │ │ │      * _S_o_r_t_e_d_D_i_g_r_a_p_h -- hashtable used in topSort
│ │ │ │      * _t_o_p_o_l_o_g_i_c_a_l_S_o_r_t -- outputs a list of vertices in a topologically sorted
│ │ │ │        order of a DAG.
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZmluZFdlaWdodENvbnN0cmFpbnRz
│ │ │  #:len=2630
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyBhIG1hdHJpeCBvZiB3ZWln
│ │ │  aHQgY29uc3RyYWludHMiLCAibGluZW51bSIgPT4gNjkxLCBJbnB1dHMgPT4ge1NQQU57VFR7Ik0i
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out
│ │ │ @@ -243,67 +243,72 @@
│ │ │        |      2                    2                                              2  2                                                                     2                 2         2  2               2       2  2  2 |
│ │ │        |t  - t   + t  t  t   - t  t   + t  t  t   + t  t  t   - t  t  t  t   + 25t  t   + 4t  t  t   - 2t  t  t  t   - 2t  t  t  t   - 2t  t  t  t   + t  t  t  t   + t  t  t  t   - 3t  t   + 3t  t  t  t   - 26t  t  t  |
│ │ │        | 6    23    16 20 22    14 22    23 22 13    23 16 19    16 22 13 19      16 19     23 20 21     20 22 13 21     16 20 19 21     23 13 19 21    22 13 19 21    16 13 19 21     20 21     20 13 19 21      13 19 21|
│ │ │        +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i12 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10
│ │ │ +o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11
│ │ │        -----------------------------------------------------------------------
│ │ │ -      11 19 19 24 -29 |
│ │ │ +      -30 -29 24 -29 |
│ │ │  
│ │ │                 1       24
│ │ │  o12 : Matrix kk  <-- kk
│ │ │  
│ │ │  i13 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38
│ │ │ +o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -16 0 -24 34 |
│ │ │ +      0 -24 34 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
│ │ │  
│ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  + 11b*c + 39c  - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c -
│ │ │ +              2             2                            2               
│ │ │ +o14 = ideal (a  - 12b*c - 4c  - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                  
│ │ │ -      10c  - 29a*d - 48b*d + 28c*d + 6d , b  + 19b*c - 8c  + 19a*d - 22b*d -
│ │ │ +         2                              2   2              2                
│ │ │ +      23c  + 19a*d + 38b*d + 20c*d - 19d , b  - 30b*c - 22c  - 29a*d - 29b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                              2
│ │ │ -      36c*d - 46d , a*c + 24b*c - 10c  - 29a*d + 11b*d - 30c*d - 12d )
│ │ │ +                   2                   2                              2
│ │ │ +      - 36c*d + 50d , a*c + 24b*c + 19c  - 29a*d + 11b*d - 10c*d - 34d )
│ │ │  
│ │ │  o14 : Ideal of S
│ │ │  
│ │ │  i15 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o15 = ideal (a  + 29b*c + 30c  + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c -
│ │ │ +              2              2                           2               
│ │ │ +o15 = ideal (a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      49c  - 4a*d + 11b*d + 32c*d - 39d , b  - 16b*c + 19c  - 47b*d + 43c*d +
│ │ │ +         2                           2   2              2                  
│ │ │ +      17c  + 3a*d - 2b*d - 8c*d - 29d , b  + 39b*c - 47c  + 19b*d - 49c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │           2                   2                              2
│ │ │ -      17d , a*c - 24b*c + 39c  + 34a*d - 38b*d + 21c*d - 12d )
│ │ │ +      47d , a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d )
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │  
│ │ │  i16 : decompose F1
│ │ │  
│ │ │ -                                    2             2                      2
│ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b  + 19b*c - 8c  + 27b*d - 48c*d - 26d ),
│ │ │ +                                    2              2                      2
│ │ │ +o16 = {ideal (a + 24b + 19c + 36d, b  - 30b*c - 22c  - 40b*d + 10c*d - 17d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 29d, b + 23d, a - 14d)}
│ │ │ +      ideal (c - 29d, b + 30d, a - 38d)}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : decompose F2
│ │ │  
│ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c -
│ │ │ +                               2                              2   2          
│ │ │ +o17 = {ideal (a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      9d)}
│ │ │ +         2                      2                   2                       
│ │ │ +      47c  + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c  + 3a*d - 2b*d - 8c*d -
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         2   2              2                           2
│ │ │ +      29d , a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d )}
│ │ │  
│ │ │  o17 : List
│ │ │  
│ │ │  i18 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out
│ │ │ @@ -103,46 +103,46 @@
│ │ │  
│ │ │  i13 : #compsJ
│ │ │  
│ │ │  o13 = 2
│ │ │  
│ │ │  i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | -48 -21 -28 -32 16 31 15 -12 14 5 -13 29 -40 -22 2 30 25 -10 3 -1 3
│ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -30 19 -16 -24 -29 -29 24 -29 -24 -36 21 -8 -38 19 39 |
│ │ │ +      24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
│ │ │  
│ │ │  i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 6 29 -50 41 50 -19 -15 -34 -28 -44 21 -45 -23 -13 -23 25 31 -18 45 22
│ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -40 17 -47 38 -25 -43 19 26 -47 -15 34 16 0 -28 -39 2 |
│ │ │ +      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
│ │ │  
│ │ │  i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2            2                              2               
│ │ │ -o16 = ideal (a  + 2b*c + 5c  + 29a*d + 31b*d - 28c*d - 48d , a*b - 24b*c +
│ │ │ +              2              2                              2               
│ │ │ +o16 = ideal (a  + 31b*c - 17c  + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                             2                   2                
│ │ │ -      3c  + 3a*d - 40b*d + 14c*d - 21d , a*c + 21b*c - 29c  + 24a*d - b*d -
│ │ │ +       2                             2                   2                  
│ │ │ +      c  + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c  + 39a*d - 18b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                             2     2  
│ │ │ -      22c*d + 16d , b  + 19b*c - 29c  - 8a*d - 30b*d + 30c*d + 15d , b*c  -
│ │ │ +                 2   2              2                              2     2  
│ │ │ +      10c*d - 14d , b  - 29b*c + 19c  - 29a*d + 24b*d + 22c*d + 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      24b*c*d + 19c d - 29a*d  + 25b*d  - 13c*d  - 32d , c  + 39b*c*d - 36c d
│ │ │ +                  2         2      2        2      3   3                2   
│ │ │ +      8b*c*d - 30c d - 36a*d  + b*d  - 14c*d  - 39d , c  - 38b*c*d + 21c d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 38a*d  - 16b*d  - 10c*d  - 12d )
│ │ │ +           2        2        2     3
│ │ │ +      22a*d  - 24b*d  + 19c*d  - 2d )
│ │ │  
│ │ │  o16 : Ideal of S
│ │ │  
│ │ │  i17 : betti res F1
│ │ │  
│ │ │               0 1 2 3
│ │ │  o17 = total: 1 6 8 3
│ │ │ @@ -150,28 +150,28 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o17 : BettiTally
│ │ │  
│ │ │  i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o18 = ideal (a  - 23b*c - 44c  - 45a*d - 19b*d - 50c*d + 6d , a*b - 25b*c +
│ │ │ +              2              2                              2               
│ │ │ +o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                   2                
│ │ │ -      45c  - 40a*d - 23b*d - 28c*d + 29d , a*c + 16b*c - 43c  + 26a*d + 22b*d
│ │ │ +         2                              2                  2                
│ │ │ +      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2     2  
│ │ │ -      - 13c*d + 50d , b  - 39b*c - 47c  + 17b*d + 25c*d - 15d , b*c  -
│ │ │ +                   2   2              2                   2     2            
│ │ │ +      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3               2 
│ │ │ -      15b*c*d - 47c d + 19a*d  + 31b*d  + 21c*d  + 41d , c  + 2b*c*d + 34c d
│ │ │ +         2         2        2        2      3   3                2         2
│ │ │ +      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 28a*d  + 38b*d  - 18c*d  - 34d )
│ │ │ +             2        2      3
│ │ │ +      - 47b*d  - 39c*d  + 40d )
│ │ │  
│ │ │  o18 : Ideal of S
│ │ │  
│ │ │  i19 : betti res F2
│ │ │  
│ │ │               0 1 2 3
│ │ │  o19 = total: 1 6 8 3
│ │ │ @@ -179,27 +179,24 @@
│ │ │            1: . 4 4 1
│ │ │            2: . 2 4 2
│ │ │  
│ │ │  o19 : BettiTally
│ │ │  
│ │ │  i20 : netList decompose F1
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 49d, b - 6d, a - 15d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 24d, b + 37d, a + 35d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 18c - 32d, a + 46c + 33d, c  - 26c*d - 15d )|
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 37c - 27d, a + 41c + 29d, c  + 15c*d + 16d )|
│ │ │ -      +-------------------------------------------------------+
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d)                                                                 |
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +      |ideal (c + 39d, b - 23d, a + 34d)                                                               |
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     2                      2   2      2                      2 |
│ │ │ +      |ideal (a - 16b - 29c + 10d, b*c - 38c  + 23b*d + 43c*d - 16d , b  + 28c  + 25b*d + 24c*d - 38d )|
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i21 : netList decompose F2
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                        2                              2   2              2                      2                   2                              2   2              2                             2   3               2         2        2        2      3     2                2         2        2        2      3 |
│ │ │ -o21 = |ideal (a*c + 16b*c - 43c  + 26a*d + 22b*d - 13c*d + 50d , b  - 39b*c - 47c  + 17b*d + 25c*d - 15d , a*b - 25b*c + 45c  - 40a*d - 23b*d - 28c*d + 29d , a  - 23b*c - 44c  - 45a*d - 19b*d - 50c*d + 6d , c  + 2b*c*d + 34c d - 28a*d  + 38b*d  - 18c*d  - 34d , b*c  - 15b*c*d - 47c d + 19a*d  + 31b*d  + 21c*d  + 41d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                       2                              2   2              2                   2                   2                              2   2              2                              2   3                2         2        2        2      3     2                2         2        2        2      3 |
│ │ │ +o21 = |ideal (a*c + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  - 47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d )|
│ │ │ +      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │                                               2
│ │ │       c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : pt = randomPointOnRationalVariety I
│ │ │  
│ │ │ -o4 = | -25 20 -30 -16 24 -36 |
│ │ │ +o4 = | 1 49 24 -23 -36 -30 |
│ │ │  
│ │ │                1       6
│ │ │  o4 : Matrix kk  <-- kk
│ │ │  
│ │ │  i5 : sub(I, pt) == 0
│ │ │  
│ │ │  o5 = true
│ │ │ @@ -200,67 +200,72 @@
│ │ │  
│ │ │  o12 = {11, 8}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22
│ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -24 -29 -38 19 -16 |
│ │ │ +      19 -10 -29 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
│ │ │  
│ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  - 45b*c + 49c  + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c -
│ │ │ +              2              2                              2              
│ │ │ +o14 = ideal (a  + 12b*c - 46c  - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                 
│ │ │ -      32c  - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c  - 38a*d - 9b*d -
│ │ │ +         2                             2                   2                
│ │ │ +      21c  - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c  - 29a*d - 13b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2
│ │ │ -      10c*d + 38d , b  + 19b*c - 22c  - 16a*d - 24b*d - 29c*d - 46d )
│ │ │ +                   2   2             2                             2
│ │ │ +      + 19c*d - 23d , b  - 29b*c - 8c  - 24a*d + 19b*d - 29c*d - 6d )
│ │ │  
│ │ │  o14 : Ideal of S
│ │ │  
│ │ │  i15 : decompose F1
│ │ │  
│ │ │ -                                  2              2                      2
│ │ │ -o15 = {ideal (a - 29b - 29c - d, b  + 19b*c - 22c  + 17b*d + 12c*d + 39d ),
│ │ │ +                                    2             2                     2
│ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b  - 29b*c - 8c  - 19b*d - 9c*d + 20d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 38d, b - 49d, a + 41d)}
│ │ │ +      ideal (c - 29d, b + 16d, a - 20d)}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39
│ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -13 -18 34 0 |
│ │ │ +      34 19 39 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
│ │ │  
│ │ │  i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                             2               
│ │ │ -o17 = ideal (a  - 7b*c + 30c  - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c -
│ │ │ +              2              2                             2               
│ │ │ +o17 = ideal (a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                
│ │ │ -      5c  + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c  - 18a*d + 21b*d
│ │ │ +         2                             2                   2                
│ │ │ +      18c  + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c  + 19a*d - 13b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2
│ │ │ -      + 19c*d - 20d , b  + 34b*c - 47c  - 39b*d + 20c*d - 35d )
│ │ │ +                  2   2              2                    2
│ │ │ +      - 39c*d - 9d , b  + 39b*c - 47c  + 21b*d + 31c*d - d )
│ │ │  
│ │ │  o17 : Ideal of S
│ │ │  
│ │ │  i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c -
│ │ │ +                               2                             2   2          
│ │ │ +o18 = {ideal (a*c + 34b*c - 18c  + 19a*d - 13b*d - 39c*d - 9d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      5d)}
│ │ │ +         2                    2                   2                         
│ │ │ +      47c  + 21b*d + 31c*d - d , a*b - 31b*c - 18c  + 44a*d + 21b*d + 6c*d +
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         2   2              2                             2
│ │ │ +      21d , a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d )}
│ │ │  
│ │ │  o18 : List
│ │ │  
│ │ │  i19 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.out
│ │ │ @@ -68,54 +68,54 @@
│ │ │  
│ │ │  o12 = {11, 8}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │  
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +o13 = || 42 -50 -50 38 -39 6 -1 47 49 -18 18 -28 -47 19 48 34 -13 11 -16 -38 39 21 -43 -39 |    |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || -44 -31 -9 -21 -42 -36 -47 -20 38 -34 -3 -43 22 16 -35 2 -48 32 -28 -15 -47 38 -47 45 ||
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 45 -29 -1 -4 42 -35 4 -13 18 -17 1 21 39 -23 50 15 -11 -11 19 47 -16 7 48 43 |         |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 35 -44 -33 -8 21 -2 -44 -20 19 -28 19 27 11 40 34 33 1 -14 35 36 11 -38 -3 46 |        |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 33 47 46 16 -22 -25 -44 -36 -30 -37 30 -25 -47 29 -41 2 -13 -41 -47 22 -23 -7 -10 15 | |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 43 1 -41 23 -42 -14 37 -50 -32 -20 -5 -49 -9 32 -18 -22 24 43 -18 30 39 27 -30 -32 |   |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 12 8 26 15 22 12 0 -5 6 17 -21 -18 -33 -49 -19 33 -20 0 -15 -48 39 0 44 -19 |          |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || -48 -27 -8 -33 -35 -16 -31 -44 -46 -49 -21 -3 -26 13 -40 4 -11 -48 36 -39 9 -39 -8 22 ||
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 34 29 30 -1 34 -22 32 -6 39 -28 -15 37 41 -30 47 -22 -6 -7 -8 43 36 -3 35 16 |         |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 10 -5 -38 -25 -21 21 14 42 19 -41 8 -35 25 -31 29 3 -49 38 -35 -9 6 40 -13 -2 |        |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +
│ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ +
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -o13 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 -39 -47 39 -43 |  |
│ │ │ +o14 = || 1 4 44 1 40 20 29 -26 42 -16 -4 50 -40 48 23 -34 -35 37 30 4 -47 27 -31 0 |           |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45 22 -47 -47 |     |
│ │ │ +      || -29 -11 2 28 -12 -21 -30 20 -1 -42 27 -45 -48 -14 -3 -45 -37 30 -31 -39 -48 -29 47 0 ||
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 -16 48 |          |
│ │ │ +      || -20 14 48 3 14 33 18 -46 24 45 -1 -29 1 31 -44 -2 -22 40 28 -49 -18 46 10 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46 11 11 -3 |       |
│ │ │ +      || 21 25 -23 -45 -48 -40 8 -7 -6 -1 14 3 -13 -44 -44 21 -41 3 30 7 13 -17 8 0 |          |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47 -23 -10 |      |
│ │ │ +      || -15 26 -17 12 0 -5 35 -11 0 -31 -39 40 49 -21 -42 28 42 -18 -29 8 30 -46 23 0 |       |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27 -32 -9 39 -30 ||
│ │ │ +      || -17 38 -49 -9 2 -1 -9 2 3 49 33 49 -46 25 1 0 -18 12 15 -28 18 -16 27 0 |             |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 -19 -33 39 44 |   |
│ │ │ +      || -50 3 -12 -21 -10 4 36 -14 19 -9 43 -48 44 -35 16 6 20 -39 23 -21 -37 -23 19 0 |      |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 -26 9 -8 |       |
│ │ │ +      || 36 -17 -27 37 -4 12 42 -16 -15 12 -3 -14 -28 38 -24 18 -9 6 -47 0 -28 47 -33 0 |      |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16 41 36 35 |       |
│ │ │ +      || 43 44 31 14 -9 -4 -4 -40 -48 45 47 17 -37 29 -9 10 -28 -33 -29 28 26 5 42 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 -2 25 6 -13 |      |
│ │ │ +      || 17 -47 -23 8 -18 -45 -25 2 44 41 -19 -37 5 16 2 38 -13 -20 30 44 4 22 -29 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │  
│ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ -
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html
│ │ │ @@ -237,52 +237,52 @@
│ │ │  o13 = 2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | -48 -21 -28 -32 16 31 15 -12 14 5 -13 29 -40 -22 2 30 25 -10 3 -1 3
│ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -30 19 -16 -24 -29 -29 24 -29 -24 -36 21 -8 -38 19 39 |
│ │ │ +      24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 6 29 -50 41 50 -19 -15 -34 -28 -44 21 -45 -23 -13 -23 25 31 -18 45 22
│ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -40 17 -47 38 -25 -43 19 26 -47 -15 34 16 0 -28 -39 2 |
│ │ │ +      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2            2                              2               
│ │ │ -o16 = ideal (a  + 2b*c + 5c  + 29a*d + 31b*d - 28c*d - 48d , a*b - 24b*c +
│ │ │ +              2              2                              2               
│ │ │ +o16 = ideal (a  + 31b*c - 17c  + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                             2                   2                
│ │ │ -      3c  + 3a*d - 40b*d + 14c*d - 21d , a*c + 21b*c - 29c  + 24a*d - b*d -
│ │ │ +       2                             2                   2                  
│ │ │ +      c  + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c  + 39a*d - 18b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                             2     2  
│ │ │ -      22c*d + 16d , b  + 19b*c - 29c  - 8a*d - 30b*d + 30c*d + 15d , b*c  -
│ │ │ +                 2   2              2                              2     2  
│ │ │ +      10c*d - 14d , b  - 29b*c + 19c  - 29a*d + 24b*d + 22c*d + 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      24b*c*d + 19c d - 29a*d  + 25b*d  - 13c*d  - 32d , c  + 39b*c*d - 36c d
│ │ │ +                  2         2      2        2      3   3                2   
│ │ │ +      8b*c*d - 30c d - 36a*d  + b*d  - 14c*d  - 39d , c  - 38b*c*d + 21c d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 38a*d  - 16b*d  - 10c*d  - 12d )
│ │ │ +           2        2        2     3
│ │ │ +      22a*d  - 24b*d  + 19c*d  - 2d )
│ │ │  
│ │ │  o16 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : betti res F1
│ │ │ @@ -296,28 +296,28 @@
│ │ │  o17 : BettiTally
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o18 = ideal (a  - 23b*c - 44c  - 45a*d - 19b*d - 50c*d + 6d , a*b - 25b*c +
│ │ │ +              2              2                              2               
│ │ │ +o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                   2                
│ │ │ -      45c  - 40a*d - 23b*d - 28c*d + 29d , a*c + 16b*c - 43c  + 26a*d + 22b*d
│ │ │ +         2                              2                  2                
│ │ │ +      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2     2  
│ │ │ -      - 13c*d + 50d , b  - 39b*c - 47c  + 17b*d + 25c*d - 15d , b*c  -
│ │ │ +                   2   2              2                   2     2            
│ │ │ +      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3               2 
│ │ │ -      15b*c*d - 47c d + 19a*d  + 31b*d  + 21c*d  + 41d , c  + 2b*c*d + 34c d
│ │ │ +         2         2        2        2      3   3                2         2
│ │ │ +      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d 
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 28a*d  + 38b*d  - 18c*d  - 34d )
│ │ │ +             2        2      3
│ │ │ +      - 47b*d  - 39c*d  + 40d )
│ │ │  
│ │ │  o18 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : betti res F2
│ │ │ @@ -336,35 +336,32 @@
│ │ │            
    What are the ideals F1 and F2?
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : netList decompose F1
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 49d, b - 6d, a - 15d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 24d, b + 37d, a + 35d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 18c - 32d, a + 46c + 33d, c  - 26c*d - 15d )|
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 37c - 27d, a + 41c + 29d, c  + 15c*d + 16d )|
│ │ │ -      +-------------------------------------------------------+
    
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d)                                                                 |
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +      |ideal (c + 39d, b - 23d, a + 34d)                                                               |
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │ +      |                                     2                      2   2      2                      2 |
│ │ │ +      |ideal (a - 16b - 29c + 10d, b*c - 38c  + 23b*d + 43c*d - 16d , b  + 28c  + 25b*d + 24c*d - 38d )|
│ │ │ +      +------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │                
    i21 : netList decompose F2
│ │ │  
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ -      |                        2                              2   2              2                      2                   2                              2   2              2                             2   3               2         2        2        2      3     2                2         2        2        2      3 |
│ │ │ -o21 = |ideal (a*c + 16b*c - 43c  + 26a*d + 22b*d - 13c*d + 50d , b  - 39b*c - 47c  + 17b*d + 25c*d - 15d , a*b - 25b*c + 45c  - 40a*d - 23b*d - 28c*d + 29d , a  - 23b*c - 44c  - 45a*d - 19b*d - 50c*d + 6d , c  + 2b*c*d + 34c d - 28a*d  + 38b*d  - 18c*d  - 34d , b*c  - 15b*c*d - 47c d + 19a*d  + 31b*d  + 21c*d  + 41d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    
│ │ │ +      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ +      |                       2                              2   2              2                   2                   2                              2   2              2                              2   3                2         2        2        2      3     2                2         2        2        2      3 |
│ │ │ +o21 = |ideal (a*c + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  - 47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d )|
│ │ │ +      +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │              
    
│ │ │          
    
│ │ │            
    We can determine what these represent.  One should be a set of 6 points, where 5 lie on a plane.  The other should be 6 points with 3 points on one line, and the other 3 points on a skew line.
    
│ │ │          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -170,117 +170,122 @@
│ │ │ │  32   13   21   33   19   31
│ │ │ │  i12 : compsJ = decompose J;
│ │ │ │  i13 : #compsJ
│ │ │ │  
│ │ │ │  o13 = 2
│ │ │ │  i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │ │  
│ │ │ │ -o14 = | -48 -21 -28 -32 16 31 15 -12 14 5 -13 29 -40 -22 2 30 25 -10 3 -1 3
│ │ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -30 19 -16 -24 -29 -29 24 -29 -24 -36 21 -8 -38 19 39 |
│ │ │ │ +      24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o14 : Matrix kk  <-- kk
│ │ │ │  i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │ │  
│ │ │ │ -o15 = | 6 29 -50 41 50 -19 -15 -34 -28 -44 21 -45 -23 -13 -23 25 31 -18 45 22
│ │ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -40 17 -47 38 -25 -43 19 26 -47 -15 34 16 0 -28 -39 2 |
│ │ │ │ +      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │ │  
│ │ │ │                 1       36
│ │ │ │  o15 : Matrix kk  <-- kk
│ │ │ │  i16 : F1 = sub(F, (vars S)|pt1)
│ │ │ │  
│ │ │ │ -              2            2                              2
│ │ │ │ -o16 = ideal (a  + 2b*c + 5c  + 29a*d + 31b*d - 28c*d - 48d , a*b - 24b*c +
│ │ │ │ +              2              2                              2
│ │ │ │ +o16 = ideal (a  + 31b*c - 17c  + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                             2                   2
│ │ │ │ -      3c  + 3a*d - 40b*d + 14c*d - 21d , a*c + 21b*c - 29c  + 24a*d - b*d -
│ │ │ │ +       2                             2                   2
│ │ │ │ +      c  + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c  + 39a*d - 18b*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2   2              2                             2     2
│ │ │ │ -      22c*d + 16d , b  + 19b*c - 29c  - 8a*d - 30b*d + 30c*d + 15d , b*c  -
│ │ │ │ +                 2   2              2                              2     2
│ │ │ │ +      10c*d - 14d , b  - 29b*c + 19c  - 29a*d + 24b*d + 22c*d + 35d , b*c  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2        2        2      3   3                2
│ │ │ │ -      24b*c*d + 19c d - 29a*d  + 25b*d  - 13c*d  - 32d , c  + 39b*c*d - 36c d
│ │ │ │ +                  2         2      2        2      3   3                2
│ │ │ │ +      8b*c*d - 30c d - 36a*d  + b*d  - 14c*d  - 39d , c  - 38b*c*d + 21c d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2        2      3
│ │ │ │ -      - 38a*d  - 16b*d  - 10c*d  - 12d )
│ │ │ │ +           2        2        2     3
│ │ │ │ +      22a*d  - 24b*d  + 19c*d  - 2d )
│ │ │ │  
│ │ │ │  o16 : Ideal of S
│ │ │ │  i17 : betti res F1
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o17 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o17 : BettiTally
│ │ │ │  i18 : F2 = sub(F, (vars S)|pt2)
│ │ │ │  
│ │ │ │ -              2              2                             2
│ │ │ │ -o18 = ideal (a  - 23b*c - 44c  - 45a*d - 19b*d - 50c*d + 6d , a*b - 25b*c +
│ │ │ │ +              2              2                              2
│ │ │ │ +o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                              2                   2
│ │ │ │ -      45c  - 40a*d - 23b*d - 28c*d + 29d , a*c + 16b*c - 43c  + 26a*d + 22b*d
│ │ │ │ +         2                              2                  2
│ │ │ │ +      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2   2              2                      2     2
│ │ │ │ -      - 13c*d + 50d , b  - 39b*c - 47c  + 17b*d + 25c*d - 15d , b*c  -
│ │ │ │ +                   2   2              2                   2     2
│ │ │ │ +      - 18c*d - 39d , b  - 47b*c - 28c  - 5b*d - c*d - 30d , b*c  - 43b*c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2         2        2        2      3   3               2
│ │ │ │ -      15b*c*d - 47c d + 19a*d  + 31b*d  + 21c*d  + 41d , c  + 2b*c*d + 34c d
│ │ │ │ +         2         2        2        2      3   3                2         2
│ │ │ │ +      19c d + 34a*d  + 21b*d  + 46c*d  + 20d , c  + 38b*c*d + 22c d - 15a*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -             2        2        2      3
│ │ │ │ -      - 28a*d  + 38b*d  - 18c*d  - 34d )
│ │ │ │ +             2        2      3
│ │ │ │ +      - 47b*d  - 39c*d  + 40d )
│ │ │ │  
│ │ │ │  o18 : Ideal of S
│ │ │ │  i19 : betti res F2
│ │ │ │  
│ │ │ │               0 1 2 3
│ │ │ │  o19 = total: 1 6 8 3
│ │ │ │            0: 1 . . .
│ │ │ │            1: . 4 4 1
│ │ │ │            2: . 2 4 2
│ │ │ │  
│ │ │ │  o19 : BettiTally
│ │ │ │  What are the ideals F1 and F2?
│ │ │ │  i20 : netList decompose F1
│ │ │ │  
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ -o20 = |ideal (c - 49d, b - 6d, a - 15d)                       |
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ -      |ideal (c + 24d, b + 37d, a + 35d)                      |
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ -      |                                      2              2 |
│ │ │ │ -      |ideal (b - 18c - 32d, a + 46c + 33d, c  - 26c*d - 15d )|
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ -      |                                      2              2 |
│ │ │ │ -      |ideal (b - 37c - 27d, a + 41c + 29d, c  + 15c*d + 16d )|
│ │ │ │ -      +-------------------------------------------------------+
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +------------------------+
│ │ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d)
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +------------------------+
│ │ │ │ +      |ideal (c + 39d, b - 23d, a + 34d)
│ │ │ │ +|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +------------------------+
│ │ │ │ +      |                                     2                      2   2      2
│ │ │ │ +2 |
│ │ │ │ +      |ideal (a - 16b - 29c + 10d, b*c - 38c  + 23b*d + 43c*d - 16d , b  + 28c
│ │ │ │ ++ 25b*d + 24c*d - 38d )|
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +------------------------+
│ │ │ │  i21 : netList decompose F2
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ----+
│ │ │ │ -      |                        2                              2   2
│ │ │ │ -2                      2                   2                              2   2
│ │ │ │ -2                             2   3               2         2        2        2
│ │ │ │ -3     2                2         2        2        2      3 |
│ │ │ │ -o21 = |ideal (a*c + 16b*c - 43c  + 26a*d + 22b*d - 13c*d + 50d , b  - 39b*c -
│ │ │ │ -47c  + 17b*d + 25c*d - 15d , a*b - 25b*c + 45c  - 40a*d - 23b*d - 28c*d + 29d ,
│ │ │ │ -a  - 23b*c - 44c  - 45a*d - 19b*d - 50c*d + 6d , c  + 2b*c*d + 34c d - 28a*d  +
│ │ │ │ -38b*d  - 18c*d  - 34d , b*c  - 15b*c*d - 47c d + 19a*d  + 31b*d  + 21c*d  + 41d
│ │ │ │ +-+
│ │ │ │ +      |                       2                              2   2
│ │ │ │ +2                   2                   2                              2   2
│ │ │ │ +2                              2   3                2         2        2
│ │ │ │ +2      3     2                2         2        2        2      3 |
│ │ │ │ +o21 = |ideal (a*c + 2b*c - 13c  + 33a*d + 16b*d - 18c*d - 39d , b  - 47b*c -
│ │ │ │ +28c  - 5b*d - c*d - 30d , a*b - 20b*c - 45c  - 35a*d + 21b*d - 34c*d + 10d , a
│ │ │ │ +- 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , c  + 38b*c*d + 22c d - 15a*d  -
│ │ │ │ +47b*d  - 39c*d  + 40d , b*c  - 43b*c*d + 19c d + 34a*d  + 21b*d  + 46c*d  + 20d
│ │ │ │  )|
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │ ----+
│ │ │ │ +-+
│ │ │ │  We can determine what these represent. One should be a set of 6 points, where 5
│ │ │ │  lie on a plane. The other should be 6 points with 3 points on one line, and the
│ │ │ │  other 3 points on a skew line.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y -- find a random point on a variety that can
│ │ │ │        be detected to be rational
│ │ │ │  ********** WWaayyss ttoo uussee nnoonnmmiinniimmaallMMaappss:: **********
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : pt = randomPointOnRationalVariety I
│ │ │  
│ │ │ -o4 = | -25 20 -30 -16 24 -36 |
│ │ │ +o4 = | 1 49 24 -23 -36 -30 |
│ │ │  
│ │ │                1       6
│ │ │  o4 : Matrix kk  <-- kk
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -323,90 +323,95 @@
│ │ │            
    There are 2 components.  We attempt to find a point on the first component
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22
│ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -24 -29 -38 19 -16 |
│ │ │ +      19 -10 -29 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  - 45b*c + 49c  + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c -
│ │ │ +              2              2                              2              
│ │ │ +o14 = ideal (a  + 12b*c - 46c  - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                 
│ │ │ -      32c  - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c  - 38a*d - 9b*d -
│ │ │ +         2                             2                   2                
│ │ │ +      21c  - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c  - 29a*d - 13b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2
│ │ │ -      10c*d + 38d , b  + 19b*c - 22c  - 16a*d - 24b*d - 29c*d - 46d )
│ │ │ +                   2   2             2                             2
│ │ │ +      + 19c*d - 23d , b  - 29b*c - 8c  - 24a*d + 19b*d - 29c*d - 6d )
│ │ │  
│ │ │  o14 : Ideal of S
    
│ │ │              
    
│ │ │                
    i15 : decompose F1
│ │ │  
│ │ │ -                                  2              2                      2
│ │ │ -o15 = {ideal (a - 29b - 29c - d, b  + 19b*c - 22c  + 17b*d + 12c*d + 39d ),
│ │ │ +                                    2             2                     2
│ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b  - 29b*c - 8c  - 19b*d - 9c*d + 20d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 38d, b - 49d, a + 41d)}
│ │ │ +      ideal (c - 29d, b + 16d, a - 20d)}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    We attempt to find a point on the second component in parameter space, and its corresponding ideal.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39
│ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -13 -18 34 0 |
│ │ │ +      34 19 39 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                             2               
│ │ │ -o17 = ideal (a  - 7b*c + 30c  - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c -
│ │ │ +              2              2                             2               
│ │ │ +o17 = ideal (a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                
│ │ │ -      5c  + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c  - 18a*d + 21b*d
│ │ │ +         2                             2                   2                
│ │ │ +      18c  + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c  + 19a*d - 13b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2
│ │ │ -      + 19c*d - 20d , b  + 34b*c - 47c  - 39b*d + 20c*d - 35d )
│ │ │ +                  2   2              2                    2
│ │ │ +      - 39c*d - 9d , b  + 39b*c - 47c  + 21b*d + 31c*d - d )
│ │ │  
│ │ │  o17 : Ideal of S
    
│ │ │              
    
│ │ │                
    i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c -
│ │ │ +                               2                             2   2          
│ │ │ +o18 = {ideal (a*c + 34b*c - 18c  + 19a*d - 13b*d - 39c*d - 9d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      5d)}
│ │ │ +         2                    2                   2                         
│ │ │ +      47c  + 21b*d + 31c*d - d , a*b - 31b*c - 18c  + 44a*d + 21b*d + 6c*d +
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         2   2              2                             2
│ │ │ +      21d , a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d )}
│ │ │  
│ │ │  o18 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    It turns out that this is the ideal of 2 skew lines, just not defined over this field.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                               2
│ │ │ │       c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : pt = randomPointOnRationalVariety I
│ │ │ │  
│ │ │ │ -o4 = | -25 20 -30 -16 24 -36 |
│ │ │ │ +o4 = | 1 49 24 -23 -36 -30 |
│ │ │ │  
│ │ │ │                1       6
│ │ │ │  o4 : Matrix kk  <-- kk
│ │ │ │  i5 : sub(I, pt) == 0
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : S = kk[a..d];
│ │ │ │ @@ -212,67 +212,72 @@
│ │ │ │  
│ │ │ │  o12 = {11, 8}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  There are 2 components. We attempt to find a point on the first component
│ │ │ │  i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │ │  
│ │ │ │ -o13 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22
│ │ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -24 -29 -38 19 -16 |
│ │ │ │ +      19 -10 -29 -29 -24 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o13 : Matrix kk  <-- kk
│ │ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │ │  
│ │ │ │                2              2                              2
│ │ │ │ -o14 = ideal (a  - 45b*c + 49c  + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c -
│ │ │ │ +o14 = ideal (a  + 12b*c - 46c  - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │           2                             2                   2
│ │ │ │ -      32c  - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c  - 38a*d - 9b*d -
│ │ │ │ +      21c  - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c  - 29a*d - 13b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2   2              2                              2
│ │ │ │ -      10c*d + 38d , b  + 19b*c - 22c  - 16a*d - 24b*d - 29c*d - 46d )
│ │ │ │ +                   2   2             2                             2
│ │ │ │ +      + 19c*d - 23d , b  - 29b*c - 8c  - 24a*d + 19b*d - 29c*d - 6d )
│ │ │ │  
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : decompose F1
│ │ │ │  
│ │ │ │ -                                  2              2                      2
│ │ │ │ -o15 = {ideal (a - 29b - 29c - d, b  + 19b*c - 22c  + 17b*d + 12c*d + 39d ),
│ │ │ │ +                                    2             2                     2
│ │ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b  - 29b*c - 8c  - 19b*d - 9c*d + 20d ),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ideal (c - 38d, b - 49d, a + 41d)}
│ │ │ │ +      ideal (c - 29d, b + 16d, a - 20d)}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  We attempt to find a point on the second component in parameter space, and its
│ │ │ │  corresponding ideal.
│ │ │ │  i16 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │ │  
│ │ │ │ -o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39
│ │ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -13 -18 34 0 |
│ │ │ │ +      34 19 39 0 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o16 : Matrix kk  <-- kk
│ │ │ │  i17 : F2 = sub(F, (vars S)|pt2)
│ │ │ │  
│ │ │ │ -              2             2                             2
│ │ │ │ -o17 = ideal (a  - 7b*c + 30c  - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c -
│ │ │ │ +              2              2                             2
│ │ │ │ +o17 = ideal (a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -        2                              2                   2
│ │ │ │ -      5c  + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c  - 18a*d + 21b*d
│ │ │ │ +         2                             2                   2
│ │ │ │ +      18c  + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c  + 19a*d - 13b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                   2   2              2                      2
│ │ │ │ -      + 19c*d - 20d , b  + 34b*c - 47c  - 39b*d + 20c*d - 35d )
│ │ │ │ +                  2   2              2                    2
│ │ │ │ +      - 39c*d - 9d , b  + 39b*c - 47c  + 21b*d + 31c*d - d )
│ │ │ │  
│ │ │ │  o17 : Ideal of S
│ │ │ │  i18 : decompose F2
│ │ │ │  
│ │ │ │ -o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c -
│ │ │ │ +                               2                             2   2
│ │ │ │ +o18 = {ideal (a*c + 34b*c - 18c  + 19a*d - 13b*d - 39c*d - 9d , b  + 39b*c -
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         2                    2                   2
│ │ │ │ +      47c  + 21b*d + 31c*d - d , a*b - 31b*c - 18c  + 44a*d + 21b*d + 6c*d +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      5d)}
│ │ │ │ +         2   2              2                             2
│ │ │ │ +      21d , a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d )}
│ │ │ │  
│ │ │ │  o18 : List
│ │ │ │  It turns out that this is the ideal of 2 skew lines, just not defined over this
│ │ │ │  field.
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This routine expects the input to represent an irreducible variety
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Points__On__Rational__Variety_lp__Ideal_cm__Z__Z_rp.html
│ │ │ @@ -191,64 +191,64 @@
│ │ │            
    There are 2 components.  We attempt to find points on each of these two components. We are successful.  This indicates that the corresponding varieties are both rational. Also, if we can find one point, we can find as many as we want.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │ +          
│ │ │ +          
│ │ │ +            
│ │ │            
│ │ │ -          
│ │ │ -            
│ │ │ -          
│ │ │          
    
│ │ │                
    i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │  
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +o13 = || 42 -50 -50 38 -39 6 -1 47 49 -18 18 -28 -47 19 48 34 -13 11 -16 -38 39 21 -43 -39 |    |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || -44 -31 -9 -21 -42 -36 -47 -20 38 -34 -3 -43 22 16 -35 2 -48 32 -28 -15 -47 38 -47 45 ||
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 45 -29 -1 -4 42 -35 4 -13 18 -17 1 21 39 -23 50 15 -11 -11 19 47 -16 7 48 43 |         |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 35 -44 -33 -8 21 -2 -44 -20 19 -28 19 27 11 40 34 33 1 -14 35 36 11 -38 -3 46 |        |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 33 47 46 16 -22 -25 -44 -36 -30 -37 30 -25 -47 29 -41 2 -13 -41 -47 22 -23 -7 -10 15 | |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 43 1 -41 23 -42 -14 37 -50 -32 -20 -5 -49 -9 32 -18 -22 24 43 -18 30 39 27 -30 -32 |   |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 12 8 26 15 22 12 0 -5 6 17 -21 -18 -33 -49 -19 33 -20 0 -15 -48 39 0 44 -19 |          |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || -48 -27 -8 -33 -35 -16 -31 -44 -46 -49 -21 -3 -26 13 -40 4 -11 -48 36 -39 9 -39 -8 22 ||
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 34 29 30 -1 34 -22 32 -6 39 -28 -15 37 41 -30 47 -22 -6 -7 -8 43 36 -3 35 16 |         |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 10 -5 -38 -25 -21 21 14 42 19 -41 8 -35 25 -31 29 3 -49 38 -35 -9 6 40 -13 -2 |        |
│ │ │ +      +-----------------------------------------------------------------------------------------+
    
│ │ │ +            
    
│ │ │ +              
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ +
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -o13 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 -39 -47 39 -43 |  |
│ │ │ +o14 = || 1 4 44 1 40 20 29 -26 42 -16 -4 50 -40 48 23 -34 -35 37 30 4 -47 27 -31 0 |           |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45 22 -47 -47 |     |
│ │ │ +      || -29 -11 2 28 -12 -21 -30 20 -1 -42 27 -45 -48 -14 -3 -45 -37 30 -31 -39 -48 -29 47 0 ||
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 -16 48 |          |
│ │ │ +      || -20 14 48 3 14 33 18 -46 24 45 -1 -29 1 31 -44 -2 -22 40 28 -49 -18 46 10 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46 11 11 -3 |       |
│ │ │ +      || 21 25 -23 -45 -48 -40 8 -7 -6 -1 14 3 -13 -44 -44 21 -41 3 30 7 13 -17 8 0 |          |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47 -23 -10 |      |
│ │ │ +      || -15 26 -17 12 0 -5 35 -11 0 -31 -39 40 49 -21 -42 28 42 -18 -29 8 30 -46 23 0 |       |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27 -32 -9 39 -30 ||
│ │ │ +      || -17 38 -49 -9 2 -1 -9 2 3 49 33 49 -46 25 1 0 -18 12 15 -28 18 -16 27 0 |             |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 -19 -33 39 44 |   |
│ │ │ +      || -50 3 -12 -21 -10 4 36 -14 19 -9 43 -48 44 -35 16 6 20 -39 23 -21 -37 -23 19 0 |      |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 -26 9 -8 |       |
│ │ │ +      || 36 -17 -27 37 -4 12 42 -16 -15 12 -3 -14 -28 38 -24 18 -9 6 -47 0 -28 47 -33 0 |      |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16 41 36 35 |       |
│ │ │ +      || 43 44 31 14 -9 -4 -4 -40 -48 45 47 17 -37 29 -9 10 -28 -33 -29 28 26 5 42 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 -2 25 6 -13 |      |
│ │ │ +      || 17 -47 -23 8 -18 -45 -25 2 44 41 -19 -37 5 16 2 38 -13 -20 30 44 4 22 -29 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
    
│ │ │              
    
│ │ │ -              
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ -
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40 37 -47 0 |   |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 |   |
│ │ │ -      +-------------------------------------------------------------------------------------+
    
│ │ │ -            
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │          
    
│ │ │            
    This routine expects the input to represent an irreducible variety
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -84,99 +84,99 @@
│ │ │ │  There are 2 components. We attempt to find points on each of these two
│ │ │ │  components. We are successful. This indicates that the corresponding varieties
│ │ │ │  are both rational. Also, if we can find one point, we can find as many as we
│ │ │ │  want.
│ │ │ │  i13 : netList randomPointsOnRationalVariety(compsJ_0, 10)
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ -----------------+
│ │ │ │ -o13 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 -
│ │ │ │ -39 -47 39 -43 |  |
│ │ │ │ +-----------------+
│ │ │ │ +o13 = || 42 -50 -50 38 -39 6 -1 47 49 -18 18 -28 -47 19 48 34 -13 11 -16 -38 39
│ │ │ │ +21 -43 -39 |    |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || -44 -31 -9 -21 -42 -36 -47 -20 38 -34 -3 -43 22 16 -35 2 -48 32 -28 -
│ │ │ │ +15 -47 38 -47 45 ||
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || 45 -29 -1 -4 42 -35 4 -13 18 -17 1 21 39 -23 50 15 -11 -11 19 47 -16 7
│ │ │ │ +48 43 |         |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || 35 -44 -33 -8 21 -2 -44 -20 19 -28 19 27 11 40 34 33 1 -14 35 36 11 -
│ │ │ │ +38 -3 46 |        |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || 33 47 46 16 -22 -25 -44 -36 -30 -37 30 -25 -47 29 -41 2 -13 -41 -47 22
│ │ │ │ +-23 -7 -10 15 | |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || 43 1 -41 23 -42 -14 37 -50 -32 -20 -5 -49 -9 32 -18 -22 24 43 -18 30
│ │ │ │ +39 27 -30 -32 |   |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || 12 8 26 15 22 12 0 -5 6 17 -21 -18 -33 -49 -19 33 -20 0 -15 -48 39 0
│ │ │ │ +44 -19 |          |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || -48 -27 -8 -33 -35 -16 -31 -44 -46 -49 -21 -3 -26 13 -40 4 -11 -48 36
│ │ │ │ +-39 9 -39 -8 22 ||
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || 34 29 30 -1 34 -22 32 -6 39 -28 -15 37 41 -30 47 -22 -6 -7 -8 43 36 -
│ │ │ │ +3 35 16 |         |
│ │ │ │ +      +------------------------------------------------------------------------
│ │ │ │ +-----------------+
│ │ │ │ +      || 10 -5 -38 -25 -21 21 14 42 19 -41 8 -35 25 -31 29 3 -49 38 -35 -9 6 40
│ │ │ │ +-13 -2 |        |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ -----------------+
│ │ │ │ -      || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45
│ │ │ │ -22 -47 -47 |     |
│ │ │ │ +-----------------+
│ │ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ │ +
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -      || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 -
│ │ │ │ -16 48 |          |
│ │ │ │ +o14 = || 1 4 44 1 40 20 29 -26 42 -16 -4 50 -40 48 23 -34 -35 37 30 4 -47 27 -
│ │ │ │ +31 0 |           |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -      || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46
│ │ │ │ -11 11 -3 |       |
│ │ │ │ +      || -29 -11 2 28 -12 -21 -30 20 -1 -42 27 -45 -48 -14 -3 -45 -37 30 -31 -
│ │ │ │ +39 -48 -29 47 0 ||
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -      || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47
│ │ │ │ --23 -10 |      |
│ │ │ │ +      || -20 14 48 3 14 33 18 -46 24 45 -1 -29 1 31 -44 -2 -22 40 28 -49 -18 46
│ │ │ │ +10 0 |         |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -      || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27
│ │ │ │ --32 -9 39 -30 ||
│ │ │ │ +      || 21 25 -23 -45 -48 -40 8 -7 -6 -1 14 3 -13 -44 -44 21 -41 3 30 7 13 -17
│ │ │ │ +8 0 |          |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -      || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 -
│ │ │ │ -19 -33 39 44 |   |
│ │ │ │ +      || -15 26 -17 12 0 -5 35 -11 0 -31 -39 40 49 -21 -42 28 42 -18 -29 8 30 -
│ │ │ │ +46 23 0 |       |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -      || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 -
│ │ │ │ -26 9 -8 |       |
│ │ │ │ +      || -17 38 -49 -9 2 -1 -9 2 3 49 33 49 -46 25 1 0 -18 12 15 -28 18 -16 27
│ │ │ │ +0 |             |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -      || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16
│ │ │ │ -41 36 35 |       |
│ │ │ │ +      || -50 3 -12 -21 -10 4 36 -14 19 -9 43 -48 44 -35 16 6 20 -39 23 -21 -37
│ │ │ │ +-23 19 0 |      |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -      || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 -
│ │ │ │ -2 25 6 -13 |      |
│ │ │ │ +      || 36 -17 -27 37 -4 12 42 -16 -15 12 -3 -14 -28 38 -24 18 -9 6 -47 0 -28
│ │ │ │ +47 -33 0 |      |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  ----------------+
│ │ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ │ -
│ │ │ │ +      || 43 44 31 14 -9 -4 -4 -40 -48 45 47 17 -37 29 -9 10 -28 -33 -29 28 26 5
│ │ │ │ +42 0 |         |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -o14 = || 38 -31 49 39 4 46 -29 -5 -39 -40 14 -11 -31 46 43 -26 4 30 -35 27 -40
│ │ │ │ -37 -47 0 |   |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -
│ │ │ │ -48 30 -48 0 ||
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1
│ │ │ │ -40 -18 0 |    |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13
│ │ │ │ -3 13 0 |     |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -
│ │ │ │ -18 30 0 |    |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46
│ │ │ │ -12 18 0 |     |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39
│ │ │ │ --37 0 |      |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28
│ │ │ │ -6 -28 0 |    |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -
│ │ │ │ -33 26 0 |    |
│ │ │ │ -      +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5
│ │ │ │ --20 4 0 |   |
│ │ │ │ +----------------+
│ │ │ │ +      || 17 -47 -23 8 -18 -45 -25 2 44 41 -19 -37 5 16 2 38 -13 -20 30 44 4 22
│ │ │ │ +-29 0 |         |
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ --------------+
│ │ │ │ +----------------+
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This routine expects the input to represent an irreducible variety
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_) -- find a random point on a variety
│ │ │ │        that can be detected to be rational
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _r_a_n_d_o_m_P_o_i_n_t_s_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y_(_I_d_e_a_l_,_Z_Z_) -- find random points on a
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html
│ │ │ @@ -355,85 +355,90 @@
│ │ │            
    We can find random points on each component, since these components are rational.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10
│ │ │ +o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11
│ │ │        -----------------------------------------------------------------------
│ │ │ -      11 19 19 24 -29 |
│ │ │ +      -30 -29 24 -29 |
│ │ │  
│ │ │                 1       24
│ │ │  o12 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i13 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38
│ │ │ +o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -16 0 -24 34 |
│ │ │ +      0 -24 34 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    
│ │ │              
    
│ │ │                
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  + 11b*c + 39c  - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c -
│ │ │ +              2             2                            2               
│ │ │ +o14 = ideal (a  - 12b*c - 4c  - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                  
│ │ │ -      10c  - 29a*d - 48b*d + 28c*d + 6d , b  + 19b*c - 8c  + 19a*d - 22b*d -
│ │ │ +         2                              2   2              2                
│ │ │ +      23c  + 19a*d + 38b*d + 20c*d - 19d , b  - 30b*c - 22c  - 29a*d - 29b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                              2
│ │ │ -      36c*d - 46d , a*c + 24b*c - 10c  - 29a*d + 11b*d - 30c*d - 12d )
│ │ │ +                   2                   2                              2
│ │ │ +      - 36c*d + 50d , a*c + 24b*c + 19c  - 29a*d + 11b*d - 10c*d - 34d )
│ │ │  
│ │ │  o14 : Ideal of S
    
│ │ │              
    
│ │ │                
    i15 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o15 = ideal (a  + 29b*c + 30c  + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c -
│ │ │ +              2              2                           2               
│ │ │ +o15 = ideal (a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      49c  - 4a*d + 11b*d + 32c*d - 39d , b  - 16b*c + 19c  - 47b*d + 43c*d +
│ │ │ +         2                           2   2              2                  
│ │ │ +      17c  + 3a*d - 2b*d - 8c*d - 29d , b  + 39b*c - 47c  + 19b*d - 49c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │           2                   2                              2
│ │ │ -      17d , a*c - 24b*c + 39c  + 34a*d - 38b*d + 21c*d - 12d )
│ │ │ +      47d , a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d )
│ │ │  
│ │ │  o15 : Ideal of S
    
│ │ │              
    
│ │ │                
    i16 : decompose F1
│ │ │  
│ │ │ -                                    2             2                      2
│ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b  + 19b*c - 8c  + 27b*d - 48c*d - 26d ),
│ │ │ +                                    2              2                      2
│ │ │ +o16 = {ideal (a + 24b + 19c + 36d, b  - 30b*c - 22c  - 40b*d + 10c*d - 17d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 29d, b + 23d, a - 14d)}
│ │ │ +      ideal (c - 29d, b + 30d, a - 38d)}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : decompose F2
│ │ │  
│ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c -
│ │ │ +                               2                              2   2          
│ │ │ +o17 = {ideal (a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      9d)}
│ │ │ +         2                      2                   2                       
│ │ │ +      47c  + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c  + 3a*d - 2b*d - 8c*d -
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         2   2              2                           2
│ │ │ +      29d , a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d )}
│ │ │  
│ │ │  o17 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Note, the general element of one component is a plane conic union a point. (The dimension of the locus of all such is: (choice of plane) + (choice of degree 2 in plane) + choice of point. This is 3 + 5 + 3 = 11.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -424,65 +424,70 @@
│ │ │ │  -----------------------------------------------------------+
│ │ │ │  This tells us that there are 2 components (at least over the given field).
│ │ │ │  Their dimensions are 11, 8.
│ │ │ │  We can find random points on each component, since these components are
│ │ │ │  rational.
│ │ │ │  i12 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │ │  
│ │ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10
│ │ │ │ +o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      11 19 19 24 -29 |
│ │ │ │ +      -30 -29 24 -29 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o12 : Matrix kk  <-- kk
│ │ │ │  i13 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │ │  
│ │ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38
│ │ │ │ +o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      -16 0 -24 34 |
│ │ │ │ +      0 -24 34 |
│ │ │ │  
│ │ │ │                 1       24
│ │ │ │  o13 : Matrix kk  <-- kk
│ │ │ │  i14 : F1 = sub(F, (vars S)|pt1)
│ │ │ │  
│ │ │ │ -              2              2                              2
│ │ │ │ -o14 = ideal (a  + 11b*c + 39c  - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c -
│ │ │ │ +              2             2                            2
│ │ │ │ +o14 = ideal (a  - 12b*c - 4c  - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2   2             2
│ │ │ │ -      10c  - 29a*d - 48b*d + 28c*d + 6d , b  + 19b*c - 8c  + 19a*d - 22b*d -
│ │ │ │ +         2                              2   2              2
│ │ │ │ +      23c  + 19a*d + 38b*d + 20c*d - 19d , b  - 30b*c - 22c  - 29a*d - 29b*d
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -                 2                   2                              2
│ │ │ │ -      36c*d - 46d , a*c + 24b*c - 10c  - 29a*d + 11b*d - 30c*d - 12d )
│ │ │ │ +                   2                   2                              2
│ │ │ │ +      - 36c*d + 50d , a*c + 24b*c + 19c  - 29a*d + 11b*d - 10c*d - 34d )
│ │ │ │  
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : F2 = sub(F, (vars S)|pt2)
│ │ │ │  
│ │ │ │ -              2              2                              2
│ │ │ │ -o15 = ideal (a  + 29b*c + 30c  + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c -
│ │ │ │ +              2              2                           2
│ │ │ │ +o15 = ideal (a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -         2                             2   2              2
│ │ │ │ -      49c  - 4a*d + 11b*d + 32c*d - 39d , b  - 16b*c + 19c  - 47b*d + 43c*d +
│ │ │ │ +         2                           2   2              2
│ │ │ │ +      17c  + 3a*d - 2b*d - 8c*d - 29d , b  + 39b*c - 47c  + 19b*d - 49c*d -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │           2                   2                              2
│ │ │ │ -      17d , a*c - 24b*c + 39c  + 34a*d - 38b*d + 21c*d - 12d )
│ │ │ │ +      47d , a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d )
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : decompose F1
│ │ │ │  
│ │ │ │ -                                    2             2                      2
│ │ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b  + 19b*c - 8c  + 27b*d - 48c*d - 26d ),
│ │ │ │ +                                    2              2                      2
│ │ │ │ +o16 = {ideal (a + 24b + 19c + 36d, b  - 30b*c - 22c  - 40b*d + 10c*d - 17d ),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ideal (c - 29d, b + 23d, a - 14d)}
│ │ │ │ +      ideal (c - 29d, b + 30d, a - 38d)}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : decompose F2
│ │ │ │  
│ │ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c -
│ │ │ │ +                               2                              2   2
│ │ │ │ +o17 = {ideal (a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d , b  + 39b*c -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      9d)}
│ │ │ │ +         2                      2                   2
│ │ │ │ +      47c  + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c  + 3a*d - 2b*d - 8c*d -
│ │ │ │ +      -----------------------------------------------------------------------
│ │ │ │ +         2   2              2                           2
│ │ │ │ +      29d , a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d )}
│ │ │ │  
│ │ │ │  o17 : List
│ │ │ │  Note, the general element of one component is a plane conic union a point. (The
│ │ │ │  dimension of the locus of all such is: (choice of plane) + (choice of degree 2
│ │ │ │  in plane) + choice of point. This is 3 + 5 + 3 = 11.
│ │ │ │  The other component consists of two skew lines. This has dimension (choice of
│ │ │ │  line) + (choice of line). This is 4 + 4 = 8. Also notice that the 2 skew lines
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  c2V0V2Fsa1RyYWNlKFpaKQ==
│ │ │  #:len=251
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzZXRXYWxrVHJhY2UsWlopLCJzZXRXYWxrVHJhY2Uo
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out
│ │ │ @@ -11,21 +11,21 @@
│ │ │  i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}];
│ │ │  
│ │ │  i4 : I2 = sub(I1, R2);
│ │ │  
│ │ │  o4 : Ideal of R2
│ │ │  
│ │ │  i5 : elapsedTime gb I2
│ │ │ - -- 3.51847s elapsed
│ │ │ + -- 2.60796s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
│ │ │  
│ │ │  i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 3.08455s elapsed
│ │ │ + -- 2.09066s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html
│ │ │ @@ -97,30 +97,30 @@
│ │ │  
│ │ │  o4 : Ideal of R2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime gb I2
│ │ │ - -- 3.51847s elapsed
│ │ │ + -- 2.60796s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    but it is faster to compute directly in the first order and then use the Groebner walk.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 3.08455s elapsed
│ │ │ + -- 2.09066s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,23 +38,23 @@
│ │ │ │  using a different weight vector and then graded reverse lexicographic we could
│ │ │ │  substitute and compute directly,
│ │ │ │  i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}];
│ │ │ │  i4 : I2 = sub(I1, R2);
│ │ │ │  
│ │ │ │  o4 : Ideal of R2
│ │ │ │  i5 : elapsedTime gb I2
│ │ │ │ - -- 3.51847s elapsed
│ │ │ │ + -- 2.60796s elapsed
│ │ │ │  
│ │ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │ │  
│ │ │ │  o5 : GroebnerBasis
│ │ │ │  but it is faster to compute directly in the first order and then use the
│ │ │ │  Groebner walk.
│ │ │ │  i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ │ - -- 3.08455s elapsed
│ │ │ │ + -- 2.09066s elapsed
│ │ │ │  
│ │ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │ │  
│ │ │ │  o6 : GroebnerBasis
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The target ring must be the same ring as the ring of the starting ideal, except
│ │ │ │  with different monomial order. The ring must be a polynomial ring over a field.
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  aWRlYWxPZlByb2plY3RpdmVQb2ludHM=
│ │ │  #:len=1240
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIGlkZWFsIG9mIHNl
│ │ │  dCBvZiBwb2ludHMiLCAibGluZW51bSIgPT4gNDI0LCBJbnB1dHMgPT4ge1NQQU57VFR7IkwifSwi
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__List_cm__List_rp.out
│ │ │ @@ -2,12 +2,12 @@
│ │ │  
│ │ │  i1 : L = {point{0,1}, point{1,2}};
│ │ │  
│ │ │  i2 : M = {point{1,0}, point{2,2}};
│ │ │  
│ │ │  i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{0, 2}, Point{2, 4}, Point{1, 0}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_ideal__Of__Projective__Points.out
│ │ │ @@ -30,17 +30,17 @@
│ │ │            2       2
│ │ │       + x*y  - 6x*z )
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : X2 = hadamardPower(X,2)
│ │ │  
│ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1},
│ │ │ +o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Point{1, 1, 0}, Point{1, 2, 0}}
│ │ │ +     Point{0, 1, 1}, Point{1, 1, 0}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │  
│ │ │  o6 = true
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_cm__List_rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │                
    i2 : M = {point{1,0}, point{2,2}};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{0, 2}, Point{2, 4}, Point{1, 0}}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  Given two sets of points $L$ and $M$ returns the list of (well-defined)
│ │ │ │  entrywise multiplication of pairs of points in the cartesian product $L\times
│ │ │ │  M$.
│ │ │ │  i1 : L = {point{0,1}, point{1,2}};
│ │ │ │  i2 : M = {point{1,0}, point{2,2}};
│ │ │ │  i3 : hadamardProduct(L,M)
│ │ │ │  
│ │ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ │ +o3 = {Point{0, 2}, Point{2, 4}, Point{1, 0}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_d_a_m_a_r_d_P_r_o_d_u_c_t_(_L_i_s_t_,_L_i_s_t_) -- Hadamard product of two sets of points
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Hadamard.m2:345:0.
│ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_ideal__Of__Projective__Points.html
│ │ │ @@ -121,17 +121,17 @@
│ │ │  o4 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : X2 = hadamardPower(X,2)
│ │ │  
│ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1},
│ │ │ +o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Point{1, 1, 0}, Point{1, 2, 0}}
│ │ │ +     Point{0, 1, 1}, Point{1, 1, 0}}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,17 +39,17 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            2       2
│ │ │ │       + x*y  - 6x*z )
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : X2 = hadamardPower(X,2)
│ │ │ │  
│ │ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1},
│ │ │ │ +o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Point{1, 1, 0}, Point{1, 2, 0}}
│ │ │ │ +     Point{0, 1, 1}, Point{1, 1, 0}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  ********** WWaayyss ttoo uussee iiddeeaallOOffPPrroojjeeccttiivveePPooiinnttss:: **********
│ │ │ │      * idealOfProjectivePoints(List,Ring)
│ │ ├── ./usr/share/doc/Macaulay2/HigherCIOperators/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  Y2lPcGVyYXRvclJlc29sdXRpb24=
│ │ │  #:len=2606
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiXCJsaWZ0IHJlc29sdXRpb24gZnJvbSBj
│ │ │  b21wbGV0ZSBpbnRlcnNlY3Rpb24gdXNpbmcgaGlnaGVyIGNpLW9wZXJhdG9yc1wiIiwgImxpbmVu
│ │ ├── ./usr/share/doc/Macaulay2/HighestWeights/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  R3JvdXBBY3Rpbmc=
│ │ │  #:len=621
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3RvcmVzIHRoZSBEeW5raW4gdHlwZSBv
│ │ │  ZiB0aGUgZ3JvdXAgYWN0aW5nIG9uIGEgcmluZyIsICJsaW5lbnVtIiA9PiA4MywgU2VlQWxzbyA9
│ │ ├── ./usr/share/doc/Macaulay2/HodgeIntegrals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  a2FwcGE=
│ │ │  #:len=1396
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTXVtZm9yZC1Nb3JpdGEtTWlsbGVyIGNs
│ │ │  YXNzZXMiLCAibGluZW51bSIgPT4gNzE1LCBJbnB1dHMgPT4ge1NQQU57VFR7ImEifSwiLCAiLFNQ
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  ZXVsZXJPcGVyYXRvcnM=
│ │ │  #:len=1959
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXVsZXIgT3BlcmF0b3JzIiwgImxpbmVu
│ │ │  dW0iID0+IDE0MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJBIn0sIiwgIixTUEFOeyJhICIsVE8ye25l
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004839s elapsed
│ │ │ - -- .000004879s elapsed
│ │ │ - -- .00000513s elapsed
│ │ │ - -- .0000051s elapsed
│ │ │ - -- .000003987s elapsed
│ │ │ + -- .000057712s elapsed
│ │ │ + -- .0000069s elapsed
│ │ │ + -- .000005695s elapsed
│ │ │ + -- .000008023s elapsed
│ │ │ + -- .000006668s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3, t_2*t_4);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : solveFrobeniusIdeal I
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000006092s elapsed
│ │ │ + -- .000011081s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
│ │ │ @@ -26,15 +26,15 @@
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │  
│ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000007755s elapsed
│ │ │ + -- .000012434s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html
│ │ │ @@ -139,19 +139,19 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004839s elapsed
│ │ │ - -- .000004879s elapsed
│ │ │ - -- .00000513s elapsed
│ │ │ - -- .0000051s elapsed
│ │ │ - -- .000003987s elapsed
│ │ │ + -- .000057712s elapsed
│ │ │ + -- .0000069s elapsed
│ │ │ + -- .000005695s elapsed
│ │ │ + -- .000008023s elapsed
│ │ │ + -- .000006668s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,19 +57,19 @@
│ │ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000004839s elapsed
│ │ │ │ - -- .000004879s elapsed
│ │ │ │ - -- .00000513s elapsed
│ │ │ │ - -- .0000051s elapsed
│ │ │ │ - -- .000003987s elapsed
│ │ │ │ + -- .000057712s elapsed
│ │ │ │ + -- .0000069s elapsed
│ │ │ │ + -- .000005695s elapsed
│ │ │ │ + -- .000008023s elapsed
│ │ │ │ + -- .000006668s elapsed
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ │ │       |   2 2   2 2                                        |
│ │ │ │  o6 = |x   x x   x                                         |
│ │ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : solveFrobeniusIdeal I
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000006092s elapsed
│ │ │ + -- .000011081s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
│ │ │ @@ -120,15 +120,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : solveFrobeniusIdeal(I, W)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000007755s elapsed
│ │ │ + -- .000012434s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  t_2*t_4);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : solveFrobeniusIdeal I
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000006092s elapsed
│ │ │ │ + -- .000011081s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1             1    2   1             1             1             1    2
│ │ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000007755s elapsed
│ │ │ │ + -- .000012434s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1             1    2   1             1             1             1    2
│ │ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  YWxsZ2VucyhER0FsZ2VicmEsWlop
│ │ │  #:len=269
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbGxnZW5zLERHQWxnZWJyYSxaWiksImFsbGdlbnMo
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket.out
│ │ │ @@ -88,112 +88,112 @@
│ │ │  
│ │ │  o13 = 600
│ │ │  
│ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │  
│ │ │  i15 : H'
│ │ │  
│ │ │ -o15 = {({T   , T   }, - T   T    - T   T    + y*T    + z*T   ), ({T   ,
│ │ │ -          1,4   2,3      1,2 2,2    1,4 2,3      3,2      3,4      1,3 
│ │ │ +o15 = {({T   , T   }, T   T    + x*T    - z*T   ), ({T   , T   }, - T   T   
│ │ │ +          1,3   2,3    1,3 2,3      3,5      3,7      1,1   2,3      1,3 2,1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    - z*T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ -       2,4    1,3 2,4      3,5      3,7      1,3   2,1      1,3 2,1  
│ │ │ +      - T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, T   T    -
│ │ │ +         1,5 2,2    1,1 2,3      3,2      3,4      1,4   2,4    1,4 2,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ -       1,5 2,2    1,1 2,3      3,2      3,4      1,1   2,2      1,1 2,2  
│ │ │ +      T   T    - z*T    + z*T   ), ({T   , T   }, - T   T    - T   T    +
│ │ │ +       1,5 2,5      3,7      3,9      1,2   2,3      1,2 2,3    1,5 2,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ -         3,3      1,5   2,4      1,2 2,3    1,5 2,4      3,5      1,3   2,5  
│ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ +         3,5      1,5   2,2      1,3 2,1    1,5 2,2    1,1 2,3      3,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ -         1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,2      1,2 2,2  
│ │ │ +      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ +         3,4      1,4   2,4    1,2 2,1    1,3 2,3    1,4 2,4      3,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T    + z*T   ), ({T   , T   }, - T   T    + T   T    -
│ │ │ -       1,4 2,3      3,2      3,4      1,4   2,3      1,4 2,3    1,3 2,5  
│ │ │ +      z*T   ), ({T   , T   }, T   T    + T   T    - z*T    + y*T    ),
│ │ │ +         3,7      1,3   2,5    1,5 2,1    1,3 2,5      3,8      3,10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ),
│ │ │ -         3,8      3,9      1,2   2,5      1,5 2,3    1,2 2,5      3,9  
│ │ │ +      ({T   , T   }, T   T    + T   T    - z*T    + y*T    ), ({T   , T   },
│ │ │ +         1,5   2,1    1,5 2,1    1,3 2,5      3,8      3,10      1,4   2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ -         1,4   2,5      1,4 2,5      3,8      1,5   2,3      1,5 2,3  
│ │ │ +      - T   T    - T   T    + z*T    + z*T    ), ({T   , T   }, - T   T    -
│ │ │ +         1,5 2,2    1,4 2,5      3,2      3,10      1,1   2,1      1,1 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T   ), ({T   , T   }, T   T    + x*T    - z*T   ), ({T   ,
│ │ │ -       1,2 2,5      3,9      1,3   2,3    1,3 2,3      3,5      3,7      1,1 
│ │ │ +      T   T    + x*T   ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ +       1,4 2,2      3,1      1,3   2,4    1,5 2,3    1,3 2,4      3,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   ,
│ │ │ -       2,3      1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4 
│ │ │ +      x*T   ), ({T   , T   }, T   T    + T   T    - z*T    + y*T   ), ({T   ,
│ │ │ +         3,6      1,4   2,1    1,4 2,1    1,3 2,2      3,1      3,3      1,5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, -
│ │ │ -       2,4    1,4 2,4    1,5 2,5      3,7      3,9      1,2   2,3    
│ │ │ +      T   }, - T   T    + y*T   ), ({T   , T   }, T   T    - T   T    +
│ │ │ +       2,5      1,5 2,5      3,8      1,4   2,1    1,4 2,1    1,1 2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    - T   T    -
│ │ │ -       1,2 2,3    1,5 2,4      3,5      1,5   2,2      1,3 2,1    1,5 2,2  
│ │ │ +      x*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   ,
│ │ │ +         3,10      1,4   2,2      1,1 2,1    1,4 2,2      3,1      1,2 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + z*T    + x*T   ), ({T   , T   }, T   T    + T   T    +
│ │ │ -       1,1 2,3      3,2      3,4      1,4   2,4    1,2 2,1    1,3 2,3  
│ │ │ +      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ +       2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T    - z*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ -       1,4 2,4      3,4      3,7      1,3   2,5    1,5 2,1    1,3 2,5  
│ │ │ +      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, T   T   
│ │ │ +       2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3   2,3    1,2 2,1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ -         3,8      3,10      1,5   2,1    1,5 2,1    1,3 2,5      3,8  
│ │ │ +      + T   T    + T   T    + y*T    - z*T   ), ({T   , T   }, T   T    +
│ │ │ +         1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,1    1,3 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T    ), ({T   , T   }, - T   T    - T   T    + z*T    + z*T    ),
│ │ │ -         3,10      1,4   2,5      1,5 2,2    1,4 2,5      3,2      3,10  
│ │ │ +      y*T    - z*T   ), ({T   , T   }, - T   T    - T   T    + z*T    +
│ │ │ +         3,1      3,2      1,5   2,2      1,5 2,2    1,4 2,5      3,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   , T   }, T   T   
│ │ │ -         1,1   2,1      1,1 2,1    1,4 2,2      3,1      1,3   2,4    1,5 2,3
│ │ │ +      z*T    ), ({T   , T   }, T   T    + T   T    - z*T    + y*T   ),
│ │ │ +         3,10      1,3   2,2    1,4 2,1    1,3 2,2      3,1      3,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      + T   T    - z*T    + x*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ -         1,3 2,4      3,5      3,6      1,4   2,1    1,4 2,1    1,3 2,2  
│ │ │ +      ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ +         1,2   2,4      1,2 2,4      3,6      1,1   2,4      1,5 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   },
│ │ │ -         3,1      3,3      1,5   2,5      1,5 2,5      3,8      1,4   2,1  
│ │ │ +      T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    + z*T   ),
│ │ │ +       1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2      3,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ -       1,4 2,1    1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2  
│ │ │ +      ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   }, -
│ │ │ +         1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ -         3,1      1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4  
│ │ │ +      T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ), ({T   ,
│ │ │ +       1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2      1,5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   ,
│ │ │ -         3,7      1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3 
│ │ │ +      T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +       2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ -       2,3    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3 
│ │ │ +      T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ +       1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,3      1,2 2,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    -
│ │ │ -       2,1    1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2  
│ │ │ +      T   T    + y*T    + z*T   ), ({T   , T   }, T   T    - z*T    +
│ │ │ +       1,4 2,3      3,2      3,4      1,3   2,4    1,3 2,4      3,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + z*T    + z*T    ), ({T   , T   }, T   T    + T   T    -
│ │ │ -       1,4 2,5      3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2  
│ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ +         3,7      1,3   2,1      1,3 2,1    1,5 2,2    1,1 2,3      3,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, -
│ │ │ -         3,1      3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4    
│ │ │ +      x*T   ), ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T   
│ │ │ +         3,4      1,1   2,2      1,1 2,2      3,3      1,5   2,4      1,2 2,3
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ -       1,5 2,1    1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2  
│ │ │ +      - T   T    + y*T   ), ({T   , T   }, - T   T    + T   T    - z*T    +
│ │ │ +         1,5 2,4      3,5      1,3   2,5      1,4 2,3    1,3 2,5      3,8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   },
│ │ │ -         3,3      1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4  
│ │ │ +      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T    + z*T   ),
│ │ │ +         3,9      1,2   2,2      1,2 2,2    1,4 2,3      3,2      3,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ),
│ │ │ -         1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2  
│ │ │ +      ({T   , T   }, - T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +         1,4   2,3      1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ -         1,5   2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3  
│ │ │ +      - T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    + x*T   ),
│ │ │ +         1,5 2,3    1,2 2,5      3,9      1,4   2,5      1,4 2,5      3,8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + T   T    - z*T    + x*T   )}
│ │ │ -       1,5 2,3    1,3 2,4      3,5      3,6
│ │ │ +      ({T   , T   }, - T   T    - T   T    + y*T   )}
│ │ │ +         1,5   2,3      1,5 2,3    1,2 2,5      3,9
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = -1
│ │ │ +o16 = 1
│ │ │  
│ │ │  o16 : S[T   ..T   , T   ..T    , T   ..T    , T   ..T    ]
│ │ │           1,1   2,5   3,1   3,10   4,1   4,24   5,1   5,55
│ │ │  
│ │ │  i17 : bracketMatrix(A,1,2)
│ │ │  
│ │ │  o17 = | 0        -T_(2,3) -T_(2,1) -T_(2,2) -T_(2,5) |
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html
│ │ │ @@ -223,115 +223,115 @@
│ │ │                
    i14 : H' = select(keys H, k->H#k != 0);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : H'
│ │ │  
│ │ │ -o15 = {({T   , T   }, - T   T    - T   T    + y*T    + z*T   ), ({T   ,
│ │ │ -          1,4   2,3      1,2 2,2    1,4 2,3      3,2      3,4      1,3 
│ │ │ +o15 = {({T   , T   }, T   T    + x*T    - z*T   ), ({T   , T   }, - T   T   
│ │ │ +          1,3   2,3    1,3 2,3      3,5      3,7      1,1   2,3      1,3 2,1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    - z*T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ -       2,4    1,3 2,4      3,5      3,7      1,3   2,1      1,3 2,1  
│ │ │ +      - T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, T   T    -
│ │ │ +         1,5 2,2    1,1 2,3      3,2      3,4      1,4   2,4    1,4 2,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ -       1,5 2,2    1,1 2,3      3,2      3,4      1,1   2,2      1,1 2,2  
│ │ │ +      T   T    - z*T    + z*T   ), ({T   , T   }, - T   T    - T   T    +
│ │ │ +       1,5 2,5      3,7      3,9      1,2   2,3      1,2 2,3    1,5 2,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ -         3,3      1,5   2,4      1,2 2,3    1,5 2,4      3,5      1,3   2,5  
│ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ +         3,5      1,5   2,2      1,3 2,1    1,5 2,2    1,1 2,3      3,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ -         1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,2      1,2 2,2  
│ │ │ +      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ +         3,4      1,4   2,4    1,2 2,1    1,3 2,3    1,4 2,4      3,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T    + z*T   ), ({T   , T   }, - T   T    + T   T    -
│ │ │ -       1,4 2,3      3,2      3,4      1,4   2,3      1,4 2,3    1,3 2,5  
│ │ │ +      z*T   ), ({T   , T   }, T   T    + T   T    - z*T    + y*T    ),
│ │ │ +         3,7      1,3   2,5    1,5 2,1    1,3 2,5      3,8      3,10  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ),
│ │ │ -         3,8      3,9      1,2   2,5      1,5 2,3    1,2 2,5      3,9  
│ │ │ +      ({T   , T   }, T   T    + T   T    - z*T    + y*T    ), ({T   , T   },
│ │ │ +         1,5   2,1    1,5 2,1    1,3 2,5      3,8      3,10      1,4   2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ -         1,4   2,5      1,4 2,5      3,8      1,5   2,3      1,5 2,3  
│ │ │ +      - T   T    - T   T    + z*T    + z*T    ), ({T   , T   }, - T   T    -
│ │ │ +         1,5 2,2    1,4 2,5      3,2      3,10      1,1   2,1      1,1 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T   ), ({T   , T   }, T   T    + x*T    - z*T   ), ({T   ,
│ │ │ -       1,2 2,5      3,9      1,3   2,3    1,3 2,3      3,5      3,7      1,1 
│ │ │ +      T   T    + x*T   ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ +       1,4 2,2      3,1      1,3   2,4    1,5 2,3    1,3 2,4      3,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   ,
│ │ │ -       2,3      1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4 
│ │ │ +      x*T   ), ({T   , T   }, T   T    + T   T    - z*T    + y*T   ), ({T   ,
│ │ │ +         3,6      1,4   2,1    1,4 2,1    1,3 2,2      3,1      3,3      1,5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, -
│ │ │ -       2,4    1,4 2,4    1,5 2,5      3,7      3,9      1,2   2,3    
│ │ │ +      T   }, - T   T    + y*T   ), ({T   , T   }, T   T    - T   T    +
│ │ │ +       2,5      1,5 2,5      3,8      1,4   2,1    1,4 2,1    1,1 2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    - T   T    -
│ │ │ -       1,2 2,3    1,5 2,4      3,5      1,5   2,2      1,3 2,1    1,5 2,2  
│ │ │ +      x*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   ,
│ │ │ +         3,10      1,4   2,2      1,1 2,1    1,4 2,2      3,1      1,2 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + z*T    + x*T   ), ({T   , T   }, T   T    + T   T    +
│ │ │ -       1,1 2,3      3,2      3,4      1,4   2,4    1,2 2,1    1,3 2,3  
│ │ │ +      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ +       2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T    - z*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ -       1,4 2,4      3,4      3,7      1,3   2,5    1,5 2,1    1,3 2,5  
│ │ │ +      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, T   T   
│ │ │ +       2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3   2,3    1,2 2,1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ -         3,8      3,10      1,5   2,1    1,5 2,1    1,3 2,5      3,8  
│ │ │ +      + T   T    + T   T    + y*T    - z*T   ), ({T   , T   }, T   T    +
│ │ │ +         1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,1    1,3 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T    ), ({T   , T   }, - T   T    - T   T    + z*T    + z*T    ),
│ │ │ -         3,10      1,4   2,5      1,5 2,2    1,4 2,5      3,2      3,10  
│ │ │ +      y*T    - z*T   ), ({T   , T   }, - T   T    - T   T    + z*T    +
│ │ │ +         3,1      3,2      1,5   2,2      1,5 2,2    1,4 2,5      3,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   , T   }, T   T   
│ │ │ -         1,1   2,1      1,1 2,1    1,4 2,2      3,1      1,3   2,4    1,5 2,3
│ │ │ +      z*T    ), ({T   , T   }, T   T    + T   T    - z*T    + y*T   ),
│ │ │ +         3,10      1,3   2,2    1,4 2,1    1,3 2,2      3,1      3,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      + T   T    - z*T    + x*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ -         1,3 2,4      3,5      3,6      1,4   2,1    1,4 2,1    1,3 2,2  
│ │ │ +      ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ +         1,2   2,4      1,2 2,4      3,6      1,1   2,4      1,5 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   },
│ │ │ -         3,1      3,3      1,5   2,5      1,5 2,5      3,8      1,4   2,1  
│ │ │ +      T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    + z*T   ),
│ │ │ +       1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2      3,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ -       1,4 2,1    1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2  
│ │ │ +      ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   }, -
│ │ │ +         1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ -         3,1      1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4  
│ │ │ +      T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ), ({T   ,
│ │ │ +       1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2      1,5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   ,
│ │ │ -         3,7      1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3 
│ │ │ +      T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +       2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ -       2,3    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3 
│ │ │ +      T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ +       1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,3      1,2 2,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    -
│ │ │ -       2,1    1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2  
│ │ │ +      T   T    + y*T    + z*T   ), ({T   , T   }, T   T    - z*T    +
│ │ │ +       1,4 2,3      3,2      3,4      1,3   2,4    1,3 2,4      3,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + z*T    + z*T    ), ({T   , T   }, T   T    + T   T    -
│ │ │ -       1,4 2,5      3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2  
│ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ +         3,7      1,3   2,1      1,3 2,1    1,5 2,2    1,1 2,3      3,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, -
│ │ │ -         3,1      3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4    
│ │ │ +      x*T   ), ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T   
│ │ │ +         3,4      1,1   2,2      1,1 2,2      3,3      1,5   2,4      1,2 2,3
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ -       1,5 2,1    1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2  
│ │ │ +      - T   T    + y*T   ), ({T   , T   }, - T   T    + T   T    - z*T    +
│ │ │ +         1,5 2,4      3,5      1,3   2,5      1,4 2,3    1,3 2,5      3,8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   },
│ │ │ -         3,3      1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4  
│ │ │ +      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T    + z*T   ),
│ │ │ +         3,9      1,2   2,2      1,2 2,2    1,4 2,3      3,2      3,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ),
│ │ │ -         1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2  
│ │ │ +      ({T   , T   }, - T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +         1,4   2,3      1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ -         1,5   2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3  
│ │ │ +      - T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    + x*T   ),
│ │ │ +         1,5 2,3    1,2 2,5      3,9      1,4   2,5      1,4 2,5      3,8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + T   T    - z*T    + x*T   )}
│ │ │ -       1,5 2,3    1,3 2,4      3,5      3,6
│ │ │ +      ({T   , T   }, - T   T    - T   T    + y*T   )}
│ │ │ +         1,5   2,3      1,5 2,3    1,2 2,5      3,9
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : H#(H'_0)
│ │ │  
│ │ │ -o16 = -1
│ │ │ +o16 = 1
│ │ │  
│ │ │  o16 : S[T   ..T   , T   ..T    , T   ..T    , T   ..T    ]
│ │ │           1,1   2,5   3,1   3,10   4,1   4,24   5,1   5,55
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -119,111 +119,111 @@
│ │ │ │  i12 : H = bracket(A,2,3);
│ │ │ │  i13 : #keys H
│ │ │ │  
│ │ │ │  o13 = 600
│ │ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │ │  i15 : H'
│ │ │ │  
│ │ │ │ -o15 = {({T   , T   }, - T   T    - T   T    + y*T    + z*T   ), ({T   ,
│ │ │ │ -          1,4   2,3      1,2 2,2    1,4 2,3      3,2      3,4      1,3
│ │ │ │ +o15 = {({T   , T   }, T   T    + x*T    - z*T   ), ({T   , T   }, - T   T
│ │ │ │ +          1,3   2,3    1,3 2,3      3,5      3,7      1,1   2,3      1,3 2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, T   T    - z*T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ │ -       2,4    1,3 2,4      3,5      3,7      1,3   2,1      1,3 2,1
│ │ │ │ +      - T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, T   T    -
│ │ │ │ +         1,5 2,2    1,1 2,3      3,2      3,4      1,4   2,4    1,4 2,4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ │ -       1,5 2,2    1,1 2,3      3,2      3,4      1,1   2,2      1,1 2,2
│ │ │ │ +      T   T    - z*T    + z*T   ), ({T   , T   }, - T   T    - T   T    +
│ │ │ │ +       1,5 2,5      3,7      3,9      1,2   2,3      1,2 2,3    1,5 2,4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ │ -         3,3      1,5   2,4      1,2 2,3    1,5 2,4      3,5      1,3   2,5
│ │ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ │ +         3,5      1,5   2,2      1,3 2,1    1,5 2,2    1,1 2,3      3,2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      - T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ │ -         1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,2      1,2 2,2
│ │ │ │ +      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ │ +         3,4      1,4   2,4    1,2 2,1    1,3 2,3    1,4 2,4      3,4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + y*T    + z*T   ), ({T   , T   }, - T   T    + T   T    -
│ │ │ │ -       1,4 2,3      3,2      3,4      1,4   2,3      1,4 2,3    1,3 2,5
│ │ │ │ +      z*T   ), ({T   , T   }, T   T    + T   T    - z*T    + y*T    ),
│ │ │ │ +         3,7      1,3   2,5    1,5 2,1    1,3 2,5      3,8      3,10
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T    + x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ),
│ │ │ │ -         3,8      3,9      1,2   2,5      1,5 2,3    1,2 2,5      3,9
│ │ │ │ +      ({T   , T   }, T   T    + T   T    - z*T    + y*T    ), ({T   , T   },
│ │ │ │ +         1,5   2,1    1,5 2,1    1,3 2,5      3,8      3,10      1,4   2,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ │ -         1,4   2,5      1,4 2,5      3,8      1,5   2,3      1,5 2,3
│ │ │ │ +      - T   T    - T   T    + z*T    + z*T    ), ({T   , T   }, - T   T    -
│ │ │ │ +         1,5 2,2    1,4 2,5      3,2      3,10      1,1   2,1      1,1 2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + y*T   ), ({T   , T   }, T   T    + x*T    - z*T   ), ({T   ,
│ │ │ │ -       1,2 2,5      3,9      1,3   2,3    1,3 2,3      3,5      3,7      1,1
│ │ │ │ +      T   T    + x*T   ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ │ +       1,4 2,2      3,1      1,3   2,4    1,5 2,3    1,3 2,4      3,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   ,
│ │ │ │ -       2,3      1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4
│ │ │ │ +      x*T   ), ({T   , T   }, T   T    + T   T    - z*T    + y*T   ), ({T   ,
│ │ │ │ +         3,6      1,4   2,1    1,4 2,1    1,3 2,2      3,1      3,3      1,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, -
│ │ │ │ -       2,4    1,4 2,4    1,5 2,5      3,7      3,9      1,2   2,3
│ │ │ │ +      T   }, - T   T    + y*T   ), ({T   , T   }, T   T    - T   T    +
│ │ │ │ +       2,5      1,5 2,5      3,8      1,4   2,1    1,4 2,1    1,1 2,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    - T   T    -
│ │ │ │ -       1,2 2,3    1,5 2,4      3,5      1,5   2,2      1,3 2,1    1,5 2,2
│ │ │ │ +      x*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   ,
│ │ │ │ +         3,10      1,4   2,2      1,1 2,1    1,4 2,2      3,1      1,2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + z*T    + x*T   ), ({T   , T   }, T   T    + T   T    +
│ │ │ │ -       1,1 2,3      3,2      3,4      1,4   2,4    1,2 2,1    1,3 2,3
│ │ │ │ +      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ │ +       2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + y*T    - z*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ │ -       1,4 2,4      3,4      3,7      1,3   2,5    1,5 2,1    1,3 2,5
│ │ │ │ +      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, T   T
│ │ │ │ +       2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3   2,3    1,2 2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T    + y*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ │ -         3,8      3,10      1,5   2,1    1,5 2,1    1,3 2,5      3,8
│ │ │ │ +      + T   T    + T   T    + y*T    - z*T   ), ({T   , T   }, T   T    +
│ │ │ │ +         1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,1    1,3 2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      y*T    ), ({T   , T   }, - T   T    - T   T    + z*T    + z*T    ),
│ │ │ │ -         3,10      1,4   2,5      1,5 2,2    1,4 2,5      3,2      3,10
│ │ │ │ +      y*T    - z*T   ), ({T   , T   }, - T   T    - T   T    + z*T    +
│ │ │ │ +         3,1      3,2      1,5   2,2      1,5 2,2    1,4 2,5      3,2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   , T   }, T   T
│ │ │ │ -         1,1   2,1      1,1 2,1    1,4 2,2      3,1      1,3   2,4    1,5 2,3
│ │ │ │ +      z*T    ), ({T   , T   }, T   T    + T   T    - z*T    + y*T   ),
│ │ │ │ +         3,10      1,3   2,2    1,4 2,1    1,3 2,2      3,1      3,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      + T   T    - z*T    + x*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ │ -         1,3 2,4      3,5      3,6      1,4   2,1    1,4 2,1    1,3 2,2
│ │ │ │ +      ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ │ +         1,2   2,4      1,2 2,4      3,6      1,1   2,4      1,5 2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   },
│ │ │ │ -         3,1      3,3      1,5   2,5      1,5 2,5      3,8      1,4   2,1
│ │ │ │ +      T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    + z*T   ),
│ │ │ │ +       1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2      3,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    - T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ │ -       1,4 2,1    1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2
│ │ │ │ +      ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   }, -
│ │ │ │ +         1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ │ -         3,1      1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4
│ │ │ │ +      T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ), ({T   ,
│ │ │ │ +       1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2      1,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   ,
│ │ │ │ -         3,7      1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3
│ │ │ │ +      T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ │ +       2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ │ -       2,3    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3
│ │ │ │ +      T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ │ +       1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,3      1,2 2,2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    -
│ │ │ │ -       2,1    1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2
│ │ │ │ +      T   T    + y*T    + z*T   ), ({T   , T   }, T   T    - z*T    +
│ │ │ │ +       1,4 2,3      3,2      3,4      1,3   2,4    1,3 2,4      3,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + z*T    + z*T    ), ({T   , T   }, T   T    + T   T    -
│ │ │ │ -       1,4 2,5      3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2
│ │ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ │ +         3,7      1,3   2,1      1,3 2,1    1,5 2,2    1,1 2,3      3,2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, -
│ │ │ │ -         3,1      3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4
│ │ │ │ +      x*T   ), ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T
│ │ │ │ +         3,4      1,1   2,2      1,1 2,2      3,3      1,5   2,4      1,2 2,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ │ -       1,5 2,1    1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2
│ │ │ │ +      - T   T    + y*T   ), ({T   , T   }, - T   T    + T   T    - z*T    +
│ │ │ │ +         1,5 2,4      3,5      1,3   2,5      1,4 2,3    1,3 2,5      3,8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   },
│ │ │ │ -         3,3      1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4
│ │ │ │ +      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T    + z*T   ),
│ │ │ │ +         3,9      1,2   2,2      1,2 2,2    1,4 2,3      3,2      3,4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      - T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ),
│ │ │ │ -         1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2
│ │ │ │ +      ({T   , T   }, - T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ │ +         1,4   2,3      1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T   , T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ │ -         1,5   2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3
│ │ │ │ +      - T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    + x*T   ),
│ │ │ │ +         1,5 2,3    1,2 2,5      3,9      1,4   2,5      1,4 2,5      3,8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + T   T    - z*T    + x*T   )}
│ │ │ │ -       1,5 2,3    1,3 2,4      3,5      3,6
│ │ │ │ +      ({T   , T   }, - T   T    - T   T    + y*T   )}
│ │ │ │ +         1,5   2,3      1,5 2,3    1,2 2,5      3,9
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : H#(H'_0)
│ │ │ │  
│ │ │ │ -o16 = -1
│ │ │ │ +o16 = 1
│ │ │ │  
│ │ │ │  o16 : S[T   ..T   , T   ..T    , T   ..T    , T   ..T    ]
│ │ │ │           1,1   2,5   3,1   3,10   4,1   4,24   5,1   5,55
│ │ │ │  From this we see that [T_5, T_6] sends T_37 to -1 in kk.
│ │ │ │  Another, often simpler view of the pairing is given by _b_r_a_c_k_e_t_M_a_t_r_i_x, where the
│ │ │ │  rows and columns correspond to the generators of Pi^d and Pi^e, and the entries
│ │ │ │  are the bracket products, interpreted as elements of Pi^{d+e}. Note the anti-
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  cmVzdHJpY3Rpb24oQXJyYW5nZW1lbnQsTGlzdCk=
│ │ │  #:len=343
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc3OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVzdHJpY3Rpb24sQXJyYW5nZW1lbnQsTGlzdCks
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_cone_lp__Arrangement_cm__Ring__Element_rp.out
│ │ │ @@ -44,15 +44,15 @@
│ │ │  
│ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │  
│ │ │  o13 : Hyperplane Arrangement 
│ │ │  
│ │ │  i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {y, x, x - y}
│ │ │ +o14 = {x - y, y, x}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
│ │ │  
│ │ │  i15 : assert isCentral cA''
│ │ │  
│ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.out
│ │ │ @@ -10,35 +10,35 @@
│ │ │  
│ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : restriction(A,1)
│ │ │  
│ │ │  o4 = {x, z, x, x - z}
│ │ │  
│ │ │  o4 : Hyperplane Arrangement 
│ │ │  
│ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {x - z, z, x}
│ │ │ +o5 = {z, x, x - z}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │  
│ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : prune image der(A,m)
│ │ │  
│ │ │        3
│ │ │  o9 = R
│ │ │ @@ -59,16 +59,16 @@
│ │ │  
│ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │  
│ │ │  i12 : A = arrangement "bracelet";
│ │ │  
│ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │  
│ │ │  o13 : Sequence
│ │ │  
│ │ │  i14 : C = restriction(A,0)
│ │ │  
│ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_trim_lp__Arrangement_rp.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  o2 = {x, x, 0, y, y, y, x + y, x + y, x + y, x + y, x + y}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : A' = trim A
│ │ │  
│ │ │ -o3 = {x + y, y, x}
│ │ │ +o3 = {y, x, x + y}
│ │ │  
│ │ │  o3 : Hyperplane Arrangement 
│ │ │  
│ │ │  i4 : assert(ring A' === R)
│ │ │  
│ │ │  i5 : assert(trim A' == A')
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_type__B_lp__Z__Z_cm__Ring_rp.out
│ │ │ @@ -33,16 +33,16 @@
│ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │         1   1    2   1    2   2
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
│ │ │  
│ │ │  i7 : ring A3
│ │ │  
│ │ │       ZZ
│ │ │  o7 = --[x ..x ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_cone_lp__Arrangement_cm__Ring__Element_rp.html
│ │ │ @@ -172,15 +172,15 @@
│ │ │  o13 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {y, x, x - y}
│ │ │ +o14 = {x - y, y, x}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : assert isCentral cA''
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  i13 : cone(A, x)
│ │ │ │  
│ │ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │ │  
│ │ │ │  o13 : Hyperplane Arrangement
│ │ │ │  i14 : cA'' = trim cone(A, x)
│ │ │ │  
│ │ │ │ -o14 = {y, x, x - y}
│ │ │ │ +o14 = {x - y, y, x}
│ │ │ │  
│ │ │ │  o14 : Hyperplane Arrangement
│ │ │ │  i15 : assert isCentral cA''
│ │ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ │ │  When the second input is a _S_y_m_b_o_l, this method creates a new ring from the
│ │ │ │  underlying ring of $A$ by adjoining the symbol as a variable and constructs the
│ │ │ │  cone in this new ring.
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : restriction(A,1)
│ │ │ @@ -118,15 +118,15 @@
│ │ │  o4 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {x - z, z, x}
│ │ │ +o5 = {z, x, x - z}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$, where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the Euler restriction is free as well, and the modules of logarithmic derivations form a short exact sequence.  See the paper of Abe, Terao and Wakefield for details.
    
│ │ │ @@ -137,15 +137,15 @@
│ │ │                
    i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : prune image der(A,m)
│ │ │ @@ -186,16 +186,16 @@
│ │ │                
    i12 : A = arrangement "bracelet";
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │  
│ │ │  o13 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : C = restriction(A,0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,36 +33,36 @@
│ │ │ │  i2 : A = arrangement {x,y,z,x-y,x-z}
│ │ │ │  
│ │ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │ │  
│ │ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : restriction(A,1)
│ │ │ │  
│ │ │ │  o4 = {x, z, x, x - z}
│ │ │ │  
│ │ │ │  o4 : Hyperplane Arrangement
│ │ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │ │  
│ │ │ │ -o5 = {x - z, z, x}
│ │ │ │ +o5 = {z, x, x - z}
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$,
│ │ │ │  where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the
│ │ │ │  Euler restriction is free as well, and the modules of _l_o_g_a_r_i_t_h_m_i_c_ _d_e_r_i_v_a_t_i_o_n_s
│ │ │ │  form a short exact sequence. See the paper of Abe, Terao and Wakefield for
│ │ │ │  details.
│ │ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │ │  
│ │ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : prune image der(A,m)
│ │ │ │  
│ │ │ │        3
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │ @@ -80,16 +80,16 @@
│ │ │ │  
│ │ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │ │  It may be the case that the Euler restriction is free, while the naive
│ │ │ │  restriction is not:
│ │ │ │  i12 : A = arrangement "bracelet";
│ │ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │ │  
│ │ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ │ │  i14 : C = restriction(A,0)
│ │ │ │  
│ │ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_trim_lp__Arrangement_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : A' = trim A
│ │ │  
│ │ │ -o3 = {x + y, y, x}
│ │ │ +o3 = {y, x, x + y}
│ │ │  
│ │ │  o3 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : assert(ring A' === R)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : A = arrangement{x,x,0_R,y,y,y,x+y,x+y,x+y,x+y,x+y}
│ │ │ │  
│ │ │ │  o2 = {x, x, 0, y, y, y, x + y, x + y, x + y, x + y, x + y}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : A' = trim A
│ │ │ │  
│ │ │ │ -o3 = {x + y, y, x}
│ │ │ │ +o3 = {y, x, x + y}
│ │ │ │  
│ │ │ │  o3 : Hyperplane Arrangement
│ │ │ │  i4 : assert(ring A' === R)
│ │ │ │  i5 : assert(trim A' == A')
│ │ │ │  i6 : assert(trim A' == A')
│ │ │ │  Some natural operations produce non-simple hyperplane arrangements.
│ │ │ │  i7 : A'' = restriction(A, y)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_type__B_lp__Z__Z_cm__Ring_rp.html
│ │ │ @@ -130,16 +130,16 @@
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : ring A3
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,16 +51,16 @@
│ │ │ │  
│ │ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │ │         1   1    2   1    2   2
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  i6 : trim A3
│ │ │ │  
│ │ │ │ -o6 = {x , x , x  + x }
│ │ │ │ -       2   1   1    2
│ │ │ │ +o6 = {x  + x , x , x }
│ │ │ │ +       1    2   2   1
│ │ │ │  
│ │ │ │  o6 : Hyperplane Arrangement
│ │ │ │  i7 : ring A3
│ │ │ │  
│ │ │ │       ZZ
│ │ │ │  o7 = --[x ..x ]
│ │ │ │        2  1   2
│ │ ├── ./usr/share/doc/Macaulay2/IncidenceCorrespondenceCohomology/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  cmVjdXJzaXZlRGl2aWRlZENvaG9tb2xvZ3k=
│ │ │  #:len=3143
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgZGltZW5zaW9uIG9yIGNo
│ │ │  YXJhY3RlciBvZiBzaGVhZiBjb2hvbW9sb2d5IG9mIHR3aXN0cyBvZiBkaXZpZGVkIHBvd2VycyBv
│ │ ├── ./usr/share/doc/Macaulay2/IntegerProgramming/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=36
│ │ │  YWRhcHRlZE1vbm9taWFsT3JkZXIoLi4uLEZpZWxkPT4uLi4p
│ │ │  #:len=286
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1thZGFwdGVkTW9ub21pYWxPcmRlcixGaWVsZF0sImFk
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  aW50ZWdyYWxDbG9zdXJlKC4uLixMaW1pdD0+Li4uKQ==
│ │ │  #:len=1120
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZG8gYSBwYXJ0aWFsIGludGVncmFsIGNs
│ │ │  b3N1cmUiLCAibGluZW51bSIgPT4gMTM5NywgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  i3 : R = S/f
│ │ │  
│ │ │  o3 = R
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : time R' = integralClosure R
│ │ │ - -- used 0.387896s (cpu); 0.303595s (thread); 0s (gc)
│ │ │ + -- used 0.495745s (cpu); 0.40616s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -83,15 +83,15 @@
│ │ │  i9 : R = S/f
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.508329s (cpu); 0.342291s (thread); 0s (gc)
│ │ │ + -- used 0.481682s (cpu); 0.369384s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -150,15 +150,15 @@
│ │ │  i15 : R = S/f
│ │ │  
│ │ │  o15 = R
│ │ │  
│ │ │  o15 : QuotientRing
│ │ │  
│ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.507202s (cpu); 0.349308s (thread); 0s (gc)
│ │ │ + -- used 0.634005s (cpu); 0.43794s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -208,15 +208,15 @@
│ │ │  i20 : R = S/f
│ │ │  
│ │ │  o20 = R
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │  
│ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.400249s (cpu); 0.317688s (thread); 0s (gc)
│ │ │ + -- used 0.514023s (cpu); 0.406189s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
│ │ │  
│ │ │  i22 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -266,15 +266,15 @@
│ │ │  i25 : R = S/f
│ │ │  
│ │ │  o25 = R
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │  
│ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc)
│ │ │ + -- used 1.05887s (cpu); 0.754982s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
│ │ │  
│ │ │  i27 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -324,15 +324,15 @@
│ │ │  i30 : R = S/f
│ │ │  
│ │ │  o30 = R
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │  
│ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.508579s (cpu); 0.336293s (thread); 0s (gc)
│ │ │ + -- used 0.690025s (cpu); 0.481698s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
│ │ │  
│ │ │  i32 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -382,15 +382,15 @@
│ │ │  i35 : R = S/f
│ │ │  
│ │ │  o35 = R
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │  
│ │ │  i36 : time R' = integralClosure R
│ │ │ - -- used 0.0428345s (cpu); 0.0428212s (thread); 0s (gc)
│ │ │ + -- used 0.0593594s (cpu); 0.0593584s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
│ │ │  
│ │ │  i37 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -432,15 +432,15 @@
│ │ │  i40 : R = S/I
│ │ │  
│ │ │  o40 = R
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │  
│ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.167523s (cpu); 0.0828249s (thread); 0s (gc)
│ │ │ + -- used 0.237179s (cpu); 0.127037s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
│ │ │  
│ │ │  i42 : icFractions R
│ │ │  
│ │ │ @@ -467,15 +467,15 @@
│ │ │  i45 : R = S/I
│ │ │  
│ │ │  o45 = R
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │  
│ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0639871s (cpu); 0.0639869s (thread); 0s (gc)
│ │ │ + -- used 0.087973s (cpu); 0.0879644s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
│ │ │  
│ │ │  i47 : icFractions R
│ │ │  
│ │ │ @@ -501,15 +501,15 @@
│ │ │  i50 : R = S/I
│ │ │  
│ │ │  o50 = R
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │  
│ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0434306s (cpu); 0.0434315s (thread); 0s (gc)
│ │ │ + -- used 0.225128s (cpu); 0.107125s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
│ │ │  
│ │ │  i52 : icFractions R
│ │ │  
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i55 : R = S/I
│ │ │  
│ │ │  o55 = R
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │  
│ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.175803s (cpu); 0.0898169s (thread); 0s (gc)
│ │ │ + -- used 0.0651322s (cpu); 0.0650082s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -633,15 +633,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc)
│ │ │ + -- used 0.0793309s (cpu); 0.0793339s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -722,15 +722,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc)
│ │ │ + -- used 0.659194s (cpu); 0.428289s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -750,15 +750,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc)
│ │ │ + -- used 0.846712s (cpu); 0.517999s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -778,20 +778,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000532358 sec #minors 4]
│ │ │ + [jacobian time .000693675 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .228455 sec  #fractions 6]
│ │ │ - [step 1:   time .246733 sec  #fractions 6]
│ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc)
│ │ │ + [step 0:   time .133476 sec  #fractions 6]
│ │ │ + [step 1:   time .333505 sec  #fractions 6]
│ │ │ + -- used 0.471731s (cpu); 0.365342s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -811,20 +811,20 @@
│ │ │  i89 : R = S/sub(I,S)
│ │ │  
│ │ │  o89 = R
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000530094 sec #minors 4]
│ │ │ + [jacobian time .000698962 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .210442 sec  #fractions 6]
│ │ │ - [step 1:   time .269174 sec  #fractions 6]
│ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc)
│ │ │ + [step 0:   time .270247 sec  #fractions 6]
│ │ │ + [step 1:   time .348455 sec  #fractions 6]
│ │ │ + -- used 0.623774s (cpu); 0.394577s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -844,20 +844,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000774762 sec #minors 1]
│ │ │ + [jacobian time .000919903 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .271111 sec  #fractions 6]
│ │ │ - [step 1:   time .874897 sec  #fractions 6]
│ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc)
│ │ │ + [step 0:   time .328666 sec  #fractions 6]
│ │ │ + [step 1:   time .943765 sec  #fractions 6]
│ │ │ + -- used 1.27766s (cpu); 0.801939s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
│ │ │  
│ │ │  i95 : icFractions R
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out
│ │ │ @@ -1,50 +1,50 @@
│ │ │  -- -*- M2-comint -*- hash: 13177954069434615273
│ │ │  
│ │ │  i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
│ │ │  
│ │ │  i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000522279 sec #minors 3]
│ │ │ + [jacobian time .000754032 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00233339 seconds
│ │ │ -      idlizer1:  .00747003 seconds
│ │ │ -      idlizer2:  .00844957 seconds
│ │ │ -      minpres:   .00811702 seconds
│ │ │ -  time .0371527 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .0033531 seconds
│ │ │ +      idlizer1:  .0125993 seconds
│ │ │ +      idlizer2:  .0154869 seconds
│ │ │ +      minpres:   .0114944 seconds
│ │ │ +  time .0583676 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00248298 seconds
│ │ │ -      idlizer1:  .0119562 seconds
│ │ │ -      idlizer2:  .0111 seconds
│ │ │ -      minpres:   .0117312 seconds
│ │ │ -  time .0484066 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00315316 seconds
│ │ │ +      idlizer1:  .0158136 seconds
│ │ │ +      idlizer2:  .0165777 seconds
│ │ │ +      minpres:   .0176091 seconds
│ │ │ +  time .0697502 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00413852 seconds
│ │ │ -      idlizer1:  .0169512 seconds
│ │ │ -      idlizer2:  .0143372 seconds
│ │ │ -      minpres:   .0137525 seconds
│ │ │ -  time .203496 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .175739 seconds
│ │ │ +      idlizer1:  .0177073 seconds
│ │ │ +      idlizer2:  .0138726 seconds
│ │ │ +      minpres:   .0165695 seconds
│ │ │ +  time .241448 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00279962 seconds
│ │ │ -      idlizer1:  .0141692 seconds
│ │ │ -      idlizer2:  .0166105 seconds
│ │ │ -      minpres:   .0188805 seconds
│ │ │ -  time .0680688 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00416642 seconds
│ │ │ +      idlizer1:  .0206092 seconds
│ │ │ +      idlizer2:  .0202314 seconds
│ │ │ +      minpres:   .0305962 seconds
│ │ │ +  time .0952017 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .0025556 seconds
│ │ │ -      idlizer1:  .00935896 seconds
│ │ │ -      idlizer2:  .0181192 seconds
│ │ │ -      minpres:   .0119314 seconds
│ │ │ -  time .0555899 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00392613 seconds
│ │ │ +      idlizer1:  .0148562 seconds
│ │ │ +      idlizer2:  .0294321 seconds
│ │ │ +      minpres:   .0174288 seconds
│ │ │ +  time .0858714 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00261851 seconds
│ │ │ -      idlizer1:  .00849894 seconds
│ │ │ -  time .0184597 sec  #fractions 5]
│ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00496138 seconds
│ │ │ +      idlizer1:  .0145131 seconds
│ │ │ +  time .0307322 sec  #fractions 5]
│ │ │ + -- used 0.587091s (cpu); 0.470111s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │  i3 : trim ideal R'
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  
│ │ │                  2      2    2        2   2 2     2
│ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time integralClosure J
│ │ │ - -- used 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ + -- used 2.19654s (cpu); 0.986132s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ + -- used 1.34763s (cpu); 0.713726s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.387896s (cpu); 0.303595s (thread); 0s (gc)
│ │ │ + -- used 0.495745s (cpu); 0.40616s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -191,15 +191,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.508329s (cpu); 0.342291s (thread); 0s (gc)
│ │ │ + -- used 0.481682s (cpu); 0.369384s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -278,15 +278,15 @@
│ │ │  
│ │ │  o15 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.507202s (cpu); 0.349308s (thread); 0s (gc)
│ │ │ + -- used 0.634005s (cpu); 0.43794s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -353,15 +353,15 @@
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.400249s (cpu); 0.317688s (thread); 0s (gc)
│ │ │ + -- used 0.514023s (cpu); 0.406189s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -428,15 +428,15 @@
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc)
│ │ │ + -- used 1.05887s (cpu); 0.754982s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -503,15 +503,15 @@
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.508579s (cpu); 0.336293s (thread); 0s (gc)
│ │ │ + -- used 0.690025s (cpu); 0.481698s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -578,15 +578,15 @@
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.0428345s (cpu); 0.0428212s (thread); 0s (gc)
│ │ │ + -- used 0.0593594s (cpu); 0.0593584s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -648,15 +648,15 @@
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.167523s (cpu); 0.0828249s (thread); 0s (gc)
│ │ │ + -- used 0.237179s (cpu); 0.127037s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -700,15 +700,15 @@
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0639871s (cpu); 0.0639869s (thread); 0s (gc)
│ │ │ + -- used 0.087973s (cpu); 0.0879644s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -751,15 +751,15 @@
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0434306s (cpu); 0.0434315s (thread); 0s (gc)
│ │ │ + -- used 0.225128s (cpu); 0.107125s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -803,15 +803,15 @@
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.175803s (cpu); 0.0898169s (thread); 0s (gc)
│ │ │ + -- used 0.0651322s (cpu); 0.0650082s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -938,15 +938,15 @@
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc)
│ │ │ + -- used 0.0793309s (cpu); 0.0793339s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1062,15 +1062,15 @@
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc)
│ │ │ + -- used 0.659194s (cpu); 0.428289s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1104,15 +1104,15 @@
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc)
│ │ │ + -- used 0.846712s (cpu); 0.517999s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1146,20 +1146,20 @@
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000532358 sec #minors 4]
│ │ │ + [jacobian time .000693675 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .228455 sec  #fractions 6]
│ │ │ - [step 1:   time .246733 sec  #fractions 6]
│ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc)
│ │ │ + [step 0:   time .133476 sec  #fractions 6]
│ │ │ + [step 1:   time .333505 sec  #fractions 6]
│ │ │ + -- used 0.471731s (cpu); 0.365342s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1193,20 +1193,20 @@
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000530094 sec #minors 4]
│ │ │ + [jacobian time .000698962 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .210442 sec  #fractions 6]
│ │ │ - [step 1:   time .269174 sec  #fractions 6]
│ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc)
│ │ │ + [step 0:   time .270247 sec  #fractions 6]
│ │ │ + [step 1:   time .348455 sec  #fractions 6]
│ │ │ + -- used 0.623774s (cpu); 0.394577s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -1243,20 +1243,20 @@
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000774762 sec #minors 1]
│ │ │ + [jacobian time .000919903 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .271111 sec  #fractions 6]
│ │ │ - [step 1:   time .874897 sec  #fractions 6]
│ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc)
│ │ │ + [step 0:   time .328666 sec  #fractions 6]
│ │ │ + [step 1:   time .943765 sec  #fractions 6]
│ │ │ + -- used 1.27766s (cpu); 0.801939s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  o2 : Ideal of S
│ │ │ │  i3 : R = S/f
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : time R' = integralClosure R
│ │ │ │ - -- used 0.387896s (cpu); 0.303595s (thread); 0s (gc)
│ │ │ │ + -- used 0.495745s (cpu); 0.40616s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = R'
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : netList (ideal R')_*
│ │ │ │  
│ │ │ │       +------------------------------------------------------------------------+
│ │ │ │ @@ -109,15 +109,15 @@
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : R = S/f
│ │ │ │  
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.508329s (cpu); 0.342291s (thread); 0s (gc)
│ │ │ │ + -- used 0.481682s (cpu); 0.369384s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = R'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -199,15 +199,15 @@
│ │ │ │  o14 : Ideal of S
│ │ │ │  i15 : R = S/f
│ │ │ │  
│ │ │ │  o15 = R
│ │ │ │  
│ │ │ │  o15 : QuotientRing
│ │ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.507202s (cpu); 0.349308s (thread); 0s (gc)
│ │ │ │ + -- used 0.634005s (cpu); 0.43794s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = R'
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  i17 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -281,15 +281,15 @@
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : R = S/f
│ │ │ │  
│ │ │ │  o20 = R
│ │ │ │  
│ │ │ │  o20 : QuotientRing
│ │ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ │ - -- used 0.400249s (cpu); 0.317688s (thread); 0s (gc)
│ │ │ │ + -- used 0.514023s (cpu); 0.406189s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = R'
│ │ │ │  
│ │ │ │  o21 : QuotientRing
│ │ │ │  i22 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -363,15 +363,15 @@
│ │ │ │  o24 : Ideal of S
│ │ │ │  i25 : R = S/f
│ │ │ │  
│ │ │ │  o25 = R
│ │ │ │  
│ │ │ │  o25 : QuotientRing
│ │ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc)
│ │ │ │ + -- used 1.05887s (cpu); 0.754982s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o26 = R'
│ │ │ │  
│ │ │ │  o26 : QuotientRing
│ │ │ │  i27 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -445,15 +445,15 @@
│ │ │ │  o29 : Ideal of S
│ │ │ │  i30 : R = S/f
│ │ │ │  
│ │ │ │  o30 = R
│ │ │ │  
│ │ │ │  o30 : QuotientRing
│ │ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ │ - -- used 0.508579s (cpu); 0.336293s (thread); 0s (gc)
│ │ │ │ + -- used 0.690025s (cpu); 0.481698s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = R'
│ │ │ │  
│ │ │ │  o31 : QuotientRing
│ │ │ │  i32 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │ @@ -527,15 +527,15 @@
│ │ │ │  o34 : Ideal of S
│ │ │ │  i35 : R = S/f
│ │ │ │  
│ │ │ │  o35 = R
│ │ │ │  
│ │ │ │  o35 : QuotientRing
│ │ │ │  i36 : time R' = integralClosure R
│ │ │ │ - -- used 0.0428345s (cpu); 0.0428212s (thread); 0s (gc)
│ │ │ │ + -- used 0.0593594s (cpu); 0.0593584s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o36 = R'
│ │ │ │  
│ │ │ │  o36 : QuotientRing
│ │ │ │  i37 : netList (ideal R')_*
│ │ │ │  
│ │ │ │        +-----------+
│ │ │ │ @@ -573,15 +573,15 @@
│ │ │ │  o39 : Ideal of S
│ │ │ │  i40 : R = S/I
│ │ │ │  
│ │ │ │  o40 = R
│ │ │ │  
│ │ │ │  o40 : QuotientRing
│ │ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.167523s (cpu); 0.0828249s (thread); 0s (gc)
│ │ │ │ + -- used 0.237179s (cpu); 0.127037s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o41 = R'
│ │ │ │  
│ │ │ │  o41 : QuotientRing
│ │ │ │  i42 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -603,15 +603,15 @@
│ │ │ │  o44 : Ideal of S
│ │ │ │  i45 : R = S/I
│ │ │ │  
│ │ │ │  o45 = R
│ │ │ │  
│ │ │ │  o45 : QuotientRing
│ │ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.0639871s (cpu); 0.0639869s (thread); 0s (gc)
│ │ │ │ + -- used 0.087973s (cpu); 0.0879644s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = R'
│ │ │ │  
│ │ │ │  o46 : QuotientRing
│ │ │ │  i47 : icFractions R
│ │ │ │  
│ │ │ │         b*d
│ │ │ │ @@ -632,15 +632,15 @@
│ │ │ │  o49 : Ideal of S
│ │ │ │  i50 : R = S/I
│ │ │ │  
│ │ │ │  o50 = R
│ │ │ │  
│ │ │ │  o50 : QuotientRing
│ │ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ │ - -- used 0.0434306s (cpu); 0.0434315s (thread); 0s (gc)
│ │ │ │ + -- used 0.225128s (cpu); 0.107125s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = R'
│ │ │ │  
│ │ │ │  o51 : QuotientRing
│ │ │ │  i52 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -662,15 +662,15 @@
│ │ │ │  o54 : Ideal of S
│ │ │ │  i55 : R = S/I
│ │ │ │  
│ │ │ │  o55 = R
│ │ │ │  
│ │ │ │  o55 : QuotientRing
│ │ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ │ - -- used 0.175803s (cpu); 0.0898169s (thread); 0s (gc)
│ │ │ │ + -- used 0.0651322s (cpu); 0.0650082s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o56 = R'
│ │ │ │  
│ │ │ │  o56 : QuotientRing
│ │ │ │  i57 : icFractions R
│ │ │ │  
│ │ │ │          2
│ │ │ │ @@ -755,15 +755,15 @@
│ │ │ │  o65 : BettiTally
│ │ │ │  i66 : R = S/I
│ │ │ │  
│ │ │ │  o66 = R
│ │ │ │  
│ │ │ │  o66 : QuotientRing
│ │ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc)
│ │ │ │ + -- used 0.0793309s (cpu); 0.0793339s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o67 = R'
│ │ │ │  
│ │ │ │  o67 : QuotientRing
│ │ │ │  i68 : icFractions R
│ │ │ │  
│ │ │ │          2    2
│ │ │ │ @@ -839,15 +839,15 @@
│ │ │ │  o76 : BettiTally
│ │ │ │  i77 : R = S/I
│ │ │ │  
│ │ │ │  o77 = R
│ │ │ │  
│ │ │ │  o77 : QuotientRing
│ │ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc)
│ │ │ │ + -- used 0.659194s (cpu); 0.428289s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o78 = R'
│ │ │ │  
│ │ │ │  o78 : QuotientRing
│ │ │ │  i79 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -863,15 +863,15 @@
│ │ │ │  o80 : PolynomialRing
│ │ │ │  i81 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o81 = R
│ │ │ │  
│ │ │ │  o81 : QuotientRing
│ │ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc)
│ │ │ │ + -- used 0.846712s (cpu); 0.517999s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o82 = R'
│ │ │ │  
│ │ │ │  o82 : QuotientRing
│ │ │ │  i83 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -887,20 +887,20 @@
│ │ │ │  o84 : PolynomialRing
│ │ │ │  i85 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o85 = R
│ │ │ │  
│ │ │ │  o85 : QuotientRing
│ │ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ │ - [jacobian time .000532358 sec #minors 4]
│ │ │ │ + [jacobian time .000693675 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .228455 sec  #fractions 6]
│ │ │ │ - [step 1:   time .246733 sec  #fractions 6]
│ │ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .133476 sec  #fractions 6]
│ │ │ │ + [step 1:   time .333505 sec  #fractions 6]
│ │ │ │ + -- used 0.471731s (cpu); 0.365342s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o86 = R'
│ │ │ │  
│ │ │ │  o86 : QuotientRing
│ │ │ │  i87 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -916,20 +916,20 @@
│ │ │ │  o88 : PolynomialRing
│ │ │ │  i89 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o89 = R
│ │ │ │  
│ │ │ │  o89 : QuotientRing
│ │ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ │ - [jacobian time .000530094 sec #minors 4]
│ │ │ │ + [jacobian time .000698962 sec #minors 4]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .210442 sec  #fractions 6]
│ │ │ │ - [step 1:   time .269174 sec  #fractions 6]
│ │ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .270247 sec  #fractions 6]
│ │ │ │ + [step 1:   time .348455 sec  #fractions 6]
│ │ │ │ + -- used 0.623774s (cpu); 0.394577s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o90 = R'
│ │ │ │  
│ │ │ │  o90 : QuotientRing
│ │ │ │  i91 : icFractions R
│ │ │ │  
│ │ │ │          2    2   2     3      2
│ │ │ │ @@ -948,20 +948,20 @@
│ │ │ │  i93 : R = S/sub(I,S)
│ │ │ │  
│ │ │ │  o93 = R
│ │ │ │  
│ │ │ │  o93 : QuotientRing
│ │ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos,
│ │ │ │  StartWithOneMinor}, Verbosity => 1)
│ │ │ │ - [jacobian time .000774762 sec #minors 1]
│ │ │ │ + [jacobian time .000919903 sec #minors 1]
│ │ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │ - [step 0:   time .271111 sec  #fractions 6]
│ │ │ │ - [step 1:   time .874897 sec  #fractions 6]
│ │ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc)
│ │ │ │ + [step 0:   time .328666 sec  #fractions 6]
│ │ │ │ + [step 1:   time .943765 sec  #fractions 6]
│ │ │ │ + -- used 1.27766s (cpu); 0.801939s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o94 = R'
│ │ │ │  
│ │ │ │  o94 : QuotientRing
│ │ │ │  i95 : icFractions R
│ │ │ │  
│ │ │ │          2    2          2   2     3      2
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html
│ │ │ @@ -76,52 +76,52 @@
│ │ │              
│ │ │                
    i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000522279 sec #minors 3]
│ │ │ + [jacobian time .000754032 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00233339 seconds
│ │ │ -      idlizer1:  .00747003 seconds
│ │ │ -      idlizer2:  .00844957 seconds
│ │ │ -      minpres:   .00811702 seconds
│ │ │ -  time .0371527 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .0033531 seconds
│ │ │ +      idlizer1:  .0125993 seconds
│ │ │ +      idlizer2:  .0154869 seconds
│ │ │ +      minpres:   .0114944 seconds
│ │ │ +  time .0583676 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00248298 seconds
│ │ │ -      idlizer1:  .0119562 seconds
│ │ │ -      idlizer2:  .0111 seconds
│ │ │ -      minpres:   .0117312 seconds
│ │ │ -  time .0484066 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00315316 seconds
│ │ │ +      idlizer1:  .0158136 seconds
│ │ │ +      idlizer2:  .0165777 seconds
│ │ │ +      minpres:   .0176091 seconds
│ │ │ +  time .0697502 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00413852 seconds
│ │ │ -      idlizer1:  .0169512 seconds
│ │ │ -      idlizer2:  .0143372 seconds
│ │ │ -      minpres:   .0137525 seconds
│ │ │ -  time .203496 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .175739 seconds
│ │ │ +      idlizer1:  .0177073 seconds
│ │ │ +      idlizer2:  .0138726 seconds
│ │ │ +      minpres:   .0165695 seconds
│ │ │ +  time .241448 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00279962 seconds
│ │ │ -      idlizer1:  .0141692 seconds
│ │ │ -      idlizer2:  .0166105 seconds
│ │ │ -      minpres:   .0188805 seconds
│ │ │ -  time .0680688 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00416642 seconds
│ │ │ +      idlizer1:  .0206092 seconds
│ │ │ +      idlizer2:  .0202314 seconds
│ │ │ +      minpres:   .0305962 seconds
│ │ │ +  time .0952017 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .0025556 seconds
│ │ │ -      idlizer1:  .00935896 seconds
│ │ │ -      idlizer2:  .0181192 seconds
│ │ │ -      minpres:   .0119314 seconds
│ │ │ -  time .0555899 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00392613 seconds
│ │ │ +      idlizer1:  .0148562 seconds
│ │ │ +      idlizer2:  .0294321 seconds
│ │ │ +      minpres:   .0174288 seconds
│ │ │ +  time .0858714 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00261851 seconds
│ │ │ -      idlizer1:  .00849894 seconds
│ │ │ -  time .0184597 sec  #fractions 5]
│ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .00496138 seconds
│ │ │ +      idlizer1:  .0145131 seconds
│ │ │ +  time .0307322 sec  #fractions 5]
│ │ │ + -- used 0.587091s (cpu); 0.470111s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,52 +12,52 @@
│ │ │ │              displayed. A value of 0 means: keep quiet.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  When the computation takes a considerable time, this function can be used to
│ │ │ │  decide if it will ever finish, or to get a feel for what is happening during
│ │ │ │  the computation.
│ │ │ │  i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
│ │ │ │  i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ │ - [jacobian time .000522279 sec #minors 3]
│ │ │ │ + [jacobian time .000754032 sec #minors 3]
│ │ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │ │  
│ │ │ │   [step 0:
│ │ │ │ -      radical (use minprimes) .00233339 seconds
│ │ │ │ -      idlizer1:  .00747003 seconds
│ │ │ │ -      idlizer2:  .00844957 seconds
│ │ │ │ -      minpres:   .00811702 seconds
│ │ │ │ -  time .0371527 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .0033531 seconds
│ │ │ │ +      idlizer1:  .0125993 seconds
│ │ │ │ +      idlizer2:  .0154869 seconds
│ │ │ │ +      minpres:   .0114944 seconds
│ │ │ │ +  time .0583676 sec  #fractions 4]
│ │ │ │   [step 1:
│ │ │ │ -      radical (use minprimes) .00248298 seconds
│ │ │ │ -      idlizer1:  .0119562 seconds
│ │ │ │ -      idlizer2:  .0111 seconds
│ │ │ │ -      minpres:   .0117312 seconds
│ │ │ │ -  time .0484066 sec  #fractions 4]
│ │ │ │ +      radical (use minprimes) .00315316 seconds
│ │ │ │ +      idlizer1:  .0158136 seconds
│ │ │ │ +      idlizer2:  .0165777 seconds
│ │ │ │ +      minpres:   .0176091 seconds
│ │ │ │ +  time .0697502 sec  #fractions 4]
│ │ │ │   [step 2:
│ │ │ │ -      radical (use minprimes) .00413852 seconds
│ │ │ │ -      idlizer1:  .0169512 seconds
│ │ │ │ -      idlizer2:  .0143372 seconds
│ │ │ │ -      minpres:   .0137525 seconds
│ │ │ │ -  time .203496 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .175739 seconds
│ │ │ │ +      idlizer1:  .0177073 seconds
│ │ │ │ +      idlizer2:  .0138726 seconds
│ │ │ │ +      minpres:   .0165695 seconds
│ │ │ │ +  time .241448 sec  #fractions 5]
│ │ │ │   [step 3:
│ │ │ │ -      radical (use minprimes) .00279962 seconds
│ │ │ │ -      idlizer1:  .0141692 seconds
│ │ │ │ -      idlizer2:  .0166105 seconds
│ │ │ │ -      minpres:   .0188805 seconds
│ │ │ │ -  time .0680688 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00416642 seconds
│ │ │ │ +      idlizer1:  .0206092 seconds
│ │ │ │ +      idlizer2:  .0202314 seconds
│ │ │ │ +      minpres:   .0305962 seconds
│ │ │ │ +  time .0952017 sec  #fractions 5]
│ │ │ │   [step 4:
│ │ │ │ -      radical (use minprimes) .0025556 seconds
│ │ │ │ -      idlizer1:  .00935896 seconds
│ │ │ │ -      idlizer2:  .0181192 seconds
│ │ │ │ -      minpres:   .0119314 seconds
│ │ │ │ -  time .0555899 sec  #fractions 5]
│ │ │ │ +      radical (use minprimes) .00392613 seconds
│ │ │ │ +      idlizer1:  .0148562 seconds
│ │ │ │ +      idlizer2:  .0294321 seconds
│ │ │ │ +      minpres:   .0174288 seconds
│ │ │ │ +  time .0858714 sec  #fractions 5]
│ │ │ │   [step 5:
│ │ │ │ -      radical (use minprimes) .00261851 seconds
│ │ │ │ -      idlizer1:  .00849894 seconds
│ │ │ │ -  time .0184597 sec  #fractions 5]
│ │ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc)
│ │ │ │ +      radical (use minprimes) .00496138 seconds
│ │ │ │ +      idlizer1:  .0145131 seconds
│ │ │ │ +  time .0307322 sec  #fractions 5]
│ │ │ │ + -- used 0.587091s (cpu); 0.470111s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = R'
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : trim ideal R'
│ │ │ │  
│ │ │ │                       3   2                     2 2    4           4
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html
│ │ │ @@ -114,29 +114,29 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time integralClosure J
│ │ │ - -- used 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ + -- used 2.19654s (cpu); 0.986132s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ + -- used 1.34763s (cpu); 0.713726s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,25 +46,25 @@
│ │ │ │  i3 : J = ideal jacobian ideal F
│ │ │ │  
│ │ │ │                  2      2    2        2   2 2     2
│ │ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time integralClosure J
│ │ │ │ - -- used 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ │ + -- used 2.19654s (cpu); 0.986132s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ │ - -- used 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ │ + -- used 1.34763s (cpu); 0.713726s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  c2NocmVpZXJHcmFwaA==
│ │ │  #:len=1712
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NocmVpZXIgZ3JhcGggb2YgYSBmaW5p
│ │ │  dGUgZ3JvdXAiLCAibGluZW51bSIgPT4gMjYyLCBJbnB1dHMgPT4ge1NQQU57VFR7IkcifSwiLCAi
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00739572s elapsed
│ │ │ + -- .003481s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (ζ ζ  + ζ ζ  + ζ ζ  + ζ ζ   + 1 + ζ   + ζ  ζ  + ζ  ζ   + ζ  ζ   + ζ  )T 
│ │ │ @@ -51,10 +51,10 @@
│ │ │           0   1
│ │ │  
│ │ │  i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000676866s elapsed
│ │ │ + -- .000770832s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out
│ │ │ @@ -23,23 +23,23 @@
│ │ │  o3 = QQ[x..z] <- <| 0 -1 0  |, | 0 -1 0 |>
│ │ │                    | 1 0  0  |  | 1 0  0 |
│ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ + -- used 0.579158s (cpu); 0.400719s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   4    4
│ │ │  o4 = {z , x  + y , x  + y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ + -- used 0.493081s (cpu); 0.406421s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -90,23 +90,23 @@
│ │ │       ------------------------------------------------------------------------
│ │ │          2 6    8
│ │ │       90y z  + z }
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ + -- used 0.163789s (cpu); 0.0693537s (thread); 0s (gc)
│ │ │  
│ │ │            3       3
│ │ │  o6 = {1, x y - x*y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ + -- used 1.58205s (cpu); 1.20675s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .422203s elapsed
│ │ │ + -- .374046s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -32,15 +32,15 @@
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .339469s elapsed
│ │ │ + -- .320883s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .735714s elapsed
│ │ │ + -- .567699s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ - -- .126449s elapsed
│ │ │ + -- .107399s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00739572s elapsed
│ │ │ + -- .003481s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (ζ ζ  + ζ ζ  + ζ ζ  + ζ ζ   + 1 + ζ   + ζ  ζ  + ζ  ζ   + ζ  ζ   + ζ  )T 
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000676866s elapsed
    
│ │ │ + -- .000770832s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    For the programmer
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │       | 0  -1 1 |
│ │ │ │  
│ │ │ │  o3 : DiagonalAction
│ │ │ │  i4 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ │ - -- .00739572s elapsed
│ │ │ │ + -- .003481s elapsed
│ │ │ │  
│ │ │ │                    -1    -1       2 2              -2    -1 -1    -2  2
│ │ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ
│ │ │ │  )T  +
│ │ │ │             0 1    1     0        0 1    0    1    1     0  1     0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │ │ @@ -57,13 +57,13 @@
│ │ │ │  
│ │ │ │  o5 : ZZ[ζ ..ζ ][T]
│ │ │ │           0   1
│ │ │ │  i6 : T.cache.?equivariantHilbert
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ │ - -- .000676866s elapsed
│ │ │ │ + -- .000770832s elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _e_q_u_i_v_a_r_i_a_n_t_H_i_l_b_e_r_t is a _s_y_m_b_o_l.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:225:0.
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html
│ │ │ @@ -97,26 +97,26 @@
│ │ │            
│ │ │          
│ │ │          
    The two algorithms used in primaryInvariants are timed. One sees that the Dade algorithm is faster, however the primary invariants output are all of degree 8 and have ugly coefficients.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ + -- used 0.579158s (cpu); 0.400719s (thread); 0s (gc)
│ │ │  
│ │ │         2   2    2   4    4
│ │ │  o4 = {z , x  + y , x  + y }
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ + -- used 0.493081s (cpu); 0.406421s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -173,26 +173,26 @@
│ │ │            
    
│ │ │          
    The extra work done by the default algorithm to ensure an optimal hsop is rewarded by needing to calculate a smaller collection of corresponding secondary invariants.  In fact, it has proved quicker overall to calculate the invariant ring based on the optimal algorithm rather than the Dade algorithm.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ + -- used 0.163789s (cpu); 0.0693537s (thread); 0s (gc)
│ │ │  
│ │ │            3       3
│ │ │  o6 = {1, x y - x*y }
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ + -- used 1.58205s (cpu); 1.20675s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  The two algorithms used in _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s are timed. One sees that the Dade
│ │ │ │  algorithm is faster, however the primary invariants output are all of degree 8
│ │ │ │  and have ugly coefficients.
│ │ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ │ - -- used 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ │ + -- used 0.579158s (cpu); 0.400719s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2   2    2   4    4
│ │ │ │  o4 = {z , x  + y , x  + y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ │ + -- used 0.493081s (cpu); 0.406421s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                     8                 7                   6 2
│ │ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     5 3                  4 4                 3 5
│ │ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -138,22 +138,22 @@
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  The extra work done by the default algorithm to ensure an optimal hsop is
│ │ │ │  rewarded by needing to calculate a smaller collection of corresponding
│ │ │ │  secondary invariants. In fact, it has proved quicker overall to calculate the
│ │ │ │  invariant ring based on the optimal algorithm rather than the Dade algorithm.
│ │ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ │ - -- used 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ │ + -- used 0.163789s (cpu); 0.0693537s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            3       3
│ │ │ │  o6 = {1, x y - x*y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ │ + -- used 1.58205s (cpu); 1.20675s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4
│ │ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime invariants S4
│ │ │ - -- .422203s elapsed
│ │ │ + -- .374046s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -122,15 +122,15 @@
│ │ │              
    		
│ │ │                
    i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .339469s elapsed
│ │ │ + -- .320883s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o3 = R <- <| 0 1 0 0 |, | 0 0 0 1 |>
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .422203s elapsed
│ │ │ │ + -- .374046s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │ │  
│ │ │ │  Warning: stopping condition not met!
│ │ │ │  Output may not generate the entire ring of invariants.
│ │ │ │  Increase value of DegreeBound.
│ │ │ │  
│ │ │ │ - -- .339469s elapsed
│ │ │ │ + -- .320883s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime invariants S4
│ │ │ - -- .735714s elapsed
│ │ │ + -- .567699s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ - -- .126449s elapsed
│ │ │ + -- .107399s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,27 +26,27 @@
│ │ │ │  o3 = R <- <| 0 1 0 0 |, | 0 0 0 1 |>
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .735714s elapsed
│ │ │ │ + -- .567699s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ │ - -- .126449s elapsed
│ │ │ │ + -- .107399s elapsed
│ │ │ │  
│ │ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InverseSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  ZnJvbUR1YWwoTWF0cml4KQ==
│ │ │  #:len=249
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzMyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmcm9tRHVhbCxNYXRyaXgpLCJmcm9tRHVhbChNYXRy
│ │ ├── ./usr/share/doc/Macaulay2/InvolutiveBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  aW52UmVkdWNlKFJpbmdFbGVtZW50LEludm9sdXRpdmVCYXNpcyk=
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaW52UmVkdWNlLFJpbmdFbGVtZW50LEludm9sdXRp
│ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  aXNJc29tb3JwaGljKE1hdHJpeCxNYXRyaXgp
│ │ │  #:len=270
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTMxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0lzb21vcnBoaWMsTWF0cml4LE1hdHJpeCksImlz
│ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out
│ │ │ @@ -156,20 +156,20 @@
│ │ │                     {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_2  x_1  x_0  |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    x_0  0    0    0    0    0    0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   -x_0^2 -x_2 x_1 |
│ │ │                     {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 1 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    x_0  -x_2 x_1  -x_3 x_2  0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   0      0    0   -x_1^2 -x_3 x_2 |
│ │ │  
│ │ │                                  40
│ │ │  o22 : S-module, subquotient of S
│ │ │  
│ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.69407s elapsed
│ │ │ + -- 1.65188s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .000029024s elapsed
│ │ │ + -- .000029398s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html
│ │ │ @@ -333,23 +333,23 @@
│ │ │                                  40
│ │ │  o22 : S-module, subquotient of S
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.69407s elapsed
│ │ │ + -- 1.65188s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .000029024s elapsed
│ │ │ + -- .000029398s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ │ ├── html2text {}
│ │ │ │ @@ -724,19 +724,19 @@
│ │ │ │  0      0    0   0      0    0   0      0    0   0      0    0   0      0    0
│ │ │ │  0      0    0   0      0    0   0      0    0   0      0    0   -x_1^2 -x_3 x_2
│ │ │ │  |
│ │ │ │  
│ │ │ │                                  40
│ │ │ │  o22 : S-module, subquotient of S
│ │ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ │ - -- 1.69407s elapsed
│ │ │ │ + -- 1.65188s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .000029024s elapsed
│ │ │ │ + -- .000029398s elapsed
│ │ │ │  
│ │ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677
│ │ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098
│ │ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393
│ │ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851
│ │ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./usr/share/doc/Macaulay2/JSON/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  TmFtZVNlcGFyYXRvcg==
│ │ │  #:len=210
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzQzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJOYW1lU2VwYXJhdG9yIiwiTmFtZVNlcGFyYXRvciIs
│ │ ├── ./usr/share/doc/Macaulay2/JSON/example-output/_from__J__S__O__N.out
│ │ │ @@ -37,19 +37,19 @@
│ │ │  
│ │ │  o8 = {1, 2, 3}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-35451-0/0.json
│ │ │ +o9 = /tmp/M2-48911-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ +o10 = /tmp/M2-48911-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -170,22 +170,22 @@
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-35451-0/0.json
    
│ │ │ +o9 = /tmp/M2-48911-0/0.json
│ │ │              
    
│ │ │                
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ +o10 = /tmp/M2-48911-0/0.json
│ │ │  
│ │ │  o10 : File
    
│ │ │              
    
│ │ │                
    i11 : fromJSON openIn jsonFile
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,18 +49,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-35451-0/0.json
│ │ │ │ +o9 = /tmp/M2-48911-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ │ +o10 = /tmp/M2-48911-0/0.json
│ │ │ │  
│ │ │ │  o10 : File
│ │ │ │  i11 : fromJSON openIn jsonFile
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o11 : List
│ │ ├── ./usr/share/doc/Macaulay2/JSONRPC/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  bWFrZVJlcXVlc3QoTGlzdCk=
│ │ │  #:len=236
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUmVxdWVzdCxMaXN0KSwibWFrZVJlcXVlc3Qo
│ │ ├── ./usr/share/doc/Macaulay2/Jets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  amV0cyhaWixBZmZpbmVWYXJpZXR5KQ==
│ │ │  #:len=2469
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGpldHMgb2YgYW4gYWZmaW5lIHZh
│ │ │  cmlldHkiLCAibGluZW51bSIgPT4gMTcxMCwgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out
│ │ │ @@ -17,24 +17,24 @@
│ │ │  o3 = ideal (y0*z0*x2 + x0*z0*y2 + x0*y0*z2 + z0*x1*y1 + y0*x1*z1 + x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1 + x0*z0*y1 + x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o3 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .0023111s elapsed
│ │ │ + -- .00306362s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i5 : elapsedTime radical J2I
│ │ │ - -- .424366s elapsed
│ │ │ + -- .297746s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : I.cache.?jet
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 : elapsedTime jets(3,I)
│ │ │ - -- .00766552s elapsed
│ │ │ + -- .011248s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i9 : I.cache.?jet
│ │ │ @@ -53,23 +53,23 @@
│ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │                                 | 2x0x1-y1       |
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i11 : elapsedTime jets(3,I)
│ │ │ - -- .0025072s elapsed
│ │ │ + -- .00355483s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i12 : elapsedTime jets(2,I)
│ │ │ - -- .00226291s elapsed
│ │ │ + -- .00332444s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │  
│ │ │  i13 : Q = R/I
│ │ │ @@ -148,15 +148,15 @@
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : f.cache.?jet
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : elapsedTime jets(3,f)
│ │ │ - -- .0113717s elapsed
│ │ │ + -- .0155914s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │                                 | t2 2t0t2+t1^2  |
│ │ │                                 | t1 2t0t1       |
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i27 : elapsedTime jets(2,f)
│ │ │ - -- .000732911s elapsed
│ │ │ + -- .00142101s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html
│ │ │ @@ -92,27 +92,27 @@
│ │ │          
    
│ │ │            
    However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is generated by the individual terms in the generators of the ideal of jets. This observation provides an alternative algorithm for computing radicals of jets of monomial ideals, which can be faster than the default radical computation in Macaulay2.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial
│ │ │ │  ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is
│ │ │ │  generated by the individual terms in the generators of the ideal of jets. This
│ │ │ │  observation provides an alternative algorithm for computing radicals of jets of
│ │ │ │  monomial ideals, which can be faster than the default radical computation in
│ │ │ │  Macaulay2.
│ │ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ │ - -- .0023111s elapsed
│ │ │ │ + -- .00306362s elapsed
│ │ │ │  
│ │ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │ │  
│ │ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  i5 : elapsedTime radical J2I
│ │ │ │ - -- .424366s elapsed
│ │ │ │ + -- .297746s elapsed
│ │ │ │  
│ │ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │  
│ │ │  o7 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -151,26 +151,26 @@
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -295,15 +295,15 @@
│ │ │  
│ │ │  o23 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .0023111s elapsed
│ │ │ + -- .00306362s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime radical J2I
│ │ │ - -- .424366s elapsed
│ │ │ + -- .297746s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime jets(3,I)
│ │ │ - -- .00766552s elapsed
│ │ │ + -- .011248s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime jets(3,I)
│ │ │ - -- .0025072s elapsed
│ │ │ + -- .00355483s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │              
    
│ │ │                
    i12 : elapsedTime jets(2,I)
│ │ │ - -- .00226291s elapsed
│ │ │ + -- .00332444s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime jets(3,f)
│ │ │ - -- .0113717s elapsed
│ │ │ + -- .0155914s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -329,15 +329,15 @@
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000732911s elapsed
│ │ │ + -- .00142101s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  o6 = ideal(x  - y)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : I.cache.?jet
│ │ │ │  
│ │ │ │  o7 = false
│ │ │ │  i8 : elapsedTime jets(3,I)
│ │ │ │ - -- .00766552s elapsed
│ │ │ │ + -- .011248s elapsed
│ │ │ │  
│ │ │ │                                                    2                 2
│ │ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i9 : I.cache.?jet
│ │ │ │  
│ │ │ │ @@ -58,22 +58,22 @@
│ │ │ │  
│ │ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │ │                                 | 2x0x1-y1       |
│ │ │ │                                 | x0^2-y0        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i11 : elapsedTime jets(3,I)
│ │ │ │ - -- .0025072s elapsed
│ │ │ │ + -- .00355483s elapsed
│ │ │ │  
│ │ │ │                                                     2                 2
│ │ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i12 : elapsedTime jets(2,I)
│ │ │ │ - -- .00226291s elapsed
│ │ │ │ + -- .00332444s elapsed
│ │ │ │  
│ │ │ │                               2                 2
│ │ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  For quotient rings, data is stored under *.jet. Each jets order gives rise to a
│ │ │ │  different quotient that is stored separately under *.jet.jetsRing (order zero
│ │ │ │ @@ -153,15 +153,15 @@
│ │ │ │  i22 : isWellDefined f
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : f.cache.?jet
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : elapsedTime jets(3,f)
│ │ │ │ - -- .0113717s elapsed
│ │ │ │ + -- .0155914s elapsed
│ │ │ │  
│ │ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3,
│ │ │ │  y3]                                                      2                    2
│ │ │ │  o24 = map (QQ[t0][t1][t2][t3], ------------------------------------------------
│ │ │ │  ----------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                                                        2
│ │ │ │  2
│ │ │ │ @@ -183,15 +183,15 @@
│ │ │ │  
│ │ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │ │                                 | t2 2t0t2+t1^2  |
│ │ │ │                                 | t1 2t0t1       |
│ │ │ │                                 | t0 t0^2        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i27 : elapsedTime jets(2,f)
│ │ │ │ - -- .000732911s elapsed
│ │ │ │ + -- .00142101s elapsed
│ │ │ │  
│ │ │ │                                     QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  2                    2
│ │ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2,
│ │ │ │  2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                              2                 2
│ │ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  Y2Fub25pY2FsSG9tb3RvcGllcyguLi4sRmluZUdyYWRpbmc9Pi4uLik=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU5Niwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2Fub25pY2FsSG9tb3RvcGllcyxGaW5lR3JhZGlu
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │        32003  0   5   0   5         32003  0   5   0   5          32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5          32003  0   5   0   5
│ │ │                                                                                                                                                                                                                                                                                           
│ │ │       0                            1                             2                              3                              4                              5                              6                              7                              8                             9
│ │ │  
│ │ │  o3 : Complex
│ │ │  
│ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0249269s elapsed
│ │ │ + -- .0319077s elapsed
│ │ │  
│ │ │  o5 = 350
│ │ │  
│ │ │  i6 : betti F_a, betti F
│ │ │  
│ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ @@ -46,19 +46,19 @@
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │  
│ │ │  o9 = 14
│ │ │  
│ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00229463s elapsed
│ │ │ - -- .00936212s elapsed
│ │ │ - -- .0239474s elapsed
│ │ │ - -- .011302s elapsed
│ │ │ - -- .00360161s elapsed
│ │ │ + -- .00356852s elapsed
│ │ │ + -- .00751063s elapsed
│ │ │ + -- .0364778s elapsed
│ │ │ + -- .0118629s elapsed
│ │ │ + -- .00430071s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out
│ │ │ @@ -3,20 +3,20 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00231006s elapsed
│ │ │ - -- .0217836s elapsed
│ │ │ - -- .0230895s elapsed
│ │ │ - -- .0101838s elapsed
│ │ │ - -- .00353337s elapsed
│ │ │ - -- .294917s elapsed
│ │ │ + -- .00351128s elapsed
│ │ │ + -- .00758549s elapsed
│ │ │ + -- .0332399s elapsed
│ │ │ + -- .0140415s elapsed
│ │ │ + -- .00437793s elapsed
│ │ │ + -- .390911s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -26,15 +26,15 @@
│ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .125196s elapsed
│ │ │ + -- .183748s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,22 +48,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00475633s elapsed
│ │ │ - -- .0968306s elapsed
│ │ │ - -- .111433s elapsed
│ │ │ - -- 1.47579s elapsed
│ │ │ - -- .551722s elapsed
│ │ │ - -- .0398804s elapsed
│ │ │ - -- .00727613s elapsed
│ │ │ - -- 6.05064s elapsed
│ │ │ + -- .0098388s elapsed
│ │ │ + -- .0207802s elapsed
│ │ │ + -- .122366s elapsed
│ │ │ + -- 1.22963s elapsed
│ │ │ + -- .529908s elapsed
│ │ │ + -- .062278s elapsed
│ │ │ + -- .00878806s elapsed
│ │ │ + -- 5.74471s elapsed
│ │ │  
│ │ │  i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out
│ │ │ @@ -3,19 +3,19 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00224642s elapsed
│ │ │ - -- .010053s elapsed
│ │ │ - -- .0365563s elapsed
│ │ │ - -- .0100498s elapsed
│ │ │ - -- .00353941s elapsed
│ │ │ + -- .00312854s elapsed
│ │ │ + -- .00868187s elapsed
│ │ │ + -- .0283381s elapsed
│ │ │ + -- .0163789s elapsed
│ │ │ + -- .00962418s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,15 +48,15 @@
│ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .196215s elapsed
│ │ │ + -- .227352s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -70,22 +70,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o6 : BettiTally
│ │ │  
│ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00490087s elapsed
│ │ │ - -- .0189491s elapsed
│ │ │ - -- .0966489s elapsed
│ │ │ - -- 1.19159s elapsed
│ │ │ - -- .524549s elapsed
│ │ │ - -- .0405632s elapsed
│ │ │ - -- .00680951s elapsed
│ │ │ - -- 5.37805s elapsed
│ │ │ + -- .00558834s elapsed
│ │ │ + -- .0212596s elapsed
│ │ │ + -- .123751s elapsed
│ │ │ + -- 1.20679s elapsed
│ │ │ + -- .50096s elapsed
│ │ │ + -- .0434345s elapsed
│ │ │ + -- .0116503s elapsed
│ │ │ + -- 5.96533s elapsed
│ │ │  
│ │ │  i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │  
│ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out
│ │ │ @@ -3,82 +3,82 @@
│ │ │  i1 : a=4,b=4
│ │ │  
│ │ │  o1 = (4, 4)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : d=carpetDet(a,b)
│ │ │ - -- .00631474s elapsed
│ │ │ - -- .0125312s elapsed
│ │ │ + -- .0081636s elapsed
│ │ │ + -- .0156894s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000299471s elapsed
│ │ │ + -- .000313777s elapsed
│ │ │  1
│ │ │ - -- .000146984s elapsed
│ │ │ + -- .000231077s elapsed
│ │ │  1
│ │ │ - -- .00012821s elapsed
│ │ │ + -- .000223958s elapsed
│ │ │  1
│ │ │ - -- .000124793s elapsed
│ │ │ + -- .00019135s elapsed
│ │ │  1
│ │ │ - -- .000129423s elapsed
│ │ │ + -- .000204637s elapsed
│ │ │  2
│ │ │ - -- .000136796s elapsed
│ │ │ + -- .000201959s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000137637s elapsed
│ │ │ + -- .000208875s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000148369s elapsed
│ │ │ + -- .000202077s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000144691s elapsed
│ │ │ + -- .000210087s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000132719s elapsed
│ │ │ + -- .000183057s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000125866s elapsed
│ │ │ + -- .000176668s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132297s elapsed
│ │ │ + -- .000181352s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140242s elapsed
│ │ │ + -- .000173832s elapsed
│ │ │  2
│ │ │ - -- .000122429s elapsed
│ │ │ + -- .000181969s elapsed
│ │ │  2
│ │ │ - -- .000133088s elapsed
│ │ │ + -- .000184566s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120466s elapsed
│ │ │ + -- .000181482s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000136004s elapsed
│ │ │ + -- .000200433s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000131376s elapsed
│ │ │ + -- .000198517s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000208069s elapsed
│ │ │ + -- .000255707s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140834s elapsed
│ │ │ + -- .000202399s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000134101s elapsed
│ │ │ + -- .000187603s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000126856s elapsed
│ │ │ + -- .000177369s elapsed
│ │ │  2
│ │ │ - -- .000129172s elapsed
│ │ │ + -- .000170942s elapsed
│ │ │  1
│ │ │ - -- .000130714s elapsed
│ │ │ + -- .000170301s elapsed
│ │ │  1
│ │ │ - -- .000156823s elapsed
│ │ │ + -- .000174209s elapsed
│ │ │  1
│ │ │ - -- .000123991s elapsed
│ │ │ + -- .000181225s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
│ │ │  
│ │ │  i3 : factor d
│ │ │  
│ │ │        32 6
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out
│ │ │ @@ -3,17 +3,17 @@
│ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │  
│ │ │  o1 = (9, 7)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : computeBound 3
│ │ │ - -- .268927s elapsed
│ │ │ - -- .33298s elapsed
│ │ │ - -- .23855s elapsed
│ │ │ - -- .286255s elapsed
│ │ │ - -- .382732s elapsed
│ │ │ - -- .298869s elapsed
│ │ │ + -- .171542s elapsed
│ │ │ + -- .202882s elapsed
│ │ │ + -- .190809s elapsed
│ │ │ + -- .253235s elapsed
│ │ │ + -- .207018s elapsed
│ │ │ + -- .282349s elapsed
│ │ │  
│ │ │  o2 = 6
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out
│ │ │ @@ -9,19 +9,19 @@
│ │ │  i2 : e=(-1,5)
│ │ │  
│ │ │  o2 = (-1, 5)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .0030227s elapsed
│ │ │ - -- .00794913s elapsed
│ │ │ - -- .023886s elapsed
│ │ │ - -- .0114692s elapsed
│ │ │ - -- .00368452s elapsed
│ │ │ + -- .00297639s elapsed
│ │ │ + -- .00708582s elapsed
│ │ │ + -- .0347633s elapsed
│ │ │ + -- .0118056s elapsed
│ │ │ + -- .00447966s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -49,15 +49,15 @@
│ │ │  i4 : keys h
│ │ │  
│ │ │  o4 = {0, 2, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .147997s elapsed
│ │ │ + -- .226802s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -77,19 +77,19 @@
│ │ │  i7 : e=(-1,5^2)
│ │ │  
│ │ │  o7 = (-1, 25)
│ │ │  
│ │ │  o7 : Sequence
│ │ │  
│ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00447939s elapsed
│ │ │ - -- .0118282s elapsed
│ │ │ - -- .043588s elapsed
│ │ │ - -- .0167169s elapsed
│ │ │ - -- .00606616s elapsed
│ │ │ + -- .00356828s elapsed
│ │ │ + -- .00885021s elapsed
│ │ │ + -- .0282208s elapsed
│ │ │ + -- .0109376s elapsed
│ │ │ + -- .0044943s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out
│ │ │ @@ -1,172 +1,172 @@
│ │ │  -- -*- M2-comint -*- hash: 1729182891690704738
│ │ │  
│ │ │  i1 : a=4
│ │ │  
│ │ │  o1 = 4
│ │ │  
│ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0201004s elapsed
│ │ │ + -- .0332412s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000054162s elapsed
│ │ │ + -- .000053742s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000094566s elapsed
│ │ │ + -- .000121111s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000071484s elapsed
│ │ │ + -- .000096801s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000081081s elapsed
│ │ │ + -- .000099984s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000092442s elapsed
│ │ │ + -- .000118218s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000092503s elapsed
│ │ │ + -- .000118291s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024937s elapsed
│ │ │ + -- .00003314s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000097393s elapsed
│ │ │ + -- .000091982s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000083987s elapsed
│ │ │ + -- .000116368s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000091982s elapsed
│ │ │ + -- .000113506s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000094778s elapsed
│ │ │ + -- .000103508s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000082534s elapsed
│ │ │ + -- .000101051s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .000077535s elapsed
│ │ │ + -- .000094853s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000072095s elapsed
│ │ │ + -- .000090902s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000067977s elapsed
│ │ │ + -- .000082303s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000022111s elapsed
│ │ │ + -- .000031057s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000071755s elapsed
│ │ │ + -- .000084881s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000026329s elapsed
│ │ │ + -- .000034504s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0249269s elapsed
│ │ │ + -- .0319077s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -146,19 +146,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00229463s elapsed
│ │ │ - -- .00936212s elapsed
│ │ │ - -- .0239474s elapsed
│ │ │ - -- .011302s elapsed
│ │ │ - -- .00360161s elapsed
│ │ │ + -- .00356852s elapsed
│ │ │ + -- .00751063s elapsed
│ │ │ + -- .0364778s elapsed
│ │ │ + -- .0118629s elapsed
│ │ │ + -- .00430071s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │       0                            1                             2
│ │ │ │  3                              4                              5
│ │ │ │  6                              7                              8
│ │ │ │  9
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ │ - -- .0249269s elapsed
│ │ │ │ + -- .0319077s elapsed
│ │ │ │  
│ │ │ │  o5 = 350
│ │ │ │  i6 : betti F_a, betti F
│ │ │ │  
│ │ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ │            6: 350      0: 1  .   .   .   .   .   .   .  . .
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  o7 = 2   3
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │ │  
│ │ │ │  o9 = 14
│ │ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ │ - -- .00229463s elapsed
│ │ │ │ - -- .00936212s elapsed
│ │ │ │ - -- .0239474s elapsed
│ │ │ │ - -- .011302s elapsed
│ │ │ │ - -- .00360161s elapsed
│ │ │ │ + -- .00356852s elapsed
│ │ │ │ + -- .00751063s elapsed
│ │ │ │ + -- .0364778s elapsed
│ │ │ │ + -- .0118629s elapsed
│ │ │ │ + -- .00430071s elapsed
│ │ │ │  
│ │ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html
│ │ │ @@ -88,20 +88,20 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00231006s elapsed
│ │ │ - -- .0217836s elapsed
│ │ │ - -- .0230895s elapsed
│ │ │ - -- .0101838s elapsed
│ │ │ - -- .00353337s elapsed
│ │ │ - -- .294917s elapsed
│ │ │ + -- .00351128s elapsed
│ │ │ + -- .00758549s elapsed
│ │ │ + -- .0332399s elapsed
│ │ │ + -- .0140415s elapsed
│ │ │ + -- .00437793s elapsed
│ │ │ + -- .390911s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .125196s elapsed
│ │ │ + -- .183748s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -145,22 +145,22 @@
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00475633s elapsed
│ │ │ - -- .0968306s elapsed
│ │ │ - -- .111433s elapsed
│ │ │ - -- 1.47579s elapsed
│ │ │ - -- .551722s elapsed
│ │ │ - -- .0398804s elapsed
│ │ │ - -- .00727613s elapsed
│ │ │ - -- 6.05064s elapsed
    
│ │ │ + -- .0098388s elapsed
│ │ │ + -- .0207802s elapsed
│ │ │ + -- .122366s elapsed
│ │ │ + -- 1.22963s elapsed
│ │ │ + -- .529908s elapsed
│ │ │ + -- .062278s elapsed
│ │ │ + -- .00878806s elapsed
│ │ │ + -- 5.74471s elapsed
│ │ │              
    
│ │ │                
    i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ │ - -- .00231006s elapsed
│ │ │ │ - -- .0217836s elapsed
│ │ │ │ - -- .0230895s elapsed
│ │ │ │ - -- .0101838s elapsed
│ │ │ │ - -- .00353337s elapsed
│ │ │ │ - -- .294917s elapsed
│ │ │ │ + -- .00351128s elapsed
│ │ │ │ + -- .00758549s elapsed
│ │ │ │ + -- .0332399s elapsed
│ │ │ │ + -- .0140415s elapsed
│ │ │ │ + -- .00437793s elapsed
│ │ │ │ + -- .390911s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │  o2 : BettiTally
│ │ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .125196s elapsed
│ │ │ │ + -- .183748s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -66,22 +66,22 @@
│ │ │ │  o5 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00475633s elapsed
│ │ │ │ - -- .0968306s elapsed
│ │ │ │ - -- .111433s elapsed
│ │ │ │ - -- 1.47579s elapsed
│ │ │ │ - -- .551722s elapsed
│ │ │ │ - -- .0398804s elapsed
│ │ │ │ - -- .00727613s elapsed
│ │ │ │ - -- 6.05064s elapsed
│ │ │ │ + -- .0098388s elapsed
│ │ │ │ + -- .0207802s elapsed
│ │ │ │ + -- .122366s elapsed
│ │ │ │ + -- 1.22963s elapsed
│ │ │ │ + -- .529908s elapsed
│ │ │ │ + -- .062278s elapsed
│ │ │ │ + -- .00878806s elapsed
│ │ │ │ + -- 5.74471s elapsed
│ │ │ │  i7 : carpetBettiTable(h,7)
│ │ │ │  
│ │ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ │ │           2: .  .   .   .    .    . 1155 1408 891 320 55  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html
│ │ │ @@ -85,19 +85,19 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00224642s elapsed
│ │ │ - -- .010053s elapsed
│ │ │ - -- .0365563s elapsed
│ │ │ - -- .0100498s elapsed
│ │ │ - -- .00353941s elapsed
│ │ │ + -- .00312854s elapsed
│ │ │ + -- .00868187s elapsed
│ │ │ + -- .0283381s elapsed
│ │ │ + -- .0163789s elapsed
│ │ │ + -- .00962418s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -139,15 +139,15 @@
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .196215s elapsed
│ │ │ + -- .227352s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -167,22 +167,22 @@
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00490087s elapsed
│ │ │ - -- .0189491s elapsed
│ │ │ - -- .0966489s elapsed
│ │ │ - -- 1.19159s elapsed
│ │ │ - -- .524549s elapsed
│ │ │ - -- .0405632s elapsed
│ │ │ - -- .00680951s elapsed
│ │ │ - -- 5.37805s elapsed
    
│ │ │ + -- .00558834s elapsed
│ │ │ + -- .0212596s elapsed
│ │ │ + -- .123751s elapsed
│ │ │ + -- 1.20679s elapsed
│ │ │ + -- .50096s elapsed
│ │ │ + -- .0434345s elapsed
│ │ │ + -- .0116503s elapsed
│ │ │ + -- 5.96533s elapsed
│ │ │              
    
│ │ │                
    i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,19 +21,19 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ │ - -- .00224642s elapsed
│ │ │ │ - -- .010053s elapsed
│ │ │ │ - -- .0365563s elapsed
│ │ │ │ - -- .0100498s elapsed
│ │ │ │ - -- .00353941s elapsed
│ │ │ │ + -- .00312854s elapsed
│ │ │ │ + -- .00868187s elapsed
│ │ │ │ + -- .0283381s elapsed
│ │ │ │ + -- .0163789s elapsed
│ │ │ │ + -- .00962418s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .196215s elapsed
│ │ │ │ + -- .227352s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -83,22 +83,22 @@
│ │ │ │  o6 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o6 : BettiTally
│ │ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00490087s elapsed
│ │ │ │ - -- .0189491s elapsed
│ │ │ │ - -- .0966489s elapsed
│ │ │ │ - -- 1.19159s elapsed
│ │ │ │ - -- .524549s elapsed
│ │ │ │ - -- .0405632s elapsed
│ │ │ │ - -- .00680951s elapsed
│ │ │ │ - -- 5.37805s elapsed
│ │ │ │ + -- .00558834s elapsed
│ │ │ │ + -- .0212596s elapsed
│ │ │ │ + -- .123751s elapsed
│ │ │ │ + -- 1.20679s elapsed
│ │ │ │ + -- .50096s elapsed
│ │ │ │ + -- .0434345s elapsed
│ │ │ │ + -- .0116503s elapsed
│ │ │ │ + -- 5.96533s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -85,82 +85,82 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .00631474s elapsed
│ │ │ - -- .0125312s elapsed
│ │ │ + -- .0081636s elapsed
│ │ │ + -- .0156894s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000299471s elapsed
│ │ │ + -- .000313777s elapsed
│ │ │  1
│ │ │ - -- .000146984s elapsed
│ │ │ + -- .000231077s elapsed
│ │ │  1
│ │ │ - -- .00012821s elapsed
│ │ │ + -- .000223958s elapsed
│ │ │  1
│ │ │ - -- .000124793s elapsed
│ │ │ + -- .00019135s elapsed
│ │ │  1
│ │ │ - -- .000129423s elapsed
│ │ │ + -- .000204637s elapsed
│ │ │  2
│ │ │ - -- .000136796s elapsed
│ │ │ + -- .000201959s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000137637s elapsed
│ │ │ + -- .000208875s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000148369s elapsed
│ │ │ + -- .000202077s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000144691s elapsed
│ │ │ + -- .000210087s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000132719s elapsed
│ │ │ + -- .000183057s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000125866s elapsed
│ │ │ + -- .000176668s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132297s elapsed
│ │ │ + -- .000181352s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140242s elapsed
│ │ │ + -- .000173832s elapsed
│ │ │  2
│ │ │ - -- .000122429s elapsed
│ │ │ + -- .000181969s elapsed
│ │ │  2
│ │ │ - -- .000133088s elapsed
│ │ │ + -- .000184566s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120466s elapsed
│ │ │ + -- .000181482s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000136004s elapsed
│ │ │ + -- .000200433s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000131376s elapsed
│ │ │ + -- .000198517s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000208069s elapsed
│ │ │ + -- .000255707s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140834s elapsed
│ │ │ + -- .000202399s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000134101s elapsed
│ │ │ + -- .000187603s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000126856s elapsed
│ │ │ + -- .000177369s elapsed
│ │ │  2
│ │ │ - -- .000129172s elapsed
│ │ │ + -- .000170942s elapsed
│ │ │  1
│ │ │ - -- .000130714s elapsed
│ │ │ + -- .000170301s elapsed
│ │ │  1
│ │ │ - -- .000156823s elapsed
│ │ │ + -- .000174209s elapsed
│ │ │  1
│ │ │ - -- .000123991s elapsed
│ │ │ + -- .000181225s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,82 +19,82 @@
│ │ │ │  determinants and return their product.
│ │ │ │  i1 : a=4,b=4
│ │ │ │  
│ │ │ │  o1 = (4, 4)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : d=carpetDet(a,b)
│ │ │ │ - -- .00631474s elapsed
│ │ │ │ - -- .0125312s elapsed
│ │ │ │ + -- .0081636s elapsed
│ │ │ │ + -- .0156894s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │ - -- .000299471s elapsed
│ │ │ │ + -- .000313777s elapsed
│ │ │ │  1
│ │ │ │ - -- .000146984s elapsed
│ │ │ │ + -- .000231077s elapsed
│ │ │ │  1
│ │ │ │ - -- .00012821s elapsed
│ │ │ │ + -- .000223958s elapsed
│ │ │ │  1
│ │ │ │ - -- .000124793s elapsed
│ │ │ │ + -- .00019135s elapsed
│ │ │ │  1
│ │ │ │ - -- .000129423s elapsed
│ │ │ │ + -- .000204637s elapsed
│ │ │ │  2
│ │ │ │ - -- .000136796s elapsed
│ │ │ │ + -- .000201959s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000137637s elapsed
│ │ │ │ + -- .000208875s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000148369s elapsed
│ │ │ │ + -- .000202077s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000144691s elapsed
│ │ │ │ + -- .000210087s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000132719s elapsed
│ │ │ │ + -- .000183057s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000125866s elapsed
│ │ │ │ + -- .000176668s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000132297s elapsed
│ │ │ │ + -- .000181352s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000140242s elapsed
│ │ │ │ + -- .000173832s elapsed
│ │ │ │  2
│ │ │ │ - -- .000122429s elapsed
│ │ │ │ + -- .000181969s elapsed
│ │ │ │  2
│ │ │ │ - -- .000133088s elapsed
│ │ │ │ + -- .000184566s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000120466s elapsed
│ │ │ │ + -- .000181482s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000136004s elapsed
│ │ │ │ + -- .000200433s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000131376s elapsed
│ │ │ │ + -- .000198517s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000208069s elapsed
│ │ │ │ + -- .000255707s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000140834s elapsed
│ │ │ │ + -- .000202399s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000134101s elapsed
│ │ │ │ + -- .000187603s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000126856s elapsed
│ │ │ │ + -- .000177369s elapsed
│ │ │ │  2
│ │ │ │ - -- .000129172s elapsed
│ │ │ │ + -- .000170942s elapsed
│ │ │ │  1
│ │ │ │ - -- .000130714s elapsed
│ │ │ │ + -- .000170301s elapsed
│ │ │ │  1
│ │ │ │ - -- .000156823s elapsed
│ │ │ │ + -- .000174209s elapsed
│ │ │ │  1
│ │ │ │ - -- .000123991s elapsed
│ │ │ │ + -- .000181225s elapsed
│ │ │ │  1
│ │ │ │  
│ │ │ │  o2 = 3131031158784
│ │ │ │  i3 : factor d
│ │ │ │  
│ │ │ │        32 6
│ │ │ │  o3 = 2  3
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -90,20 +90,20 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .268927s elapsed
│ │ │ - -- .33298s elapsed
│ │ │ - -- .23855s elapsed
│ │ │ - -- .286255s elapsed
│ │ │ - -- .382732s elapsed
│ │ │ - -- .298869s elapsed
│ │ │ + -- .171542s elapsed
│ │ │ + -- .202882s elapsed
│ │ │ + -- .190809s elapsed
│ │ │ + -- .253235s elapsed
│ │ │ + -- .207018s elapsed
│ │ │ + -- .282349s elapsed
│ │ │  
│ │ │  o2 = 6
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  classes mod k. We conjecture that c=k^2-k.
│ │ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │ │  
│ │ │ │  o1 = (9, 7)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : computeBound 3
│ │ │ │ - -- .268927s elapsed
│ │ │ │ - -- .33298s elapsed
│ │ │ │ - -- .23855s elapsed
│ │ │ │ - -- .286255s elapsed
│ │ │ │ - -- .382732s elapsed
│ │ │ │ - -- .298869s elapsed
│ │ │ │ + -- .171542s elapsed
│ │ │ │ + -- .202882s elapsed
│ │ │ │ + -- .190809s elapsed
│ │ │ │ + -- .253235s elapsed
│ │ │ │ + -- .207018s elapsed
│ │ │ │ + -- .282349s elapsed
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_l_a_t_i_v_e_E_q_u_a_t_i_o_n_s -- compute the relative quadrics
│ │ │ │  ********** WWaayyss ttoo uussee ccoommppuutteeBBoouunndd:: **********
│ │ │ │      * computeBound(ZZ)
│ │ │ │      * computeBound(ZZ,ZZ,ZZ)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html
│ │ │ @@ -95,19 +95,19 @@
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .0030227s elapsed
│ │ │ - -- .00794913s elapsed
│ │ │ - -- .023886s elapsed
│ │ │ - -- .0114692s elapsed
│ │ │ - -- .00368452s elapsed
│ │ │ + -- .00297639s elapsed
│ │ │ + -- .00708582s elapsed
│ │ │ + -- .0347633s elapsed
│ │ │ + -- .0118056s elapsed
│ │ │ + -- .00447966s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .147997s elapsed
│ │ │ + -- .226802s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -183,19 +183,19 @@
│ │ │  
│ │ │  o7 : Sequence
    
│ │ │              
    
│ │ │                
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00447939s elapsed
│ │ │ - -- .0118282s elapsed
│ │ │ - -- .043588s elapsed
│ │ │ - -- .0167169s elapsed
│ │ │ - -- .00606616s elapsed
│ │ │ + -- .00356828s elapsed
│ │ │ + -- .00885021s elapsed
│ │ │ + -- .0282208s elapsed
│ │ │ + -- .0109376s elapsed
│ │ │ + -- .0044943s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : e=(-1,5)
│ │ │ │  
│ │ │ │  o2 = (-1, 5)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .0030227s elapsed
│ │ │ │ - -- .00794913s elapsed
│ │ │ │ - -- .023886s elapsed
│ │ │ │ - -- .0114692s elapsed
│ │ │ │ - -- .00368452s elapsed
│ │ │ │ + -- .00297639s elapsed
│ │ │ │ + -- .00708582s elapsed
│ │ │ │ + -- .0347633s elapsed
│ │ │ │ + -- .0118056s elapsed
│ │ │ │ + -- .00447966s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  o3 : HashTable
│ │ │ │  i4 : keys h
│ │ │ │  
│ │ │ │  o4 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ │ - -- .147997s elapsed
│ │ │ │ + -- .226802s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -94,19 +94,19 @@
│ │ │ │  these mistakes.
│ │ │ │  i7 : e=(-1,5^2)
│ │ │ │  
│ │ │ │  o7 = (-1, 25)
│ │ │ │  
│ │ │ │  o7 : Sequence
│ │ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00447939s elapsed
│ │ │ │ - -- .0118282s elapsed
│ │ │ │ - -- .043588s elapsed
│ │ │ │ - -- .0167169s elapsed
│ │ │ │ - -- .00606616s elapsed
│ │ │ │ + -- .00356828s elapsed
│ │ │ │ + -- .00885021s elapsed
│ │ │ │ + -- .0282208s elapsed
│ │ │ │ + -- .0109376s elapsed
│ │ │ │ + -- .0044943s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html
│ │ │ @@ -83,172 +83,172 @@
│ │ │  
│ │ │  o1 = 4
    
│ │ │              
    
│ │ │                
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0201004s elapsed
│ │ │ + -- .0332412s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000054162s elapsed
│ │ │ + -- .000053742s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000094566s elapsed
│ │ │ + -- .000121111s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000071484s elapsed
│ │ │ + -- .000096801s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000081081s elapsed
│ │ │ + -- .000099984s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000092442s elapsed
│ │ │ + -- .000118218s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000092503s elapsed
│ │ │ + -- .000118291s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024937s elapsed
│ │ │ + -- .00003314s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000097393s elapsed
│ │ │ + -- .000091982s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000083987s elapsed
│ │ │ + -- .000116368s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000091982s elapsed
│ │ │ + -- .000113506s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000094778s elapsed
│ │ │ + -- .000103508s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000082534s elapsed
│ │ │ + -- .000101051s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .000077535s elapsed
│ │ │ + -- .000094853s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000072095s elapsed
│ │ │ + -- .000090902s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000067977s elapsed
│ │ │ + -- .000082303s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000022111s elapsed
│ │ │ + -- .000031057s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000071755s elapsed
│ │ │ + -- .000084881s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000026329s elapsed
│ │ │ + -- .000034504s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,172 +19,172 @@
│ │ │ │  grading. Viewed as a resolution over QQ(e_1,e_2), this resolution is non-
│ │ │ │  minimal and carries further gradings. We decompose the crucial map of the a-th
│ │ │ │  strand into blocks, compute their determinants, and factor the product.
│ │ │ │  i1 : a=4
│ │ │ │  
│ │ │ │  o1 = 4
│ │ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ │ - -- .0201004s elapsed
│ │ │ │ + -- .0332412s elapsed
│ │ │ │  (number of blocks= , 18)
│ │ │ │  (size of the matrices, Tally{1 => 4})
│ │ │ │                               2 => 6
│ │ │ │                               3 => 2
│ │ │ │                               4 => 6
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      7: 1 1
│ │ │ │ - -- .000054162s elapsed
│ │ │ │ + -- .000053742s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . 2
│ │ │ │ - -- .000094566s elapsed
│ │ │ │ + -- .000121111s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . .
│ │ │ │      9: . 2
│ │ │ │ - -- .000071484s elapsed
│ │ │ │ + -- .000096801s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │      7: 2 .
│ │ │ │      8: 1 .
│ │ │ │      9: . 1
│ │ │ │     10: . 2
│ │ │ │ - -- .000081081s elapsed
│ │ │ │ + -- .000099984s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (-3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      7: 1 .
│ │ │ │      8: 1 .
│ │ │ │      9: 2 2
│ │ │ │     10: . 1
│ │ │ │     11: . 1
│ │ │ │ - -- .000092442s elapsed
│ │ │ │ + -- .000118218s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      8: 1 .
│ │ │ │      9: 2 1
│ │ │ │     10: 1 2
│ │ │ │     11: . 1
│ │ │ │ - -- .000092503s elapsed
│ │ │ │ + -- .000118291s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      9: 1 1
│ │ │ │ - -- .000024937s elapsed
│ │ │ │ + -- .00003314s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      9: 1 1
│ │ │ │     10: 1 1
│ │ │ │ - -- .000097393s elapsed
│ │ │ │ + -- .000091982s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000083987s elapsed
│ │ │ │ + -- .000116368s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 2 1
│ │ │ │     11: 1 2
│ │ │ │     12: . 1
│ │ │ │ - -- .000091982s elapsed
│ │ │ │ + -- .000113506s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 1 .
│ │ │ │     11: 2 2
│ │ │ │     12: . 1
│ │ │ │     13: . 1
│ │ │ │ - -- .000094778s elapsed
│ │ │ │ + -- .000103508s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000082534s elapsed
│ │ │ │ + -- .000101051s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │     10: 2 .
│ │ │ │     11: 1 .
│ │ │ │     12: . 1
│ │ │ │     13: . 2
│ │ │ │ - -- .000077535s elapsed
│ │ │ │ + -- .000094853s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     10: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000072095s elapsed
│ │ │ │ + -- .000090902s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     11: 2 .
│ │ │ │     12: . .
│ │ │ │     13: . 2
│ │ │ │ - -- .000067977s elapsed
│ │ │ │ + -- .000082303s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000022111s elapsed
│ │ │ │ + -- .000031057s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     12: 2 .
│ │ │ │     13: . 2
│ │ │ │ - -- .000071755s elapsed
│ │ │ │ + -- .000084881s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     13: 1 1
│ │ │ │ - -- .000026329s elapsed
│ │ │ │ + -- .000034504s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │  
│ │ │ │         6      32    32
│ │ │ │  o2 = (3 , (e )  (e )  )
│ │ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Surfaces/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  cHJvamVjdChWaXNpYmxlTGlzdCxFbWJlZGRlZEszc3VyZmFjZSk=
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQ4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwcm9qZWN0LFZpc2libGVMaXN0LEVtYmVkZGVkSzNz
│ │ ├── ./usr/share/doc/Macaulay2/Kronecker/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  a3JvbmVja2VyTm9ybWFsRm9ybShNYXRyaXgp
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIwNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoa3JvbmVja2VyTm9ybWFsRm9ybSxNYXRyaXgpLCJr
│ │ ├── ./usr/share/doc/Macaulay2/KustinMiller/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=37
│ │ │  a3VzdGluTWlsbGVyQ29tcGxleCguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=577
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3B0aW9uIHRvIHByaW50IGludGVybWVk
│ │ │  aWF0ZSBkYXRhIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7VE97bmV3IERvY3VtZW50VGFn
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  UmVhbFhE
│ │ │  #:len=499
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidXNlIGV4dGVuZGVkIGV4cG9uZW50IHJl
│ │ │  YWwgbnVtYmVycyIsIERlc2NyaXB0aW9uID0+IChUVHsiUmVhbFhEIn0sIiAtLSBhIHN0cmF0ZWd5
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -7,55 +7,55 @@
│ │ │  
│ │ │  i2 : m = syz m1;
│ │ │  
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i3 : time LLL m;
│ │ │ - -- used 0.00867687s (cpu); 0.00867114s (thread); 0s (gc)
│ │ │ + -- used 0.00981584s (cpu); 0.00981256s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc)
│ │ │ + -- used 0.0296589s (cpu); 0.0296704s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc)
│ │ │ + -- used 0.133861s (cpu); 0.13387s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc)
│ │ │ + -- used 0.0124944s (cpu); 0.0125021s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc)
│ │ │ + -- used 0.0631345s (cpu); 0.0631465s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc)
│ │ │ + -- used 0.0738717s (cpu); 0.073877s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc)
│ │ │ + -- used 0.36991s (cpu); 0.369917s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc)
│ │ │ + -- used 0.165858s (cpu); 0.165571s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -144,78 +144,78 @@
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i3 : time LLL m;
│ │ │ - -- used 0.00867687s (cpu); 0.00867114s (thread); 0s (gc)
│ │ │ + -- used 0.00981584s (cpu); 0.00981256s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc)
│ │ │ + -- used 0.0296589s (cpu); 0.0296704s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc)
│ │ │ + -- used 0.133861s (cpu); 0.13387s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc)
│ │ │ + -- used 0.0124944s (cpu); 0.0125021s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc)
│ │ │ + -- used 0.0631345s (cpu); 0.0631465s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc)
│ │ │ + -- used 0.0738717s (cpu); 0.073877s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc)
│ │ │ + -- used 0.36991s (cpu); 0.369917s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc)
│ │ │ + -- used 0.165858s (cpu); 0.165571s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -115,50 +115,50 @@
│ │ │ │                50       50
│ │ │ │  o1 : Matrix ZZ   <-- ZZ
│ │ │ │  i2 : m = syz m1;
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │ │  i3 : time LLL m;
│ │ │ │ - -- used 0.00867687s (cpu); 0.00867114s (thread); 0s (gc)
│ │ │ │ + -- used 0.00981584s (cpu); 0.00981256s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc)
│ │ │ │ + -- used 0.0296589s (cpu); 0.0296704s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc)
│ │ │ │ + -- used 0.133861s (cpu); 0.13387s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc)
│ │ │ │ + -- used 0.0124944s (cpu); 0.0125021s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc)
│ │ │ │ + -- used 0.0631345s (cpu); 0.0631465s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc)
│ │ │ │ + -- used 0.0738717s (cpu); 0.073877s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc)
│ │ │ │ + -- used 0.36991s (cpu); 0.369917s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc)
│ │ │ │ + -- used 0.165858s (cpu); 0.165571s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 50       47
│ │ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  For most of the options, the columns do not need to be linearly independent.
│ │ │ │  The strategies CohenEngine and CohenTopLevel currently require the columns to
│ │ │ │  be linearly independent.
│ │ ├── ./usr/share/doc/Macaulay2/LanguageServer/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  c3RhcnQoTFNQU2VydmVyKQ==
│ │ │  #:len=246
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjYsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KHN0YXJ0LExTUFNlcnZlciksInN0YXJ0KExTUFNlcnZl
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  YXJlSXNvbW9ycGhpYyhNYXRyaXgsTWF0cml4KQ==
│ │ │  #:len=289
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhcmVJc29tb3JwaGljLE1hdHJpeCxNYXRyaXgpLCJh
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out
│ │ │ @@ -16,14 +16,14 @@
│ │ │  
│ │ │                3       8
│ │ │  o4 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i5 : P = convexHull(M);
│ │ │  
│ │ │  i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.74582s (cpu); 0.506968s (thread); 0s (gc)
│ │ │ + -- used 1.38825s (cpu); 0.629678s (thread); 0s (gc)
│ │ │  
│ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc)
│ │ │ + -- used 0.511725s (cpu); 0.334933s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html
│ │ │ @@ -125,21 +125,21 @@
│ │ │              
    		
│ │ │                
    i5 : P = convexHull(M);
    
│ │ │              
    
│ │ │                
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.74582s (cpu); 0.506968s (thread); 0s (gc)
    
│ │ │ + -- used 1.38825s (cpu); 0.629678s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc)
    
│ │ │ + -- used 0.511725s (cpu); 0.334933s (thread); 0s (gc)
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use areIsomorphic:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,17 +35,17 @@
│ │ │ │       | 0 0 1 0 1 0 1 1 |
│ │ │ │       | 0 0 0 1 0 1 1 1 |
│ │ │ │  
│ │ │ │                3       8
│ │ │ │  o4 : Matrix ZZ  <-- ZZ
│ │ │ │  i5 : P = convexHull(M);
│ │ │ │  i6 : time areIsomorphic(P,P);
│ │ │ │ - -- used 0.74582s (cpu); 0.506968s (thread); 0s (gc)
│ │ │ │ + -- used 1.38825s (cpu); 0.629678s (thread); 0s (gc)
│ │ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc)
│ │ │ │ + -- used 0.511725s (cpu); 0.334933s (thread); 0s (gc)
│ │ │ │  ********** WWaayyss ttoo uussee aarreeIIssoommoorrpphhiicc:: **********
│ │ │ │      * areIsomorphic(Matrix,Matrix)
│ │ │ │      * areIsomorphic(Polyhedron,Polyhedron)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _a_r_e_I_s_o_m_o_r_p_h_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/LexIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  TGV4SWRlYWxz
│ │ │  #:len=470
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciB3b3JraW5nIHdp
│ │ │  dGggbGV4IGlkZWFscyIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsiIixFTXsiTGV4
│ │ ├── ./usr/share/doc/Macaulay2/LieAlgebraRepresentations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=43
│ │ │  c3ltbWV0cmljUG93ZXIoWlosTGllQWxnZWJyYVJlcHJlc2VudGF0aW9uKQ==
│ │ │  #:len=1614
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIGV4cGxpY2l0IGFj
│ │ │  dGlvbiBvbiAkXFxvcGVyYXRvcm5hbWV7U3ltfV5kIFYkIGZvciBhICRcXG1hdGhmcmFre2d9JC1t
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  c3VwcG9ydE9mVG9yKENvbXBsZXgp
│ │ │  #:len=277
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE0Mywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3VwcG9ydE9mVG9yLENvbXBsZXgpLCJzdXBwb3J0
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out
│ │ │ @@ -29,21 +29,21 @@
│ │ │  i5 : findRegion({{0,0},{4,4}},M,f)
│ │ │  
│ │ │  o5 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .134906s elapsed
│ │ │ + -- .106253s elapsed
│ │ │  
│ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
│ │ │ - -- .0127315s elapsed
│ │ │ + -- .0185218s elapsed
│ │ │  
│ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out
│ │ │ @@ -30,21 +30,21 @@
│ │ │  i5 : apply(L, d -> isLinearComplex res prune truncate(d,M))
│ │ │  
│ │ │  o5 = {true, true}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 3.72413s elapsed
│ │ │ + -- 4.10984s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .115325s elapsed
│ │ │ + -- .0587915s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html
│ │ │ @@ -130,25 +130,25 @@
│ │ │          
    
│ │ │            
    If some degrees d are known to satisfy f(d,M), then they can be specified using the option Inner in order to expedite the computation.  Similarly, degrees not above those given in Outer will be assumed not to satisfy f(d,M). If f takes options these can also be given to findRegion.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .134906s elapsed
│ │ │ + -- .106253s elapsed
│ │ │  
│ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
│ │ │ - -- .0127315s elapsed
│ │ │ + -- .0185218s elapsed
│ │ │  
│ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,22 +48,22 @@
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  If some degrees d are known to satisfy f(d,M), then they can be specified using
│ │ │ │  the option Inner in order to expedite the computation. Similarly, degrees not
│ │ │ │  above those given in Outer will be assumed not to satisfy f(d,M). If f takes
│ │ │ │  options these can also be given to findRegion.
│ │ │ │  i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ │ - -- .134906s elapsed
│ │ │ │ + -- .106253s elapsed
│ │ │ │  
│ │ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{
│ │ │ │  {1,1}})
│ │ │ │ - -- .0127315s elapsed
│ │ │ │ + -- .0185218s elapsed
│ │ │ │  
│ │ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** CCoonnttrriibbuuttoorrss **********
│ │ │ │  Mahrud Sayrafi contributed to the code for this function.
│ │ │ │  ********** CCaavveeaatt **********
│ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html
│ │ │ @@ -128,25 +128,25 @@
│ │ │          
    
│ │ │            
    The output is a list of the minimal multidegrees $d$ such that the sum of the positive coordinates of $b-d$ is at most $i$ for all degrees $b$ appearing in the i-th step of the resolution of $M$.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 3.72413s elapsed
│ │ │ + -- 4.10984s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .115325s elapsed
│ │ │ + -- .0587915s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │  o5 = {true, true}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  The output is a list of the minimal multidegrees $d$ such that the sum of the
│ │ │ │  positive coordinates of $b-d$ is at most $i$ for all degrees $b$ appearing in
│ │ │ │  the i-th step of the resolution of $M$.
│ │ │ │  i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ │ - -- 3.72413s elapsed
│ │ │ │ + -- 4.10984s elapsed
│ │ │ │  
│ │ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : elapsedTime linearTruncationsBound M
│ │ │ │ - -- .115325s elapsed
│ │ │ │ + -- .0587915s elapsed
│ │ │ │  
│ │ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  In general linearTruncationsBound will not find the minimal degrees where $M$
│ │ │ │  has a linear resolution but will be faster than repeatedly truncating $M$.
│ │ ├── ./usr/share/doc/Macaulay2/LocalRings/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  TG9jYWxSaW5ncw==
│ │ │  #:len=5280
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTG9jYWxpemF0aW9ucyBvZiBwb2x5bm9t
│ │ │  aWFsIHJpbmdzIGF0IHByaW1lIGlkZWFscyIsICJsaW5lbnVtIiA9PiA2MiwgImZpbGVuYW1lIiA9
│ │ ├── ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  
│ │ │  o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 |
│ │ │  
│ │ │                                1
│ │ │  o4 : RP-module, quotient of RP
│ │ │  
│ │ │  i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .192462s elapsed
│ │ │ + -- .207122s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : oo//sum
│ │ │  
│ │ │ @@ -44,21 +44,21 @@
│ │ │  
│ │ │                2   3
│ │ │  o10 = ideal (x , y )
│ │ │  
│ │ │  o10 : Ideal of RP
│ │ │  
│ │ │  i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0128718s elapsed
│ │ │ + -- .0186687s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .376491s elapsed
│ │ │ + -- .3197s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │                                1
│ │ │  o4 : RP-module, quotient of RP
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .192462s elapsed
│ │ │ + -- .207122s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -168,25 +168,25 @@
│ │ │  
│ │ │  o10 : Ideal of RP
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0128718s elapsed
│ │ │ + -- .0186687s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .376491s elapsed
│ │ │ + -- .3197s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  i4 : M = RP^1/I
│ │ │ │  
│ │ │ │  o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 |
│ │ │ │  
│ │ │ │                                1
│ │ │ │  o4 : RP-module, quotient of RP
│ │ │ │  i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ │ - -- .192462s elapsed
│ │ │ │ + -- .207122s elapsed
│ │ │ │  
│ │ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : oo//sum
│ │ │ │  
│ │ │ │  o6 = 27
│ │ │ │ @@ -65,21 +65,21 @@
│ │ │ │  i10 : q = ideal"x2,y3"
│ │ │ │  
│ │ │ │                2   3
│ │ │ │  o10 = ideal (x , y )
│ │ │ │  
│ │ │ │  o10 : Ideal of RP
│ │ │ │  i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ │ - -- .0128718s elapsed
│ │ │ │ + -- .0186687s elapsed
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ │ - -- .376491s elapsed
│ │ │ │ + -- .3197s elapsed
│ │ │ │  
│ │ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Hilbert-Samuel function with respect to a parameter ideal other than the
│ │ │ │  maximal ideal can be slower.
│ │ ├── ./usr/share/doc/Macaulay2/M0nbar/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  dGV4KEN1cnZlQ2xhc3NSZXByZXNlbnRhdGl2ZU0wbmJhcik=
│ │ │  #:len=923
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydCB0byBUZVggZm9ybWF0Iiwg
│ │ │  ImxpbmVudW0iID0+IDE1ODQsIElucHV0cyA9PiB7U1BBTntUVHsiQyJ9LCIsICIsU1BBTnsiYW4g
│ │ ├── ./usr/share/doc/Macaulay2/MCMApproximations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  Y29BcHByb3hpbWF0aW9uKC4uLixUb3RhbD0+Li4uKQ==
│ │ │  #:len=293
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tjb0FwcHJveGltYXRpb24sVG90YWxdLCJjb0FwcHJv
│ │ ├── ./usr/share/doc/Macaulay2/MRDI/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  c2F2ZU1SREkoLi4uLEZpbGVOYW1lPT4uLi4p
│ │ │  #:len=232
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTc3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzYXZlTVJESSxGaWxlTmFtZV0sInNhdmVNUkRJKC4u
│ │ ├── ./usr/share/doc/Macaulay2/MRDI/example-output/_load__M__R__D__I.out
│ │ │ @@ -57,15 +57,15 @@
│ │ │  
│ │ │  i9 : I === J
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o10 = /tmp/M2-40008-0/0.mrdi
│ │ │ +o10 = /tmp/M2-56075-0/0.mrdi
│ │ │  
│ │ │  i11 : saveMRDI(I, FileName => fn)
│ │ │  
│ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │        "Ideal"}, "data": [[[["0", "0", "2", "0"], ["1", "1"]], [["0", "1",
│ │ │        "0", "1"], ["-1", "1"]]], [[["0", "1", "1", "0"], ["1", "1"]], [["1",
│ │ │        "0", "0", "1"], ["-1", "1"]]], [[["0", "2", "0", "0"], ["1", "1"]],
│ │ ├── ./usr/share/doc/Macaulay2/MRDI/example-output/_save__M__R__D__I.out
│ │ │ @@ -39,15 +39,15 @@
│ │ │       "1"]]]], "_ns": {"Macaulay2": ["https://macaulay2.com", "1.26.06"]},
│ │ │       "_refs": {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params":
│ │ │       {"_type": "Ring", "data": "QQ"}, "name": "PolynomialRing"}, "data":
│ │ │       {"variables": ["x", "y", "z"]}}}}
│ │ │  
│ │ │  i7 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o7 = /tmp/M2-40027-0/0.mrdi
│ │ │ +o7 = /tmp/M2-56114-0/0.mrdi
│ │ │  
│ │ │  i8 : saveMRDI(f, FileName => fn)
│ │ │  
│ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │       "RingElement"}, "data": [[["2", "0", "0"], ["1", "1"]], [["0", "1",
│ │ │       "1"], ["1", "1"]], [["1", "0", "0"], ["-3", "1"]]], "_ns": {"Macaulay2":
│ │ │       ["https://macaulay2.com", "1.26.06"]}, "_refs":
│ │ │ @@ -69,22 +69,22 @@
│ │ │                  "_type" => ZZ
│ │ │                  "data" => 5
│ │ │  
│ │ │  o11 : HashTable
│ │ │  
│ │ │  i12 : methods {"Macaulay2", saveMRDI}
│ │ │  
│ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ -      {1 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ -      {2 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ -      {3 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ -      {4 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ -      {5 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ -      {6 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ -      {7 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ +      {1 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ +      {2 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ +      {3 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ +      {4 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ +      {5 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ +      {6 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ +      {7 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │  
│ │ │  o12 : NumberedVerticalList
│ │ │  
│ │ │  i13 : methods {"Oscar", saveMRDI}
│ │ │  
│ │ │  o13 = {0 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │        {1 => ({Oscar, saveMRDI}, Ring)          }
│ │ ├── ./usr/share/doc/Macaulay2/MRDI/html/_load__M__R__D__I.html
│ │ │ @@ -176,15 +176,15 @@
│ │ │            
    Objects can be loaded from a file as well using get.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o10 = /tmp/M2-40008-0/0.mrdi
    
│ │ │ +o10 = /tmp/M2-56075-0/0.mrdi
│ │ │              
    
│ │ │                
    i11 : saveMRDI(I, FileName => fn)
│ │ │  
│ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : I === J
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  Objects can be loaded from a file as well using _g_e_t.
│ │ │ │  i10 : fn = temporaryFileName() | ".mrdi"
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-40008-0/0.mrdi
│ │ │ │ +o10 = /tmp/M2-56075-0/0.mrdi
│ │ │ │  i11 : saveMRDI(I, FileName => fn)
│ │ │ │  
│ │ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ │        "Ideal"}, "data": [[[["0", "0", "2", "0"], ["1", "1"]], [["0", "1",
│ │ │ │        "0", "1"], ["-1", "1"]]], [[["0", "1", "1", "0"], ["1", "1"]], [["1",
│ │ │ │        "0", "0", "1"], ["-1", "1"]]], [[["0", "2", "0", "0"], ["1", "1"]],
│ │ │ │        [["1", "0", "1", "0"], ["-1", "1"]]]], "_ns": {"Macaulay2":
│ │ ├── ./usr/share/doc/Macaulay2/MRDI/html/_save__M__R__D__I.html
│ │ │ @@ -160,15 +160,15 @@
│ │ │            
    The output can be written directly to a file using the FileName option.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o7 = /tmp/M2-40027-0/0.mrdi
    
│ │ │ +o7 = /tmp/M2-56114-0/0.mrdi
│ │ │              
    
│ │ │                
    i8 : saveMRDI(f, FileName => fn)
│ │ │  
│ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ @@ -220,22 +220,22 @@
│ │ │            
    To see which types have built-in save methods for a given namespace, call methods as follows.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : methods {"Macaulay2", saveMRDI}
│ │ │  
│ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ -      {1 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ -      {2 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ -      {3 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ -      {4 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ -      {5 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ -      {6 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ -      {7 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ +      {1 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ +      {2 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ +      {3 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ +      {4 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ +      {5 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ +      {6 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ +      {7 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │  
│ │ │  o12 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i13 : methods {"Oscar", saveMRDI}
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,15 +66,15 @@
│ │ │ │       "1"]]]], "_ns": {"Macaulay2": ["https://macaulay2.com", "1.26.06"]},
│ │ │ │       "_refs": {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params":
│ │ │ │       {"_type": "Ring", "data": "QQ"}, "name": "PolynomialRing"}, "data":
│ │ │ │       {"variables": ["x", "y", "z"]}}}}
│ │ │ │  The output can be written directly to a file using the _F_i_l_e_N_a_m_e option.
│ │ │ │  i7 : fn = temporaryFileName() | ".mrdi"
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-40027-0/0.mrdi
│ │ │ │ +o7 = /tmp/M2-56114-0/0.mrdi
│ │ │ │  i8 : saveMRDI(f, FileName => fn)
│ │ │ │  
│ │ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ │       "RingElement"}, "data": [[["2", "0", "0"], ["1", "1"]], [["0", "1",
│ │ │ │       "1"], ["1", "1"]], [["1", "0", "0"], ["-3", "1"]]], "_ns": {"Macaulay2":
│ │ │ │       ["https://macaulay2.com", "1.26.06"]}, "_refs":
│ │ │ │       {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params": {"_type":
│ │ │ │ @@ -98,22 +98,22 @@
│ │ │ │                  "data" => 5
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  To see which types have built-in save methods for a given namespace, call
│ │ │ │  _m_e_t_h_o_d_s as follows.
│ │ │ │  i12 : methods {"Macaulay2", saveMRDI}
│ │ │ │  
│ │ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ │ -      {1 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ │ -      {2 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ │ -      {3 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ │ -      {4 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ │ -      {5 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ │ -      {6 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ │ -      {7 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ │ +      {1 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ │ +      {2 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ │ +      {3 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ │ +      {4 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ │ +      {5 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ │ +      {6 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ │ +      {7 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ │  
│ │ │ │  o12 : NumberedVerticalList
│ │ │ │  i13 : methods {"Oscar", saveMRDI}
│ │ │ │  
│ │ │ │  o13 = {0 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ │        {1 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ │        {2 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  dW5pbnN0YWxsUGFja2FnZShTdHJpbmcp
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjg5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh1bmluc3RhbGxQYWNrYWdlLFN0cmluZyksInVuaW5z
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Command.out
│ │ │ @@ -5,12 +5,12 @@
│ │ │  i2 : f
│ │ │  
│ │ │  o2 = 1073741824
│ │ │  
│ │ │  i3 : (c = Command "date";)
│ │ │  
│ │ │  i4 : c
│ │ │ -Sat Oct 24 03:11:57 UTC 2026
│ │ │ +Mon Oct 26 10:44:58 UTC 2026
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Database.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 9579076464446459296
│ │ │  
│ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-12569-0/0.dbm
│ │ │ +o1 = /tmp/M2-14359-0/0.dbm
│ │ │  
│ │ │  i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ +o2 = /tmp/M2-14359-0/0.dbm
│ │ │  
│ │ │  o2 : Database
│ │ │  
│ │ │  i3 : x#"first" = "hi there"
│ │ │  
│ │ │  o3 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 1731899428494721487
│ │ │  
│ │ │  i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 69379379194        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 69207431898        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 219217920
│ │ │ -               "numGCs" => 1481
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 255086592
│ │ │ +               "numGCs" => 1447
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
│ │ │  
│ │ │  i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 219217920
│ │ │ +o2 = 255086592
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out
│ │ │ @@ -40,20 +40,20 @@
│ │ │  o6 : PolynomialRing
│ │ │  
│ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ + -- used 0.00706833s (cpu); 0.00706626s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ + -- used 0.0571337s (cpu); 0.0570479s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : numgens J
│ │ │  
│ │ │  o10 = 1067
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal.out
│ │ │ @@ -77,15 +77,15 @@
│ │ │  o11 = monomialIdeal (a*b*c, a d, b*c*d)
│ │ │  
│ │ │  o11 : MonomialIdeal of R
│ │ │  
│ │ │  i12 : standardPairs I
│ │ │  
│ │ │                                                                             
│ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1,
│ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
│ │ │        -----------------------------------------------------------------------
│ │ │                             2
│ │ │        {b, a}}, {b, {c}}, {b , {c}}}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : independentSets I
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__List.out
│ │ │ @@ -37,19 +37,19 @@
│ │ │  i10 : s = new MutableList
│ │ │  
│ │ │  o10 = MutableList{}
│ │ │  
│ │ │  o10 : MutableList
│ │ │  
│ │ │  i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at each step
│ │ │ - -- .00605048s elapsed
│ │ │ + -- .00588647s elapsed
│ │ │  
│ │ │  i12 : t = new MutableList from 1000
│ │ │  
│ │ │  o12 = MutableList{...1000...}
│ │ │  
│ │ │  o12 : MutableList
│ │ │  
│ │ │  i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ - -- .000712171s elapsed
│ │ │ + -- .000397922s elapsed
│ │ │  
│ │ │  i14 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutex.out
│ │ │ @@ -8,40 +8,31 @@
│ │ │  
│ │ │  o2 = sayhello
│ │ │  
│ │ │  o2 : FunctionClosure
│ │ │  
│ │ │  i3 : T = apply(10, i -> schedule(() -> sayhello i))
│ │ │  
│ │ │ -o3 = {<>, <>, <>, <>, <>, <>, <>, <>, <>, <>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     task done>>, <>, <>, <>, <>, <>,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     <>}
│ │ │ +     <>, <>, <>}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : while not all(T, isReady) do null
│ │ │  
│ │ │  i5 : stack sort lines msgs
│ │ │  
│ │ │  o5 = hello from thread #0
│ │ │       hello from thread #1
│ │ │       hello from thread #2
│ │ │       hello from thread #3
│ │ │       hello from thread #4
│ │ │ -     hello from thread #5
│ │ │ -     hello from thread #6
│ │ │ -     hello from thread #7
│ │ │  
│ │ │  i6 : m = new Mutex
│ │ │  
│ │ │  o6 = m
│ │ │  
│ │ │  o6 : Mutex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out
│ │ │ @@ -3,13 +3,13 @@
│ │ │  i1 : M = random(RR^200, RR^200);
│ │ │  
│ │ │                  200         200
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
│ │ │  
│ │ │  i2 : time SVD(M);
│ │ │ - -- used 0.0228821s (cpu); 0.0228783s (thread); 0s (gc)
│ │ │ + -- used 0.0586664s (cpu); 0.0586605s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc)
│ │ │ + -- used 0.0401241s (cpu); 0.0401408s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : A = set {1,2};
│ │ │  
│ │ │  i2 : R = QQ[a..d];
│ │ │  
│ │ │  i3 : B = set{a^2-b*c,b*d}
│ │ │  
│ │ │ -           2
│ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ +                2
│ │ │ +o3 = set {b*d, a  - b*c}
│ │ │  
│ │ │  o3 : Set
│ │ │  
│ │ │  i4 : toList B
│ │ │  
│ │ │ -       2
│ │ │ -o4 = {a  - b*c, b*d}
│ │ │ +            2
│ │ │ +o4 = {b*d, a  - b*c}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : member(1,A)
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out
│ │ │ @@ -347,15 +347,15 @@
│ │ │                 | b e h k n q |
│ │ │                 | c f i l o r |
│ │ │  
│ │ │                               3
│ │ │  o58 : R-module, quotient of R
│ │ │  
│ │ │  i59 : time C = resolution M
│ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ + -- used 0.00415729s (cpu); 0.00415383s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o59 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out
│ │ │ @@ -14,10 +14,10 @@
│ │ │  
│ │ │  i4 : peek read f
│ │ │  
│ │ │  o4 = "hi there"
│ │ │  
│ │ │  i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
│ │ │ +o5 = true
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out
│ │ │ @@ -1,9 +1,9 @@
│ │ │  -- -*- M2-comint -*- hash: 1330379359420
│ │ │  
│ │ │  i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0003750952360409743
│ │ │ +o1 = .0003905923410533509
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : R = QQ[x,y,z];
│ │ │  
│ │ │  i3 : M = coker vars R;
│ │ │  
│ │ │  i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .0035172s elapsed
│ │ │ + -- .00447657s elapsed
│ │ │  
│ │ │  o4 = 3
│ │ │  
│ │ │  i5 : elapsedTime pdim' M
│ │ │ - -- 9.22e-7s elapsed
│ │ │ + -- .000002713s elapsed
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : peek M.cache
│ │ │  
│ │ │  o6 = CacheTable{cache => MutableHashTable{}                                              }
│ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_can__Use__Hilbert__Hint.out
│ │ │ @@ -211,15 +211,15 @@
│ │ │        -----------------------------------------------------------------------
│ │ │             5
│ │ │        + 12w )
│ │ │  
│ │ │  o17 : Ideal of Rlex
│ │ │  
│ │ │  i18 : elapsedTime g1 = gens gb Ilex;
│ │ │ - -- 1.68769s elapsed
│ │ │ + -- 2.38888s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o18 : Matrix Rlex  <-- Rlex
│ │ │  
│ │ │  i19 : Ilex = ideal(Ilex_*) -- clear out the previous Groebner basis
│ │ │  
│ │ │                  5      4       4       4       3 2      3         3     
│ │ │ @@ -293,15 +293,15 @@
│ │ │        -----------------------------------------------------------------------
│ │ │             5
│ │ │        + 12w )
│ │ │  
│ │ │  o19 : Ideal of Rlex
│ │ │  
│ │ │  i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf);
│ │ │ - -- .850458s elapsed
│ │ │ + -- 1.08468s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o20 : Matrix Rlex  <-- Rlex
│ │ │  
│ │ │  i21 : g1 == g2
│ │ │  
│ │ │  o21 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  
│ │ │  o4 = <>
│ │ │  
│ │ │  o4 : Task
│ │ │  
│ │ │  i5 : n
│ │ │  
│ │ │ -o5 = 336394
│ │ │ +o5 = 720838
│ │ │  
│ │ │  i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
│ │ │  
│ │ │  i7 : t
│ │ │  
│ │ │  o7 = <>
│ │ │  
│ │ │  o7 : Task
│ │ │  
│ │ │  i8 : n
│ │ │  
│ │ │ -o8 = 929987
│ │ │ +o8 = 1699814
│ │ │  
│ │ │  i9 : isReady t
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : cancelTask t
│ │ │  
│ │ │ @@ -53,22 +53,22 @@
│ │ │  
│ │ │  o12 = <>
│ │ │  
│ │ │  o12 : Task
│ │ │  
│ │ │  i13 : n
│ │ │  
│ │ │ -o13 = 930227
│ │ │ +o13 = 1700147
│ │ │  
│ │ │  i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
│ │ │  
│ │ │  i15 : n
│ │ │  
│ │ │ -o15 = 930227
│ │ │ +o15 = 1700147
│ │ │  
│ │ │  i16 : isReady t
│ │ │  
│ │ │  o16 = false
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8535510246140175278
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11296-0/0
│ │ │ +o1 = /tmp/M2-11766-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11296-0/0
│ │ │ +o2 = /tmp/M2-11766-0/0
│ │ │  
│ │ │  i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-11296-0/0/
│ │ │ +o3 = /tmp/M2-11766-0/0/
│ │ │  
│ │ │  i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-11296-0/0/
│ │ │ +o4 = /tmp/M2-11766-0/0/
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out
│ │ │ @@ -4,51 +4,51 @@
│ │ │  
│ │ │  o1 = FirstPackage
│ │ │  
│ │ │  o1 : Package
│ │ │  
│ │ │  i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .313321s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .219073s elapsed
│ │ │  
│ │ │  i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .282764s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283285s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .192358s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .207897s elapsed
│ │ │  
│ │ │  i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .275704s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .193146s elapsed
│ │ │  
│ │ │  i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .246295s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .21342s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .194179s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .206836s elapsed
│ │ │  
│ │ │  i6 : tests(1, "FirstPackage")
│ │ │  
│ │ │  o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]
│ │ │  
│ │ │  o6 : TestInput
│ │ │  
│ │ │  i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .302726s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .217212s elapsed
│ │ │  
│ │ │  i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │       {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o8 : NumberedVerticalList
│ │ │  
│ │ │  i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .28407s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .282385s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .190655s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .198866s elapsed
│ │ │  
│ │ │  i10 : tests "FirstPackage"
│ │ │  
│ │ │  o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │        {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │  
│ │ │  i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283414s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .199975s elapsed
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 10365735446967377456
│ │ │  
│ │ │  i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close
│ │ │   ba 
│ │ │   ad 
│ │ │  
│ │ │  o2 = !grep a
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ -     6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n"
│ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +     6.12.100-1 (2026-07-30) x86_64 GNU/Linux\n"
│ │ │  
│ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ │  
│ │ │  o4 = !grep -E '^in'
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │ @@ -74,15 +74,15 @@
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : isReady f
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : while not isReady f do (sleep 1; << "." << flush)
│ │ │ -......
│ │ │ +.....
│ │ │  i11 : read f
│ │ │  
│ │ │  o11 = the answer is 4
│ │ │  
│ │ │  i12 : isReady f
│ │ │  
│ │ │  o12 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │                  ZZ
│ │ │  o23 : Ideal of ----[x..z, w]
│ │ │                 1277
│ │ │  
│ │ │  i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f57805dd000
│ │ │ +   -- registering gb 5 at 0x7f0008d49000
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -177,15 +177,15 @@
│ │ │  
│ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │  
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
│ │ │  
│ │ │  i33 : time betti gb f
│ │ │ - -- used 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ + -- used 0.231793s (cpu); 0.229499s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -208,15 +208,15 @@
│ │ │  
│ │ │              3    5     8     9    12     14    17
│ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │  
│ │ │  o35 : ZZ[T]
│ │ │  
│ │ │  i36 : time betti gb f
│ │ │ - -- used 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ + -- used 0.00800088s (cpu); 0.00892148s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__Directory_lp__String_cm__String_rp.out
│ │ │ @@ -1,76 +1,76 @@
│ │ │  -- -*- M2-comint -*- hash: 11422793294564310273
│ │ │  
│ │ │  i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12075-0/0/
│ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12075-0/1/
│ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12075-0/0/a/
│ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12075-0/0/b/
│ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
│ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ -     /tmp/M2-12075-0/0/a/f
│ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │  
│ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │  
│ │ │  i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ -      /tmp/M2-12075-0/1/b/c/g
│ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ +      /tmp/M2-13345-0/1/b/c/g
│ │ │  
│ │ │  i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
│ │ │  
│ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out
│ │ │ @@ -1,41 +1,41 @@
│ │ │  -- -*- M2-comint -*- hash: 11539475420155775110
│ │ │  
│ │ │  i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11841-0/0
│ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11841-0/1
│ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
│ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │  
│ │ │  i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │  
│ │ │  i9 : get dst
│ │ │  
│ │ │  o9 = hi there
│ │ │  
│ │ │  i10 : removeFile src
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 15508153783232232453
│ │ │  
│ │ │  i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 611.770723009
│ │ │ +o1 = 679.2678651149999
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │  
│ │ │  i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 613.313432871
│ │ │ +o3 = 680.676774634
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.542709862000038
│ │ │ +o4 = 1.408909519000076
│ │ │  
│ │ │  o4 : RR (of precision 53)
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out
│ │ │ @@ -1,24 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 3660839476107967259
│ │ │  
│ │ │  i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1792811657
│ │ │ +o1 = 1793011624
│ │ │  
│ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.81193979063632
│ │ │ +o2 = 56.81827649260931
│ │ │  
│ │ │  o2 : RR (of precision 53)
│ │ │  
│ │ │  i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 9.743277487635794
│ │ │ +o3 = 9.819317911311686
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : run "date"
│ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ +Mon Oct 26 10:47:04 UTC 2026
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1330565958025
│ │ │  
│ │ │  i1 : elapsedTime sleep 1
│ │ │ - -- 1.00015s elapsed
│ │ │ + -- 1.00023s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 1731106803207298715
│ │ │  
│ │ │  i1 : elapsedTiming sleep 1
│ │ │  
│ │ │  o1 = 0
│ │ │ -     -- 1.00011 seconds
│ │ │ +     -- 1.00014 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00011, 0}
│ │ │ +o2 = Time{1.00014, 0}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │                 3                   3     2               3
│ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time leadTerm gens gb I
│ │ │ - -- used 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ + -- used 0.482245s (cpu); 0.207912s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │                 3                   3     2               3
│ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ + -- used 0.105157s (cpu); 0.105157s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -154,15 +154,15 @@
│ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │  
│ │ │  i11 : I1 = substitute(I,R1);
│ │ │  
│ │ │  o11 : Ideal of R1
│ │ │  
│ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ + -- used 0.0340178s (cpu); 0.034025s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -228,15 +228,15 @@
│ │ │  
│ │ │                     3             3     2         3
│ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │  
│ │ │  o16 : RingMap A <-- B
│ │ │  
│ │ │  i17 : time G = kernel F
│ │ │ - -- used 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ + -- used 0.451116s (cpu); 0.184461s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -297,23 +297,23 @@
│ │ │  i19 : use ring I
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : PolynomialRing
│ │ │  
│ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ + -- used 0.00204952s (cpu); 0.00204688s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
│ │ │  
│ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ + -- used 0.0362494s (cpu); 0.036266s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-11022-0/96-rundir/
│ │ │ +             source directory => /tmp/M2-11242-0/96-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
│ │ │  
│ │ │  i7 : dictionaryPath
│ │ │  
│ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 7475038936570224899
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11391-0/0
│ │ │ +o1 = /tmp/M2-11961-0/0
│ │ │  
│ │ │  i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ +o3 = /tmp/M2-11961-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : fileExists fn
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Length.out
│ │ │ @@ -1,28 +1,28 @@
│ │ │  -- -*- M2-comint -*- hash: 1216695447195237994
│ │ │  
│ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-13078-0/0
│ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o1 : File
│ │ │  
│ │ │  i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-13078-0/0
│ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
│ │ │  
│ │ │  i6 : get filename
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__File_rp.out
│ │ │ @@ -1,25 +1,25 @@
│ │ │  -- -*- M2-comint -*- hash: 11202140621123993633
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12284-0/0
│ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
│ │ │  
│ │ │  i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : removeFile fn
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 4782570202197464532
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11860-0/0
│ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode fn
│ │ │  
│ │ │  o3 = 420
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__File_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 17473878267845575442
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11725-0/0
│ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │  
│ │ │  o3 = 511
│ │ │  
│ │ │ @@ -18,15 +18,15 @@
│ │ │  
│ │ │  i5 : fileMode f
│ │ │  
│ │ │  o5 = 511
│ │ │  
│ │ │  i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-11725-0/0
│ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : fileMode fn
│ │ │  
│ │ │  o7 = 511
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 16772784390799334723
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12905-0/0
│ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : m = fileMode fn
│ │ │  
│ │ │  o3 = 420
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331310711075
│ │ │  
│ │ │  i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 111
│ │ │ +o1 = 119
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.out
│ │ │ @@ -29,15 +29,15 @@
│ │ │       {4} | 0     x2-3  y3-1  |
│ │ │  
│ │ │               3      3
│ │ │  o6 : Matrix R  <-- R
│ │ │  
│ │ │  i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f466ed708c0
│ │ │ +   -- registering gb 0 at 0x7f3b4b7c98c0
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_get.out
│ │ │ @@ -10,11 +10,11 @@
│ │ │  
│ │ │  o2 = hi there
│ │ │  
│ │ │  i3 : removeFile "test-file"
│ │ │  
│ │ │  i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sat Oct 24 03:12:48 UTC 2026
│ │ │ +o4 = Mon Oct 26 10:45:43 UTC 2026
│ │ │  
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Directory.out
│ │ │ @@ -2,19 +2,19 @@
│ │ │  
│ │ │  i1 : isDirectory "."
│ │ │  
│ │ │  o1 = true
│ │ │  
│ │ │  i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11213-0/0
│ │ │ +o2 = /tmp/M2-11603-0/0
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ +o3 = /tmp/M2-11603-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : isDirectory fn
│ │ │  
│ │ │  o4 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out
│ │ │ @@ -75,15 +75,15 @@
│ │ │  o17 = false
│ │ │  
│ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │  
│ │ │  o18 = false
│ │ │  
│ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 3.67323s elapsed
│ │ │ + -- 5.39915s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
│ │ │  
│ │ │  i20 : facs = facs//toList/toList
│ │ │  
│ │ │ @@ -97,17 +97,17 @@
│ │ │  
│ │ │  i22 : m3 = nextPrime (m^3)
│ │ │  
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │  
│ │ │  i23 : elapsedTime isPrime m3
│ │ │ - -- .05718s elapsed
│ │ │ + -- .0688154s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000110026s elapsed
│ │ │ + -- .000244807s elapsed
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 4782205245758053629
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-13116-0/0
│ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : isRegularFile fn
│ │ │  
│ │ │  o3 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_lock_lp__Mutex_cm__Function_rp.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i2 : f = i -> x += 1;
│ │ │  
│ │ │  i3 : parallelApply(1..1000, f);
│ │ │  
│ │ │  i4 : x
│ │ │  
│ │ │ -o4 = 333
│ │ │ +o4 = 344
│ │ │  
│ │ │  i5 : x = 0
│ │ │  
│ │ │  o5 = 0
│ │ │  
│ │ │  i6 : f = lock f;
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_make__Directory_lp__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 5113372159204571746
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11593-0/0
│ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │  
│ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
│ │ │ +o2 = /tmp/M2-12363-0/0/a/b/c
│ │ │  
│ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │  
│ │ │  i5 : removeDirectory (dir|"/a")
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331887830690
│ │ │  
│ │ │  i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
│ │ │ +o1 = 17
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out
│ │ │ @@ -3,31 +3,31 @@
│ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │  
│ │ │  o1 = fib
│ │ │  
│ │ │  o1 : FunctionClosure
│ │ │  
│ │ │  i2 : time fib 28
│ │ │ - -- used 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ + -- used 1.15321s (cpu); 0.653719s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
│ │ │  
│ │ │  i3 : fib = memoize fib
│ │ │  
│ │ │  o3 = fib
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : time fib 28
│ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ + -- used 7.6966e-05s (cpu); 7.3392e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
│ │ │  
│ │ │  i5 : time fib 28
│ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ + -- used 3.961e-06s (cpu); 3.503e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
│ │ │  
│ │ │  i6 : fib = memoize( n -> fib(n-1) + fib(n-2) )
│ │ │  
│ │ │  o6 = fib
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out
│ │ │ @@ -17,20 +17,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ +     {20 => (codim, BettiTally)                                   }
│ │ │       {21 => (dual, BettiTally)                                    }
│ │ │       {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ -     {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ +     {23 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ +     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │  
│ │ │  i2 : methods resolution
│ │ │  
│ │ │  o2 = {0 => (freeResolution, Ideal)        }
│ │ │ @@ -60,20 +60,20 @@
│ │ │  
│ │ │  o4 = {0 => (++, Module, Module)}
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ -     {3 => (contract, Matrix, Matrix)                            }
│ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ +o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ +     {1 => (diff, Matrix, Matrix)                                }
│ │ │ +     {2 => (-, Matrix, Matrix)                                   }
│ │ │ +     {3 => (contract', Matrix, Matrix)                           }
│ │ │ +     {4 => (contract, Matrix, Matrix)                            }
│ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -84,22 +84,22 @@
│ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {30 => (intersect, Matrix, Matrix)                          }
│ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.20053s elapsed
│ │ │ + -- 1.97379s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -26,44 +26,44 @@
│ │ │  o3 : BettiTally
│ │ │  
│ │ │  i4 : I = ideal I_*;
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .517063s elapsed
│ │ │ + -- .838305s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : I = ideal I_*;
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0258497s elapsed
│ │ │ + -- .0477547s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
│ │ │  
│ │ │  i8 : I = ideal I_*;
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │  
│ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- .826979s elapsed
│ │ │ + -- 1.27651s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out
│ │ │ @@ -1,22 +1,22 @@
│ │ │  -- -*- M2-comint -*- hash: 15555226809509933135
│ │ │  
│ │ │  i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11612-0/0/
│ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │  
│ │ │  i2 : mkdir p
│ │ │  
│ │ │  i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ +o4 = /tmp/M2-12402-0/0/foo
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : get fn
│ │ │  
│ │ │  o5 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_move__File_lp__String_cm__String_rp.out
│ │ │ @@ -1,31 +1,31 @@
│ │ │  -- -*- M2-comint -*- hash: 4857944042471093218
│ │ │  
│ │ │  i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11448-0/0
│ │ │ +o1 = /tmp/M2-12078-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11448-0/1
│ │ │ +o2 = /tmp/M2-12078-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ +o3 = /tmp/M2-12078-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
│ │ │ +--moving: /tmp/M2-12078-0/0 -> /tmp/M2-12078-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ +--backup file created: /tmp/M2-12078-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-11448-0/1.bak
│ │ │ +o6 = /tmp/M2-12078-0/1.bak
│ │ │  
│ │ │  i7 : removeFile bak
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_nanosleep.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1331114612441
│ │ │  
│ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500147s elapsed
│ │ │ + -- .500173s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_options_lp__Function_rp.out
│ │ │ @@ -20,60 +20,60 @@
│ │ │  
│ │ │  o3 = OptionTable{Generic => false}
│ │ │  
│ │ │  o3 : OptionTable
│ │ │  
│ │ │  i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ -     {1 => (codim, Ideal)         }
│ │ │ -     {2 => (codim, PolynomialRing)}
│ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ -     {5 => (codim, Module)        }
│ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ -     {7 => (codim, BettiTally)    }
│ │ │ +o4 = {0 => (codim, Module)        }
│ │ │ +     {1 => (codim, CoherentSheaf) }
│ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ +     {3 => (codim, Variety)       }
│ │ │ +     {4 => (codim, Ideal)         }
│ │ │ +     {5 => (codim, PolynomialRing)}
│ │ │ +     {6 => (codim, QuotientRing)  }
│ │ │ +     {7 => (codim, MonomialIdeal) }
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : options oo
│ │ │  
│ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │       {1 => (OptionTable{Generic => false})}
│ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ +     {2 => (OptionTable{})                }
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ -     {7 => (OptionTable{})                }
│ │ │ +     {7 => (OptionTable{Generic => false})}
│ │ │  
│ │ │  o5 : NumberedVerticalList
│ │ │  
│ │ │  i6 : methods intersect
│ │ │  
│ │ │  o6 = {0 => (intersect, List)                          }
│ │ │       {1 => (intersect, Module)                        }
│ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ -     {6 => (intersect, Set)                           }
│ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {8 => (intersect, Set)                           }
│ │ │       {9 => (intersect, Ideal)                         }
│ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │       {11 => (intersect, CCi)                          }
│ │ │       {12 => (intersect, RRi)                          }
│ │ │       {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │       {14 => (intersect, Cone, Cone)                   }
│ │ │       {15 => (intersect, Matrix)                       }
│ │ │       {16 => (intersect, Module, Module)               }
│ │ │       {17 => (intersect, CCi, CCi)                     }
│ │ │ -     {18 => (intersect, CCi, RRi)                     }
│ │ │ -     {19 => (intersect, RRi, CCi)                     }
│ │ │ +     {18 => (intersect, RRi, CCi)                     }
│ │ │ +     {19 => (intersect, CCi, RRi)                     }
│ │ │       {20 => (intersect, Sequence)                     }
│ │ │  
│ │ │  o6 : NumberedVerticalList
│ │ │  
│ │ │  i7 : options 0
│ │ │  
│ │ │  o7 = OptionTable{}
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out
│ │ │ @@ -5,26 +5,26 @@
│ │ │  o1 = {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : L = shuffle toList (1..10000);
│ │ │  
│ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .659735s elapsed
│ │ │ + -- .949337s elapsed
│ │ │  
│ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .211134s elapsed
│ │ │ + -- .204447s elapsed
│ │ │  
│ │ │  i5 : allowableThreads
│ │ │  
│ │ │  o5 = 5
│ │ │  
│ │ │  i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
│ │ │ +o6 = 17
│ │ │  
│ │ │  i7 : R = QQ[x,y,z];
│ │ │  
│ │ │  i8 : I = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z)
│ │ │  
│ │ │               2     2            2     2             2
│ │ │  o8 = ideal (x  + 2y  - y - 2z, x  - 8y  + 10z - 1, x  - 7y*z)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │  i3 : S = ring I
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : elapsedTime minimalBetti I
│ │ │ - -- 1.15969s elapsed
│ │ │ + -- 1.85279s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -84,15 +84,15 @@
│ │ │  o4 : BettiTally
│ │ │  
│ │ │  i5 : I = ideal I_*;
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.14895s elapsed
│ │ │ + -- 1.84267s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : numTBBThreads = 1
│ │ │  
│ │ │  o8 = 1
│ │ │  
│ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.62269s elapsed
│ │ │ + -- 1.86033s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -132,15 +132,15 @@
│ │ │  o11 = 0
│ │ │  
│ │ │  i12 : I = ideal I_*;
│ │ │  
│ │ │  o12 : Ideal of S
│ │ │  
│ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.37571s elapsed
│ │ │ + -- 2.28683s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
│ │ │ @@ -150,15 +150,15 @@
│ │ │  o14 = 1
│ │ │  
│ │ │  i15 : I = ideal I_*;
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │  
│ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 1.51803s elapsed
│ │ │ + -- 2.2938s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
│ │ │ @@ -174,43 +174,43 @@
│ │ │  o18 : PolynomialRing
│ │ │  
│ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │  
│ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.09157s elapsed
│ │ │ + -- 4.53922s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
│ │ │  
│ │ │  i21 : numTBBThreads = 1
│ │ │  
│ │ │  o21 = 1
│ │ │  
│ │ │  i22 : I = ideal I_*;
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │  
│ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.21927s elapsed
│ │ │ + -- 10.2962s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
│ │ │  
│ │ │  i24 : numTBBThreads = 10
│ │ │  
│ │ │  o24 = 10
│ │ │  
│ │ │  i25 : I = ideal I_*;
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.08357s elapsed
│ │ │ + -- 4.30526s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
│ │ │  
│ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │  
│ │ │  o27 = AssociativeAlgebras
│ │ │ @@ -233,15 +233,15 @@
│ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │  
│ │ │                  ZZ
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │  
│ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- .823549s elapsed
│ │ │ + -- 1.18493s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
│ │ │  
│ │ │  i32 : I = ideal I_*
│ │ │  
│ │ │ @@ -253,14 +253,14 @@
│ │ │                 101
│ │ │  
│ │ │  i33 : numTBBThreads = 1
│ │ │  
│ │ │  o33 = 1
│ │ │  
│ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.45319s elapsed
│ │ │ + -- 2.07506s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
│ │ │  
│ │ │  i35 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out
│ │ │ @@ -146,65 +146,65 @@
│ │ │  o26 : ZZ[T]
│ │ │  
│ │ │  i27 : gbTrace = 3
│ │ │  
│ │ │  o27 = 3
│ │ │  
│ │ │  i28 : time poincare I
│ │ │ - -- used 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00271247s (cpu); 1.302e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
│ │ │  
│ │ │  i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ +   -- registering gb 16 at 0x7f2922d09c40
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00927416s (cpu); 0.0124893s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
│ │ │  
│ │ │  i30 : R = QQ[a..d];
│ │ │  
│ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ +   -- registering gb 17 at 0x7f2922d09a80
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │     -- nsaved = 0
│ │ │     -- 
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ +   -- registering gb 18 at 0x7f2922d098c0
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00399734s (cpu); 0.00481081s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
│ │ │  
│ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ @@ -254,27 +254,27 @@
│ │ │  
│ │ │  i36 : gbTrace = 3
│ │ │  
│ │ │  o36 = 3
│ │ │  
│ │ │  i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ +   -- registering gb 19 at 0x7f2922d09700
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
│ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0599994s (cpu); 0.0573032s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
│ │ │  
│ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │  
│ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_printing_spto_spa_spfile.out
│ │ │ @@ -12,19 +12,19 @@
│ │ │  
│ │ │  o2 = stdio
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11803-0/0
│ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : get fn
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │ @@ -49,27 +49,27 @@
│ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │  o8 = stdio
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11803-0/0
│ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o9 : File
│ │ │  
│ │ │  i10 : get fn
│ │ │  
│ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │        + 1
│ │ │  
│ │ │  i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11803-0/0
│ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o11 : File
│ │ │  
│ │ │  i12 : get fn
│ │ │  
│ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │        1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_process__I__D.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1330513630563
│ │ │  
│ │ │  i1 : processID()
│ │ │  
│ │ │ -o1 = 11022
│ │ │ +o1 = 11242
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out
│ │ │ @@ -9,35 +9,35 @@
│ │ │  
│ │ │                4       5
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22  
│ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61     
│ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52 
│ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ -     1     .0041s  elapsed total                    
│ │ │ +     1     93.17   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ +     1     90.46   ../../m2/matrix1.m2:275:22-275:43
│ │ │ +     1     44.54   ../../m2/matrix1.m2:183:25-183:52
│ │ │ +     1     31.45   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ +     1     30.27   ../../m2/matrix1.m2:130:10-145:16
│ │ │ +     1     21.85   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ +     1     21.76   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ +     1     20.45   ../../m2/set.m2:129:5-129:61     
│ │ │ +     1     3.26    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ +     1     2.37    ../../m2/matrix1.m2:131:13-131:78
│ │ │ +     1     2.08    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ +     1     1.58    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ +     1     1.46    ../../m2/matrix1.m2:137:20-137:64
│ │ │ +     1     1.34    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ +     1     1.16    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ +     1     1.07    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ +     20    1.03    ../../m2/matrix1.m2:181:14-182:67
│ │ │ +     1     1.02    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ +     20    .90     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ +     1     .74     ../../m2/modules.m2:282:4-282:52 
│ │ │ +     1     .0040s  elapsed total                    
│ │ │  
│ │ │  i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │       ../../m2/lists.m2:148:24-148:32
│ │ │       ../../m2/lists.m2:148:34-148:58
│ │ │       ../../m2/matrix.m2:30:5-30:35
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │  i5 : codim I, degree I
│ │ │  
│ │ │  o5 = (2, 10)
│ │ │  
│ │ │  o5 : Sequence
│ │ │  
│ │ │  i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ + -- used 0.842s (cpu); 0.205593s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : R=kk[x_0..x_5];
│ │ │ @@ -33,15 +33,15 @@
│ │ │  i9 : codim I, degree I
│ │ │  
│ │ │  o9 = (3, 10)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ + -- used 0.499404s (cpu); 0.284665s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : p=10007,kk=ZZ/p,R=kk[x_0..x_2]
│ │ │ @@ -58,12 +58,12 @@
│ │ │  
│ │ │  i14 : I=ideal random(n,R);
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ + -- used 8.3979s (cpu); 3.04804s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
│ │ │  
│ │ │  i16 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_read__Directory.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 20910736704070514
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12474-0/0
│ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-12474-0/0
│ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │  
│ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ +o3 = /tmp/M2-14164-0/0/foo
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : removeFile fn
│ │ │  
│ │ │  i6 : removeDirectory dir
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reading_spfiles.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 13513555104200944796
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11978-0/0
│ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  i2 : fn << "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3" << endl << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11978-0/0
│ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : get fn
│ │ │  
│ │ │  o3 = z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
│ │ │       +8*y^3
│ │ │ @@ -38,15 +38,15 @@
│ │ │  
│ │ │  o6 : Expression of class Product
│ │ │  
│ │ │  i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11978-0/0
│ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : get fn
│ │ │  
│ │ │  o8 = sample = 2^100
│ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_readlink.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 4408639611478781130
│ │ │  
│ │ │  i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-12734-0/0
│ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │  
│ │ │  i2 : symlinkFile ("foo", p)
│ │ │  
│ │ │  i3 : readlink p
│ │ │  
│ │ │  o3 = foo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out
│ │ │ @@ -1,39 +1,39 @@
│ │ │  -- -*- M2-comint -*- hash: 324072347213224656
│ │ │  
│ │ │  i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-11022-0/91-rundir/
│ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │  
│ │ │  i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12753-0/0
│ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12753-0/1
│ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │  
│ │ │  i4 : symlinkFile(p,q)
│ │ │  
│ │ │  i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12753-0/0
│ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12753-0/0
│ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i8 : removeFile p
│ │ │  
│ │ │  i9 : removeFile q
│ │ │  
│ │ │  i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
│ │ │ +o10 = /tmp/M2-11242-0/91-rundir/
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out
│ │ │ @@ -1,17 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 1729384374372662693
│ │ │  
│ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
│ │ │  
│ │ │  i2 : collectGarbage() 
│ │ │ ---finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (1)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (3)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (4)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (5)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (7)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (9)[6]: -- finalizing sequence #7 --
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8532980310097060089
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11650-0/0
│ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11650-0/0
│ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │  
│ │ │  i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : removeDirectory dir
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__Path.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731420232148149387
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11116-0/0
│ │ │ +o1 = /tmp/M2-11406-0/0
│ │ │  
│ │ │  i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-11116-0/0
│ │ │ +o2 = /tmp/M2-11406-0/0
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__U__R__I.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731420231525572968
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12417-0/0
│ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │  
│ │ │  i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-12417-0/0
│ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_saving_sppolynomials_spand_spmatrices_spin_spfiles.out
│ │ │ @@ -25,19 +25,19 @@
│ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │  
│ │ │                               1
│ │ │  o4 : R-module, submodule of R
│ │ │  
│ │ │  i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-12265-0/0
│ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : get f
│ │ │  
│ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_serial__Number.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 5271760183816554957
│ │ │  
│ │ │  i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1367325
│ │ │ +o1 = 1367343
│ │ │  
│ │ │  i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1367327
│ │ │ +o2 = 1367345
│ │ │  
│ │ │  i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out
│ │ │ @@ -189,18 +189,18 @@
│ │ │  o25 = 40
│ │ │  
│ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │  
│ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │  
│ │ │  i30 : time X = solve(A,B);
│ │ │ - -- used 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
│ │ │ + -- used 0.000213924s (cpu); 0.000199003s (thread); 0s (gc)
│ │ │  
│ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
│ │ │ + -- used 9.4969e-05s (cpu); 9.4516e-05s (thread); 0s (gc)
│ │ │  
│ │ │  i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │  
│ │ │  o32 : RR (of precision 53)
│ │ │  
│ │ │ @@ -209,18 +209,18 @@
│ │ │  o33 = 100
│ │ │  
│ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │  
│ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │  
│ │ │  i38 : time X = solve(A,B);
│ │ │ - -- used 0.232067s (cpu); 0.232067s (thread); 0s (gc)
│ │ │ + -- used 0.14571s (cpu); 0.145712s (thread); 0s (gc)
│ │ │  
│ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
│ │ │ + -- used 0.1453s (cpu); 0.145313s (thread); 0s (gc)
│ │ │  
│ │ │  i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │  
│ │ │  o40 : RR (of precision 100)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__Directory_lp__String_cm__String_rp.out
│ │ │ @@ -1,60 +1,60 @@
│ │ │  -- -*- M2-comint -*- hash: 2989513528213557691
│ │ │  
│ │ │  i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12037-0/0/
│ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12037-0/1/
│ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12037-0/0/a/
│ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12037-0/0/b/
│ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
│ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │  
│ │ │  i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
│ │ │  
│ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │  
│ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │  
│ │ │  o12 = rm
│ │ │  
│ │ │  o12 : FunctionClosure
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__File.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 9343844672940306595
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12094-0/0
│ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │  
│ │ │  i2 : symlinkFile("qwert", fn)
│ │ │  
│ │ │  i3 : fileExists fn
│ │ │  
│ │ │  o3 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_temporary__File__Name.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731926531291302106
│ │ │  
│ │ │  i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-13097-0/0.tex
│ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │  
│ │ │  i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-13097-0/1.html
│ │ │ +o2 = /tmp/M2-15437-0/1.html
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1332435500723
│ │ │  
│ │ │  i1 : time 3^30
│ │ │ - -- used 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ + -- used 2.075e-05s (cpu); 6.622e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 1730988300469098603
│ │ │  
│ │ │  i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000013355 seconds
│ │ │ +     -- .000024414 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000013355, 205891132094649}
│ │ │ +o2 = Time{.000024414, 205891132094649}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.out
│ │ │ @@ -14,26 +14,26 @@
│ │ │  
│ │ │            4     5    6     9     10     11    14     15    16    20
│ │ │  o3 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │  
│ │ │  o3 : ZZ[T]
│ │ │  
│ │ │  i4 : elapsedTime Mgb = gb M
│ │ │ - -- 1.4495s elapsed
│ │ │ + -- 2.6727s elapsed
│ │ │  
│ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o4 : GroebnerBasis
│ │ │  
│ │ │  i5 : M = ideal M_*;
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .415018s elapsed
│ │ │ + -- .778775s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o6 : GroebnerBasis
│ │ │  
│ │ │  i7 : S = degreesRing R
│ │ │  
│ │ │ @@ -55,14 +55,14 @@
│ │ │  o9 : S
│ │ │  
│ │ │  i10 : M = ideal M_*;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .517879s elapsed
│ │ │ + -- .872512s elapsed
│ │ │  
│ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o11 : GroebnerBasis
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out
│ │ │ @@ -36,15 +36,15 @@
│ │ │                 "memtailor version" => 1.4
│ │ │                 "mpfi version" => 1.5.5
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.1
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ +               "operating system release" => 6.12.100+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism DirectSummands CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC SimplicialModules MatrixFactorizations PathSignatures MacaulayPosets MRDI EliminationTemplates WittVectors Padic WeierstrassSemigroups ResultantComplexes EuclideanDistanceDegree NeuralIdeals TestAudit LanguageServer
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.14.9
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2023.1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │              
    		
│ │ │                
    i3 : (c = Command "date";)
    
│ │ │              
    
│ │ │                
    i4 : c
│ │ │ -Sat Oct 24 03:11:57 UTC 2026
│ │ │ +Mon Oct 26 10:44:58 UTC 2026
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  in a file), then it gets executed with empty argument list.
│ │ │ │  i1 : (f = Command ( () -> 2^30 );)
│ │ │ │  i2 : f
│ │ │ │  
│ │ │ │  o2 = 1073741824
│ │ │ │  i3 : (c = Command "date";)
│ │ │ │  i4 : c
│ │ │ │ -Sat Oct 24 03:11:57 UTC 2026
│ │ │ │ +Mon Oct 26 10:44:58 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_u_n -- run an external command
│ │ │ │      * _A_f_t_e_r_E_v_a_l -- top level method applied after evaluation
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa ccoommmmaanndd:: **********
│ │ │ │      * code(Command) -- see _c_o_d_e -- display source code
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html
│ │ │ @@ -57,22 +57,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  A database file is just like a hash table, except both the keys and values have to be strings.  In this example we create a database file, store a few entries, remove an entry with remove, close the file, and then remove the file.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │                               2
│ │ │ │  o11 = monomialIdeal (a*b*c, a d, b*c*d)
│ │ │ │  
│ │ │ │  o11 : MonomialIdeal of R
│ │ │ │  i12 : standardPairs I
│ │ │ │  
│ │ │ │  
│ │ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1,
│ │ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                             2
│ │ │ │        {b, a}}, {b, {c}}, {b , {c}}}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : independentSets I
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutable__List.html
│ │ │ @@ -145,30 +145,30 @@
│ │ │  
│ │ │  o10 : MutableList
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-12569-0/0.dbm
    
│ │ │ +o1 = /tmp/M2-14359-0/0.dbm
│ │ │              
    
│ │ │                
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ +o2 = /tmp/M2-14359-0/0.dbm
│ │ │  
│ │ │  o2 : Database
    
│ │ │              
    
│ │ │                
    i3 : x#"first" = "hi there"
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,18 +7,18 @@
│ │ │ │  ************ DDaattaabbaassee ---- tthhee ccllaassss ooff aallll ddaattaabbaassee ffiilleess ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A database file is just like a hash table, except both the keys and values have
│ │ │ │  to be strings. In this example we create a database file, store a few entries,
│ │ │ │  remove an entry with _r_e_m_o_v_e, close the file, and then remove the file.
│ │ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12569-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-14359-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-14359-0/0.dbm
│ │ │ │  
│ │ │ │  o2 : Database
│ │ │ │  i3 : x#"first" = "hi there"
│ │ │ │  
│ │ │ │  o3 = hi there
│ │ │ │  i4 : x#"first"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -58,33 +58,33 @@
│ │ │          
    Description
    
│ │ │          
    Macaulay2 uses the Hans Boehm garbage collector to reclaim unused memory.  The function GCstats provides information about its status, such as the total number of bytes allocated, the current heap size, the number of garbage collections done, the number of threads used in each collection, the total cpu time spent in garbage collection, etc.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 69379379194        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 69207431898        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 219217920
│ │ │ -               "numGCs" => 1481
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 255086592
│ │ │ +               "numGCs" => 1447
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
    
│ │ │              
    
│ │ │          
    The value returned is a hash table, from which individual bits of information can be easily extracted, as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 219217920
    
│ │ │ +o2 = 255086592
│ │ │              
    
│ │ │          
    Any entries whose keys are all upper case give the values of environment variables affecting the operation of the garbage collector that have been specified by the user.
    
│ │ │          
    For further information about the individual items in the table, we refer the user to the source code and documentation of the garbage collector.
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,28 @@
│ │ │ │  Macaulay2 uses the Hans Boehm _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r to reclaim unused memory. The
│ │ │ │  function GCstats provides information about its status, such as the total
│ │ │ │  number of bytes allocated, the current heap size, the number of garbage
│ │ │ │  collections done, the number of threads used in each collection, the total cpu
│ │ │ │  time spent in garbage collection, etc.
│ │ │ │  i1 : s = GCstats()
│ │ │ │  
│ │ │ │ -o1 = HashTable{"bytesAlloc" => 69379379194        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 69207431898        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 219217920
│ │ │ │ -               "numGCs" => 1481
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 255086592
│ │ │ │ +               "numGCs" => 1447
│ │ │ │ +               "numGCThreads" => 16
│ │ │ │  
│ │ │ │  o1 : HashTable
│ │ │ │  The value returned is a hash table, from which individual bits of information
│ │ │ │  can be easily extracted, as follows.
│ │ │ │  i2 : s#"heapSize"
│ │ │ │  
│ │ │ │ -o2 = 219217920
│ │ │ │ +o2 = 255086592
│ │ │ │  Any entries whose keys are all upper case give the values of environment
│ │ │ │  variables affecting the operation of the garbage collector that have been
│ │ │ │  specified by the user.
│ │ │ │  For further information about the individual items in the table, we refer the
│ │ │ │  user to the source code and documentation of the garbage collector.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Minimal__Generators.html
│ │ │ @@ -133,23 +133,23 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ + -- used 0.00706833s (cpu); 0.00706626s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
    
│ │ │                
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ + -- used 0.0571337s (cpu); 0.0570479s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : numgens J
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,19 +46,19 @@
│ │ │ │  o6 = R
│ │ │ │  
│ │ │ │  o6 : PolynomialRing
│ │ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ │ - -- used 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ │ + -- used 0.00706833s (cpu); 0.00706626s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ │ + -- used 0.0571337s (cpu); 0.0570479s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : numgens J
│ │ │ │  
│ │ │ │  o10 = 1067
│ │ │ │  i11 : numgens K
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Monomial__Ideal.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │              
    
│ │ │                
    i12 : standardPairs I
│ │ │  
│ │ │                                                                             
│ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1,
│ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
│ │ │        -----------------------------------------------------------------------
│ │ │                             2
│ │ │        {b, a}}, {b, {c}}, {b , {c}}}
│ │ │  
│ │ │  o12 : List
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at each step
│ │ │ - -- .00605048s elapsed
    
│ │ │ + -- .00588647s elapsed
│ │ │              
    
│ │ │                
    i12 : t = new MutableList from 1000
│ │ │  
│ │ │  o12 = MutableList{...1000...}
│ │ │  
│ │ │  o12 : MutableList
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ - -- .000712171s elapsed
    
│ │ │ + -- .000397922s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,22 +43,22 @@
│ │ │ │  i10 : s = new MutableList
│ │ │ │  
│ │ │ │  o10 = MutableList{}
│ │ │ │  
│ │ │ │  o10 : MutableList
│ │ │ │  i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at
│ │ │ │  each step
│ │ │ │ - -- .00605048s elapsed
│ │ │ │ + -- .00588647s elapsed
│ │ │ │  i12 : t = new MutableList from 1000
│ │ │ │  
│ │ │ │  o12 = MutableList{...1000...}
│ │ │ │  
│ │ │ │  o12 : MutableList
│ │ │ │  i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ │ - -- .000712171s elapsed
│ │ │ │ + -- .000397922s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _B_a_s_i_c_L_i_s_t -- the class of all basic lists
│ │ │ │  ******** MMeennuu ********
│ │ │ │      * _B_a_g -- the class of all bags
│ │ │ │  ********** TTyyppeess ooff mmuuttaabbllee lliisstt:: **********
│ │ │ │      * _B_a_g -- the class of all bags
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa mmuuttaabbllee lliisstt:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutex.html
│ │ │ @@ -78,25 +78,19 @@
│ │ │  o2 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i3 : T = apply(10, i -> schedule(() -> sayhello i))
│ │ │  
│ │ │ -o3 = {<<task, result available, task done>>, <<task, result available, task
│ │ │ +o3 = {<<task, created>>, <<task, created>>, <<task, created>>, <<task,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     done>>, <<task, result available, task done>>, <<task, result available,
│ │ │ +     created>>, <<task, created>>, <<task, created>>, <<task, created>>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     task done>>, <<task, result available, task done>>, <<task, result
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     available, task done>>, <<task, result available, task done>>, <<task,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     result available, task done>>, <<task, result available, task done>>,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     <<task, result available, task done>>}
│ │ │ +     <<task, created>>, <<task, created>>, <<task, created>>}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : while not all(T, isReady) do null
    
│ │ │ @@ -106,18 +100,15 @@
│ │ │                  		
│ │ │                
    i5 : stack sort lines msgs
│ │ │  
│ │ │  o5 = hello from thread #0
│ │ │       hello from thread #1
│ │ │       hello from thread #2
│ │ │       hello from thread #3
│ │ │ -     hello from thread #4
│ │ │ -     hello from thread #5
│ │ │ -     hello from thread #6
│ │ │ -     hello from thread #7
    
│ │ │ +     hello from thread #4
│ │ │              
    
│ │ │          
    
│ │ │            
    We likely ended up with fewer than the expected number of 10 messages. We can get around this issue by using a mutex to lock the string so that only one thread can modify it at a time.
    
│ │ │          
    
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,38 +23,29 @@
│ │ │ │  i2 : sayhello = i -> msgs |= "hello from thread #" | toString i | newline
│ │ │ │  
│ │ │ │  o2 = sayhello
│ │ │ │  
│ │ │ │  o2 : FunctionClosure
│ │ │ │  i3 : T = apply(10, i -> schedule(() -> sayhello i))
│ │ │ │  
│ │ │ │ -o3 = {<>, <>, <>, <>, <>, <>, <>, <>, <>, <>,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     task done>>, <>, <>, <>, <>, <>,
│ │ │ │ -     ------------------------------------------------------------------------
│ │ │ │ -     <>}
│ │ │ │ +     <>, <>, <>}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : while not all(T, isReady) do null
│ │ │ │  i5 : stack sort lines msgs
│ │ │ │  
│ │ │ │  o5 = hello from thread #0
│ │ │ │       hello from thread #1
│ │ │ │       hello from thread #2
│ │ │ │       hello from thread #3
│ │ │ │       hello from thread #4
│ │ │ │ -     hello from thread #5
│ │ │ │ -     hello from thread #6
│ │ │ │ -     hello from thread #7
│ │ │ │  We likely ended up with fewer than the expected number of 10 messages. We can
│ │ │ │  get around this issue by using a mutex to lock the string so that only one
│ │ │ │  thread can modify it at a time.
│ │ │ │  i6 : m = new Mutex
│ │ │ │  
│ │ │ │  o6 = m
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html
│ │ │ @@ -73,21 +73,21 @@
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : time SVD(M);
│ │ │ - -- used 0.0228821s (cpu); 0.0228783s (thread); 0s (gc)
    
│ │ │ + -- used 0.0586664s (cpu); 0.0586605s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc)
    
│ │ │ + -- used 0.0401241s (cpu); 0.0401408s (thread); 0s (gc)
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    
│ │ │            
    Functions with optional argument named DivideConquer:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,17 +11,17 @@
│ │ │ │  For large matrices, this algorithm is often much faster.
│ │ │ │  i1 : M = random(RR^200, RR^200);
│ │ │ │  
│ │ │ │                  200         200
│ │ │ │  o1 : Matrix RR      <-- RR
│ │ │ │                53          53
│ │ │ │  i2 : time SVD(M);
│ │ │ │ - -- used 0.0228821s (cpu); 0.0228783s (thread); 0s (gc)
│ │ │ │ + -- used 0.0586664s (cpu); 0.0586605s (thread); 0s (gc)
│ │ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc)
│ │ │ │ + -- used 0.0401241s (cpu); 0.0401408s (thread); 0s (gc)
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd DDiivviiddeeCCoonnqquueerr:: **********
│ │ │ │      * _S_V_D_(_._._._,_D_i_v_i_d_e_C_o_n_q_u_e_r_=_>_._._._) -- whether to use the LAPACK divide and
│ │ │ │        conquer SVD algorithm
│ │ │ │  ********** FFuurrtthheerr iinnffoorrmmaattiioonn **********
│ │ │ │      * Default value: _t_r_u_e
│ │ │ │      * Function: _S_V_D -- singular value decomposition of a matrix
│ │ │ │      * Option key: _D_i_v_i_d_e_C_o_n_q_u_e_r -- an optional argument
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Set.html
│ │ │ @@ -67,28 +67,28 @@
│ │ │                
    i2 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i3 : B = set{a^2-b*c,b*d}
│ │ │  
│ │ │ -           2
│ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ +                2
│ │ │ +o3 = set {b*d, a  - b*c}
│ │ │  
│ │ │  o3 : Set
    
│ │ │              
    
│ │ │  Set operations, such as membership, union, intersection, difference, Cartesian product, Cartesian power, and subset are available. For example,        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : toList B
│ │ │  
│ │ │ -       2
│ │ │ -o4 = {a  - b*c, b*d}
│ │ │ +            2
│ │ │ +o4 = {b*d, a  - b*c}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : member(1,A)
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,24 +7,24 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Elements of sets may be any immutable object, such as integers, ring elements
│ │ │ │  and lists of such. Ideals may also be elements of sets.
│ │ │ │  i1 : A = set {1,2};
│ │ │ │  i2 : R = QQ[a..d];
│ │ │ │  i3 : B = set{a^2-b*c,b*d}
│ │ │ │  
│ │ │ │ -           2
│ │ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ │ +                2
│ │ │ │ +o3 = set {b*d, a  - b*c}
│ │ │ │  
│ │ │ │  o3 : Set
│ │ │ │  Set operations, such as _m_e_m_b_e_r_s_h_i_p, _u_n_i_o_n, _i_n_t_e_r_s_e_c_t_i_o_n, _d_i_f_f_e_r_e_n_c_e, _C_a_r_t_e_s_i_a_n
│ │ │ │  _p_r_o_d_u_c_t, _C_a_r_t_e_s_i_a_n_ _p_o_w_e_r, and _s_u_b_s_e_t are available. For example,
│ │ │ │  i4 : toList B
│ │ │ │  
│ │ │ │ -       2
│ │ │ │ -o4 = {a  - b*c, b*d}
│ │ │ │ +            2
│ │ │ │ +o4 = {b*d, a  - b*c}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : member(1,A)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : member(-b*c+a^2,B)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_a_spfirst_sp__Macaulay2_spsession.html
│ │ │ @@ -827,15 +827,15 @@
│ │ │          
    
│ │ │            
    We may use resolution to produce a projective resolution of it, and time to report the time required.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i59 : time C = resolution M
│ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ + -- used 0.00415729s (cpu); 0.00415383s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o59 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -386,15 +386,15 @@
│ │ │ │                 | c f i l o r |
│ │ │ │  
│ │ │ │                               3
│ │ │ │  o58 : R-module, quotient of R
│ │ │ │  We may use _r_e_s_o_l_u_t_i_o_n to produce a projective resolution of it, and _t_i_m_e to
│ │ │ │  report the time required.
│ │ │ │  i59 : time C = resolution M
│ │ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ │ + -- used 0.00415729s (cpu); 0.00415383s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3      6      15      18      6
│ │ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │ │  
│ │ │ │        0      1      2       3       4
│ │ │ │  
│ │ │ │  o59 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  o4 = "hi there"
│ │ │              
    
│ │ │                
    i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
    
│ │ │ +o5 = true
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,13 +23,13 @@
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : peek read f
│ │ │ │  
│ │ │ │  o4 = "hi there"
│ │ │ │  i5 : atEndOfFile f
│ │ │ │  
│ │ │ │ -o5 = false
│ │ │ │ +o5 = true
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _a_t_E_n_d_O_f_F_i_l_e_(_F_i_l_e_) -- test for end of file
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_files.m2:374:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  Produces an accurate timing for the code contained in the string s.  The value returned is the number of seconds.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0003750952360409743
│ │ │ +o1 = .0003905923410533509
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │  The snippet of code provided will be run enough times to register meaningfully on the clock, and the garbage collector will be called beforehand.      
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds it takes to evaluate the code
│ │ │ │              in s
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Produces an accurate timing for the code contained in the string s. The value
│ │ │ │  returned is the number of seconds.
│ │ │ │  i1 : benchmark "sqrt 2p100000"
│ │ │ │  
│ │ │ │ -o1 = .0003750952360409743
│ │ │ │ +o1 = .0003905923410533509
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  The snippet of code provided will be run enough times to register meaningfully
│ │ │ │  on the clock, and the garbage collector will be called beforehand.
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _b_e_n_c_h_m_a_r_k is a _f_u_n_c_t_i_o_n_ _c_l_o_s_u_r_e.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html
│ │ │ @@ -74,23 +74,23 @@
│ │ │                
    i3 : M = coker vars R;
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .0035172s elapsed
│ │ │ + -- .00447657s elapsed
│ │ │  
│ │ │  o4 = 3
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime pdim' M
│ │ │ - -- 9.22e-7s elapsed
│ │ │ + -- .000002713s elapsed
│ │ │  
│ │ │  o5 = 3
    
│ │ │              
    
│ │ │                
    i6 : peek M.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,19 +8,19 @@
│ │ │ │  Here is a simple example of caching a computation in a _C_a_c_h_e_T_a_b_l_e, using the
│ │ │ │  augmented null coalescing operator _?_?_=.
│ │ │ │  i1 : pdim' = M -> M.cache.pdim' ??= ( printerr "computing pdim'"; pdim M );
│ │ │ │  i2 : R = QQ[x,y,z];
│ │ │ │  i3 : M = coker vars R;
│ │ │ │  i4 : elapsedTime pdim' M
│ │ │ │   -- computing pdim'
│ │ │ │ - -- .0035172s elapsed
│ │ │ │ + -- .00447657s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- 9.22e-7s elapsed
│ │ │ │ + -- .000002713s elapsed
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : peek M.cache
│ │ │ │  
│ │ │ │  o6 = CacheTable{cache => MutableHashTable{}
│ │ │ │  }
│ │ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_can__Use__Hilbert__Hint.html
│ │ │ @@ -347,15 +347,15 @@
│ │ │  
│ │ │  o17 : Ideal of Rlex
    
│ │ │              
    
│ │ │                
    i18 : elapsedTime g1 = gens gb Ilex;
│ │ │ - -- 1.68769s elapsed
│ │ │ + -- 2.38888s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o18 : Matrix Rlex  <-- Rlex
    
│ │ │              
    
│ │ │ @@ -435,15 +435,15 @@
│ │ │  
│ │ │  o19 : Ideal of Rlex
│ │ │              
    
│ │ │                
    i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf);
│ │ │ - -- .850458s elapsed
│ │ │ + -- 1.08468s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o20 : Matrix Rlex  <-- Rlex
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -221,15 +221,15 @@
│ │ │ │        18y*z w  + 42y*z*w  + 23y*w  - 28z  + 15z w + 18z w  - 16z w  - 46z*w
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │             5
│ │ │ │        + 12w )
│ │ │ │  
│ │ │ │  o17 : Ideal of Rlex
│ │ │ │  i18 : elapsedTime g1 = gens gb Ilex;
│ │ │ │ - -- 1.68769s elapsed
│ │ │ │ + -- 2.38888s elapsed
│ │ │ │  
│ │ │ │                   1         372
│ │ │ │  o18 : Matrix Rlex  <-- Rlex
│ │ │ │  i19 : Ilex = ideal(Ilex_*) -- clear out the previous Groebner basis
│ │ │ │  
│ │ │ │                  5      4       4       4       3 2      3         3
│ │ │ │  o19 = ideal (24x  - 36x y - 30x z - 29x w + 19x y  + 19x y*z - 10x y*w -
│ │ │ │ @@ -301,15 +301,15 @@
│ │ │ │        18y*z w  + 42y*z*w  + 23y*w  - 28z  + 15z w + 18z w  - 16z w  - 46z*w
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │             5
│ │ │ │        + 12w )
│ │ │ │  
│ │ │ │  o19 : Ideal of Rlex
│ │ │ │  i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf);
│ │ │ │ - -- .850458s elapsed
│ │ │ │ + -- 1.08468s elapsed
│ │ │ │  
│ │ │ │                   1         372
│ │ │ │  o20 : Matrix Rlex  <-- Rlex
│ │ │ │  i21 : g1 == g2
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  ********** CCaavveeaatt **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  o4 : Task
│ │ │              
    
│ │ │                
    i5 : n
│ │ │  
│ │ │ -o5 = 336394
    
│ │ │ +o5 = 720838
│ │ │              
    
│ │ │                
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    
│ │ │ @@ -132,15 +132,15 @@
│ │ │  o7 : Task
│ │ │              
    
│ │ │                
    i8 : n
│ │ │  
│ │ │ -o8 = 929987
    
│ │ │ +o8 = 1699814
│ │ │              
    
│ │ │                
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    
│ │ │ @@ -168,29 +168,29 @@
│ │ │  o12 : Task
│ │ │              
    
│ │ │                
    i13 : n
│ │ │  
│ │ │ -o13 = 930227
    
│ │ │ +o13 = 1700147
│ │ │              
    
│ │ │                
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    
│ │ │              
    
│ │ │                
    i15 : n
│ │ │  
│ │ │ -o15 = 930227
    
│ │ │ +o15 = 1700147
│ │ │              
    
│ │ │                
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,26 +28,26 @@
│ │ │ │  i4 : t
│ │ │ │  
│ │ │ │  o4 = <>
│ │ │ │  
│ │ │ │  o4 : Task
│ │ │ │  i5 : n
│ │ │ │  
│ │ │ │ -o5 = 336394
│ │ │ │ +o5 = 720838
│ │ │ │  i6 : sleep 1
│ │ │ │  
│ │ │ │  o6 = 0
│ │ │ │  i7 : t
│ │ │ │  
│ │ │ │  o7 = <>
│ │ │ │  
│ │ │ │  o7 : Task
│ │ │ │  i8 : n
│ │ │ │  
│ │ │ │ -o8 = 929987
│ │ │ │ +o8 = 1699814
│ │ │ │  i9 : isReady t
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  i10 : cancelTask t
│ │ │ │  i11 : sleep 2
│ │ │ │  stdio:2:25:(3):[1]: error: interrupted
│ │ │ │  
│ │ │ │ @@ -55,21 +55,21 @@
│ │ │ │  i12 : t
│ │ │ │  
│ │ │ │  o12 = <>
│ │ │ │  
│ │ │ │  o12 : Task
│ │ │ │  i13 : n
│ │ │ │  
│ │ │ │ -o13 = 930227
│ │ │ │ +o13 = 1700147
│ │ │ │  i14 : sleep 1
│ │ │ │  
│ │ │ │  o14 = 0
│ │ │ │  i15 : n
│ │ │ │  
│ │ │ │ -o15 = 930227
│ │ │ │ +o15 = 1700147
│ │ │ │  i16 : isReady t
│ │ │ │  
│ │ │ │  o16 = false
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _c_a_n_c_e_l_T_a_s_k_(_T_a_s_k_) -- stop a task
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html
│ │ │ @@ -76,36 +76,36 @@
│ │ │            
    Change the current working directory to dir.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11296-0/0
    
│ │ │ +o1 = /tmp/M2-11766-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11296-0/0
    
│ │ │ +o2 = /tmp/M2-11766-0/0
│ │ │              
    
│ │ │                
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-11296-0/0/
    
│ │ │ +o3 = /tmp/M2-11766-0/0/
│ │ │              
    
│ │ │                
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-11296-0/0/
    
│ │ │ +o4 = /tmp/M2-11766-0/0/
│ │ │              
    
│ │ │          
    
│ │ │            
    If dir is omitted, then the current working directory is changed to the user's home directory.
    
│ │ │          
    
│ │ │        
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,24 +11,24 @@
│ │ │ │            o dir, a _s_t_r_i_n_g,
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the new working directory;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Change the current working directory to dir.
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11296-0/0
│ │ │ │ +o1 = /tmp/M2-11766-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11296-0/0
│ │ │ │ +o2 = /tmp/M2-11766-0/0
│ │ │ │  i3 : changeDirectory dir
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11296-0/0/
│ │ │ │ +o3 = /tmp/M2-11766-0/0/
│ │ │ │  i4 : currentDirectory()
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11296-0/0/
│ │ │ │ +o4 = /tmp/M2-11766-0/0/
│ │ │ │  If dir is omitted, then the current working directory is changed to the user's
│ │ │ │  home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_D_i_r_e_c_t_o_r_y -- current working directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_h_a_n_g_e_D_i_r_e_c_t_o_r_y is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html
│ │ │ @@ -100,40 +100,40 @@
│ │ │  o1 : Package
│ │ │              
    
│ │ │                
    i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .313321s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .219073s elapsed
│ │ │              
    
│ │ │                
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .282764s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283285s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .192358s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .207897s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    Alternatively, if the package is installed somewhere accessible, one can do the following.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .275704s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .193146s elapsed
│ │ │              
    
│ │ │                
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .246295s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .21342s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .194179s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .206836s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    A TestInput object (or a list of such objects) can also be run directly.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -145,15 +145,15 @@
│ │ │  
│ │ │  o6 : TestInput
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .302726s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .217212s elapsed
│ │ │              
    
│ │ │                
    i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ @@ -161,16 +161,16 @@
│ │ │  
│ │ │  o8 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .28407s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .282385s elapsed
    
│ │ │ + -- capturing check(0, "FirstPackage")        -- .190655s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .198866s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    If only an integer is passed as an argument, then the test with that index from the last call to tests is run.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -183,15 +183,15 @@
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283414s elapsed
    
│ │ │ + -- capturing check(1, "FirstPackage")        -- .199975s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,52 +42,52 @@
│ │ │ │  
│ │ │ │  o1 = FirstPackage
│ │ │ │  
│ │ │ │  o1 : Package
│ │ │ │  i2 : check_1 FirstPackage
│ │ │ │   -- warning: reloading FirstPackage; recreate instances of types from this
│ │ │ │  package
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .313321s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .219073s elapsed
│ │ │ │  i3 : check FirstPackage
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .282764s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .283285s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .192358s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .207897s elapsed
│ │ │ │  Alternatively, if the package is installed somewhere accessible, one can do the
│ │ │ │  following.
│ │ │ │  i4 : check_1 "FirstPackage"
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .275704s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .193146s elapsed
│ │ │ │  i5 : check "FirstPackage"
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .246295s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .21342s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .194179s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .206836s elapsed
│ │ │ │  A _T_e_s_t_I_n_p_u_t object (or a list of such objects) can also be run directly.
│ │ │ │  i6 : tests(1, "FirstPackage")
│ │ │ │  
│ │ │ │  o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]
│ │ │ │  
│ │ │ │  o6 : TestInput
│ │ │ │  i7 : check oo
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .302726s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .217212s elapsed
│ │ │ │  i8 : tests "FirstPackage"
│ │ │ │  
│ │ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ │       {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │ │  
│ │ │ │  o8 : NumberedVerticalList
│ │ │ │  i9 : check oo
│ │ │ │ - -- capturing check(0, "FirstPackage")        -- .28407s elapsed
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .282385s elapsed
│ │ │ │ + -- capturing check(0, "FirstPackage")        -- .190655s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .198866s elapsed
│ │ │ │  If only an integer is passed as an argument, then the test with that index from
│ │ │ │  the last call to _t_e_s_t_s is run.
│ │ │ │  i10 : tests "FirstPackage"
│ │ │ │  
│ │ │ │  o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ │        {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │ │  
│ │ │ │  o10 : NumberedVerticalList
│ │ │ │  i11 : check 1
│ │ │ │ - -- capturing check(1, "FirstPackage")        -- .283414s elapsed
│ │ │ │ + -- capturing check(1, "FirstPackage")        -- .199975s elapsed
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Currently, if the package was only partially loaded because the documentation
│ │ │ │  was obtainable from a database (see _b_e_g_i_n_D_o_c_u_m_e_n_t_a_t_i_o_n), then the package will
│ │ │ │  be reloaded, this time completely, to ensure that all tests are considered;
│ │ │ │  this may affect user objects of types declared by the package, as they may be
│ │ │ │  not usable by the new instance of the package. In a future version, either the
│ │ │ │  tests and the documentation will both be cached, or neither will.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html
│ │ │ @@ -55,15 +55,15 @@
│ │ │      
    
│ │ │        
    communicating with programs
    
│ │ │        
    
│ │ │  The most naive way to interact with another program is simply to run it, let it communicate directly with the user, and wait for it to finish.  This is done with the run command.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │  To run a program and provide it with input, one way is use the operator <<, with a file name whose first character is an exclamation point; the rest of the file name will be taken as the command to run, as in the following example.        
│ │ │            
│ │ │ @@ -79,16 +79,16 @@
│ │ │            
│ │ │          
    
│ │ │  More often, one wants to write Macaulay2 code to obtain and manipulate the output from the other program.  If the program requires no input data, then we can use get with a file name whose first character is an exclamation point.  In the following example, we also peek at the string to see whether it includes a newline character.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ -     6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n"
    
│ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +     6.12.100-1 (2026-07-30) x86_64 GNU/Linux\n"
│ │ │              
    
│ │ │  Bidirectional communication with a program is also possible.  We use openInOut to create a file that serves as a bidirectional connection to a program.  That file is called an input output file.  In this example we open a connection to the unix utility grep and use it to locate the symbol names in Macaulay2 that begin with in.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,17 +31,17 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal(x, y, z)
│ │ │ │  
│ │ │ │  o2 = ideal (x, y, z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6)
│ │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/
│ │ │ │ -M2-38527-0/0-out.ms
│ │ │ │ -Initial seed for pseudo-random number generator is 1792814436
│ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54126-0/0-in.ms -o /tmp/
│ │ │ │ +M2-54126-0/0-out.ms
│ │ │ │ +Initial seed for pseudo-random number generator is 1793013995
│ │ │ │  
│ │ │ │  --------------- INPUT DATA ---------------
│ │ │ │  #variables                       3
│ │ │ │  #equations                       3
│ │ │ │  #invalid equations               0
│ │ │ │  field characteristic             0
│ │ │ │  homogeneous input?               1
│ │ │ │ @@ -52,15 +52,15 @@
│ │ │ │  initial hash table size     131072 (2^17)
│ │ │ │  max pair selection             ALL
│ │ │ │  reduce gb                        1
│ │ │ │  #threads                         6
│ │ │ │  info level                       2
│ │ │ │  generate pbm files               0
│ │ │ │  ------------------------------------------
│ │ │ │ -Initial prime = 1099986659
│ │ │ │ +Initial prime = 1102609337
│ │ │ │  
│ │ │ │  Legend for f4 information
│ │ │ │  --------------------------------------------------------
│ │ │ │  deg       current degree of pairs selected in this round
│ │ │ │  sel       number of pairs selected in this round
│ │ │ │  pairs     total number of pairs in pair list
│ │ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ │ @@ -82,18 +82,18 @@
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │  overall(elapsed)        0.00 sec
│ │ │ │  overall(cpu)            0.00 sec
│ │ │ │  select                  0.00 sec   0.0%
│ │ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ │ -update                  0.00 sec  96.6%
│ │ │ │ -convert                 0.00 sec   0.6%
│ │ │ │ -linear algebra          0.00 sec   0.2%
│ │ │ │ +symbolic prep.          0.00 sec   0.4%
│ │ │ │ +update                  0.00 sec  71.6%
│ │ │ │ +convert                 0.00 sec   2.5%
│ │ │ │ +linear algebra          0.00 sec   1.5%
│ │ │ │  reduce gb               0.00 sec   0.0%
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  size of basis                     3
│ │ │ │  #terms in basis                   3
│ │ │ │  #pairs reduced                    0
│ │ │ │ @@ -107,18 +107,18 @@
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │ │  [3]
│ │ │ │  #polynomials to lift              3
│ │ │ │  -----------------------------------------
│ │ │ │ -New prime = 1120193111
│ │ │ │ +New prime = 1300419397
│ │ │ │  
│ │ │ │  ---------------- TIMINGS ----------------
│ │ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │ │  learning phase                  0.00 Gops/sec
│ │ │ │  application phase               0.00 Gops/sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  multi-modular steps
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------------
│ │ │ │ @@ -137,15 +137,15 @@
│ │ │ │  CRT     (elapsed)               0.00 sec
│ │ │ │  ratrecon(elapsed)               0.00 sec
│ │ │ │  -----------------------------------------
│ │ │ │  
│ │ │ │  
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │ -msolve overall time           0.08 sec (elapsed) /  0.27 sec (cpu)
│ │ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----
│ │ │ │  
│ │ │ │  o3 = | z y x |
│ │ │ │  
│ │ │ │               1      3
│ │ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/MultiGradedRationalMap/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  ZGVncmVlT2ZNYXAoSWRlYWwp
│ │ │  #:len=283
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjk3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkZWdyZWVPZk1hcCxJZGVhbCksImRlZ3JlZU9mTWFw
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedBGG/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  ZGVncmVlKERpZmZlcmVudGlhbE1vZHVsZSk=
│ │ │  #:len=1348
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgZGVncmVlIG9mIHRo
│ │ │  ZSBkaWZmZXJlbnRpYWwiLCAibGluZW51bSIgPT4gNDgzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkQi
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  TXVsdGlncmFkZWRJbXBsaWNpdGl6YXRpb24=
│ │ │  #:len=1401
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUGFja2FnZSBmb3IgbGV2YXJhZ2luZyBt
│ │ │  dWx0aWdyYWRpbmdzIHRvIHNvbHZlIGltcGxpY2l0aXphdGlvbiBwcm9ibGVtcyIsIERlc2NyaXB0
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out
│ │ │ @@ -23,19 +23,19 @@
│ │ │  o4 : RingMap S <-- R
│ │ │  
│ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00225028s elapsed
│ │ │ + -- .00259946s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00849938s elapsed
│ │ │ + -- .011003s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html
│ │ │ @@ -122,19 +122,19 @@
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,23 +20,23 @@
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │ │  
│ │ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime jMult I
│ │ │ │ - -- .0242277s elapsed
│ │ │ │ + -- .0413968s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  i4 : elapsedTime monjMult I
│ │ │ │ - -- .159695s elapsed
│ │ │ │ + -- .0901594s elapsed
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ │ - -- .270321s elapsed
│ │ │ │ + -- .164849s elapsed
│ │ │ │  
│ │ │ │  o5 = HashTable{2 => 3}
│ │ │ │                 3 => 2
│ │ │ │  
│ │ │ │  o5 : HashTable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : f = openInOut "!grep -E '^in'"
│ │ │ @@ -167,15 +167,15 @@
│ │ │  
│ │ │  o9 = false
    
│ │ │              
    
│ │ │                
    i10 : while not isReady f do (sleep 1; << "." << flush)
│ │ │ -......
    
│ │ │ +.....
│ │ │              
    
│ │ │                
    i11 : read f
│ │ │  
│ │ │  o11 = the answer is 4
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -5,16 +5,16 @@
│ │ │ │  _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c
│ │ │ │  ===============================================================================
│ │ │ │  ************ ccoommmmuunniiccaattiinngg wwiitthh pprrooggrraammss ************
│ │ │ │  The most naive way to interact with another program is simply to run it, let it
│ │ │ │  communicate directly with the user, and wait for it to finish. This is done
│ │ │ │  with the _r_u_n command.
│ │ │ │  i1 : run "uname -a"
│ │ │ │ -Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1
│ │ │ │ -(2026-10-15) x86_64 GNU/Linux
│ │ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-
│ │ │ │ +1 (2026-07-30) x86_64 GNU/Linux
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  To run a program and provide it with input, one way is use the operator _<_<,
│ │ │ │  with a file name whose first character is an exclamation point; the rest of the
│ │ │ │  file name will be taken as the command to run, as in the following example.
│ │ │ │  i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close
│ │ │ │   ba
│ │ │ │ @@ -26,16 +26,16 @@
│ │ │ │  More often, one wants to write Macaulay2 code to obtain and manipulate the
│ │ │ │  output from the other program. If the program requires no input data, then we
│ │ │ │  can use _g_e_t with a file name whose first character is an exclamation point. In
│ │ │ │  the following example, we also peek at the string to see whether it includes a
│ │ │ │  newline character.
│ │ │ │  i3 : peek get "!uname -a"
│ │ │ │  
│ │ │ │ -o3 = "Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ -     6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n"
│ │ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +     6.12.100-1 (2026-07-30) x86_64 GNU/Linux\n"
│ │ │ │  Bidirectional communication with a program is also possible. We use _o_p_e_n_I_n_O_u_t
│ │ │ │  to create a file that serves as a bidirectional connection to a program. That
│ │ │ │  file is called an input output file. In this example we open a connection to
│ │ │ │  the unix utility grep and use it to locate the symbol names in Macaulay2 that
│ │ │ │  begin with in.
│ │ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ │ │  
│ │ │ │ @@ -97,15 +97,15 @@
│ │ │ │  o8 = !sleep 5; echo -n the answer is 4
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : isReady f
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  i10 : while not isReady f do (sleep 1; << "." << flush)
│ │ │ │ -......
│ │ │ │ +.....
│ │ │ │  i11 : read f
│ │ │ │  
│ │ │ │  o11 = the answer is 4
│ │ │ │  i12 : isReady f
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : atEndOfFile f
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_sp__Groebner_spbases.html
│ │ │ @@ -274,15 +274,15 @@
│ │ │                 1277
│ │ │              
    
│ │ │                
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f57805dd000
│ │ │ +   -- registering gb 5 at 0x7f0008d49000
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -378,15 +378,15 @@
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i33 : time betti gb f
│ │ │ - -- used 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ + -- used 0.231793s (cpu); 0.229499s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -422,15 +422,15 @@
│ │ │  
│ │ │  o35 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i36 : time betti gb f
│ │ │ - -- used 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ + -- used 0.00800088s (cpu); 0.00892148s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -142,15 +142,15 @@
│ │ │ │  o23 = ideal (x*y - z , y  - w )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o23 : Ideal of ----[x..z, w]
│ │ │ │                 1277
│ │ │ │  i24 : gb I
│ │ │ │  
│ │ │ │ -   -- registering gb 5 at 0x7f57805dd000
│ │ │ │ +   -- registering gb 5 at 0x7f0008d49000
│ │ │ │  
│ │ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │ │     -- number of monomials                = 8
│ │ │ │     -- #reduction steps = 2
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │ @@ -215,15 +215,15 @@
│ │ │ │  
│ │ │ │  o31 : ZZ[T]
│ │ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │ │  
│ │ │ │                1      4
│ │ │ │  o32 : Matrix R  <-- R
│ │ │ │  i33 : time betti gb f
│ │ │ │ - -- used 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ │ + -- used 0.231793s (cpu); 0.229499s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o33 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ │ │ @@ -247,15 +247,15 @@
│ │ │ │  i35 : poincare cokernel f = (1-T^3)*(1-T^3)*(1-T^5)*(1-T^6) -- cache poincare
│ │ │ │  
│ │ │ │              3    5     8     9    12     14    17
│ │ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : time betti gb f
│ │ │ │ - -- used 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ │ + -- used 0.00800088s (cpu); 0.00892148s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o36 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,112 +85,112 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,41 +22,41 @@
│ │ │ │            o npaths, an _i_n_t_e_g_e_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Output is verbose. For other dynamic strategies, see _M_o_n_o_d_r_o_m_y_S_o_l_v_e_r_O_p_t_i_o_n_s.
│ │ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ │ ---backup directory created: /tmp/M2-25512-0/2
│ │ │ │ - -- .00616338s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-33165-0/2
│ │ │ │ + -- .0042495s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00900117s elapsed
│ │ │ │ + -- .00425265s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000649635s elapsed
│ │ │ │ + -- .000841071s elapsed
│ │ │ │  number of paths tracked: 2
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .00324889s elapsed
│ │ │ │ + -- .00421182s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00435076s elapsed
│ │ │ │ + -- .00499457s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000544598s elapsed
│ │ │ │ + -- .000718755s elapsed
│ │ │ │  number of paths tracked: 4
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .00428387s elapsed
│ │ │ │ + -- .00396059s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00298132s elapsed
│ │ │ │ + -- .0044697s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000466532s elapsed
│ │ │ │ + -- .000681483s elapsed
│ │ │ │  number of paths tracked: 6
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .0044506s elapsed
│ │ │ │ + -- .00462941s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00309306s elapsed
│ │ │ │ + -- .00409509s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000457295s elapsed
│ │ │ │ + -- .000827698s elapsed
│ │ │ │  number of paths tracked: 8
│ │ │ │  found 1 points in the fiber so far
│ │ │ │  
│ │ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html
│ │ │ @@ -96,22 +96,22 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12075-0/0/
    
│ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12075-0/1/
    
│ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12075-0/0/a/
    
│ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12075-0/0/b/
    
│ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ -     /tmp/M2-12075-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ -      /tmp/M2-12075-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ +      /tmp/M2-13345-0/1/b/c/g
│ │ │              
    
│ │ │                
    i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,68 +25,68 @@
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o a copy of the directory tree rooted at src is created, rooted at
│ │ │ │              dst
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12075-0/0/
│ │ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12075-0/1/
│ │ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12075-0/0/a/
│ │ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12075-0/0/b/
│ │ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ │ -     /tmp/M2-12075-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ │ -      /tmp/M2-12075-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ │ +      /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i13 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o13 = ho there
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o14 = rm
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__File_lp__String_cm__String_rp.html
│ │ │ @@ -83,65 +83,65 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11841-0/0
    
│ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11841-0/1
    
│ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
    
│ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
    
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
    
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i9 : get dst
│ │ │  
│ │ │  o9 = hi there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,37 +18,37 @@
│ │ │ │            o Verbose => a _B_o_o_l_e_a_n_ _v_a_l_u_e, default value false, whether to report
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o the file may be copied
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11841-0/0
│ │ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11841-0/1
│ │ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
│ │ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │ │  i9 : get dst
│ │ │ │  
│ │ │ │  o9 = hi there
│ │ │ │  i10 : removeFile src
│ │ │ │  i11 : removeFile dst
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_D_i_r_e_c_t_o_r_y
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html
│ │ │ @@ -69,38 +69,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 611.770723009
│ │ │ +o1 = 679.2678651149999
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 613.313432871
│ │ │ +o3 = 680.676774634
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.542709862000038
│ │ │ +o4 = 1.408909519000076
│ │ │  
│ │ │  o4 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,26 +9,26 @@
│ │ │ │              cpuTime()
│ │ │ │      * Outputs:
│ │ │ │            o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds of cpu time used since the
│ │ │ │              program was started
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : t1 = cpuTime()
│ │ │ │  
│ │ │ │ -o1 = 611.770723009
│ │ │ │ +o1 = 679.2678651149999
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 613.313432871
│ │ │ │ +o3 = 680.676774634
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 1.542709862000038
│ │ │ │ +o4 = 1.408909519000076
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_current__Time.html
│ │ │ @@ -69,48 +69,48 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1792811657
    
│ │ │ +o1 = 1793011624
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.81193979063632
│ │ │ +o2 = 56.81827649260931
│ │ │  
│ │ │  o2 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    We can also compute how many months account for the fractional part of that number.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 9.743277487635794
│ │ │ +o3 = 9.819317911311686
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ +Mon Oct 26 10:47:04 UTC 2026
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,31 +9,31 @@
│ │ │ │              currentTime()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the current time, in seconds since 00:00:00 1970-01-01
│ │ │ │              UTC, the beginning of the epoch
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : currentTime()
│ │ │ │  
│ │ │ │ -o1 = 1792811657
│ │ │ │ +o1 = 1793011624
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 56.81193979063632
│ │ │ │ +o2 = 56.81827649260931
│ │ │ │  
│ │ │ │  o2 : RR (of precision 53)
│ │ │ │  We can also compute how many months account for the fractional part of that
│ │ │ │  number.
│ │ │ │  i3 : 12 * (oo - floor oo)
│ │ │ │  
│ │ │ │ -o3 = 9.743277487635794
│ │ │ │ +o3 = 9.819317911311686
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ │ +Mon Oct 26 10:47:04 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_u_r_r_e_n_t_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:1849:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00015s elapsed
│ │ │ + -- 1.00023s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │  ************ eellaappsseeddTTiimmee ---- ttiimmee aa ccoommppuuttaattiioonn iinncclluuddiinngg ttiimmee eellaappsseedd ************
│ │ │ │      *   Usage:
│ │ │ │              elapsedTime e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the
│ │ │ │  value of e.
│ │ │ │  i1 : elapsedTime sleep 1
│ │ │ │ - -- 1.00015s elapsed
│ │ │ │ + -- 1.00023s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _G_C_s_t_a_t_s -- information about the status of the garbage collector
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Timing.html
│ │ │ @@ -59,24 +59,24 @@
│ │ │  elapsedTiming e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of time elapsed, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTiming sleep 1
│ │ │  
│ │ │  o1 = 0
│ │ │ -     -- 1.00011 seconds
│ │ │ +     -- 1.00014 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00011, 0}
    
│ │ │ +o2 = Time{1.00014, 0}
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  where t is the number of seconds of time elapsed, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : elapsedTiming sleep 1
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │ -     -- 1.00011 seconds
│ │ │ │ +     -- 1.00014 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00011, 0}
│ │ │ │ +o2 = Time{1.00014, 0}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elimination_spof_spvariables.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : time leadTerm gens gb I
│ │ │ - -- used 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ + -- used 0.482245s (cpu); 0.207912s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,15 +168,15 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ + -- used 0.105157s (cpu); 0.105157s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -251,15 +251,15 @@
│ │ │  
│ │ │  o11 : Ideal of R1
    
│ │ │              
    
│ │ │                
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ + -- used 0.0340178s (cpu); 0.034025s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -343,15 +343,15 @@
│ │ │  
│ │ │  o16 : RingMap A <-- B
    
│ │ │              
    
│ │ │                
    i17 : time G = kernel F
│ │ │ - -- used 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ + -- used 0.451116s (cpu); 0.184461s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -424,26 +424,26 @@
│ │ │  
│ │ │  o19 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ + -- used 0.00204952s (cpu); 0.00204688s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
    
│ │ │              
    
│ │ │                
    i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ + -- used 0.0362494s (cpu); 0.036266s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  i2 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time leadTerm gens gb I
│ │ │ │ - -- used 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ │ + -- used 0.482245s (cpu); 0.207912s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  i7 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ │ - -- used 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ │ + -- used 0.105157s (cpu); 0.105157s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -161,15 +161,15 @@
│ │ │ │  Sometimes giving the variables different degrees will speed up the
│ │ │ │  computations. Here, we set the degrees of x, y, and z to be the total degrees.
│ │ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │ │  i11 : I1 = substitute(I,R1);
│ │ │ │  
│ │ │ │  o11 : Ideal of R1
│ │ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ │ - -- used 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ │ + -- used 0.0340178s (cpu); 0.034025s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7
│ │ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2
│ │ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -232,15 +232,15 @@
│ │ │ │  i16 : F = map(A,B,{s^3+s*t+1, t^3+3*t^2+t, s*t^3})
│ │ │ │  
│ │ │ │                     3             3     2         3
│ │ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │ │  
│ │ │ │  o16 : RingMap A <-- B
│ │ │ │  i17 : time G = kernel F
│ │ │ │ - -- used 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ │ + -- used 0.451116s (cpu); 0.184461s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -301,22 +301,22 @@
│ │ │ │  involve the variables s and t.
│ │ │ │  i19 : use ring I
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : PolynomialRing
│ │ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ │ - -- used 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ │ + -- used 0.00204952s (cpu); 0.00204688s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9    9      7    3
│ │ │ │  o20 = x*t  - t  - z*t  - z
│ │ │ │  
│ │ │ │  o20 : R
│ │ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ │ - -- used 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ │ + -- used 0.0362494s (cpu); 0.036266s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2
│ │ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8
│ │ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html
│ │ │ @@ -159,15 +159,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-11022-0/96-rundir/
│ │ │ +             source directory => /tmp/M2-11242-0/96-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
    
│ │ │              
    
│ │ │                
    i7 : dictionaryPath
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,15 +77,15 @@
│ │ │ │                                      Version => 0.0
│ │ │ │               package prefix => /usr/
│ │ │ │               PackageIsLoaded => true
│ │ │ │               pkgname => Foo
│ │ │ │               private dictionary => Foo#"private dictionary"
│ │ │ │               processed documentation => MutableHashTable{}
│ │ │ │               raw documentation => MutableHashTable{}
│ │ │ │ -             source directory => /tmp/M2-11022-0/96-rundir/
│ │ │ │ +             source directory => /tmp/M2-11242-0/96-rundir/
│ │ │ │               source file => stdio
│ │ │ │               test inputs => MutableList{}
│ │ │ │  i7 : dictionaryPath
│ │ │ │  
│ │ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Truncations.Dictionary, Polyhedra.Dictionary, Saturation.Dictionary,
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
│ │ │ @@ -73,29 +73,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -515,27 +515,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11391-0/0
    
│ │ │ +o1 = /tmp/M2-11961-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ +o3 = /tmp/M2-11961-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : fileExists fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether a file with the filename or path fn exists
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11391-0/0
│ │ │ │ +o1 = /tmp/M2-11961-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ │ +o3 = /tmp/M2-11961-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : fileExists fn
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : removeFile fn
│ │ │ │  If fn refers to a symbolic link, then whether the file exists is determined by
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Length.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-13078-0/0
│ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -90,24 +90,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-13078-0/0
    
│ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │              
    
│ │ │                
    i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,28 +12,28 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the length of the file f or the file whose name is f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The length of an open output file is determined from the internal count of the
│ │ │ │  number of bytes written so far.
│ │ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-13078-0/0
│ │ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-13078-0/0
│ │ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │ │  i5 : fileLength filename
│ │ │ │  
│ │ │ │  o5 = 8
│ │ │ │  i6 : get filename
│ │ │ │  
│ │ │ │  o6 = hi there
│ │ │ │  i7 : length oo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__File_rp.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -430,15 +430,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12284-0/0
    
│ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,26 +11,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Outputs:
│ │ │ │            o the mode of the open file f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12284-0/0
│ │ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _f_i_l_e_M_o_d_e_(_F_i_l_e_) -- get file mode
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__String_rp.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11860-0/0
    
│ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the mode of the file located at the filename or path fn
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11860-0/0
│ │ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__File_rp.html
│ │ │ @@ -78,22 +78,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -250,23 +250,23 @@
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11725-0/0
    
│ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -113,15 +113,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-11725-0/0
│ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,30 +12,30 @@
│ │ │ │            o mo, an _i_n_t_e_g_e_r
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the open file f is set to mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11725-0/0
│ │ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │ │  
│ │ │ │  o3 = 511
│ │ │ │  i4 : fileMode(m,f)
│ │ │ │  i5 : fileMode f
│ │ │ │  
│ │ │ │  o5 = 511
│ │ │ │  i6 : close f
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11725-0/0
│ │ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : fileMode fn
│ │ │ │  
│ │ │ │  o7 = 511
│ │ │ │  i8 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__String_rp.html
│ │ │ @@ -78,22 +78,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12905-0/0
    
│ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the file located at the filename or path fn is set to
│ │ │ │              mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12905-0/0
│ │ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : fileMode(m|7,fn)
│ │ │ │  i5 : fileMode fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Time.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning of the epoch, so the number of seconds ago a file or directory was modified may be found by using the following code.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 111
    
│ │ │ +o1 = 119
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │              returns null if no error occurs
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning
│ │ │ │  of the epoch, so the number of seconds ago a file or directory was modified may
│ │ │ │  be found by using the following code.
│ │ │ │  i1 : currentTime() - fileTime "."
│ │ │ │  
│ │ │ │ -o1 = 111
│ │ │ │ +o1 = 119
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │      * _f_i_l_e_ _m_a_n_i_p_u_l_a_t_i_o_n -- Unix file manipulation functions
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _f_i_l_e_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  o6 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f466ed708c0
│ │ │ +   -- registering gb 0 at 0x7f3b4b7c98c0
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       {3} | x2-3  0     -z4+2 |
│ │ │ │       {4} | 0     x2-3  y3-1  |
│ │ │ │  
│ │ │ │               3      3
│ │ │ │  o6 : Matrix R  <-- R
│ │ │ │  i7 : syz f
│ │ │ │  
│ │ │ │ -   -- registering gb 0 at 0x7f466ed708c0
│ │ │ │ +   -- registering gb 0 at 0x7f3b4b7c98c0
│ │ │ │  
│ │ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb
│ │ │ │  elements = 3
│ │ │ │     -- number of monomials                = 9
│ │ │ │     -- #reduction steps = 6
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_get.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │                
    i3 : removeFile "test-file"
    
│ │ │              
    
│ │ │                
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sat Oct 24 03:12:48 UTC 2026
    
│ │ │ +o4 = Mon Oct 26 10:45:43 UTC 2026
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  o1 : File
│ │ │ │  i2 : get "test-file"
│ │ │ │  
│ │ │ │  o2 = hi there
│ │ │ │  i3 : removeFile "test-file"
│ │ │ │  i4 : get "!date"
│ │ │ │  
│ │ │ │ -o4 = Sat Oct 24 03:12:48 UTC 2026
│ │ │ │ +o4 = Mon Oct 26 10:45:43 UTC 2026
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d -- read from a file
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ │ │      * _c_l_o_s_e -- close a file
│ │ │ │      * _F_i_l_e_ _<_<_ _T_h_i_n_g -- print to a file
│ │ │ │  ********** WWaayyss ttoo uussee ggeett:: **********
│ │ │ │      * get(File)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │                 defaultPrecision => 53
│ │ │ │                 engineDebugLevel => 0
│ │ │ │                 errorDepth => 0
│ │ │ │                 gbTrace => 0
│ │ │ │                 interpreterDepth => 1
│ │ │ │                 lineNumber => 2
│ │ │ │                 loadDepth => 3
│ │ │ │ -               maxAllowableThreads => 7
│ │ │ │ +               maxAllowableThreads => 17
│ │ │ │                 maxExponent => 1073741823
│ │ │ │                 minExponent => -1073741824
│ │ │ │                 numTBBThreads => 0
│ │ │ │                 o1 => 2432902008176640000
│ │ │ │                 oo => 2432902008176640000
│ │ │ │                 printingAccuracy => -1
│ │ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Directory.html
│ │ │ @@ -80,22 +80,22 @@
│ │ │  o1 = true
    
│ │ │              
    
│ │ │                
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11213-0/0
    
│ │ │ +o2 = /tmp/M2-11603-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ +o3 = /tmp/M2-11603-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : isDirectory fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : isDirectory "."
│ │ │ │  
│ │ │ │  o1 = true
│ │ │ │  i2 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11213-0/0
│ │ │ │ +o2 = /tmp/M2-11603-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ │ +o3 = /tmp/M2-11603-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : isDirectory fn
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html
│ │ │ @@ -216,15 +216,15 @@
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 3.67323s elapsed
│ │ │ + -- 5.39915s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .05718s elapsed
│ │ │ + -- .0688154s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000110026s elapsed
│ │ │ + -- .000244807s elapsed
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │  i17 : isPrime (m*m1)
│ │ │ │  
│ │ │ │  o17 = false
│ │ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ │ - -- 3.67323s elapsed
│ │ │ │ + -- 5.39915s elapsed
│ │ │ │  
│ │ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │ │  
│ │ │ │  o19 : Expression of class Product
│ │ │ │  i20 : facs = facs//toList/toList
│ │ │ │  
│ │ │ │  o20 = {{1000000000000000000000000000057, 1},
│ │ │ │ @@ -98,19 +98,19 @@
│ │ │ │  o20 : List
│ │ │ │  i21 : assert(set facs === set {{m,1}, {m1,1}})
│ │ │ │  i22 : m3 = nextPrime (m^3)
│ │ │ │  
│ │ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │ │        00000000000000185613
│ │ │ │  i23 : elapsedTime isPrime m3
│ │ │ │ - -- .05718s elapsed
│ │ │ │ + -- .0688154s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000110026s elapsed
│ │ │ │ + -- .000244807s elapsed
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_P_r_i_m_e_(_Z_Z_) -- whether a integer or polynomial is prime
│ │ │ │      * _f_a_c_t_o_r_(_Z_Z_) -- factor a ring element
│ │ │ │      * _n_e_x_t_P_r_i_m_e_(_N_u_m_b_e_r_) -- compute the smallest prime greater than or equal to
│ │ │ │        a given number
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Regular__File.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  In UNIX, a regular file is one that is not special in some way.  Special files include symbolic links and directories.  A regular file is a sequence of bytes stored permanently in a file system.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-13116-0/0
    
│ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : isRegularFile fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a regular file
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  In UNIX, a regular file is one that is not special in some way. Special files
│ │ │ │  include symbolic links and directories. A regular file is a sequence of bytes
│ │ │ │  stored permanently in a file system.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-13116-0/0
│ │ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_lock_lp__Mutex_cm__Function_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │                
    i3 : parallelApply(1..1000, f);
    
│ │ │              
    
│ │ │                
    i4 : x
│ │ │  
│ │ │ -o4 = 333
    
│ │ │ +o4 = 344
│ │ │              
    
│ │ │          
    
│ │ │            
    You likely see that x ended up less than the expected value of 1000.  This is an example of a race condition, as multiple threads were trying to modify the global variable x at the same time.
    
│ │ │            
    Let's try again using lock.
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  i1 : x = 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : f = i -> x += 1;
│ │ │ │  i3 : parallelApply(1..1000, f);
│ │ │ │  i4 : x
│ │ │ │  
│ │ │ │ -o4 = 333
│ │ │ │ +o4 = 344
│ │ │ │  You likely see that x ended up less than the expected value of 1000. This is an
│ │ │ │  example of a rraaccee ccoonnddiittiioonn, as multiple threads were trying to modify the
│ │ │ │  global variable x at the same time.
│ │ │ │  Let's try again using lock.
│ │ │ │  i5 : x = 0
│ │ │ │  
│ │ │ │  o5 = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -81,22 +81,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the name of the newly made directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is made, with as many new path components as needed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11593-0/0
│ │ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-12363-0/0/a/b/c
│ │ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │ │  i5 : removeDirectory (dir|"/a")
│ │ │ │  A filename starting with ~/ will have the tilde replaced by the home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_k_d_i_r
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11593-0/0
    
│ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-12363-0/0/a/b/c
│ │ │              
    
│ │ │                
    i3 : removeDirectory (dir|"/a/b/c")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
    
│ │ │ +o1 = 17
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,15 +9,15 @@
│ │ │ │      *   Usage:
│ │ │ │              maxAllowableThreads
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the maximum number to which _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s can be set
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : maxAllowableThreads
│ │ │ │  
│ │ │ │ -o1 = 7
│ │ │ │ +o1 = 17
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_a_x_A_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s is an _i_n_t_e_g_e_r.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_threads.m2:502:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -66,15 +66,15 @@
│ │ │  
│ │ │  o1 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i2 : time fib 28
│ │ │ - -- used 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ + -- used 1.15321s (cpu); 0.653719s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -83,23 +83,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ + -- used 7.6966e-05s (cpu); 7.3392e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ + -- used 3.961e-06s (cpu); 3.503e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
    
│ │ │              
    
│ │ │          
    An optional second argument to memoize provides a list of initial values, each of the form x => v, where v is the value to be provided for the argument x.
    
│ │ │          
    Alternatively, values can be provided after defining the memoized function using the syntax f x = v. A slightly more efficient implementation of the above would be
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,28 +11,28 @@
│ │ │ │  arguments are presented.
│ │ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │ │  
│ │ │ │  o1 = fib
│ │ │ │  
│ │ │ │  o1 : FunctionClosure
│ │ │ │  i2 : time fib 28
│ │ │ │ - -- used 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ │ + -- used 1.15321s (cpu); 0.653719s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ │ + -- used 7.6966e-05s (cpu); 7.3392e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.961e-06s (cpu); 3.503e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 514229
│ │ │ │  An optional second argument to memoize provides a list of initial values, each
│ │ │ │  of the form x => v, where v is the value to be provided for the argument x.
│ │ │ │  Alternatively, values can be provided after defining the memoized function
│ │ │ │  using the syntax f x = v. A slightly more efficient implementation of the above
│ │ │ │  would be
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_methods.html
│ │ │ @@ -94,20 +94,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ +     {20 => (codim, BettiTally)                                   }
│ │ │       {21 => (dual, BettiTally)                                    }
│ │ │       {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ -     {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ +     {23 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ +     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │                
    
│ │ │ @@ -193,20 +193,20 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ -     {3 => (contract, Matrix, Matrix)                            }
│ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ +o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ +     {1 => (diff, Matrix, Matrix)                                }
│ │ │ +     {2 => (-, Matrix, Matrix)                                   }
│ │ │ +     {3 => (contract', Matrix, Matrix)                           }
│ │ │ +     {4 => (contract, Matrix, Matrix)                            }
│ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -217,22 +217,22 @@
│ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {30 => (intersect, Matrix, Matrix)                          }
│ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,20 +29,20 @@
│ │ │ │       {12 => (poincare, BettiTally)                                }
│ │ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │ │       {14 => (degree, BettiTally)                                  }
│ │ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │ │       {17 => (regularity, BettiTally)                              }
│ │ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ │ -     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ +     {19 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ +     {20 => (codim, BettiTally)                                   }
│ │ │ │       {21 => (dual, BettiTally)                                    }
│ │ │ │       {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ -     {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ │ +     {23 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ +     {24 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (freeResolution, Ideal)        }
│ │ │ │       {1 => (freeResolution, MonomialIdeal)}
│ │ │ │ @@ -85,20 +85,20 @@
│ │ │ │      * Inputs:
│ │ │ │            o X, a _t_y_p_e
│ │ │ │            o Y, a _t_y_p_e
│ │ │ │      * Outputs:
│ │ │ │            o a _v_e_r_t_i_c_a_l_ _l_i_s_t of those methods associated with
│ │ │ │  i5 : methods( Matrix, Matrix )
│ │ │ │  
│ │ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ │ -     {3 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ │ +o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ │ +     {1 => (diff, Matrix, Matrix)                                }
│ │ │ │ +     {2 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {3 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {4 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ │ @@ -109,22 +109,22 @@
│ │ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ │ -     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {30 => (intersect, Matrix, Matrix)                          }
│ │ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.20053s elapsed
│ │ │ + -- 1.97379s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -130,15 +130,15 @@
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .517063s elapsed
│ │ │ + -- .838305s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │ @@ -151,15 +151,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0258497s elapsed
│ │ │ + -- .0477547s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
    
│ │ │                
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- .826979s elapsed
│ │ │ + -- 1.27651s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ │ - -- 1.20053s elapsed
│ │ │ │ + -- 1.97379s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -60,40 +60,40 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  One can compute smaller parts of the Betti table, by using _D_e_g_r_e_e_L_i_m_i_t and/or
│ │ │ │  _L_e_n_g_t_h_L_i_m_i_t.
│ │ │ │  i4 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ │ - -- .517063s elapsed
│ │ │ │ + -- .838305s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7
│ │ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │ │           0: 1  .   .   .   .    .   .   .
│ │ │ │           1: . 35 140 189  84    .   .   .
│ │ │ │           2: .  .   . 196 735 1080 735 196
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ │ - -- .0258497s elapsed
│ │ │ │ + -- .0477547s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3  4
│ │ │ │  o7 = total: 1 35 140 189 84
│ │ │ │           0: 1  .   .   .  .
│ │ │ │           1: . 35 140 189 84
│ │ │ │  
│ │ │ │  o7 : BettiTally
│ │ │ │  i8 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ │ - -- .826979s elapsed
│ │ │ │ + -- 1.27651s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5
│ │ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │ │           0: 1  .   .   .   .    .
│ │ │ │           1: . 35 140 189  84    .
│ │ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_mkdir.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -96,15 +96,15 @@
│ │ │  o3 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11612-0/0/
    
│ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ +o4 = /tmp/M2-12402-0/0/foo
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,22 +12,22 @@
│ │ │ │      * Consequences:
│ │ │ │            o a directory will be created at the path p
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Only one directory will be made, so the components of the path p other than the
│ │ │ │  last must already exist.
│ │ │ │  i1 : p = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11612-0/0/
│ │ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ │ +o4 = /tmp/M2-12402-0/0/foo
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : removeFile fn
│ │ │ │  i7 : removeDirectory p
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_move__File_lp__String_cm__String_rp.html
│ │ │ @@ -86,52 +86,52 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,32 +20,32 @@
│ │ │ │            o the name of the backup file if one was created, or _n_u_l_l
│ │ │ │      * Consequences:
│ │ │ │            o the file will be moved by creating a new link to the file and
│ │ │ │              removing the old one
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11448-0/0
│ │ │ │ +o1 = /tmp/M2-12078-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11448-0/1
│ │ │ │ +o2 = /tmp/M2-12078-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ │ +o3 = /tmp/M2-12078-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
│ │ │ │ +--moving: /tmp/M2-12078-0/0 -> /tmp/M2-12078-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-12078-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11448-0/1.bak
│ │ │ │ +o6 = /tmp/M2-12078-0/1.bak
│ │ │ │  i7 : removeFile bak
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_F_i_l_e
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * moveFile(String)
│ │ │ │      * _m_o_v_e_F_i_l_e_(_S_t_r_i_n_g_,_S_t_r_i_n_g_)
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_nanosleep.html
│ │ │ @@ -56,15 +56,15 @@
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11448-0/0
    
│ │ │ +o1 = /tmp/M2-12078-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11448-0/1
    
│ │ │ +o2 = /tmp/M2-12078-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ +o3 = /tmp/M2-12078-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
    
│ │ │ +--moving: /tmp/M2-12078-0/0 -> /tmp/M2-12078-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ +--backup file created: /tmp/M2-12078-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-11448-0/1.bak
    
│ │ │ +o6 = /tmp/M2-12078-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500147s elapsed
│ │ │ + -- .500173s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -4,15 +4,15 @@
│ │ │ │  [q                   ]
│ │ │ │  _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c
│ │ │ │  ===============================================================================
│ │ │ │  ************ nnaannoosslleeeepp ---- sslleeeepp ffoorr aa ggiivveenn nnuummbbeerr ooff nnaannoosseeccoonnddss ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  nanosleep n -- sleeps for n nanoseconds.
│ │ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ │ - -- .500147s elapsed
│ │ │ │ + -- .500173s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_l_e_e_p -- sleep for a while
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _n_a_n_o_s_l_e_e_p is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_options_lp__Function_rp.html
│ │ │ @@ -108,66 +108,66 @@
│ │ │  o3 : OptionTable
    
│ │ │              
    
│ │ │                
    i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ -     {1 => (codim, Ideal)         }
│ │ │ -     {2 => (codim, PolynomialRing)}
│ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ -     {5 => (codim, Module)        }
│ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ -     {7 => (codim, BettiTally)    }
│ │ │ +o4 = {0 => (codim, Module)        }
│ │ │ +     {1 => (codim, CoherentSheaf) }
│ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ +     {3 => (codim, Variety)       }
│ │ │ +     {4 => (codim, Ideal)         }
│ │ │ +     {5 => (codim, PolynomialRing)}
│ │ │ +     {6 => (codim, QuotientRing)  }
│ │ │ +     {7 => (codim, MonomialIdeal) }
│ │ │  
│ │ │  o4 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i5 : options oo
│ │ │  
│ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │       {1 => (OptionTable{Generic => false})}
│ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ +     {2 => (OptionTable{})                }
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ -     {7 => (OptionTable{})                }
│ │ │ +     {7 => (OptionTable{Generic => false})}
│ │ │  
│ │ │  o5 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i6 : methods intersect
│ │ │  
│ │ │  o6 = {0 => (intersect, List)                          }
│ │ │       {1 => (intersect, Module)                        }
│ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ -     {6 => (intersect, Set)                           }
│ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {8 => (intersect, Set)                           }
│ │ │       {9 => (intersect, Ideal)                         }
│ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │       {11 => (intersect, CCi)                          }
│ │ │       {12 => (intersect, RRi)                          }
│ │ │       {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │       {14 => (intersect, Cone, Cone)                   }
│ │ │       {15 => (intersect, Matrix)                       }
│ │ │       {16 => (intersect, Module, Module)               }
│ │ │       {17 => (intersect, CCi, CCi)                     }
│ │ │ -     {18 => (intersect, CCi, RRi)                     }
│ │ │ -     {19 => (intersect, RRi, CCi)                     }
│ │ │ +     {18 => (intersect, RRi, CCi)                     }
│ │ │ +     {19 => (intersect, CCi, RRi)                     }
│ │ │       {20 => (intersect, Sequence)                     }
│ │ │  
│ │ │  o6 : NumberedVerticalList
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,58 +34,58 @@
│ │ │ │  i3 : options(codim, Ideal)
│ │ │ │  
│ │ │ │  o3 = OptionTable{Generic => false}
│ │ │ │  
│ │ │ │  o3 : OptionTable
│ │ │ │  i4 : methods codim
│ │ │ │  
│ │ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ │ -     {1 => (codim, Ideal)         }
│ │ │ │ -     {2 => (codim, PolynomialRing)}
│ │ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ │ -     {5 => (codim, Module)        }
│ │ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ │ -     {7 => (codim, BettiTally)    }
│ │ │ │ +o4 = {0 => (codim, Module)        }
│ │ │ │ +     {1 => (codim, CoherentSheaf) }
│ │ │ │ +     {2 => (codim, BettiTally)    }
│ │ │ │ +     {3 => (codim, Variety)       }
│ │ │ │ +     {4 => (codim, Ideal)         }
│ │ │ │ +     {5 => (codim, PolynomialRing)}
│ │ │ │ +     {6 => (codim, QuotientRing)  }
│ │ │ │ +     {7 => (codim, MonomialIdeal) }
│ │ │ │  
│ │ │ │  o4 : NumberedVerticalList
│ │ │ │  i5 : options oo
│ │ │ │  
│ │ │ │  o5 = {0 => (OptionTable{Generic => false})}
│ │ │ │       {1 => (OptionTable{Generic => false})}
│ │ │ │ -     {2 => (OptionTable{Generic => false})}
│ │ │ │ +     {2 => (OptionTable{})                }
│ │ │ │       {3 => (OptionTable{Generic => false})}
│ │ │ │       {4 => (OptionTable{Generic => false})}
│ │ │ │       {5 => (OptionTable{Generic => false})}
│ │ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ │ -     {7 => (OptionTable{})                }
│ │ │ │ +     {7 => (OptionTable{Generic => false})}
│ │ │ │  
│ │ │ │  o5 : NumberedVerticalList
│ │ │ │  i6 : methods intersect
│ │ │ │  
│ │ │ │  o6 = {0 => (intersect, List)                          }
│ │ │ │       {1 => (intersect, Module)                        }
│ │ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ │ -     {6 => (intersect, Set)                           }
│ │ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ │ +     {8 => (intersect, Set)                           }
│ │ │ │       {9 => (intersect, Ideal)                         }
│ │ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │ │       {11 => (intersect, CCi)                          }
│ │ │ │       {12 => (intersect, RRi)                          }
│ │ │ │       {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │ │       {14 => (intersect, Cone, Cone)                   }
│ │ │ │       {15 => (intersect, Matrix)                       }
│ │ │ │       {16 => (intersect, Module, Module)               }
│ │ │ │       {17 => (intersect, CCi, CCi)                     }
│ │ │ │ -     {18 => (intersect, CCi, RRi)                     }
│ │ │ │ -     {19 => (intersect, RRi, CCi)                     }
│ │ │ │ +     {18 => (intersect, RRi, CCi)                     }
│ │ │ │ +     {19 => (intersect, CCi, RRi)                     }
│ │ │ │       {20 => (intersect, Sequence)                     }
│ │ │ │  
│ │ │ │  o6 : NumberedVerticalList
│ │ │ │  i7 : options 0
│ │ │ │  
│ │ │ │  o7 = OptionTable{}
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │              
│ │ │                
    i2 : L = shuffle toList (1..10000);
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .659735s elapsed
    
│ │ │ + -- .949337s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .211134s elapsed
    
│ │ │ + -- .204447s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    You will have to try it on your examples to see how much they speed up.
    
│ │ │            
    Warning: Threads computing in parallel can give wrong answers if their code is not "thread safe", meaning they make modifications to memory without ensuring the modifications get safely communicated to other threads. (Thread safety can slow computations some.) Currently, modifications to Macaulay2 variables and mutable hash tables are thread safe, but not changes inside mutable lists. Also, access to external libraries such as singular, etc., may not currently be thread safe.
    
│ │ │            
    The rest of this document describes how to control parallel tasks more directly.
    
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o5 = 5
    
│ │ │              
    
│ │ │                
    i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
    
│ │ │ +o6 = 17
│ │ │              
    
│ │ │          
    
│ │ │            
    To run a function in another thread use schedule, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,17 +17,17 @@
│ │ │ │  big computation. If the list is long, it will be split into chunks for each
│ │ │ │  core, reducing the overhead. But the speedup is still limited by the different
│ │ │ │  threads competing for memory, including cpu caches; it is like running
│ │ │ │  Macaulay2 on a computer that is running other big programs at the same time. We
│ │ │ │  can see this using _e_l_a_p_s_e_d_T_i_m_e.
│ │ │ │  i2 : L = shuffle toList (1..10000);
│ │ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ │ - -- .659735s elapsed
│ │ │ │ + -- .949337s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .211134s elapsed
│ │ │ │ + -- .204447s elapsed
│ │ │ │  You will have to try it on your examples to see how much they speed up.
│ │ │ │  Warning: Threads computing in parallel can give wrong answers if their code is
│ │ │ │  not "thread safe", meaning they make modifications to memory without ensuring
│ │ │ │  the modifications get safely communicated to other threads. (Thread safety can
│ │ │ │  slow computations some.) Currently, modifications to Macaulay2 variables and
│ │ │ │  mutable hash tables are thread safe, but not changes inside mutable lists.
│ │ │ │  Also, access to external libraries such as singular, etc., may not currently be
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │  _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s, and may be examined and changed as follows. (_a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s
│ │ │ │  is temporarily increased if necessary inside _p_a_r_a_l_l_e_l_A_p_p_l_y.)
│ │ │ │  i5 : allowableThreads
│ │ │ │  
│ │ │ │  o5 = 5
│ │ │ │  i6 : allowableThreads = maxAllowableThreads
│ │ │ │  
│ │ │ │ -o6 = 7
│ │ │ │ +o6 = 17
│ │ │ │  To run a function in another thread use _s_c_h_e_d_u_l_e, as in the following example.
│ │ │ │  i7 : R = QQ[x,y,z];
│ │ │ │  i8 : I = ideal(x^2 + 2*y^2 - y - 2*z, x^2 - 8*y^2 + 10*z - 1, x^2 - 7*y*z)
│ │ │ │  
│ │ │ │               2     2            2     2             2
│ │ │ │  o8 = ideal (x  + 2y  - y - 2z, x  - 8y  + 10z - 1, x  - 7y*z)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 1.15969s elapsed
│ │ │ + -- 1.85279s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -165,15 +165,15 @@
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.14895s elapsed
│ │ │ + -- 1.84267s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -195,15 +195,15 @@
│ │ │  
│ │ │  o8 = 1
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.62269s elapsed
│ │ │ + -- 1.86033s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -236,15 +236,15 @@
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.37571s elapsed
│ │ │ + -- 2.28683s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
    
│ │ │ @@ -263,15 +263,15 @@
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │              
    
│ │ │                
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 1.51803s elapsed
│ │ │ + -- 2.2938s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
    
│ │ │ @@ -304,15 +304,15 @@
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │              
    
│ │ │                
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.09157s elapsed
│ │ │ + -- 4.53922s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -327,15 +327,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.21927s elapsed
│ │ │ + -- 10.2962s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -350,15 +350,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.08357s elapsed
│ │ │ + -- 4.30526s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -401,15 +401,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- .823549s elapsed
│ │ │ + -- 1.18493s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.45319s elapsed
│ │ │ + -- 2.07506s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -93,30 +93,30 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i3 : S = ring I
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : elapsedTime minimalBetti I
│ │ │ │ - -- 1.15969s elapsed
│ │ │ │ + -- 1.85279s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │           4: .  .   .   .   .    .   .   .   .  .  1
│ │ │ │  
│ │ │ │  o4 : BettiTally
│ │ │ │  i5 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ │ - -- 1.14895s elapsed
│ │ │ │ + -- 1.84267s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -126,15 +126,15 @@
│ │ │ │  i7 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o8 = 1
│ │ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ │ - -- 1.62269s elapsed
│ │ │ │ + -- 1.86033s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -149,15 +149,15 @@
│ │ │ │  i11 : numTBBThreads = 0
│ │ │ │  
│ │ │ │  o11 = 0
│ │ │ │  i12 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.37571s elapsed
│ │ │ │ + -- 2.28683s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -168,15 +168,15 @@
│ │ │ │  i14 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 1.51803s elapsed
│ │ │ │ + -- 2.2938s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -195,37 +195,37 @@
│ │ │ │  o18 = S
│ │ │ │  
│ │ │ │  o18 : PolynomialRing
│ │ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │ │  
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.09157s elapsed
│ │ │ │ + -- 4.53922s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o20 : Matrix S  <-- S
│ │ │ │  i21 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o21 = 1
│ │ │ │  i22 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o22 : Ideal of S
│ │ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 7.21927s elapsed
│ │ │ │ + -- 10.2962s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o23 : Matrix S  <-- S
│ │ │ │  i24 : numTBBThreads = 10
│ │ │ │  
│ │ │ │  o24 = 10
│ │ │ │  i25 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.08357s elapsed
│ │ │ │ + -- 4.30526s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o26 : Matrix S  <-- S
│ │ │ │  For Gröbner basis computation in associative algebras, ParallelizeByDegree is
│ │ │ │  not relevant. In this case, use numTBBThreads to control the amount of
│ │ │ │  parallelism.
│ │ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │ │ @@ -246,15 +246,15 @@
│ │ │ │                 2                         2                         2
│ │ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- .823549s elapsed
│ │ │ │ + -- 1.18493s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  i32 : I = ideal I_*
│ │ │ │  
│ │ │ │                 2                         2                         2
│ │ │ │ @@ -263,15 +263,15 @@
│ │ │ │                  ZZ
│ │ │ │  o32 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i33 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o33 = 1
│ │ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.45319s elapsed
│ │ │ │ + -- 2.07506s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_i_m_a_l_B_e_t_t_i -- minimal betti numbers of (the minimal free resolution of)
│ │ │ │        a homogeneous ideal or module
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_poincare.html
│ │ │ @@ -375,36 +375,36 @@
│ │ │  
│ │ │  o27 = 3
│ │ │              
    
│ │ │                
    i28 : time poincare I
│ │ │ - -- used 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00271247s (cpu); 1.302e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ +   -- registering gb 16 at 0x7f2922d09c40
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00927416s (cpu); 0.0124893s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -416,15 +416,15 @@
│ │ │                
    i30 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ +   -- registering gb 17 at 0x7f2922d09a80
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -433,24 +433,24 @@
│ │ │  o31 : Ideal of R
    
│ │ │              
    
│ │ │                
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ +   -- registering gb 18 at 0x7f2922d098c0
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00399734s (cpu); 0.00481081s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ +   -- registering gb 19 at 0x7f2922d09700
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
│ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0599994s (cpu); 0.0573032s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -177,66 +177,66 @@
│ │ │ │  o26 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o26 : ZZ[T]
│ │ │ │  i27 : gbTrace = 3
│ │ │ │  
│ │ │ │  o27 = 3
│ │ │ │  i28 : time poincare I
│ │ │ │ - -- used 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00271247s (cpu); 1.302e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ │ +   -- registering gb 16 at 0x7f2922d09c40
│ │ │ │  
│ │ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 4186
│ │ │ │     -- #reduction steps = 38
│ │ │ │     -- #spairs done = 11
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 29
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00927416s (cpu); 0.0124893s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      11
│ │ │ │  o29 : Matrix R  <-- R
│ │ │ │  In this case, the savings is minimal, but often it can be dramatic. Another
│ │ │ │  important situation is to compute a Gröbner basis using a different monomial
│ │ │ │  order.
│ │ │ │  i30 : R = QQ[a..d];
│ │ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │ │  
│ │ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ │ +   -- registering gb 17 at 0x7f2922d09a80
│ │ │ │  
│ │ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │ │     -- number of monomials                = 0
│ │ │ │     -- #reduction steps = 0
│ │ │ │     -- #spairs done = 0
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │     -- nsaved = 0
│ │ │ │     --
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time p = poincare I
│ │ │ │  
│ │ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ │ +   -- registering gb 18 at 0x7f2922d098c0
│ │ │ │  
│ │ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 267
│ │ │ │     -- #reduction steps = 236
│ │ │ │     -- #spairs done = 30
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 20
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00399734s (cpu); 0.00481081s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o32 : ZZ[T]
│ │ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ │  
│ │ │ │ @@ -281,30 +281,30 @@
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : gbTrace = 3
│ │ │ │  
│ │ │ │  o36 = 3
│ │ │ │  i37 : time gens gb J;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ │ +   -- registering gb 19 at 0x7f2922d09700
│ │ │ │  
│ │ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}
│ │ │ │  (3,9)m
│ │ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m
│ │ │ │  {24}(1,3)m
│ │ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements
│ │ │ │  = 39
│ │ │ │     -- number of monomials                = 1051
│ │ │ │     -- #reduction steps = 284
│ │ │ │     -- #spairs done = 53
│ │ │ │     -- ncalls = 46
│ │ │ │     -- nloop = 54
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0599994s (cpu); 0.0573032s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      39
│ │ │ │  o37 : Matrix S  <-- S
│ │ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │ │  
│ │ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_printing_spto_spa_spfile.html
│ │ │ @@ -102,22 +102,22 @@
│ │ │  o2 : File
│ │ │              
    
│ │ │                
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11803-0/0
    
│ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -156,15 +156,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11803-0/0
│ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -174,15 +174,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11803-0/0
│ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11803-0/0
│ │ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : R = QQ[x]
│ │ │ │  
│ │ │ │ @@ -66,25 +66,25 @@
│ │ │ │   10      9      8       7       6       5       4       3      2
│ │ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │ │  o8 = stdio
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : fn << f << close
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11803-0/0
│ │ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  o9 : File
│ │ │ │  i10 : get fn
│ │ │ │  
│ │ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │ │        + 1
│ │ │ │  i11 : fn << toExternalString f << close
│ │ │ │  
│ │ │ │ -o11 = /tmp/M2-11803-0/0
│ │ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  o11 : File
│ │ │ │  i12 : get fn
│ │ │ │  
│ │ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │ │        1
│ │ │ │  i13 : value get fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_process__I__D.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : processID()
│ │ │  
│ │ │ -o1 = 11022
    
│ │ │ +o1 = 11242
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,15 +8,15 @@
│ │ │ │      *   Usage:
│ │ │ │              processID()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the process identifier of the current Macaulay2 process
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : processID()
│ │ │ │  
│ │ │ │ -o1 = 11022
│ │ │ │ +o1 = 11242
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_r_o_u_p_I_D -- the process group identifier
│ │ │ │      * _s_e_t_G_r_o_u_p_I_D -- set the process group identifier
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _p_r_o_c_e_s_s_I_D is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_profile.html
│ │ │ @@ -96,35 +96,35 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o9 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -178,15 +178,15 @@
│ │ │  o14 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22  
│ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61     
│ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52 
│ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ -     1     .0041s  elapsed total                    
    
│ │ │ +     1     93.17   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ +     1     90.46   ../../m2/matrix1.m2:275:22-275:43
│ │ │ +     1     44.54   ../../m2/matrix1.m2:183:25-183:52
│ │ │ +     1     31.45   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ +     1     30.27   ../../m2/matrix1.m2:130:10-145:16
│ │ │ +     1     21.85   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ +     1     21.76   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ +     1     20.45   ../../m2/set.m2:129:5-129:61     
│ │ │ +     1     3.26    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ +     1     2.37    ../../m2/matrix1.m2:131:13-131:78
│ │ │ +     1     2.08    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ +     1     1.58    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ +     1     1.46    ../../m2/matrix1.m2:137:20-137:64
│ │ │ +     1     1.34    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ +     1     1.16    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ +     1     1.07    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ +     20    1.03    ../../m2/matrix1.m2:181:14-182:67
│ │ │ +     1     1.02    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ +     20    .90     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ +     1     .74     ../../m2/modules.m2:282:4-282:52 
│ │ │ +     1     .0040s  elapsed total                    
│ │ │              
    
│ │ │                
    i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,35 +25,35 @@
│ │ │ │                4       5
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  Afterwards, running profileSummary and coverageSummary produces easy to read
│ │ │ │  tables summarizing the accumulated data so far in different ways.
│ │ │ │  i2 : profileSummary
│ │ │ │  
│ │ │ │  o2 = #run  %time   position
│ │ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58
│ │ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72
│ │ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42
│ │ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22
│ │ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61
│ │ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29
│ │ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11
│ │ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16
│ │ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73
│ │ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91
│ │ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74
│ │ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46
│ │ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52
│ │ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71
│ │ │ │ -     1     .0041s  elapsed total
│ │ │ │ +     1     93.17   ../../m2/matrix1.m2:273:4-276:58
│ │ │ │ +     1     90.46   ../../m2/matrix1.m2:275:22-275:43
│ │ │ │ +     1     44.54   ../../m2/matrix1.m2:183:25-183:52
│ │ │ │ +     1     31.45   ../../m2/matrix1.m2:104:5-146:72
│ │ │ │ +     1     30.27   ../../m2/matrix1.m2:130:10-145:16
│ │ │ │ +     1     21.85   ../../m2/matrix1.m2:171:4-171:42
│ │ │ │ +     1     21.76   ../../m2/matrix1.m2:35:10-39:22
│ │ │ │ +     1     20.45   ../../m2/set.m2:129:5-129:61
│ │ │ │ +     1     3.26    ../../m2/matrix1.m2:102:5-102:29
│ │ │ │ +     1     2.37    ../../m2/matrix1.m2:131:13-131:78
│ │ │ │ +     1     2.08    ../../m2/matrix1.m2:86:5-99:11
│ │ │ │ +     1     1.58    ../../m2/matrix1.m2:275:7-275:16
│ │ │ │ +     1     1.46    ../../m2/matrix1.m2:137:20-137:64
│ │ │ │ +     1     1.34    ../../m2/matrix1.m2:270:4-271:73
│ │ │ │ +     1     1.16    ../../m2/matrix1.m2:101:5-101:91
│ │ │ │ +     1     1.07    ../../m2/matrix1.m2:88:10-88:46
│ │ │ │ +     20    1.03    ../../m2/matrix1.m2:181:14-182:67
│ │ │ │ +     1     1.02    ../../m2/matrix1.m2:172:4-174:74
│ │ │ │ +     20    .90     ../../m2/matrix1.m2:37:43-37:71
│ │ │ │ +     1     .74     ../../m2/modules.m2:282:4-282:52
│ │ │ │ +     1     .0040s  elapsed total
│ │ │ │  i3 : coverageSummary
│ │ │ │  
│ │ │ │  o3 = covered lines:
│ │ │ │       ../../m2/lists.m2:148:24-148:32
│ │ │ │       ../../m2/lists.m2:148:34-148:58
│ │ │ │       ../../m2/matrix.m2:30:5-30:35
│ │ │ │       ../../m2/matrix.m2:31:5-31:46
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
    
│ │ │                
    i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ + -- used 0.842s (cpu); 0.205593s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ + -- used 0.499404s (cpu); 0.284665s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ + -- used 8.3979s (cpu); 3.04804s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : codim I, degree I
│ │ │ │  
│ │ │ │  o5 = (2, 10)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ │ + -- used 0.842s (cpu); 0.205593s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │ │               2      3   1     3   0     3
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : R=kk[x_0..x_5];
│ │ │ │  i8 : I=minors(3,random(R^5,R^{3:-1}));
│ │ │ │ @@ -45,15 +45,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : codim I, degree I
│ │ │ │  
│ │ │ │  o9 = (3, 10)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ │ + -- used 0.499404s (cpu); 0.284665s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  The claim that $63 \%$ of the intersections contain a K-rational point can be
│ │ │ │  experimentally tested:
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o13 : RR (of precision 53)
│ │ │ │  i14 : I=ideal random(n,R);
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c-
│ │ │ │  >degree c);
│ │ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ │ - -- used 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ │ + -- used 8.3979s (cpu); 3.04804s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 58
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommKKRRaattiioonnaallPPooiinntt:: **********
│ │ │ │      * randomKRationalPoint(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_read__Directory.html
│ │ │ @@ -73,38 +73,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12474-0/0
    
│ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-12474-0/0
    
│ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ +o3 = /tmp/M2-14164-0/0/foo
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,26 +10,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Outputs:
│ │ │ │            o a _l_i_s_t, the list of filenames stored in the directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12474-0/0
│ │ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12474-0/0
│ │ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ │ +o3 = /tmp/M2-14164-0/0/foo
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : readDirectory dir
│ │ │ │  
│ │ │ │ -o4 = {., .., foo}
│ │ │ │ +o4 = {.., ., foo}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : removeFile fn
│ │ │ │  i6 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_m_o_v_e_D_i_r_e_c_t_o_r_y -- remove a directory
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reading_spfiles.html
│ │ │ @@ -57,22 +57,22 @@
│ │ │        
    
│ │ │  Sometimes a file will contain a single expression whose value you wish to have access to.  For example, it might be a polynomial produced by another program.  The function get can be used to obtain the entire contents of a file as a single string.  We illustrate this here with a file whose name is expression.        
    
│ │ │  First we create the file by writing the desired text to it.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11978-0/0
    
│ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3" << endl << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11978-0/0
│ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │ @@ -121,15 +121,15 @@
│ │ │  Often a file will contain code written in the Macaulay2 language. Let's create such a file.        
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11978-0/0
│ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │  Now verify that it contains the desired text with get.        
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,20 +8,20 @@
│ │ │ │  Sometimes a file will contain a single expression whose value you wish to have
│ │ │ │  access to. For example, it might be a polynomial produced by another program.
│ │ │ │  The function _g_e_t can be used to obtain the entire contents of a file as a
│ │ │ │  single string. We illustrate this here with a file whose name is expression.
│ │ │ │  First we create the file by writing the desired text to it.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11978-0/0
│ │ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │ │  i2 : fn <<
│ │ │ │  "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3"
│ │ │ │  << endl << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11978-0/0
│ │ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  Now we get the contents of the file, as a single string.
│ │ │ │  i3 : get fn
│ │ │ │  
│ │ │ │  o3 = z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
│ │ │ │       +8*y^3
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o6 : Expression of class Product
│ │ │ │  Often a file will contain code written in the Macaulay2 language. Let's create
│ │ │ │  such a file.
│ │ │ │  i7 : fn << "sample = 2^100
│ │ │ │       print sample
│ │ │ │       " << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11978-0/0
│ │ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  Now verify that it contains the desired text with _g_e_t.
│ │ │ │  i8 : get fn
│ │ │ │  
│ │ │ │  o8 = sample = 2^100
│ │ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_readlink.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the resolved path to a symbolic link, or null if the file
│ │ │ │              was not a symbolic link.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : p = temporaryFileName ()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12734-0/0
│ │ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │ │  i2 : symlinkFile ("foo", p)
│ │ │ │  i3 : readlink p
│ │ │ │  
│ │ │ │  o3 = foo
│ │ │ │  i4 : removeFile p
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_l_p_a_t_h -- convert a filename to one passing through no symbolic links
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
│ │ │ @@ -73,57 +73,57 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-12734-0/0
    
│ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -135,15 +135,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-11022-0/91-rundir/
    
│ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12753-0/0
    
│ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12753-0/1
    
│ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12753-0/0
    
│ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12753-0/0
    
│ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
    
│ │ │ +o10 = /tmp/M2-11242-0/91-rundir/
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  Every component of the path must exist in the file system and be accessible to the user. Terminal slashes will be dropped.  Warning: under most operating systems, the value returned is an absolute path (one starting at the root of the file system), but under Solaris, this system call may, in certain circumstances, return a relative path when given a relative path.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,39 +12,39 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename, or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, a pathname to fn passing through no symbolic links, and
│ │ │ │              ending with a slash if fn refers to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : realpath "."
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11022-0/91-rundir/
│ │ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12753-0/0
│ │ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12753-0/1
│ │ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12753-0/0
│ │ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12753-0/0
│ │ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
│ │ │ │ +o10 = /tmp/M2-11242-0/91-rundir/
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Every component of the path must exist in the file system and be accessible to
│ │ │ │  the user. Terminal slashes will be dropped. Warning: under most operating
│ │ │ │  systems, the value returned is an absolute path (one starting at the root of
│ │ │ │  the file system), but under Solaris, this system call may, in certain
│ │ │ │  circumstances, return a relative path when given a relative path.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html
│ │ │ @@ -81,24 +81,23 @@
│ │ │                  		
│ │ │                
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │              
    
│ │ │                
    i2 : collectGarbage() 
│ │ │ ---finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
    
│ │ │ +--finalization: (1)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (3)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (4)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (5)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (7)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (9)[6]: -- finalizing sequence #7 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  This function should mainly be used for debugging.  Having a large number of finalizers might degrade the performance of the program.  Moreover, registering two or more objects that are members of a circular chain of pointers for finalization will result in a memory leak, with none of the objects in the chain being freed, even if nothing else points to any of them.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,24 +14,23 @@
│ │ │ │      * Consequences:
│ │ │ │            o A finalizer is registered with the garbage collector to print a
│ │ │ │              string when that object is collected as garbage
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-
│ │ │ │  - finalizing sequence #"|i|" --"))
│ │ │ │  i2 : collectGarbage()
│ │ │ │ ---finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (1)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ │ +--finalization: (3)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (4)[7]: -- finalizing sequence #8 --
│ │ │ │ +--finalization: (5)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (6)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (7)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (8)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (9)[6]: -- finalizing sequence #7 --
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This function should mainly be used for debugging. Having a large number of
│ │ │ │  finalizers might degrade the performance of the program. Moreover, registering
│ │ │ │  two or more objects that are members of a circular chain of pointers for
│ │ │ │  finalization will result in a memory leak, with none of the objects in the
│ │ │ │  chain being freed, even if nothing else points to any of them.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remove__Directory.html
│ │ │ @@ -76,29 +76,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11650-0/0
    
│ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11650-0/0
    
│ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │              
    
│ │ │                
    i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : removeDirectory dir
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is removed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11650-0/0
│ │ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11650-0/0
│ │ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │ │  i3 : readDirectory dir
│ │ │ │  
│ │ │ │ -o3 = {., ..}
│ │ │ │ +o3 = {.., .}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d_D_i_r_e_c_t_o_r_y -- read the contents of a directory
│ │ │ │      * _m_a_k_e_D_i_r_e_c_t_o_r_y -- make a directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__Path.html
│ │ │ @@ -70,22 +70,22 @@
│ │ │          
    Description
    
│ │ │          
    This string may be concatenated with an absolute path to get one understandable by external programs.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11116-0/0
    
│ │ │ +o1 = /tmp/M2-11406-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-11116-0/0
    
│ │ │ +o2 = /tmp/M2-11406-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by external programs, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by external programs.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11116-0/0
│ │ │ │ +o1 = /tmp/M2-11406-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11116-0/0
│ │ │ │ +o2 = /tmp/M2-11406-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_U_R_I
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_P_a_t_h is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2025:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -70,22 +70,22 @@
│ │ │          
      Description
      
│ │ │          
      This string may be concatenated with an absolute path to get one understandable by an external browser.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12417-0/0
      
│ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-12417-0/0
      
│ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by an external browser, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by an external browser.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12417-0/0
│ │ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-12417-0/0
│ │ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_P_a_t_h
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_U_R_I is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2041:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -95,22 +95,22 @@
│ │ │  o4 : R-module, submodule of R
│ │ │              
    
│ │ │                
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-12265-0/0
    
│ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : get f
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,18 +28,18 @@
│ │ │ │  
│ │ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o4 : R-module, submodule of R
│ │ │ │  i5 : f = temporaryFileName()
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12265-0/0
│ │ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : get f
│ │ │ │  
│ │ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ │ │  i8 : (p',m',M') = value get f
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_serial__Number.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1367325
    
│ │ │ +o1 = 1367343
│ │ │              
    
│ │ │                
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1367327
    
│ │ │ +o2 = 1367345
│ │ │              
    
│ │ │                
    i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,18 +10,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o x
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the serial number of x
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : serialNumber asdf
│ │ │ │  
│ │ │ │ -o1 = 1367325
│ │ │ │ +o1 = 1367343
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1367327
│ │ │ │ +o2 = 1367345
│ │ │ │  i3 : serialNumber ZZ
│ │ │ │  
│ │ │ │  o3 = 1000050
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _s_e_r_i_a_l_N_u_m_b_e_r is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html
│ │ │ @@ -371,21 +371,21 @@
│ │ │                  		
│ │ │                
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
    
│ │ │ + -- used 0.000213924s (cpu); 0.000199003s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
    
│ │ │ + -- used 9.4969e-05s (cpu); 9.4516e-05s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │ @@ -416,21 +416,21 @@
│ │ │              
    		
│ │ │                
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.232067s (cpu); 0.232067s (thread); 0s (gc)
    
│ │ │ + -- used 0.14571s (cpu); 0.145712s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
    
│ │ │ + -- used 0.1453s (cpu); 0.145313s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,33 +192,33 @@
│ │ │ │  i24 : printingPrecision = 4;
│ │ │ │  i25 : N = 40
│ │ │ │  
│ │ │ │  o25 = 40
│ │ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │ │  i30 : time X = solve(A,B);
│ │ │ │ - -- used 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
│ │ │ │ + -- used 0.000213924s (cpu); 0.000199003s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
│ │ │ │ + -- used 9.4969e-05s (cpu); 9.4516e-05s (thread); 0s (gc)
│ │ │ │  i32 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o32 = 5.111850690840453e-15
│ │ │ │  
│ │ │ │  o32 : RR (of precision 53)
│ │ │ │  Over higher precision RR or CC, these routines will be much slower than the
│ │ │ │  lower precision LAPACK routines.
│ │ │ │  i33 : N = 100
│ │ │ │  
│ │ │ │  o33 = 100
│ │ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │ │  i38 : time X = solve(A,B);
│ │ │ │ - -- used 0.232067s (cpu); 0.232067s (thread); 0s (gc)
│ │ │ │ + -- used 0.14571s (cpu); 0.145712s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
│ │ │ │ + -- used 0.1453s (cpu); 0.145313s (thread); 0s (gc)
│ │ │ │  i40 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ │  
│ │ │ │  o40 : RR (of precision 100)
│ │ │ │  Giving the option ClosestFit=>true, in the case when the field is RR or CC,
│ │ │ │  uses a least squares algorithm to find a best fit solution.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,93 +85,93 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12037-0/0/
    
│ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12037-0/1/
    
│ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12037-0/0/a/
    
│ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12037-0/0/b/
    
│ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │              
    
│ │ │                
    i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
    
│ │ │              
    
│ │ │                
    i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │              
    
│ │ │  Now we remove the files and directories we created.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,53 +30,53 @@
│ │ │ │            o The directory tree rooted at src is duplicated by a directory tree
│ │ │ │              rooted at dst. The files in the source tree are represented by
│ │ │ │              relative symbolic links in the destination tree to the original
│ │ │ │              files in the source tree.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12037-0/0/
│ │ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12037-0/1/
│ │ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12037-0/0/a/
│ │ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12037-0/0/b/
│ │ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │ │  i10 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o10 = ho there
│ │ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o12 = rm
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : scan(reverse findFiles src, rm)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__File.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o dst, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o a symbolic link at the location in the directory tree specified by
│ │ │ │              dst is created, pointing to src
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12094-0/0
│ │ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The file name is so unique that even with various suffixes appended, no collision with existing files will occur.  The files will be removed when the program terminates, unless it terminates as the result of an error.        
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12094-0/0
    
│ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-13097-0/0.tex
    
│ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-13097-0/1.html
    
│ │ │ +o2 = /tmp/M2-15437-0/1.html
│ │ │              
    
│ │ │          
    This function will work under Unix, and also under Windows if you have a directory on the same drive called /tmp.
    
│ │ │          
    If the name of the temporary file will be given to an external program, it may be necessary to concatenate it with rootPath or rootURI to enable the external program to find the file.
    
│ │ │          
    The temporary file name is derived from the value of the environment variable TMPDIR, if it has one.
    
│ │ │          
    If fork is used, then the parent and child Macaulay2 processes will each remove their own temporary files upon termination, with the parent removing any files created before fork was called.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a unique temporary file name.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The file name is so unique that even with various suffixes appended, no
│ │ │ │  collision with existing files will occur. The files will be removed when the
│ │ │ │  program terminates, unless it terminates as the result of an error.
│ │ │ │  i1 : temporaryFileName () | ".tex"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-13097-0/0.tex
│ │ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-13097-0/1.html
│ │ │ │ +o2 = /tmp/M2-15437-0/1.html
│ │ │ │  This function will work under Unix, and also under Windows if you have a
│ │ │ │  directory on the same drive called /tmp.
│ │ │ │  If the name of the temporary file will be given to an external program, it may
│ │ │ │  be necessary to concatenate it with _r_o_o_t_P_a_t_h or _r_o_o_t_U_R_I to enable the external
│ │ │ │  program to find the file.
│ │ │ │  The temporary file name is derived from the value of the environment variable
│ │ │ │  TMPDIR, if it has one.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value of e.  The time used by the the current thread and garbage collection during the evaluation of e is also shown.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time 3^30
│ │ │ - -- used 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ + -- used 2.075e-05s (cpu); 6.622e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │      *   Usage:
│ │ │ │              time e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value
│ │ │ │  of e. The time used by the the current thread and garbage collection during the
│ │ │ │  evaluation of e is also shown.
│ │ │ │  i1 : time 3^30
│ │ │ │ - -- used 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ │ + -- used 2.075e-05s (cpu); 6.622e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_timing.html
│ │ │ @@ -59,24 +59,24 @@
│ │ │  timing e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of cpu timing used, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000013355 seconds
│ │ │ +     -- .000024414 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000013355, 205891132094649}
    
│ │ │ +o2 = Time{.000024414, 205891132094649}
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  is the number of seconds of cpu timing used, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : timing 3^30
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │ -     -- .000013355 seconds
│ │ │ │ +     -- .000024414 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000013355, 205891132094649}
│ │ │ │ +o2 = Time{.000024414, 205891132094649}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o3 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime Mgb = gb M
│ │ │ - -- 1.4495s elapsed
│ │ │ + -- 2.6727s elapsed
│ │ │  
│ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o4 : GroebnerBasis
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .415018s elapsed
│ │ │ + -- .778775s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o6 : GroebnerBasis
    
│ │ │              
    
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .517879s elapsed
│ │ │ + -- .872512s elapsed
│ │ │  
│ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o11 : GroebnerBasis
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,24 +29,24 @@
│ │ │ │  i3 : hf = poincare ideal (a^4,b^5,c^5,d^6)
│ │ │ │  
│ │ │ │            4     5    6     9     10     11    14     15    16    20
│ │ │ │  o3 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │ │  
│ │ │ │  o3 : ZZ[T]
│ │ │ │  i4 : elapsedTime Mgb = gb M
│ │ │ │ - -- 1.4495s elapsed
│ │ │ │ + -- 2.6727s elapsed
│ │ │ │  
│ │ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o4 : GroebnerBasis
│ │ │ │  i5 : M = ideal M_*;
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ │ - -- .415018s elapsed
│ │ │ │ + -- .778775s elapsed
│ │ │ │  
│ │ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o6 : GroebnerBasis
│ │ │ │  However, obtaining the Hilbert function is not always easy to provide in this
│ │ │ │  way. In this case, one must work with the _d_e_g_r_e_e_s_R_i_n_g of the ring in question.
│ │ │ │  i7 : S = degreesRing R
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  o9 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │ │  
│ │ │ │  o9 : S
│ │ │ │  i10 : M = ideal M_*;
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ │ - -- .517879s elapsed
│ │ │ │ + -- .872512s elapsed
│ │ │ │  
│ │ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o11 : GroebnerBasis
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_a_n_U_s_e_H_i_l_b_e_r_t_H_i_n_t -- whether certain Groebner computations can make use
│ │ │ │        of the Hilbert function
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │                 "memtailor version" => 1.4
│ │ │                 "mpfi version" => 1.5.5
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.1
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ +               "operating system release" => 6.12.100+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism DirectSummands CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC SimplicialModules MatrixFactorizations PathSignatures MacaulayPosets MRDI EliminationTemplates WittVectors Padic WeierstrassSemigroups ResultantComplexes EuclideanDistanceDegree NeuralIdeals TestAudit LanguageServer
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.14.9
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2023.1
│ │ │ ├── html2text {}
│ │ │ │ @@ -67,15 +67,15 @@
│ │ │ │                 "memtailor version" => 1.4
│ │ │ │                 "mpfi version" => 1.5.5
│ │ │ │                 "mpfr version" => 4.2.2
│ │ │ │                 "mpsolve version" => 3.2.2
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.11.1
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ │ +               "operating system release" => 6.12.100+deb13-cloud-amd64
│ │ │ │                 "operating system" => Linux
│ │ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic
│ │ │ │  Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition
│ │ │ │  FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato
│ │ │ │  ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure
│ │ │ │  HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra
│ │ │ │  Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes
│ │ ├── ./usr/share/doc/Macaulay2/MacaulayPosets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  UG9zZXRNYXA=
│ │ │  #:len=820
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNsYXNzIG9mIGFsbCBtb3JwaGlz
│ │ │  bXMgYmV0d2VlbiBwb3NldHMiLCAibGluZW51bSIgPT4gMTYyNCwgU2VlQWxzbyA9PiBESVZ7SEVB
│ │ ├── ./usr/share/doc/Macaulay2/MapleInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  c3RvcmU=
│ │ │  #:len=651
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU3RvcmUgcmVzdWx0IG9mIGEgTWFwbGUg
│ │ │  Y29tcHV0YXRpb24gaW4gYSBmaWxlLiIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsi
│ │ ├── ./usr/share/doc/Macaulay2/Markov/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  Z2F1c3NJZGVhbChSaW5nLExpc3Qp
│ │ │  #:len=239
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjE4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnYXVzc0lkZWFsLFJpbmcsTGlzdCksImdhdXNzSWRl
│ │ ├── ./usr/share/doc/Macaulay2/Markov/example-output/___Markov.out
│ │ │ @@ -70,15 +70,15 @@
│ │ │       |   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2|   1,2,2,2 2,2,2,1    1,2,2,1 2,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │       |- p       p        + p       p       |- p       p        + p       p       |
│ │ │       |   1,1,2,1 1,2,1,1    1,1,1,1 1,2,2,1|   1,1,2,2 1,2,1,2    1,1,1,2 1,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │  
│ │ │  i8 : time netList primaryDecomposition J
│ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ + -- used 2.46031s (cpu); 1.74217s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ ├── ./usr/share/doc/Macaulay2/Markov/html/index.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │          
    
│ │ │            
    This ideal has 5 primary components.  The first is the one that has statistical significance. The significance of the other components is still poorly understood.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i8 : time netList primaryDecomposition J
│ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ + -- used 2.46031s (cpu); 1.74217s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,15 +102,15 @@
│ │ │ │  1,2,2,2|
│ │ │ │       +-------------------------------------+-----------------------------------
│ │ │ │  --+
│ │ │ │  This ideal has 5 primary components. The first is the one that has statistical
│ │ │ │  significance. The significance of the other components is still poorly
│ │ │ │  understood.
│ │ │ │  i8 : time netList primaryDecomposition J
│ │ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ │ + -- used 2.46031s (cpu); 1.74217s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MatchingFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  YWxnZWJyYWljTWF0cm9pZENpcmN1aXRz
│ │ │  #:len=1461
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIGJhc2VzIG9mIHRoZSBhbGdlYnJh
│ │ │  aWMgbWF0cm9pZCIsICJsaW5lbnVtIiA9PiAxOTI3LCBJbnB1dHMgPT4ge1NQQU57VFR7IkwifSwi
│ │ ├── ./usr/share/doc/Macaulay2/MatrixFactorizations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  SG9tKFpaZEZhY3Rvcml6YXRpb25NYXAsWlpkRmFjdG9yaXphdGlvbik=
│ │ │  #:len=413
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMzMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoSG9tLFpaZEZhY3Rvcml6YXRpb25NYXAsWlpkRmFj
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aXNBU01VbmlvbihMaXN0KQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNBU01VbmlvbixMaXN0KSwiaXNBU01VbmlvbihM
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_sp__A__S__M_spvarieties.out
│ │ │ @@ -212,17 +212,17 @@
│ │ │        | 1 -1 1 |
│ │ │        | 0 1  0 |
│ │ │  
│ │ │                 3       3
│ │ │  o22 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i23 : time schubertRegularity B
│ │ │ - -- used 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ + -- used 0.128722s (cpu); 0.0464251s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ + -- used 0.0216843s (cpu); 0.0216949s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_spmatrix_sp__Schubert_spvarieties.out
│ │ │ @@ -178,17 +178,17 @@
│ │ │        z   z   z   , z   z   z    - z   z   , z   z   z    - z   z   )
│ │ │         1,2 1,3 2,4   1,2 1,4 2,2    1,2 2,4   1,2 1,3 2,2    1,2 2,3
│ │ │  
│ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │                     1,1   5,5
│ │ │  
│ │ │  i16 : time schubertRegularity p
│ │ │ - -- used 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ + -- used 0.000449496s (cpu); 0.000429659s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ + -- used 0.0283083s (cpu); 0.0283158s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
│ │ │  
│ │ │  i18 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_grothendieck__Polynomial.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : w = {2,1,4,3}
│ │ │  
│ │ │  o1 = {2, 1, 4, 3}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ + -- used 0.00509149s (cpu); 0.00509282s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
│ │ │  
│ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ + -- used 0.00297381s (cpu); 0.00297647s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html
│ │ │ @@ -388,23 +388,23 @@
│ │ │          
    
│ │ │            
    Additionally, this package facilitates investigating homological invariants of ASM ideals such as regularity (schubertRegularity) and codimension (schubertCodim). efficiently by computing the associated invariants for their antidiagonal initial ideals, which are known to be squarefree by [Wei17]. Therefore the extremal Betti numbers (which encode regularity, depth, and projective dimension) of ASM ideals coincide with those of their antidiagonal initial ideals by [CV20].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i23 : time schubertRegularity B
│ │ │ - -- used 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ + -- used 0.128722s (cpu); 0.0464251s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
    
│ │ │              
    
│ │ │                
    i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ + -- used 0.0216843s (cpu); 0.0216949s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Functions for investigating ASM varieties
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -244,19 +244,19 @@
│ │ │ │  ASM ideals such as regularity (_s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y) and codimension
│ │ │ │  (_s_c_h_u_b_e_r_t_C_o_d_i_m). efficiently by computing the associated invariants for their
│ │ │ │  antidiagonal initial ideals, which are known to be squarefree by [Wei17].
│ │ │ │  Therefore the extremal Betti numbers (which encode regularity, depth, and
│ │ │ │  projective dimension) of ASM ideals coincide with those of their antidiagonal
│ │ │ │  initial ideals by [CV20].
│ │ │ │  i23 : time schubertRegularity B
│ │ │ │ - -- used 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ │ + -- used 0.128722s (cpu); 0.0464251s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ │ + -- used 0.0216843s (cpu); 0.0216949s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = 1
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg AASSMM vvaarriieettiieess **********
│ │ │ │      * _i_s_P_a_r_t_i_a_l_A_S_M_(_M_a_t_r_i_x_) -- whether a matrix is a partial alternating sign
│ │ │ │        matrix
│ │ │ │      * _p_a_r_t_i_a_l_A_S_M_T_o_A_S_M_(_M_a_t_r_i_x_) -- extend a partial alternating sign matrix to an
│ │ │ │        alternating sign matrix
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_spmatrix_sp__Schubert_spvarieties.html
│ │ │ @@ -320,23 +320,23 @@
│ │ │          
    
│ │ │            
    Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW24] via schubertRegularityschubertCodim.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : time schubertRegularity p
│ │ │ - -- used 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ + -- used 0.000449496s (cpu); 0.000429659s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
    
│ │ │              
    
│ │ │                
    i17 : time regularity comodule I
│ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ + -- used 0.0283083s (cpu); 0.0283158s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Functions for investigating matrix Schubert varieties
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -545,19 +545,19 @@
│ │ │ │  
│ │ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │ │                     1,1   5,5
│ │ │ │  Finally, this package contains functions for investigating homological
│ │ │ │  invariants of matrix Schubert varieties efficiently through combinatorial
│ │ │ │  algorithms produced in [PSW24] via _s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y_s_c_h_u_b_e_r_t_C_o_d_i_m.
│ │ │ │  i16 : time schubertRegularity p
│ │ │ │ - -- used 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ │ + -- used 0.000449496s (cpu); 0.000429659s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ │ + -- used 0.0283083s (cpu); 0.0283158s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 5
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg mmaattrriixx SScchhuubbeerrtt vvaarriieettiieess **********
│ │ │ │      * _a_n_t_i_D_i_a_g_I_n_i_t_(_L_i_s_t_) -- compute the (unique) antidiagonal initial ideal of
│ │ │ │        an ASM ideal
│ │ │ │      * _r_a_n_k_T_a_b_l_e_(_L_i_s_t_) -- compute a table of rank conditions that determines the
│ │ │ │        corresponding ASM or matrix Schubert variety
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
│ │ │ @@ -85,28 +85,28 @@
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │                
    i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ + -- used 0.00509149s (cpu); 0.00509282s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
    
│ │ │              
    
│ │ │                
    i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ + -- used 0.00297381s (cpu); 0.00297647s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,24 +19,24 @@
│ │ │ │  PipeDream.
│ │ │ │  i1 : w = {2,1,4,3}
│ │ │ │  
│ │ │ │  o1 = {2, 1, 4, 3}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time grothendieckPolynomial w
│ │ │ │ - -- used 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ │ + -- used 0.00509149s (cpu); 0.00509282s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o2 : QQ[x ..x ]
│ │ │ │           1   4
│ │ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ │ - -- used 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ │ + -- used 0.00297381s (cpu); 0.00297647s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o3 : QQ[x ..x ]
│ │ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  ZmxhdHMoTWF0cm9pZCxaWixTdHJpbmcp
│ │ │  #:len=255
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxhdHMsTWF0cm9pZCxaWixTdHJpbmcpLCJmbGF0
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out
│ │ │ @@ -51,20 +51,20 @@
│ │ │  i9 : keys R10.cache
│ │ │  
│ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time isWellDefined R10
│ │ │ - -- used 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ + -- used 0.0605118s (cpu); 0.0605134s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ + -- used 0.0469636s (cpu); 0.0469726s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -76,15 +76,15 @@
│ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │        -----------------------------------------------------------------------
│ │ │        groundSet, dual, storedRepresentation}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : time fVector R10
│ │ │ - -- used 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ + -- used 0.000236297s (cpu); 0.000234036s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_all__Minors.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │  
│ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │  
│ │ │  o2 : Matroid
│ │ │  
│ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0437144s elapsed
│ │ │ + -- .0455575s elapsed
│ │ │  
│ │ │  i4 : #L
│ │ │  
│ │ │  o4 = 64
│ │ │  
│ │ │  i5 : netList L_{0..4}
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Isos.out
│ │ │ @@ -33,14 +33,14 @@
│ │ │  i6 : F7 = specificMatroid "fano"
│ │ │  
│ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │  
│ │ │  o6 : Matroid
│ │ │  
│ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
│ │ │ + -- used 0.0496341s (cpu); 0.0496294s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : #autF7
│ │ │  
│ │ │  o8 = 168
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out
│ │ │ @@ -9,12 +9,12 @@
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : time hasMinor(M6, M5)
│ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ + -- used 2.28301s (cpu); 1.54248s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_isomorphism_lp__Matroid_cm__Matroid_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │  
│ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │  
│ │ │  o4 : Matroid
│ │ │  
│ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ + -- used 0.0278925s (cpu); 0.0278917s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -56,15 +56,15 @@
│ │ │  i7 : N = relabel M6
│ │ │  
│ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │  
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ + -- used 3.48107s (cpu); 2.97728s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_quick__Isomorphism__Test.out
│ │ │ @@ -37,15 +37,15 @@
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ + -- used 0.000859008s (cpu); 0.000855324s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
│ │ │  
│ │ │  i11 : value oo === false
│ │ │  
│ │ │  o11 = true
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_set__Representation.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  i5 : keys M.cache
│ │ │  
│ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime fVector M
│ │ │ - -- .0127952s elapsed
│ │ │ + -- .0128398s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -153,23 +153,23 @@
│ │ │  
│ │ │  o9 : List
│ │ │              
    
│ │ │                
    i10 : time isWellDefined R10
│ │ │ - -- used 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ + -- used 0.0605118s (cpu); 0.0605134s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │              
    
│ │ │                
    i11 : time fVector R10
│ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ + -- used 0.0469636s (cpu); 0.0469726s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -187,15 +187,15 @@
│ │ │  
│ │ │  o12 : List
    
│ │ │              
    
│ │ │                
    i13 : time fVector R10
│ │ │ - -- used 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ + -- used 0.000236297s (cpu); 0.000234036s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -71,19 +71,19 @@
│ │ │ │  o8 : Matroid
│ │ │ │  i9 : keys R10.cache
│ │ │ │  
│ │ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  i10 : time isWellDefined R10
│ │ │ │ - -- used 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ │ + -- used 0.0605118s (cpu); 0.0605134s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ │ + -- used 0.0469636s (cpu); 0.0469726s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ │ │ @@ -93,15 +93,15 @@
│ │ │ │  
│ │ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        groundSet, dual, storedRepresentation}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : time fVector R10
│ │ │ │ - -- used 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ │ + -- used 0.000236297s (cpu); 0.000234036s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_all__Minors.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : Matroid
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0437144s elapsed
    
│ │ │ + -- .0455575s elapsed
│ │ │              
    
│ │ │                
    i4 : #L
│ │ │  
│ │ │  o4 = 64
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o1 : Matroid
│ │ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │ │  
│ │ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │ │  
│ │ │ │  o2 : Matroid
│ │ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ │ - -- .0437144s elapsed
│ │ │ │ + -- .0455575s elapsed
│ │ │ │  i4 : #L
│ │ │ │  
│ │ │ │  o4 = 64
│ │ │ │  i5 : netList L_{0..4}
│ │ │ │  
│ │ │ │       +----------+-------+
│ │ │ │  o5 = |set {5, 3}|set {2}|
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_get__Isos.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o6 : Matroid
│ │ │              
    
│ │ │                
    i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
    
│ │ │ + -- used 0.0496341s (cpu); 0.0496294s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : #autF7
│ │ │  
│ │ │  o8 = 168
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  symmetric group S_7:
│ │ │ │  i6 : F7 = specificMatroid "fano"
│ │ │ │  
│ │ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │ │  
│ │ │ │  o6 : Matroid
│ │ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ │ - -- used 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
│ │ │ │ + -- used 0.0496341s (cpu); 0.0496294s (thread); 0s (gc)
│ │ │ │  i8 : #autF7
│ │ │ │  
│ │ │ │  o8 = 168
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ │ │        isomorphic matroids
│ │ │ │      * _q_u_i_c_k_I_s_o_m_o_r_p_h_i_s_m_T_e_s_t -- quick checks for isomorphism between matroids
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o2 = false
│ │ │              
    
│ │ │                
    i3 : time hasMinor(M6, M5)
│ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ + -- used 2.28301s (cpu); 1.54248s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │       elements, a "matroid" of rank 5 on 15 elements)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : time hasMinor(M6, M5)
│ │ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ │ + -- used 2.28301s (cpu); 1.54248s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r -- minor of matroid
│ │ │ │      * _i_s_B_i_n_a_r_y -- whether a matroid is representable over F_2
│ │ │ │  ********** WWaayyss ttoo uussee hhaassMMiinnoorr:: **********
│ │ │ │      * hasMinor(Matroid,Matroid)
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html
│ │ │ @@ -123,15 +123,15 @@
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │              
    
│ │ │                
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ + -- used 0.0278925s (cpu); 0.0278917s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -169,15 +169,15 @@
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │              
    
│ │ │                
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ + -- used 3.48107s (cpu); 2.97728s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │ │  
│ │ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │ │  
│ │ │ │  o4 : Matroid
│ │ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ │ - -- used 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ │ + -- used 0.0278925s (cpu); 0.0278917s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{0 => 1}
│ │ │ │                 1 => 0
│ │ │ │                 2 => 3
│ │ │ │                 3 => 2
│ │ │ │                 4 => 6
│ │ │ │                 5 => 5
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  o6 : HashTable
│ │ │ │  i7 : N = relabel M6
│ │ │ │  
│ │ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ │ + -- used 3.48107s (cpu); 2.97728s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HashTable{0 => 11 }
│ │ │ │                 1 => 0
│ │ │ │                 2 => 1
│ │ │ │                 3 => 6
│ │ │ │                 4 => 9
│ │ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_quick__Isomorphism__Test.html
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ + -- used 0.000859008s (cpu); 0.000855324s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
    
│ │ │              
    
│ │ │                
    i11 : value oo === false
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o7 = a "matroid" of rank 7 on 11 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ │ - -- used 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ │ + -- used 0.000859008s (cpu); 0.000855324s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : value oo === false
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_set__Representation.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime fVector M
│ │ │ - -- .0127952s elapsed
│ │ │ + -- .0128398s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  o4 : Matrix QQ  <-- QQ
│ │ │ │  i5 : keys M.cache
│ │ │ │  
│ │ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime fVector M
│ │ │ │ - -- .0127952s elapsed
│ │ │ │ + -- .0128398s elapsed
│ │ │ │  
│ │ │ │  o6 = HashTable{0 => 1 }
│ │ │ │                 1 => 6
│ │ │ │                 2 => 15
│ │ │ │                 3 => 20
│ │ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/MergeTeX/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  bWVyZ2VUZVgoLi4uLFBhdGg9Pi4uLik=
│ │ │  #:len=236
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttZXJnZVRlWCxQYXRoXSwibWVyZ2VUZVgoLi4uLFBh
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  ZGVjb21wb3NlKElkZWFsLFN0cmF0ZWd5PT4uLi4p
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1soZGVjb21wb3NlLElkZWFsKSxTdHJhdGVneV0sImRl
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out
│ │ │ @@ -5,16 +5,16 @@
│ │ │  i2 : R = ZZ/101[w..z];
│ │ │  
│ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .001266)   #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0370962)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000434127)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000036223)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000630212
│ │ │ - -- .0384912s elapsed
│ │ │ + --  Time taken : .000855779
│ │ │ + -- .0324191s elapsed
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical.out
│ │ │ @@ -30,21 +30,21 @@
│ │ │  
│ │ │               2        2   3     2
│ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000442257s elapsed
│ │ │ + -- .000568649s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0121996s elapsed
│ │ │ + -- .0158824s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical__Containment.out
│ │ │ @@ -29,22 +29,22 @@
│ │ │  o5 = 840
│ │ │  
│ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .151496s elapsed
│ │ │ + -- .0916603s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00157389s elapsed
│ │ │ + -- .00202296s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00123922s elapsed
│ │ │ + -- .00155956s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .001266)   #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0370962)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000434127)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000036223)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000630212
│ │ │ - -- .0384912s elapsed
    
│ │ │ + --  Time taken : .000855779
│ │ │ + -- .0324191s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,23 +11,23 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim =
│ │ │ │ +  Strategy: Linear            (time .001266)   #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0370962)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .000434127)  #primes = 0 #prunedViaCodim =
│ │ │ │  0
│ │ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim =
│ │ │ │ +  Strategy: DecomposeMonomials(time .000036223)  #primes = 1 #prunedViaCodim =
│ │ │ │  0
│ │ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ │ - --  Time taken : .000630212
│ │ │ │ - -- .0384912s elapsed
│ │ │ │ + --  Time taken : .000855779
│ │ │ │ + -- .0324191s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n_(_._._._,_S_t_r_a_t_e_g_y_=_>_._._._)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _H_y_b_r_i_d is a _s_e_l_f_ _i_n_i_t_i_a_l_i_z_i_n_g_ _t_y_p_e, with ancestor classes _L_i_s_t <
│ │ │ │  _V_i_s_i_b_l_e_L_i_s_t < _B_a_s_i_c_L_i_s_t < _T_h_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -136,25 +136,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000442257s elapsed
│ │ │ + -- .000568649s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0121996s elapsed
│ │ │ + -- .0158824s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,21 +62,21 @@
│ │ │ │  i5 : I = intersect(ideal(a^2,b^2,c), ideal(a,b^3,c^2))
│ │ │ │  
│ │ │ │               2        2   3     2
│ │ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ │ - -- .000442257s elapsed
│ │ │ │ + -- .000568649s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0121996s elapsed
│ │ │ │ + -- .0158824s elapsed
│ │ │ │  
│ │ │ │  o7 = ideal (c, b, a)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  For another example, see _P_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n.
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  Eisenbud, Huneke, Vasconcelos, Invent. Math. 110 207-235 (1992).
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html
│ │ │ @@ -130,31 +130,31 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .151496s elapsed
│ │ │ + -- .0916603s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00157389s elapsed
│ │ │ + -- .00202296s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00123922s elapsed
│ │ │ + -- .00155956s elapsed
│ │ │  
│ │ │  o9 = false
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,23 +50,23 @@
│ │ │ │  i5 : D = product(I_*/degree/sum)
│ │ │ │  
│ │ │ │  o5 = 840
│ │ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ │ - -- .151496s elapsed
│ │ │ │ + -- .0916603s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00157389s elapsed
│ │ │ │ + -- .00202296s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .00123922s elapsed
│ │ │ │ + -- .00155956s elapsed
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_d_i_c_a_l -- the radical of an ideal
│ │ │ │  ********** WWaayyss ttoo uussee rraaddiiccaallCCoonnttaaiinnmmeenntt:: **********
│ │ │ │      * radicalContainment(Ideal,Ideal)
│ │ │ │      * radicalContainment(RingElement,Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/Miura/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  c2NhbGFyTXVsdGlwbGljYXRpb24=
│ │ │  #:len=1318
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQWRkIFJlZHVjZWQgSWRlYWwgTXVsdGlw
│ │ │  bGUgVGltZXMiLCAibGluZW51bSIgPT4gMjAwLCBJbnB1dHMgPT4ge1NQQU57VFR7IkoifSwiLCAi
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bXVsdGlSZWVzSWRlYWwoSWRlYWwp
│ │ │  #:len=280
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjA3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0aVJlZXNJZGVhbCxJZGVhbCksIm11bHRpUmVl
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out
│ │ │ @@ -57,29 +57,29 @@
│ │ │  i9 : J = ideal vars U
│ │ │  
│ │ │  o9 = ideal (a, b, c)
│ │ │  
│ │ │  o9 : Ideal of U
│ │ │  
│ │ │  i10 : time multiReesIdeal J
│ │ │ - -- used 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ + -- used 0.36721s (cpu); 0.151515s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │  
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
│ │ │  
│ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ + -- used 0.0773044s (cpu); 0.0241347s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o9 : Ideal of U
│ │ │              
    
│ │ │                
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ + -- used 0.36721s (cpu); 0.151515s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ @@ -195,15 +195,15 @@
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    
│ │ │              
    
│ │ │                
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ + -- used 0.0773044s (cpu); 0.0241347s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,28 +79,28 @@
│ │ │ │  i8 : U = T/minors(2,m);
│ │ │ │  i9 : J = ideal vars U
│ │ │ │  
│ │ │ │  o9 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o9 : Ideal of U
│ │ │ │  i10 : time multiReesIdeal J
│ │ │ │ - -- used 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ │ + -- used 0.36721s (cpu); 0.151515s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │ │  
│ │ │ │  o10 : Ideal of U[X ..X ]
│ │ │ │                    0   2
│ │ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ │ - -- used 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ │ + -- used 0.0773044s (cpu); 0.0241347s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  ZGVmb3JtTUNNTW9kdWxl
│ │ │  #:len=2237
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmVyc2FsIGRlZm9ybWF0aW9uIG9mIE1D
│ │ │  TS1tb2R1bGUgb24gaHlwZXJzdXJmYWNlIiwgRGVzY3JpcHRpb24gPT4gKCJUaGlzIGlzIHRoZSBt
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  
│ │ │  o7 = image | x2 y2 |
│ │ │  
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
│ │ │  
│ │ │  i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ + -- used 0.892138s (cpu); 0.483869s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -70,15 +70,15 @@
│ │ │  o10 = cokernel | x2 y2  |
│ │ │                 | -y -x2 |
│ │ │  
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
│ │ │  
│ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ + -- used 0.895496s (cpu); 0.63875s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    
│ │ │              
    
│ │ │                
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ + -- used 0.892138s (cpu); 0.483869s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -191,15 +191,15 @@
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    
│ │ │              
    
│ │ │                
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ + -- used 0.895496s (cpu); 0.63875s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ │ ├── html2text {}
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │  i7 : N0 = module ideal (x^2,y^2)
│ │ │ │  
│ │ │ │  o7 = image | x2 y2 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o7 : R-module, submodule of R
│ │ │ │  i8 : (S,N) = time deformMCMModule N0
│ │ │ │ - -- used 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ │ + -- used 0.892138s (cpu); 0.483869s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │ │                    {8} | xxi_4-y+xi_3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │ │       -x2-xxi_2-xi_1                                           |
│ │ │ │  
│ │ │ │ @@ -103,15 +103,15 @@
│ │ │ │  
│ │ │ │  o10 = cokernel | x2 y2  |
│ │ │ │                 | -y -x2 |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o10 : R-module, quotient of R
│ │ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ │ - -- used 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ │ + -- used 0.895496s (cpu); 0.63875s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │ │                      | xxi_4-y+xi_3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  c3BhcnNlTW9ub2Ryb215U29sdmUoLi4uLE51bWJlck9mUmVwZWF0cz0+Li4uKQ==
│ │ │  #:len=345
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDAwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzcGFyc2VNb25vZHJvbXlTb2x2ZSxOdW1iZXJPZlJl
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out
│ │ │ @@ -3,41 +3,41 @@
│ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │  
│ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │  
│ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │  
│ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ ---backup directory created: /tmp/M2-25512-0/2
│ │ │ - -- .00616338s elapsed
│ │ │ +--backup directory created: /tmp/M2-33165-0/2
│ │ │ + -- .0042495s elapsed
│ │ │    H01: 1
│ │ │ - -- .00900117s elapsed
│ │ │ + -- .00425265s elapsed
│ │ │    H10: 1
│ │ │ - -- .000649635s elapsed
│ │ │ + -- .000841071s elapsed
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00324889s elapsed
│ │ │ + -- .00421182s elapsed
│ │ │    H01: 1
│ │ │ - -- .00435076s elapsed
│ │ │ + -- .00499457s elapsed
│ │ │    H10: 1
│ │ │ - -- .000544598s elapsed
│ │ │ + -- .000718755s elapsed
│ │ │  number of paths tracked: 4
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00428387s elapsed
│ │ │ + -- .00396059s elapsed
│ │ │    H01: 1
│ │ │ - -- .00298132s elapsed
│ │ │ + -- .0044697s elapsed
│ │ │    H10: 1
│ │ │ - -- .000466532s elapsed
│ │ │ + -- .000681483s elapsed
│ │ │  number of paths tracked: 6
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .0044506s elapsed
│ │ │ + -- .00462941s elapsed
│ │ │    H01: 1
│ │ │ - -- .00309306s elapsed
│ │ │ + -- .00409509s elapsed
│ │ │    H10: 1
│ │ │ - -- .000457295s elapsed
│ │ │ + -- .000827698s elapsed
│ │ │  number of paths tracked: 8
│ │ │  found 1 points in the fiber so far
│ │ │  
│ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │  
│ │ │  o4 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_get__Track__Time.out
│ │ │ @@ -6,16 +6,16 @@
│ │ │  i2 : declareVariable \ {A,B,C,D,X,Y};
│ │ │  
│ │ │  i3 : F = gateSystem(matrix{{A,B,C,D}},matrix{{X,Y}},matrix{{A*(X-1)^2-B}, {C*(Y+2)^2+D}});
│ │ │  
│ │ │  i4 : (p0,x0) = createSeedPair F;
│ │ │  
│ │ │  i5 : time (V,npaths) = monodromySolve(F,p0,{x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false);
│ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc)
│ │ │ + -- used 0.0824846s (cpu); 0.0824756s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : getTrackTime V.Graph
│ │ │  
│ │ │ -o6 = .006001457999999999
│ │ │ +o6 = .01447032
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -101,41 +101,41 @@
│ │ │              
    		
│ │ │                
    i3 : (p0, x0) = createSeedPair polys;
    
│ │ │              
    
│ │ │                
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ ---backup directory created: /tmp/M2-25512-0/2
│ │ │ - -- .00616338s elapsed
│ │ │ +--backup directory created: /tmp/M2-33165-0/2
│ │ │ + -- .0042495s elapsed
│ │ │    H01: 1
│ │ │ - -- .00900117s elapsed
│ │ │ + -- .00425265s elapsed
│ │ │    H10: 1
│ │ │ - -- .000649635s elapsed
│ │ │ + -- .000841071s elapsed
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00324889s elapsed
│ │ │ + -- .00421182s elapsed
│ │ │    H01: 1
│ │ │ - -- .00435076s elapsed
│ │ │ + -- .00499457s elapsed
│ │ │    H10: 1
│ │ │ - -- .000544598s elapsed
│ │ │ + -- .000718755s elapsed
│ │ │  number of paths tracked: 4
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00428387s elapsed
│ │ │ + -- .00396059s elapsed
│ │ │    H01: 1
│ │ │ - -- .00298132s elapsed
│ │ │ + -- .0044697s elapsed
│ │ │    H10: 1
│ │ │ - -- .000466532s elapsed
│ │ │ + -- .000681483s elapsed
│ │ │  number of paths tracked: 6
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .0044506s elapsed
│ │ │ + -- .00462941s elapsed
│ │ │    H01: 1
│ │ │ - -- .00309306s elapsed
│ │ │ + -- .00409509s elapsed
│ │ │    H10: 1
│ │ │ - -- .000457295s elapsed
│ │ │ + -- .000827698s elapsed
│ │ │  number of paths tracked: 8
│ │ │  found 1 points in the fiber so far
│ │ │  
│ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │                  		
│ │ │                
    i4 : (p0,x0) = createSeedPair F;
    
│ │ │              
    
│ │ │                
    i5 : time (V,npaths) = monodromySolve(F,p0,{x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false);
│ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc)
    
│ │ │ + -- used 0.0824846s (cpu); 0.0824756s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : getTrackTime V.Graph
│ │ │  
│ │ │ -o6 = .006001457999999999
│ │ │ +o6 = .01447032
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    
│ │ │            
    time measures the elapsed time for the whole call, including graph construction, and other overhead. getTrackTime reports only the cumulative time spent tracking paths along edges of the homotopy graph.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,18 +16,18 @@
│ │ │ │   -- setting random seed to 0
│ │ │ │  i2 : declareVariable \ {A,B,C,D,X,Y};
│ │ │ │  i3 : F = gateSystem(matrix{{A,B,C,D}},matrix{{X,Y}},matrix{{A*(X-1)^2-B}, {C*
│ │ │ │  (Y+2)^2+D}});
│ │ │ │  i4 : (p0,x0) = createSeedPair F;
│ │ │ │  i5 : time (V,npaths) = monodromySolve(F,p0,
│ │ │ │  {x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false);
│ │ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc)
│ │ │ │ + -- used 0.0824846s (cpu); 0.0824756s (thread); 0s (gc)
│ │ │ │  i6 : getTrackTime V.Graph
│ │ │ │  
│ │ │ │ -o6 = .006001457999999999
│ │ │ │ +o6 = .01447032
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  time measures the elapsed time for the whole call, including graph
│ │ │ │  construction, and other overhead. getTrackTime reports only the cumulative time
│ │ │ │  spent tracking paths along edges of the homotopy graph.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/MonomialAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  ZGVjb21wb3NlSG9tb2dlbmVvdXNNQShMaXN0KQ==
│ │ │  #:len=308
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY1Nywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZGVjb21wb3NlSG9tb2dlbmVvdXNNQSxMaXN0KSwi
│ │ ├── ./usr/share/doc/Macaulay2/MonomialIntegerPrograms/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  R3JhZGVkQmV0dGlz
│ │ │  #:len=431
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiB7fSwgImxpbmVudW0iID0+IDExNDYs
│ │ │  IHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkdyYWRlZEJldHRp
│ │ ├── ./usr/share/doc/Macaulay2/MonomialOrbits/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  aGlsYmVydFJlcHJlc2VudGF0aXZlcw==
│ │ │  #:len=3397
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZCByZXByZXNlbnRhdGl2ZXMgb2Yg
│ │ │  bW9ub21pYWwgaWRlYWxzIHVuZGVyIHBlcm11dGF0aW9ucyBvZiB0aGUgdmFyaWFibGVzIiwgImxp
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  bXNvbHZlUlVSKElkZWFsKQ==
│ │ │  #:len=227
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQ0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtc29sdmVSVVIsSWRlYWwpLCJtc29sdmVSVVIoSWRl
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out
│ │ │ @@ -9,16 +9,16 @@
│ │ │  i2 : I = ideal(x, y, z)
│ │ │  
│ │ │  o2 = ideal (x, y, z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/M2-38527-0/0-out.ms
│ │ │ -Initial seed for pseudo-random number generator is 1792814436
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54126-0/0-in.ms -o /tmp/M2-54126-0/0-out.ms
│ │ │ +Initial seed for pseudo-random number generator is 1793013995
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -29,15 +29,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1099986659
│ │ │ +Initial prime = 1102609337
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -54,18 +54,18 @@
│ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │  overall(elapsed)        0.00 sec
│ │ │  overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.00 sec  96.6%
│ │ │ -convert                 0.00 sec   0.6%
│ │ │ -linear algebra          0.00 sec   0.2%
│ │ │ +symbolic prep.          0.00 sec   0.4%
│ │ │ +update                  0.00 sec  71.6%
│ │ │ +convert                 0.00 sec   2.5%
│ │ │ +linear algebra          0.00 sec   1.5%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -79,18 +79,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1120193111
│ │ │ +New prime = 1300419397
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -106,15 +106,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.08 sec (elapsed) /  0.27 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/index.html
│ │ │ @@ -83,16 +83,16 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/M2-38527-0/0-out.ms
│ │ │ -Initial seed for pseudo-random number generator is 1792814436
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54126-0/0-in.ms -o /tmp/M2-54126-0/0-out.ms
│ │ │ +Initial seed for pseudo-random number generator is 1793013995
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -103,15 +103,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1099986659
│ │ │ +Initial prime = 1102609337
│ │ │  
│ │ │  Legend for f4 information
│ │ │  --------------------------------------------------------
│ │ │  deg       current degree of pairs selected in this round
│ │ │  sel       number of pairs selected in this round
│ │ │  pairs     total number of pairs in pair list
│ │ │  mat       matrix dimensions (# rows x # columns)
│ │ │ @@ -128,18 +128,18 @@
│ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │  overall(elapsed)        0.00 sec
│ │ │  overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.00 sec  96.6%
│ │ │ -convert                 0.00 sec   0.6%
│ │ │ -linear algebra          0.00 sec   0.2%
│ │ │ +symbolic prep.          0.00 sec   0.4%
│ │ │ +update                  0.00 sec  71.6%
│ │ │ +convert                 0.00 sec   2.5%
│ │ │ +linear algebra          0.00 sec   1.5%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -153,18 +153,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1120193111
│ │ │ +New prime = 1300419397
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ +multi-mod overall(elapsed)      0.00 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -180,15 +180,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.08 sec (elapsed) /  0.27 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.04 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00225028s elapsed
│ │ │ + -- .00259946s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00849938s elapsed
│ │ │ + -- .011003s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,19 +51,19 @@
│ │ │ │  o4 : RingMap S <-- R
│ │ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 6
│ │ │ │  number of distinct multidegrees = 6
│ │ │ │ - -- .00225028s elapsed
│ │ │ │ + -- .00259946s elapsed
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ │ │ - -- .00849938s elapsed
│ │ │ │ + -- .011003s elapsed
│ │ │ │  
│ │ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  Z3JHcihJZGVhbCk=
│ │ │  #:len=249
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzc4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnckdyLElkZWFsKSwiZ3JHcihJZGVhbCkiLCJNdWx0
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │  
│ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime jMult I
│ │ │ - -- .0242277s elapsed
│ │ │ + -- .0413968s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .159695s elapsed
│ │ │ + -- .0901594s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .270321s elapsed
│ │ │ + -- .164849s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out
│ │ │ @@ -10,12 +10,12 @@
│ │ │  
│ │ │               2        3
│ │ │  o2 = ideal (x , x*y, y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .0952293s elapsed
│ │ │ + -- .0894602s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out
│ │ │ @@ -13,17 +13,17 @@
│ │ │       ------------------------------------------------------------------------
│ │ │        10 11   8 12   9 11   10 10   11 9   12 8
│ │ │       x  y  , x y  , x y  , x  y  , x  y , x  y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monjMult I
│ │ │ - -- .189553s elapsed
│ │ │ + -- .124999s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.87284s elapsed
│ │ │ + -- 1.61816s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -93,31 +93,31 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime jMult I
│ │ │ - -- .0242277s elapsed
│ │ │ + -- .0413968s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime monjMult I
│ │ │ - -- .159695s elapsed
│ │ │ + -- .0901594s elapsed
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .270321s elapsed
│ │ │ + -- .164849s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .0952293s elapsed
│ │ │ + -- .0894602s elapsed
│ │ │  
│ │ │  o3 = 2
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : I = ideal"x2,xy,y3"
│ │ │ │  
│ │ │ │               2        3
│ │ │ │  o2 = ideal (x , x*y, y )
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ │ - -- .0952293s elapsed
│ │ │ │ + -- .0894602s elapsed
│ │ │ │  
│ │ │ │  o3 = 2
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _N_P -- the Newton polyhedron of a monomial ideal
│ │ │ │  ********** WWaayyss ttoo uussee mmoonnAAnnaallyyttiiccSSpprreeaadd:: **********
│ │ │ │      * monAnalyticSpread(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html
│ │ │ @@ -97,23 +97,23 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime monjMult I
│ │ │ - -- .189553s elapsed
│ │ │ + -- .124999s elapsed
│ │ │  
│ │ │  o3 = 192
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime jMult I
│ │ │ - -- 1.87284s elapsed
│ │ │ + -- 1.61816s elapsed
│ │ │  
│ │ │  o4 = 192
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  o2 = ideal (x y  , x y  , x y  , x y  , x y  , x y  , x y  , x y  , x y  ,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        10 11   8 12   9 11   10 10   11 9   12 8
│ │ │ │       x  y  , x y  , x y  , x  y  , x  y , x  y )
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime monjMult I
│ │ │ │ - -- .189553s elapsed
│ │ │ │ + -- .124999s elapsed
│ │ │ │  
│ │ │ │  o3 = 192
│ │ │ │  i4 : elapsedTime jMult I
│ │ │ │ - -- 1.87284s elapsed
│ │ │ │ + -- 1.61816s elapsed
│ │ │ │  
│ │ │ │  o4 = 192
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal
│ │ │ │      * _j_M_u_l_t -- the j-multiplicity of an ideal
│ │ │ │      * _m_o_n_R_e_d_u_c_t_i_o_n -- the minimal monomial reduction of a monomial ideal
│ │ │ │      * _N_P -- the Newton polyhedron of a monomial ideal
│ │ ├── ./usr/share/doc/Macaulay2/MultiplierIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  bG9nQ2Fub25pY2FsVGhyZXNob2xkKENlbnRyYWxBcnJhbmdlbWVudCk=
│ │ │  #:len=361
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3MCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobG9nQ2Fub25pY2FsVGhyZXNob2xkLENlbnRyYWxB
│ │ ├── ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  TXVsdElkZWFs
│ │ │  #:len=1363
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIG11bHRpcGxpZXIg
│ │ │  aWRlYWwgb2YgYSBnaXZlbiBudW1iZXIuIiwgImxpbmVudW0iID0+IDY0OCwgSW5wdXRzID0+IHtT
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  dGV4TWF0aChSQVQp
│ │ │  #:len=209
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDE5OCwgInVuZG9jdW1lbnRlZCIgPT4g
│ │ │  dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4TWF0
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │  o4 : ProjectiveVariety, curve in PP^8
│ │ │  
│ │ │  i5 : ? X
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 
│ │ │  
│ │ │  i6 : time f = X ===> Y;
│ │ │ - -- used 3.4873s (cpu); 1.91387s (thread); 0s (gc)
│ │ │ + -- used 4.27157s (cpu); 2.31752s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
│ │ │  
│ │ │  i7 : f X
│ │ │  
│ │ │  o7 = Y
│ │ │  
│ │ │ @@ -38,15 +38,15 @@
│ │ │  o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │  
│ │ │  i10 : W = random({{2},{1}},Y);
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │  
│ │ │  i11 : time g = V ===> W;
│ │ │ - -- used 3.66337s (cpu); 2.10606s (thread); 0s (gc)
│ │ │ + -- used 4.86775s (cpu); 2.64097s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
│ │ │  
│ │ │  i12 : g||W
│ │ │  
│ │ │  o12 = multi-rational map consisting of one single rational map
│ │ │        source variety: 6-dimensional subvariety of PP^8 cut out by 2 hypersurfaces of degrees 1^1 2^1 
│ │ │ @@ -129,15 +129,15 @@
│ │ │  o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9
│ │ │  
│ │ │  i16 : ? Z
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │  
│ │ │  i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 8.41326s (cpu); 5.16056s (thread); 0s (gc)
│ │ │ + -- used 7.81943s (cpu); 5.38503s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
│ │ │  
│ │ │  i18 : h || GG_K(1,4)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random({1,1},ring target Phi));
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │  
│ │ │  i4 : time X = Phi^* Y;
│ │ │ - -- used 5.22987s (cpu); 3.71826s (thread); 0s (gc)
│ │ │ + -- used 4.9519s (cpu); 4.02468s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │  
│ │ │  i5 : dim X, degree X, degrees X
│ │ │  
│ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out
│ │ │ @@ -11,26 +11,26 @@
│ │ │  o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^7 x PP^7)
│ │ │  
│ │ │  i4 : Z = source Phi;
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
│ │ │  
│ │ │  i5 : time Phi Z;
│ │ │ - -- used 0.20103s (cpu); 0.125952s (thread); 0s (gc)
│ │ │ + -- used 0.287633s (cpu); 0.160083s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
│ │ │  
│ │ │  i6 : dim oo, degree oo, degrees oo
│ │ │  
│ │ │  o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 2.02267s (cpu); 1.47252s (thread); 0s (gc)
│ │ │ + -- used 2.51512s (cpu); 1.64975s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
│ │ │  
│ │ │  i8 : dim oo, degree oo, degrees oo
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out
│ │ │ @@ -11,22 +11,22 @@
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of
│ │ │       target variety: hypersurface in PP^4 defined by a form of degree 2
│ │ │       ------------------------------------------------------------------------
│ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │  
│ │ │  i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 4.01388s (cpu); 2.8819s (thread); 0s (gc)
│ │ │ + -- used 4.61632s (cpu); 2.8649s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc)
│ │ │ + -- used 0.544741s (cpu); 0.367379s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time degree Phi
│ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc)
│ │ │ + -- used 0.248759s (cpu); 0.249351s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out
│ │ │ @@ -3,12 +3,12 @@
│ │ │  i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2};
│ │ │  
│ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : time degree Phi
│ │ │ - -- used 0.43902s (cpu); 0.366735s (thread); 0s (gc)
│ │ │ + -- used 0.506866s (cpu); 0.406099s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out
│ │ │ @@ -1,52 +1,52 @@
│ │ │  -- -*- M2-comint -*- hash: 11533721324852072161
│ │ │  
│ │ │  i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5)
│ │ │  
│ │ │  i2 : time ? Phi
│ │ │ - -- used 0.00332575s (cpu); 0.000166912s (thread); 0s (gc)
│ │ │ + -- used 0.00163981s (cpu); 0.000165572s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │  
│ │ │  i3 : image Phi;
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │  
│ │ │  i4 : time ? Phi
│ │ │ - -- used 0.00316389s (cpu); 0.000237795s (thread); 0s (gc)
│ │ │ + -- used 0.000270867s (cpu); 0.000272717s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │  
│ │ │  i5 : time describe Phi
│ │ │ - -- used 1.3359s (cpu); 1.00639s (thread); 0s (gc)
│ │ │ + -- used 1.32481s (cpu); 1.06433s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       multidegree: {51, 51, 51, 51, 51}
│ │ │       degree: 1
│ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
│ │ │  
│ │ │  i6 : time ? Phi
│ │ │ - -- used 0.000157475s (cpu); 0.000414858s (thread); 0s (gc)
│ │ │ + -- used 0.000159938s (cpu); 0.000514033s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out
│ │ │ @@ -3,45 +3,45 @@
│ │ │  i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)
│ │ │  
│ │ │  o1 = Phi
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0281486s (cpu); 0.0267017s (thread); 0s (gc)
│ │ │ + -- used 0.0404278s (cpu); 0.0280394s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : Phi1;
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4)
│ │ │  
│ │ │  i4 : Phi2;
│ │ │  
│ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.134338s (cpu); 0.060805s (thread); 0s (gc)
│ │ │ + -- used 0.172221s (cpu); 0.0723836s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
│ │ │  
│ │ │  i6 : Phi21;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i7 : Phi22;
│ │ │  
│ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.124474s (cpu); 0.123543s (thread); 0s (gc)
│ │ │ + -- used 0.160714s (cpu); 0.143483s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : Phi211;
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out
│ │ │ @@ -11,25 +11,25 @@
│ │ │  o3 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i4 : Phi = rationalMap {f,g};
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │  
│ │ │  i5 : time Z = image Phi;
│ │ │ - -- used 0.103109s (cpu); 0.102376s (thread); 0s (gc)
│ │ │ + -- used 0.148354s (cpu); 0.134531s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │  
│ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │  
│ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 5.47152s (cpu); 2.82366s (thread); 0s (gc)
│ │ │ + -- used 10.5565s (cpu); 3.0039s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │  
│ │ │  i8 : assert(Z == Z')
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out
│ │ │ @@ -4,25 +4,25 @@
│ │ │  
│ │ │  i2 : -- map defined by the cubics through the secant variety to the rational normal curve of degree 6
│ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal PP_K([6],2));
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │  
│ │ │  i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.418413s (cpu); 0.331336s (thread); 0s (gc)
│ │ │ + -- used 0.48341s (cpu); 0.372566s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
│ │ │  
│ │ │  i4 : assert(Phi * Psi == 1)
│ │ │  
│ │ │  i5 : Phi' = Phi || Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │  
│ │ │  i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.45295s (cpu); 1.03172s (thread); 0s (gc)
│ │ │ + -- used 1.59155s (cpu); 1.29345s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │  
│ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out
│ │ │ @@ -7,33 +7,33 @@
│ │ │  
│ │ │  i2 : -- we see Phi as a dominant map
│ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time inverse Phi;
│ │ │ - -- used 0.157948s (cpu); 0.0767566s (thread); 0s (gc)
│ │ │ + -- used 0.205323s (cpu); 0.100852s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
│ │ │  
│ │ │  i4 : Psi = last graph Phi;
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
│ │ │  
│ │ │  i5 : time inverse Psi;
│ │ │ - -- used 0.186263s (cpu); 0.107471s (thread); 0s (gc)
│ │ │ + -- used 0.307174s (cpu); 0.14909s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i6 : Eta = first graph Psi;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │  
│ │ │  i7 : time inverse Eta;
│ │ │ - -- used 0.48562s (cpu); 0.313057s (thread); 0s (gc)
│ │ │ + -- used 0.706961s (cpu); 0.401679s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │  
│ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │  
│ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out
│ │ │ @@ -6,32 +6,32 @@
│ │ │  o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
│ │ │  
│ │ │  i3 : Phi = rationalMap {f,f};
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │  
│ │ │  i4 : time isIsomorphism Phi
│ │ │ - -- used 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00209948s (cpu); 1.1327e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
│ │ │  
│ │ │  i6 : time isIsomorphism Psi
│ │ │ - -- used 0.396692s (cpu); 0.213364s (thread); 0s (gc)
│ │ │ + -- used 0.72621s (cpu); 0.312779s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : Eta = first graph Psi;
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2)
│ │ │  
│ │ │  i8 : time isIsomorphism Eta
│ │ │ - -- used 1.74335s (cpu); 0.941708s (thread); 0s (gc)
│ │ │ + -- used 2.19717s (cpu); 1.13093s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out
│ │ │ @@ -3,24 +3,24 @@
│ │ │  i1 : ZZ/300007[a..e], f = first graph rationalMap ideal(c^2-b*d,b*c-a*d,b^2-a*c,e), g = rationalMap submatrix(matrix f,{0..2});
│ │ │  
│ │ │  i2 : Phi = rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │  
│ │ │  i3 : time isMorphism Phi
│ │ │ - -- used 0.386427s (cpu); 0.269828s (thread); 0s (gc)
│ │ │ + -- used 0.46239s (cpu); 0.294334s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc)
│ │ │ + -- used 0.279277s (cpu); 0.123968s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
│ │ │  
│ │ │  i5 : time isMorphism Psi
│ │ │ - -- used 4.42649s (cpu); 3.16319s (thread); 0s (gc)
│ │ │ + -- used 4.59518s (cpu); 3.83013s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : assert((not o3) and o5)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linearly__Normal__Embedding.out
│ │ │ @@ -3,24 +3,24 @@
│ │ │  i1 : K = ZZ/333331;
│ │ │  
│ │ │  i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │  
│ │ │  i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.00800583s (cpu); 0.00919368s (thread); 0s (gc)
│ │ │ + -- used 0.0120029s (cpu); 0.0124938s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (automorphism of X)
│ │ │  
│ │ │  i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3
│ │ │  
│ │ │  i5 : time g = linearlyNormalEmbedding Y;
│ │ │ - -- used 0.709324s (cpu); 0.523301s (thread); 0s (gc)
│ │ │ + -- used 0.630571s (cpu); 0.514714s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │  
│ │ │  i6 : assert(isIsomorphism g)
│ │ │  
│ │ │  i7 : describe g
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2};
│ │ │  
│ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │  
│ │ │  i3 : time multidegree Phi
│ │ │ - -- used 0.453684s (cpu); 0.376266s (thread); 0s (gc)
│ │ │ + -- used 0.541031s (cpu); 0.431948s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (degree source Phi,degree image Phi)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 16199733219210081214
│ │ │  
│ │ │  i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4);
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │  
│ │ │  i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00102199s (cpu); 0.00133739s (thread); 0s (gc)
│ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc)
│ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc)
│ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc)
│ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc)
│ │ │ + -- used 0.00190693s (cpu); 0.00168437s (thread); 0s (gc)
│ │ │ + -- used 0.291065s (cpu); 0.197186s (thread); 0s (gc)
│ │ │ + -- used 0.43834s (cpu); 0.254514s (thread); 0s (gc)
│ │ │ + -- used 0.277805s (cpu); 0.186268s (thread); 0s (gc)
│ │ │ + -- used 0.253413s (cpu); 0.157809s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc)
│ │ │ + -- used 0.279069s (cpu); 0.114092s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out
│ │ │ @@ -3,26 +3,26 @@
│ │ │  i1 : K = ZZ/1000003;
│ │ │  
│ │ │  i2 : X = PP_K^({1,1,2},{3,2,3});
│ │ │  
│ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i3 : time p := point X
│ │ │ - -- used 0.0200872s (cpu); 0.0193746s (thread); 0s (gc)
│ │ │ + -- used 0.0380632s (cpu); 0.0265688s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i4 : Y = random({2,1,2},X);
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i5 : time q = point Y
│ │ │ - -- used 2.20856s (cpu); 1.14282s (thread); 0s (gc)
│ │ │ + -- used 1.92256s (cpu); 1.20772s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │  
│ │ │  i6 : assert(isSubset(p,X) and isSubset(q,Y))
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  o4 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i5 : Phi = rationalMap {f,g,h};
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4)
│ │ │  
│ │ │  i6 : time segre Phi;
│ │ │ - -- used 0.774927s (cpu); 0.592841s (thread); 0s (gc)
│ │ │ + -- used 1.96491s (cpu); 0.848059s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │  
│ │ │  i7 : describe segre Phi
│ │ │  
│ │ │  o7 = rational map defined by forms of degree 6
│ │ │       source variety: PP^4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3)
│ │ │  
│ │ │  o1 = Phi
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
│ │ │  
│ │ │  i2 : time describe Phi
│ │ │ - -- used 0.410659s (cpu); 0.270055s (thread); 0s (gc)
│ │ │ + -- used 0.379541s (cpu); 0.190044s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time f = X ===> Y;
│ │ │ - -- used 3.4873s (cpu); 1.91387s (thread); 0s (gc)
│ │ │ + -- used 4.27157s (cpu); 2.31752s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : f X
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time g = V ===> W;
│ │ │ - -- used 3.66337s (cpu); 2.10606s (thread); 0s (gc)
│ │ │ + -- used 4.86775s (cpu); 2.64097s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : g||W
│ │ │ @@ -257,15 +257,15 @@
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 8.41326s (cpu); 5.16056s (thread); 0s (gc)
│ │ │ + -- used 7.81943s (cpu); 5.38503s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  take(N,-2));
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^8
│ │ │ │  i5 : ? X
│ │ │ │  
│ │ │ │  o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15
│ │ │ │  i6 : time f = X ===> Y;
│ │ │ │ - -- used 3.4873s (cpu); 1.91387s (thread); 0s (gc)
│ │ │ │ + -- used 4.27157s (cpu); 2.31752s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (automorphism of PP^8)
│ │ │ │  i7 : f X
│ │ │ │  
│ │ │ │  o7 = Y
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, curve in PP^8
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  i9 : V = random({{2},{1}},X);
│ │ │ │  
│ │ │ │  o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │ │  i10 : W = random({{2},{1}},Y);
│ │ │ │  
│ │ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
│ │ │ │  i11 : time g = V ===> W;
│ │ │ │ - -- used 3.66337s (cpu); 2.10606s (thread); 0s (gc)
│ │ │ │ + -- used 4.86775s (cpu); 2.64097s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 : MultirationalMap (automorphism of PP^8)
│ │ │ │  i12 : g||W
│ │ │ │  
│ │ │ │  o12 = multi-rational map consisting of one single rational map
│ │ │ │        source variety: 6-dimensional subvariety of PP^8 cut out by 2
│ │ │ │  hypersurfaces of degrees 1^1 2^1
│ │ │ │ @@ -144,15 +144,15 @@
│ │ │ │  i15 : Z = projectiveVariety pfaffians(4,A);
│ │ │ │  
│ │ │ │  o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9
│ │ │ │  i16 : ? Z
│ │ │ │  
│ │ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │ │  i17 : time h = Z ===> GG_K(1,4)
│ │ │ │ - -- used 8.41326s (cpu); 5.16056s (thread); 0s (gc)
│ │ │ │ + -- used 7.81943s (cpu); 5.38503s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = h
│ │ │ │  
│ │ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
│ │ │ │  i18 : h || GG_K(1,4)
│ │ │ │  
│ │ │ │  o18 = multi-rational map consisting of one single rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time X = Phi^* Y;
│ │ │ - -- used 5.22987s (cpu); 3.71826s (thread); 0s (gc)
│ │ │ + -- used 4.9519s (cpu); 4.02468s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : dim X, degree X, degrees X
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random(
│ │ │ │  {1,1},ring target Phi));
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │ │  i4 : time X = Phi^* Y;
│ │ │ │ - -- used 5.22987s (cpu); 3.71826s (thread); 0s (gc)
│ │ │ │ + -- used 4.9519s (cpu); 4.02468s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension
│ │ │ │  2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-
│ │ │ │  degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │ │  i5 : dim X, degree X, degrees X
│ │ │ │  
│ │ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time Phi Z;
│ │ │ - -- used 0.20103s (cpu); 0.125952s (thread); 0s (gc)
│ │ │ + -- used 0.287633s (cpu); 0.160083s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : dim oo, degree oo, degrees oo
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o6 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 2.02267s (cpu); 1.47252s (thread); 0s (gc)
│ │ │ + -- used 2.51512s (cpu); 1.64975s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,24 +25,24 @@
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 to PP^7 x PP^7)
│ │ │ │  i4 : Z = source Phi;
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
│ │ │ │  i5 : time Phi Z;
│ │ │ │ - -- used 0.20103s (cpu); 0.125952s (thread); 0s (gc)
│ │ │ │ + -- used 0.287633s (cpu); 0.160083s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
│ │ │ │  i6 : dim oo, degree oo, degrees oo
│ │ │ │  
│ │ │ │  o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  i7 : time Phi (point Z + point Z + point Z)
│ │ │ │ - -- used 2.02267s (cpu); 1.47252s (thread); 0s (gc)
│ │ │ │ + -- used 2.51512s (cpu); 1.64975s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of
│ │ │ │  multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
│ │ │ │  i8 : dim oo, degree oo, degrees oo
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_cm__Option_rp.html
│ │ │ @@ -98,31 +98,31 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 4.01388s (cpu); 2.8819s (thread); 0s (gc)
│ │ │ + -- used 4.61632s (cpu); 2.8649s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc)
│ │ │ + -- used 0.544741s (cpu); 0.367379s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time degree Phi
│ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc)
│ │ │ + -- used 0.248759s (cpu); 0.249351s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    Note, as in the example above, that calculation times may vary depending on the strategy used.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of
│ │ │ │       target variety: hypersurface in PP^4 defined by a form of degree 2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1
│ │ │ │  i4 : time degree(Phi,Strategy=>"random point")
│ │ │ │ - -- used 4.01388s (cpu); 2.8819s (thread); 0s (gc)
│ │ │ │ + -- used 4.61632s (cpu); 2.8649s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc)
│ │ │ │ + -- used 0.544741s (cpu); 0.367379s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time degree Phi
│ │ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc)
│ │ │ │ + -- used 0.248759s (cpu); 0.249351s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = 2
│ │ │ │  Note, as in the example above, that calculation times may vary depending on the
│ │ │ │  strategy used.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- degree of a multi-rational map
│ │ │ │      * _d_e_g_r_e_e_M_a_p_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time degree Phi
│ │ │ - -- used 0.43902s (cpu); 0.366735s (thread); 0s (gc)
│ │ │ + -- used 0.506866s (cpu); 0.406099s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3,
│ │ │ │  x_3^2};
│ │ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : time degree Phi
│ │ │ │ - -- used 0.43902s (cpu); 0.366735s (thread); 0s (gc)
│ │ │ │ + -- used 0.506866s (cpu); 0.406099s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = 1
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_,_O_p_t_i_o_n_) -- degree of a multi-rational map using a
│ │ │ │        probabilistic approach
│ │ │ │      * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ │ │      * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time ? Phi
│ │ │ - -- used 0.00332575s (cpu); 0.000166912s (thread); 0s (gc)
│ │ │ + -- used 0.00163981s (cpu); 0.000165572s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    
│ │ │              
│ │ │ @@ -101,27 +101,27 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time ? Phi
│ │ │ - -- used 0.00316389s (cpu); 0.000237795s (thread); 0s (gc)
│ │ │ + -- used 0.000270867s (cpu); 0.000272717s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time describe Phi
│ │ │ - -- used 1.3359s (cpu); 1.00639s (thread); 0s (gc)
│ │ │ + -- used 1.32481s (cpu); 1.06433s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │ @@ -131,15 +131,15 @@
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time ? Phi
│ │ │ - -- used 0.000157475s (cpu); 0.000414858s (thread); 0s (gc)
│ │ │ + -- used 0.000159938s (cpu); 0.000514033s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,36 +16,36 @@
│ │ │ │  ? Phi is a lite version of describe Phi. The latter has a different behavior
│ │ │ │  than _d_e_s_c_r_i_b_e_(_R_a_t_i_o_n_a_l_M_a_p_), since it performs computations.
│ │ │ │  i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^4 x PP^5)
│ │ │ │  i2 : time ? Phi
│ │ │ │ - -- used 0.00332575s (cpu); 0.000166912s (thread); 0s (gc)
│ │ │ │ + -- used 0.00163981s (cpu); 0.000165572s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i3 : image Phi;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │ │  i4 : time ? Phi
│ │ │ │ - -- used 0.00316389s (cpu); 0.000237795s (thread); 0s (gc)
│ │ │ │ + -- used 0.000270867s (cpu); 0.000272717s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i5 : time describe Phi
│ │ │ │ - -- used 1.3359s (cpu); 1.00639s (thread); 0s (gc)
│ │ │ │ + -- used 1.32481s (cpu); 1.06433s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       multidegree: {51, 51, 51, 51, 51}
│ │ │ │       degree: 1
│ │ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │ │       coefficient ring: ZZ/65521
│ │ │ │  i6 : time ? Phi
│ │ │ │ - -- used 0.000157475s (cpu); 0.000414858s (thread); 0s (gc)
│ │ │ │ + -- used 0.000159938s (cpu); 0.000514033s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0281486s (cpu); 0.0267017s (thread); 0s (gc)
│ │ │ + -- used 0.0404278s (cpu); 0.0280394s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.134338s (cpu); 0.060805s (thread); 0s (gc)
│ │ │ + -- used 0.172221s (cpu); 0.0723836s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.124474s (cpu); 0.123543s (thread); 0s (gc)
│ │ │ + -- used 0.160714s (cpu); 0.143483s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,43 +19,43 @@
│ │ │ │  Phi)^-1 * (last graph Phi) == Phi are always satisfied.
│ │ │ │  i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)
│ │ │ │  
│ │ │ │  o1 = Phi
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │ │  i2 : time (Phi1,Phi2) = graph Phi
│ │ │ │ - -- used 0.0281486s (cpu); 0.0267017s (thread); 0s (gc)
│ │ │ │ + -- used 0.0404278s (cpu); 0.0280394s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = (Phi1, Phi2)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : Phi1;
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^4)
│ │ │ │  i4 : Phi2;
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of
│ │ │ │  PP^4 x PP^5 to hypersurface in PP^5)
│ │ │ │  i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ │ - -- used 0.134338s (cpu); 0.060805s (thread); 0s (gc)
│ │ │ │ + -- used 0.172221s (cpu); 0.0723836s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = (Phi21, Phi22)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : Phi21;
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i7 : Phi22;
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of
│ │ │ │  PP^4 x PP^5 x PP^5 to hypersurface in PP^5)
│ │ │ │  i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ │ - -- used 0.124474s (cpu); 0.123543s (thread); 0s (gc)
│ │ │ │ + -- used 0.160714s (cpu); 0.143483s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = (Phi211, Phi212)
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : Phi211;
│ │ │ │  
│ │ │ │  o9 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time Z = image Phi;
│ │ │ - -- used 0.103109s (cpu); 0.102376s (thread); 0s (gc)
│ │ │ + -- used 0.148354s (cpu); 0.134531s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : dim Z, degree Z, degrees Z
│ │ │ @@ -120,15 +120,15 @@
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3,
│ │ │ │  x_3^2};
│ │ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : time multidegree Phi
│ │ │ │ - -- used 0.453684s (cpu); 0.376266s (thread); 0s (gc)
│ │ │ │ + -- used 0.541031s (cpu); 0.431948s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {66, 46, 31, 20}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (degree source Phi,degree image Phi)
│ │ │ │  
│ │ │ │  o4 = (66, 20)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html
│ │ │ @@ -82,29 +82,29 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 5.47152s (cpu); 2.82366s (thread); 0s (gc)
│ │ │ + -- used 10.5565s (cpu); 3.0039s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
    
│ │ │                
    i8 : assert(Z == Z')
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,26 +23,26 @@
│ │ │ │  3*x_2^2+2*x_1*x_3+x_0*x_4, 2*x_1*x_2-2*x_0*x_3, -x_1^2+x_0*x_2};
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i4 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │ │  i5 : time Z = image Phi;
│ │ │ │ - -- used 0.103109s (cpu); 0.102376s (thread); 0s (gc)
│ │ │ │ + -- used 0.148354s (cpu); 0.134531s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │ │  
│ │ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  Alternatively, the calculation can be performed using the Segre embedding as
│ │ │ │  follows:
│ │ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ │ - -- used 5.47152s (cpu); 2.82366s (thread); 0s (gc)
│ │ │ │ + -- used 10.5565s (cpu); 3.0039s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i8 : assert(Z == Z')
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y -- direct image via a multi-
│ │ │ │        rational map
│ │ │ │      * _i_m_a_g_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- closure of the image of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │              
    
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.418413s (cpu); 0.331336s (thread); 0s (gc)
│ │ │ + -- used 0.48341s (cpu); 0.372566s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │              
    
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │              
    
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.45295s (cpu); 1.03172s (thread); 0s (gc)
│ │ │ + -- used 1.59155s (cpu); 1.29345s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
    
│ │ │              
    
│ │ │                
    i7 : assert(Phi' * Psi' == 1)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,23 +24,23 @@
│ │ │ │  i2 : -- map defined by the cubics through the secant variety to the rational
│ │ │ │  normal curve of degree 6
│ │ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal
│ │ │ │  PP_K([6],2));
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │ │  i3 : time Psi = inverse2 Phi;
│ │ │ │ - -- used 0.418413s (cpu); 0.331336s (thread); 0s (gc)
│ │ │ │ + -- used 0.48341s (cpu); 0.372566s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
│ │ │ │  i4 : assert(Phi * Psi == 1)
│ │ │ │  i5 : Phi' = Phi || Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │ │  i6 : time Psi' = inverse2 Phi';
│ │ │ │ - -- used 1.45295s (cpu); 1.03172s (thread); 0s (gc)
│ │ │ │ + -- used 1.59155s (cpu); 1.29345s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _<_=_=_>_ _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p -- equality of multi-rational maps
│ │ │ │        with checks on internal data
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html
│ │ │ @@ -93,45 +93,45 @@
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.157948s (cpu); 0.0767566s (thread); 0s (gc)
│ │ │ + -- used 0.205323s (cpu); 0.100852s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
    
│ │ │              
    
│ │ │                
    i4 : Psi = last graph Phi;
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i5 : time inverse Psi;
│ │ │ - -- used 0.186263s (cpu); 0.107471s (thread); 0s (gc)
│ │ │ + -- used 0.307174s (cpu); 0.14909s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │              
    
│ │ │                
    i6 : Eta = first graph Psi;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │              
    
│ │ │                
    i7 : time inverse Eta;
│ │ │ - -- used 0.48562s (cpu); 0.313057s (thread); 0s (gc)
│ │ │ + -- used 0.706961s (cpu); 0.401679s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
    
│ │ │              
    
│ │ │                
    i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,32 +24,32 @@
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from PP^4 to PP^5)
│ │ │ │  i2 : -- we see Phi as a dominant map
│ │ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │ │  i3 : time inverse Phi;
│ │ │ │ - -- used 0.157948s (cpu); 0.0767566s (thread); 0s (gc)
│ │ │ │ + -- used 0.205323s (cpu); 0.100852s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i4 : Psi = last graph Phi;
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to hypersurface in PP^5)
│ │ │ │  i5 : time inverse Psi;
│ │ │ │ - -- used 0.186263s (cpu); 0.107471s (thread); 0s (gc)
│ │ │ │ + -- used 0.307174s (cpu); 0.14909s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-
│ │ │ │  dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i6 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i7 : time inverse Eta;
│ │ │ │ - -- used 0.48562s (cpu); 0.313057s (thread); 0s (gc)
│ │ │ │ + -- used 0.706961s (cpu); 0.401679s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ │ │  i10 : assert(Eta * Eta^-1 == 1 and Eta^-1 * Eta == 1)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html
│ │ │ @@ -88,45 +88,45 @@
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │              
    
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00209948s (cpu); 1.1327e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │              
    
│ │ │                
    i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
    
│ │ │              
    
│ │ │                
    i6 : time isIsomorphism Psi
│ │ │ - -- used 0.396692s (cpu); 0.213364s (thread); 0s (gc)
│ │ │ + -- used 0.72621s (cpu); 0.312779s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = false
    
│ │ │              
    
│ │ │                
    i7 : Eta = first graph Psi;
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2)
    
│ │ │              
    
│ │ │                
    i8 : time isIsomorphism Eta
│ │ │ - -- used 1.74335s (cpu); 0.941708s (thread); 0s (gc)
│ │ │ + -- used 2.19717s (cpu); 1.13093s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : assert(o8 and (not o6) and (not o4))
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,31 +17,31 @@
│ │ │ │       ZZ/33331[a..d]; f = rationalMap {c^2-b*d,b*c-a*d,b^2-a*c};
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
│ │ │ │  i3 : Phi = rationalMap {f,f};
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │ │  i4 : time isIsomorphism Phi
│ │ │ │ - -- used 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00209948s (cpu); 1.1327e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : Psi = first graph Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to
│ │ │ │  PP^3)
│ │ │ │  i6 : time isIsomorphism Psi
│ │ │ │ - -- used 0.396692s (cpu); 0.213364s (thread); 0s (gc)
│ │ │ │ + -- used 0.72621s (cpu); 0.312779s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x
│ │ │ │  PP^3 to threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i8 : time isIsomorphism Eta
│ │ │ │ - -- used 1.74335s (cpu); 0.941708s (thread); 0s (gc)
│ │ │ │ + -- used 2.19717s (cpu); 1.13093s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a multi-rational map is a
│ │ │ │        morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html
│ │ │ @@ -85,31 +85,31 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │              
    
│ │ │                
    i3 : time isMorphism Phi
│ │ │ - -- used 0.386427s (cpu); 0.269828s (thread); 0s (gc)
│ │ │ + -- used 0.46239s (cpu); 0.294334s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc)
│ │ │ + -- used 0.279277s (cpu); 0.123968s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
    
│ │ │              
    
│ │ │                
    i5 : time isMorphism Psi
│ │ │ - -- used 4.42649s (cpu); 3.16319s (thread); 0s (gc)
│ │ │ + -- used 4.59518s (cpu); 3.83013s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │                
    i6 : assert((not o3) and o5)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,24 +17,24 @@
│ │ │ │  i1 : ZZ/300007[a..e], f = first graph rationalMap ideal(c^2-b*d,b*c-a*d,b^2-
│ │ │ │  a*c,e), g = rationalMap submatrix(matrix f,{0..2});
│ │ │ │  i2 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 to PP^4 x PP^2)
│ │ │ │  i3 : time isMorphism Phi
│ │ │ │ - -- used 0.386427s (cpu); 0.269828s (thread); 0s (gc)
│ │ │ │ + -- used 0.46239s (cpu); 0.294334s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc)
│ │ │ │ + -- used 0.279277s (cpu); 0.123968s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
│ │ │ │  i5 : time isMorphism Psi
│ │ │ │ - -- used 4.42649s (cpu); 3.16319s (thread); 0s (gc)
│ │ │ │ + -- used 4.59518s (cpu); 3.83013s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : assert((not o3) and o5)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_I_s_o_m_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a birational map is an
│ │ │ │        isomorphism
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_R_a_t_i_o_n_a_l_M_a_p_) -- whether a rational map is a morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html
│ │ │ @@ -84,30 +84,30 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │              
    
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.00800583s (cpu); 0.00919368s (thread); 0s (gc)
│ │ │ + -- used 0.0120029s (cpu); 0.0124938s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (automorphism of X)
    
│ │ │              
    
│ │ │                
    i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3
    
│ │ │              
    
│ │ │                
    i5 : time g = linearlyNormalEmbedding Y;
│ │ │ - -- used 0.709324s (cpu); 0.523301s (thread); 0s (gc)
│ │ │ + -- used 0.630571s (cpu); 0.514714s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
    
│ │ │              
    
│ │ │                
    i6 : assert(isIsomorphism g)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,23 +13,23 @@
│ │ │ │              is a linear projection
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : K = ZZ/333331;
│ │ │ │  i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │ │  i3 : time f = linearlyNormalEmbedding X;
│ │ │ │ - -- used 0.00800583s (cpu); 0.00919368s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120029s (cpu); 0.0124938s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (automorphism of X)
│ │ │ │  i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an
│ │ │ │  isomorphic projection of X in PP^3
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3
│ │ │ │  i5 : time g = linearlyNormalEmbedding Y;
│ │ │ │ - -- used 0.709324s (cpu); 0.523301s (thread); 0s (gc)
│ │ │ │ + -- used 0.630571s (cpu); 0.514714s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │ │  i6 : assert(isIsomorphism g)
│ │ │ │  i7 : describe g
│ │ │ │  
│ │ │ │  o7 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: curve in PP^3 cut out by 6 hypersurfaces of degree 4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multirational__Map_rp.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │              
    
│ │ │                
    i3 : time multidegree Phi
│ │ │ - -- used 0.453684s (cpu); 0.376266s (thread); 0s (gc)
│ │ │ + -- used 0.541031s (cpu); 0.431948s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00102199s (cpu); 0.00133739s (thread); 0s (gc)
│ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc)
│ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc)
│ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc)
│ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc)
│ │ │ + -- used 0.00190693s (cpu); 0.00168437s (thread); 0s (gc)
│ │ │ + -- used 0.291065s (cpu); 0.197186s (thread); 0s (gc)
│ │ │ + -- used 0.43834s (cpu); 0.254514s (thread); 0s (gc)
│ │ │ + -- used 0.277805s (cpu); 0.186268s (thread); 0s (gc)
│ │ │ + -- used 0.253413s (cpu); 0.157809s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc)
    
│ │ │ + -- used 0.279069s (cpu); 0.114092s (thread); 0s (gc)
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    References
    
│ │ │  ArXiv preprint: Computations with rational maps between multi-projective varieties.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,25 +17,25 @@
│ │ │ │  This is calculated by means of the inverse image of an appropriate random
│ │ │ │  subvariety of the target.
│ │ │ │  i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4);
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^5)
│ │ │ │  i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ │ - -- used 0.00102199s (cpu); 0.00133739s (thread); 0s (gc)
│ │ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc)
│ │ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc)
│ │ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc)
│ │ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc)
│ │ │ │ + -- used 0.00190693s (cpu); 0.00168437s (thread); 0s (gc)
│ │ │ │ + -- used 0.291065s (cpu); 0.197186s (thread); 0s (gc)
│ │ │ │ + -- used 0.43834s (cpu); 0.254514s (thread); 0s (gc)
│ │ │ │ + -- used 0.277805s (cpu); 0.186268s (thread); 0s (gc)
│ │ │ │ + -- used 0.253413s (cpu); 0.157809s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time assert(oo == multidegree Phi)
│ │ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc)
│ │ │ │ + -- used 0.279069s (cpu); 0.114092s (thread); 0s (gc)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  ArXiv preprint: _C_o_m_p_u_t_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _m_a_p_s_ _b_e_t_w_e_e_n_ _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e
│ │ │ │  _v_a_r_i_e_t_i_e_s.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational
│ │ │ │        map
│ │ │ │      * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s_(_R_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_point_lp__Multiprojective__Variety_rp.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time p := point X
│ │ │ - -- used 0.0200872s (cpu); 0.0193746s (thread); 0s (gc)
│ │ │ + -- used 0.0380632s (cpu); 0.0265688s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time q = point Y
│ │ │ - -- used 2.20856s (cpu); 1.14282s (thread); 0s (gc)
│ │ │ + -- used 1.92256s (cpu); 1.20772s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,25 +14,25 @@
│ │ │ │            o a _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e_ _v_a_r_i_e_t_y, a random rational point on $X$
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : K = ZZ/1000003;
│ │ │ │  i2 : X = PP_K^({1,1,2},{3,2,3});
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9
│ │ │ │  i3 : time p := point X
│ │ │ │ - -- used 0.0200872s (cpu); 0.0193746s (thread); 0s (gc)
│ │ │ │ + -- used 0.0380632s (cpu); 0.0265688s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],
│ │ │ │  [3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │ │  i4 : Y = random({2,1,2},X);
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9
│ │ │ │  i5 : time q = point Y
│ │ │ │ - -- used 2.20856s (cpu); 1.14282s (thread); 0s (gc)
│ │ │ │ + -- used 1.92256s (cpu); 1.20772s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = q
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
│ │ │ │  i6 : assert(isSubset(p,X) and isSubset(q,Y))
│ │ │ │  The list of homogeneous coordinates can be obtained with the operator |-.
│ │ │ │  i7 : |- p
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time segre Phi;
│ │ │ - -- used 0.774927s (cpu); 0.592841s (thread); 0s (gc)
│ │ │ + -- used 1.96491s (cpu); 0.848059s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : describe segre Phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o4 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i5 : Phi = rationalMap {f,g,h};
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x
│ │ │ │  PP^4)
│ │ │ │  i6 : time segre Phi;
│ │ │ │ - -- used 0.774927s (cpu); 0.592841s (thread); 0s (gc)
│ │ │ │ + -- used 1.96491s (cpu); 0.848059s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │ │  i7 : describe segre Phi
│ │ │ │  
│ │ │ │  o7 = rational map defined by forms of degree 6
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^149
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time describe Phi
│ │ │ - -- used 0.410659s (cpu); 0.270055s (thread); 0s (gc)
│ │ │ + -- used 0.379541s (cpu); 0.190044s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3)
│ │ │ │  
│ │ │ │  o1 = Phi
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to
│ │ │ │  threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i2 : time describe Phi
│ │ │ │ - -- used 0.410659s (cpu); 0.270055s (thread); 0s (gc)
│ │ │ │ + -- used 0.379541s (cpu); 0.190044s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of
│ │ │ │  multi-degree (1,1)
│ │ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces
│ │ │ │  of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2
│ │ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ ├── ./usr/share/doc/Macaulay2/NAGtypes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  UG9seVNwYWNl
│ │ │  #:len=827
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwb2x5bm9taWFsIHZlY3RvciBzdWJz
│ │ │  cGFjZSIsICJsaW5lbnVtIiA9PiA4NTcsIFNlZUFsc28gPT4gRElWe0hFQURFUjJ7IlNlZSBhbHNv
│ │ ├── ./usr/share/doc/Macaulay2/NCAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  UVEgJSBOQ0dyb2VibmVyQmFzaXM=
│ │ │  #:len=317
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY3MCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sICUsUVEsTkNHcm9lYm5lckJhc2lzKSwi
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  Z2VuZXJhdGVSYW5kb21HcmFwaHMoWlosWlosWlop
│ │ │  #:len=275
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2VuZXJhdGVSYW5kb21HcmFwaHMsWlosWlosWlop
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
│ │ │ @@ -26,22 +26,22 @@
│ │ │  
│ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true};
│ │ │  
│ │ │  i8 : prob = n -> log(n)/n;
│ │ │  
│ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96,
│ │ │ +o9 = (75, 80, 95, 88, 91, 92, 98, 91, 98, 97, 94, 96, 97, 94, 98, 99, 98, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99)
│ │ │ +     99, 97, 100, 98, 97, 98, 98, 98, 98, 100, 99)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0,
│ │ │ +o10 = (18, 8, 3, 4, 3, 1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 0, 0, 2, 3, 1, 1, 0, 1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 0, 0, 0)
│ │ │ +      1, 0, 1, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Graphs.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ +o2 = {D]?, DQC, Dus, DCO, DqW}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 1729831171060067675
│ │ │  
│ │ │  i1 : R = QQ[a..e];
│ │ │  
│ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, c}, {a, d}, {b, e}, {d, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out
│ │ │ @@ -13,13 +13,13 @@
│ │ │  i3 : graphComplement "Dhc"
│ │ │  
│ │ │  o3 = DUW
│ │ │  
│ │ │  i4 : G = generateBipartiteGraphs 7;
│ │ │  
│ │ │  i5 : time graphComplement G;
│ │ │ - -- used 0.000443261s (cpu); 0.000438352s (thread); 0s (gc)
│ │ │ + -- used 0.00070963s (cpu); 0.00069308s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc)
│ │ │ + -- used 0.0758999s (cpu); 0.0730277s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -122,28 +122,28 @@
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96,
│ │ │ +o9 = (75, 80, 95, 88, 91, 92, 98, 91, 98, 97, 94, 96, 97, 94, 98, 99, 98, 97,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99)
│ │ │ +     99, 97, 100, 98, 97, 98, 98, 98, 98, 100, 99)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0,
│ │ │ +o10 = (18, 8, 3, 4, 3, 1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 0, 0, 2, 3, 1, 1, 0, 1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 0, 0, 0)
│ │ │ +      1, 0, 1, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,25 +38,25 @@
│ │ │ │  connected, at least as $n$ tends to infinity.
│ │ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" =>
│ │ │ │  true};
│ │ │ │  i8 : prob = n -> log(n)/n;
│ │ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o9 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96,
│ │ │ │ +o9 = (75, 80, 95, 88, 91, 92, 98, 91, 98, 97, 94, 96, 97, 94, 98, 99, 98, 97,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99)
│ │ │ │ +     99, 97, 100, 98, 97, 98, 98, 98, 98, 100, 99)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0,
│ │ │ │ +o10 = (18, 8, 3, 4, 3, 1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 0, 0, 2, 3, 1, 1, 0, 1,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      1, 0, 0, 0, 0, 0)
│ │ │ │ +      1, 0, 1, 0, 0, 0)
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with
│ │ │ │        filterGraphs and countGraphs
│ │ │ │      * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given
│ │ │ │        properties
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ +o2 = {D]?, DQC, Dus, DCO, DqW}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ │ +o2 = {D]?, DQC, Dus, DCO, DqW}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -92,23 +92,23 @@
│ │ │                
    i1 : R = QQ[a..e];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, c}, {a, d}, {b, e}, {d, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,23 +24,23 @@
│ │ │ │  vertices with a given regularity. Note that some graphs may be isomorphic.
│ │ │ │  If a _P_o_l_y_n_o_m_i_a_l_R_i_n_g $R$ is supplied instead, then the number of vertices is the
│ │ │ │  number of generators. Moreover, the nauty-based strings are automatically
│ │ │ │  converted to instances of the class _G_r_a_p_h in $R$.
│ │ │ │  i1 : R = QQ[a..e];
│ │ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │ │  
│ │ │ │ -o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │ │ +o2 = {Graph{"edges" => {{a, c}, {b, c}, {a, d}, {b, e}, {d, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}},
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ │ +     Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The number of vertices $n$ must be positive as nauty cannot handle graphs with
│ │ │ │  zero vertices.
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html
│ │ │ @@ -121,21 +121,21 @@
│ │ │              
│ │ │                
    i4 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time graphComplement G;
│ │ │ - -- used 0.000443261s (cpu); 0.000438352s (thread); 0s (gc)
    
│ │ │ + -- used 0.00070963s (cpu); 0.00069308s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc)
    
│ │ │ + -- used 0.0758999s (cpu); 0.0730277s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o3 = DUW
│ │ │ │  Batch calls can be performed considerably faster when using the List input
│ │ │ │  format. However, care should be taken as the returned list is entirely in
│ │ │ │  Graph6 or Sparse6 format.
│ │ │ │  i4 : G = generateBipartiteGraphs 7;
│ │ │ │  i5 : time graphComplement G;
│ │ │ │ - -- used 0.000443261s (cpu); 0.000438352s (thread); 0s (gc)
│ │ │ │ + -- used 0.00070963s (cpu); 0.00069308s (thread); 0s (gc)
│ │ │ │  i6 : time (graphComplement \ G);
│ │ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc)
│ │ │ │ + -- used 0.0758999s (cpu); 0.0730277s (thread); 0s (gc)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_m_p_l_e_m_e_n_t_G_r_a_p_h -- returns the complement of a graph or hypergraph
│ │ │ │  ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: **********
│ │ │ │      * graphComplement(Graph)
│ │ │ │      * graphComplement(List)
│ │ │ │      * graphComplement(String)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  YWRkRWRnZXMoLi4uLE5vTmV3NUN5Y2xlcz0+Li4uKQ==
│ │ │  #:len=261
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1thZGRFZGdlcyxOb05ldzVDeWNsZXNdLCJhZGRFZGdl
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
│ │ │ @@ -26,22 +26,22 @@
│ │ │  
│ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true};
│ │ │  
│ │ │  i8 : prob = n -> log(n)/n;
│ │ │  
│ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96,
│ │ │ +o9 = (68, 82, 93, 85, 93, 91, 95, 94, 94, 97, 97, 95, 96, 96, 98, 97, 96, 99,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98)
│ │ │ +     97, 94, 97, 99, 97, 98, 96, 100, 100, 97, 99)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,
│ │ │ +o10 = (24, 12, 4, 5, 1, 2, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 1, 0, 0, 0, 1)
│ │ │ +      0, 1, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DJc, DRK, DhS, D_s, DkS}
│ │ │ +o2 = {D]g, DKC, Dfc, Dyw, DM{}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │  
│ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out
│ │ │ @@ -1,9 +1,9 @@
│ │ │  -- -*- M2-comint -*- hash: 1331287392268
│ │ │  
│ │ │  i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DkK, DUW, DLo}
│ │ │ +o1 = {DRo, D[S, DqK}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out
│ │ │ @@ -13,13 +13,13 @@
│ │ │             4 => {2, 1}
│ │ │  
│ │ │  o2 : Graph
│ │ │  
│ │ │  i3 : G = generateBipartiteGraphs 7;
│ │ │  
│ │ │  i4 : time graphComplement G;
│ │ │ - -- used 0.000506259s (cpu); 0.000493143s (thread); 0s (gc)
│ │ │ + -- used 0.000651875s (cpu); 0.000633689s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : time (graphComplement \ G);
│ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc)
│ │ │ + -- used 0.0785912s (cpu); 0.0757277s (thread); 0s (gc)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -122,28 +122,28 @@
│ │ │                
      i8 : prob = n -> log(n)/n;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96,
│ │ │ +o9 = (68, 82, 93, 85, 93, 91, 95, 94, 94, 97, 97, 95, 96, 96, 98, 97, 96, 99,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98)
│ │ │ +     97, 94, 97, 99, 97, 98, 96, 100, 100, 97, 99)
│ │ │  
│ │ │  o9 : Sequence
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,
│ │ │ +o10 = (24, 12, 4, 5, 1, 2, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 1, 0, 0, 0, 1)
│ │ │ +      0, 1, 0, 0, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,25 +38,25 @@
│ │ │ │  connected, at least as $n$ tends to infinity.
│ │ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" =>
│ │ │ │  true};
│ │ │ │  i8 : prob = n -> log(n)/n;
│ │ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o9 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96,
│ │ │ │ +o9 = (68, 82, 93, 85, 93, 91, 95, 94, 94, 97, 97, 95, 96, 96, 98, 97, 96, 99,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98)
│ │ │ │ +     97, 94, 97, 99, 97, 98, 96, 100, 100, 97, 99)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2),
│ │ │ │  connected))
│ │ │ │  
│ │ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,
│ │ │ │ +o10 = (24, 12, 4, 5, 1, 2, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      0, 1, 0, 0, 0, 1)
│ │ │ │ +      0, 1, 0, 0, 0, 0)
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with
│ │ │ │        filterGraphs and countGraphs
│ │ │ │      * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given
│ │ │ │        properties
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DJc, DRK, DhS, D_s, DkS}
│ │ │ +o2 = {D]g, DKC, Dfc, Dyw, DM{}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {DJc, DRK, DhS, D_s, DkS}
│ │ │ │ +o2 = {D]g, DKC, Dfc, Dyw, DM{}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │            
    This method generates a specified number of random graphs on a given number of vertices with a given regularity. Note that some graphs may be isomorphic.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DkK, DUW, DLo}
│ │ │ +o1 = {DRo, D[S, DqK}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o G, a _l_i_s_t, the randomly generated regular graphs
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method generates a specified number of random graphs on a given number of
│ │ │ │  vertices with a given regularity. Note that some graphs may be isomorphic.
│ │ │ │  i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │ │  
│ │ │ │ -o1 = {DkK, DUW, DLo}
│ │ │ │ +o1 = {DRo, D[S, DqK}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The number of vertices $n$ must be positive as nauty cannot handle graphs with
│ │ │ │  zero vertices.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_e_n_e_r_a_t_e_R_a_n_d_o_m_G_r_a_p_h_s -- generates random graphs on a given number of
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html
│ │ │ @@ -115,21 +115,21 @@
│ │ │              
│ │ │                
    i3 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time graphComplement G;
│ │ │ - -- used 0.000506259s (cpu); 0.000493143s (thread); 0s (gc)
    
│ │ │ + -- used 0.000651875s (cpu); 0.000633689s (thread); 0s (gc)    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc)
    
│ │ │ + -- used 0.0785912s (cpu); 0.0757277s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use graphComplement:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,17 +38,17 @@
│ │ │ │  
│ │ │ │  o2 : Graph
│ │ │ │  Batch calls can be performed considerably faster when using the List input
│ │ │ │  format. However, care should be taken as the returned list is entirely in
│ │ │ │  Graph6 or Sparse6 format.
│ │ │ │  i3 : G = generateBipartiteGraphs 7;
│ │ │ │  i4 : time graphComplement G;
│ │ │ │ - -- used 0.000506259s (cpu); 0.000493143s (thread); 0s (gc)
│ │ │ │ + -- used 0.000651875s (cpu); 0.000633689s (thread); 0s (gc)
│ │ │ │  i5 : time (graphComplement \ G);
│ │ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc)
│ │ │ │ + -- used 0.0785912s (cpu); 0.0757277s (thread); 0s (gc)
│ │ │ │  ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: **********
│ │ │ │      * graphComplement(Graph)
│ │ │ │      * graphComplement(List)
│ │ │ │      * graphComplement(String)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _g_r_a_p_h_C_o_m_p_l_e_m_e_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/NeuralIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  aXNDYW5vbmljYWwoLi4uLFBvbGFyaXplZD0+Li4uKQ==
│ │ │  #:len=270
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTI5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0Nhbm9uaWNhbCxQb2xhcml6ZWRdLCJpc0Nhbm9u
│ │ ├── ./usr/share/doc/Macaulay2/NoetherNormalization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  bm9ldGhlck5vcm1hbGl6YXRpb24oLi4uLFZlcmJvc2U9Pi4uLik=
│ │ │  #:len=326
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzIwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tub2V0aGVyTm9ybWFsaXphdGlvbixWZXJib3NlXSwi
│ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  bm9ldGhlcmlhbk9wZXJhdG9ycyhJZGVhbCxJZGVhbCk=
│ │ │  #:len=2583
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTm9ldGhlcmlhbiBvcGVyYXRvcnMgb2Yg
│ │ │  YSBwcmltYXJ5IGNvbXBvbmVudCIsICJsaW5lbnVtIiA9PiAyNzQ5LCBJbnB1dHMgPT4ge1NQQU57
│ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out
│ │ │ @@ -47,15 +47,15 @@
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : isPrimary Q
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .167714s elapsed
│ │ │ + -- .112192s elapsed
│ │ │  
│ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │       ------------------------------------------------------------------------
│ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │  
│ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  
│ │ │  o5 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .167714s elapsed
│ │ │ + -- .112192s elapsed
│ │ │  
│ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │       ------------------------------------------------------------------------
│ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │         1 2 3    2 3
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : isPrimary Q
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ │ - -- .167714s elapsed
│ │ │ │ + -- .112192s elapsed
│ │ │ │  
│ │ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/NonPrincipalTestIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  aXNJbnZlcnRpYmxlSWRlYWwoSWRlYWwp
│ │ │  #:len=308
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIwNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNJbnZlcnRpYmxlSWRlYWwsSWRlYWwpLCJpc0lu
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  VG9yaWNEaXZpc29yID09IFpa
│ │ │  #:len=349
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTQ0MSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sID09LFRvcmljRGl2aXNvcixaWiksIlRv
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out
│ │ │ @@ -78,15 +78,15 @@
│ │ │  i13 : for i to dim X list hilbertFunction (i, A1)
│ │ │  
│ │ │  o13 = {1, 2, 3, 3, 2, 1}
│ │ │  
│ │ │  o13 : List
│ │ │  
│ │ │  i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.212369s (cpu); 0.212369s (thread); 0s (gc)
│ │ │ + -- used 0.28206s (cpu); 0.282059s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : A2 = intersectionRing Y;
│ │ │  
│ │ │  i16 : assert (# rays Y === numgens A2)
│ │ │  
│ │ │  i17 : ideal A2
│ │ │  
│ │ │ @@ -110,19 +110,19 @@
│ │ │          2                 2   2                       2   2                             2             2                      2     3      2
│ │ │        (t  + t t , t t  + t , t  + t t , t t , t t  + t , t  - t t  - 3t t   + t t   + 2t  , - t t  + t  + 2t t  , t t , - t t   + t  , t t  )
│ │ │          3    3 5   3 5    5   5    5 6   3 6   5 6    6   8    8 9     8 10    9 10     10     8 9    9     9 10   8 9     8 10    10   8 10
│ │ │  
│ │ │  o18 : QuotientRing
│ │ │  
│ │ │  i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00146338s (cpu); 0.00146077s (thread); 0s (gc)
│ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc)
│ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc)
│ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc)
│ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc)
│ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00184098s (cpu); 0.0018382s (thread); 0s (gc)
│ │ │ + -- used 0.000126861s (cpu); 0.000127156s (thread); 0s (gc)
│ │ │ + -- used 9.7322e-05s (cpu); 9.7573e-05s (thread); 0s (gc)
│ │ │ + -- used 9.4768e-05s (cpu); 9.5022e-05s (thread); 0s (gc)
│ │ │ + -- used 8.8519e-05s (cpu); 8.8688e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000101221s (cpu); 0.0001014s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 :
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_is__Well__Defined_lp__Normal__Toric__Variety_rp.out
│ │ │ @@ -1,29 +1,29 @@
│ │ │  -- -*- M2-comint -*- hash: 16408385764843695632
│ │ │  
│ │ │  i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1))
│ │ │  
│ │ │  i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1792812560
│ │ │ + -- setting random seed to 1793012443
│ │ │  
│ │ │  i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {2, 4, 6}
│ │ │ +o3 = {0, 3, 4}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : assert isWellDefined kleinschmidt (4,a)
│ │ │  
│ │ │  i5 : q = sort apply (5, j -> random (1,9));
│ │ │  
│ │ │  i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
│ │ │  
│ │ │  i7 : q
│ │ │  
│ │ │ -o7 = {1, 5, 6, 9, 9}
│ │ │ +o7 = {1, 6, 7, 7, 9}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : assert isWellDefined weightedProjectiveSpace q
│ │ │  
│ │ │  i9 : X = new MutableHashTable;
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out
│ │ │ @@ -6,61 +6,61 @@
│ │ │  
│ │ │  o2 = 5*PP2
│ │ │            0
│ │ │  
│ │ │  o2 : ToricDivisor on PP2
│ │ │  
│ │ │  i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .033906s elapsed
│ │ │ + -- .0332415s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │        0 1 2   0 1 2   0 1 2   0 2   1   0 1   0 1   0 1   0 1   0
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .0131253s elapsed
│ │ │ + -- .0014707s elapsed
│ │ │  
│ │ │  i5 : FF2 = hirzebruchSurface 2;
│ │ │  
│ │ │  i6 : D2 = 2*FF2_0 + 3 * FF2_1
│ │ │  
│ │ │  o6 = 2*FF2  + 3*FF2
│ │ │            0        1
│ │ │  
│ │ │  o6 : ToricDivisor on FF2
│ │ │  
│ │ │  i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .134092s elapsed
│ │ │ + -- .082907s elapsed
│ │ │  
│ │ │         2     3 2     3     2 3
│ │ │  o7 = {x x , x x , x x x , x x }
│ │ │         1 3   1 2   0 1 2   0 1
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2))
│ │ │ - -- .00103874s elapsed
│ │ │ + -- .00144736s elapsed
│ │ │  
│ │ │  i9 : X = kleinschmidt (5, {1,2,3});
│ │ │  
│ │ │  i10 : D3 = 3*X_0 + 5*X_1
│ │ │  
│ │ │  o10 = 3*X  + 5*X
│ │ │           0      1
│ │ │  
│ │ │  o10 : ToricDivisor on X
│ │ │  
│ │ │  i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 46.8661s elapsed
│ │ │ + -- 34.7961s elapsed
│ │ │  
│ │ │  o11 = 7909
│ │ │  
│ │ │  i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .183473s elapsed
│ │ │ + -- .0336445s elapsed
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out
│ │ │ @@ -24,19 +24,19 @@
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : X = normalToricVariety F;
│ │ │  
│ │ │  i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F)
│ │ │  
│ │ │  i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc)
│ │ │ + -- used 3.1996e-05s (cpu); 2.4054e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = X1
│ │ │  
│ │ │  o6 : NormalToricVariety
│ │ │  
│ │ │  i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ - -- used 0.0871239s (cpu); 0.08713s (thread); 0s (gc)
│ │ │ + -- used 0.0571382s (cpu); 0.0571486s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out
│ │ │ @@ -88,15 +88,15 @@
│ │ │  o18 = | 0 1 0 |
│ │ │        | 0 0 1 |
│ │ │  
│ │ │                 2       3
│ │ │  o18 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.023116s (cpu); 0.0231153s (thread); 0s (gc)
│ │ │ + -- used 0.0310338s (cpu); 0.0310381s (thread); 0s (gc)
│ │ │  
│ │ │  i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc)
│ │ │ + -- used 0.00329262s (cpu); 0.00330037s (thread); 0s (gc)
│ │ │  
│ │ │  i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html
│ │ │ @@ -212,15 +212,15 @@
│ │ │          
    
│ │ │            
    We end with a slightly larger example.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -259,20 +259,20 @@
│ │ │  
│ │ │  o18 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.212369s (cpu); 0.212369s (thread); 0s (gc)
    
│ │ │ + -- used 0.28206s (cpu); 0.282059s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i15 : A2 = intersectionRing Y;
    
│ │ │              
    
│ │ │                
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00146338s (cpu); 0.00146077s (thread); 0s (gc)
│ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc)
│ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc)
│ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc)
│ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc)
│ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00184098s (cpu); 0.0018382s (thread); 0s (gc)
│ │ │ + -- used 0.000126861s (cpu); 0.000127156s (thread); 0s (gc)
│ │ │ + -- used 9.7322e-05s (cpu); 9.7573e-05s (thread); 0s (gc)
│ │ │ + -- used 9.4768e-05s (cpu); 9.5022e-05s (thread); 0s (gc)
│ │ │ + -- used 8.8519e-05s (cpu); 8.8688e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000101221s (cpu); 0.0001014s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │  
│ │ │  o19 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -96,15 +96,15 @@
│ │ │ │  i13 : for i to dim X list hilbertFunction (i, A1)
│ │ │ │  
│ │ │ │  o13 = {1, 2, 3, 3, 2, 1}
│ │ │ │  
│ │ │ │  o13 : List
│ │ │ │  We end with a slightly larger example.
│ │ │ │  i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ │ - -- used 0.212369s (cpu); 0.212369s (thread); 0s (gc)
│ │ │ │ + -- used 0.28206s (cpu); 0.282059s (thread); 0s (gc)
│ │ │ │  i15 : A2 = intersectionRing Y;
│ │ │ │  i16 : assert (# rays Y === numgens A2)
│ │ │ │  i17 : ideal A2
│ │ │ │  
│ │ │ │  o17 = ideal (t t , t t , t t , t t , t t , t t , t t , t t , t t t  ,
│ │ │ │                2 3   2 5   4 5   3 6   4 6   1 7   7 9   8 9   0 1 10
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -129,20 +129,20 @@
│ │ │ │        (t  + t t , t t  + t , t  + t t , t t , t t  + t , t  - t t  - 3t t   + t
│ │ │ │  t   + 2t  , - t t  + t  + 2t t  , t t , - t t   + t  , t t  )
│ │ │ │          3    3 5   3 5    5   5    5 6   3 6   5 6    6   8    8 9     8 10
│ │ │ │  9 10     10     8 9    9     9 10   8 9     8 10    10   8 10
│ │ │ │  
│ │ │ │  o18 : QuotientRing
│ │ │ │  i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ │ - -- used 0.00146338s (cpu); 0.00146077s (thread); 0s (gc)
│ │ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc)
│ │ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00184098s (cpu); 0.0018382s (thread); 0s (gc)
│ │ │ │ + -- used 0.000126861s (cpu); 0.000127156s (thread); 0s (gc)
│ │ │ │ + -- used 9.7322e-05s (cpu); 9.7573e-05s (thread); 0s (gc)
│ │ │ │ + -- used 9.4768e-05s (cpu); 9.5022e-05s (thread); 0s (gc)
│ │ │ │ + -- used 8.8519e-05s (cpu); 8.8688e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.000101221s (cpu); 0.0001014s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = {1, 6, 13, 13, 6, 1}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _w_o_r_k_i_n_g_ _w_i_t_h_ _s_h_e_a_v_e_s -- information about coherent sheaves and total
│ │ │ │        coordinate rings (a.k.a. Cox rings)
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_is__Well__Defined_lp__Normal__Toric__Variety_rp.html
│ │ │ @@ -98,22 +98,22 @@
│ │ │          
    
│ │ │            
    The second examples show that a randomly selected Kleinschmidt toric variety and a weighted projective space are also well-defined.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1792812560
    
│ │ │ + -- setting random seed to 1793012443
│ │ │              
    
│ │ │                
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {2, 4, 6}
│ │ │ +o3 = {0, 3, 4}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : assert isWellDefined kleinschmidt (4,a)
    
│ │ │ @@ -131,15 +131,15 @@
│ │ │                
    i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
    
│ │ │              
    
│ │ │                
    i7 : q
│ │ │  
│ │ │ -o7 = {1, 5, 6, 9, 9}
│ │ │ +o7 = {1, 6, 7, 7, 9}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : assert isWellDefined weightedProjectiveSpace q
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,27 +28,27 @@
│ │ │ │      * the intersection of the cones associated to two elements of coneList is a
│ │ │ │        face of each cone.
│ │ │ │  The first examples illustrate that small projective spaces are well-defined.
│ │ │ │  i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1))
│ │ │ │  The second examples show that a randomly selected Kleinschmidt toric variety
│ │ │ │  and a weighted projective space are also well-defined.
│ │ │ │  i2 : setRandomSeed (currentTime ());
│ │ │ │ - -- setting random seed to 1792812560
│ │ │ │ + -- setting random seed to 1793012443
│ │ │ │  i3 : a = sort apply (3, i -> random (7))
│ │ │ │  
│ │ │ │ -o3 = {2, 4, 6}
│ │ │ │ +o3 = {0, 3, 4}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : assert isWellDefined kleinschmidt (4,a)
│ │ │ │  i5 : q = sort apply (5, j -> random (1,9));
│ │ │ │  i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j
│ │ │ │  -> random (1,9));
│ │ │ │  i7 : q
│ │ │ │  
│ │ │ │ -o7 = {1, 5, 6, 9, 9}
│ │ │ │ +o7 = {1, 6, 7, 7, 9}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : assert isWellDefined weightedProjectiveSpace q
│ │ │ │  The next ten examples illustrate various ways that two lists can fail to define
│ │ │ │  a normal toric variety. By making the current debugging level greater than one,
│ │ │ │  one gets some addition information about the nature of the failure.
│ │ │ │  i9 : X = new MutableHashTable;
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o2 : ToricDivisor on PP2
│ │ │              
    
│ │ │                
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .033906s elapsed
│ │ │ + -- .0332415s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .0131253s elapsed
    
│ │ │ + -- .0014707s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    Toric varieties of Picard-rank 2 are slightly more interesting.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -143,27 +143,27 @@
│ │ │  
│ │ │  o6 : ToricDivisor on FF2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -176,23 +176,23 @@
│ │ │  
│ │ │  o10 : ToricDivisor on X
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .134092s elapsed
│ │ │ + -- .082907s elapsed
│ │ │  
│ │ │         2     3 2     3     2 3
│ │ │  o7 = {x x , x x , x x x , x x }
│ │ │         1 3   1 2   0 1 2   0 1
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2))
│ │ │ - -- .00103874s elapsed
    
│ │ │ + -- .00144736s elapsed
│ │ │              
    
│ │ │                
    i9 : X = kleinschmidt (5, {1,2,3});
    
│ │ │              
    
│ │ │                
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 46.8661s elapsed
│ │ │ + -- 34.7961s elapsed
│ │ │  
│ │ │  o11 = 7909
    
│ │ │              
    
│ │ │                
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .183473s elapsed
    
│ │ │ + -- .0336445s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    By exploiting latticePoints, this method function avoids using the basis function.
    
│ │ │          
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,61 +27,61 @@
│ │ │ │  i2 : D1 = 5*PP2_0
│ │ │ │  
│ │ │ │  o2 = 5*PP2
│ │ │ │            0
│ │ │ │  
│ │ │ │  o2 : ToricDivisor on PP2
│ │ │ │  i3 : M1 = elapsedTime monomials D1
│ │ │ │ - -- .033906s elapsed
│ │ │ │ + -- .0332415s elapsed
│ │ │ │  
│ │ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4
│ │ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │ │        0 1 2   0 1 2   0 1 2   0 2   1   0 1   0 1   0 1   0 1   0
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring
│ │ │ │  variety D1))
│ │ │ │ - -- .0131253s elapsed
│ │ │ │ + -- .0014707s elapsed
│ │ │ │  Toric varieties of Picard-rank 2 are slightly more interesting.
│ │ │ │  i5 : FF2 = hirzebruchSurface 2;
│ │ │ │  i6 : D2 = 2*FF2_0 + 3 * FF2_1
│ │ │ │  
│ │ │ │  o6 = 2*FF2  + 3*FF2
│ │ │ │            0        1
│ │ │ │  
│ │ │ │  o6 : ToricDivisor on FF2
│ │ │ │  i7 : M2 = elapsedTime monomials D2
│ │ │ │ - -- .134092s elapsed
│ │ │ │ + -- .082907s elapsed
│ │ │ │  
│ │ │ │         2     3 2     3     2 3
│ │ │ │  o7 = {x x , x x , x x x , x x }
│ │ │ │         1 3   1 2   0 1 2   0 1
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring
│ │ │ │  variety D2))
│ │ │ │ - -- .00103874s elapsed
│ │ │ │ + -- .00144736s elapsed
│ │ │ │  i9 : X = kleinschmidt (5, {1,2,3});
│ │ │ │  i10 : D3 = 3*X_0 + 5*X_1
│ │ │ │  
│ │ │ │  o10 = 3*X  + 5*X
│ │ │ │           0      1
│ │ │ │  
│ │ │ │  o10 : ToricDivisor on X
│ │ │ │  i11 : m3 = elapsedTime # monomials D3
│ │ │ │ - -- 46.8661s elapsed
│ │ │ │ + -- 34.7961s elapsed
│ │ │ │  
│ │ │ │  o11 = 7909
│ │ │ │  i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety
│ │ │ │  D3))
│ │ │ │ - -- .183473s elapsed
│ │ │ │ + -- .0336445s elapsed
│ │ │ │  By exploiting _l_a_t_t_i_c_e_P_o_i_n_t_s, this method function avoids using the _b_a_s_i_s
│ │ │ │  function.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _w_o_r_k_i_n_g_ _w_i_t_h_ _d_i_v_i_s_o_r_s -- information about toric divisors and their
│ │ │ │        related groups
│ │ │ │      * _r_i_n_g_(_N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_) -- make the total coordinate ring (a.k.a. Cox
│ │ │ │        ring)
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html
│ │ │ @@ -130,25 +130,25 @@
│ │ │          
    
│ │ │            
    The recommended method for creating a NormalToricVariety from a fan is normalToricVariety(List,List).  In fact, this package avoids using objects from the Polyhedra package whenever possible.   Here is a trivial example, namely projective 2-space, illustrating the substantial increase in time resulting from the use of a Polyhedra fan.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,22 +48,22 @@
│ │ │ │  i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F)
│ │ │ │  The recommended method for creating a _N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y from a fan is
│ │ │ │  _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_L_i_s_t_,_L_i_s_t_). In fact, this package avoids using objects from
│ │ │ │  the _P_o_l_y_h_e_d_r_a package whenever possible. Here is a trivial example, namely
│ │ │ │  projective 2-space, illustrating the substantial increase in time resulting
│ │ │ │  from the use of a _P_o_l_y_h_e_d_r_a fan.
│ │ │ │  i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ │ - -- used 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc)
│ │ │ │ + -- used 3.1996e-05s (cpu); 2.4054e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = X1
│ │ │ │  
│ │ │ │  o6 : NormalToricVariety
│ │ │ │  i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull
│ │ │ │  matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ │ - -- used 0.0871239s (cpu); 0.08713s (thread); 0s (gc)
│ │ │ │ + -- used 0.0571382s (cpu); 0.0571486s (thread); 0s (gc)
│ │ │ │  i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors
│ │ │ │      * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y -- make a normal toric variety
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_F_a_n_) -- make a normal toric variety from a 'Polyhedra'
│ │ │ │        fan
│ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html
│ │ │ @@ -238,21 +238,21 @@
│ │ │                 2       3
│ │ │  o18 : Matrix ZZ  <-- ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,17 +102,17 @@
│ │ │ │  
│ │ │ │  o18 = | 0 1 0 |
│ │ │ │        | 0 0 1 |
│ │ │ │  
│ │ │ │                 2       3
│ │ │ │  o18 : Matrix ZZ  <-- ZZ
│ │ │ │  i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ │ - -- used 0.023116s (cpu); 0.0231153s (thread); 0s (gc)
│ │ │ │ + -- used 0.0310338s (cpu); 0.0310381s (thread); 0s (gc)
│ │ │ │  i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc)
│ │ │ │ + -- used 0.00329262s (cpu); 0.00330037s (thread); 0s (gc)
│ │ │ │  i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors
│ │ │ │      * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_M_a_t_r_i_x_) -- make a normal toric variety from a polytope
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_P_o_l_y_h_e_d_r_o_n_) -- make a normal toric variety from a
│ │ │ │        'Polyhedra' polyhedron
│ │ ├── ./usr/share/doc/Macaulay2/Normaliz/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  aW50Y2xNb25JZGVhbChJZGVhbCxSaW5nRWxlbWVudCk=
│ │ │  #:len=2753
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibm9ybWFsaXphdGlvbiBvZiBSZWVzIGFs
│ │ │  Z2VicmEiLCAibGluZW51bSIgPT4gMTg3NiwgSW5wdXRzID0+IHtTUEFOe1NQQU57ImFuICIsVE8y
│ │ ├── ./usr/share/doc/Macaulay2/NumericSolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  am9yZGFuRm9ybShNYXRyaXgp
│ │ │  #:len=263
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODE0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhqb3JkYW5Gb3JtLE1hdHJpeCksImpvcmRhbkZvcm0o
│ │ ├── ./usr/share/doc/Macaulay2/NumericalAlgebraicGeometry/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=43
│ │ │  aXNTdWJzZXQoTnVtZXJpY2FsVmFyaWV0eSxOdW1lcmljYWxWYXJpZXR5KQ==
│ │ │  #:len=1399
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2sgY29udGFpbm1lbnQiLCAibGlu
│ │ │  ZW51bSIgPT4gNzg5LCBJbnB1dHMgPT4ge1NQQU57VFR7IlYifSwiLCAiLFNQQU57ImEgIixUTzJ7
│ │ ├── ./usr/share/doc/Macaulay2/NumericalCertification/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  aW50ZXJ2YWxDQ2koUlJpLE51bWJlcik=
│ │ │  #:len=261
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDQsICJ1bmRvY3VtZW50ZWQiID0+IHRy
│ │ │  dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KGludGVydmFs
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=42
│ │ │  bnVtZXJpY2FsSGlsYmVydEZ1bmN0aW9uKFJpbmdNYXAsSWRlYWwsWlop
│ │ │  #:len=367
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDUwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhudW1lcmljYWxIaWxiZXJ0RnVuY3Rpb24sUmluZ01h
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out
│ │ │ @@ -21,21 +21,21 @@
│ │ │  
│ │ │  i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue == 0
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00691559 seconds
│ │ │ +     -- used .00876802 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .005626 seconds
│ │ │ +     -- used .00710886 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00273274 seconds
│ │ │ +     -- used .00340547 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000418905 seconds
│ │ │ +     -- used .000435962 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
│ │ │  
│ │ │  i6 : extractImageEquations(T, AttemptZZ => true)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out
│ │ │ @@ -11,21 +11,21 @@
│ │ │  o2 = | s3 s2t st2 t3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00350208 seconds
│ │ │ +     -- used .00415945 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00258173 seconds
│ │ │ +     -- used .00315149 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00100088 seconds
│ │ │ +     -- used .00123137 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000278743 seconds
│ │ │ +     -- used .000287794 seconds
│ │ │  
│ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │  
│ │ │                            1                   3
│ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │                 53  0   3           53  0   3
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out
│ │ │ @@ -11,40 +11,40 @@
│ │ │  o2 = | s3 s2t st2 t3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0119364 seconds
│ │ │ +     -- used .0150904 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0107729 seconds
│ │ │ +     -- used .0181986 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00683732 seconds
│ │ │ +     -- used .00855493 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000713648 seconds
│ │ │ +     -- used .00134518 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
│ │ │  
│ │ │  i4 : R = CC[x_(1,1)..x_(2,4)];
│ │ │  
│ │ │  i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*;
│ │ │  
│ │ │  i6 : S = numericalImageSample(F, ideal 0_R, 60);
│ │ │  
│ │ │  i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00320623 seconds
│ │ │ +     -- used .00443695 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00838546 seconds
│ │ │ +     -- used .0137956 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000876183 seconds
│ │ │ +     -- used .00139674 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out
│ │ │ @@ -20,12 +20,12 @@
│ │ │  
│ │ │  i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5))));
│ │ │  
│ │ │               1      70
│ │ │  o8 : Matrix R  <-- R
│ │ │  
│ │ │  i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.117425s (cpu); 0.11742s (thread); 0s (gc)
│ │ │ + -- used 0.0850717s (cpu); 0.0850636s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_real__Point.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : I = I1 + I2;
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .642071s elapsed
│ │ │ + -- .543686s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
│ │ │  
│ │ │  i8 : matrix pack(5, p#Coordinates)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html
│ │ │ @@ -105,21 +105,21 @@
│ │ │  o4 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,21 +33,21 @@
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue
│ │ │ │  == 0
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .00691559 seconds
│ │ │ │ +     -- used .00876802 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .005626 seconds
│ │ │ │ +     -- used .00710886 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00273274 seconds
│ │ │ │ +     -- used .00340547 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000418905 seconds
│ │ │ │ +     -- used .000435962 seconds
│ │ │ │  
│ │ │ │  o5 = a "numerical interpolation table", indicating
│ │ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │ │  
│ │ │ │  o5 : NumericalInterpolationTable
│ │ │ │  i6 : extractImageEquations(T, AttemptZZ => true)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html
│ │ │ @@ -107,21 +107,21 @@
│ │ │  o2 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,21 +38,21 @@
│ │ │ │  
│ │ │ │  o2 = | s3 s2t st2 t3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .00350208 seconds
│ │ │ │ +     -- used .00415945 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .00258173 seconds
│ │ │ │ +     -- used .00315149 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00100088 seconds
│ │ │ │ +     -- used .00123137 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000278743 seconds
│ │ │ │ +     -- used .000287794 seconds
│ │ │ │  
│ │ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │ │  
│ │ │ │                            1                   3
│ │ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │ │                 53  0   3           53  0   3
│ │ │ │  Here is how to do the same computation symbolically.
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html
│ │ │ @@ -112,21 +112,21 @@
│ │ │  o2 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -152,19 +152,19 @@
│ │ │                
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,39 +57,39 @@
│ │ │ │  
│ │ │ │  o2 = | s3 s2t st2 t3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │ │  Sampling image points ...
│ │ │ │ -     -- used .0119364 seconds
│ │ │ │ +     -- used .0150904 seconds
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .0107729 seconds
│ │ │ │ +     -- used .0181986 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00683732 seconds
│ │ │ │ +     -- used .00855493 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000713648 seconds
│ │ │ │ +     -- used .00134518 seconds
│ │ │ │  
│ │ │ │  o3 = a "numerical interpolation table", indicating
│ │ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │ │  
│ │ │ │  o3 : NumericalInterpolationTable
│ │ │ │  The following example computes the dimension of Plücker quadrics in the
│ │ │ │  defining ideal of the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, in the
│ │ │ │  ambient space $P^5$.
│ │ │ │  i4 : R = CC[x_(1,1)..x_(2,4)];
│ │ │ │  i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*;
│ │ │ │  i6 : S = numericalImageSample(F, ideal 0_R, 60);
│ │ │ │  i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │ │  Creating interpolation matrix ...
│ │ │ │ -     -- used .00320623 seconds
│ │ │ │ +     -- used .00443695 seconds
│ │ │ │  Performing normalization preconditioning ...
│ │ │ │ -     -- used .00838546 seconds
│ │ │ │ +     -- used .0137956 seconds
│ │ │ │  Computing numerical kernel ...
│ │ │ │ -     -- used .000876183 seconds
│ │ │ │ +     -- used .00139674 seconds
│ │ │ │  
│ │ │ │  o7 = a "numerical interpolation table", indicating
│ │ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │ │  
│ │ │ │  o7 : NumericalInterpolationTable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _N_u_m_e_r_i_c_a_l_I_n_t_e_r_p_o_l_a_t_i_o_n_T_a_b_l_e -- the class of all
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html
│ │ │ @@ -145,15 +145,15 @@
│ │ │               1      70
│ │ │  o8 : Matrix R  <-- R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc)
│ │ │ + -- used 3.1996e-05s (cpu); 2.4054e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = X1
│ │ │  
│ │ │  o6 : NormalToricVariety
    
│ │ │              
    
│ │ │                
    i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ - -- used 0.0871239s (cpu); 0.08713s (thread); 0s (gc)
    
│ │ │ + -- used 0.0571382s (cpu); 0.0571486s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    
│ │ │              
    
│ │ │                
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.023116s (cpu); 0.0231153s (thread); 0s (gc)
    
│ │ │ + -- used 0.0310338s (cpu); 0.0310381s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc)
    
│ │ │ + -- used 0.00329262s (cpu); 0.00330037s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
    
│ │ │              
    
│ │ │                
    i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00691559 seconds
│ │ │ +     -- used .00876802 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .005626 seconds
│ │ │ +     -- used .00710886 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00273274 seconds
│ │ │ +     -- used .00340547 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000418905 seconds
│ │ │ +     -- used .000435962 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00350208 seconds
│ │ │ +     -- used .00415945 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00258173 seconds
│ │ │ +     -- used .00315149 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00100088 seconds
│ │ │ +     -- used .00123137 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000278743 seconds
│ │ │ +     -- used .000287794 seconds
│ │ │  
│ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │  
│ │ │                            1                   3
│ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │                 53  0   3           53  0   3
    
│ │ │              
    
│ │ │                
    i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0119364 seconds
│ │ │ +     -- used .0150904 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0107729 seconds
│ │ │ +     -- used .0181986 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00683732 seconds
│ │ │ +     -- used .00855493 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000713648 seconds
│ │ │ +     -- used .00134518 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00320623 seconds
│ │ │ +     -- used .00443695 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00838546 seconds
│ │ │ +     -- used .0137956 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000876183 seconds
│ │ │ +     -- used .00139674 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
    
│ │ │              
    
│ │ │                
    i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.117425s (cpu); 0.11742s (thread); 0s (gc)
│ │ │ + -- used 0.0850717s (cpu); 0.0850636s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  201-222. We numerically verify this below.
│ │ │ │  i7 : R = CC[a_(1,1)..a_(14,5)];
│ │ │ │  i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5))));
│ │ │ │  
│ │ │ │               1      70
│ │ │ │  o8 : Matrix R  <-- R
│ │ │ │  i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ │ - -- used 0.117425s (cpu); 0.11742s (thread); 0s (gc)
│ │ │ │ + -- used 0.0850717s (cpu); 0.0850636s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = 69
│ │ │ │  ********** WWaayyss ttoo uussee nnuummeerriiccaallIImmaaggeeDDiimm:: **********
│ │ │ │      * numericalImageDim(List,Ideal)
│ │ │ │      * numericalImageDim(List,Ideal,Point)
│ │ │ │      * numericalImageDim(Matrix,Ideal)
│ │ │ │      * numericalImageDim(Matrix,Ideal,Point)
│ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .642071s elapsed
│ │ │ + -- .543686s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  i5 : I2 = ideal apply(entries transpose A, row -> sum(row, v -> v^2) - 1);
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : I = I1 + I2;
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ │ - -- .642071s elapsed
│ │ │ │ + -- .543686s elapsed
│ │ │ │  
│ │ │ │  o7 = p
│ │ │ │  
│ │ │ │  o7 : Point
│ │ │ │  i8 : matrix pack(5, p#Coordinates)
│ │ │ │  
│ │ │ │  o8 = | .722359  .289465  -.295808  .591752  -.454678 |
│ │ ├── ./usr/share/doc/Macaulay2/NumericalLinearAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  bnVtZXJpY2FsS2VybmVsKC4uLixUb2xlcmFuY2U9Pi4uLik=
│ │ │  #:len=352
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tudW1lcmljYWxLZXJuZWwsVG9sZXJhbmNlXSwibnVt
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  UGllcmlSb290Q291bnQoLi4uLFZlcmJvc2U9Pi4uLik=
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmVxdWVzdCB2ZXJib3NlIGZlZWRiYWNr
│ │ │  IiwgRGVzY3JpcHRpb24gPT4ge30sICJsaW5lbnVtIiA9PiAzMzMsIHN5bWJvbCBEb2N1bWVudFRh
│ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  c2VtaWdyb3VwRnJvbU11KExpc3Qp
│ │ │  #:len=289
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzAwNCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2VtaWdyb3VwRnJvbU11LExpc3QpLCJzZW1pZ3Jv
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  dG9TdHJpbmcoUG9seW5vbWlhbE9JQWxnZWJyYSk=
│ │ │  #:len=1044
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheSBhIHBvbHlub21pYWwgT0kt
│ │ │  YWxnZWJyYSBpbiBjb25kZW5zZWQgZm9ybSIsICJsaW5lbnVtIiA9PiAxNTE2LCBJbnB1dHMgPT4g
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.0877009s (cpu); 0.0877049s (thread); 0s (gc)
│ │ │ + -- used 0.1087s (cpu); 0.108701s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  o5 : OIResolution
│ │ │  
│ │ │  i6 : C.dd_0
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution_sp_us_sp__Z__Z.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.224314s (cpu); 0.152688s (thread); 0s (gc)
│ │ │ + -- used 0.258087s (cpu); 0.144204s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : C_0
│ │ │  
│ │ │  o6 = Basis symbol: e0
│ │ │       Basis element widths: {2}
│ │ │       Degree shifts: {-2}
│ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Top__Nonminimal.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.459073s (cpu); 0.290457s (thread); 0s (gc)
│ │ │ + -- used 0.55674s (cpu); 0.344168s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe__Full.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0985988s (cpu); 0.0985983s (thread); 0s (gc)
│ │ │ + -- used 0.146821s (cpu); 0.146821s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │                  Basis element widths: {2}
│ │ │                  Degree shifts: {-2}
│ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__O__I__Resolution_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0855533s (cpu); 0.0855538s (thread); 0s (gc)
│ │ │ + -- used 0.0974196s (cpu); 0.097417s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : describe C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │                  Basis element widths: {2}
│ │ │                  Degree shifts: {-2}
│ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.357846s (cpu); 0.284765s (thread); 0s (gc)
│ │ │ + -- used 0.41913s (cpu); 0.330559s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__O__I__G__B.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │  
│ │ │  i10 : isOIGB {b1, b2}
│ │ │  
│ │ │  o10 = false
│ │ │  
│ │ │  i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.167028s (cpu); 0.0579109s (thread); 0s (gc)
│ │ │ + -- used 0.184268s (cpu); 0.070284s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │  
│ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │  
│ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │  
│ │ │  i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0295377s (cpu); 0.0295415s (thread); 0s (gc)
│ │ │ + -- used 0.045918s (cpu); 0.0459179s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.368596s (cpu); 0.179956s (thread); 0s (gc)
│ │ │ + -- used 0.45764s (cpu); 0.222971s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : net C
│ │ │  
│ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__G__B.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │  
│ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │  
│ │ │  i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.0272125s (cpu); 0.0272119s (thread); 0s (gc)
│ │ │ + -- used 0.0357822s (cpu); 0.0357814s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Res.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │  
│ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │  
│ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.652096s (cpu); 0.338501s (thread); 0s (gc)
│ │ │ + -- used 0.825231s (cpu); 0.478245s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │  
│ │ │  i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
│ │ │  
│ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
│ │ │  
│ │ │  i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.133994s (cpu); 0.133994s (thread); 0s (gc)
│ │ │ + -- used 0.15313s (cpu); 0.152842s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2 
│ │ │       ------------------------------------------------------------------------
│ │ │        2                  2
│ │ │       x   x   e        - x   x   e       }
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.0877009s (cpu); 0.0877049s (thread); 0s (gc)
│ │ │ + -- used 0.1087s (cpu); 0.108701s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │  complex, use _i_s_C_o_m_p_l_e_x. To get the $n$th differential in an OI-resolution C,
│ │ │ │  use C.dd_n.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1)
│ │ │ │ - -- used 0.0877009s (cpu); 0.0877049s (thread); 0s (gc)
│ │ │ │ + -- used 0.1087s (cpu); 0.108701s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : C.dd_0
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution_sp_us_sp__Z__Z.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.224314s (cpu); 0.152688s (thread); 0s (gc)
    
│ │ │ + -- used 0.258087s (cpu); 0.144204s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : C_0
│ │ │  
│ │ │  o6 = Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Returns the free OI-module of $C$ in homological degree $n$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.224314s (cpu); 0.152688s (thread); 0s (gc)
│ │ │ │ + -- used 0.258087s (cpu); 0.144204s (thread); 0s (gc)
│ │ │ │  i6 : C_0
│ │ │ │  
│ │ │ │  o6 = Basis symbol: e0
│ │ │ │       Basis element widths: {2}
│ │ │ │       Degree shifts: {-2}
│ │ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │       Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.459073s (cpu); 0.290457s (thread); 0s (gc)
│ │ │ + -- used 0.55674s (cpu); 0.344168s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │  homological degree $n-1$ to be minimized. Therefore, use TopNonminimal => true
│ │ │ │  for no minimization of the basis in degree $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.459073s (cpu); 0.290457s (thread); 0s (gc)
│ │ │ │ + -- used 0.55674s (cpu); 0.344168s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd TTooppNNoonnmmiinniimmaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0985988s (cpu); 0.0985983s (thread); 0s (gc)
    
│ │ │ + -- used 0.146821s (cpu); 0.146821s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  Displays the free OI-modules and describes the differentials of an OI-
│ │ │ │  resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.0985988s (cpu); 0.0985983s (thread); 0s (gc)
│ │ │ │ + -- used 0.146821s (cpu); 0.146821s (thread); 0s (gc)
│ │ │ │  i6 : describeFull C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0855533s (cpu); 0.0855538s (thread); 0s (gc)
    
│ │ │ + -- used 0.0974196s (cpu); 0.097417s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : describe C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Displays the free OI-modules and differentials of an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.0855533s (cpu); 0.0855538s (thread); 0s (gc)
│ │ │ │ + -- used 0.0974196s (cpu); 0.097417s (thread); 0s (gc)
│ │ │ │  i6 : describe C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Complex.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.357846s (cpu); 0.284765s (thread); 0s (gc)
│ │ │ + -- used 0.41913s (cpu); 0.330559s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  option must be either true or false, depending on whether one wants debug
│ │ │ │  information printed.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.357846s (cpu); 0.284765s (thread); 0s (gc)
│ │ │ │ + -- used 0.41913s (cpu); 0.330559s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : isComplex C
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html
│ │ │ @@ -121,15 +121,15 @@
│ │ │  
│ │ │  o10 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.167028s (cpu); 0.0579109s (thread); 0s (gc)
│ │ │ + -- used 0.184268s (cpu); 0.070284s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : isOIGB {b1, b2}
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.167028s (cpu); 0.0579109s (thread); 0s (gc)
│ │ │ │ + -- used 0.184268s (cpu); 0.070284s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │              
│ │ │                
    i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0295377s (cpu); 0.0295415s (thread); 0s (gc)
│ │ │ + -- used 0.045918s (cpu); 0.0459179s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.0295377s (cpu); 0.0295415s (thread); 0s (gc)
│ │ │ │ + -- used 0.045918s (cpu); 0.0459179s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.368596s (cpu); 0.179956s (thread); 0s (gc)
    
│ │ │ + -- used 0.45764s (cpu); 0.222971s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : net C
│ │ │  
│ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  Displays the basis element widths and degree shifts of the free OI-modules in
│ │ │ │  an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.368596s (cpu); 0.179956s (thread); 0s (gc)
│ │ │ │ + -- used 0.45764s (cpu); 0.222971s (thread); 0s (gc)
│ │ │ │  i6 : net C
│ │ │ │  
│ │ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_e_t_(_O_I_R_e_s_o_l_u_t_i_o_n_) -- display an OI-resolution
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.0272125s (cpu); 0.0272119s (thread); 0s (gc)
│ │ │ + -- used 0.0357822s (cpu); 0.0357814s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i9 : time oiGB {b1, b2}
│ │ │ │ - -- used 0.0272125s (cpu); 0.0272119s (thread); 0s (gc)
│ │ │ │ + -- used 0.0357822s (cpu); 0.0357814s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   x   x   e           - x   x   x   e          }
│ │ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.652096s (cpu); 0.338501s (thread); 0s (gc)
│ │ │ + -- used 0.825231s (cpu); 0.478245s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  Therefore, use TopNonminimal => true for no minimization of the basis in degree
│ │ │ │  $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.652096s (cpu); 0.338501s (thread); 0s (gc)
│ │ │ │ + -- used 0.825231s (cpu); 0.478245s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** WWaayyss ttoo uussee ooiiRReess:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.133994s (cpu); 0.133994s (thread); 0s (gc)
│ │ │ + -- used 0.15313s (cpu); 0.152842s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2 
│ │ │       ------------------------------------------------------------------------
│ │ │        2                  2
│ │ │       x   x   e        - x   x   e       }
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
│ │ │ │  i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ │ - -- used 0.133994s (cpu); 0.133994s (thread); 0s (gc)
│ │ │ │ + -- used 0.15313s (cpu); 0.152842s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2                  2
│ │ │ │       x   x   e        - x   x   e       }
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  Z3JhZGVkTW9kdWxlKE1vZHVsZSk=
│ │ │  #:len=294
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODEsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KGdyYWRlZE1vZHVsZSxNb2R1bGUpLCJncmFkZWRNb2R1
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Fast__Nonminimal.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.66263s elapsed
│ │ │ + -- 2.0742s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │  
│ │ │  i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- 1.80662s elapsed
│ │ │ + -- 1.64838s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.7181s elapsed
│ │ │ + -- 2.11431s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out
│ │ │ @@ -36,17 +36,17 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ + -- used 1.09254s (cpu); 0.998408s (thread); 0s (gc)
│ │ │ + -- used 1.19254s (cpu); 0.981584s (thread); 0s (gc)
│ │ │ + -- used 0.10505s (cpu); 0.107124s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html
│ │ │ @@ -94,28 +94,28 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.66263s elapsed
│ │ │ + -- 2.0742s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- 1.80662s elapsed
│ │ │ + -- 1.64838s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 1.66263s elapsed
│ │ │ │ + -- 2.0742s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ │ │  
│ │ │ │  
│ │ │ │       0      1       2        3        4         5         6         7         8
│ │ │ │  9        10      11
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ │ - -- 1.80662s elapsed
│ │ │ │ + -- 1.64838s elapsed
│ │ │ │  
│ │ │ │        1      35      140      385      819      1080      819      385      140
│ │ │ │  35      1
│ │ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S
│ │ │ │  <-- S   <-- S  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.7181s elapsed
│ │ │ + -- 2.11431s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 1.7181s elapsed
│ │ │ │ + -- 2.11431s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html
│ │ │ @@ -117,17 +117,17 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ + -- used 1.09254s (cpu); 0.998408s (thread); 0s (gc)
│ │ │ + -- used 1.19254s (cpu); 0.981584s (thread); 0s (gc)
│ │ │ + -- used 0.10505s (cpu); 0.107124s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,17 +50,17 @@
│ │ │ │            << res M << endl << endl;
│ │ │ │            break;
│ │ │ │            ) else (
│ │ │ │            << "-- computation interrupted" << endl;
│ │ │ │            status M.cache.resolution;
│ │ │ │            << "-- continuing the computation" << endl;
│ │ │ │            ))
│ │ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ │ + -- used 1.09254s (cpu); 0.998408s (thread); 0s (gc)
│ │ │ │ + -- used 1.19254s (cpu); 0.981584s (thread); 0s (gc)
│ │ │ │ + -- used 0.10505s (cpu); 0.107124s (thread); 0s (gc)
│ │ │ │  -- computation started:
│ │ │ │  -- computation interrupted
│ │ │ │  -- continuing the computation
│ │ │ │  -- computation interrupted
│ │ │ │  -- continuing the computation
│ │ │ │  -- computation complete
│ │ │ │   4      11      89      122      40
│ │ ├── ./usr/share/doc/Macaulay2/OnlineLookup/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  b2Vpcw==
│ │ │  #:len=753
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT0VJUyBsb29rdXAiLCBEZXNjcmlwdGlv
│ │ │  biA9PiAoRElWe1BBUkF7VEVYeyJUaGlzIGZ1bmN0aW9uIGxvb2tzIHVwIHRoZSBhcmd1bWVudCAo
│ │ ├── ./usr/share/doc/Macaulay2/OpenMath/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  T3Blbk1hdGg=
│ │ │  #:len=478
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3Blbk1hdGggc3VwcG9ydCIsICJsaW5l
│ │ │  bnVtIiA9PiA4NSwgU2VlQWxzbyA9PiBESVZ7SEVBREVSMnsiU2VlIGFsc28ifSxVTHtMSXtUT0h7
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  b3NjUmluZyguLi4sQ29lZmZpY2llbnRSaW5nPT4uLi4p
│ │ │  #:len=277
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tvc2NSaW5nLENvZWZmaWNpZW50UmluZ10sIm9zY1Jp
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out
│ │ │ @@ -182,25 +182,25 @@
│ │ │  o15 = 4
│ │ │  
│ │ │  i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .284219s elapsed
│ │ │ - -- .401541s elapsed
│ │ │ - -- .867569s elapsed
│ │ │ - -- .245534s elapsed
│ │ │ - -- .354798s elapsed
│ │ │ - -- .429779s elapsed
│ │ │ - -- .666427s elapsed
│ │ │ - -- .418819s elapsed
│ │ │ - -- .478799s elapsed
│ │ │ - -- .409628s elapsed
│ │ │ - -- .282237s elapsed
│ │ │ + -- .335419s elapsed
│ │ │ + -- .331055s elapsed
│ │ │ + -- .497901s elapsed
│ │ │ + -- .262321s elapsed
│ │ │ + -- .281588s elapsed
│ │ │ + -- .335308s elapsed
│ │ │ + -- .619586s elapsed
│ │ │ + -- .493291s elapsed
│ │ │ + -- .552481s elapsed
│ │ │ + -- .405523s elapsed
│ │ │ + -- .257239s elapsed
│ │ │  
│ │ │  i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │        +---------------+---------------+
│ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ @@ -242,75 +242,75 @@
│ │ │  o22 = 15
│ │ │  
│ │ │  i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .539129s elapsed
│ │ │ - -- .645006s elapsed
│ │ │ - -- 1.05918s elapsed
│ │ │ - -- 1.23488s elapsed
│ │ │ - -- .813801s elapsed
│ │ │ - -- 1.01888s elapsed
│ │ │ - -- 1.18556s elapsed
│ │ │ - -- 1.19237s elapsed
│ │ │ - -- .809367s elapsed
│ │ │ - -- .688287s elapsed
│ │ │ - -- .34871s elapsed
│ │ │ - -- .470565s elapsed
│ │ │ - -- .796532s elapsed
│ │ │ - -- .592092s elapsed
│ │ │ - -- .870412s elapsed
│ │ │ - -- 1.45151s elapsed
│ │ │ - -- 1.08533s elapsed
│ │ │ - -- .929525s elapsed
│ │ │ - -- 1.53034s elapsed
│ │ │ - -- 1.33678s elapsed
│ │ │ - -- .760184s elapsed
│ │ │ - -- 1.28518s elapsed
│ │ │ - -- 1.71666s elapsed
│ │ │ - -- 1.22854s elapsed
│ │ │ - -- .674943s elapsed
│ │ │ - -- .894164s elapsed
│ │ │ - -- 1.57127s elapsed
│ │ │ - -- .743101s elapsed
│ │ │ - -- .756756s elapsed
│ │ │ - -- 1.13138s elapsed
│ │ │ - -- 1.25775s elapsed
│ │ │ - -- 1.00643s elapsed
│ │ │ - -- .580285s elapsed
│ │ │ - -- 1.47228s elapsed
│ │ │ - -- 1.02334s elapsed
│ │ │ - -- 1.53993s elapsed
│ │ │ - -- .942582s elapsed
│ │ │ - -- 1.21686s elapsed
│ │ │ - -- 1.59056s elapsed
│ │ │ - -- 1.01279s elapsed
│ │ │ - -- .781942s elapsed
│ │ │ - -- 1.63071s elapsed
│ │ │ - -- 1.80895s elapsed
│ │ │ - -- 2.00814s elapsed
│ │ │ - -- 1.15234s elapsed
│ │ │ - -- 1.32068s elapsed
│ │ │ - -- 1.58339s elapsed
│ │ │ - -- 1.20196s elapsed
│ │ │ - -- 1.10322s elapsed
│ │ │ - -- 1.215s elapsed
│ │ │ - -- 1.22183s elapsed
│ │ │ - -- .920406s elapsed
│ │ │ - -- .900219s elapsed
│ │ │ - -- 1.32942s elapsed
│ │ │ - -- .609367s elapsed
│ │ │ - -- 1.44447s elapsed
│ │ │ - -- 1.35121s elapsed
│ │ │ - -- 1.47404s elapsed
│ │ │ - -- .758204s elapsed
│ │ │ - -- .533039s elapsed
│ │ │ - -- .40806s elapsed
│ │ │ + -- .430481s elapsed
│ │ │ + -- .478611s elapsed
│ │ │ + -- 1.07401s elapsed
│ │ │ + -- 1.27661s elapsed
│ │ │ + -- .798695s elapsed
│ │ │ + -- .973423s elapsed
│ │ │ + -- 1.11411s elapsed
│ │ │ + -- 1.19183s elapsed
│ │ │ + -- .785875s elapsed
│ │ │ + -- .809735s elapsed
│ │ │ + -- .445107s elapsed
│ │ │ + -- .503298s elapsed
│ │ │ + -- .553849s elapsed
│ │ │ + -- .852018s elapsed
│ │ │ + -- .979527s elapsed
│ │ │ + -- 1.49724s elapsed
│ │ │ + -- 1.06241s elapsed
│ │ │ + -- .962888s elapsed
│ │ │ + -- 1.36874s elapsed
│ │ │ + -- 1.18341s elapsed
│ │ │ + -- .85027s elapsed
│ │ │ + -- .959929s elapsed
│ │ │ + -- 1.56453s elapsed
│ │ │ + -- 1.45758s elapsed
│ │ │ + -- .563328s elapsed
│ │ │ + -- .777612s elapsed
│ │ │ + -- 1.55993s elapsed
│ │ │ + -- .793068s elapsed
│ │ │ + -- .682856s elapsed
│ │ │ + -- 1.03098s elapsed
│ │ │ + -- 1.19399s elapsed
│ │ │ + -- 1.03686s elapsed
│ │ │ + -- .599697s elapsed
│ │ │ + -- 1.17951s elapsed
│ │ │ + -- .826155s elapsed
│ │ │ + -- 1.2777s elapsed
│ │ │ + -- 1.07736s elapsed
│ │ │ + -- 1.29144s elapsed
│ │ │ + -- 1.5116s elapsed
│ │ │ + -- .793485s elapsed
│ │ │ + -- .818472s elapsed
│ │ │ + -- 1.33471s elapsed
│ │ │ + -- 1.7221s elapsed
│ │ │ + -- 1.85756s elapsed
│ │ │ + -- 1.21867s elapsed
│ │ │ + -- 1.24148s elapsed
│ │ │ + -- 1.49977s elapsed
│ │ │ + -- 1.2667s elapsed
│ │ │ + -- 1.14228s elapsed
│ │ │ + -- 1.1876s elapsed
│ │ │ + -- 1.33001s elapsed
│ │ │ + -- .979043s elapsed
│ │ │ + -- .955915s elapsed
│ │ │ + -- 1.09069s elapsed
│ │ │ + -- .786395s elapsed
│ │ │ + -- 1.28427s elapsed
│ │ │ + -- 1.3899s elapsed
│ │ │ + -- 1.4169s elapsed
│ │ │ + -- .806374s elapsed
│ │ │ + -- .52043s elapsed
│ │ │ + -- .426372s elapsed
│ │ │  
│ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │  o24 = |codimensions            |degrees                 |
│ │ │        +------------------------+------------------------+
│ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out
│ │ │ @@ -39,15 +39,15 @@
│ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │       ------------------------------------------------------------------------
│ │ │       .954}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .859s elapsed
│ │ │ + -- 1.04s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -60,15 +60,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │ - -- .901s elapsed
│ │ │ + -- 1.09s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │  
│ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  
│ │ │  i5 : printingPrecision = 3
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ - -- .646s elapsed
│ │ │ + -- .746s elapsed
│ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97}
│ │ │ - -- .578s elapsed
│ │ │ - -- .694s elapsed
│ │ │ - -- .859s elapsed
│ │ │ + -- .672s elapsed
│ │ │ + -- .972s elapsed
│ │ │ + -- 1.08s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ @@ -69,20 +69,20 @@
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │ - -- .966s elapsed
│ │ │ - -- 1.19s elapsed
│ │ │ + -- 1.35s elapsed
│ │ │ + -- 1.35s elapsed
│ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, 47, 48, 53, 59, 60, 61}
│ │ │ - -- 1.27s elapsed
│ │ │ + -- 1.73s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34}
│ │ │ - -- 1.07s elapsed
│ │ │ - -- 1.05s elapsed
│ │ │ + -- 1.51s elapsed
│ │ │ + -- 1.53s elapsed
│ │ │  warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ - -- 1.21s elapsed
│ │ │ + -- 1.8s elapsed
│ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, 75, 76, 77}
│ │ │ - -- 1.03s elapsed
│ │ │ + -- 1.5s elapsed
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 1729328129346969841
│ │ │  
│ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │  
│ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21}
│ │ │ - -- .215945s elapsed
│ │ │ + -- .232354s elapsed
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │             2 => {1, 3}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 3}
│ │ │  
│ │ │  o1 : Graph
│ │ │  
│ │ │  i2 : showExoticSolutions G
│ │ │ - -- .722313s elapsed
│ │ │ + -- 1.06031s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -48,14 +48,14 @@
│ │ │             2 => {1, 3, 4}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 2, 3}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : showExoticSolutions G
│ │ │ - -- .986724s elapsed
│ │ │ + -- 1.37758s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html
│ │ │ @@ -300,25 +300,25 @@
│ │ │            
│ │ │              
│ │ │                
    i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .284219s elapsed
│ │ │ - -- .401541s elapsed
│ │ │ - -- .867569s elapsed
│ │ │ - -- .245534s elapsed
│ │ │ - -- .354798s elapsed
│ │ │ - -- .429779s elapsed
│ │ │ - -- .666427s elapsed
│ │ │ - -- .418819s elapsed
│ │ │ - -- .478799s elapsed
│ │ │ - -- .409628s elapsed
│ │ │ - -- .282237s elapsed
    
│ │ │ + -- .335419s elapsed
│ │ │ + -- .331055s elapsed
│ │ │ + -- .497901s elapsed
│ │ │ + -- .262321s elapsed
│ │ │ + -- .281588s elapsed
│ │ │ + -- .335308s elapsed
│ │ │ + -- .619586s elapsed
│ │ │ + -- .493291s elapsed
│ │ │ + -- .552481s elapsed
│ │ │ + -- .405523s elapsed
│ │ │ + -- .257239s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │ @@ -385,75 +385,75 @@
│ │ │            
│ │ │              
│ │ │                
    i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .539129s elapsed
│ │ │ - -- .645006s elapsed
│ │ │ - -- 1.05918s elapsed
│ │ │ - -- 1.23488s elapsed
│ │ │ - -- .813801s elapsed
│ │ │ - -- 1.01888s elapsed
│ │ │ - -- 1.18556s elapsed
│ │ │ - -- 1.19237s elapsed
│ │ │ - -- .809367s elapsed
│ │ │ - -- .688287s elapsed
│ │ │ - -- .34871s elapsed
│ │ │ - -- .470565s elapsed
│ │ │ - -- .796532s elapsed
│ │ │ - -- .592092s elapsed
│ │ │ - -- .870412s elapsed
│ │ │ - -- 1.45151s elapsed
│ │ │ - -- 1.08533s elapsed
│ │ │ - -- .929525s elapsed
│ │ │ - -- 1.53034s elapsed
│ │ │ - -- 1.33678s elapsed
│ │ │ - -- .760184s elapsed
│ │ │ - -- 1.28518s elapsed
│ │ │ - -- 1.71666s elapsed
│ │ │ - -- 1.22854s elapsed
│ │ │ - -- .674943s elapsed
│ │ │ - -- .894164s elapsed
│ │ │ - -- 1.57127s elapsed
│ │ │ - -- .743101s elapsed
│ │ │ - -- .756756s elapsed
│ │ │ - -- 1.13138s elapsed
│ │ │ - -- 1.25775s elapsed
│ │ │ - -- 1.00643s elapsed
│ │ │ - -- .580285s elapsed
│ │ │ - -- 1.47228s elapsed
│ │ │ - -- 1.02334s elapsed
│ │ │ - -- 1.53993s elapsed
│ │ │ - -- .942582s elapsed
│ │ │ - -- 1.21686s elapsed
│ │ │ - -- 1.59056s elapsed
│ │ │ - -- 1.01279s elapsed
│ │ │ - -- .781942s elapsed
│ │ │ - -- 1.63071s elapsed
│ │ │ - -- 1.80895s elapsed
│ │ │ - -- 2.00814s elapsed
│ │ │ - -- 1.15234s elapsed
│ │ │ - -- 1.32068s elapsed
│ │ │ - -- 1.58339s elapsed
│ │ │ - -- 1.20196s elapsed
│ │ │ - -- 1.10322s elapsed
│ │ │ - -- 1.215s elapsed
│ │ │ - -- 1.22183s elapsed
│ │ │ - -- .920406s elapsed
│ │ │ - -- .900219s elapsed
│ │ │ - -- 1.32942s elapsed
│ │ │ - -- .609367s elapsed
│ │ │ - -- 1.44447s elapsed
│ │ │ - -- 1.35121s elapsed
│ │ │ - -- 1.47404s elapsed
│ │ │ - -- .758204s elapsed
│ │ │ - -- .533039s elapsed
│ │ │ - -- .40806s elapsed
    
│ │ │ + -- .430481s elapsed
│ │ │ + -- .478611s elapsed
│ │ │ + -- 1.07401s elapsed
│ │ │ + -- 1.27661s elapsed
│ │ │ + -- .798695s elapsed
│ │ │ + -- .973423s elapsed
│ │ │ + -- 1.11411s elapsed
│ │ │ + -- 1.19183s elapsed
│ │ │ + -- .785875s elapsed
│ │ │ + -- .809735s elapsed
│ │ │ + -- .445107s elapsed
│ │ │ + -- .503298s elapsed
│ │ │ + -- .553849s elapsed
│ │ │ + -- .852018s elapsed
│ │ │ + -- .979527s elapsed
│ │ │ + -- 1.49724s elapsed
│ │ │ + -- 1.06241s elapsed
│ │ │ + -- .962888s elapsed
│ │ │ + -- 1.36874s elapsed
│ │ │ + -- 1.18341s elapsed
│ │ │ + -- .85027s elapsed
│ │ │ + -- .959929s elapsed
│ │ │ + -- 1.56453s elapsed
│ │ │ + -- 1.45758s elapsed
│ │ │ + -- .563328s elapsed
│ │ │ + -- .777612s elapsed
│ │ │ + -- 1.55993s elapsed
│ │ │ + -- .793068s elapsed
│ │ │ + -- .682856s elapsed
│ │ │ + -- 1.03098s elapsed
│ │ │ + -- 1.19399s elapsed
│ │ │ + -- 1.03686s elapsed
│ │ │ + -- .599697s elapsed
│ │ │ + -- 1.17951s elapsed
│ │ │ + -- .826155s elapsed
│ │ │ + -- 1.2777s elapsed
│ │ │ + -- 1.07736s elapsed
│ │ │ + -- 1.29144s elapsed
│ │ │ + -- 1.5116s elapsed
│ │ │ + -- .793485s elapsed
│ │ │ + -- .818472s elapsed
│ │ │ + -- 1.33471s elapsed
│ │ │ + -- 1.7221s elapsed
│ │ │ + -- 1.85756s elapsed
│ │ │ + -- 1.21867s elapsed
│ │ │ + -- 1.24148s elapsed
│ │ │ + -- 1.49977s elapsed
│ │ │ + -- 1.2667s elapsed
│ │ │ + -- 1.14228s elapsed
│ │ │ + -- 1.1876s elapsed
│ │ │ + -- 1.33001s elapsed
│ │ │ + -- .979043s elapsed
│ │ │ + -- .955915s elapsed
│ │ │ + -- 1.09069s elapsed
│ │ │ + -- .786395s elapsed
│ │ │ + -- 1.28427s elapsed
│ │ │ + -- 1.3899s elapsed
│ │ │ + -- 1.4169s elapsed
│ │ │ + -- .806374s elapsed
│ │ │ + -- .52043s elapsed
│ │ │ + -- .426372s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -180,25 +180,25 @@
│ │ │ │  
│ │ │ │  o15 = 4
│ │ │ │  i16 : for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .284219s elapsed
│ │ │ │ - -- .401541s elapsed
│ │ │ │ - -- .867569s elapsed
│ │ │ │ - -- .245534s elapsed
│ │ │ │ - -- .354798s elapsed
│ │ │ │ - -- .429779s elapsed
│ │ │ │ - -- .666427s elapsed
│ │ │ │ - -- .418819s elapsed
│ │ │ │ - -- .478799s elapsed
│ │ │ │ - -- .409628s elapsed
│ │ │ │ - -- .282237s elapsed
│ │ │ │ + -- .335419s elapsed
│ │ │ │ + -- .331055s elapsed
│ │ │ │ + -- .497901s elapsed
│ │ │ │ + -- .262321s elapsed
│ │ │ │ + -- .281588s elapsed
│ │ │ │ + -- .335308s elapsed
│ │ │ │ + -- .619586s elapsed
│ │ │ │ + -- .493291s elapsed
│ │ │ │ + -- .552481s elapsed
│ │ │ │ + -- .405523s elapsed
│ │ │ │ + -- .257239s elapsed
│ │ │ │  i17 : netList oo
│ │ │ │  
│ │ │ │        +---------------+---------------+
│ │ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │ @@ -233,75 +233,75 @@
│ │ │ │  
│ │ │ │  o22 = 15
│ │ │ │  i23 : allcomps = for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .539129s elapsed
│ │ │ │ - -- .645006s elapsed
│ │ │ │ - -- 1.05918s elapsed
│ │ │ │ - -- 1.23488s elapsed
│ │ │ │ - -- .813801s elapsed
│ │ │ │ - -- 1.01888s elapsed
│ │ │ │ - -- 1.18556s elapsed
│ │ │ │ - -- 1.19237s elapsed
│ │ │ │ - -- .809367s elapsed
│ │ │ │ - -- .688287s elapsed
│ │ │ │ - -- .34871s elapsed
│ │ │ │ - -- .470565s elapsed
│ │ │ │ - -- .796532s elapsed
│ │ │ │ - -- .592092s elapsed
│ │ │ │ - -- .870412s elapsed
│ │ │ │ - -- 1.45151s elapsed
│ │ │ │ - -- 1.08533s elapsed
│ │ │ │ - -- .929525s elapsed
│ │ │ │ - -- 1.53034s elapsed
│ │ │ │ - -- 1.33678s elapsed
│ │ │ │ - -- .760184s elapsed
│ │ │ │ - -- 1.28518s elapsed
│ │ │ │ - -- 1.71666s elapsed
│ │ │ │ - -- 1.22854s elapsed
│ │ │ │ - -- .674943s elapsed
│ │ │ │ - -- .894164s elapsed
│ │ │ │ - -- 1.57127s elapsed
│ │ │ │ - -- .743101s elapsed
│ │ │ │ - -- .756756s elapsed
│ │ │ │ - -- 1.13138s elapsed
│ │ │ │ - -- 1.25775s elapsed
│ │ │ │ - -- 1.00643s elapsed
│ │ │ │ - -- .580285s elapsed
│ │ │ │ - -- 1.47228s elapsed
│ │ │ │ - -- 1.02334s elapsed
│ │ │ │ - -- 1.53993s elapsed
│ │ │ │ - -- .942582s elapsed
│ │ │ │ - -- 1.21686s elapsed
│ │ │ │ - -- 1.59056s elapsed
│ │ │ │ - -- 1.01279s elapsed
│ │ │ │ - -- .781942s elapsed
│ │ │ │ - -- 1.63071s elapsed
│ │ │ │ - -- 1.80895s elapsed
│ │ │ │ - -- 2.00814s elapsed
│ │ │ │ - -- 1.15234s elapsed
│ │ │ │ - -- 1.32068s elapsed
│ │ │ │ - -- 1.58339s elapsed
│ │ │ │ - -- 1.20196s elapsed
│ │ │ │ - -- 1.10322s elapsed
│ │ │ │ - -- 1.215s elapsed
│ │ │ │ - -- 1.22183s elapsed
│ │ │ │ - -- .920406s elapsed
│ │ │ │ - -- .900219s elapsed
│ │ │ │ - -- 1.32942s elapsed
│ │ │ │ - -- .609367s elapsed
│ │ │ │ - -- 1.44447s elapsed
│ │ │ │ - -- 1.35121s elapsed
│ │ │ │ - -- 1.47404s elapsed
│ │ │ │ - -- .758204s elapsed
│ │ │ │ - -- .533039s elapsed
│ │ │ │ - -- .40806s elapsed
│ │ │ │ + -- .430481s elapsed
│ │ │ │ + -- .478611s elapsed
│ │ │ │ + -- 1.07401s elapsed
│ │ │ │ + -- 1.27661s elapsed
│ │ │ │ + -- .798695s elapsed
│ │ │ │ + -- .973423s elapsed
│ │ │ │ + -- 1.11411s elapsed
│ │ │ │ + -- 1.19183s elapsed
│ │ │ │ + -- .785875s elapsed
│ │ │ │ + -- .809735s elapsed
│ │ │ │ + -- .445107s elapsed
│ │ │ │ + -- .503298s elapsed
│ │ │ │ + -- .553849s elapsed
│ │ │ │ + -- .852018s elapsed
│ │ │ │ + -- .979527s elapsed
│ │ │ │ + -- 1.49724s elapsed
│ │ │ │ + -- 1.06241s elapsed
│ │ │ │ + -- .962888s elapsed
│ │ │ │ + -- 1.36874s elapsed
│ │ │ │ + -- 1.18341s elapsed
│ │ │ │ + -- .85027s elapsed
│ │ │ │ + -- .959929s elapsed
│ │ │ │ + -- 1.56453s elapsed
│ │ │ │ + -- 1.45758s elapsed
│ │ │ │ + -- .563328s elapsed
│ │ │ │ + -- .777612s elapsed
│ │ │ │ + -- 1.55993s elapsed
│ │ │ │ + -- .793068s elapsed
│ │ │ │ + -- .682856s elapsed
│ │ │ │ + -- 1.03098s elapsed
│ │ │ │ + -- 1.19399s elapsed
│ │ │ │ + -- 1.03686s elapsed
│ │ │ │ + -- .599697s elapsed
│ │ │ │ + -- 1.17951s elapsed
│ │ │ │ + -- .826155s elapsed
│ │ │ │ + -- 1.2777s elapsed
│ │ │ │ + -- 1.07736s elapsed
│ │ │ │ + -- 1.29144s elapsed
│ │ │ │ + -- 1.5116s elapsed
│ │ │ │ + -- .793485s elapsed
│ │ │ │ + -- .818472s elapsed
│ │ │ │ + -- 1.33471s elapsed
│ │ │ │ + -- 1.7221s elapsed
│ │ │ │ + -- 1.85756s elapsed
│ │ │ │ + -- 1.21867s elapsed
│ │ │ │ + -- 1.24148s elapsed
│ │ │ │ + -- 1.49977s elapsed
│ │ │ │ + -- 1.2667s elapsed
│ │ │ │ + -- 1.14228s elapsed
│ │ │ │ + -- 1.1876s elapsed
│ │ │ │ + -- 1.33001s elapsed
│ │ │ │ + -- .979043s elapsed
│ │ │ │ + -- .955915s elapsed
│ │ │ │ + -- 1.09069s elapsed
│ │ │ │ + -- .786395s elapsed
│ │ │ │ + -- 1.28427s elapsed
│ │ │ │ + -- 1.3899s elapsed
│ │ │ │ + -- 1.4169s elapsed
│ │ │ │ + -- .806374s elapsed
│ │ │ │ + -- .52043s elapsed
│ │ │ │ + -- .426372s elapsed
│ │ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │ │  
│ │ │ │        +------------------------+------------------------+
│ │ │ │  o24 = |codimensions            |degrees                 |
│ │ │ │        +------------------------+------------------------+
│ │ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ │ │        +------------------------+------------------------+
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .859s elapsed
│ │ │ + -- 1.04s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -136,15 +136,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │ - -- .901s elapsed
│ │ │ + -- 1.09s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       .954}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ │ - -- .859s elapsed
│ │ │ │ + -- 1.04s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │ │                                                  1 => {0, 2}
│ │ │ │                                                  2 => {1, 3}
│ │ │ │                                                  3 => {2, 4}
│ │ │ │                                                  4 => {0, 3}
│ │ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ │ @@ -64,15 +64,15 @@
│ │ │ │  +---+---+---+---+
│ │ │ │  |72 |144|216|288|
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │  |288|216|144|72 |
│ │ │ │  +---+---+---+---+
│ │ │ │ - -- .901s elapsed
│ │ │ │ + -- 1.09s elapsed
│ │ │ │  
│ │ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Computing the (linearly) stable solutions for K5C5 takes a minute or two:
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html
│ │ │ @@ -120,19 +120,19 @@
│ │ │  o5 = 3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ - -- .646s elapsed
│ │ │ + -- .746s elapsed
│ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97}
│ │ │ - -- .578s elapsed
│ │ │ - -- .694s elapsed
│ │ │ - -- .859s elapsed
│ │ │ + -- .672s elapsed
│ │ │ + -- .972s elapsed
│ │ │ + -- 1.08s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ @@ -145,25 +145,25 @@
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │ - -- .966s elapsed
│ │ │ - -- 1.19s elapsed
│ │ │ + -- 1.35s elapsed
│ │ │ + -- 1.35s elapsed
│ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, 47, 48, 53, 59, 60, 61}
│ │ │ - -- 1.27s elapsed
│ │ │ + -- 1.73s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34}
│ │ │ - -- 1.07s elapsed
│ │ │ - -- 1.05s elapsed
│ │ │ + -- 1.51s elapsed
│ │ │ + -- 1.53s elapsed
│ │ │  warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ - -- 1.21s elapsed
│ │ │ + -- 1.8s elapsed
│ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, 75, 76, 77}
│ │ │ - -- 1.03s elapsed
    
│ │ │ + -- 1.5s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,21 +48,21 @@
│ │ │ │  i5 : printingPrecision = 3
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53,
│ │ │ │  54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79,
│ │ │ │  80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ │ - -- .646s elapsed
│ │ │ │ + -- .746s elapsed
│ │ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61,
│ │ │ │  62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88,
│ │ │ │  91, 94, 95, 97}
│ │ │ │ - -- .578s elapsed
│ │ │ │ - -- .694s elapsed
│ │ │ │ - -- .859s elapsed
│ │ │ │ + -- .672s elapsed
│ │ │ │ + -- .972s elapsed
│ │ │ │ + -- 1.08s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │ │                                                  1 => {3, 4}
│ │ │ │                                                  2 => {0, 4}
│ │ │ │                                                  3 => {0, 1}
│ │ │ │                                                  4 => {2, 1}
│ │ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ │ @@ -75,24 +75,24 @@
│ │ │ │  +---+---+---+---+
│ │ │ │  |0  |0  |0  |0  |
│ │ │ │  +---+---+---+---+
│ │ │ │  |216|72 |288|144|
│ │ │ │  +---+---+---+---+
│ │ │ │  |144|288|72 |216|
│ │ │ │  +---+---+---+---+
│ │ │ │ - -- .966s elapsed
│ │ │ │ - -- 1.19s elapsed
│ │ │ │ + -- 1.35s elapsed
│ │ │ │ + -- 1.35s elapsed
│ │ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46,
│ │ │ │  47, 48, 53, 59, 60, 61}
│ │ │ │ - -- 1.27s elapsed
│ │ │ │ + -- 1.73s elapsed
│ │ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27,
│ │ │ │  28, 29, 30, 31, 32, 33, 34}
│ │ │ │ - -- 1.07s elapsed
│ │ │ │ - -- 1.05s elapsed
│ │ │ │ + -- 1.51s elapsed
│ │ │ │ + -- 1.53s elapsed
│ │ │ │  warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ │ - -- 1.21s elapsed
│ │ │ │ + -- 1.8s elapsed
│ │ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74,
│ │ │ │  75, 76, 77}
│ │ │ │ - -- 1.03s elapsed
│ │ │ │ + -- 1.5s elapsed
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0.
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │                
    i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : getLinearlyStableSolutions(G)
│ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21}
│ │ │ - -- .215945s elapsed
│ │ │ + -- .232354s elapsed
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  of each oscillator is given by the Kuramoto model. The linear stability of a
│ │ │ │  solution is determined by the eigenvalues of the Jacobian matrix of the system
│ │ │ │  evaluated at the solution.
│ │ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15,
│ │ │ │  16, 17, 18, 19, 21}
│ │ │ │ - -- .215945s elapsed
│ │ │ │ + -- .232354s elapsed
│ │ │ │  
│ │ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _f_i_n_d_R_e_a_l_S_o_l_u_t_i_o_n_s -- find real solutions, at least one per component for
│ │ │ │        well-conditioned systems
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o1 : Graph
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : showExoticSolutions G
│ │ │ - -- .722313s elapsed
│ │ │ + -- 1.06031s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : showExoticSolutions G
│ │ │ - -- .986724s elapsed
│ │ │ + -- 1.37758s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │             1 => {0, 2}
│ │ │ │             2 => {1, 3}
│ │ │ │             3 => {2, 4}
│ │ │ │             4 => {0, 3}
│ │ │ │  
│ │ │ │  o1 : Graph
│ │ │ │  i2 : showExoticSolutions G
│ │ │ │ - -- .722313s elapsed
│ │ │ │ + -- 1.06031s elapsed
│ │ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │ │                                                  1 => {0, 2}
│ │ │ │                                                  2 => {1, 3}
│ │ │ │                                                  3 => {2, 4}
│ │ │ │                                                  4 => {0, 3}
│ │ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │             1 => {0, 2}
│ │ │ │             2 => {1, 3, 4}
│ │ │ │             3 => {2, 4}
│ │ │ │             4 => {0, 2, 3}
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : showExoticSolutions G
│ │ │ │ - -- .986724s elapsed
│ │ │ │ + -- 1.37758s elapsed
│ │ │ │  
│ │ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_e_t_L_i_n_e_a_r_l_y_S_t_a_b_l_e_S_o_l_u_t_i_o_n_s -- Compute linearly stable solutions for the
│ │ │ │        Kuramoto oscillator system associated to a graph
│ │ ├── ./usr/share/doc/Macaulay2/PHCpack/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  ZmFjdG9yV2l0bmVzc1NldCguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=595
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uIHRvIHNwZWNpZnkgd2hldGhl
│ │ │  ciBhZGRpdGlvbmFsIG91dHB1dCBpcyB3YW50ZWQiLCAibGluZW51bSIgPT4gNDEyLCAiZmlsZW5h
│ │ ├── ./usr/share/doc/Macaulay2/PackageCitations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  UGFja2FnZUNpdGF0aW9ucw==
│ │ │  #:len=1928
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZhY2lsaXRhdGluZyBj
│ │ │  aXRhdGlvbiBvZiBNYWNhdWxheTIgcGFja2FnZXMiLCAibGluZW51bSIgPT4gMjY1LCBTZWVBbHNv
│ │ ├── ./usr/share/doc/Macaulay2/PackageTemplate/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  c2Vjb25kRnVuY3Rpb24=
│ │ │  #:len=379
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBzaWxseSBzZWNvbmQgZnVuY3Rpb24i
│ │ │  LCBEZXNjcmlwdGlvbiA9PiAoIlRoaXMgZnVuY3Rpb24gaXMgcHJvdmlkZWQgYnkgdGhlIHBhY2th
│ │ ├── ./usr/share/doc/Macaulay2/Padic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  aW52ZXJzZShQYWRpY051bWJlcik=
│ │ │  #:len=1089
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibXVsdGlwbGljYXRpdmUgaW52ZXJzZSBv
│ │ │  ZiBhIHAtYWRpYyBudW1iZXIiLCAibGluZW51bSIgPT4gNzU4LCBJbnB1dHMgPT4ge1NQQU57VFR7
│ │ ├── ./usr/share/doc/Macaulay2/Parametrization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  cGFyYW1ldHJpemVDb25pYw==
│ │ │  #:len=1904
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3B0aW9uIHdoZXRoZXIgdG8gcmF0aW9u
│ │ │  YWxseSBwYXJhbWV0cml6ZSBjb25pY3MuIiwgImxpbmVudW0iID0+IDE2NzYsICJmaWxlbmFtZSIg
│ │ ├── ./usr/share/doc/Macaulay2/Parsing/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  Y2hhckFuYWx5emVy
│ │ │  #:len=752
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBsZXhpY2FsIGFuYWx5emVyIHRoYXQg
│ │ │  cHJvdmlkZXMgY2hhcmFjdGVycyBmcm9tIGEgc3RyaW5nIG9uZSBhdCBhIHRpbWUiLCAibGluZW51
│ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=7
│ │ │  YW1iaWVudA==
│ │ │  #:len=256
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZXhwb3J0ZWQgZnJvbSB0aGUgTkNBTGdl
│ │ │  YnJhIHBhY2thZ2UuIiwgRGVzY3JpcHRpb24gPT4ge30sICJsaW5lbnVtIiA9PiAxNTksIHN5bWJv
│ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out
│ │ │ @@ -80,20 +80,20 @@
│ │ │  i19 : needsPackage "MultigradedImplicitization";
│ │ │  
│ │ │  i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .0157139s elapsed
│ │ │ + -- .0113357s elapsed
│ │ │  WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.
│ │ │  computing total degree: 2
│ │ │  number of monomials = 45
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .717745s elapsed
│ │ │ + -- .594891s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
│ │ │  
│ │ │  i21 : dim I
│ │ │  
│ │ │  o21 = 5
│ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html
│ │ │ @@ -213,20 +213,20 @@
│ │ │            
│ │ │              
│ │ │                
    i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .0157139s elapsed
│ │ │ + -- .0113357s elapsed
│ │ │  WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.
│ │ │  computing total degree: 2
│ │ │  number of monomials = 45
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .717745s elapsed
│ │ │ + -- .594891s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : dim I
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,21 +77,21 @@
│ │ │ │  i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix
│ │ │ │  {toList(9:1)}), S);
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 9
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .0157139s elapsed
│ │ │ │ + -- .0113357s elapsed
│ │ │ │  WARNING: There are linear relations. You may want to reduce the number of
│ │ │ │  variables to speed up the computation.
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 45
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .717745s elapsed
│ │ │ │ + -- .594891s elapsed
│ │ │ │  
│ │ │ │  o20 : Ideal of S
│ │ │ │  i21 : dim I
│ │ │ │  
│ │ │ │  o21 = 5
│ │ │ │  i22 : isPrime I
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  cmFuZG9tRXh0ZW5zaW9uKE1hdHJpeCxNYXRyaXgp
│ │ │  #:len=301
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzIxMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmFuZG9tRXh0ZW5zaW9uLE1hdHJpeCxNYXRyaXgp
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │  i3 : g=3
│ │ │  
│ │ │  o3 = 3
│ │ │  
│ │ │  i4 : kk= ZZ/101;
│ │ │  
│ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- .260086s elapsed
│ │ │ + -- .270421s elapsed
│ │ │  
│ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │  
│ │ │  o6 : CliffordModule
│ │ │  
│ │ │ @@ -67,30 +67,30 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .898827s elapsed
│ │ │ + -- .432392s elapsed
│ │ │  
│ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │            -4: 16  .     -2: 32  .
│ │ │            -3: 16  .     -1:  .  .
│ │ │            -2:  .  .      0:  .  .
│ │ │            -1:  . 16      1:  . 32
│ │ │             0:  . 16
│ │ │  
│ │ │  o12 : Sequence
│ │ │  
│ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 30.2671s elapsed
│ │ │ + -- 18.6434s elapsed
│ │ │  
│ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │  o14 : Matrix S   <-- S
│ │ │  
│ │ │  i15 : r=2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_search__Ulrich.out
│ │ │ @@ -46,30 +46,30 @@
│ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o11 = CliffordModule{...6...}
│ │ │  
│ │ │  o11 : CliffordModule
│ │ │  
│ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- 1.14431s elapsed
│ │ │ + -- .692644s elapsed
│ │ │  
│ │ │  i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │  o13 = total: 8 16 8
│ │ │            0: 8 16 8
│ │ │  
│ │ │  o13 : BettiTally
│ │ │  
│ │ │  i14 : ann Ulr == ideal qs
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.91837s elapsed
│ │ │ + -- 2.26056s elapsed
│ │ │  
│ │ │  i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
│ │ │ @@ -133,15 +133,15 @@
│ │ │              
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- .260086s elapsed
    
│ │ │ + -- .270421s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │ @@ -177,15 +177,15 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .898827s elapsed
    
│ │ │ + -- .432392s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o12 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 30.2671s elapsed
    
│ │ │ + -- 18.6434s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │ ├── html2text {}
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │              -- will give an Ulrich bundle, with betti table
│ │ │ │              -- 16 32 16
│ │ │ │  i3 : g=3
│ │ │ │  
│ │ │ │  o3 = 3
│ │ │ │  i4 : kk= ZZ/101;
│ │ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ │ - -- .260086s elapsed
│ │ │ │ + -- .270421s elapsed
│ │ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o6 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o6 : CliffordModule
│ │ │ │  i7 : Mor = vectorBundleOnE M.evenCenter;
│ │ │ │  i8 : Mor1= vectorBundleOnE M.oddCenter;
│ │ │ │ @@ -75,29 +75,29 @@
│ │ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │ │            V = vectorBundleOnE m12;
│ │ │ │            Ul=tensorProduct(Mor,V);
│ │ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │ │            d0=unique degrees target Ul.yAction;
│ │ │ │            d1=unique degrees target Ul1.yAction;
│ │ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ │ - -- .898827s elapsed
│ │ │ │ + -- .432392s elapsed
│ │ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │ │  
│ │ │ │                 0  1          0  1
│ │ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │ │            -4: 16  .     -2: 32  .
│ │ │ │            -3: 16  .     -1:  .  .
│ │ │ │            -2:  .  .      0:  .  .
│ │ │ │            -1:  . 16      1:  . 32
│ │ │ │             0:  . 16
│ │ │ │  
│ │ │ │  o12 : Sequence
│ │ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the
│ │ │ │  actions of generators
│ │ │ │ - -- 30.2671s elapsed
│ │ │ │ + -- 18.6434s elapsed
│ │ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │ │  
│ │ │ │                32      32
│ │ │ │  o14 : Matrix S   <-- S
│ │ │ │  i15 : r=2
│ │ │ │  
│ │ │ │  o15 = 2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o11 : CliffordModule
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- 1.14431s elapsed
    
│ │ │ + -- .692644s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -190,15 +190,15 @@
│ │ │  
│ │ │  o14 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.91837s elapsed
    
│ │ │ + -- 2.26056s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,27 +64,27 @@
│ │ │ │  o10 : Matrix S  <-- S
│ │ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o11 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o11 : CliffordModule
│ │ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ │ - -- 1.14431s elapsed
│ │ │ │ + -- .692644s elapsed
│ │ │ │  i13 : betti freeResolution Ulr
│ │ │ │  
│ │ │ │               0  1 2
│ │ │ │  o13 = total: 8 16 8
│ │ │ │            0: 8 16 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : ann Ulr == ideal qs
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ │ - -- 2.91837s elapsed
│ │ │ │ + -- 2.26056s elapsed
│ │ │ │  i16 : betti freeResolution Ulr3
│ │ │ │  
│ │ │ │                0  1  2
│ │ │ │  o16 = total: 12 24 12
│ │ │ │            0: 12 24 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/Permanents/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  cnlzZXI=
│ │ │  #:len=2107
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBwZXJtYW5lbnQgdXNpbmcg
│ │ │  UnlzZXIncyBmb3JtdWxhIiwgImxpbmVudW0iID0+IDUyMywgSW5wdXRzID0+IHtTUEFOe1RUeyJN
│ │ ├── ./usr/share/doc/Macaulay2/Permutations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZGVzY2VudHMoUGVybXV0YXRpb24p
│ │ │  #:len=276
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkZXNjZW50cyxQZXJtdXRhdGlvbiksImRlc2NlbnRz
│ │ ├── ./usr/share/doc/Macaulay2/PhylogeneticTrees/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  cGh5bG9Ub3JpY1F1YWRz
│ │ │  #:len=2848
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSB0aGUgcXVhZHJhdGljIGlu
│ │ │  dmFyaWFudHMgb2YgYSBncm91cC1iYXNlZCBwaHlsb2dlbmV0aWMgbW9kZWwiLCAibGluZW51bSIg
│ │ ├── ./usr/share/doc/Macaulay2/PieriMaps/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  ZGlzcGxheUxSSW1hZ2UoTGlzdCk=
│ │ │  #:len=258
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkaXNwbGF5TFJJbWFnZSxMaXN0KSwiZGlzcGxheUxS
│ │ ├── ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  YWRkaXRpb24oSWRlYWwsSWRlYWwsSWRlYWwp
│ │ │  #:len=295
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRpdGlvbixJZGVhbCxJZGVhbCxJZGVhbCksImFk
│ │ ├── ./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  cmFuZG9tUG9pbnRzTWF0KFJpbmcsWlop
│ │ │  #:len=255
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODgzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21Qb2ludHNNYXQsUmluZyxaWiksInJhbmRv
│ │ ├── ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out
│ │ │ @@ -66,17 +66,17 @@
│ │ │  i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3}
│ │ │  
│ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.34235s elapsed
│ │ │ + -- 1.8324s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.80041s elapsed
│ │ │ + -- 4.68795s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -182,21 +182,21 @@
│ │ │  
│ │ │  o9 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.34235s elapsed
    
│ │ │ + -- 1.8324s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.80041s elapsed
    
│ │ │ + -- 4.68795s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : G==H
│ │ │  
│ │ │  o12 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,17 +81,17 @@
│ │ │ │  o8 : Matrix K  <-- K
│ │ │ │  i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3}
│ │ │ │  
│ │ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ │ - -- 2.34235s elapsed
│ │ │ │ + -- 1.8324s elapsed
│ │ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ │ - -- 5.80041s elapsed
│ │ │ │ + -- 4.68795s elapsed
│ │ │ │  i12 : G==H
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  For reduced points, this function may be a bit slower than _a_f_f_i_n_e_P_o_i_n_t_s.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _a_f_f_i_n_e_F_a_t_P_o_i_n_t_s_B_y_I_n_t_e_r_s_e_c_t_i_o_n_(_M_a_t_r_i_x_,_L_i_s_t_,_R_i_n_g_) -- computes ideal of fat
│ │ ├── ./usr/share/doc/Macaulay2/Polyhedra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  bWF4Q29uZXM=
│ │ │  #:len=1185
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg
│ │ │  Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gODQzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkYi
│ │ ├── ./usr/share/doc/Macaulay2/Polymake/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  UG9seW1ha2U=
│ │ │  #:len=610
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciBpbnRlcmZhY2lu
│ │ │  ZyB3aXRoIHBvbHltYWtlIiwgRGVzY3JpcHRpb24gPT4gKEVNeyJQb2x5bWFrZSJ9LCIgaXMgYSBw
│ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  c3RhbmRhcmRSb29rTnVtYmVy
│ │ │  #:len=1159
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU3RhbmRhcmQgcm9vayBudW1iZXIgb2Yg
│ │ │  YSBjb2xsZWN0aW9uIG9mIGNlbGxzIiwgImxpbmVudW0iID0+IDE1MzQsIElucHV0cyA9PiB7U1BB
│ │ ├── ./usr/share/doc/Macaulay2/Posets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  bWF4aW1hbEFudGljaGFpbnM=
│ │ │  #:len=1127
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYWxsIG1heGltYWwgYW50
│ │ │  aWNoYWlucyBvZiBhIHBvc2V0IiwgImxpbmVudW0iID0+IDQ5ODgsIElucHV0cyA9PiB7U1BBTntU
│ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out
│ │ │ @@ -31,27 +31,27 @@
│ │ │  o5 = CacheTable{name => P}
│ │ │  
│ │ │  i6 : C == P
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : time isDistributive C
│ │ │ - -- used 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc)
│ │ │ + -- used 1.5072e-05s (cpu); 7.174e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : time isDistributive P
│ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc)
│ │ │ + -- used 8.61041s (cpu); 5.25771s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : C' = dual C;
│ │ │  
│ │ │  i10 : time isDistributive C'
│ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc)
│ │ │ + -- used 1.0861e-05s (cpu); 8.147e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : peek C'.cache
│ │ │  
│ │ │  o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}                                      }
│ │ │                   coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4}, {6, 5}, {7, 6}, {8, 7}, {9, 8}}
│ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out
│ │ │ @@ -7,22 +7,22 @@
│ │ │  o2 = Partition{4, 2}
│ │ │  
│ │ │  o2 : Partition
│ │ │  
│ │ │  i3 : D = dominanceLattice 6;
│ │ │  
│ │ │  i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc)
│ │ │ + -- used 0.532824s (cpu); 0.307911s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
│ │ │  
│ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc)
│ │ │ + -- used 1.7517e-05s (cpu); 1.5606e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
│ │ │  
│ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html
│ │ │ @@ -112,23 +112,23 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time isDistributive C
│ │ │ - -- used 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc)
│ │ │ + -- used 1.5072e-05s (cpu); 7.174e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time isDistributive P
│ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc)
│ │ │ + -- used 8.61041s (cpu); 5.25771s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    We also know that the dual of a distributive lattice is again a distributive lattice.  Other information is copied when possible.
    
│ │ │ @@ -138,15 +138,15 @@
│ │ │              
│ │ │                
    i9 : C' = dual C;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time isDistributive C'
│ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc)
│ │ │ + -- used 1.0861e-05s (cpu); 8.147e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : peek C'.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,26 +41,26 @@
│ │ │ │  i5 : peek P.cache
│ │ │ │  
│ │ │ │  o5 = CacheTable{name => P}
│ │ │ │  i6 : C == P
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : time isDistributive C
│ │ │ │ - -- used 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.5072e-05s (cpu); 7.174e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc)
│ │ │ │ + -- used 8.61041s (cpu); 5.25771s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  We also know that the dual of a distributive lattice is again a distributive
│ │ │ │  lattice. Other information is copied when possible.
│ │ │ │  i9 : C' = dual C;
│ │ │ │  i10 : time isDistributive C'
│ │ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.0861e-05s (cpu); 8.147e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : peek C'.cache
│ │ │ │  
│ │ │ │  o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}
│ │ │ │  }
│ │ │ │                   coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4},
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html
│ │ │ @@ -107,25 +107,25 @@
│ │ │              
│ │ │                
    i3 : D = dominanceLattice 6;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc)
│ │ │ + -- used 0.532824s (cpu); 0.307911s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc)
│ │ │ + -- used 1.7517e-05s (cpu); 1.5606e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,21 +30,21 @@
│ │ │ │  
│ │ │ │  o2 : Partition
│ │ │ │  The conjugate of $l$ has the same property, but with chains replaced by
│ │ │ │  _a_n_t_i_c_h_a_i_n_s. Because of this, it is often better to count via antichains instead
│ │ │ │  of chains. This can be done by passing "antichains" as the Strategy.
│ │ │ │  i3 : D = dominanceLattice 6;
│ │ │ │  i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc)
│ │ │ │ + -- used 0.532824s (cpu); 0.307911s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.7517e-05s (cpu); 1.5606e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o5 : Partition
│ │ │ │  The Greene-Kleitman partition of the $n$ _c_h_a_i_n is the partition of $n$ with $1$
│ │ │ │  part.
│ │ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/PositivityToricBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  ZHJhd1BhcmxpYW1lbnQyRHRpa3o=
│ │ │  #:len=2626
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmlzdWFsaXNlcyB0aGUgcGFybGlhbWVu
│ │ │  dCBvZiBwb2x5dG9wZXMgZm9yIGEgdmVjdG9yIGJ1bmRsZSBvbiBhIHRvcmljIHN1cmZhY2UgdXNp
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  cHJpbWFyeUNvbXBvbmVudCguLi4sU3RyYXRlZ3k9Pi4uLik=
│ │ │  #:len=316
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzcxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcmltYXJ5Q29tcG9uZW50LFN0cmF0ZWd5XSwicHJp
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated__Primes.out
│ │ │ @@ -125,24 +125,24 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o20 : Set
│ │ │  
│ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o21 : Set
│ │ │  
│ │ │  i22 : assert(M1 === M2)
│ │ │  
│ │ │  i23 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out
│ │ │ @@ -24,35 +24,35 @@
│ │ │                | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │  
│ │ │  i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .217729s elapsed
│ │ │ + -- .16538s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
│ │ │  
│ │ │  i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0174897s elapsed
│ │ │ + -- .0225017s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
│ │ │  
│ │ │  i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .0178098s elapsed
│ │ │ + -- .0232471s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       x x )
│ │ │        0 4
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .0318276s elapsed
│ │ │ + -- .0451594s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : R = QQ[h,l,s,x,y,z]
│ │ │ @@ -41,15 +41,15 @@
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00758549s elapsed
│ │ │ + -- .00905121s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html
│ │ │ @@ -291,28 +291,28 @@
│ │ │  o19 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o20 : Set
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o21 : Set
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i22 : assert(M1 === M2)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -155,24 +155,24 @@
│ │ │ │  o19 = {ideal (a, e), ideal (a, b, c), ideal (a, b, d), ideal (a, b, c, d),
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │ │  
│ │ │ │ -o20 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │ │  
│ │ │ │  o20 : Set
│ │ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │ │  
│ │ │ │ -o21 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │ │  
│ │ │ │  o21 : Set
│ │ │ │  i22 : assert(M1 === M2)
│ │ │ │  The method using Ext modules comes from Eisenbud-Huneke-Vasconcelos, Invent.
│ │ │ │  Math 110 (1992) 207-235.
│ │ │ │  Original author (for ideals): _C_._ _Y_a_c_k_e_l. Updated for modules by J. Chen.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html
│ │ │ @@ -112,41 +112,41 @@
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .217729s elapsed
│ │ │ + -- .16538s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0174897s elapsed
│ │ │ + -- .0225017s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .0178098s elapsed
│ │ │ + -- .0232471s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,39 +41,39 @@
│ │ │ │  |
│ │ │ │                | 0            0             0            0              x_1^5-
│ │ │ │  x_0x_2^4 |
│ │ │ │  
│ │ │ │                              3
│ │ │ │  o3 : R-module, quotient of R
│ │ │ │  i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ │ - -- .217729s elapsed
│ │ │ │ + -- .16538s elapsed
│ │ │ │  
│ │ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0
│ │ │ │  0              |)
│ │ │ │                    | 1 0 |  | 0            0             0            x_1^3-
│ │ │ │  x_0x_2^2 0              |
│ │ │ │                    | 0 1 |  | 0            0             0            0
│ │ │ │  x_1^5-x_0x_2^4 |
│ │ │ │  
│ │ │ │                                 3
│ │ │ │  o4 : R-module, subquotient of R
│ │ │ │  i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ │ - -- .0174897s elapsed
│ │ │ │ + -- .0225017s elapsed
│ │ │ │  
│ │ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0
│ │ │ │  0              |)
│ │ │ │                    | 0 0 |  | 0            0             0            x_1^3-
│ │ │ │  x_0x_2^2 0              |
│ │ │ │                    | 0 1 |  | 0            0             0            0
│ │ │ │  x_1^5-x_0x_2^4 |
│ │ │ │  
│ │ │ │                                 3
│ │ │ │  o5 : R-module, subquotient of R
│ │ │ │  i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ │ - -- .0178098s elapsed
│ │ │ │ + -- .0232471s elapsed
│ │ │ │  
│ │ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0
│ │ │ │  0              |)
│ │ │ │                    | 0 1 |  | 0            0             0            x_1^3-
│ │ │ │  x_0x_2^2 0              |
│ │ │ │                    | 0 0 |  | 0            0             0            0
│ │ │ │  x_1^5-x_0x_2^4 |
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .0318276s elapsed
│ │ │ + -- .0451594s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00758549s elapsed
│ │ │ + -- .00905121s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │               2 7   0 7   3 6   2 6   1 6   0 6   2 5   0 5   3 4   2 4   1 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x )
│ │ │ │        0 4
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : elapsedTime regSeqInIdeal I
│ │ │ │ - -- .0318276s elapsed
│ │ │ │ + -- .0451594s elapsed
│ │ │ │  
│ │ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  If I is the unit ideal, then an ideal of variables of the ring is returned.
│ │ │ │  If the codimension of I is already known, then one can specify this, along with
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │       l , s )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ │ - -- .00758549s elapsed
│ │ │ │ + -- .00905121s elapsed
│ │ │ │  
│ │ │ │                     2                3    2     2    8    3    2     2
│ │ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_d_i_c_a_l -- the radical of an ideal
│ │ ├── ./usr/share/doc/Macaulay2/Probability/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  dERpc3RyaWJ1dGlvbihOdW1iZXIp
│ │ │  #:len=261
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIyOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodERpc3RyaWJ1dGlvbixOdW1iZXIpLCJ0RGlzdHJp
│ │ ├── ./usr/share/doc/Macaulay2/PseudomonomialPrimaryDecomposition/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  UHNldWRvbW9ub21pYWxQcmltYXJ5RGVjb21wb3NpdGlvbg==
│ │ │  #:len=1413
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJpbWFyeSBkZWNvbXBvc2l0aW9uIG9m
│ │ │  IGEgc3F1YXJlIGZyZWUgcHNldWRvbW9ub21pYWwgaWRlYWwiLCBEZXNjcmlwdGlvbiA9PiAoRElW
│ │ ├── ./usr/share/doc/Macaulay2/Pullback/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  aW50ZXJuYWxVc2VEaXJlY3RTdW0oUmluZyxSaW5nKQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpbnRlcm5hbFVzZURpcmVjdFN1bSxSaW5nLFJpbmcp
│ │ ├── ./usr/share/doc/Macaulay2/PushForward/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  cHVzaEZ3ZChNb2R1bGUp
│ │ │  #:len=267
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwdXNoRndkLE1vZHVsZSksInB1c2hGd2QoTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/Python/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=39
│ │ │  YWRkUHlUb00yRnVuY3Rpb24oTGlzdCxGdW5jdGlvbixTdHJpbmcp
│ │ │  #:len=293
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDcxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRQeVRvTTJGdW5jdGlvbixMaXN0LEZ1bmN0aW9u
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_iterator_lp__Python__Object_rp.out
│ │ │ @@ -10,12 +10,12 @@
│ │ │  
│ │ │  o2 = range(0, 3)
│ │ │  
│ │ │  o2 : PythonObject of class range
│ │ │  
│ │ │  i3 : i = iterator x
│ │ │  
│ │ │ -o3 = 
│ │ │ +o3 = 
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_next_lp__Python__Object_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  o2 = range(0, 3)
│ │ │  
│ │ │  o2 : PythonObject of class range
│ │ │  
│ │ │  i3 : i = iterator x
│ │ │  
│ │ │ -o3 = 
│ │ │ +o3 = 
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
│ │ │  
│ │ │  i4 : next i
│ │ │  
│ │ │  o4 = 0
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_python__Run__Script.out
│ │ │ @@ -1,22 +1,22 @@
│ │ │  -- -*- M2-comint -*- hash: 447449196062331972
│ │ │  
│ │ │  i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-32830-0/0.py
│ │ │ +o1 = /tmp/M2-44257-0/0.py
│ │ │  
│ │ │  i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-32830-0/0.py
│ │ │ +o2 = /tmp/M2-44257-0/0.py
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-32830-0/0.py
│ │ │ +o3 = /tmp/M2-44257-0/0.py
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : get pyfile
│ │ │  
│ │ │  o4 = import math
│ │ │       x = math.sin(3.4)
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_to__Python.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │  
│ │ │  o12 = m2sqrt
│ │ │  
│ │ │  o12 : FunctionClosure
│ │ │  
│ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = 
│ │ │ +o13 = 
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │  
│ │ │  i14 : pysqrt 2
│ │ │  calling Macaulay2 code from Python!
│ │ │  
│ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_use_lp__Python__Context_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │  
│ │ │  o7 : Symbol
│ │ │  
│ │ │  i8 : use ctx
│ │ │  
│ │ │  i9 : f
│ │ │  
│ │ │ -o9 =  at 0x7f5ec19a1f30>
│ │ │ +o9 =  at 0x7f13a89f5f30>
│ │ │  
│ │ │  o9 : PythonObject of class function
│ │ │  
│ │ │  i10 : x
│ │ │  
│ │ │  o10 = 5
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : PythonObject of class range
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f5ec1813cc0>
│ │ │ +o3 = <range_iterator object at 0x7f13a8867d80>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _n_e_x_t_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- retrieve the next item from a python iterator
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _i_t_e_r_a_t_o_r_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- get iterator of iterable python object
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_next_lp__Python__Object_rp.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │  o2 : PythonObject of class range
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f5ec190e9d0>
│ │ │ +o3 = <range_iterator object at 0x7f13a8961e90>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : next i
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  i4 : next i
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  
│ │ │ │  o4 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_python__Run__Script.html
│ │ │ @@ -81,31 +81,31 @@
│ │ │            
    The return value is a Python dictionary containing all the variables defined in the global scope.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-32830-0/0.py
    
│ │ │ +o1 = /tmp/M2-44257-0/0.py
│ │ │              
    
│ │ │                
    i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-32830-0/0.py
│ │ │ +o2 = /tmp/M2-44257-0/0.py
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-32830-0/0.py
│ │ │ +o3 = /tmp/M2-44257-0/0.py
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : get pyfile
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,23 +16,23 @@
│ │ │ │  Execute a sequence of statements as if they were read from a Python file. This
│ │ │ │  is for multi-line code that might contain definitions, control structures,
│ │ │ │  imports, etc. It is great for running Python code from a file.
│ │ │ │  The return value is a Python dictionary containing all the variables defined in
│ │ │ │  the global scope.
│ │ │ │  i1 : pyfile = temporaryFileName() | ".py"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-32830-0/0.py
│ │ │ │ +o1 = /tmp/M2-44257-0/0.py
│ │ │ │  i2 : pyfile << "import math" << endl
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-32830-0/0.py
│ │ │ │ +o2 = /tmp/M2-44257-0/0.py
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-32830-0/0.py
│ │ │ │ +o3 = /tmp/M2-44257-0/0.py
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : get pyfile
│ │ │ │  
│ │ │ │  o4 = import math
│ │ │ │       x = math.sin(3.4)
│ │ │ │  i5 : pythonRunScript oo
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -186,15 +186,15 @@
│ │ │  o12 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7f5ec17e36f0>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7f13a88376f0>
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
    
│ │ │              
    
│ │ │                
    i14 : pysqrt 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │            sqrt x)
│ │ │ │  
│ │ │ │  o12 = m2sqrt
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │ │  
│ │ │ │ -o13 = 
│ │ │ │ +o13 = 
│ │ │ │  
│ │ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │ │  i14 : pysqrt 2
│ │ │ │  calling Macaulay2 code from Python!
│ │ │ │  
│ │ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_use_lp__Python__Context_rp.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │                
    i8 : use ctx
    
│ │ │              
    
│ │ │                
    i9 : f
│ │ │  
│ │ │ -o9 = <function <lambda> at 0x7f5ec19a1f30>
│ │ │ +o9 = <function <lambda> at 0x7f13a89f5f30>
│ │ │  
│ │ │  o9 : PythonObject of class function
    
│ │ │              
    
│ │ │                
    i10 : x
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o7 = y
│ │ │ │  
│ │ │ │  o7 : Symbol
│ │ │ │  i8 : use ctx
│ │ │ │  i9 : f
│ │ │ │  
│ │ │ │ -o9 =  at 0x7f5ec19a1f30>
│ │ │ │ +o9 =  at 0x7f13a89f5f30>
│ │ │ │  
│ │ │ │  o9 : PythonObject of class function
│ │ │ │  i10 : x
│ │ │ │  
│ │ │ │  o10 = 5
│ │ │ │  
│ │ │ │  o10 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/QthPower/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  bWluaW1pemF0aW9u
│ │ │  #:len=2755
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hhbmdlIHRvIGEgYmV0dGVyIE5vZXRo
│ │ │  ZXIgbm9ybWFsaXphdGlvbiBzdWdnZXN0ZWQgYnkgdGhlIGluZHVjZWQgd2VpZ2h0cyIsICJsaW5l
│ │ ├── ./usr/share/doc/Macaulay2/QuadraticIdealExamplesByRoos/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aGlnaGVyRGVwdGhUYWJsZQ==
│ │ │  #:len=793
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ3JlYXRlcyBoYXNodGFibGUgb2YgSmFu
│ │ │  LUVyaWsgUm9vcycgZXhhbXBsZXMgb2YgcXVhZHJhdGljIGlkZWFscyB3aXRoIHBvc2l0aXZlIGRl
│ │ ├── ./usr/share/doc/Macaulay2/Quasidegrees/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  cXVhc2lkZWdyZWVzTG9jYWxDb2hvbW9sb2d5
│ │ │  #:len=3881
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgcXVhc2lkZWdyZWUg
│ │ │  c2V0cyBvZiBsb2NhbCBjb2hvbW9sb2d5IG1vZHVsZXMiLCAibGluZW51bSIgPT4gNzg1LCBJbnB1
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  W1FRXQ==
│ │ │  #:len=544
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUXVhdGVybmFyeSBRdWFydGljIEZvcm1z
│ │ │  IGFuZCBHb3JlbnN0ZWluIHJpbmdzIChLYXB1c3RrYSwgS2FwdXN0a2EsIFJhbmVzdGFkLCBTY2hl
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
│ │ │ @@ -180,15 +180,15 @@
│ │ │  i21 : L = trim groebnerStratum F;
│ │ │  
│ │ │  o21 : Ideal of T
│ │ │  
│ │ │  i22 : assert(dim L == 18)
│ │ │  
│ │ │  i23 : elapsedTime isPrime L
│ │ │ - -- 3.65241s elapsed
│ │ │ + -- 2.6268s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : I = pointsIdeal randomPoints(S, 6)
│ │ │  
│ │ │                               2                              2   2          
│ │ │  o24 = ideal (a*c - 7b*c - 49c  + 40a*d - 42b*d + 12c*d + 28d , b  - 36b*c -
│ │ │ @@ -302,15 +302,15 @@
│ │ │  o38 = true
│ │ │  
│ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │  
│ │ │  o39 : Ideal of T
│ │ │  
│ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.58155s elapsed
│ │ │ + -- 2.15372s elapsed
│ │ │  
│ │ │  i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
│ │ │  
│ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html
│ │ │ @@ -349,15 +349,15 @@
│ │ │              
    		
│ │ │                
    i22 : assert(dim L == 18)
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime L
│ │ │ - -- 3.65241s elapsed
│ │ │ + -- 2.6268s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The Schreyer resolution and minimal Betti numbers
    
│ │ │ @@ -561,15 +561,15 @@
│ │ │  
│ │ │  o39 : Ideal of T    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.58155s elapsed
    
│ │ │ + -- 2.15372s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -251,15 +251,15 @@
│ │ │ │        |      31         33        32       34        35        36    |
│ │ │ │        +--------------------------------------------------------------+
│ │ │ │  i21 : L = trim groebnerStratum F;
│ │ │ │  
│ │ │ │  o21 : Ideal of T
│ │ │ │  i22 : assert(dim L == 18)
│ │ │ │  i23 : elapsedTime isPrime L
│ │ │ │ - -- 3.65241s elapsed
│ │ │ │ + -- 2.6268s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  ********** TThhee SScchhrreeyyeerr rreessoolluuttiioonn aanndd mmiinniimmaall BBeettttii nnuummbbeerrss **********
│ │ │ │  Schreyer's construction of a nonminimal free resolution starts with a Groebner
│ │ │ │  basis. First, one constructs the SScchhrreeyyeerr ffrraammee (see La Scala, Stillman). This
│ │ │ │  is determined solely from the initial ideal $J$ and its minimal generators (but
│ │ │ │  depends on some choices of ordering, but otherwise is combinatorial). This
│ │ │ │ @@ -415,15 +415,15 @@
│ │ │ │  We now compute the locus in $V(L)$ where the Betti diagram has no cancellation.
│ │ │ │  This is a closed subscheme of $V(L)$, which is a closed subscheme of the
│ │ │ │  Hilbert scheme. Notice that there are two components.
│ │ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │ │  
│ │ │ │  o39 : Ideal of T
│ │ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ │ - -- 2.58155s elapsed
│ │ │ │ + -- 2.15372s elapsed
│ │ │ │  i41 : #compsL441
│ │ │ │  
│ │ │ │  o41 = 2
│ │ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ │ │  
│ │ │ │  o42 = {16, 14}
│ │ ├── ./usr/share/doc/Macaulay2/QuillenSuslin/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  aG9ycm9ja3M=
│ │ │  #:len=4664
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBsb2NhbCBzb2x1dGlv
│ │ │  biB0byB0aGUgdW5pbW9kdWxhciByb3cgcHJvYmxlbSBvdmVyIGEgbG9jYWxpemF0aW9uIGF0IGEg
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  bmV3IFJPYmplY3QgZnJvbSBDQw==
│ │ │  #:len=1343
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY3JlYXRlIGFuIFIgY29tcGxleCB2ZWN0
│ │ │  b3IgZnJvbSBhIGNvbXBsZXggbnVtYmVyIiwgImxpbmVudW0iID0+IDI1NSwgSW5wdXRzID0+IHtT
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/example-output/___R__Value.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │  
│ │ │  o3 = [1] 120
│ │ │  
│ │ │  o3 : RObject of type double
│ │ │  
│ │ │  i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │  
│ │ │ -o4 = 
│ │ │ +o4 = 
│ │ │  
│ │ │  o4 : RObject of type environment
│ │ │  
│ │ │  i5 : RValue("choose(n, k)", Environment => env)
│ │ │  
│ │ │  o5 = [1] 120
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/example-output/_new_sp__R__Object_spfrom_sp__Hash__Table.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 5466188237138105394
│ │ │  
│ │ │  i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │  
│ │ │ -o1 = 
│ │ │ +o1 = 
│ │ │  
│ │ │  o1 : RObject of type environment
│ │ │  
│ │ │  i2 : RValue("x^y", Environment => env)
│ │ │  
│ │ │  o2 = [1] 25
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/html/___R__Value.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │            
    The Environment option specifies the R environment in which to evaluate the code.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │  
│ │ │ -o4 = <environment: 0x7ff03f169410>
│ │ │ +o4 = <environment: 0x7fc42f16acd0>
│ │ │  
│ │ │  o4 : RObject of type environment
    
│ │ │              
    
│ │ │                
    i5 : RValue("choose(n, k)", Environment => env)
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │  o3 = [1] 120
│ │ │ │  
│ │ │ │  o3 : RObject of type double
│ │ │ │  The Environment option specifies the R environment in which to evaluate the
│ │ │ │  code.
│ │ │ │  i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │ │  
│ │ │ │ -o4 = 
│ │ │ │ +o4 = 
│ │ │ │  
│ │ │ │  o4 : RObject of type environment
│ │ │ │  i5 : RValue("choose(n, k)", Environment => env)
│ │ │ │  
│ │ │ │  o5 = [1] 120
│ │ │ │  
│ │ │ │  o5 : RObject of type double
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/html/_new_sp__R__Object_spfrom_sp__Hash__Table.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
    Converts a Macaulay2 HashTable with String keys to an R environment, with each key-value pair becoming a variable binding.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │  
│ │ │ -o1 = <environment: 0x7ff03ea9ce20>
│ │ │ +o1 = <environment: 0x7fc42ea9e6e0>
│ │ │  
│ │ │  o1 : RObject of type environment
    
│ │ │              
    
│ │ │                
    i2 : RValue("x^y", Environment => env)
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _R_ _o_b_j_e_c_t, an R environment;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Converts a Macaulay2 _H_a_s_h_T_a_b_l_e with _S_t_r_i_n_g keys to an R eennvviirroonnmmeenntt, with each
│ │ │ │  key-value pair becoming a variable binding.
│ │ │ │  i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │ │  
│ │ │ │ -o1 = 
│ │ │ │ +o1 = 
│ │ │ │  
│ │ │ │  o1 : RObject of type environment
│ │ │ │  i2 : RValue("x^y", Environment => env)
│ │ │ │  
│ │ │ │  o2 = [1] 25
│ │ │ │  
│ │ │ │  o2 : RObject of type double
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bWF4aW1hbEVudHJ5KENvbXBsZXgp
│ │ │  #:len=270
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDAwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYXhpbWFsRW50cnksQ29tcGxleCksIm1heGltYWxF
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out
│ │ │ @@ -7,42 +7,42 @@
│ │ │  
│ │ │  o2 = (10, 20)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00359595, .00152225)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00227775, .000928276)
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00729653, .0492751)
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ +o5 = {(.00804378, .015448), (.00745882, .00511358), (.00789833, .00986211),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ +     (.0103156, .0144462), (.00886839, .0213945), (.00670768, .0154332),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ +     (.00642232, .0102041), (.00549724, .00883622), (.00456062, .00737522),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00953055, .0147996)}
│ │ │ +     (.00821477, .0111171)}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .0101318
│ │ │ +o6 = .007398754600000057
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .01648367429999995
│ │ │ +o7 = .01192303080000006
│ │ │  
│ │ │  o7 : RR (of precision 53)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html
│ │ │ @@ -98,57 +98,57 @@
│ │ │  o2 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00359595, .00152225)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00227775, .000928276)
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
    
│ │ │                
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00729653, .0492751)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │              
    
│ │ │                
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ +o5 = {(.00804378, .015448), (.00745882, .00511358), (.00789833, .00986211),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ +     (.0103156, .0144462), (.00886839, .0213945), (.00670768, .0154332),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ +     (.00642232, .0102041), (.00549724, .00883622), (.00456062, .00737522),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00953055, .0147996)}
│ │ │ +     (.00821477, .0111171)}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .0101318
│ │ │ +o6 = .007398754600000057
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .01648367429999995
│ │ │ +o7 = .01192303080000006
│ │ │  
│ │ │  o7 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,41 +25,41 @@
│ │ │ │  i2 : r=10,n=20
│ │ │ │  
│ │ │ │  o2 = (10, 20)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │ │  
│ │ │ │ -o3 = ({5, 2.91596e52, 9}, .00359595, .00152225)
│ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00227775, .000928276)
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │ │  
│ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ │ +o4 = ({50, 2.30853e454, 98}, .00729653, .0492751)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │ │  
│ │ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ │ +o5 = {(.00804378, .015448), (.00745882, .00511358), (.00789833, .00986211),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ │ +     (.0103156, .0144462), (.00886839, .0213945), (.00670768, .0154332),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ │ +     (.00642232, .0102041), (.00549724, .00883622), (.00456062, .00737522),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00953055, .0147996)}
│ │ │ │ +     (.00821477, .0111171)}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │ │  
│ │ │ │ -o6 = .0101318
│ │ │ │ +o6 = .007398754600000057
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │ │  
│ │ │ │ -o7 = .01648367429999995
│ │ │ │ +o7 = .01192303080000006
│ │ │ │  
│ │ │ │  o7 : RR (of precision 53)
│ │ │ │  ********** WWaayyss ttoo uussee tteessttTTiimmeeFFoorrLLLLLLoonnSSyyzzyyggiieess:: **********
│ │ │ │      * testTimeForLLLonSyzygies(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _t_e_s_t_T_i_m_e_F_o_r_L_L_L_o_n_S_y_z_y_g_i_e_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  a25vd25VbmlyYXRpb25hbENvbXBvbmVudE9mU3BhY2VDdXJ2ZXM=
│ │ │  #:len=2280
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2sgd2hldGhlciB0aGVyZSBpcyBh
│ │ │  IHVuaXJhdGlvbmFsIGNvbnN0cnVjdGlvbiBmb3IgYSBjb21wb25lbnQgb2YgdGhlIEhpbGJlcnQg
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/example-output/_canonical__Curve.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  i2 : g=14;
│ │ │  
│ │ │  i3 : FF=ZZ/10007;
│ │ │  
│ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │  
│ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 9.30535s (cpu); 6.17515s (thread); 0s (gc)
│ │ │ + -- used 9.39943s (cpu); 7.15344s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/example-output/_random__Curve__Genus14__Degree18in__P6.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │   -- setting random seed to 10206284518
│ │ │  
│ │ │  i2 : FF=ZZ/10007;
│ │ │  
│ │ │  i3 : S=FF[x_0..x_6];
│ │ │  
│ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ + -- used 2.0695s (cpu); 1.72098s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : betti res I
│ │ │  
│ │ │              0  1  2  3  4 5
│ │ │  o5 = total: 1 13 45 56 25 2
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/html/_canonical__Curve.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │              
    		
│ │ │                
    i4 : R=FF[x_0..x_(g-1)];
    
│ │ │              
    
│ │ │                
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 9.30535s (cpu); 6.17515s (thread); 0s (gc)
│ │ │ + -- used 9.39943s (cpu); 7.15344s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra.
│ │ │ │  i1 : setRandomSeed "alpha";
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : g=14;
│ │ │ │  i3 : FF=ZZ/10007;
│ │ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ │ - -- used 9.30535s (cpu); 6.17515s (thread); 0s (gc)
│ │ │ │ + -- used 9.39943s (cpu); 7.15344s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              0  1
│ │ │ │  o5 = total: 1 66
│ │ │ │           0: 1  .
│ │ │ │           1: . 66
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/html/_random__Curve__Genus14__Degree18in__P6.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │                  		
│ │ │                
    i3 : S=FF[x_0..x_6];
    
│ │ │              
    
│ │ │                
    i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ + -- used 2.0695s (cpu); 1.72098s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : betti res I
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  fields of the chosen finite characteristic 10007, for fields of characteristic
│ │ │ │  0 by semi-continuity, and, hence, for all but finitely many primes $p$.
│ │ │ │  i1 : setRandomSeed("alpha");
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : FF=ZZ/10007;
│ │ │ │  i3 : S=FF[x_0..x_6];
│ │ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ │ - -- used 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ │ + -- used 2.0695s (cpu); 1.72098s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : betti res I
│ │ │ │  
│ │ │ │              0  1  2  3  4 5
│ │ │ │  o5 = total: 1 13 45 56 25 2
│ │ │ │           0: 1  .  .  .  . .
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  c21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1KC4uLixQcmludGluZz0+Li4uKQ==
│ │ │  #:len=370
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTI4Mywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1LFBy
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 11549527689790345152
│ │ │  
│ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ + -- used 1.49504s (cpu); 1.28292s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
│ │ │  
│ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │            
    If the option Printing is set to true then printings about the current step in the construction are displayed.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  For g<=10 the curves are constructed via plane models.
│ │ │ │  For g<=13 the curves are constructed via space models.
│ │ │ │  For g=14 the curves are constructed by Verra's method.
│ │ │ │  For g=15 the curves are constructed via matrix factorizations.
│ │ │ │  If the option Printing is set to true then printings about the current step in
│ │ │ │  the construction are displayed.
│ │ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ │ - -- used 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ │ + -- used 1.49504s (cpu); 1.28292s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o1 : Ideal of --[t ..t  ]
│ │ │ │                 5  0   10
│ │ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ │ │  
│ │ │ │  o2 = (2, 11, 20)
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  cmFuZG9tU2hlbGxhYmxlSWRlYWxDaGFpbg==
│ │ │  #:len=1799
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUHJvZHVjZXMgYSBjaGFpbiBvZiBpZGVh
│ │ │  bHMgZnJvbSBhIHJhbmRvbSBzaGVsbGluZyIsICJsaW5lbnVtIiA9PiA3NjAsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out
│ │ │ @@ -1,24 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812652
│ │ │ + -- setting random seed to 1793012525
│ │ │  
│ │ │ -o1 = 1792812652
│ │ │ +o1 = 1793012525
│ │ │  
│ │ │  i2 : kk=ZZ/101;
│ │ │  
│ │ │  i3 : S=kk[vars(0..5)];
│ │ │  
│ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 4.13607s (cpu); 2.14846s (thread); 0s (gc)
│ │ │ + -- used 4.74489s (cpu); 2.39212s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 37 }
│ │ │ +o4 = Tally{4 => 44 }
│ │ │             5 => 221
│ │ │ -           6 => 158
│ │ │ -           7 => 75
│ │ │ -           8 => 8
│ │ │ -           9 => 1
│ │ │ +           6 => 160
│ │ │ +           7 => 62
│ │ │ +           8 => 13
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5959465567197821046
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812658
│ │ │ + -- setting random seed to 1793012529
│ │ │  
│ │ │ -o1 = 1792812658
│ │ │ +o1 = 1793012529
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,12 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = b c
│ │ │ +o4 = a*b*c
│ │ │  
│ │ │  o4 : S
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 8876340562021865447
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812665
│ │ │ + -- setting random seed to 1793012535
│ │ │  
│ │ │ -o1 = 1792812665
│ │ │ +o1 = 1793012535
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -22,18 +22,18 @@
│ │ │  o4 = {3, 5, 7}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*c*d)
│ │ │ +o5 = ideal(b*c*d)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ +o6 = ideal (b*c, d*e, a*e, c*d, a*d)
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 10504911213508281315
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812666
│ │ │ + -- setting random seed to 1793012537
│ │ │  
│ │ │ -o1 = 1792812666
│ │ │ +o1 = 1793012537
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -15,15 +15,15 @@
│ │ │  
│ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │  
│ │ │  o3 : MonomialIdeal of S
│ │ │  
│ │ │  i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : setRandomSeed(1)
│ │ │   -- setting random seed to 1
│ │ │  
│ │ │  o5 = 1
│ │ │ @@ -37,15 +37,15 @@
│ │ │  i7 : J = monomialIdeal 0_S
│ │ │  
│ │ │  o7 = monomialIdeal ()
│ │ │  
│ │ │  o7 : MonomialIdeal of S
│ │ │  
│ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc)
│ │ │ + -- used 4.62792s (cpu); 3.47182s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -76,17 +76,17 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
    
│ │ │                
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ + -- used 1.49504s (cpu); 1.28292s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812658
│ │ │ + -- setting random seed to 1793012529
│ │ │  
│ │ │ -o1 = 1792812658
    
│ │ │ +o1 = 1793012529
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -103,16 +103,15 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = b c
│ │ │ +o4 = a*b*c
│ │ │  
│ │ │  o4 : S
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,31 +11,30 @@
│ │ │ │            o d, an _i_n_t_e_g_e_r, non-negative
│ │ │ │            o S, a _r_i_n_g, polynomial ring
│ │ │ │      * Outputs:
│ │ │ │            o m, a _r_i_n_g_ _e_l_e_m_e_n_t, monomial of S
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Chooses a random monomial.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1792812658
│ │ │ │ + -- setting random seed to 1793012529
│ │ │ │  
│ │ │ │ -o1 = 1792812658
│ │ │ │ +o1 = 1793012529
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : randomMonomial(3,S)
│ │ │ │  
│ │ │ │ -      2
│ │ │ │ -o4 = b c
│ │ │ │ +o4 = a*b*c
│ │ │ │  
│ │ │ │  o4 : S
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_M_o_n_o_m_i_a_l_I_d_e_a_l -- random monomial ideal with given degree generators
│ │ │ │      * _r_a_n_d_o_m_S_q_u_a_r_e_F_r_e_e_M_o_n_o_m_i_a_l_I_d_e_a_l -- random square-free monomial ideal with
│ │ │ │        given degree generators
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommMMoonnoommiiaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html
│ │ │ @@ -76,17 +76,17 @@
│ │ │          
    
│ │ │            
    Choose a random square-free monomial ideal whose generators have the degrees specified by the list or sequence L.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812665
│ │ │ + -- setting random seed to 1793012535
│ │ │  
│ │ │ -o1 = 1792812665
    
│ │ │ +o1 = 1793012535
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -113,24 +113,24 @@
│ │ │              
    
│ │ │                
    i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*c*d)
│ │ │ +o5 = ideal(b*c*d)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ +o6 = ideal (b*c, d*e, a*e, c*d, a*d)
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,17 +13,17 @@
│ │ │ │      * Outputs:
│ │ │ │            o I, an _i_d_e_a_l, square-free monomial ideal with generators of
│ │ │ │              specified degrees
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Choose a random square-free monomial ideal whose generators have the degrees
│ │ │ │  specified by the list or sequence L.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1792812665
│ │ │ │ + -- setting random seed to 1793012535
│ │ │ │  
│ │ │ │ -o1 = 1792812665
│ │ │ │ +o1 = 1793012535
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a..e]
│ │ │ │  
│ │ │ │ @@ -34,20 +34,20 @@
│ │ │ │  
│ │ │ │  o4 = {3, 5, 7}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │ │  low degree gens generated everything
│ │ │ │  
│ │ │ │ -o5 = ideal(a*c*d)
│ │ │ │ +o5 = ideal(b*c*d)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ │ +o6 = ideal (b*c, d*e, a*e, c*d, a*d)
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The ideal is constructed degree by degree, starting from the lowest degree
│ │ │ │  specified. If there are not enough monomials of the next specified degree that
│ │ │ │  are not already in the ideal, the function prints a warning and returns an
│ │ │ │  ideal containing all the generators of that degree.
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html
│ │ │ @@ -84,17 +84,17 @@
│ │ │            
    With probability p the routine takes the Alexander dual of I; the default value of p is .05, and it can be set with the option AlexanderProbility.
    
│ │ │            
    Otherwise uses the Metropolis algorithm to produce a random walk on the space of square-free ideals. Note that there are a LOT of square-free ideals; these are the Dedekind numbers, and the sequence (with 1,2,3,4,5,6,7,8 variables) begins 3,6,20,168,7581, 7828354, 2414682040998, 56130437228687557907788. (see the Online Encyclopedia of Integer Sequences for more information). Given I in a polynomial ring S, we make a list ISocgens of the square-free minimal monomial generators of the socle of S/(squares+I) and a list of minimal generators Igens of I. A candidate "next" ideal J is formed as follows: We choose randomly from the union of these lists; if a socle element is chosen, it's added to I; if a minimal generator is chosen, it's replaced by the square-free part of the maximal ideal times it. the chance of making the given move is then 1/(#ISocgens+#Igens), and the chance of making the move back would be the similar quantity for J, so we make the move or not depending on whether random RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812666
│ │ │ + -- setting random seed to 1793012537
│ │ │  
│ │ │ -o1 = 1792812666
    
│ │ │ +o1 = 1793012537
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -111,15 +111,15 @@
│ │ │  o3 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    With 4 variables and 168 possible monomial ideals, a run of 5000 takes less than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000 takes about 2 seconds.
    
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc)
    
│ │ │ + -- used 4.62792s (cpu); 3.47182s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,30 +35,30 @@
│ │ │ │  choose randomly from the union of these lists; if a socle element is chosen,
│ │ │ │  it's added to I; if a minimal generator is chosen, it's replaced by the square-
│ │ │ │  free part of the maximal ideal times it. the chance of making the given move is
│ │ │ │  then 1/(#ISocgens+#Igens), and the chance of making the move back would be the
│ │ │ │  similar quantity for J, so we make the move or not depending on whether random
│ │ │ │  RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1792812666
│ │ │ │ + -- setting random seed to 1793012537
│ │ │ │  
│ │ │ │ -o1 = 1792812666
│ │ │ │ +o1 = 1793012537
│ │ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : J = monomialIdeal"ab,ad, bcd"
│ │ │ │  
│ │ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │ │  
│ │ │ │  o3 : MonomialIdeal of S
│ │ │ │  i4 : randomSquareFreeStep J
│ │ │ │  
│ │ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}}
│ │ │ │ +o4 = {monomialIdeal (a*b, a*d, b*d), {a*b, a*d, b*d}, {c*d, b*c, a*c}}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  With 4 variables and 168 possible monomial ideals, a run of 5000 takes less
│ │ │ │  than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000
│ │ │ │  takes about 2 seconds.
│ │ │ │  i5 : setRandomSeed(1)
│ │ │ │   -- setting random seed to 1
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │  i7 : J = monomialIdeal 0_S
│ │ │ │  
│ │ │ │  o7 = monomialIdeal ()
│ │ │ │  
│ │ │ │  o7 : MonomialIdeal of S
│ │ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs
│ │ │ │  (J,AlexanderProbability => .01));
│ │ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc)
│ │ │ │ + -- used 4.62792s (cpu); 3.47182s (thread); 0s (gc)
│ │ │ │  i9 : #T
│ │ │ │  
│ │ │ │  o9 = 168
│ │ │ │  i10 : T
│ │ │ │  
│ │ │ │  o10 = Tally{monomialIdeal () => 45                            }
│ │ │ │              monomialIdeal (a*b*c, a*b*d) => 33
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html
│ │ │ @@ -59,17 +59,17 @@
│ │ │          
    
│ │ │            
    This package can be used to make experiments, trying many ideals, perhaps over small fields. For example...what would you expect the regularities of "typical" monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a bunch of examples -- it's fast. Here we do only 500 -- this takes about a second on a fast machine -- but with a little patience, thousands can be done conveniently.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -77,22 +77,21 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812652
│ │ │ + -- setting random seed to 1793012525
│ │ │  
│ │ │ -o1 = 1792812652
    
│ │ │ +o1 = 1793012525
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101;
    
│ │ │              
    
│ │ │                
    i3 : S=kk[vars(0..5)];
    
│ │ │              
    
│ │ │                
    i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 4.13607s (cpu); 2.14846s (thread); 0s (gc)
│ │ │ + -- used 4.74489s (cpu); 2.39212s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 37 }
│ │ │ +o4 = Tally{4 => 44 }
│ │ │             5 => 221
│ │ │ -           6 => 158
│ │ │ -           7 => 75
│ │ │ -           8 => 8
│ │ │ -           9 => 1
│ │ │ +           6 => 160
│ │ │ +           7 => 62
│ │ │ +           8 => 13
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │          
    
│ │ │            
    How does this compare with the case of binomial ideals? or pure binomial ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger numbers of examples. Click the link "Finding Extreme Examples" below to see some other, more elaborate ways to search.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,27 @@
│ │ │ │  This package can be used to make experiments, trying many ideals, perhaps over
│ │ │ │  small fields. For example...what would you expect the regularities of "typical"
│ │ │ │  monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a
│ │ │ │  bunch of examples -- it's fast. Here we do only 500 -- this takes about a
│ │ │ │  second on a fast machine -- but with a little patience, thousands can be done
│ │ │ │  conveniently.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1792812652
│ │ │ │ + -- setting random seed to 1793012525
│ │ │ │  
│ │ │ │ -o1 = 1792812652
│ │ │ │ +o1 = 1793012525
│ │ │ │  i2 : kk=ZZ/101;
│ │ │ │  i3 : S=kk[vars(0..5)];
│ │ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ │ - -- used 4.13607s (cpu); 2.14846s (thread); 0s (gc)
│ │ │ │ + -- used 4.74489s (cpu); 2.39212s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -o4 = Tally{4 => 37 }
│ │ │ │ +o4 = Tally{4 => 44 }
│ │ │ │             5 => 221
│ │ │ │ -           6 => 158
│ │ │ │ -           7 => 75
│ │ │ │ -           8 => 8
│ │ │ │ -           9 => 1
│ │ │ │ +           6 => 160
│ │ │ │ +           7 => 62
│ │ │ │ +           8 => 13
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  How does this compare with the case of binomial ideals? or pure binomial
│ │ │ │  ideals? We invite the reader to experiment, replacing "randomMonomialIdeal"
│ │ │ │  above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger
│ │ │ │  numbers of examples. Click the link "Finding Extreme Examples" below to see
│ │ │ │  some other, more elaborate ways to search.
│ │ ├── ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  VmFyaWFibGVOYW1l
│ │ │  #:len=1241
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl
│ │ │  IHRoZSBpbmRleGVkIHZhcmlhYmxlIG5hbWUgZm9yIHRoZSBwb2x5bm9taWFsIHJpbmciLCAibGlu
│ │ ├── ./usr/share/doc/Macaulay2/RandomObjects/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  QXR0ZW1wdHM=
│ │ │  #:len=502
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibnVtYmVyIG9mIGF0dGVtcHRzIGluIHRo
│ │ │  ZSBjb25zdHJ1Y3Rpb24gb2YgYSByYW5kb20gb2JqZWN0IiwgImxpbmVudW0iID0+IDI2MiwgImZp
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  ZmluZEFOb25aZXJvTWlub3IoLi4uLFBvaW50Q2hlY2tBdHRlbXB0cz0+Li4uKQ==
│ │ │  #:len=315
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg4OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEFOb25aZXJvTWlub3IsUG9pbnRDaGVja0F0
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out
│ │ │ @@ -5,17 +5,17 @@
│ │ │  i2 : S=kk[y_0..y_8];
│ │ │  
│ │ │  i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2;
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 2.00843s elapsed
│ │ │ + -- 1.62638s elapsed
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : elapsedTime dim I
│ │ │ - -- 3.91258s elapsed
│ │ │ + -- 3.94261s elapsed
│ │ │  
│ │ │  o5 = 4
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  i8 : i = 0;
│ │ │  
│ │ │  i9 : J = I;
│ │ │  
│ │ │  o9 : Ideal of T
│ │ │  
│ │ │  i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.88102s elapsed
│ │ │ + -- 2.45189s elapsed
│ │ │  
│ │ │  i11 : dim J
│ │ │  
│ │ │  o11 = 1
│ │ │  
│ │ │  i12 : i
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out
│ │ │ @@ -27,24 +27,24 @@
│ │ │  i6 : S=ZZ/103[y_0..y_30];
│ │ │  
│ │ │  i7 : I=minors(2,random(S^3,S^{3:-1}));
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 4.05108s elapsed
│ │ │ + -- 3.51543s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 3.02229s elapsed
│ │ │ + -- 2.7863s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html
│ │ │ @@ -100,23 +100,23 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
    
│ │ │                
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 2.00843s elapsed
│ │ │ + -- 1.62638s elapsed
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime dim I
│ │ │ - -- 3.91258s elapsed
│ │ │ + -- 3.94261s elapsed
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The user may set the MinimumFieldSize to ensure that the field being worked over is big enough.  For instance, there are relatively few linear spaces over a field of characteristic 2, and this can cause incorrect results to be provided.  If no size is provided, the function tries to guess a good size based on ambient ring.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,19 +32,19 @@
│ │ │ │  examples, the built in dim function is much faster.
│ │ │ │  i1 : kk=ZZ/101;
│ │ │ │  i2 : S=kk[y_0..y_8];
│ │ │ │  i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2;
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : elapsedTime dimViaBezout(I)
│ │ │ │ - -- 2.00843s elapsed
│ │ │ │ + -- 1.62638s elapsed
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : elapsedTime dim I
│ │ │ │ - -- 3.91258s elapsed
│ │ │ │ + -- 3.94261s elapsed
│ │ │ │  
│ │ │ │  o5 = 4
│ │ │ │  The user may set the MinimumFieldSize to ensure that the field being worked
│ │ │ │  over is big enough. For instance, there are relatively few linear spaces over a
│ │ │ │  field of characteristic 2, and this can cause incorrect results to be provided.
│ │ │ │  If no size is provided, the function tries to guess a good size based on
│ │ │ │  ambient ring.
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o9 : Ideal of T
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.88102s elapsed
    
│ │ │ + -- 2.45189s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : dim J
│ │ │  
│ │ │  o11 = 1
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │  o7 : Matrix T  <-- T
│ │ │ │  i8 : i = 0;
│ │ │ │  i9 : J = I;
│ │ │ │  
│ │ │ │  o9 : Ideal of T
│ │ │ │  i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J =
│ │ │ │  extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ │ - -- 2.88102s elapsed
│ │ │ │ + -- 2.45189s elapsed
│ │ │ │  i11 : dim J
│ │ │ │  
│ │ │ │  o11 = 1
│ │ │ │  i12 : i
│ │ │ │  
│ │ │ │  o12 = 4
│ │ │ │  In this particular example, there tend to be about 5 associated primes when
│ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html
│ │ │ @@ -149,27 +149,27 @@
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 4.05108s elapsed
│ │ │ + -- 3.51543s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 3.02229s elapsed
│ │ │ + -- 2.7863s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,24 +66,24 @@
│ │ │ │  first in rings with more variables.
│ │ │ │  i6 : S=ZZ/103[y_0..y_30];
│ │ │ │  i7 : I=minors(2,random(S^3,S^{3:-1}));
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection,
│ │ │ │  DecompositionStrategy=>MultiplicationTable)
│ │ │ │ - -- 4.05108s elapsed
│ │ │ │ + -- 3.51543s elapsed
│ │ │ │  
│ │ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection,
│ │ │ │  DecompositionStrategy=>Decompose)
│ │ │ │ - -- 3.02229s elapsed
│ │ │ │ + -- 2.7863s elapsed
│ │ │ │  
│ │ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommPPooiinnttss:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  aXNCaXJhdGlvbmFsT250b0ltYWdlKC4uLixBc3N1bWVEb21pbmFudD0+Li4uKQ==
│ │ │  #:len=321
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgwNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaXNCaXJhdGlvbmFsT250b0ltYWdlLEFzc3VtZURv
│ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │  i12 : Q=QQ[x,y,z,t,u];
│ │ │  
│ │ │  i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
│ │ │  
│ │ │  o13 : RingMap Q <-- Q
│ │ │  
│ │ │  i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.341115s (cpu); 0.337449s (thread); 0s (gc)
│ │ │ + -- used 0.611841s (cpu); 0.436221s (thread); 0s (gc)
│ │ │  
│ │ │                                  125   124      120 5    124    100 25     104 20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120        8 115 2        12 110 3         16 105 4         20 100 5          24 95 6          28 90 7           32 85 8           36 80 9           40 75 10           44 70 11           48 65 12           52 60 13           56 55 14           60 50 15           64 45 16           68 40 17          72 35 18          76 30 19         80 25 20         84 20 21        88 15 22       92 10 23      96 5 24    100 25     24 100        28 95          32 90 2         36 85 3          40 80 4          44 75 5           48 70 6           52 65 7           56 60 8           60 55 9           64 50 10           68 45 11           72 40 12           76 35 13           80 30 14          84 25 15          88 20 16         92 15 17        96 10 18        100 5 19      104 20       48 75 2       52 70   2        56 65 2 2        60 60 3 2         64 55 4 2         68 50 5 2         72 45 6 2         76 40 7 2         80 35 8 2         84 30 9 2         88 25 10 2         92 20 11 2        96 15 12 2        100 10 13 2       104 5 14 2      108 15 2      72 50 3       76 45   3       80 40 2 3        84 35 3 3        88 30 4 3        92 25 5 3        96 20 6 3        100 15 7 3       104 10 8 3       108 5 9 3      112 10 3     96 25 4      100 20   4      104 15 2 4      108 10 3 4      112 5 4 4     116 5 4    120 5    124
│ │ │  o14 = Proj Q - - - > Proj Q   {x   , x   y, - x   y  + x   z, x   y   - 5x   y  z + 10x   y  z  - 10x   y  z  + 5x   y z  - x   z  + x   t, - y    + 25x y   z - 300x y   z  + 2300x  y   z  - 12650x  y   z  + 53130x  y   z  - 177100x  y  z  + 480700x  y  z  - 1081575x  y  z  + 2042975x  y  z  - 3268760x  y  z   + 4457400x  y  z   - 5200300x  y  z   + 5200300x  y  z   - 4457400x  y  z   + 3268760x  y  z   - 2042975x  y  z   + 1081575x  y  z   - 480700x  y  z   + 177100x  y  z   - 53130x  y  z   + 12650x  y  z   - 2300x  y  z   + 300x  y  z   - 25x  y z   + x   z   - 5x  y   t + 100x  y  z*t - 950x  y  z t + 5700x  y  z t - 24225x  y  z t + 77520x  y  z t - 193800x  y  z t + 387600x  y  z t - 629850x  y  z t + 839800x  y  z t - 923780x  y  z  t + 839800x  y  z  t - 629850x  y  z  t + 387600x  y  z  t - 193800x  y  z  t + 77520x  y  z  t - 24225x  y  z  t + 5700x  y  z  t - 950x  y  z  t + 100x   y z  t - 5x   z  t - 10x  y  t  + 150x  y  z*t  - 1050x  y  z t  + 4550x  y  z t  - 13650x  y  z t  + 30030x  y  z t  - 50050x  y  z t  + 64350x  y  z t  - 64350x  y  z t  + 50050x  y  z t  - 30030x  y  z  t  + 13650x  y  z  t  - 4550x  y  z  t  + 1050x   y  z  t  - 150x   y z  t  + 10x   z  t  - 10x  y  t  + 100x  y  z*t  - 450x  y  z t  + 1200x  y  z t  - 2100x  y  z t  + 2520x  y  z t  - 2100x  y  z t  + 1200x   y  z t  - 450x   y  z t  + 100x   y z t  - 10x   z  t  - 5x  y  t  + 25x   y  z*t  - 50x   y  z t  + 50x   y  z t  - 25x   y z t  + 5x   z t  - x   t  + x   u}
│ │ │  
│ │ │  o14 : RationalMapping
│ │ │  
│ │ │  i15 : R=QQ[x,y,z,t]/(z-2*t);
│ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html
│ │ │ @@ -194,15 +194,15 @@
│ │ │  
│ │ │  o13 : RingMap Q <-- Q
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.341115s (cpu); 0.337449s (thread); 0s (gc)
│ │ │ + -- used 0.611841s (cpu); 0.436221s (thread); 0s (gc)
│ │ │  
│ │ │                                  125   124      120 5    124    100 25     104 20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120        8 115 2        12 110 3         16 105 4         20 100 5          24 95 6          28 90 7           32 85 8           36 80 9           40 75 10           44 70 11           48 65 12           52 60 13           56 55 14           60 50 15           64 45 16           68 40 17          72 35 18          76 30 19         80 25 20         84 20 21        88 15 22       92 10 23      96 5 24    100 25     24 100        28 95          32 90 2         36 85 3          40 80 4          44 75 5           48 70 6           52 65 7           56 60 8           60 55 9           64 50 10           68 45 11           72 40 12           76 35 13           80 30 14          84 25 15          88 20 16         92 15 17        96 10 18        100 5 19      104 20       48 75 2       52 70   2        56 65 2 2        60 60 3 2         64 55 4 2         68 50 5 2         72 45 6 2         76 40 7 2         80 35 8 2         84 30 9 2         88 25 10 2         92 20 11 2        96 15 12 2        100 10 13 2       104 5 14 2      108 15 2      72 50 3       76 45   3       80 40 2 3        84 35 3 3        88 30 4 3        92 25 5 3        96 20 6 3        100 15 7 3       104 10 8 3       108 5 9 3      112 10 3     96 25 4      100 20   4      104 15 2 4      108 10 3 4      112 5 4 4     116 5 4    120 5    124
│ │ │  o14 = Proj Q - - - > Proj Q   {x   , x   y, - x   y  + x   z, x   y   - 5x   y  z + 10x   y  z  - 10x   y  z  + 5x   y z  - x   z  + x   t, - y    + 25x y   z - 300x y   z  + 2300x  y   z  - 12650x  y   z  + 53130x  y   z  - 177100x  y  z  + 480700x  y  z  - 1081575x  y  z  + 2042975x  y  z  - 3268760x  y  z   + 4457400x  y  z   - 5200300x  y  z   + 5200300x  y  z   - 4457400x  y  z   + 3268760x  y  z   - 2042975x  y  z   + 1081575x  y  z   - 480700x  y  z   + 177100x  y  z   - 53130x  y  z   + 12650x  y  z   - 2300x  y  z   + 300x  y  z   - 25x  y z   + x   z   - 5x  y   t + 100x  y  z*t - 950x  y  z t + 5700x  y  z t - 24225x  y  z t + 77520x  y  z t - 193800x  y  z t + 387600x  y  z t - 629850x  y  z t + 839800x  y  z t - 923780x  y  z  t + 839800x  y  z  t - 629850x  y  z  t + 387600x  y  z  t - 193800x  y  z  t + 77520x  y  z  t - 24225x  y  z  t + 5700x  y  z  t - 950x  y  z  t + 100x   y z  t - 5x   z  t - 10x  y  t  + 150x  y  z*t  - 1050x  y  z t  + 4550x  y  z t  - 13650x  y  z t  + 30030x  y  z t  - 50050x  y  z t  + 64350x  y  z t  - 64350x  y  z t  + 50050x  y  z t  - 30030x  y  z  t  + 13650x  y  z  t  - 4550x  y  z  t  + 1050x   y  z  t  - 150x   y z  t  + 10x   z  t  - 10x  y  t  + 100x  y  z*t  - 450x  y  z t  + 1200x  y  z t  - 2100x  y  z t  + 2520x  y  z t  - 2100x  y  z t  + 1200x   y  z t  - 450x   y  z t  + 100x   y z t  - 10x   z  t  - 5x  y  t  + 25x   y  z*t  - 50x   y  z t  + 50x   y  z t  - 25x   y z t  + 5x   z t  - x   t  + x   u}
│ │ │  
│ │ │  o14 : RationalMapping
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  o11 : Ideal of blowUpSubvar
│ │ │ │  The next example is a birational map on $\mathbb{P}^4$.
│ │ │ │  i12 : Q=QQ[x,y,z,t,u];
│ │ │ │  i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
│ │ │ │  
│ │ │ │  o13 : RingMap Q <-- Q
│ │ │ │  i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ │ - -- used 0.341115s (cpu); 0.337449s (thread); 0s (gc)
│ │ │ │ + -- used 0.611841s (cpu); 0.436221s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                  125   124      120 5    124    100 25     104
│ │ │ │  20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120
│ │ │ │  8 115 2        12 110 3         16 105 4         20 100 5          24 95 6
│ │ │ │  28 90 7           32 85 8           36 80 9           40 75 10           44 70
│ │ │ │  11           48 65 12           52 60 13           56 55 14           60 50 15
│ │ │ │  64 45 16           68 40 17          72 35 18          76 30 19         80 25
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  U29ydEdlbnM=
│ │ │  #:len=246
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTb3J0R2VucyIsIlNvcnRHZW5zIiwiUmF0aW9uYWxQ
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  bmV0KFByb2plY3RpdmVQb2ludCk=
│ │ │  #:len=205
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTcsICJ1bmRvY3VtZW50ZWQiID0+IHRy
│ │ │  dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KG5ldCxQcm9q
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out
│ │ │ @@ -48,15 +48,15 @@
│ │ │               0    1    2    3    4    5    6    7    8    9    10
│ │ │  
│ │ │                  ZZ
│ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │                 101  0   10
│ │ │  
│ │ │  i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00513051s (cpu); 0.00512835s (thread); 0s (gc)
│ │ │ + -- used 0.00507743s (cpu); 0.00507847s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
│ │ │  
│ │ │  i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z);
│ │ │  
│ │ │  o16 : Ideal of QQ[x..z]
│ │ │  
│ │ │ @@ -142,23 +142,23 @@
│ │ │  
│ │ │  i31 : nodes = I + ideal jacobian I;
│ │ │  
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 1.28715s (cpu); 1.09718s (thread); 0s (gc)
│ │ │ + -- used 1.2359s (cpu); 0.975535s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : #oo
│ │ │  
│ │ │  o33 = 31
│ │ │  
│ │ │  i34 : nodes' = baseChange_32003 nodes;
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │  
│ │ │  i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.282073s (cpu); 0.221929s (thread); 0s (gc)
│ │ │ + -- used 0.353955s (cpu); 0.269327s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
│ │ │  
│ │ │  i36 :
│ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html
│ │ │ @@ -183,15 +183,15 @@
│ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │                 101  0   10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00513051s (cpu); 0.00512835s (thread); 0s (gc)
│ │ │ + -- used 0.00507743s (cpu); 0.00507847s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Over number fields
    
│ │ │ @@ -353,15 +353,15 @@
│ │ │  o31 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 1.28715s (cpu); 1.09718s (thread); 0s (gc)
    
│ │ │ + -- used 1.2359s (cpu); 0.975535s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    
│ │ │ @@ -378,15 +378,15 @@
│ │ │  
│ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.282073s (cpu); 0.221929s (thread); 0s (gc)
│ │ │ + -- used 0.353955s (cpu); 0.269327s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -89,15 +89,15 @@
│ │ │ │  o13 = ideal(u  + u  + u  + u  + u  + u  + u  + u  + u  + u  + u  )
│ │ │ │               0    1    2    3    4    5    6    7    8    9    10
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o13 : Ideal of ---[u ..u  ]
│ │ │ │                 101  0   10
│ │ │ │  i14 : time rationalPoints(I, Amount => true)
│ │ │ │ - -- used 0.00513051s (cpu); 0.00512835s (thread); 0s (gc)
│ │ │ │ + -- used 0.00507743s (cpu); 0.00507847s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 110462212541120451001
│ │ │ │  ****** OOvveerr nnuummbbeerr ffiieellddss ******
│ │ │ │  Over a number field one can use the option Bound to specify a maximal
│ │ │ │  multiplicative height given by $(x_0:\dots:x_n)\mapsto \prod_{v}\max_i|x_i|_v ^
│ │ │ │  {d_v/d}$ (this is also available as a method _g_l_o_b_a_l_H_e_i_g_h_t).
│ │ │ │  i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z);
│ │ │ │ @@ -197,24 +197,24 @@
│ │ │ │  
│ │ │ │  o30 : Ideal of R
│ │ │ │  i31 : nodes = I + ideal jacobian I;
│ │ │ │  
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ │ - -- used 1.28715s (cpu); 1.09718s (thread); 0s (gc)
│ │ │ │ + -- used 1.2359s (cpu); 0.975535s (thread); 0s (gc)
│ │ │ │  i33 : #oo
│ │ │ │  
│ │ │ │  o33 = 31
│ │ │ │  Still it runs a lot faster when reduced to a positive characteristic.
│ │ │ │  i34 : nodes' = baseChange_32003 nodes;
│ │ │ │  
│ │ │ │  o34 : Ideal of GF 1048969271299456081[x..z, w]
│ │ │ │  i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ │ - -- used 0.282073s (cpu); 0.221929s (thread); 0s (gc)
│ │ │ │ + -- used 0.353955s (cpu); 0.269327s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o35 = 31
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  For a number field other than QQ, the enumeration of elements with bounded
│ │ │ │  height depends on an algorithm by Doyle–Krumm, which is currently only
│ │ │ │  implemented in Sage.
│ │ │ │  ******** MMeennuu ********
│ │ ├── ./usr/share/doc/Macaulay2/ReactionNetworks/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  bW9kaWZpY2F0aW9uT2ZUd29TdWJzdHJhdGVzSCgp
│ │ │  #:len=344
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20gezE6KG1vZGlmaWNhdGlvbk9mVHdvU3Vic3RyYXRlc0gp
│ │ ├── ./usr/share/doc/Macaulay2/RealRoots/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  dHJhY2VDb3VudChJZGVhbCk=
│ │ │  #:len=241
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTAyMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodHJhY2VDb3VudCxJZGVhbCksInRyYWNlQ291bnQo
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  aXNMaW5lYXJUeXBlKE1vZHVsZSk=
│ │ │  #:len=257
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE5OSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNMaW5lYXJUeXBlLE1vZHVsZSksImlzTGluZWFy
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/___Plane__Curve__Singularities.out
│ │ │ @@ -331,15 +331,15 @@
│ │ │                 2 2    2   2             2 2    2
│ │ │        - p w , p y  - p , p w y - p p , p w  - p )
│ │ │           2 1   0      1   0 0     1 2   0 0    2
│ │ │  
│ │ │  o47 : Ideal of B2
│ │ │  
│ │ │  i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 1.07953s (cpu); 0.841221s (thread); 0s (gc)
│ │ │ + -- used 1.09188s (cpu); 0.855329s (thread); 0s (gc)
│ │ │  
│ │ │                   ZZ
│ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │                 32003  0   2   0   1
│ │ │  
│ │ │  i49 : saturate(sing2, sub(irrelTot, ring sing2))
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_expected__Rees__Ideal.out
│ │ │ @@ -58,15 +58,15 @@
│ │ │  o5 : Matrix S  <-- S
│ │ │  
│ │ │  i6 : I = minors(n-1, M);
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ - -- used 1.37298s (cpu); 0.841764s (thread); 0s (gc)
│ │ │ + -- used 1.58915s (cpu); 1.09766s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
│ │ │  
│ │ │  i8 : kk = ZZ/101;
│ │ │  
│ │ │  i9 : S = kk[x,y,z];
│ │ │ @@ -77,19 +77,19 @@
│ │ │  o10 : Matrix S  <-- S
│ │ │  
│ │ │  i11 : I = minors(3,m);
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │  
│ │ │  i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.79026s (cpu); 1.2643s (thread); 0s (gc)
│ │ │ + -- used 2.35562s (cpu); 1.8092s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
│ │ │  
│ │ │  i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc)
│ │ │ + -- used 2.34144s (cpu); 1.75105s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of S[w ..w ]
│ │ │                    0   3
│ │ │  
│ │ │  i14 :
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out
│ │ │ @@ -13,21 +13,21 @@
│ │ │                  3    2
│ │ │       - x x x , x  - x x )
│ │ │          0 2 4   1    0 4
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0295188s (cpu); 0.0275558s (thread); 0s (gc)
│ │ │ + -- used 0.0861655s (cpu); 0.0363373s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
│ │ │  
│ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc)
│ │ │ + -- used 0.191094s (cpu); 0.17887s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
│ │ │  
│ │ │  i6 : S=kk[a,b,c]
│ │ │  
│ │ │  o6 = S
│ │ │ @@ -47,21 +47,21 @@
│ │ │  
│ │ │               2        2
│ │ │  o8 = ideal (a , a*b, b )
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │  
│ │ │  i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.023238s (cpu); 0.0222073s (thread); 0s (gc)
│ │ │ + -- used 0.0802411s (cpu); 0.030119s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
│ │ │  
│ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc)
│ │ │ + -- used 0.0243107s (cpu); 0.0118956s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of S[w ..w ]
│ │ │                    0   2
│ │ │  
│ │ │  i11 : transpose gens I1
│ │ │  
│ │ │  o11 = {-1, -3} | aw_1-bw_2    |
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/___Plane__Curve__Singularities.html
│ │ │ @@ -592,15 +592,15 @@
│ │ │          
    
│ │ │            
    We compute the singular locus once again:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -325,15 +325,15 @@
│ │ │ │                 2 2    2   2             2 2    2
│ │ │ │        - p w , p y  - p , p w y - p p , p w  - p )
│ │ │ │           2 1   0      1   0 0     1 2   0 0    2
│ │ │ │  
│ │ │ │  o47 : Ideal of B2
│ │ │ │  We compute the singular locus once again:
│ │ │ │  i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ │ - -- used 1.07953s (cpu); 0.841221s (thread); 0s (gc)
│ │ │ │ + -- used 1.09188s (cpu); 0.855329s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │ │                 32003  0   2   0   1
│ │ │ │  i49 : saturate(sing2, sub(irrelTot, ring sing2))
│ │ │ │  
│ │ │ │  o49 = ideal 1
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_expected__Rees__Ideal.html
│ │ │ @@ -156,15 +156,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 1.07953s (cpu); 0.841221s (thread); 0s (gc)
│ │ │ + -- used 1.09188s (cpu); 0.855329s (thread); 0s (gc)
│ │ │  
│ │ │                   ZZ
│ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │                 32003  0   2   0   1
    
│ │ │              
    
│ │ │                
    i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ - -- used 1.37298s (cpu); 0.841764s (thread); 0s (gc)
│ │ │ + -- used 1.58915s (cpu); 1.09766s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    
│ │ │              
    
│ │ │ @@ -190,24 +190,24 @@
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │              
    
│ │ │                
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.79026s (cpu); 1.2643s (thread); 0s (gc)
│ │ │ + -- used 2.35562s (cpu); 1.8092s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    
│ │ │              
    
│ │ │                
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc)
│ │ │ + -- used 2.34144s (cpu); 1.75105s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of S[w ..w ]
│ │ │                    0   3
    
│ │ │              
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -86,34 +86,34 @@
│ │ │ │  
│ │ │ │               5      4
│ │ │ │  o5 : Matrix S  <-- S
│ │ │ │  i6 : I = minors(n-1, M);
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ │ - -- used 1.37298s (cpu); 0.841764s (thread); 0s (gc)
│ │ │ │ + -- used 1.58915s (cpu); 1.09766s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of S[w ..w ]
│ │ │ │                   0   4
│ │ │ │  i8 : kk = ZZ/101;
│ │ │ │  i9 : S = kk[x,y,z];
│ │ │ │  i10 : m = random(S^3, S^{4:-2});
│ │ │ │  
│ │ │ │                3      4
│ │ │ │  o10 : Matrix S  <-- S
│ │ │ │  i11 : I = minors(3,m);
│ │ │ │  
│ │ │ │  o11 : Ideal of S
│ │ │ │  i12 : time reesIdeal (I, I_0);
│ │ │ │ - -- used 1.79026s (cpu); 1.2643s (thread); 0s (gc)
│ │ │ │ + -- used 2.35562s (cpu); 1.8092s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of S[w ..w ]
│ │ │ │                    0   3
│ │ │ │  i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc)
│ │ │ │ + -- used 2.34144s (cpu); 1.75105s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of S[w ..w ]
│ │ │ │                    0   3
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_y_m_m_e_t_r_i_c_A_l_g_e_b_r_a_I_d_e_a_l -- Ideal of the symmetric algebra of an ideal or
│ │ │ │        module
│ │ │ │      * _j_a_c_o_b_i_a_n_D_u_a_l -- Computes the 'jacobian dual', part of a method of finding
│ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html
│ │ │ @@ -115,24 +115,24 @@
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0295188s (cpu); 0.0275558s (thread); 0s (gc)
│ │ │ + -- used 0.0861655s (cpu); 0.0363373s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc)
│ │ │ + -- used 0.191094s (cpu); 0.17887s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │ @@ -169,24 +169,24 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.023238s (cpu); 0.0222073s (thread); 0s (gc)
│ │ │ + -- used 0.0802411s (cpu); 0.030119s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc)
│ │ │ + -- used 0.0243107s (cpu); 0.0118956s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of S[w ..w ]
│ │ │                    0   2
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,20 +51,20 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                  3    2
│ │ │ │       - x x x , x  - x x )
│ │ │ │          0 2 4   1    0 4
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time V1 = reesIdeal i;
│ │ │ │ - -- used 0.0295188s (cpu); 0.0275558s (thread); 0s (gc)
│ │ │ │ + -- used 0.0861655s (cpu); 0.0363373s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S[w ..w ]
│ │ │ │                   0   6
│ │ │ │  i5 : time V2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc)
│ │ │ │ + -- used 0.191094s (cpu); 0.17887s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : Ideal of S[w ..w ]
│ │ │ │                   0   6
│ │ │ │  The following example shows how we handle degrees
│ │ │ │  i6 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o6 = S
│ │ │ │ @@ -81,20 +81,20 @@
│ │ │ │  i8 : i=minors(2,m)
│ │ │ │  
│ │ │ │               2        2
│ │ │ │  o8 = ideal (a , a*b, b )
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : time I1 = reesIdeal i;
│ │ │ │ - -- used 0.023238s (cpu); 0.0222073s (thread); 0s (gc)
│ │ │ │ + -- used 0.0802411s (cpu); 0.030119s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of S[w ..w ]
│ │ │ │                   0   2
│ │ │ │  i10 : time I2 = reesIdeal(i,i_0);
│ │ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc)
│ │ │ │ + -- used 0.0243107s (cpu); 0.0118956s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of S[w ..w ]
│ │ │ │                    0   2
│ │ │ │  i11 : transpose gens I1
│ │ │ │  
│ │ │ │  o11 = {-1, -3} | aw_1-bw_2    |
│ │ │ │        {-1, -3} | aw_0-bw_1    |
│ │ ├── ./usr/share/doc/Macaulay2/ReflexivePolytopesDB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=7
│ │ │  S1NFbnRyeQ==
│ │ │  #:len=2438
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZW50cnkgZnJvbSB0aGUgS3JldXpl
│ │ │  ci1Ta2Fya2UgZGF0YWJhc2Ugb2YgZGltZW5zaW9uIDMgYW5kIDQgcmVmbGV4aXZlIHBvbHl0b3Bl
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  UmVndWxhcml0eQ==
│ │ │  #:len=797
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBDYXN0ZWxudW92by1NdW1m
│ │ │  b3JkIHJlZ3VsYXJpdHkgb2YgYSBob21vZ2VuZW91cyBpZGVhbCIsIERlc2NyaXB0aW9uID0+IChQ
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/example-output/_m__Regularity.out
│ │ │ @@ -71,15 +71,15 @@
│ │ │       x x x , x  + x x  - x x  - x x x , x  + x  - x x )
│ │ │        0 1 3   0    0 1    1 2    0 2 5   0    2    0 5
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .3009249333333335
│ │ │ +o8 = .2968416257777775
│ │ │  
│ │ │  o8 : RR (of precision 53)
│ │ │  
│ │ │  i9 : R = QQ[x_0..x_5]
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │ @@ -87,17 +87,17 @@
│ │ │  
│ │ │  i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5, x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3);
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .04256394509756095
│ │ │ +o11 = .06197565907142862
│ │ │  
│ │ │  o11 : RR (of precision 53)
│ │ │  
│ │ │  i12 : time regularity I2  
│ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc)
│ │ │ + -- used 0.0048805s (cpu); 0.00488294s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │  o7 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .3009249333333335
│ │ │ +o8 = .2968416257777775
│ │ │  
│ │ │  o8 : RR (of precision 53)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    This is an example where regularity is faster than mRegularity.
    
│ │ │          
│ │ │ @@ -209,23 +209,23 @@
│ │ │  o10 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .04256394509756095
│ │ │ +o11 = .06197565907142862
│ │ │  
│ │ │  o11 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i12 : time regularity I2  
│ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc)
│ │ │ + -- used 0.0048805s (cpu); 0.00488294s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
    
│ │ │              
    
│ │ │          
    This symbol is provided by the package Regularity.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -94,34 +94,34 @@
│ │ │ │                3    2        2            3    3      2
│ │ │ │       x x x , x  + x x  - x x  - x x x , x  + x  - x x )
│ │ │ │        0 1 3   0    0 1    1 2    0 2 5   0    2    0 5
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : benchmark "mRegularity I1"
│ │ │ │  
│ │ │ │ -o8 = .3009249333333335
│ │ │ │ +o8 = .2968416257777775
│ │ │ │  
│ │ │ │  o8 : RR (of precision 53)
│ │ │ │  This is an example where regularity is faster than mRegularity.
│ │ │ │  i9 : R = QQ[x_0..x_5]
│ │ │ │  
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5,
│ │ │ │  x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3);
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : benchmark " mRegularity I2"
│ │ │ │  
│ │ │ │ -o11 = .04256394509756095
│ │ │ │ +o11 = .06197565907142862
│ │ │ │  
│ │ │ │  o11 : RR (of precision 53)
│ │ │ │  i12 : time regularity I2
│ │ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc)
│ │ │ │ + -- used 0.0048805s (cpu); 0.00488294s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = 4
│ │ │ │  This symbol is provided by the package Regularity.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_g_u_l_a_r_i_t_y -- compute the Castelnuovo-Mumford regularity
│ │ │ │  ********** WWaayyss ttoo uussee mmRReegguullaarriittyy:: **********
│ │ │ │      * mRegularity(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  Y2Fub25pY2FsTXVsdGlwbGllcnM=
│ │ │  #:len=1705
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNhbm9uaWNhbCBt
│ │ │  dWx0aXBsaWVycyBvZiBhIHJhdGlvbmFsIGN1cnZlcyB3aXRoIG5vZGVzIiwgImxpbmVudW0iID0+
│ │ ├── ./usr/share/doc/Macaulay2/ResLengthThree/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bXVsdFRhYmxlT25lVHdvKFJpbmcp
│ │ │  #:len=273
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjYyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0VGFibGVPbmVUd28sUmluZyksIm11bHRUYWJs
│ │ ├── ./usr/share/doc/Macaulay2/ResidualIntersections/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Z2VuZXJpY1Jlc2lkdWFs
│ │ │  #:len=1739
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgZ2VuZXJpYyByZXNpZHVh
│ │ │  bCBpbnRlcnNlY3Rpb25zIG9mIGFuIGlkZWFsIiwgImxpbmVudW0iID0+IDg4OSwgSW5wdXRzID0+
│ │ ├── ./usr/share/doc/Macaulay2/ResolutionsOfStanleyReisnerRings/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  YmFyeWNlbnRyaWNTdWJkaXZpc2lvbihTaW1wbGljaWFsQ29tcGxleCk=
│ │ │  #:len=381
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiYXJ5Y2VudHJpY1N1YmRpdmlzaW9uLFNpbXBsaWNp
│ │ ├── ./usr/share/doc/Macaulay2/ResultantComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  bWl4ZWRTdWJkaXZpc2lvbg==
│ │ │  #:len=1366
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBjb2hlcmVudCBtaXhl
│ │ │  ZCBzdWItZGl2aXNpb24iLCAibGluZW51bSIgPT4gNDc5LCBJbnB1dHMgPT4ge1NQQU57VFR7Imdy
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  aHVyd2l0ekZvcm0oLi4uLFNpbmd1bGFyTG9jdXM9Pi4uLik=
│ │ │  #:len=277
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMyOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaHVyd2l0ekZvcm0sU2luZ3VsYXJMb2N1c10sImh1
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out
│ │ │ @@ -5,18 +5,18 @@
│ │ │  o2 = ideal(x x  - x x )
│ │ │              0 1    2 3
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │  
│ │ │  i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.187761s (cpu); 0.10659s (thread); 0s (gc)
│ │ │ + -- used 0.194923s (cpu); 0.0992972s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.297065s (cpu); 0.151907s (thread); 0s (gc)
│ │ │ + -- used 0.307658s (cpu); 0.124306s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -37,17 +37,17 @@
│ │ │                                                2   2
│ │ │       4x   x   x   x    - 2x   x   x   x    + x   x   ))
│ │ │         0,0 0,1 1,2 1,3     0,2 0,3 1,2 1,3    0,2 1,3
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.288499s (cpu); 0.151777s (thread); 0s (gc)
│ │ │ + -- used 0.324907s (cpu); 0.152657s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
│ │ │  
│ │ │  i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc)
│ │ │ + -- used 0.350297s (cpu); 0.158133s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  o2 = ideal (x  + x  + x  + x , x x  + x x  + x x )
│ │ │               0    1    2    3   0 1    1 2    2 3
│ │ │  
│ │ │  o2 : Ideal of P3
│ │ │  
│ │ │  i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.0407572s (cpu); 0.040758s (thread); 0s (gc)
│ │ │ + -- used 0.0571286s (cpu); 0.057133s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -72,15 +72,15 @@
│ │ │  o5 = ideal (x  - x x , x  - x x x , x x  - x x )
│ │ │               1    0 2   2    0 1 3   1 2    0 3
│ │ │  
│ │ │  o5 : Ideal of P3
│ │ │  
│ │ │  i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.0401433s (cpu); 0.0401425s (thread); 0s (gc)
│ │ │ + -- used 0.0596789s (cpu); 0.0594294s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -117,24 +117,24 @@
│ │ │  o9 = ideal(x x  + x x )
│ │ │              0 1    2 3
│ │ │  
│ │ │  o9 : Ideal of P3
│ │ │  
│ │ │  i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.160178s (cpu); 0.106293s (thread); 0s (gc)
│ │ │ + -- used 0.243972s (cpu); 0.155972s (thread); 0s (gc)
│ │ │  
│ │ │                       2                              2
│ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3  
│ │ │        -----------------------------------------------------------------------
│ │ │        x     x     )
│ │ │         0,2,3 1,2,3
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
│ │ │ - -- used 0.0594214s (cpu); 0.0594176s (thread); 0s (gc)
│ │ │ + -- used 0.0975502s (cpu); 0.0975584s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  
│ │ │                 ZZ
│ │ │  o2 : Ideal of ----[x ..x ]
│ │ │                3331  0   5
│ │ │  
│ │ │  i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 5.50368s (cpu); 4.92753s (thread); 0s (gc)
│ │ │ + -- used 6.29748s (cpu); 5.83234s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -143,19 +143,19 @@
│ │ │                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            3331  0,1,2   0,1,3   0,2,3   1,2,3   0,1,4   0,2,4   1,2,4   0,3,4   1,3,4   2,3,4   0,1,5   0,2,5   1,2,5   0,3,5   1,3,5   2,3,5   0,4,5   1,4,5   2,4,5   3,4,5
│ │ │  o3 : -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │       (x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      - x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x      - x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     , x     x      - x     x      + x     x     )
│ │ │         2,3,5 1,4,5    1,3,5 2,4,5    1,2,5 3,4,5   2,3,4 1,4,5    1,3,4 2,4,5    1,2,4 3,4,5   2,3,5 0,4,5    0,3,5 2,4,5    0,2,5 3,4,5   1,3,5 0,4,5    0,3,5 1,4,5    0,1,5 3,4,5   1,2,5 0,4,5    0,2,5 1,4,5    0,1,5 2,4,5   2,3,4 0,4,5    0,3,4 2,4,5    0,2,4 3,4,5   1,3,4 0,4,5    0,3,4 1,4,5    0,1,4 3,4,5   1,2,4 0,4,5    0,2,4 1,4,5    0,1,4 2,4,5   1,2,3 0,4,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 1,3,5    1,3,4 2,3,5    1,2,3 3,4,5   1,2,5 0,3,5    0,2,5 1,3,5    0,1,5 2,3,5   2,3,4 0,3,5    0,3,4 2,3,5    0,2,3 3,4,5   1,3,4 0,3,5    0,3,4 1,3,5    0,1,3 3,4,5   1,2,4 0,3,5    0,2,4 1,3,5    0,1,4 2,3,5    0,1,2 3,4,5   1,2,3 0,3,5    0,2,3 1,3,5    0,1,3 2,3,5   2,3,4 1,2,5    1,2,4 2,3,5    1,2,3 2,4,5   1,3,4 1,2,5    1,2,4 1,3,5    1,2,3 1,4,5   0,3,4 1,2,5    0,2,4 1,3,5    0,1,4 2,3,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 0,2,5    0,2,4 2,3,5    0,2,3 2,4,5   1,3,4 0,2,5    0,2,4 1,3,5    0,2,3 1,4,5    0,1,2 3,4,5   0,3,4 0,2,5    0,2,4 0,3,5    0,2,3 0,4,5   1,2,4 0,2,5    0,2,4 1,2,5    0,1,2 2,4,5   1,2,3 0,2,5    0,2,3 1,2,5    0,1,2 2,3,5   2,3,4 0,1,5    0,1,4 2,3,5    0,1,3 2,4,5    0,1,2 3,4,5   1,3,4 0,1,5    0,1,4 1,3,5    0,1,3 1,4,5   0,3,4 0,1,5    0,1,4 0,3,5    0,1,3 0,4,5   1,2,4 0,1,5    0,1,4 1,2,5    0,1,2 1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5    0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
│ │ │  
│ │ │  i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.1406s (cpu); 0.999558s (thread); 0s (gc)
│ │ │ + -- used 1.37339s (cpu); 1.285s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc)
│ │ │ + -- used 0.325274s (cpu); 0.242849s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │           2               2                        2     2               2  
│ │ │  o6 = {a x  + a x x  + a x  + a x x  + a x x  + a x , b x  + b x x  + b x  +
│ │ │         0 0    1 0 1    3 1    2 0 2    4 1 2    5 2   0 0    1 0 1    3 1  
│ │ │ @@ -164,12 +164,12 @@
│ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.341473s (cpu); 0.215038s (thread); 0s (gc)
│ │ │ + -- used 0.327091s (cpu); 0.238136s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres))
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out
│ │ │ @@ -4,30 +4,30 @@
│ │ │  
│ │ │          2              2
│ │ │  o2 = a*x  + b*x*y + c*y
│ │ │  
│ │ │  o2 : ZZ[a..c][x..y]
│ │ │  
│ │ │  i3 : time discriminant F
│ │ │ - -- used 0.00924086s (cpu); 0.00924015s (thread); 0s (gc)
│ │ │ + -- used 0.0120981s (cpu); 0.0120971s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
│ │ │  
│ │ │  i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3
│ │ │  
│ │ │          3      2         2      3
│ │ │  o5 = a*x  + b*x y + c*x*y  + d*y
│ │ │  
│ │ │  o5 : ZZ[a..d][x..y]
│ │ │  
│ │ │  i6 : time discriminant F
│ │ │ - -- used 0.0098778s (cpu); 0.0098795s (thread); 0s (gc)
│ │ │ + -- used 0.012931s (cpu); 0.0129333s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
│ │ │  
│ │ │  i7 : x=symbol x; R=ZZ/331[x_0..x_3]
│ │ │ @@ -59,15 +59,15 @@
│ │ │                  4        3      4             4        3      4
│ │ │  o12 = (t  + t )x  - t x x  + t x  + (t  - t )x  + t x x  + t x
│ │ │          0    1  0    1 0 1    0 1     0    1  2    1 2 3    0 3
│ │ │  
│ │ │  o12 : R'
│ │ │  
│ │ │  i13 : time D=discriminant pencil
│ │ │ - -- used 0.44332s (cpu); 0.443247s (thread); 0s (gc)
│ │ │ + -- used 0.451077s (cpu); 0.450833s (thread); 0s (gc)
│ │ │  
│ │ │             108      106 2       102 6      100 8       98 10       96 12  
│ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │             0        0   1       0   1      0   1       0  1        0  1   
│ │ │        -----------------------------------------------------------------------
│ │ │            94 14       92 16      90 18      88 20      86 22       84 24  
│ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │       x x )
│ │ │        0 3
│ │ │  
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
│ │ │  
│ │ │  i2 : time V' = dualVariety V
│ │ │ - -- used 0.202374s (cpu); 0.134305s (thread); 0s (gc)
│ │ │ + -- used 0.229693s (cpu); 0.143067s (thread); 0s (gc)
│ │ │  
│ │ │              2                 2    2
│ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   5
│ │ │  
│ │ │  i3 : time V == dualVariety V'
│ │ │ - -- used 0.310587s (cpu); 0.159166s (thread); 0s (gc)
│ │ │ + -- used 0.352237s (cpu); 0.192348s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331)
│ │ │  
│ │ │          3      2          2      3      2                   2          2  
│ │ │  o4 = a x  + a x x  + a x x  + a x  + a x x  + a x x x  + a x x  + a x x  +
│ │ │ @@ -38,22 +38,22 @@
│ │ │        8 1 2    9 2
│ │ │  
│ │ │        ZZ
│ │ │  o4 : ----[a ..a ][x ..x ]
│ │ │       3331  0   9   0   2
│ │ │  
│ │ │  i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.162855s (cpu); 0.0858796s (thread); 0s (gc)
│ │ │ + -- used 0.188296s (cpu); 0.108424s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
│ │ │  
│ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc)
│ │ │ + -- used 0.885793s (cpu); 0.803403s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
│ │ │  
│ │ │  i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  o1 = ideal (x  - x x , x x  - x x , x  - x x )
│ │ │               2    1 3   1 2    0 3   1    0 2
│ │ │  
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │  
│ │ │  i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.141834s (cpu); 0.0752672s (thread); 0s (gc)
│ │ │ + -- used 0.161606s (cpu); 0.0770098s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -56,15 +56,15 @@
│ │ │       x   x   x   x    - 2x   x   x   x    - x   x   x   x    + x   x
│ │ │        0,0 0,1 1,1 1,3     0,0 0,2 1,1 1,3    0,0 0,1 1,2 1,3    0,0 1,3
│ │ │  
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
│ │ │  
│ │ │  i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.177092s (cpu); 0.069066s (thread); 0s (gc)
│ │ │ + -- used 0.268877s (cpu); 0.103267s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │  o4 : QQ[a   ..a   ]
│ │ │           0,0   1,1
│ │ │  
│ │ │  i5 : w = chowForm C;
│ │ │  
│ │ │  i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.108209s (cpu); 0.0419396s (thread); 0s (gc)
│ │ │ + -- used 0.138234s (cpu); 0.05342s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -130,14 +130,14 @@
│ │ │             2      3      2
│ │ │       2x   x    - x    + x   )}
│ │ │         0,0 1,1    1,1    1,0
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.115818s (cpu); 0.0462516s (thread); 0s (gc)
│ │ │ + -- used 0.140737s (cpu); 0.0565381s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │       + -p  + -p p  + 7p p  + 6p p  + -p p  + --p )
│ │ │         4 3   9 0 4     1 4     2 4   9 3 4   10 4
│ │ │  
│ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │  
│ │ │  i2 : time hurwitzForm Q
│ │ │ - -- used 0.125887s (cpu); 0.0563778s (thread); 0s (gc)
│ │ │ + -- used 0.152292s (cpu); 0.0679301s (thread); 0s (gc)
│ │ │  
│ │ │                2                                 2                      
│ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │                0,1            0,1 0,2            0,2           0,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │                                 2                                          
│ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out
│ │ │ @@ -26,15 +26,15 @@
│ │ │       QQ[p   ..p   , p   , p   , p   , p   ]
│ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │  o1 : --------------------------------------
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
│ │ │  
│ │ │  i2 : time isCoisotropic w
│ │ │ - -- used 0.00877231s (cpu); 0.00877035s (thread); 0s (gc)
│ │ │ + -- used 0.0148241s (cpu); 0.0147164s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : -- random quadric in G(1,3)
│ │ │       w' = random(2,Grass(1,3))
│ │ │  
│ │ │        3 2     3           7 2                 5           7 2     10        
│ │ │ @@ -56,12 +56,12 @@
│ │ │       QQ[p   ..p   , p   , p   , p   , p   ]
│ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │  o3 : --------------------------------------
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
│ │ │  
│ │ │  i4 : time isCoisotropic w'
│ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc)
│ │ │ + -- used 0.010694s (cpu); 0.0106331s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out
│ │ │ @@ -31,12 +31,12 @@
│ │ │            4        5
│ │ │  
│ │ │                  ZZ
│ │ │  o3 : Ideal of -----[x ..x ]
│ │ │                33331  0   5
│ │ │  
│ │ │  i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0181849s (cpu); 0.0181858s (thread); 0s (gc)
│ │ │ + -- used 0.0221501s (cpu); 0.0221494s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │                     2          2        2      3
│ │ │       c x x x  + c x x  + c x x  + c x x  + c x }
│ │ │        4 0 1 2    7 1 2    5 0 2    8 1 2    9 2
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00373056s (cpu); 0.00372864s (thread); 0s (gc)
│ │ │ + -- used 0.00509889s (cpu); 0.0051005s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -78,15 +78,15 @@
│ │ │       10   2   7   2   5 3
│ │ │       --p p  + -p p  + -p }
│ │ │        9 0 2   8 1 2   6 2
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00252047s (cpu); 0.00251818s (thread); 0s (gc)
│ │ │ + -- used 0.00479387s (cpu); 0.00479374s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
│ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
│ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
│ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
│ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
│ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out
│ │ │ @@ -9,29 +9,29 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │  o3 : Ideal of P4
│ │ │  
│ │ │  i4 : time p = plucker L
│ │ │ - -- used 0.00473681s (cpu); 0.00473472s (thread); 0s (gc)
│ │ │ + -- used 0.00722967s (cpu); 0.00722836s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │       11804x   , x    + 14854x   )
│ │ │             3,4   0,1         3,4
│ │ │  
│ │ │  o4 : Ideal of G'1'4
│ │ │  
│ │ │  i5 : time L' = plucker p
│ │ │ - -- used 0.121088s (cpu); 0.0718194s (thread); 0s (gc)
│ │ │ + -- used 0.168733s (cpu); 0.0721714s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -40,25 +40,25 @@
│ │ │  i6 : assert(L' == L)
│ │ │  
│ │ │  i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve
│ │ │  
│ │ │  o7 : Ideal of G'1'4
│ │ │  
│ │ │  i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.148823s (cpu); 0.0922017s (thread); 0s (gc)
│ │ │ + -- used 0.155673s (cpu); 0.0671198s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
│ │ │  
│ │ │  i9 : (codim W,degree W)
│ │ │  
│ │ │  o9 = (2, 5)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.244058s (cpu); 0.244015s (thread); 0s (gc)
│ │ │ + -- used 0.175743s (cpu); 0.175745s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of G'1'4
│ │ │  
│ │ │  i11 : assert(Y' == Y)
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │       3     2    9    7     2    9        3       1    8    4
│ │ │       -b)y*w  + (-a + -b)z*w  + (-a + 2b)w , 2x + -y + -z + -w}
│ │ │       4          8    8          7                4    3    5
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.562085s (cpu); 0.289584s (thread); 0s (gc)
│ │ │ + -- used 0.676111s (cpu); 0.294737s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -56,15 +56,15 @@
│ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o3 : QQ[a..b]
│ │ │  
│ │ │  i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.273183s (cpu); 0.131953s (thread); 0s (gc)
│ │ │ + -- used 0.311132s (cpu); 0.13478s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -77,15 +77,15 @@
│ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o4 : QQ[a..b]
│ │ │  
│ │ │  i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.302008s (cpu); 0.301983s (thread); 0s (gc)
│ │ │ + -- used 0.341704s (cpu); 0.341469s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -98,15 +98,15 @@
│ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │       -------------------------------a*b  - ---------------------------b
│ │ │            1119954511872000000000                895963609497600000
│ │ │  
│ │ │  o5 : QQ[a..b]
│ │ │  
│ │ │  i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.625765s (cpu); 0.559323s (thread); 0s (gc)
│ │ │ + -- used 0.773305s (cpu); 0.690286s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │         2               2                            3      2         4
│ │ │  o2 = {x  + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y  + t*x y*z + u*z }
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : time resultant F
│ │ │ - -- used 0.123723s (cpu); 0.0492639s (thread); 0s (gc)
│ │ │ + -- used 0.128718s (cpu); 0.0481717s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -64,15 +64,15 @@
│ │ │            3
│ │ │       + c x }
│ │ │          9 2
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time resultant F
│ │ │ - -- used 3.07394s (cpu); 2.19358s (thread); 0s (gc)
│ │ │ + -- used 2.73125s (cpu); 2.07551s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1690,12 +1690,12 @@
│ │ │                            2     2               2                        2
│ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time # terms resultant F
│ │ │ - -- used 0.644433s (cpu); 0.456975s (thread); 0s (gc)
│ │ │ + -- used 0.619252s (cpu); 0.453477s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 21894
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │                 1 2    0 3     1 3    0 4     3    2 4
│ │ │  
│ │ │  o2 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │  
│ │ │  i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.0398049s (cpu); 0.039807s (thread); 0s (gc)
│ │ │ + -- used 0.054413s (cpu); 0.0544143s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -26,15 +26,15 @@
│ │ │                                                            0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  o3 : ----------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │       (p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   )
│ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
│ │ │  
│ │ │  i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.16271s (cpu); 0.0952426s (thread); 0s (gc)
│ │ │ + -- used 0.190642s (cpu); 0.11278s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -68,32 +68,32 @@
│ │ │                                                                                0,1,2   0,1,3   0,2,3   1,2,3   0,1,4   0,2,4   1,2,4   0,3,4   1,3,4   2,3,4
│ │ │  o4 : ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │       (p     p      - p     p      + p     p     , p     p      - p     p      + p     p     , p     p      - p     p      + p     p     , p     p      - p     p      + p     p     , p     p      - p     p      + p     p     )
│ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
│ │ │  
│ │ │  i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.117742s (cpu); 0.0528496s (thread); 0s (gc)
│ │ │ + -- used 0.152329s (cpu); 0.0669285s (thread); 0s (gc)
│ │ │  
│ │ │                2                                             2
│ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │        0,1,3,4 0,2,3,4    0,1,2,4 0,2,3,4 1,2,3,4    0,1,2,3 1,2,3,4
│ │ │  
│ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
│ │ │  
│ │ │  i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ - -- used 0.130131s (cpu); 0.0621371s (thread); 0s (gc)
│ │ │ + -- used 0.149774s (cpu); 0.0692121s (thread); 0s (gc)
│ │ │  
│ │ │              2               2
│ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │              1 2    0 1 3    0 4
│ │ │  
│ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │  
│ │ │  i7 : -- we then can recover S
│ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ - -- used 0.190994s (cpu); 0.117998s (thread); 0s (gc)
│ │ │ + -- used 0.234863s (cpu); 0.153028s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html
│ │ │ @@ -91,24 +91,24 @@
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.187761s (cpu); 0.10659s (thread); 0s (gc)
    
│ │ │ + -- used 0.194923s (cpu); 0.0992972s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    In the next example, we calculate the defining ideal of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its dual variety.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.297065s (cpu); 0.151907s (thread); 0s (gc)
│ │ │ + -- used 0.307658s (cpu); 0.124306s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -135,26 +135,26 @@
│ │ │            
    
│ │ │          
    If the option Duality is set to true, then the method applies the so-called "dual Cayley trick".
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,20 +38,20 @@
│ │ │ │  
│ │ │ │  o2 = ideal(x x  - x x )
│ │ │ │              0 1    2 3
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │ │                    0   3
│ │ │ │  i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ │ - -- used 0.187761s (cpu); 0.10659s (thread); 0s (gc)
│ │ │ │ + -- used 0.194923s (cpu); 0.0992972s (thread); 0s (gc)
│ │ │ │  In the next example, we calculate the defining ideal of $\mathbb
│ │ │ │  {P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its
│ │ │ │  dual variety.
│ │ │ │  i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ │ - -- used 0.297065s (cpu); 0.151907s (thread); 0s (gc)
│ │ │ │ + -- used 0.307658s (cpu); 0.124306s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │  
│ │ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x
│ │ │ │ @@ -73,18 +73,18 @@
│ │ │ │       4x   x   x   x    - 2x   x   x   x    + x   x   ))
│ │ │ │         0,0 0,1 1,2 1,3     0,2 0,3 1,2 1,3    0,2 1,3
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  If the option Duality is set to true, then the method applies the so-called
│ │ │ │  "dual Cayley trick".
│ │ │ │  i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ │ - -- used 0.288499s (cpu); 0.151777s (thread); 0s (gc)
│ │ │ │ + -- used 0.324907s (cpu); 0.152657s (thread); 0s (gc)
│ │ │ │  i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
│ │ │ │  i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc)
│ │ │ │ + -- used 0.350297s (cpu); 0.158133s (thread); 0s (gc)
│ │ │ │  i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_u_a_l_V_a_r_i_e_t_y -- projective dual variety
│ │ │ │  ********** WWaayyss ttoo uussee ccaayylleeyyTTrriicckk:: **********
│ │ │ │      * cayleyTrick(Ideal,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_a_y_l_e_y_T_r_i_c_k is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : Ideal of P3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.288499s (cpu); 0.151777s (thread); 0s (gc)
    
│ │ │ + -- used 0.324907s (cpu); 0.152657s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    
│ │ │              
    
│ │ │                
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc)
    
│ │ │ + -- used 0.350297s (cpu); 0.158133s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    
│ │ │              
    
│ │ │                
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.0407572s (cpu); 0.040758s (thread); 0s (gc)
│ │ │ + -- used 0.0571286s (cpu); 0.057133s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -167,15 +167,15 @@
│ │ │  o5 : Ideal of P3
    
│ │ │              
    
│ │ │                
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.0401433s (cpu); 0.0401425s (thread); 0s (gc)
│ │ │ + -- used 0.0596789s (cpu); 0.0594294s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -227,30 +227,30 @@
│ │ │  o9 : Ideal of P3
    
│ │ │              
    
│ │ │                
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.160178s (cpu); 0.106293s (thread); 0s (gc)
│ │ │ + -- used 0.243972s (cpu); 0.155972s (thread); 0s (gc)
│ │ │  
│ │ │                       2                              2
│ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3  
│ │ │        -----------------------------------------------------------------------
│ │ │        x     x     )
│ │ │         0,2,3 1,2,3
│ │ │  
│ │ │  o10 : Sequence
    
│ │ │              
    
│ │ │                
    i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
│ │ │ - -- used 0.0594214s (cpu); 0.0594176s (thread); 0s (gc)
│ │ │ + -- used 0.0975502s (cpu); 0.0975584s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    
│ │ │              
    
│ │ │          
    Note that chowEquations(W,0) is not the same as chowEquations W.
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │               2    2    2    2
│ │ │ │  o2 = ideal (x  + x  + x  + x , x x  + x x  + x x )
│ │ │ │               0    1    2    3   0 1    1 2    2 3
│ │ │ │  
│ │ │ │  o2 : Ideal of P3
│ │ │ │  i3 : -- Chow equations of C
│ │ │ │       time eqsC = chowEquations chowForm C
│ │ │ │ - -- used 0.0407572s (cpu); 0.040758s (thread); 0s (gc)
│ │ │ │ + -- used 0.0571286s (cpu); 0.057133s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2    2 2    2 2    4                2      2 2   2
│ │ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │               2          3              2    2
│ │ │ │  o5 = ideal (x  - x x , x  - x x x , x x  - x x )
│ │ │ │               1    0 2   2    0 1 3   1 2    0 3
│ │ │ │  
│ │ │ │  o5 : Ideal of P3
│ │ │ │  i6 : -- Chow equations of D
│ │ │ │       time eqsD = chowEquations chowForm D
│ │ │ │ - -- used 0.0401433s (cpu); 0.0401425s (thread); 0s (gc)
│ │ │ │ + -- used 0.0596789s (cpu); 0.0594294s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2
│ │ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            2      3 2   3        3 2   4         2         2 2       3
│ │ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ │ @@ -135,27 +135,27 @@
│ │ │ │  o9 = ideal(x x  + x x )
│ │ │ │              0 1    2 3
│ │ │ │  
│ │ │ │  o9 : Ideal of P3
│ │ │ │  i10 : -- tangential Chow forms of Q
│ │ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm
│ │ │ │  (Q,1),tangentialChowForm(Q,2))
│ │ │ │ - -- used 0.160178s (cpu); 0.106293s (thread); 0s (gc)
│ │ │ │ + -- used 0.243972s (cpu); 0.155972s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                       2                              2
│ │ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x     x     )
│ │ │ │         0,2,3 1,2,3
│ │ │ │  
│ │ │ │  o10 : Sequence
│ │ │ │  i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations
│ │ │ │  (W2,2))
│ │ │ │ - -- used 0.0594214s (cpu); 0.0594176s (thread); 0s (gc)
│ │ │ │ + -- used 0.0975502s (cpu); 0.0975584s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  Note that chowEquations(W,0) is not the same as chowEquations W.
│ │ │ │  ********** WWaayyss ttoo uussee cchhoowwEEqquuaattiioonnss:: **********
│ │ │ │      * chowEquations(RingElement)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_h_o_w_E_q_u_a_t_i_o_n_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │                3331  0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 5.50368s (cpu); 4.92753s (thread); 0s (gc)
│ │ │ + -- used 6.29748s (cpu); 5.83234s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -232,22 +232,22 @@
│ │ │         2,3,5 1,4,5    1,3,5 2,4,5    1,2,5 3,4,5   2,3,4 1,4,5    1,3,4 2,4,5    1,2,4 3,4,5   2,3,5 0,4,5    0,3,5 2,4,5    0,2,5 3,4,5   1,3,5 0,4,5    0,3,5 1,4,5    0,1,5 3,4,5   1,2,5 0,4,5    0,2,5 1,4,5    0,1,5 2,4,5   2,3,4 0,4,5    0,3,4 2,4,5    0,2,4 3,4,5   1,3,4 0,4,5    0,3,4 1,4,5    0,1,4 3,4,5   1,2,4 0,4,5    0,2,4 1,4,5    0,1,4 2,4,5   1,2,3 0,4,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 1,3,5    1,3,4 2,3,5    1,2,3 3,4,5   1,2,5 0,3,5    0,2,5 1,3,5    0,1,5 2,3,5   2,3,4 0,3,5    0,3,4 2,3,5    0,2,3 3,4,5   1,3,4 0,3,5    0,3,4 1,3,5    0,1,3 3,4,5   1,2,4 0,3,5    0,2,4 1,3,5    0,1,4 2,3,5    0,1,2 3,4,5   1,2,3 0,3,5    0,2,3 1,3,5    0,1,3 2,3,5   2,3,4 1,2,5    1,2,4 2,3,5    1,2,3 2,4,5   1,3,4 1,2,5    1,2,4 1,3,5    1,2,3 1,4,5   0,3,4 1,2,5    0,2,4 1,3,5    0,1,4 2,3,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 0,2,5    0,2,4 2,3,5    0,2,3 2,4,5   1,3,4 0,2,5    0,2,4 1,3,5    0,2,3 1,4,5    0,1,2 3,4,5   0,3,4 0,2,5    0,2,4 0,3,5    0,2,3 0,4,5   1,2,4 0,2,5    0,2,4 1,2,5    0,1,2 2,4,5   1,2,3 0,2,5    0,2,3 1,2,5    0,1,2 2,3,5   2,3,4 0,1,5    0,1,4 2,3,5    0,1,3 2,4,5    0,1,2 3,4,5   1,3,4 0,1,5    0,1,4 1,3,5    0,1,3 1,4,5   0,3,4 0,1,5    0,1,4 0,3,5    0,1,3 0,4,5   1,2,4 0,1,5    0,1,4 1,2,5    0,1,2 1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5    0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.1406s (cpu); 0.999558s (thread); 0s (gc)
    
│ │ │ + -- used 1.37339s (cpu); 1.285s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc)
    
│ │ │ + -- used 0.325274s (cpu); 0.242849s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │ @@ -262,15 +262,15 @@
│ │ │  o6 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.341473s (cpu); 0.215038s (thread); 0s (gc)
    
│ │ │ + -- used 0.327091s (cpu); 0.238136s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres))
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o2 : Ideal of ----[x ..x ]
│ │ │ │                3331  0   5
│ │ │ │  i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an
│ │ │ │  affine chart of the Grassmannian)
│ │ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ │ - -- used 5.50368s (cpu); 4.92753s (thread); 0s (gc)
│ │ │ │ + -- used 6.29748s (cpu); 5.83234s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        4               2              2     2               2
│ │ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2     2              2
│ │ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ │ @@ -234,33 +234,33 @@
│ │ │ │  1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5
│ │ │ │  0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4
│ │ │ │  1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4
│ │ │ │  0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3
│ │ │ │  0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
│ │ │ │  i4 : -- equivalently (but faster)...
│ │ │ │       time assert(ChowV === chowForm f)
│ │ │ │ - -- used 1.1406s (cpu); 0.999558s (thread); 0s (gc)
│ │ │ │ + -- used 1.37339s (cpu); 1.285s (thread); 0s (gc)
│ │ │ │  i5 : -- X-resultant of V
│ │ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc)
│ │ │ │ + -- used 0.325274s (cpu); 0.242849s (thread); 0s (gc)
│ │ │ │  i6 : -- three generic ternary quadrics
│ │ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │ │  
│ │ │ │           2               2                        2     2               2
│ │ │ │  o6 = {a x  + a x x  + a x  + a x x  + a x x  + a x , b x  + b x x  + b x  +
│ │ │ │         0 0    1 0 1    3 1    2 0 2    4 1 2    5 2   0 0    1 0 1    3 1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                            2     2               2                        2
│ │ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : -- resultant of the three forms
│ │ │ │       time resF = resultant F;
│ │ │ │ - -- used 0.341473s (cpu); 0.215038s (thread); 0s (gc)
│ │ │ │ + -- used 0.327091s (cpu); 0.238136s (thread); 0s (gc)
│ │ │ │  i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring
│ │ │ │  Xres))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety
│ │ │ │      * _h_u_r_w_i_t_z_F_o_r_m -- Hurwitz form of a projective variety
│ │ │ │  ********** WWaayyss ttoo uussee cchhoowwFFoorrmm:: **********
│ │ │ │      * chowForm(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : ZZ[a..c][x..y]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time discriminant F
│ │ │ - -- used 0.00924086s (cpu); 0.00924015s (thread); 0s (gc)
│ │ │ + -- used 0.0120981s (cpu); 0.0120971s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    
│ │ │              
│ │ │            
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o5 : ZZ[a..d][x..y]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time discriminant F
│ │ │ - -- used 0.0098778s (cpu); 0.0098795s (thread); 0s (gc)
│ │ │ + -- used 0.012931s (cpu); 0.0129333s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
    
│ │ │              
│ │ │            
│ │ │ @@ -170,15 +170,15 @@
│ │ │  
│ │ │  o12 : R'
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.44332s (cpu); 0.443247s (thread); 0s (gc)
│ │ │ + -- used 0.451077s (cpu); 0.450833s (thread); 0s (gc)
│ │ │  
│ │ │             108      106 2       102 6      100 8       98 10       96 12  
│ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │             0        0   1       0   1      0   1       0  1        0  1   
│ │ │        -----------------------------------------------------------------------
│ │ │            94 14       92 16      90 18      88 20      86 22       84 24  
│ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,28 +23,28 @@
│ │ │ │  i1 : ZZ[a,b,c][x,y]; F = a*x^2+b*x*y+c*y^2
│ │ │ │  
│ │ │ │          2              2
│ │ │ │  o2 = a*x  + b*x*y + c*y
│ │ │ │  
│ │ │ │  o2 : ZZ[a..c][x..y]
│ │ │ │  i3 : time discriminant F
│ │ │ │ - -- used 0.00924086s (cpu); 0.00924015s (thread); 0s (gc)
│ │ │ │ + -- used 0.0120981s (cpu); 0.0120971s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2
│ │ │ │  o3 = - b  + 4a*c
│ │ │ │  
│ │ │ │  o3 : ZZ[a..c]
│ │ │ │  i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3
│ │ │ │  
│ │ │ │          3      2         2      3
│ │ │ │  o5 = a*x  + b*x y + c*x*y  + d*y
│ │ │ │  
│ │ │ │  o5 : ZZ[a..d][x..y]
│ │ │ │  i6 : time discriminant F
│ │ │ │ - -- used 0.0098778s (cpu); 0.0098795s (thread); 0s (gc)
│ │ │ │ + -- used 0.012931s (cpu); 0.0129333s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2       3     3                   2 2
│ │ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │ │  
│ │ │ │  o6 : ZZ[a..d]
│ │ │ │  The next example illustrates how computing the intersection of a pencil
│ │ │ │  generated by two degree $d$ forms $F(x_0,\ldots,x_n), G(x_0,\ldots,x_n)$ with
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  
│ │ │ │                  4        3      4             4        3      4
│ │ │ │  o12 = (t  + t )x  - t x x  + t x  + (t  - t )x  + t x x  + t x
│ │ │ │          0    1  0    1 0 1    0 1     0    1  2    1 2 3    0 3
│ │ │ │  
│ │ │ │  o12 : R'
│ │ │ │  i13 : time D=discriminant pencil
│ │ │ │ - -- used 0.44332s (cpu); 0.443247s (thread); 0s (gc)
│ │ │ │ + -- used 0.451077s (cpu); 0.450833s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             108      106 2       102 6      100 8       98 10       96 12
│ │ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │ │             0        0   1       0   1      0   1       0  1        0  1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            94 14       92 16      90 18      88 20      86 22       84 24
│ │ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html
│ │ │ @@ -95,28 +95,28 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.202374s (cpu); 0.134305s (thread); 0s (gc)
│ │ │ + -- used 0.229693s (cpu); 0.143067s (thread); 0s (gc)
│ │ │  
│ │ │              2                 2    2
│ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : time V == dualVariety V'
│ │ │ - -- used 0.310587s (cpu); 0.159166s (thread); 0s (gc)
│ │ │ + -- used 0.352237s (cpu); 0.192348s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    In the next example, we verify that the discriminant of a generic ternary cubic form coincides with the dual variety of the 3-th Veronese embedding of the plane, which is a hypersurface of degree 12 in $\mathbb{P}^9$
    
│ │ │          
│ │ │ @@ -136,25 +136,25 @@
│ │ │  o4 : ----[a ..a ][x ..x ]
│ │ │       3331  0   9   0   2
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,24 +31,24 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x )
│ │ │ │        0 3
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │ │                    0   5
│ │ │ │  i2 : time V' = dualVariety V
│ │ │ │ - -- used 0.202374s (cpu); 0.134305s (thread); 0s (gc)
│ │ │ │ + -- used 0.229693s (cpu); 0.143067s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              2                 2    2
│ │ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │ │                    0   5
│ │ │ │  i3 : time V == dualVariety V'
│ │ │ │ - -- used 0.310587s (cpu); 0.159166s (thread); 0s (gc)
│ │ │ │ + -- used 0.352237s (cpu); 0.192348s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  In the next example, we verify that the discriminant of a generic ternary cubic
│ │ │ │  form coincides with the dual variety of the 3-th Veronese embedding of the
│ │ │ │  plane, which is a hypersurface of degree 12 in $\mathbb{P}^9$
│ │ │ │  i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331)
│ │ │ │  
│ │ │ │ @@ -60,21 +60,21 @@
│ │ │ │       a x x  + a x
│ │ │ │        8 1 2    9 2
│ │ │ │  
│ │ │ │        ZZ
│ │ │ │  o4 : ----[a ..a ][x ..x ]
│ │ │ │       3331  0   9   0   2
│ │ │ │  i5 : time discF = ideal discriminant F;
│ │ │ │ - -- used 0.162855s (cpu); 0.0858796s (thread); 0s (gc)
│ │ │ │ + -- used 0.188296s (cpu); 0.108424s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o5 : Ideal of ----[a ..a ]
│ │ │ │                3331  0   9
│ │ │ │  i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc)
│ │ │ │ + -- used 0.885793s (cpu); 0.803403s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                 ZZ
│ │ │ │  o6 : Ideal of ----[x ..x ]
│ │ │ │                3331  0   9
│ │ │ │  i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z)
│ │ │ │  
│ │ │ │  o7 = true
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html
│ │ │ @@ -90,15 +90,15 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.162855s (cpu); 0.0858796s (thread); 0s (gc)
│ │ │ + -- used 0.188296s (cpu); 0.108424s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    
│ │ │              
    
│ │ │                
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc)
│ │ │ + -- used 0.885793s (cpu); 0.803403s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    
│ │ │              
    
│ │ │                
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.141834s (cpu); 0.0752672s (thread); 0s (gc)
│ │ │ + -- used 0.161606s (cpu); 0.0770098s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -143,15 +143,15 @@
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    
│ │ │              
    
│ │ │                
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.177092s (cpu); 0.069066s (thread); 0s (gc)
│ │ │ + -- used 0.268877s (cpu); 0.103267s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -184,15 +184,15 @@
│ │ │              
    		
│ │ │                
    i5 : w = chowForm C;
    
│ │ │              
    
│ │ │                
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.108209s (cpu); 0.0419396s (thread); 0s (gc)
│ │ │ + -- used 0.138234s (cpu); 0.05342s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -232,15 +232,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.115818s (cpu); 0.0462516s (thread); 0s (gc)
│ │ │ + -- used 0.140737s (cpu); 0.0565381s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │               2                       2
│ │ │ │  o1 = ideal (x  - x x , x x  - x x , x  - x x )
│ │ │ │               2    1 3   1 2    0 3   1    0 2
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │ │                    0   3
│ │ │ │  i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ │ - -- used 0.141834s (cpu); 0.0752672s (thread); 0s (gc)
│ │ │ │ + -- used 0.161606s (cpu); 0.0770098s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3   3          2   2              2       2          2   3
│ │ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2       2               2   2
│ │ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ │ @@ -75,15 +75,15 @@
│ │ │ │            2       2       2           2      2           2      3   3
│ │ │ │       x   x   x   x    - 2x   x   x   x    - x   x   x   x    + x   x
│ │ │ │        0,0 0,1 1,1 1,3     0,0 0,2 1,1 1,3    0,0 0,1 1,2 1,3    0,0 1,3
│ │ │ │  
│ │ │ │  o2 : QQ[x   ..x   ]
│ │ │ │           0,0   1,3
│ │ │ │  i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ │ - -- used 0.177092s (cpu); 0.069066s (thread); 0s (gc)
│ │ │ │ + -- used 0.268877s (cpu); 0.103267s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3          2                         2
│ │ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2      3      2
│ │ │ │       2x   x    + x    + x
│ │ │ │ @@ -105,15 +105,15 @@
│ │ │ │  o4 : QQ[a   ..a   ]
│ │ │ │           0,0   1,1
│ │ │ │  As another application, we check that the singular locus of the Chow form of
│ │ │ │  the twisted cubic has dimension 2 (on each standard chart).
│ │ │ │  i5 : w = chowForm C;
│ │ │ │  i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel
│ │ │ │  (w,AffineChartGrass=>s))
│ │ │ │ - -- used 0.108209s (cpu); 0.0419396s (thread); 0s (gc)
│ │ │ │ + -- used 0.138234s (cpu); 0.05342s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                     3          2          3                       2
│ │ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                           2      2            2   3               2
│ │ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ │ @@ -149,15 +149,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2      3      2
│ │ │ │       2x   x    - x    + x   )}
│ │ │ │         0,0 1,1    1,1    1,0
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time apply(U,u->dim singularLocus u)
│ │ │ │ - -- used 0.115818s (cpu); 0.0462516s (thread); 0s (gc)
│ │ │ │ + -- used 0.140737s (cpu); 0.0565381s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  ********** WWaayyss ttoo uussee ffrroommPPlluucckkeerrTTooSSttiieeffeell:: **********
│ │ │ │      * fromPluckerToStiefel(Ideal)
│ │ │ │      * fromPluckerToStiefel(Matrix)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │                    0   4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.125887s (cpu); 0.0563778s (thread); 0s (gc)
│ │ │ + -- used 0.152292s (cpu); 0.0679301s (thread); 0s (gc)
│ │ │  
│ │ │                2                                 2                      
│ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │                0,1            0,1 0,2            0,2           0,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │                                 2                                          
│ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │         5 2   7                       2        3 2
│ │ │ │       + -p  + -p p  + 7p p  + 6p p  + -p p  + --p )
│ │ │ │         4 3   9 0 4     1 4     2 4   9 3 4   10 4
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i2 : time hurwitzForm Q
│ │ │ │ - -- used 0.125887s (cpu); 0.0563778s (thread); 0s (gc)
│ │ │ │ + -- used 0.152292s (cpu); 0.0679301s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                 2
│ │ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │ │                0,1            0,1 0,2            0,2           0,1 1,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                 2
│ │ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time isCoisotropic w
│ │ │ - -- used 0.00877231s (cpu); 0.00877035s (thread); 0s (gc)
│ │ │ + -- used 0.0148241s (cpu); 0.0147164s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- random quadric in G(1,3)
│ │ │ @@ -145,15 +145,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc)
│ │ │ + -- used 0.010694s (cpu); 0.0106331s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -44,15 +44,15 @@
│ │ │ │  
│ │ │ │       QQ[p   ..p   , p   , p   , p   , p   ]
│ │ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │ │  o1 : --------------------------------------
│ │ │ │           p   p    - p   p    + p   p
│ │ │ │            1,2 0,3    0,2 1,3    0,1 2,3
│ │ │ │  i2 : time isCoisotropic w
│ │ │ │ - -- used 0.00877231s (cpu); 0.00877035s (thread); 0s (gc)
│ │ │ │ + -- used 0.0148241s (cpu); 0.0147164s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  i3 : -- random quadric in G(1,3)
│ │ │ │       w' = random(2,Grass(1,3))
│ │ │ │  
│ │ │ │        3 2     3           7 2                 5           7 2     10
│ │ │ │  o3 = --p    + -p   p    + -p    + 5p   p    + -p   p    + -p    + --p   p
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │  
│ │ │ │       QQ[p   ..p   , p   , p   , p   , p   ]
│ │ │ │           0,1   0,2   1,2   0,3   1,3   2,3
│ │ │ │  o3 : --------------------------------------
│ │ │ │           p   p    - p   p    + p   p
│ │ │ │            1,2 0,3    0,2 1,3    0,1 2,3
│ │ │ │  i4 : time isCoisotropic w'
│ │ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc)
│ │ │ │ + -- used 0.010694s (cpu); 0.0106331s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  ********** WWaayyss ttoo uussee iissCCooiissoottrrooppiicc:: **********
│ │ │ │      * isCoisotropic(RingElement)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _i_s_C_o_i_s_o_t_r_o_p_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │  o3 : Ideal of -----[x ..x ]
│ │ │                33331  0   5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0181849s (cpu); 0.0181858s (thread); 0s (gc)
│ │ │ + -- used 0.0221501s (cpu); 0.0221494s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │       2380x  + 9482x )
│ │ │ │            4        5
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o3 : Ideal of -----[x ..x ]
│ │ │ │                33331  0   5
│ │ │ │  i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ │ - -- used 0.0181849s (cpu); 0.0181858s (thread); 0s (gc)
│ │ │ │ + -- used 0.0221501s (cpu); 0.0221494s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety
│ │ │ │      * _p_l_u_c_k_e_r -- get the Plücker coordinates of a linear subspace
│ │ │ │  ********** WWaayyss ttoo uussee iissIInnCCooiissoottrrooppiicc:: **********
│ │ │ │      * isInCoisotropic(Ideal,Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html
│ │ │ @@ -92,15 +92,15 @@
│ │ │  
│ │ │  o1 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00373056s (cpu); 0.00372864s (thread); 0s (gc)
│ │ │ + -- used 0.00509889s (cpu); 0.0051005s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -163,15 +163,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00252047s (cpu); 0.00251818s (thread); 0s (gc)
│ │ │ + -- used 0.00479387s (cpu); 0.00479374s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
│ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
│ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
│ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
│ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
│ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     2          2        2      3
│ │ │ │       c x x x  + c x x  + c x x  + c x x  + c x }
│ │ │ │        4 0 1 2    7 1 2    5 0 2    8 1 2    9 2
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00373056s (cpu); 0.00372864s (thread); 0s (gc)
│ │ │ │ + -- used 0.00509889s (cpu); 0.0051005s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0
│ │ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0
│ │ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0
│ │ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0
│ │ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0
│ │ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       10   2   7   2   5 3
│ │ │ │       --p p  + -p p  + -p }
│ │ │ │        9 0 2   8 1 2   6 2
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00252047s (cpu); 0.00251818s (thread); 0s (gc)
│ │ │ │ + -- used 0.00479387s (cpu); 0.00479374s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0
│ │ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0
│ │ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0
│ │ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0
│ │ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0
│ │ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : Ideal of P4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time p = plucker L
│ │ │ - -- used 0.00473681s (cpu); 0.00473472s (thread); 0s (gc)
│ │ │ + -- used 0.00722967s (cpu); 0.00722836s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time L' = plucker p
│ │ │ - -- used 0.121088s (cpu); 0.0718194s (thread); 0s (gc)
│ │ │ + -- used 0.168733s (cpu); 0.0721714s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.148823s (cpu); 0.0922017s (thread); 0s (gc)
│ │ │ + -- used 0.155673s (cpu); 0.0671198s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : (codim W,degree W)
│ │ │ @@ -163,15 +163,15 @@
│ │ │            
│ │ │          
│ │ │          
    In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.244058s (cpu); 0.244015s (thread); 0s (gc)
│ │ │ + -- used 0.175743s (cpu); 0.175745s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of G'1'4
    
│ │ │              
    
│ │ │                
    i11 : assert(Y' == Y)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,28 +28,28 @@
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │  o3 : Ideal of P4
│ │ │ │  i4 : time p = plucker L
│ │ │ │ - -- used 0.00473681s (cpu); 0.00473472s (thread); 0s (gc)
│ │ │ │ + -- used 0.00722967s (cpu); 0.00722836s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11804x   , x    + 14854x   )
│ │ │ │             3,4   0,1         3,4
│ │ │ │  
│ │ │ │  o4 : Ideal of G'1'4
│ │ │ │  i5 : time L' = plucker p
│ │ │ │ - -- used 0.121088s (cpu); 0.0718194s (thread); 0s (gc)
│ │ │ │ + -- used 0.168733s (cpu); 0.0721714s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │ @@ -60,26 +60,26 @@
│ │ │ │  $W\subset\mathbb{P}^n$ swept out by the linear spaces corresponding to points
│ │ │ │  of $Y$. As an example, we now compute a surface scroll $W\subset\mathbb{P}^4$
│ │ │ │  over an elliptic curve $Y\subset\mathbb{G}(1,\mathbb{P}^4)$.
│ │ │ │  i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve
│ │ │ │  
│ │ │ │  o7 : Ideal of G'1'4
│ │ │ │  i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ │ - -- used 0.148823s (cpu); 0.0922017s (thread); 0s (gc)
│ │ │ │ + -- used 0.155673s (cpu); 0.0671198s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of P4
│ │ │ │  i9 : (codim W,degree W)
│ │ │ │  
│ │ │ │  o9 = (2, 5)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb
│ │ │ │  {P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
│ │ │ │  i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ │ - -- used 0.244058s (cpu); 0.244015s (thread); 0s (gc)
│ │ │ │ + -- used 0.175743s (cpu); 0.175745s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of G'1'4
│ │ │ │  i11 : assert(Y' == Y)
│ │ │ │  WWaarrnniinngg: Notice that, by default, the computation is done on a randomly chosen
│ │ │ │  affine chart on the Grassmannian. To change this behavior, you can use the
│ │ │ │  _A_f_f_i_n_e_C_h_a_r_t_G_r_a_s_s option.
│ │ │ │  ********** WWaayyss ttoo uussee pplluucckkeerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o2 : List
│ │ │              
    
│ │ │                
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.562085s (cpu); 0.289584s (thread); 0s (gc)
│ │ │ + -- used 0.676111s (cpu); 0.294737s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.273183s (cpu); 0.131953s (thread); 0s (gc)
│ │ │ + -- used 0.311132s (cpu); 0.13478s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -161,15 +161,15 @@
│ │ │  
│ │ │  o4 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.302008s (cpu); 0.301983s (thread); 0s (gc)
│ │ │ + -- used 0.341704s (cpu); 0.341469s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o5 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.625765s (cpu); 0.559323s (thread); 0s (gc)
│ │ │ + -- used 0.773305s (cpu); 0.690286s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,15 +58,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       3     2    9    7     2    9        3       1    8    4
│ │ │ │       -b)y*w  + (-a + -b)z*w  + (-a + 2b)w , 2x + -y + -z + -w}
│ │ │ │       4          8    8          7                4    3    5
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ │ - -- used 0.562085s (cpu); 0.289584s (thread); 0s (gc)
│ │ │ │ + -- used 0.676111s (cpu); 0.294737s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o3 : QQ[a..b]
│ │ │ │  i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ │ - -- used 0.273183s (cpu); 0.131953s (thread); 0s (gc)
│ │ │ │ + -- used 0.311132s (cpu); 0.13478s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o4 : QQ[a..b]
│ │ │ │  i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ │ - -- used 0.302008s (cpu); 0.301983s (thread); 0s (gc)
│ │ │ │ + -- used 0.341704s (cpu); 0.341469s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -118,15 +118,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o5 : QQ[a..b]
│ │ │ │  i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ │ - -- used 0.625765s (cpu); 0.559323s (thread); 0s (gc)
│ │ │ │ + -- used 0.773305s (cpu); 0.690286s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time resultant F
│ │ │ - -- used 0.123723s (cpu); 0.0492639s (thread); 0s (gc)
│ │ │ + -- used 0.128718s (cpu); 0.0481717s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time resultant F
│ │ │ - -- used 3.07394s (cpu); 2.19358s (thread); 0s (gc)
│ │ │ + -- used 2.73125s (cpu); 2.07551s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1795,15 +1795,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time # terms resultant F
│ │ │ - -- used 0.644433s (cpu); 0.456975s (thread); 0s (gc)
│ │ │ + -- used 0.619252s (cpu); 0.453477s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 21894
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │  i2 : F = {x^2+3*t*y*z-u*z^2,(t+3*u-1)*x-y,-t*x*y^3+t*x^2*y*z+u*z^4}
│ │ │ │  
│ │ │ │         2               2                            3      2         4
│ │ │ │  o2 = {x  + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y  + t*x y*z + u*z }
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time resultant F
│ │ │ │ - -- used 0.123723s (cpu); 0.0492639s (thread); 0s (gc)
│ │ │ │ + -- used 0.128718s (cpu); 0.0481717s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                6 7          5 8       11          10 2         9 3
│ │ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            3
│ │ │ │       + c x }
│ │ │ │          9 2
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time resultant F
│ │ │ │ - -- used 3.07394s (cpu); 2.19358s (thread); 0s (gc)
│ │ │ │ + -- used 2.73125s (cpu); 2.07551s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2
│ │ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2
│ │ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ │ @@ -1712,15 +1712,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                            2     2               2                        2
│ │ │ │       b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
│ │ │ │        2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time # terms resultant F
│ │ │ │ - -- used 0.644433s (cpu); 0.456975s (thread); 0s (gc)
│ │ │ │ + -- used 0.619252s (cpu); 0.453477s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = 21894
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_h_o_w_F_o_r_m -- Chow form of a projective variety
│ │ │ │      * _d_i_s_c_r_i_m_i_n_a_n_t_(_R_i_n_g_E_l_e_m_e_n_t_)
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * resultant(List)
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │                    0   4
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.0398049s (cpu); 0.039807s (thread); 0s (gc)
│ │ │ + -- used 0.054413s (cpu); 0.0544143s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -123,15 +123,15 @@
│ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.16271s (cpu); 0.0952426s (thread); 0s (gc)
│ │ │ + -- used 0.190642s (cpu); 0.11278s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -168,43 +168,43 @@
│ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.117742s (cpu); 0.0528496s (thread); 0s (gc)
│ │ │ + -- used 0.152329s (cpu); 0.0669285s (thread); 0s (gc)
│ │ │  
│ │ │                2                                             2
│ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │        0,1,3,4 0,2,3,4    0,1,2,4 0,2,3,4 1,2,3,4    0,1,2,3 1,2,3,4
│ │ │  
│ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ - -- used 0.130131s (cpu); 0.0621371s (thread); 0s (gc)
│ │ │ + -- used 0.149774s (cpu); 0.0692121s (thread); 0s (gc)
│ │ │  
│ │ │              2               2
│ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │              1 2    0 1 3    0 4
│ │ │  
│ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │                    0   4
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : -- we then can recover S
│ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ - -- used 0.190994s (cpu); 0.117998s (thread); 0s (gc)
    
│ │ │ + -- used 0.234863s (cpu); 0.153028s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o2 = ideal (- p p  + p p , - p p  + p p , - p  + p p )
│ │ │ │                 1 2    0 3     1 3    0 4     3    2 4
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │ │       time tangentialChowForm(S,0)
│ │ │ │ - -- used 0.0398049s (cpu); 0.039807s (thread); 0s (gc)
│ │ │ │ + -- used 0.054413s (cpu); 0.0544143s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2                                                       2
│ │ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2
│ │ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │  - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p
│ │ │ │  p   )
│ │ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4
│ │ │ │  0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1
│ │ │ │  2,3
│ │ │ │  i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │ │       time tangentialChowForm(S,1)
│ │ │ │ - -- used 0.16271s (cpu); 0.0952426s (thread); 0s (gc)
│ │ │ │ + -- used 0.190642s (cpu); 0.11278s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2     2        2     2               3        2     2
│ │ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │               3         3               3
│ │ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ │ @@ -138,35 +138,35 @@
│ │ │ │  p      + p     p     , p     p      - p     p      + p     p     )
│ │ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4
│ │ │ │  0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3
│ │ │ │  1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
│ │ │ │  i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent
│ │ │ │  hyperplanes to S)
│ │ │ │       time tangentialChowForm(S,2)
│ │ │ │ - -- used 0.117742s (cpu); 0.0528496s (thread); 0s (gc)
│ │ │ │ + -- used 0.152329s (cpu); 0.0669285s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                             2
│ │ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │ │        0,1,3,4 0,2,3,4    0,1,2,4 0,2,3,4 1,2,3,4    0,1,2,3 1,2,3,4
│ │ │ │  
│ │ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
│ │ │ │  i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ │ - -- used 0.130131s (cpu); 0.0621371s (thread); 0s (gc)
│ │ │ │ + -- used 0.149774s (cpu); 0.0692121s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              2               2
│ │ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │ │              1 2    0 1 3    0 4
│ │ │ │  
│ │ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i7 : -- we then can recover S
│ │ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ │ - -- used 0.190994s (cpu); 0.117998s (thread); 0s (gc)
│ │ │ │ + -- used 0.234863s (cpu); 0.153028s (thread); 0s (gc)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_C_o_i_s_o_t_r_o_p_i_c -- whether a hypersurface of a Grassmannian is a tangential
│ │ │ │        Chow form
│ │ │ │      * _c_h_o_w_F_o_r_m -- Chow form of a projective variety
│ │ │ │  ********** WWaayyss ttoo uussee ttaannggeennttiiaallCChhoowwFFoorrmm:: **********
│ │ │ │      * tangentialChowForm(Ideal,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  aXNFeHRlcm5hbE0yUGFyZW50
│ │ │  #:len=1043
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW5kaWNhdGUgaWYgdGhpcyBwcm9jZXNz
│ │ │  IGlzIGEgcGFyZW50IHByb2Nlc3Mgb3Igbm90IiwgImxpbmVudW0iID0+IDg5NCwgImZpbGVuYW1l
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │  locked memory(kbytes) 8192
│ │ │ -process              63815
│ │ │ +process              63520
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 2927978066455787395
│ │ │  
│ │ │  i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-24375-0/0.m2
│ │ │ +o1 = /tmp/M2-31230-0/0.m2
│ │ │  
│ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-24375-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/1.m2" >"/tmp/M2-31230-0/1.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │                 "output file" => null
│ │ │ @@ -33,97 +33,97 @@
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : h#"exit code"===0
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/2.m2" >"/tmp/M2-31230-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │  i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31230-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-24375-0/2.out
│ │ │ +                "output file" => /tmp/M2-31230-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 5
│ │ │ +                "time used" => 3
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
│ │ │  
│ │ │  i12 : fileExists(h#"output file")
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : fileExists(h#"answer file")
│ │ │  
│ │ │  o13 = false
│ │ │  
│ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/3.m2" >"/tmp/M2-31230-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │  i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31230-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-24375-0/3.out
│ │ │ +                "output file" => /tmp/M2-31230-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │                  "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
│ │ │  
│ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file")
│ │ │  
│ │ │  o16 = 
│ │ │  
│ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file")
│ │ │  
│ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1') >"/tmp/M2-24375-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/4.m2" >"/tmp/M2-31230-0/4.out" 2>&1') >"/tmp/M2-31230-0/4.stat" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 7.49
│ │ │ -              System time (seconds): 0.10
│ │ │ -              Percent of CPU this job got: 93%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/4.m2" >"/tmp/M2-31230-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 7.53
│ │ │ +              System time (seconds): 0.28
│ │ │ +              Percent of CPU this job got: 118%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.57
│ │ │                Average shared text size (kbytes): 0
│ │ │                Average unshared data size (kbytes): 0
│ │ │                Average stack size (kbytes): 0
│ │ │                Average total size (kbytes): 0
│ │ │ -              Maximum resident set size (kbytes): 255852
│ │ │ +              Maximum resident set size (kbytes): 340328
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 10158
│ │ │ -              Voluntary context switches: 2973
│ │ │ -              Involuntary context switches: 2382
│ │ │ +              Minor (reclaiming a frame) page faults: 13361
│ │ │ +              Voluntary context switches: 14728
│ │ │ +              Involuntary context switches: 2759
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 24
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
│ │ │  
│ │ │  
│ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │  
│ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/6.m2" >"/tmp/M2-31230-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : R=QQ[x,y];
│ │ │  
│ │ │  i23 : v=coker random(R^2,R^{3:-1})
│ │ │ @@ -131,54 +131,54 @@
│ │ │  o23 = cokernel | 9/2x+9/4y 7/9x+7/10y 7x+3/7y |
│ │ │                 | 3/4x+7/4y 7/10x+7/3y 6/7x+6y |
│ │ │  
│ │ │                               2
│ │ │  o23 : R-module, quotient of R
│ │ │  
│ │ │  i24 : h=runExternalM2(fn,identity,v)
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/7.m2" >"/tmp/M2-31230-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31230-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-24375-0/7.out
│ │ │ +                "output file" => /tmp/M2-31230-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │                  "time used" => 4
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
│ │ │  
│ │ │  i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-31230-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-24375-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-31230-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-31230-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
│ │ │  
│ │ │  
│ │ │  i26 : fn<"/tmp/M2-24375-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/8.m2" >"/tmp/M2-31230-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
│ │ │  
│ │ │  i28 : v=R;
│ │ │  
│ │ │  i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/9.m2" >"/tmp/M2-31230-0/9.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  i30 : h#value
│ │ │  
│ │ │  o30 = QQ[x..y]
│ │ │  
│ │ │  o30 : PolynomialRing
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html
│ │ │ @@ -80,15 +80,15 @@
│ │ │  time(seconds)        700
│ │ │  file(blocks)         unlimited
│ │ │  data(kbytes)         unlimited
│ │ │  stack(kbytes)        8192
│ │ │  coredump(blocks)     unlimited
│ │ │  memory(kbytes)       850000
│ │ │  locked memory(kbytes) 8192
│ │ │ -process              63815
│ │ │ +process              63520
│ │ │  nofiles              512
│ │ │  vmemory(kbytes)      unlimited
│ │ │  locks                unlimited
│ │ │  rtprio               0
│ │ │  
│ │ │  o1 = 0
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  time(seconds)        700
│ │ │ │  file(blocks)         unlimited
│ │ │ │  data(kbytes)         unlimited
│ │ │ │  stack(kbytes)        8192
│ │ │ │  coredump(blocks)     unlimited
│ │ │ │  memory(kbytes)       850000
│ │ │ │  locked memory(kbytes) 8192
│ │ │ │ -process              63815
│ │ │ │ +process              63520
│ │ │ │  nofiles              512
│ │ │ │  vmemory(kbytes)      unlimited
│ │ │ │  locks                unlimited
│ │ │ │  rtprio               0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  This starts a new shell and executes the command given, which in this case
│ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │            
      For example, we can write a few functions to a temporary file:
      
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -120,15 +120,15 @@
│ │ │          
    
│ │ │            
    and then call them:
    
│ │ │          
    
│ │ │          
    
│ │ │                
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-24375-0/0.m2
    
│ │ │ +o1 = /tmp/M2-31230-0/0.m2
│ │ │              
    
│ │ │                
    i2 : fn<</// square = (x) -> (stderr<<"Running"<<endl; sleep(1); x^2); ///<<endl;
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : h=runExternalM2(fn,"square",(4));
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/1.m2" >"/tmp/M2-24375-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/1.m2" >"/tmp/M2-31230-0/1.out" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i7 : h
│ │ │  
│ │ │ @@ -162,29 +162,29 @@
│ │ │            
    
│ │ │            
    An abnormal program exit will have a nonzero exit code; also, the value will be null, the output file should exist, but the answer file may not exist unless the routine finished successfully.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -348,20 +348,20 @@
│ │ │          
│ │ │          
    
│ │ │                
    i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/2.m2" >"/tmp/M2-31230-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31230-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-24375-0/2.out
│ │ │ +                "output file" => /tmp/M2-31230-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 5
│ │ │ +                "time used" => 3
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    
│ │ │              
    
│ │ │ @@ -204,27 +204,27 @@
│ │ │          
    
│ │ │            
    Here, we use resource limits to limit the routine to 2 seconds of computational time, while the system is asked to use 10 seconds of computational time:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -246,40 +246,40 @@
│ │ │            
    
│ │ │            
    We can get quite a lot of detail on the resources used with the KeepStatistics command:
    
│ │ │          
│ │ │          
    
│ │ │                
    i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/3.m2" >"/tmp/M2-31230-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    
│ │ │              
    
│ │ │                
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31230-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-24375-0/3.out
│ │ │ +                "output file" => /tmp/M2-31230-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │                  "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -293,15 +293,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1') >"/tmp/M2-24375-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/4.m2" >"/tmp/M2-31230-0/4.out" 2>&1') >"/tmp/M2-31230-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
    
│ │ │                
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 7.49
│ │ │ -              System time (seconds): 0.10
│ │ │ -              Percent of CPU this job got: 93%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/4.m2" >"/tmp/M2-31230-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 7.53
│ │ │ +              System time (seconds): 0.28
│ │ │ +              Percent of CPU this job got: 118%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.57
│ │ │                Average shared text size (kbytes): 0
│ │ │                Average unshared data size (kbytes): 0
│ │ │                Average stack size (kbytes): 0
│ │ │                Average total size (kbytes): 0
│ │ │ -              Maximum resident set size (kbytes): 255852
│ │ │ +              Maximum resident set size (kbytes): 340328
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 10158
│ │ │ -              Voluntary context switches: 2973
│ │ │ -              Involuntary context switches: 2382
│ │ │ +              Minor (reclaiming a frame) page faults: 13361
│ │ │ +              Voluntary context switches: 14728
│ │ │ +              Involuntary context switches: 2759
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 24
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
    
│ │ │              
    
│ │ │                
    i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
    
│ │ │              
    
│ │ │                
    i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/6.m2" >"/tmp/M2-31230-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -323,21 +323,21 @@
│ │ │                               2
│ │ │  o23 : R-module, quotient of R
│ │ │              
    
│ │ │                
    i24 : h=runExternalM2(fn,identity,v)
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/7.m2" >"/tmp/M2-31230-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31230-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-24375-0/7.out
│ │ │ +                "output file" => /tmp/M2-31230-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │                  "time used" => 4
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-31230-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-24375-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-31230-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-31230-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -372,15 +372,15 @@
│ │ │              
    		
│ │ │                
    i26 : fn<<///R=QQ[x,y];///<<endl<<flush;
    
│ │ │              
    
│ │ │                
    i27 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/8.m2" >"/tmp/M2-24375-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/8.m2" >"/tmp/M2-31230-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -392,15 +392,15 @@
│ │ │              
│ │ │                
    i28 : v=R;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31230-0/9.m2" >"/tmp/M2-31230-0/9.out" 2>&1 ))
│ │ │  Finished running.
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : h#value
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,25 +45,25 @@
│ │ │ │  the output file (unless it was deleted), the name of the answer file (unless it
│ │ │ │  was deleted), any statistics recorded about the resource usage, and the value
│ │ │ │  returned by the function func. If the child process terminates abnormally, then
│ │ │ │  usually the exit code is nonzero and the value returned is _n_u_l_l.
│ │ │ │  For example, we can write a few functions to a temporary file:
│ │ │ │  i1 : fn=temporaryFileName()|".m2"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-24375-0/0.m2
│ │ │ │ +o1 = /tmp/M2-31230-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-24375-0/1.out" 2>&1 ))
│ │ │ │ +M2-31230-0/1.m2" >"/tmp/M2-31230-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │ @@ -79,47 +79,47 @@
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  An abnormal program exit will have a nonzero exit code; also, the value will be
│ │ │ │  null, the output file should exist, but the answer file may not exist unless
│ │ │ │  the routine finished successfully.
│ │ │ │  i10 : h=runExternalM2(fn,"justexit",());
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 ))
│ │ │ │ +M2-31230-0/2.m2" >"/tmp/M2-31230-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31230-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-24375-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-31230-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 5
│ │ │ │ +                "time used" => 3
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : fileExists(h#"answer file")
│ │ │ │  
│ │ │ │  o13 = false
│ │ │ │  Here, we use _r_e_s_o_u_r_c_e_ _l_i_m_i_t_s to limit the routine to 2 seconds of computational
│ │ │ │  time, while the system is asked to use 10 seconds of computational time:
│ │ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ │  Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -
│ │ │ │ -q  <"/tmp/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-31230-0/3.m2" >"/tmp/M2-31230-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31230-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-24375-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-31230-0/3.out
│ │ │ │                  "return code" => 9
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o15 : HashTable
│ │ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get
│ │ │ │ @@ -128,110 +128,110 @@
│ │ │ │  o16 =
│ │ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get
│ │ │ │  (h#"answer file")
│ │ │ │  We can get quite a lot of detail on the resources used with the _K_e_e_p_S_t_a_t_i_s_t_i_c_s
│ │ │ │  command:
│ │ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ │  Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop -
│ │ │ │ --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-24375-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-31230-0/4.m2" >"/tmp/M2-31230-0/4.out" 2>&1')
│ │ │ │ +>"/tmp/M2-31230-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 7.49
│ │ │ │ -              System time (seconds): 0.10
│ │ │ │ -              Percent of CPU this job got: 93%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15
│ │ │ │ +--silent  -q  <"/tmp/M2-31230-0/4.m2" >"/tmp/M2-31230-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 7.53
│ │ │ │ +              System time (seconds): 0.28
│ │ │ │ +              Percent of CPU this job got: 118%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.57
│ │ │ │                Average shared text size (kbytes): 0
│ │ │ │                Average unshared data size (kbytes): 0
│ │ │ │                Average stack size (kbytes): 0
│ │ │ │                Average total size (kbytes): 0
│ │ │ │ -              Maximum resident set size (kbytes): 255852
│ │ │ │ +              Maximum resident set size (kbytes): 340328
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 10158
│ │ │ │ -              Voluntary context switches: 2973
│ │ │ │ -              Involuntary context switches: 2382
│ │ │ │ +              Minor (reclaiming a frame) page faults: 13361
│ │ │ │ +              Voluntary context switches: 14728
│ │ │ │ +              Involuntary context switches: 2759
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 24
│ │ │ │                Socket messages sent: 0
│ │ │ │                Socket messages received: 0
│ │ │ │                Signals delivered: 0
│ │ │ │                Page size (bytes): 4096
│ │ │ │                Exit status: 0
│ │ │ │  We can handle most kinds of objects as return values, although _M_u_t_a_b_l_e_M_a_t_r_i_x
│ │ │ │  does not work. Here, we use the built-in _i_d_e_n_t_i_t_y function:
│ │ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 ))
│ │ │ │ +M2-31230-0/6.m2" >"/tmp/M2-31230-0/6.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  Some care is required, however:
│ │ │ │  i22 : R=QQ[x,y];
│ │ │ │  i23 : v=coker random(R^2,R^{3:-1})
│ │ │ │  
│ │ │ │  o23 = cokernel | 9/2x+9/4y 7/9x+7/10y 7x+3/7y |
│ │ │ │                 | 3/4x+7/4y 7/10x+7/3y 6/7x+6y |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o23 : R-module, quotient of R
│ │ │ │  i24 : h=runExternalM2(fn,identity,v)
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 ))
│ │ │ │ +M2-31230-0/7.m2" >"/tmp/M2-31230-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31230-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-24375-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-31230-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │                  "time used" => 4
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-31230-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-24375-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-31230-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.ans",identity (cokernel
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-31230-0/7.ans",identity (cokernel
│ │ │ │  (map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/
│ │ │ │  4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to
│ │ │ │  objects:
│ │ │ │                    R (of class Symbol)
│ │ │ │              ^     2 (of class ZZ)
│ │ │ │  Keep in mind that the object you are passing must make sense in the context of
│ │ │ │  the file containing your function! For instance, here we need to define the
│ │ │ │  ring:
│ │ │ │  i26 : fn<"/tmp/M2-24375-0/8.out" 2>&1 ))
│ │ │ │ +M2-31230-0/8.m2" >"/tmp/M2-31230-0/8.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o27 = true
│ │ │ │  This problem can be avoided by following some _s_u_g_g_e_s_t_i_o_n_s_ _f_o_r_ _u_s_i_n_g
│ │ │ │  _R_u_n_E_x_t_e_r_n_a_l_M_2.
│ │ │ │  The objects may unavoidably lose some internal references, though:
│ │ │ │  i28 : v=R;
│ │ │ │  i29 : h=runExternalM2(fn,identity,v);
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 ))
│ │ │ │ +M2-31230-0/9.m2" >"/tmp/M2-31230-0/9.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i30 : h#value
│ │ │ │  
│ │ │ │  o30 = QQ[x..y]
│ │ │ │  
│ │ │ │  o30 : PolynomialRing
│ │ │ │  i31 : v===h#value
│ │ ├── ./usr/share/doc/Macaulay2/SCMAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsTW9kdWxlKElkZWFsKQ==
│ │ │  #:len=266
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjYW5vbmljYWxNb2R1bGUsSWRlYWwpLCJjYW5vbmlj
│ │ ├── ./usr/share/doc/Macaulay2/SCSCP/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  UmVtb3RlT2JqZWN0IGFuZCBSZW1vdGVPYmplY3Q=
│ │ │  #:len=217
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMwLCAidW5kb2N1bWVudGVkIiA9PiB0
│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wg
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  dHJhbnNwb3NlKEdhdGVNYXRyaXgp
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjU1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0cmFuc3Bvc2UsR2F0ZU1hdHJpeCksInRyYW5zcG9z
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out
│ │ │ @@ -30,23 +30,23 @@
│ │ │                                              )
│ │ │  
│ │ │                            "variable positions" => {-1}
│ │ │  
│ │ │  o5 : InterpretedSLProgram
│ │ │  
│ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ + -- used 0.000333386s (cpu); 0.000324504s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ + -- used 5.40536s (cpu); 3.9351s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -109,29 +109,29 @@
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ + -- used 0.000333386s (cpu); 0.000324504s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : ZZ[y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ + -- used 5.40536s (cpu); 3.9351s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,21 +38,21 @@
│ │ │ │                                              output nodes: 1
│ │ │ │                                              )
│ │ │ │  
│ │ │ │                            "variable positions" => {-1}
│ │ │ │  
│ │ │ │  o5 : InterpretedSLProgram
│ │ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ │ - -- used 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ │ + -- used 0.000333386s (cpu); 0.000324504s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ │ + -- used 5.40536s (cpu); 3.9351s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c2xFcXVpdmFyaWFudFZlY3RvckJ1bmRsZQ==
│ │ │  #:len=4962
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBTTC1lcXVpdmFyaWFu
│ │ │  dCB2ZWN0b3IgYnVuZGxlIG92ZXIgc29tZSBwcm9qZWN0aXZlIHNwYWNlIiwgImxpbmVudW0iID0+
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  RXhhbXBsZSBmaXJzdCBvcmRlciBkZWZvcm1hdGlvbg==
│ │ │  #:len=3174
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXhhbXBsZSBhY2Nlc3NpbmcgdGhlIGRh
│ │ │  dGEgc3RvcmVkIGluIGEgZmlyc3Qgb3JkZXIgZGVmb3JtYXRpb24uIiwgImxpbmVudW0iID0+IDQy
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/example-output/___Co__Complex.out
│ │ │ @@ -56,15 +56,15 @@
│ │ │  
│ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i8 : cC=complement C
│ │ │  
│ │ │  o8 = 2: x x x  x x x  x x x  x x x  x x x  x x x  x x x  x x x  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i9 : dualize cC
│ │ │  
│ │ │  o9 = 2: v v v  v v v  v v v  v v v  v v v  v v v  v v v  v v v
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/html/___Co__Complex.html
│ │ │ @@ -163,15 +163,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : cC=complement C
│ │ │  
│ │ │  o8 = 2: x x x  x x x  x x x  x x x  x x x  x x x  x x x  x x x  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,15 +82,15 @@
│ │ │ │           0 1 2  0 1 4  0 3 4  1 2 3  1 2 5  1 4 5  2 3 4  3 4 5
│ │ │ │  
│ │ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │ │  i8 : cC=complement C
│ │ │ │  
│ │ │ │  o8 = 2: x x x  x x x  x x x  x x x  x x x  x x x  x x x  x x x
│ │ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │ │  
│ │ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │ │  i9 : dualize cC
│ │ │ │  
│ │ │ │  o9 = 2: v v v  v v v  v v v  v v v  v v v  v v v  v v v  v v v
│ │ │ │           3 4 5  2 3 5  1 2 5  0 4 5  0 3 4  0 2 3  0 1 5  0 1 2
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  cHJvamVjdFRvQ29tcGxleChDb21wbGV4LEhhc2hUYWJsZSk=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE4NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocHJvamVjdFRvQ29tcGxleCxDb21wbGV4LEhhc2hU
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : r={5,11,3,2}
│ │ │  
│ │ │  o3 = {5, 11, 3, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .0570737s elapsed
│ │ │ + -- .0374058s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : Complex
│ │ │ @@ -41,15 +41,15 @@
│ │ │         53        53         53         53        53
│ │ │                                                  
│ │ │       -1        0          1          2         3
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00246826s elapsed
│ │ │ + -- .00465262s elapsed
│ │ │  
│ │ │  i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │                 0 => 3
│ │ │                 1 => 5
│ │ │                 2 => 2
│ │ │ @@ -85,15 +85,15 @@
│ │ │  i12 : maximalEntry complex errors
│ │ │  
│ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00494602s elapsed
│ │ │ + -- .0100539s elapsed
│ │ │  
│ │ │  i14 : hL === h
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : SigmaL =source U;
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : r={4,3,3}
│ │ │  
│ │ │  o3 = {4, 3, 3}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0717096s elapsed
│ │ │ + -- .0265459s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : Complex
│ │ │ @@ -41,25 +41,25 @@
│ │ │         53        53         53         53
│ │ │                                        
│ │ │       0         1          2          3
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000541143s elapsed
│ │ │ + -- .000631871s elapsed
│ │ │  
│ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │                  3 => 2
│ │ │  
│ │ │  o7 : Sequence
│ │ │  
│ │ │  i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ - -- .00894283s elapsed
│ │ │ + -- .00148166s elapsed
│ │ │  
│ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │  
│ │ │  o8 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,5}
│ │ │  
│ │ │  o4 = {4, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .0727341s elapsed
│ │ │ + -- .029239s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0562647s elapsed
│ │ │ + -- .0324154s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i4 : r={4,3,3}
│ │ │  
│ │ │  o4 = {4, 3, 3}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00316014s elapsed
│ │ │ + -- .00358455s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .0570737s elapsed
│ │ │ + -- .0374058s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : Complex
    
│ │ │ @@ -145,15 +145,15 @@
│ │ │  
│ │ │  o6 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00246826s elapsed
    
│ │ │ + -- .00465262s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -207,15 +207,15 @@
│ │ │  
│ │ │  o12 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00494602s elapsed
    
│ │ │ + -- .0100539s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : hL === h
│ │ │  
│ │ │  o14 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │  o2 : List
│ │ │ │  i3 : r={5,11,3,2}
│ │ │ │  
│ │ │ │  o3 = {5, 11, 3, 2}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ │ - -- .0570737s elapsed
│ │ │ │ + -- .0374058s elapsed
│ │ │ │  
│ │ │ │         6       19       19       7       3
│ │ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3       4
│ │ │ │  
│ │ │ │  o4 : Complex
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  o6 = RR    <-- RR     <-- RR     <-- RR    <-- RR
│ │ │ │         53        53         53         53        53
│ │ │ │  
│ │ │ │       -1        0          1          2         3
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ │ - -- .00246826s elapsed
│ │ │ │ + -- .00465262s elapsed
│ │ │ │  i8 : h
│ │ │ │  
│ │ │ │  o8 = HashTable{-1 => 1}
│ │ │ │                 0 => 3
│ │ │ │                 1 => 5
│ │ │ │                 2 => 2
│ │ │ │                 3 => 1
│ │ │ │ @@ -99,15 +99,15 @@
│ │ │ │  1)*Sigma.dd_ell*transpose U_ell);
│ │ │ │  i12 : maximalEntry complex errors
│ │ │ │  
│ │ │ │  o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ │ - -- .00494602s elapsed
│ │ │ │ + -- .0100539s elapsed
│ │ │ │  i14 : hL === h
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : SigmaL =source U;
│ │ │ │  i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i)
│ │ │ │  
│ │ │ │  o16 = {1.77636e-14, 6.39488e-14, 8.52651e-14, 3.55271e-15}
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0717096s elapsed
│ │ │ + -- .0265459s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : Complex
    
│ │ │ @@ -147,28 +147,28 @@
│ │ │  
│ │ │  o6 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000541143s elapsed
│ │ │ + -- .000631871s elapsed
│ │ │  
│ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │                  3 => 2
│ │ │  
│ │ │  o7 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ - -- .00894283s elapsed
│ │ │ + -- .00148166s elapsed
│ │ │  
│ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o2 : List
│ │ │ │  i3 : r={4,3,3}
│ │ │ │  
│ │ │ │  o3 = {4, 3, 3}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ │ - -- .0717096s elapsed
│ │ │ │ + -- .0265459s elapsed
│ │ │ │  
│ │ │ │         5       10       11       5
│ │ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o4 : Complex
│ │ │ │ @@ -63,24 +63,24 @@
│ │ │ │  o6 = RR    <-- RR     <-- RR     <-- RR
│ │ │ │         53        53         53         53
│ │ │ │  
│ │ │ │       0         1          2          3
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ │ - -- .000541143s elapsed
│ │ │ │ + -- .000631871s elapsed
│ │ │ │  
│ │ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │ │                  3 => 2
│ │ │ │  
│ │ │ │  o7 : Sequence
│ │ │ │  i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ │ - -- .00894283s elapsed
│ │ │ │ + -- .00148166s elapsed
│ │ │ │  
│ │ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .0727341s elapsed
│ │ │ + -- .029239s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  o3 : List
│ │ │ │  i4 : r={4,3,5}
│ │ │ │  
│ │ │ │  o4 = {4, 3, 5}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ │ - -- .0727341s elapsed
│ │ │ │ + -- .029239s elapsed
│ │ │ │  
│ │ │ │         6       10       13       8
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0562647s elapsed
│ │ │ + -- .0324154s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o3 : List
│ │ │ │  i4 : r={4,3,3}
│ │ │ │  
│ │ │ │  o4 = {4, 3, 3}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ │ - -- .0562647s elapsed
│ │ │ │ + -- .0324154s elapsed
│ │ │ │  
│ │ │ │         6       10       11       5
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00316014s elapsed
│ │ │ + -- .00358455s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o3 : List
│ │ │ │  i4 : r={4,3,3}
│ │ │ │  
│ │ │ │  o4 = {4, 3, 3}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ │ - -- .00316014s elapsed
│ │ │ │ + -- .00358455s elapsed
│ │ │ │  
│ │ │ │         6       10       11       5
│ │ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │ │  
│ │ │ │       0       1        2        3
│ │ │ │  
│ │ │ │  o5 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SagbiGbDetection/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  d2VpZ2h0VmVjdG9yc1JlYWxpemluZ1NBR0JJ
│ │ │  #:len=1819
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIG1haW4gZnVuY3Rpb24gZm9yIGRl
│ │ │  dGVjdGluZyBTQUdCSSBiYXNlcyIsICJsaW5lbnVtIiA9PiAyMTUsIElucHV0cyA9PiB7U1BBTntU
│ │ ├── ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  c2F0dXJhdGUoSWRlYWwsTGlzdCk=
│ │ │  #:len=258
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzYXR1cmF0ZSxJZGVhbCxMaXN0KSwic2F0dXJhdGUo
│ │ ├── ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -37,33 +37,33 @@
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5));
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.412639s (cpu); 0.315292s (thread); 0s (gc)
│ │ │ + -- used 0.40623s (cpu); 0.40604s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc)
│ │ │ + -- used 0.880953s (cpu); 0.758358s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │  
│ │ │  i9 : S = ZZ/101[vars(0..4)];
│ │ │  
│ │ │  i10 : I = ideal vars S;
│ │ │  
│ │ │  o10 : Ideal of S
│ │ │  
│ │ │  i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.021082s (cpu); 0.021086s (thread); 0s (gc)
│ │ │ + -- used 0.0346668s (cpu); 0.0346674s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
│ │ │  
│ │ │  i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc)
│ │ │ + -- used 0.0132098s (cpu); 0.0132147s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -129,23 +129,23 @@
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.412639s (cpu); 0.315292s (thread); 0s (gc)
│ │ │ + -- used 0.40623s (cpu); 0.40604s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc)
│ │ │ + -- used 0.880953s (cpu); 0.758358s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Strategy => Quotient is faster in other cases:
    
│ │ │ @@ -162,23 +162,23 @@
│ │ │  
│ │ │  o10 : Ideal of S
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.021082s (cpu); 0.021086s (thread); 0s (gc)
│ │ │ + -- used 0.0346668s (cpu); 0.0346674s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc)
│ │ │ + -- used 0.0132098s (cpu); 0.0132147s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -52,32 +52,32 @@
│ │ │ │  i5 : I = monomialCurveIdeal(S, 1..n-1);
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5));
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ │ - -- used 0.412639s (cpu); 0.315292s (thread); 0s (gc)
│ │ │ │ + -- used 0.40623s (cpu); 0.40604s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc)
│ │ │ │ + -- used 0.880953s (cpu); 0.758358s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  Strategy => Quotient is faster in other cases:
│ │ │ │  i9 : S = ZZ/101[vars(0..4)];
│ │ │ │  i10 : I = ideal vars S;
│ │ │ │  
│ │ │ │  o10 : Ideal of S
│ │ │ │  i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ │ - -- used 0.021082s (cpu); 0.021086s (thread); 0s (gc)
│ │ │ │ + -- used 0.0346668s (cpu); 0.0346674s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 : Ideal of S
│ │ │ │  i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc)
│ │ │ │ + -- used 0.0132098s (cpu); 0.0132147s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  For further information see for example Exercise 15.41 in Eisenbud's
│ │ │ │  Commutative Algebra with a View Towards Algebraic Geometry.
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd SSttrraatteeggyy:: **********
│ │ │ │      * addHook(...,Strategy=>...) -- see _a_d_d_H_o_o_k -- add a hook function to an
│ │ ├── ./usr/share/doc/Macaulay2/Schubert2/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  aW5jbHVzaW9uKC4uLixTdWJEaW1lbnNpb249Pi4uLik=
│ │ │  #:len=260
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg1Nywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW5jbHVzaW9uLFN1YkRpbWVuc2lvbl0sImluY2x1
│ │ ├── ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out
│ │ │ @@ -40,23 +40,23 @@
│ │ │            )
│ │ │  
│ │ │  o6 = f
│ │ │  
│ │ │  o6 : FunctionClosure
│ │ │  
│ │ │  i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00475218s (cpu); 0.00474919s (thread); 0s (gc)
│ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc)
│ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc)
│ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc)
│ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc)
│ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc)
│ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc)
│ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc)
│ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc)
│ │ │ + -- used 0.00729239s (cpu); 0.00728955s (thread); 0s (gc)
│ │ │ + -- used 0.00973807s (cpu); 0.0097495s (thread); 0s (gc)
│ │ │ + -- used 0.0152385s (cpu); 0.0152519s (thread); 0s (gc)
│ │ │ + -- used 0.0274754s (cpu); 0.0274885s (thread); 0s (gc)
│ │ │ + -- used 0.0459443s (cpu); 0.0458903s (thread); 0s (gc)
│ │ │ + -- used 0.093228s (cpu); 0.0931512s (thread); 0s (gc)
│ │ │ + -- used 0.115207s (cpu); 0.115217s (thread); 0s (gc)
│ │ │ + -- used 0.178213s (cpu); 0.178225s (thread); 0s (gc)
│ │ │ + -- used 0.462698s (cpu); 0.319267s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o7 : List
│ │ ├── ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html
│ │ │ @@ -131,23 +131,23 @@
│ │ │  
│ │ │  o6 : FunctionClosure
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00475218s (cpu); 0.00474919s (thread); 0s (gc)
│ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc)
│ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc)
│ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc)
│ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc)
│ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc)
│ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc)
│ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc)
│ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc)
│ │ │ + -- used 0.00729239s (cpu); 0.00728955s (thread); 0s (gc)
│ │ │ + -- used 0.00973807s (cpu); 0.0097495s (thread); 0s (gc)
│ │ │ + -- used 0.0152385s (cpu); 0.0152519s (thread); 0s (gc)
│ │ │ + -- used 0.0274754s (cpu); 0.0274885s (thread); 0s (gc)
│ │ │ + -- used 0.0459443s (cpu); 0.0458903s (thread); 0s (gc)
│ │ │ + -- used 0.093228s (cpu); 0.0931512s (thread); 0s (gc)
│ │ │ + -- used 0.115207s (cpu); 0.115217s (thread); 0s (gc)
│ │ │ + -- used 0.178213s (cpu); 0.178225s (thread); 0s (gc)
│ │ │ + -- used 0.462698s (cpu); 0.319267s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,23 +56,23 @@
│ │ │ │            integral chern symmetricPower_(2*n-3) last bundles G
│ │ │ │            )
│ │ │ │  
│ │ │ │  o6 = f
│ │ │ │  
│ │ │ │  o6 : FunctionClosure
│ │ │ │  i7 : for n from 2 to 10 list time f n
│ │ │ │ - -- used 0.00475218s (cpu); 0.00474919s (thread); 0s (gc)
│ │ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc)
│ │ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc)
│ │ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc)
│ │ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc)
│ │ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc)
│ │ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc)
│ │ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc)
│ │ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc)
│ │ │ │ + -- used 0.00729239s (cpu); 0.00728955s (thread); 0s (gc)
│ │ │ │ + -- used 0.00973807s (cpu); 0.0097495s (thread); 0s (gc)
│ │ │ │ + -- used 0.0152385s (cpu); 0.0152519s (thread); 0s (gc)
│ │ │ │ + -- used 0.0274754s (cpu); 0.0274885s (thread); 0s (gc)
│ │ │ │ + -- used 0.0459443s (cpu); 0.0458903s (thread); 0s (gc)
│ │ │ │ + -- used 0.093228s (cpu); 0.0931512s (thread); 0s (gc)
│ │ │ │ + -- used 0.115207s (cpu); 0.115217s (thread); 0s (gc)
│ │ │ │ + -- used 0.178213s (cpu); 0.178225s (thread); 0s (gc)
│ │ │ │ + -- used 0.462698s (cpu); 0.319267s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  Note: in characteristic zero, using Bertini's theorem, the numbers computed can
│ │ ├── ./usr/share/doc/Macaulay2/SchurComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  c2NodXJDb21wbGV4
│ │ │  #:len=3519
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NodXIgZnVuY3RvcnMgb2YgY2hhaW4g
│ │ │  Y29tcGxleGVzIiwgImxpbmVudW0iID0+IDU5NywgSW5wdXRzID0+IHtTUEFOe1RUeyJsYW1iZGEi
│ │ ├── ./usr/share/doc/Macaulay2/SchurFunctors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  aXNXZXlsU3RhbmRhcmQoV2V5bEZpbGxpbmcp
│ │ │  #:len=281
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTcwMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNXZXlsU3RhbmRhcmQsV2V5bEZpbGxpbmcpLCJp
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  c2NodXJSZXNvbHV0aW9uKFJpbmdFbGVtZW50LExpc3Qp
│ │ │  #:len=286
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjU4Miwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2NodXJSZXNvbHV0aW9uLFJpbmdFbGVtZW50LExp
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/___Eor__H.out
│ │ │ @@ -33,13 +33,13 @@
│ │ │  i6 : toS fe == toS fh
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : R = symmetricRing(QQ,12);
│ │ │  
│ │ │  i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false);
│ │ │ - -- .00389276s elapsed
│ │ │ + -- .00570493s elapsed
│ │ │  
│ │ │  i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false);
│ │ │ - -- .000092754s elapsed
│ │ │ + -- .000245582s elapsed
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/___Memoize.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 9113092561535051477
│ │ │  
│ │ │  i1 : R = symmetricRing(QQ, 10);
│ │ │  
│ │ │  i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000475089s elapsed
│ │ │ + -- .000478703s elapsed
│ │ │  
│ │ │  i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000014267s elapsed
│ │ │ + -- .000017481s elapsed
│ │ │  
│ │ │  i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000441827s elapsed
│ │ │ + -- .000555639s elapsed
│ │ │  
│ │ │  i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000013796s elapsed
│ │ │ + -- .000019531s elapsed
│ │ │  
│ │ │  i6 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000013405s elapsed
│ │ │ + -- .00001579s elapsed
│ │ │  
│ │ │  i7 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000012954s elapsed
│ │ │ + -- .000014093s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi.out
│ │ │ @@ -50,18 +50,18 @@
│ │ │  i11 : toS fe == toS fh
│ │ │  
│ │ │  o11 = true
│ │ │  
│ │ │  i12 : R = symmetricRing(QQ,6);
│ │ │  
│ │ │  i13 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000476681s elapsed
│ │ │ + -- .000460969s elapsed
│ │ │  
│ │ │  i14 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000014307s elapsed
│ │ │ + -- .000020852s elapsed
│ │ │  
│ │ │  i15 : R = symmetricRing(QQ,5);
│ │ │  
│ │ │  i16 : S = schurRing R;
│ │ │  
│ │ │  i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1})
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.out
│ │ │ @@ -3,16 +3,16 @@
│ │ │  i1 : R = symmetricRing(QQ,6);
│ │ │  
│ │ │  i2 : jacobiTrudi({4,3,2,1},R,Memoize => true) == jacobiTrudi({4,3,2,1},R,Memoize => false)
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .000459589s elapsed
│ │ │ + -- .000535335s elapsed
│ │ │  
│ │ │  i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .00001618s elapsed
│ │ │ + -- .000022922s elapsed
│ │ │  
│ │ │  i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false);
│ │ │ - -- .000383086s elapsed
│ │ │ + -- .000496529s elapsed
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/___Eor__H.html
│ │ │ @@ -143,21 +143,21 @@
│ │ │              
│ │ │                
    i7 : R = symmetricRing(QQ,12);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false);
│ │ │ - -- .00389276s elapsed
    
│ │ │ + -- .00570493s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false);
│ │ │ - -- .000092754s elapsed
    
│ │ │ + -- .000245582s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │        
    
│ │ │        
    
│ │ │          
    
│ │ │            
    Functions with optional argument named EorH:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,17 +46,17 @@
│ │ │ │  o6 = true
│ │ │ │  When the conjugate partition is much longer than lambda itself, the "H"-branch
│ │ │ │  requires a smaller determinant and runs measurably faster. For example on
│ │ │ │  lambda = (10) the conjugate is $(1^{10})$, so "H" only sets up a 1x1
│ │ │ │  determinant:
│ │ │ │  i7 : R = symmetricRing(QQ,12);
│ │ │ │  i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false);
│ │ │ │ - -- .00389276s elapsed
│ │ │ │ + -- .00570493s elapsed
│ │ │ │  i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false);
│ │ │ │ - -- .000092754s elapsed
│ │ │ │ + -- .000245582s elapsed
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd EEoorrHH:: **********
│ │ │ │      * jacobiTrudi(...,EorH=>...)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _E_o_r_H is a _s_y_m_b_o_l.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/SchurRings.m2:6153:0.
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/___Memoize.html
│ │ │ @@ -66,59 +66,59 @@
│ │ │              
│ │ │                
    i1 : R = symmetricRing(QQ, 10);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000475089s elapsed
    
│ │ │ + -- .000478703s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000014267s elapsed
    
│ │ │ + -- .000017481s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    
│ │ │            
    The cache is attached to the ring R.  After one partition is memoized, subsequent calls with a different partition perform the full Jacobi-Trudi determinant expansion, then cache it as well:
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000441827s elapsed
    
│ │ │ + -- .000555639s elapsed
│ │ │              
    
│ │ │                
    i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000013796s elapsed
    
│ │ │ + -- .000019531s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    
│ │ │            
    Without Memoize => true, each call recomputes the determinant from scratch; for large partitions this can be substantially more expensive than a single cached lookup.
    
│ │ │            
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i6 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000013405s elapsed
    
│ │ │ + -- .00001579s elapsed
│ │ │              
    
│ │ │                
    i7 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000012954s elapsed
    
│ │ │ + -- .000014093s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,31 +7,31 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This is an optional argument for the _j_a_c_o_b_i_T_r_u_d_i function, allowing one to
│ │ │ │  store its values in order to speed up computations. When Memoize => true, every
│ │ │ │  computed value is cached in a hash table on the symmetric ring, so repeated
│ │ │ │  calls on the same partition return the cached value in constant time.
│ │ │ │  i1 : R = symmetricRing(QQ, 10);
│ │ │ │  i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ │ - -- .000475089s elapsed
│ │ │ │ + -- .000478703s elapsed
│ │ │ │  i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ │ - -- .000014267s elapsed
│ │ │ │ + -- .000017481s elapsed
│ │ │ │  The cache is attached to the ring R. After one partition is memoized,
│ │ │ │  subsequent calls with a different partition perform the full Jacobi-Trudi
│ │ │ │  determinant expansion, then cache it as well:
│ │ │ │  i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ │ - -- .000441827s elapsed
│ │ │ │ + -- .000555639s elapsed
│ │ │ │  i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ │ - -- .000013796s elapsed
│ │ │ │ + -- .000019531s elapsed
│ │ │ │  Without Memoize => true, each call recomputes the determinant from scratch; for
│ │ │ │  large partitions this can be substantially more expensive than a single cached
│ │ │ │  lookup.
│ │ │ │  i6 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ │ - -- .000013405s elapsed
│ │ │ │ + -- .00001579s elapsed
│ │ │ │  i7 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ │ - -- .000012954s elapsed
│ │ │ │ + -- .000014093s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _j_a_c_o_b_i_T_r_u_d_i -- Jacobi-Trudi determinant
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd MMeemmooiizzee:: **********
│ │ │ │      * _j_a_c_o_b_i_T_r_u_d_i_(_._._._,_M_e_m_o_i_z_e_=_>_._._._) -- Store values of the jacobiTrudi
│ │ │ │        function.
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _M_e_m_o_i_z_e is a _s_y_m_b_o_l.
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi.html
│ │ │ @@ -186,21 +186,21 @@
│ │ │              
│ │ │                
      i12 : R = symmetricRing(QQ,6);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i13 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000476681s elapsed
      
│ │ │ + -- .000460969s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i14 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000014307s elapsed
      
│ │ │ + -- .000020852s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      
│ │ │            
      Passing a partition through toSymm applied to the corresponding Schur label reproduces the Jacobi-Trudi output:
      
│ │ │            
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -67,17 +67,17 @@
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  The routine caches intermediate subdeterminants on the ring via _j_a_c_o_b_i_T_r_u_d_i
│ │ │ │  _(_._._._,_M_e_m_o_i_z_e_=_>_._._._), so a second call on a large partition returns almost
│ │ │ │  instantly:
│ │ │ │  i12 : R = symmetricRing(QQ,6);
│ │ │ │  i13 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ │ - -- .000476681s elapsed
│ │ │ │ + -- .000460969s elapsed
│ │ │ │  i14 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ │ - -- .000014307s elapsed
│ │ │ │ + -- .000020852s elapsed
│ │ │ │  Passing a partition through _t_o_S_y_m_m applied to the corresponding Schur label
│ │ │ │  reproduces the Jacobi-Trudi output:
│ │ │ │  i15 : R = symmetricRing(QQ,5);
│ │ │ │  i16 : S = schurRing R;
│ │ │ │  i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1})
│ │ │ │  
│ │ │ │  o17 = true
│ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.html
│ │ │ @@ -85,27 +85,27 @@
│ │ │  
│ │ │  o2 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .000459589s elapsed
      
│ │ │ + -- .000535335s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .00001618s elapsed
      
│ │ │ + -- .000022922s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false);
│ │ │ - -- .000383086s elapsed
      
│ │ │ + -- .000496529s elapsed
│ │ │              
│ │ │            
│ │ │          
│ │ │        
      
│ │ │        
      
│ │ │          
      
│ │ │            
      Functions with optional argument named Memoize:
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,19 +17,19 @@
│ │ │ │  extra memory in the ring.
│ │ │ │  i1 : R = symmetricRing(QQ,6);
│ │ │ │  i2 : jacobiTrudi({4,3,2,1},R,Memoize => true) == jacobiTrudi(
│ │ │ │  {4,3,2,1},R,Memoize => false)
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ │ - -- .000459589s elapsed
│ │ │ │ + -- .000535335s elapsed
│ │ │ │  i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ │ - -- .00001618s elapsed
│ │ │ │ + -- .000022922s elapsed
│ │ │ │  i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false);
│ │ │ │ - -- .000383086s elapsed
│ │ │ │ + -- .000496529s elapsed
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd MMeemmooiizzee:: **********
│ │ │ │      * _j_a_c_o_b_i_T_r_u_d_i_(_._._._,_M_e_m_o_i_z_e_=_>_._._._) -- Store values of the jacobiTrudi
│ │ │ │        function.
│ │ │ │  ********** FFuurrtthheerr iinnffoorrmmaattiioonn **********
│ │ │ │      * Default value: _t_r_u_e
│ │ │ │      * Function: _j_a_c_o_b_i_T_r_u_d_i -- Jacobi-Trudi determinant
│ │ │ │      * Option key: _M_e_m_o_i_z_e -- Option to record values of the jacobiTrudi
│ │ ├── ./usr/share/doc/Macaulay2/SchurVeronese/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  bWFrZUJldHRpVGFsbHk=
│ │ │  #:len=1334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydHMgYSBoYXNoIHRhYmxlIHJl
│ │ │  cHJlc2VudGluZyBhIEJldHRpIHRhYmxlIHRvIGEgQmV0dGkgdGFsbHkiLCAibGluZW51bSIgPT4g
│ │ ├── ./usr/share/doc/Macaulay2/SectionRing/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  bVJlZ3VsYXI=
│ │ │  #:len=1136
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibVJlZ3VsYXIoRixHKSBjb21wdXRlcyB0
│ │ │  aGUgcmVndWxhcml0eSBvZiBGIHdpdGggcmVzcGVjdCB0byBHIChnbG9iYWxseSBnZW5lcmF0ZWQp
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  bWFrZVByb2R1Y3RSaW5nKFJpbmcsTGlzdCk=
│ │ │  #:len=277
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUHJvZHVjdFJpbmcsUmluZyxMaXN0KSwibWFr
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out
│ │ │ @@ -53,15 +53,15 @@
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : X=((W)*ideal(y)+ideal(f));
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.54047s (cpu); 3.61426s (thread); 0s (gc)
│ │ │ + -- used 7.92871s (cpu); 4.33338s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
│ │ │  
│ │ │  i12 : print "we could confirm this with the computation:"
│ │ │  we could confirm this with the computation:
│ │ │  
│ │ │  i13 : B=ideal(x)*ideal(y)*ideal(z)
│ │ │ @@ -71,12 +71,12 @@
│ │ │        b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g,
│ │ │        -----------------------------------------------------------------------
│ │ │        c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 54.8103s (cpu); 50.9154s (thread); 0s (gc)
│ │ │ + -- used 80.6332s (cpu); 74.9558s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 :
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out
│ │ │ @@ -23,24 +23,24 @@
│ │ │  i5 : A = makeChowRing(R)
│ │ │  
│ │ │  o5 = A
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.261889s (cpu); 0.158635s (thread); 0s (gc)
│ │ │ + -- used 2.10497s (cpu); 0.465057s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
│ │ │  
│ │ │  i7 : time segre(X,Y,A)
│ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc)
│ │ │ + -- used 0.224959s (cpu); 0.158837s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2     2             2
│ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │  
│ │ │  o7 : A
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html
│ │ │ @@ -167,15 +167,15 @@
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.54047s (cpu); 3.61426s (thread); 0s (gc)
│ │ │ + -- used 7.92871s (cpu); 4.33338s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i12 : print "we could confirm this with the computation:"
│ │ │ @@ -194,15 +194,15 @@
│ │ │  
│ │ │  o13 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 54.8103s (cpu); 50.9154s (thread); 0s (gc)
│ │ │ + -- used 80.6332s (cpu); 74.9558s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,30 +68,30 @@
│ │ │ │  i9 : Y=ideal (z_0*W_0-z_1*W_1)+ideal(f);
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : X=((W)*ideal(y)+ideal(f));
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : time isComponentContained(X,Y)
│ │ │ │ - -- used 4.54047s (cpu); 3.61426s (thread); 0s (gc)
│ │ │ │ + -- used 7.92871s (cpu); 4.33338s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  i12 : print "we could confirm this with the computation:"
│ │ │ │  we could confirm this with the computation:
│ │ │ │  i13 : B=ideal(x)*ideal(y)*ideal(z)
│ │ │ │  
│ │ │ │  o13 = ideal (a*d*g, a*d*h, a*d*i, a*e*g, a*e*h, a*e*i, a*f*g, a*f*h, a*f*i,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ │ - -- used 54.8103s (cpu); 50.9154s (thread); 0s (gc)
│ │ │ │ + -- used 80.6332s (cpu); 74.9558s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  ********** WWaayyss ttoo uussee iissCCoommppoonneennttCCoonnttaaiinneedd:: **********
│ │ │ │      * isComponentContained(Ideal,Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _i_s_C_o_m_p_o_n_e_n_t_C_o_n_t_a_i_n_e_d is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html
│ │ │ @@ -123,27 +123,27 @@
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.261889s (cpu); 0.158635s (thread); 0s (gc)
│ │ │ + -- used 2.10497s (cpu); 0.465057s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i7 : time segre(X,Y,A)
│ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc)
│ │ │ + -- used 0.224959s (cpu); 0.158837s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2     2             2
│ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │  
│ │ │  o7 : A
      
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,23 +48,23 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : A = makeChowRing(R)
│ │ │ │  
│ │ │ │  o5 = A
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : time s = segreDimX(X,Y,A)
│ │ │ │ - -- used 0.261889s (cpu); 0.158635s (thread); 0s (gc)
│ │ │ │ + -- used 2.10497s (cpu); 0.465057s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2             2
│ │ │ │  o6 = 2H  + 4H H  + 2H
│ │ │ │         1     1 2     2
│ │ │ │  
│ │ │ │  o6 : A
│ │ │ │  i7 : time segre(X,Y,A)
│ │ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc)
│ │ │ │ + -- used 0.224959s (cpu); 0.158837s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2     2         2     2             2
│ │ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │ │  
│ │ │ │  o7 : A
│ │ │ │  ********** WWaayyss ttoo uussee sseeggrreeDDiimmXX:: **********
│ │ ├── ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  b3B0aW1pemUoLi4uLFZlcmJvc2l0eT0+Li4uKQ==
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tvcHRpbWl6ZSxWZXJib3NpdHldLCJvcHRpbWl6ZSgu
│ │ ├── ./usr/share/doc/Macaulay2/Seminormalization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  YmV0dGVyTm9ybWFsaXphdGlvbk1hcA==
│ │ │  #:len=1762
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibm9ybWFsaXplcyBub24gZG9tYWlucyIs
│ │ │  ICJsaW5lbnVtIiA9PiA4ODEsIElucHV0cyA9PiB7U1BBTntUVHsiUyJ9LCIsICIsU1BBTnsiYSAi
│ │ ├── ./usr/share/doc/Macaulay2/Serialization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  U2VyaWFsaXphdGlvbg==
│ │ │  #:len=554
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV2ZXJzaWJsZSBjb252ZXJzaW9uIG9m
│ │ │  IGFsbCBNYWNhdWxheTIgb2JqZWN0cyB0byBzdHJpbmdzIiwgRGVzY3JpcHRpb24gPT4gMTooRElW
│ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  YXJYaXYoU3RyaW5nLFN0cmluZyk=
│ │ │  #:len=236
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhclhpdixTdHJpbmcsU3RyaW5nKSwiYXJYaXYoU3Ry
│ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331702921222
│ │ │  
│ │ │  i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .36758s elapsed
│ │ │ - -- capturing check(1, "SimpleDoc")           -- .401467s elapsed
│ │ │ + -- capturing check(0, "SimpleDoc")           -- .30005s elapsed
│ │ │ + -- capturing check(1, "SimpleDoc")           -- .243981s elapsed
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html
│ │ │ @@ -79,16 +79,16 @@
│ │ │          
      
│ │ │            
      The check method executes all package tests defined this way.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .36758s elapsed
│ │ │ - -- capturing check(1, "SimpleDoc")           -- .401467s elapsed
      
│ │ │ + -- capturing check(0, "SimpleDoc")           -- .30005s elapsed
│ │ │ + -- capturing check(1, "SimpleDoc")           -- .243981s elapsed
│ │ │              
      
│ │ │        
      
│ │ │        
      
│ │ │          
      See also
      
│ │ │          
        
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,16 +10,16 @@
│ │ │ │  The variable testExample is a _S_t_r_i_n_g containing an example of the use of _T_E_S_T
│ │ │ │  to write a test case.
│ │ │ │  TEST /// -* test for simpleDocFrob *-
│ │ │ │    assert(simpleDocFrob(2,matrix{{1,2}}) == matrix{{1,2,0,0},{0,0,1,2}})
│ │ │ │  ///
│ │ │ │  The _c_h_e_c_k method executes all package tests defined this way.
│ │ │ │  i1 : check SimpleDoc
│ │ │ │ - -- capturing check(0, "SimpleDoc")           -- .36758s elapsed
│ │ │ │ - -- capturing check(1, "SimpleDoc")           -- .401467s elapsed
│ │ │ │ + -- capturing check(0, "SimpleDoc")           -- .30005s elapsed
│ │ │ │ + -- capturing check(1, "SimpleDoc")           -- .243981s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_E_S_T -- add a test for a package
│ │ │ │      * _c_h_e_c_k -- perform tests of a package
│ │ │ │      * _p_a_c_k_a_g_e_T_e_m_p_l_a_t_e -- a template for a package
│ │ │ │      * _d_o_c_E_x_a_m_p_l_e -- an example of a documentation string
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _t_e_s_t_E_x_a_m_p_l_e is a _s_t_r_i_n_g.
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  ZWxlbWVudGFyeUNvbGxhcHNl
│ │ │  #:len=365
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU1Miwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiZWxlbWVudGFyeUNvbGxhcHNlIiwiZWxlbWVudGFy
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialDecomposability/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  c2hlbGxpbmdPcmRlciguLi4sUmFuZG9tPT4uLi4p
│ │ │  #:len=311
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODkzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzaGVsbGluZ09yZGVyLFJhbmRvbV0sInNoZWxsaW5n
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=44
│ │ │  bm9ybWFsaXplKFNpbXBsaWNpYWxNb2R1bGUsWlosQ2hlY2tTdW09Pi4uLik=
│ │ │  #:len=375
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg0Nywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbKG5vcm1hbGl6ZSxTaW1wbGljaWFsTW9kdWxlLFpa
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/___Simplicial__Modules.out
│ │ │ @@ -111,25 +111,25 @@
│ │ │  o7 = R  <-- R  <-- R   <-- R  <-- ...
│ │ │                              
│ │ │       0      1      2       3
│ │ │  
│ │ │  o7 : SimplicialModule
│ │ │  
│ │ │  i8 : elapsedTime simplicialModule(K,6) --specify top degree 6
│ │ │ - -- .0671636s elapsed
│ │ │ + -- .0881016s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o8 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o8 : SimplicialModule
│ │ │  
│ │ │  i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true)
│ │ │ - -- .195242s elapsed
│ │ │ + -- .212057s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o9 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o9 : SimplicialModule
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_ext__Power.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │  o2 = Q  <-- Q  <-- Q
│ │ │                      
│ │ │       0      1      2
│ │ │  
│ │ │  o2 : Complex
│ │ │  
│ │ │  i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ - -- 7.53161s elapsed
│ │ │ + -- 6.51779s elapsed
│ │ │  
│ │ │        1      18      63      91      60      15
│ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │                                               
│ │ │       1      2       3       4       5       6
│ │ │  
│ │ │  o3 : Complex
│ │ │ @@ -93,15 +93,15 @@
│ │ │            1                   2
│ │ │       2 : Q  <--------------- Q  : 2
│ │ │                 {2} | 0 b |
│ │ │  
│ │ │  o7 : ComplexMap
│ │ │  
│ │ │  i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ - -- .604017s elapsed
│ │ │ + -- .547073s elapsed
│ │ │  
│ │ │            3                             6
│ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │                 {1} | a b 0 0 0  0  |
│ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_exterior__Inclusion.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │  o2 = Q  <-- Q  <-- Q  <-- Q
│ │ │                             
│ │ │       0      1      2      3
│ │ │  
│ │ │  o2 : Complex
│ │ │  
│ │ │  i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .210615s elapsed
│ │ │ + -- .246475s elapsed
│ │ │  
│ │ │  i4 : isWellDefined phi
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : isCommutative phi
│ │ │  
│ │ │ @@ -160,15 +160,15 @@
│ │ │  o10 = Q  <-- Q  <-- Q  <-- Q
│ │ │                              
│ │ │        0      1      2      3
│ │ │  
│ │ │  o10 : Complex
│ │ │  
│ │ │  i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .21521s elapsed
│ │ │ + -- .265663s elapsed
│ │ │  
│ │ │  i12 : isWellDefined phi
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : isCommutative phi
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_forget__Degeneracy_lp__Simplicial__Module_rp.out
│ │ │ @@ -21,15 +21,15 @@
│ │ │  o3 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q    <-- Q   <-- ...
│ │ │                                                       
│ │ │       0      1      2       3       4       5        6
│ │ │  
│ │ │  o3 : SimplicialModule
│ │ │  
│ │ │  i4 : elapsedTime S**S
│ │ │ - -- .487245s elapsed
│ │ │ + -- .479054s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o4 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o4 : SimplicialModule
│ │ │ @@ -40,15 +40,15 @@
│ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q    <-- Q   <-- ...
│ │ │                                                       
│ │ │       0      1      2       3       4       5        6
│ │ │  
│ │ │  o5 : SimplicialModule
│ │ │  
│ │ │  i6 : elapsedTime fS**fS --faster when degeneracy is ignored
│ │ │ - -- .463629s elapsed
│ │ │ + -- .372732s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o6 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o6 : SimplicialModule
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_normalize_lp__Simplicial__Module_cm__Z__Z_rp.out
│ │ │ @@ -114,35 +114,35 @@
│ │ │  o10 = R   <-- R   <-- R    <-- R    <-- R    <-- R    <-- R    <-- R     <-- R     <-- R     <-- R    <-- ...
│ │ │                                                                                                    
│ │ │        0       1       2        3        4        5        6        7         8         9         10
│ │ │  
│ │ │  o10 : SimplicialModule
│ │ │  
│ │ │  i11 : elapsedTime prune normalize S10
│ │ │ - -- 4.94576s elapsed
│ │ │ + -- 4.62985s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o11 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o11 : Complex
│ │ │  
│ │ │  i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times slower; becomes significant for larger ranks
│ │ │ - -- 8.29592s elapsed
│ │ │ + -- 7.60824s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o12 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o12 : Complex
│ │ │  
│ │ │  i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER!
│ │ │ - -- .0571616s elapsed
│ │ │ + -- .0409263s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o13 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o13 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_schur__Map.out
│ │ │ @@ -14,25 +14,25 @@
│ │ │  o3 = Q  <-- Q  <-- Q  <-- Q   <-- Q  <-- ...
│ │ │                                     
│ │ │       0      1      2      3       4
│ │ │  
│ │ │  o3 : SimplicialModule
│ │ │  
│ │ │  i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ - -- .142323s elapsed
│ │ │ + -- .159586s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o4 : SimplicialModule
│ │ │  
│ │ │  i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ - -- .309043s elapsed
│ │ │ + -- .322986s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o5 : SimplicialModule
│ │ │ @@ -78,15 +78,15 @@
│ │ │  o10 = Q
│ │ │  
│ │ │  o10 : PolynomialRing
│ │ │  
│ │ │  i11 : K = koszulComplex vars Q;
│ │ │  
│ │ │  i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ - -- 1.00439s elapsed
│ │ │ + -- .996059s elapsed
│ │ │  
│ │ │         1      9      36      74      81
│ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o12 : Complex
│ │ │ @@ -273,15 +273,15 @@
│ │ │             1                   2
│ │ │        2 : Q  <--------------- Q  : 2
│ │ │                  {2} | 0 b |
│ │ │  
│ │ │  o30 : ComplexMap
│ │ │  
│ │ │  i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ - -- .912527s elapsed
│ │ │ + -- 1.10973s elapsed
│ │ │  
│ │ │             1             1
│ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │                  | 1 |
│ │ │  
│ │ │             5                                       9
│ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_tensor__L__E__S.out
│ │ │ @@ -62,15 +62,15 @@
│ │ │  o7 = Q  <-- Q  <-- Q  <-- Q
│ │ │                             
│ │ │       0      1      2      3
│ │ │  
│ │ │  o7 : Complex
│ │ │  
│ │ │  i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ - -- 2.28436s elapsed
│ │ │ + -- 2.23931s elapsed
│ │ │  
│ │ │                                                                                                                                                                                                                                                                                                     25      50      25
│ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b | <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0 | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-- cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ │                                                                      | 0 b a |              | 0 b a |              | 0 0 b a |              | 0 0 b a 0 0 0 0 |              | 0   0   0 0 b a 0  0  |                                                                                                              
│ │ │       0                    1                    2     3                                                                                     | 0 0 0 0 b a 0 0 |              | ab2 a2b 0 0 0 0 b2 a2 |     9                              10                        11                             12      13      14
│ │ │                                                             4                      5                      6                                 | 0 0 0 0 0 0 b a |      
│ │ │                                                                                                                                                                     8
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_ext__Power.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o2 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ - -- 7.53161s elapsed
│ │ │ + -- 6.51779s elapsed
│ │ │  
│ │ │        1      18      63      91      60      15
│ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │                                               
│ │ │       1      2       3       4       5       6
│ │ │  
│ │ │  o3 : Complex
        
│ │ │ @@ -203,15 +203,15 @@
│ │ │  
│ │ │  o7 : ComplexMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ - -- .604017s elapsed
│ │ │ + -- .547073s elapsed
│ │ │  
│ │ │            3                             6
│ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │                 {1} | a b 0 0 0  0  |
│ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │        1      2      1
│ │ │ │  o2 = Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2
│ │ │ │  
│ │ │ │  o2 : Complex
│ │ │ │  i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ │ - -- 7.53161s elapsed
│ │ │ │ + -- 6.51779s elapsed
│ │ │ │  
│ │ │ │        1      18      63      91      60      15
│ │ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │ │  
│ │ │ │       1      2       3       4       5       6
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │ @@ -115,15 +115,15 @@
│ │ │ │  
│ │ │ │            1                   2
│ │ │ │       2 : Q  <--------------- Q  : 2
│ │ │ │                 {2} | 0 b |
│ │ │ │  
│ │ │ │  o7 : ComplexMap
│ │ │ │  i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ │ - -- .604017s elapsed
│ │ │ │ + -- .547073s elapsed
│ │ │ │  
│ │ │ │            3                             6
│ │ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │ │                 {1} | a b 0 0 0  0  |
│ │ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_exterior__Inclusion.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 : Complex
        
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .210615s elapsed
        
│ │ │ + -- .246475s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i4 : isWellDefined phi
│ │ │  
│ │ │  o4 = true
        
│ │ │ @@ -278,15 +278,15 @@
│ │ │  
│ │ │  o10 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .21521s elapsed
        
│ │ │ + -- .265663s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i12 : isWellDefined phi
│ │ │  
│ │ │  o12 = true
        
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │        1      3      3      1
│ │ │ │  o2 = Q  <-- Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2      3
│ │ │ │  
│ │ │ │  o2 : Complex
│ │ │ │  i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ │ - -- .210615s elapsed
│ │ │ │ + -- .246475s elapsed
│ │ │ │  i4 : isWellDefined phi
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : isCommutative phi
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : prune coker phi
│ │ │ │ @@ -176,15 +176,15 @@
│ │ │ │         1      3      3      1
│ │ │ │  o10 = Q  <-- Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │        0      1      2      3
│ │ │ │  
│ │ │ │  o10 : Complex
│ │ │ │  i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ │ - -- .21521s elapsed
│ │ │ │ + -- .265663s elapsed
│ │ │ │  i12 : isWellDefined phi
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : isCommutative phi
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : for i to 2 list prune HH_i source phi
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Degeneracy_lp__Simplicial__Module_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : SimplicialModule
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i4 : elapsedTime S**S
│ │ │ - -- .487245s elapsed
│ │ │ + -- .479054s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o4 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o4 : SimplicialModule
        
│ │ │ @@ -134,15 +134,15 @@
│ │ │  
│ │ │  o5 : SimplicialModule
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i6 : elapsedTime fS**fS --faster when degeneracy is ignored
│ │ │ - -- .463629s elapsed
│ │ │ + -- .372732s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o6 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o6 : SimplicialModule
        
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │        1      5      15      35      70      126      210
│ │ │ │  o3 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q    <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4       5        6
│ │ │ │  
│ │ │ │  o3 : SimplicialModule
│ │ │ │  i4 : elapsedTime S**S
│ │ │ │ - -- .487245s elapsed
│ │ │ │ + -- .479054s elapsed
│ │ │ │  
│ │ │ │        1      25      225      1225      4900      15876      44100
│ │ │ │  o4 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │ │  
│ │ │ │       0      1       2        3         4         5          6
│ │ │ │  
│ │ │ │  o4 : SimplicialModule
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │        1      5      15      35      70      126      210
│ │ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- Q    <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4       5        6
│ │ │ │  
│ │ │ │  o5 : SimplicialModule
│ │ │ │  i6 : elapsedTime fS**fS --faster when degeneracy is ignored
│ │ │ │ - -- .463629s elapsed
│ │ │ │ + -- .372732s elapsed
│ │ │ │  
│ │ │ │        1      25      225      1225      4900      15876      44100
│ │ │ │  o6 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │ │  
│ │ │ │       0      1       2        3         4         5          6
│ │ │ │  
│ │ │ │  o6 : SimplicialModule
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_normalize_lp__Simplicial__Module_cm__Z__Z_rp.html
│ │ │ @@ -237,28 +237,28 @@
│ │ │  
│ │ │  o10 : SimplicialModule
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i11 : elapsedTime prune normalize S10
│ │ │ - -- 4.94576s elapsed
│ │ │ + -- 4.62985s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o11 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o11 : Complex
        
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times slower; becomes significant for larger ranks
│ │ │ - -- 8.29592s elapsed
│ │ │ + -- 7.60824s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o12 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o12 : Complex
        
│ │ │ @@ -268,15 +268,15 @@
│ │ │          
        
│ │ │            
        The user may also specify the top homological degree to compute the normalization up to. Note that this can help speed up computational time; if the user knows the normalization should have a shorter length, then they should specify this upper bound in the syntax:
        
│ │ │          
        
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
        
│ │ │                
        i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER!
│ │ │ - -- .0571616s elapsed
│ │ │ + -- .0409263s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o13 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o13 : Complex
        
│ │ │ ├── html2text {}
│ │ │ │ @@ -156,38 +156,38 @@
│ │ │ │  
│ │ │ │  
│ │ │ │        0       1       2        3        4        5        6        7         8
│ │ │ │  9         10
│ │ │ │  
│ │ │ │  o10 : SimplicialModule
│ │ │ │  i11 : elapsedTime prune normalize S10
│ │ │ │ - -- 4.94576s elapsed
│ │ │ │ + -- 4.62985s elapsed
│ │ │ │  
│ │ │ │         10      30      30      10
│ │ │ │  o11 = R   <-- R   <-- R   <-- R
│ │ │ │  
│ │ │ │        0       1       2       3
│ │ │ │  
│ │ │ │  o11 : Complex
│ │ │ │  i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times
│ │ │ │  slower; becomes significant for larger ranks
│ │ │ │ - -- 8.29592s elapsed
│ │ │ │ + -- 7.60824s elapsed
│ │ │ │  
│ │ │ │         10      30      30      10
│ │ │ │  o12 = R   <-- R   <-- R   <-- R
│ │ │ │  
│ │ │ │        0       1       2       3
│ │ │ │  
│ │ │ │  o12 : Complex
│ │ │ │  The user may also specify the top homological degree to compute the
│ │ │ │  normalization up to. Note that this can help speed up computational time; if
│ │ │ │  the user knows the normalization should have a shorter length, then they should
│ │ │ │  specify this upper bound in the syntax:
│ │ │ │  i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER!
│ │ │ │ - -- .0571616s elapsed
│ │ │ │ + -- .0409263s elapsed
│ │ │ │  
│ │ │ │         10      30      30      10
│ │ │ │  o13 = R   <-- R   <-- R   <-- R
│ │ │ │  
│ │ │ │        0       1       2       3
│ │ │ │  
│ │ │ │  o13 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_schur__Map.html
│ │ │ @@ -110,28 +110,28 @@
│ │ │  
│ │ │  o3 : SimplicialModule
│ │ │              
        
│ │ │                
        i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ - -- .142323s elapsed
│ │ │ + -- .159586s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o4 : SimplicialModule
        
│ │ │              
        
│ │ │                
        i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ - -- .309043s elapsed
│ │ │ + -- .322986s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o5 : SimplicialModule
        
│ │ │ @@ -208,15 +208,15 @@
│ │ │                      		
│ │ │                
        i11 : K = koszulComplex vars Q;
        
│ │ │              
        
│ │ │                
        i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ - -- 1.00439s elapsed
│ │ │ + -- .996059s elapsed
│ │ │  
│ │ │         1      9      36      74      81
│ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o12 : Complex
        
│ │ │ @@ -470,15 +470,15 @@
│ │ │  
│ │ │  o30 : ComplexMap
│ │ │              
        
│ │ │                
        i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ - -- .912527s elapsed
│ │ │ + -- 1.10973s elapsed
│ │ │  
│ │ │             1             1
│ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │                  | 1 |
│ │ │  
│ │ │             5                                       9
│ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,24 +42,24 @@
│ │ │ │        1      3      6      10      15
│ │ │ │  o3 = Q  <-- Q  <-- Q  <-- Q   <-- Q  <-- ...
│ │ │ │  
│ │ │ │       0      1      2      3       4
│ │ │ │  
│ │ │ │  o3 : SimplicialModule
│ │ │ │  i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ │ - -- .142323s elapsed
│ │ │ │ + -- .159586s elapsed
│ │ │ │  
│ │ │ │        1      6      21      55      120
│ │ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4
│ │ │ │  
│ │ │ │  o4 : SimplicialModule
│ │ │ │  i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ │ - -- .309043s elapsed
│ │ │ │ + -- .322986s elapsed
│ │ │ │  
│ │ │ │        1      6      21      55      120
│ │ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4
│ │ │ │  
│ │ │ │  o5 : SimplicialModule
│ │ │ │ @@ -106,15 +106,15 @@
│ │ │ │  i10 : Q = ZZ/101[a..c]
│ │ │ │  
│ │ │ │  o10 = Q
│ │ │ │  
│ │ │ │  o10 : PolynomialRing
│ │ │ │  i11 : K = koszulComplex vars Q;
│ │ │ │  i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ │ - -- 1.00439s elapsed
│ │ │ │ + -- .996059s elapsed
│ │ │ │  
│ │ │ │         1      9      36      74      81
│ │ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │ │  
│ │ │ │        0      1      2       3       4
│ │ │ │  
│ │ │ │  o12 : Complex
│ │ │ │ @@ -299,15 +299,15 @@
│ │ │ │  
│ │ │ │             1                   2
│ │ │ │        2 : Q  <--------------- Q  : 2
│ │ │ │                  {2} | 0 b |
│ │ │ │  
│ │ │ │  o30 : ComplexMap
│ │ │ │  i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ │ - -- .912527s elapsed
│ │ │ │ + -- 1.10973s elapsed
│ │ │ │  
│ │ │ │             1             1
│ │ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │ │                  | 1 |
│ │ │ │  
│ │ │ │             5                                       9
│ │ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_tensor__L__E__S.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o7 : Complex
        
│ │ │              
        
│ │ │                
        i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ - -- 2.28436s elapsed
│ │ │ + -- 2.23931s elapsed
│ │ │  
│ │ │                                                                                                                                                                                                                                                                                                     25      50      25
│ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b | <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0 | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-- cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ │                                                                      | 0 b a |              | 0 b a |              | 0 0 b a |              | 0 0 b a 0 0 0 0 |              | 0   0   0 0 b a 0  0  |                                                                                                              
│ │ │       0                    1                    2     3                                                                                     | 0 0 0 0 b a 0 0 |              | ab2 a2b 0 0 0 0 b2 a2 |     9                              10                        11                             12      13      14
│ │ │                                                             4                      5                      6                                 | 0 0 0 0 0 0 b a |      
│ │ │                                                                                                                                                                     8
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,15 +114,15 @@
│ │ │ │        1      2      2      1
│ │ │ │  o7 = Q  <-- Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2      3
│ │ │ │  
│ │ │ │  o7 : Complex
│ │ │ │  i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ │ - -- 2.28436s elapsed
│ │ │ │ + -- 2.23931s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  25      50      25
│ │ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b |
│ │ │ │  <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0
│ │ │ │  | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-
│ │ │ │  - cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/index.html
│ │ │ @@ -211,28 +211,28 @@
│ │ │  
│ │ │  o7 : SimplicialModule
        
│ │ │              
        
│ │ │                
        i8 : elapsedTime simplicialModule(K,6) --specify top degree 6
│ │ │ - -- .0671636s elapsed
│ │ │ + -- .0881016s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o8 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o8 : SimplicialModule
        
│ │ │              
        
│ │ │                
        i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true)
│ │ │ - -- .195242s elapsed
│ │ │ + -- .212057s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o9 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o9 : SimplicialModule
        
│ │ │ ├── html2text {}
│ │ │ │ @@ -135,24 +135,24 @@
│ │ │ │        1      4      10      20
│ │ │ │  o7 = R  <-- R  <-- R   <-- R  <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3
│ │ │ │  
│ │ │ │  o7 : SimplicialModule
│ │ │ │  i8 : elapsedTime simplicialModule(K,6) --specify top degree 6
│ │ │ │ - -- .0671636s elapsed
│ │ │ │ + -- .0881016s elapsed
│ │ │ │  
│ │ │ │        1      4      10      20      35      56      84
│ │ │ │  o8 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4       5       6
│ │ │ │  
│ │ │ │  o8 : SimplicialModule
│ │ │ │  i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true)
│ │ │ │ - -- .195242s elapsed
│ │ │ │ + -- .212057s elapsed
│ │ │ │  
│ │ │ │        1      4      10      20      35      56      84
│ │ │ │  o9 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4       5       6
│ │ │ │  
│ │ │ │  o9 : SimplicialModule
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialPosets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  aXNCb29sZWFu
│ │ │  #:len=1246
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRGV0ZXJtaW5lIGlmIGEgcG9zZXQgaXMg
│ │ │  YSBib29sZWFuIGFsZ2VicmEuIiwgImxpbmVudW0iID0+IDMzNywgSW5wdXRzID0+IHtTUEFOe1RU
│ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  dW5pdmVyc2FsSWRlYWw=
│ │ │  #:len=2182
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIHVuaXZlcnNhbCBy
│ │ │  ZWFsaXphdGlvbiBpZGVhbCBvZiBhIG1hdHJvaWQiLCAibGluZW51bSIgPT4gMjE0MiwgSW5wdXRz
│ │ ├── ./usr/share/doc/Macaulay2/SpaceCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aXNTbW9vdGhBQ01CZXR0aQ==
│ │ │  #:len=902
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIHdoZXRoZXIgYSBCZXR0aSB0
│ │ │  YWJsZSBpcyB0aGF0IG9mIGEgc21vb3RoIEFDTSBjdXJ2ZSIsICJsaW5lbnVtIiA9PiAxMjY5LCBJ
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=47
│ │ │  TXVsdGlkaW1lbnNpb25hbE1hdHJpeCAqIE11bHRpZGltZW5zaW9uYWxNYXRyaXg=
│ │ │  #:len=1779
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJvZHVjdCBvZiBtdWx0aWRpbWVuc2lv
│ │ │  bmFsIG1hdHJpY2VzIiwgImxpbmVudW0iID0+IDE0MjUsIElucHV0cyA9PiB7U1BBTntUVHsiTSJ9
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : n = {2,3,2}
│ │ │  
│ │ │  o1 = {2, 3, 2}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time degreeDeterminant n
│ │ │ - -- used 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc)
│ │ │ + -- used 9.9482e-05s (cpu); 8.8863e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
│ │ │  
│ │ │  i3 : M = genericMultidimensionalMatrix n;
│ │ │  warning: clearing value of symbol x2 to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 1368   or with command line option   --debug 1368
│ │ │  warning: clearing value of symbol x1 to allow access to subscripted variables based on it
│ │ │ @@ -19,14 +19,14 @@
│ │ │  warning: clearing value of symbol x0 to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 6010   or with command line option   --debug 6010
│ │ │  
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
│ │ │  
│ │ │  i4 : time degree determinant M
│ │ │ - -- used 0.0324525s (cpu); 0.0316018s (thread); 0s (gc)
│ │ │ + -- used 0.193335s (cpu); 0.0736924s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 17130321902108223178
│ │ │  
│ │ │  i1 : (d,n) := (2,3);
│ │ │  
│ │ │  i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.44512s (cpu); 0.234802s (thread); 0s (gc)
│ │ │ + -- used 0.752065s (cpu); 0.371948s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = Disc
│ │ │  
│ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |)
│ │ │                                                             | 0 0 0 1 1 2 0 0 1 0 |
│ │ │                                                             | 0 1 2 0 1 0 0 1 0 0 |
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out
│ │ │ @@ -9,18 +9,18 @@
│ │ │                   2
│ │ │       c x x  + c x  + c x  + c x  + c )
│ │ │        4 1 2    2 2    3 1    1 2    0
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.0911372s (cpu); 0.0911356s (thread); 0s (gc)
│ │ │ + -- used 0.187156s (cpu); 0.186959s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc)
│ │ │ + -- used 0.52541s (cpu); 0.452734s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc)
│ │ │ + -- used 0.46726s (cpu); 0.406394s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : assert(o2 == o3 and o3 == o4)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2,
│ │ │       ------------------------------------------------------------------------
│ │ │       3}}}}
│ │ │  
│ │ │  o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ
│ │ │  
│ │ │  i2 : time det M
│ │ │ - -- used 0.0864658s (cpu); 0.0851857s (thread); 0s (gc)
│ │ │ + -- used 0.381325s (cpu); 0.144017s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
│ │ │  
│ │ │  i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │  
│ │ │  o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5,
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -24,13 +24,13 @@
│ │ │       7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6,
│ │ │       ------------------------------------------------------------------------
│ │ │       4, 8, 4, 2}}}}}
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
│ │ │  
│ │ │  i4 : time det M
│ │ │ - -- used 0.778868s (cpu); 0.689399s (thread); 0s (gc)
│ │ │ + -- used 0.730364s (cpu); 0.730304s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │       9257139493926586400187927813888
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_generic__Skew__Multidimensional__Matrix.out
│ │ │ @@ -1,22 +1,22 @@
│ │ │  -- -*- M2-comint -*- hash: 6013064849487134497
│ │ │  
│ │ │  i1 : genericSkewMultidimensionalMatrix(3,4)
│ │ │  
│ │ │  o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ │ -                              1    0        1       2        0   2           
│ │ │ +                              3    2        3       0        2   0           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a ,
│ │ │ -         1   0                    1          3      0      3              1 
│ │ │ +         3   2                    3          1      2      1              3 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a ,
│ │ │ -         2     1         3                    2    3                 0    2 
│ │ │ +         0     3         1                    0    1                 2    0 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           0       3        2   3
│ │ │ +           2       1        0   1
│ │ │  
│ │ │  o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ]
│ │ │                                                        0   3
│ │ │  
│ │ │  i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101)
│ │ │  
│ │ │  o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Discriminant.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       a     x y z  + a     x y z  + a     x y z
│ │ │        1,1,1 1 1 1    1,2,0 1 2 0    1,2,1 1 2 1
│ │ │  
│ │ │  o1 : ZZ[a     ..a     ][x ..x , y ..y , z ..z ]
│ │ │           0,0,0   1,2,1   0   1   0   2   0   1
│ │ │  
│ │ │  i2 : time sparseDiscriminant f
│ │ │ - -- used 3.23258s (cpu); 2.80646s (thread); 0s (gc)
│ │ │ + -- used 3.79963s (cpu); 3.18936s (thread); 0s (gc)
│ │ │  
│ │ │                                                     2                        
│ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │              2     2                                2            
│ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 16228363821945730064
│ │ │  
│ │ │  i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ - -- used 0.477656s (cpu); 0.408514s (thread); 0s (gc)
│ │ │ + -- used 0.729712s (cpu); 0.553105s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = Res
│ │ │  
│ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |})
│ │ │                                                              | 0 0 1 1 |  | 1 0 1 2 |  | 0 1 0 1 |
│ │ │  
│ │ │  i2 : QQ[c_(1,1)..c_(3,4)][x,y];
│ │ │ @@ -18,15 +18,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       c   x*y + c   x + c   y + c   )
│ │ │        3,3       3,4     3,2     3,1
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : time Res(f,g,h)
│ │ │ - -- used 0.00964993s (cpu); 0.00964869s (thread); 0s (gc)
│ │ │ + -- used 0.0126041s (cpu); 0.0126038s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -730,30 +730,30 @@
│ │ │  
│ │ │  o4 : QQ[c   ..c   ]
│ │ │           1,1   3,4
│ │ │  
│ │ │  i5 : assert(Res(f,g,h) == sparseResultant(f,g,h))
│ │ │  
│ │ │  i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0310848s (cpu); 0.0302486s (thread); 0s (gc)
│ │ │ + -- used 0.121571s (cpu); 0.0520682s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
│ │ │  
│ │ │  i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
│ │ │  
│ │ │  i8 : (f,g,h) = (a_0 + a_1*x + a_2*y + a_3*x*y, b_0 + b_1*x + b_2*y + b_3*x*y, c_0 + c_1*x + c_2*y + c_3*x*y)
│ │ │  
│ │ │  o8 = (a x*y + a x + a y + a , b x*y + b x + b y + b , c x*y + c x + c y + c )
│ │ │         3       1     2     0   3       1     2     0   3       1     2     0
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : time Res(f,g,h)
│ │ │ - -- used 0.00326741s (cpu); 0.00326931s (thread); 0s (gc)
│ │ │ + -- used 0.00484116s (cpu); 0.00484402s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ @@ -822,15 +822,15 @@
│ │ │                    2
│ │ │        c x x  + c x  + c x  + c x  + c )
│ │ │         4 1 2    2 2    3 1    1 2    0
│ │ │  
│ │ │  o11 : Sequence
│ │ │  
│ │ │  i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.277694s (cpu); 0.202525s (thread); 0s (gc)
│ │ │ + -- used 0.366177s (cpu); 0.233939s (thread); 0s (gc)
│ │ │  
│ │ │  i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │  
│ │ │          2 2                   2    2                               2 2
│ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │  
│ │ │  o1 : List
│ │ │              
        
│ │ │                
        i2 : time degreeDeterminant n
│ │ │ - -- used 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc)
│ │ │ + -- used 9.9482e-05s (cpu); 8.8863e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
        
│ │ │              
        
│ │ │                
        i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -103,15 +103,15 @@
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
        
│ │ │              
        
│ │ │                
        i4 : time degree determinant M
│ │ │ - -- used 0.0324525s (cpu); 0.0316018s (thread); 0s (gc)
│ │ │ + -- used 0.193335s (cpu); 0.0736924s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
        
│ │ │              
        
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : n = {2,3,2}
│ │ │ │  
│ │ │ │  o1 = {2, 3, 2}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time degreeDeterminant n
│ │ │ │ - -- used 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc)
│ │ │ │ + -- used 9.9482e-05s (cpu); 8.8863e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  i3 : M = genericMultidimensionalMatrix n;
│ │ │ │  warning: clearing value of symbol x2 to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 1368   or with command line option   --
│ │ │ │  debug 1368
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 6010   or with command line option   --
│ │ │ │  debug 6010
│ │ │ │  
│ │ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │ │                                                        0,0,0   1,2,1
│ │ │ │  i4 : time degree determinant M
│ │ │ │ - -- used 0.0324525s (cpu); 0.0316018s (thread); 0s (gc)
│ │ │ │ + -- used 0.193335s (cpu); 0.0736924s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = {6}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_r_m_i_n_a_n_t_(_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x_) -- hyperdeterminant of a
│ │ │ │        multidimensional matrix
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │              
│ │ │                
        i1 : (d,n) := (2,3);
        
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.44512s (cpu); 0.234802s (thread); 0s (gc)
│ │ │ + -- used 0.752065s (cpu); 0.371948s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = Disc
│ │ │  
│ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |)
│ │ │                                                             | 0 0 0 1 1 2 0 0 1 0 |
│ │ │                                                             | 0 1 2 0 1 0 0 1 0 0 |
        
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o for (d,n), this is the same as _s_p_a_r_s_e_D_i_s_c_r_i_m_i_n_a_n_t _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x
│ │ │ │              ""ggeenneerriicc ppoollyynnoommiiaall ooff ddeeggrreeee dd iinn nn vvaarriiaabblleess"";;
│ │ │ │            o for f, this is the same as _a_f_f_i_n_e_D_i_s_c_r_i_m_i_n_a_n_t(f).
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : (d,n) := (2,3);
│ │ │ │  i2 : time Disc = denseDiscriminant(d,n)
│ │ │ │ - -- used 0.44512s (cpu); 0.234802s (thread); 0s (gc)
│ │ │ │ + -- used 0.752065s (cpu); 0.371948s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = Disc
│ │ │ │  
│ │ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1
│ │ │ │  2 |)
│ │ │ │                                                             | 0 0 0 1 1 2 0 0 1
│ │ │ │  0 |
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html
│ │ │ @@ -95,27 +95,27 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.0911372s (cpu); 0.0911356s (thread); 0s (gc)
        
│ │ │ + -- used 0.187156s (cpu); 0.186959s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc)
        
│ │ │ + -- used 0.52541s (cpu); 0.452734s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc)
        
│ │ │ + -- used 0.46726s (cpu); 0.406394s (thread); 0s (gc)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i5 : assert(o2 == o3 and o3 == o4)
        
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,20 +28,20 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                   2
│ │ │ │       c x x  + c x  + c x  + c x  + c )
│ │ │ │        4 1 2    2 2    3 1    1 2    0
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ │ - -- used 0.0911372s (cpu); 0.0911356s (thread); 0s (gc)
│ │ │ │ + -- used 0.187156s (cpu); 0.186959s (thread); 0s (gc)
│ │ │ │  i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay
│ │ │ │  formula
│ │ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc)
│ │ │ │ + -- used 0.52541s (cpu); 0.452734s (thread); 0s (gc)
│ │ │ │  i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc)
│ │ │ │ + -- used 0.46726s (cpu); 0.406394s (thread); 0s (gc)
│ │ │ │  i5 : assert(o2 == o3 and o3 == o4)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_p_a_r_s_e_R_e_s_u_l_t_a_n_t -- sparse resultant (A-resultant)
│ │ │ │      * _a_f_f_i_n_e_R_e_s_u_l_t_a_n_t -- affine resultant
│ │ │ │      * _d_e_n_s_e_D_i_s_c_r_i_m_i_n_a_n_t -- dense discriminant (classical discriminant)
│ │ │ │      * _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x -- exponents in one or more polynomials
│ │ │ │      * _g_e_n_e_r_i_c_L_a_u_r_e_n_t_P_o_l_y_n_o_m_i_a_l_s -- generic (Laurent) polynomials
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │  
│ │ │  o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i2 : time det M
│ │ │ - -- used 0.0864658s (cpu); 0.0851857s (thread); 0s (gc)
│ │ │ + -- used 0.381325s (cpu); 0.144017s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
        
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
        
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
        i4 : time det M
│ │ │ - -- used 0.778868s (cpu); 0.689399s (thread); 0s (gc)
│ │ │ + -- used 0.730364s (cpu); 0.730304s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │       9257139493926586400187927813888
        
│ │ │              
│ │ │            
│ │ │          
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       3}}}}
│ │ │ │  
│ │ │ │  o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ
│ │ │ │  i2 : time det M
│ │ │ │ - -- used 0.0864658s (cpu); 0.0851857s (thread); 0s (gc)
│ │ │ │ + -- used 0.381325s (cpu); 0.144017s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 9698337990421512192
│ │ │ │  i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ │  
│ │ │ │  o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       6}}}, {{{3, 5, 7, 7, 9}, {4, 5, 0, 4, 3}}, {{1, 8, 9, 1, 2}, {9, 6, 6,
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       4, 8, 4, 2}}}}}
│ │ │ │  
│ │ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
│ │ │ │  i4 : time det M
│ │ │ │ - -- used 0.778868s (cpu); 0.689399s (thread); 0s (gc)
│ │ │ │ + -- used 0.730364s (cpu); 0.730304s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │ │       9257139493926586400187927813888
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x -- the class of all multidimensional matrices
│ │ │ │      * _d_e_g_r_e_e_D_e_t_e_r_m_i_n_a_n_t -- degree of the hyperdeterminant of a generic
│ │ │ │        multidimensional matrix
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_generic__Skew__Multidimensional__Matrix.html
│ │ │ @@ -80,24 +80,24 @@
│ │ │          
      An $n$-dimensional matrix $M$ is skew symmetric if for every permutation $s$ of the set $\{0,\ldots,n-1\}$ we have permute(M,s) == sign(s)*M.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : genericSkewMultidimensionalMatrix(3,4)
│ │ │  
│ │ │  o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ │ -                              1    0        1       2        0   2           
│ │ │ +                              3    2        3       0        2   0           
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a ,
│ │ │ -         1   0                    1          3      0      3              1 
│ │ │ +         3   2                    3          1      2      1              3 
│ │ │       ------------------------------------------------------------------------
│ │ │       0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a ,
│ │ │ -         2     1         3                    2    3                 0    2 
│ │ │ +         0     3         1                    0    1                 2    0 
│ │ │       ------------------------------------------------------------------------
│ │ │       0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}}
│ │ │ -           0       3        2   3
│ │ │ +           2       1        0   1
│ │ │  
│ │ │  o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ]
│ │ │                                                        0   3
      
│ │ │              
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,24 +17,24 @@
│ │ │ │              $d\times\cdots\times d$ ($n$ times).
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  An $n$-dimensional matrix $M$ is skew symmetric if for every permutation $s$ of
│ │ │ │  the set $\{0,\ldots,n-1\}$ we have _p_e_r_m_u_t_e(M,s) == sign(s)*M.
│ │ │ │  i1 : genericSkewMultidimensionalMatrix(3,4)
│ │ │ │  
│ │ │ │  o1 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ │ │ -                              1    0        1       2        0   2
│ │ │ │ +                              3    2        3       0        2   0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       0, a , a }, {0, 0, 0, 0}, {-a , 0, 0, -a }, {-a , 0, a , 0}}, {{0, -a ,
│ │ │ │ -         1   0                    1          3      0      3              1
│ │ │ │ +         3   2                    3          1      2      1              3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       0, a }, {a , 0, 0, a }, {0, 0, 0, 0}, {-a , -a , 0, 0}}, {{0, -a , -a ,
│ │ │ │ -         2     1         3                    2    3                 0    2
│ │ │ │ +         0     3         1                    0    1                 2    0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       0}, {a , 0, -a , 0}, {a , a , 0, 0}, {0, 0, 0, 0}}}
│ │ │ │ -           0       3        2   3
│ │ │ │ +           2       1        0   1
│ │ │ │  
│ │ │ │  o1 : 3-dimensional matrix of shape 4 x 4 x 4 over QQ[a ..a ]
│ │ │ │                                                        0   3
│ │ │ │  i2 : genericSkewMultidimensionalMatrix(3,4,CoefficientRing=>ZZ/101)
│ │ │ │  
│ │ │ │  o2 = {{{0, 0, 0, 0}, {0, 0, -a , -a }, {0, a , 0, -a }, {0, a , a , 0}}, {{0,
│ │ │ │                                1    0        1       2        0   2
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o1 : ZZ[a     ..a     ][x ..x , y ..y , z ..z ]
│ │ │           0,0,0   1,2,1   0   1   0   2   0   1
│ │ │              
      
│ │ │                
      i2 : time sparseDiscriminant f
│ │ │ - -- used 3.23258s (cpu); 2.80646s (thread); 0s (gc)
│ │ │ + -- used 3.79963s (cpu); 3.18936s (thread); 0s (gc)
│ │ │  
│ │ │                                                     2                        
│ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │              2     2                                2            
│ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a     x y z  + a     x y z  + a     x y z
│ │ │ │        1,1,1 1 1 1    1,2,0 1 2 0    1,2,1 1 2 1
│ │ │ │  
│ │ │ │  o1 : ZZ[a     ..a     ][x ..x , y ..y , z ..z ]
│ │ │ │           0,0,0   1,2,1   0   1   0   2   0   1
│ │ │ │  i2 : time sparseDiscriminant f
│ │ │ │ - -- used 3.23258s (cpu); 2.80646s (thread); 0s (gc)
│ │ │ │ + -- used 3.79963s (cpu); 3.18936s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                     2
│ │ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2     2                                2
│ │ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │          
      Description
      
│ │ │          
      Alternatively, one can apply the method directly to the list of Laurent polynomials $f_0,\ldots,f_n$. In this case, the matrices $A_0,\ldots,A_n$ are automatically determined by exponentsMatrix. If you want require that $A_0=\cdots=A_n$, then use the option Unmixed=>true (this could be faster). Below we consider some examples.
      
│ │ │          
      In the first example, we calculate the sparse (mixed) resultant associated to the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$. Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
      
│ │ │                
      i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ - -- used 0.477656s (cpu); 0.408514s (thread); 0s (gc)
│ │ │ + -- used 0.729712s (cpu); 0.553105s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = Res
│ │ │  
│ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |})
│ │ │                                                              | 0 0 1 1 |  | 1 0 1 2 |  | 0 1 0 1 |
      
│ │ │              
      
│ │ │                
      i4 : time Res(f,g,h)
│ │ │ - -- used 0.00964993s (cpu); 0.00964869s (thread); 0s (gc)
│ │ │ + -- used 0.0126041s (cpu); 0.0126038s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -830,15 +830,15 @@
│ │ │            
      
│ │ │          
      In the second example, we calculate the sparse unmixed resultant associated to the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0 + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over $\mathbb{Z}/3331$.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,15 +68,15 @@
│ │ │ │  
│ │ │ │  o4 = | 0 1 |
│ │ │ │       | 2 3 |
│ │ │ │       | 4 |
│ │ │ │  
│ │ │ │  o4 : YoungTableau
│ │ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ │ - -- used 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ │ + -- used 0.00176516s (cpu); 0.0017588s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 2
│ │ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o5 : R
│ │ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ │ - -- used 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ │ + -- used 0.00158703s (cpu); 0.00158705s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 2
│ │ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │ @@ -108,15 +108,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o6 : R
│ │ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ │ - -- used 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ │ + -- used 0.00250532s (cpu); 0.00250609s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )
│ │ │ │  (x )(x ))
│ │ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4
│ │ │ │  2   0
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  2*x_2*x_4-x_3*x_4-
│ │ │ │  2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-
│ │ │ │  3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i3 : time toGrass X
│ │ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ │ + -- used 5.99311s (cpu); 3.58623s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces
│ │ │ │  of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │ │  i3 : time toGrass X
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ │ + -- used 6.50661s (cpu); 3.80633s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0310848s (cpu); 0.0302486s (thread); 0s (gc)
│ │ │ + -- used 0.121571s (cpu); 0.0520682s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
      
│ │ │              
      
│ │ │ @@ -854,15 +854,15 @@
│ │ │  
│ │ │  o8 : Sequence
│ │ │              
      
│ │ │                
      i9 : time Res(f,g,h)
│ │ │ - -- used 0.00326741s (cpu); 0.00326931s (thread); 0s (gc)
│ │ │ + -- used 0.00484116s (cpu); 0.00484402s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ @@ -943,15 +943,15 @@
│ │ │  
│ │ │  o11 : Sequence
      
│ │ │              
      
│ │ │                
      i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.277694s (cpu); 0.202525s (thread); 0s (gc)
      
│ │ │ + -- used 0.366177s (cpu); 0.233939s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │  
│ │ │          2 2                   2    2                               2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  In the first example, we calculate the sparse (mixed) resultant associated to
│ │ │ │  the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$.
│ │ │ │  Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{
│ │ │ │  (1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{
│ │ │ │  (2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
│ │ │ │  i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},
│ │ │ │  {1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ │ - -- used 0.477656s (cpu); 0.408514s (thread); 0s (gc)
│ │ │ │ + -- used 0.729712s (cpu); 0.553105s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = Res
│ │ │ │  
│ │ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1
│ │ │ │  2 2 |, | 0 0 1 1 |})
│ │ │ │                                                              | 0 0 1 1 |  | 1 0
│ │ │ │  1 2 |  | 0 1 0 1 |
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │         1,3       1,2       1,4     1,1   2,2        2,3       2,4     2,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       c   x*y + c   x + c   y + c   )
│ │ │ │        3,3       3,4     3,2     3,1
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : time Res(f,g,h)
│ │ │ │ - -- used 0.00964993s (cpu); 0.00964869s (thread); 0s (gc)
│ │ │ │ + -- used 0.0126041s (cpu); 0.0126038s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2                       4      2   2               4
│ │ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3       2       3               2                   3
│ │ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ │ @@ -771,29 +771,29 @@
│ │ │ │  In the second example, we calculate the sparse unmixed resultant associated to
│ │ │ │  the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials
│ │ │ │  $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0
│ │ │ │  + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over
│ │ │ │  $\mathbb{Z}/3331$.
│ │ │ │  i6 : time Res = sparseResultant(matrix{{0,0,1,1},
│ │ │ │  {0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ │ - -- used 0.0310848s (cpu); 0.0302486s (thread); 0s (gc)
│ │ │ │ + -- used 0.121571s (cpu); 0.0520682s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over
│ │ │ │  ZZ/3331)
│ │ │ │                                                               | 0 1 0 1 |
│ │ │ │  i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
│ │ │ │  i8 : (f,g,h) = (a_0 + a_1*x + a_2*y + a_3*x*y, b_0 + b_1*x + b_2*y + b_3*x*y,
│ │ │ │  c_0 + c_1*x + c_2*y + c_3*x*y)
│ │ │ │  
│ │ │ │  o8 = (a x*y + a x + a y + a , b x*y + b x + b y + b , c x*y + c x + c y + c )
│ │ │ │         3       1     2     0   3       1     2     0   3       1     2     0
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : time Res(f,g,h)
│ │ │ │ - -- used 0.00326741s (cpu); 0.00326931s (thread); 0s (gc)
│ │ │ │ + -- used 0.00484116s (cpu); 0.00484402s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2     2            2            2        2 2    2
│ │ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                           2                       2
│ │ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ │ @@ -863,15 +863,15 @@
│ │ │ │                    2
│ │ │ │        c x x  + c x  + c x  + c x  + c )
│ │ │ │         4 1 2    2 2    3 1    1 2    0
│ │ │ │  
│ │ │ │  o11 : Sequence
│ │ │ │  i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant
│ │ │ │  (f,g,h,Unmixed=>true));
│ │ │ │ - -- used 0.277694s (cpu); 0.202525s (thread); 0s (gc)
│ │ │ │ + -- used 0.366177s (cpu); 0.233939s (thread); 0s (gc)
│ │ │ │  i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │ │  
│ │ │ │          2 2                   2    2                               2 2
│ │ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  c2NodXJQb2x5bm9taWFsKC4uLixBc0V4cHJlc3Npb249Pi4uLik=
│ │ │  #:len=287
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMwNSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2NodXJQb2x5bm9taWFsLEFzRXhwcmVzc2lvbl0s
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o4 = | 0 1 |
│ │ │       | 2 3 |
│ │ │       | 4 |
│ │ │  
│ │ │  o4 : YoungTableau
│ │ │  
│ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ + -- used 0.00176516s (cpu); 0.0017588s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -46,15 +46,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o5 : R
│ │ │  
│ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ + -- used 0.00158703s (cpu); 0.00158705s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -67,15 +67,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o6 : R
│ │ │  
│ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ + -- used 0.00250532s (cpu); 0.00250609s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : tal := tally apply (H,h->conjugacyClass h);
│ │ │  
│ │ │  i4 : partis = partitions 6;
│ │ │  
│ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ + -- used 0.562529s (cpu); 0.445372s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out
│ │ │ @@ -20,15 +20,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.815181s (cpu); 0.539564s (thread); 0s (gc)
│ │ │ + -- used 1.03113s (cpu); 0.732527s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │  o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        }
│ │ │                                                        3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
│ │ │                                                      2 3 2 2     4 2 3 2     2 2 2 3     4 3 2   2   2 2 3   2   2 3   2 2   2   3 2 2   2 2   3 2   4   2 3 2   2 2 2   3   4 2   2 3   2   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     1 3 2 2     1 2 3 2     2 3 2 2     1 3 2 2     2 2 3 2     1 2 3 2     1 3 2 2     2 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 3 2     2 2 2 3     4 2 2 3     2 2 2 3     2 2 2 3     4 2 2 3     2 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 3 2   2   1 2 3   2   2 3 2   2   1 3 2   2   2 2 3   2   1 2 3   2   1 3 2   2   2 3 2   2   1 3 2   2   1 2 3   2   2 2 3   2   1 2 3   2   2 3   2 2   2   3 2 2   4 3   2 2   2 3   2 2   4   3 2 2   2   3 2 2   2 3   2 2   4 3   2 2   2 3   2 2   2   3 2 2   4   3 2 2   2   3 2 2   1 2   3 2   1   2 3 2   2 2   3 2   1 2   3 2   2   2 3 2   1   2 3 2   1 2   3 2   2 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   2 2 2   3   4 2 2   3   2 2 2   3   2 2 2   3   4 2 2   3   2 2 2   3   1 2   2 3   1   2 2 3   2 2   2 3   1 2   2 3   2   2 2 3   1   2 2 3   1 2   2 3   2 2   2 3   1 2   2 3   1   2 2 3   2   2 2 3   1   2 2 3
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -130,15 +130,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
      
│ │ │              
      
│ │ │                
      i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ + -- used 0.00176516s (cpu); 0.0017588s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -154,15 +154,15 @@
│ │ │  
│ │ │  o5 : R
      
│ │ │              
      
│ │ │                
      i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ + -- used 0.00158703s (cpu); 0.00158705s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o6 : R
      
│ │ │              
      
│ │ │                
      i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ + -- used 0.00250532s (cpu); 0.00250609s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
      
│ │ │              
      
│ │ │                
      i4 : partis = partitions 6;
      
│ │ │              
      
│ │ │                
      i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ + -- used 0.562529s (cpu); 0.445372s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  representations of $H$ in each irreducible representation of $S_6$. We take
│ │ │ │  into account that there are multiple copies of each representation by
│ │ │ │  multiplying the values with the number of copies which is given by the
│ │ │ │  hookLengthFormula.
│ │ │ │  i4 : partis = partitions 6;
│ │ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity
│ │ │ │  (tal,p))
│ │ │ │ - -- used 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ │ + -- used 0.562529s (cpu); 0.445372s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │ │                 Partition{2, 2, 2} => 1
│ │ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.815181s (cpu); 0.539564s (thread); 0s (gc)
      
│ │ │ + -- used 1.03113s (cpu); 0.732527s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,15 +56,15 @@
│ │ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ │ - -- used 0.815181s (cpu); 0.539564s (thread); 0s (gc)
│ │ │ │ + -- used 1.03113s (cpu); 0.732527s (thread); 0s (gc)
│ │ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │ │  
│ │ │ │                                                        2 2 2       4 2   2     2
│ │ │ │  2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2
│ │ │ │  1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2
│ │ │ │  1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2
│ │ │ │  2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  ZGV0ZWN0Q29uZ3J1ZW5jZSguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=296
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTUsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2RldGVjdENvbmdydWVuY2UsVmVyYm9zZV0sImRldGVj
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out
│ │ │ @@ -45,15 +45,15 @@
│ │ │    -- top 1, degrees: 1^1 2^3 3^3 
│ │ │    -- top 2, degrees: 2^4 3^3 
│ │ │    -- top 3, degrees: 2^3 3^4 
│ │ │    -- top 4, degrees: 2^3 3^3 4^1 
│ │ │    -- top 5, degrees: 2^3 3^3 5^1 
│ │ │    -- top 6, degrees: 2^3 3^3 6^1 
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Complete intersection of 3 quadrics in PP^7
│ │ │       of discriminant 31 = det| 8 1 |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Cubic__Fourfold_rp.out
│ │ │ @@ -42,15 +42,15 @@
│ │ │  -- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
│ │ │    -- top 1, degrees: 1^4 2^1 
│ │ │    -- top 2, degrees: 1^3 2^2 
│ │ │    -- top 3, degrees: 1^3 2^1 3^1 
│ │ │    -- top 4, degrees: 1^3 2^1 4^1 
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Special cubic fourfold of discriminant 14
│ │ │       containing a rational surface of degree 4 and sectional genus 0
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -48,15 +48,15 @@
│ │ │    -- top 3, degrees: 1^1 2^4 3^2 
│ │ │    -- top 4, degrees: 1^1 2^4 4^2 
│ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │  -- computing normalization of the surface image
│ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ + ✦ associated K3 successfully completed in 5 seconds (cpu: 7 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface of degree 2 and sectional genus 0
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -8,28 +8,28 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ + -- used 3.71194s (cpu); 2.26721s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │  
│ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │  
│ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │  
│ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ + -- used 0.605192s (cpu); 0.403906s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       containing a surface of degree 9 and sectional genus 2
│ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ + -- used 23.3875s (cpu); 9.42841s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │  
│ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ @@ -29,15 +29,15 @@
│ │ │  i5 : p := point Y -- random point on Y
│ │ │  
│ │ │  o5 = point of coordinates [14360, -1933, -494, -6471, -10457, -2246, -11879, -12725, 1]
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │  
│ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ + -- used 0.909716s (cpu); 0.503881s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │  
│ │ │  i7 : S = surface X;
│ │ │  
│ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ + -- used 0.619581s (cpu); 0.384104s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Cubic__Fourfold_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = cubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ + -- used 1.0497s (cpu); 0.693887s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │  
│ │ │  i3 : X = gushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ + -- used 4.72968s (cpu); 3.42974s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : ? X
│ │ │  
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
│ │ │  
│ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ + -- used 1.1664s (cpu); 0.779962s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_polarized__K3surface.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- surface was already irreducible
│ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 15 seconds (cpu: 21 seconds)
│ │ │  
│ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -108,16 +108,16 @@
│ │ │  -- computing p2^*(H_PP^2)
│ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │    -- computing image on K3 surface...
│ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │  -- constructing lattice polarization...
│ │ │  -- verifying self-intersection of the curve...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 6 seconds)
│ │ │ +-- total time (K3 surface + polarization): 18 seconds (cpu: 27 seconds)
│ │ │  
│ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice intersection matrix on Ũ: | 14 7 |
│ │ │ @@ -205,15 +205,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- removing 1 components of degrees {3}
│ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 6 seconds (cpu: 10 seconds)
│ │ │  
│ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -226,15 +226,15 @@
│ │ │  -- available strategies: "SpecialCurve", "MapFromW", "MapFromU", "MapFromW-Virtual", "MapFromU-Virtual"
│ │ │  -- special curves already detected on U
│ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1 
│ │ │  -- constructing lattice polarization...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ +-- total time (K3 surface + polarization): 6 seconds (cpu: 10 seconds)
│ │ │  
│ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice intersection matrix on Ũ: | 6 5  |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : x := gens ring PP_(ZZ/33331)^8;
│ │ │  
│ │ │  i2 : X = gushelMukaiFourfold(ideal(x_6-x_7, x_5, x_3-x_4, x_1, x_0-x_4, x_2*x_7-x_4*x_8), ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ + -- used 5.99311s (cpu); 3.58623s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8);
│ │ │  
│ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ + -- used 6.50661s (cpu); 3.80633s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = cubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time f = unirationalParametrization X;
│ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ + -- used 1.29151s (cpu); 0.798655s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │  
│ │ │  i5 : degreeSequence f
│ │ │  
│ │ │  o5 = {[10]}
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │    -- top 1, degrees: 1^1 2^3 3^3 
│ │ │    -- top 2, degrees: 2^4 3^3 
│ │ │    -- top 3, degrees: 2^3 3^4 
│ │ │    -- top 4, degrees: 2^3 3^3 4^1 
│ │ │    -- top 5, degrees: 2^3 3^3 5^1 
│ │ │    -- top 6, degrees: 2^3 3^3 6^1 
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │    -- top 1, degrees: 1^1 2^3 3^3
│ │ │ │    -- top 2, degrees: 2^4 3^3
│ │ │ │    -- top 3, degrees: 2^3 3^4
│ │ │ │    -- top 4, degrees: 2^3 3^3 4^1
│ │ │ │    -- top 5, degrees: 2^3 3^3 5^1
│ │ │ │    -- top 6, degrees: 2^3 3^3 6^1
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Complete intersection of 3 quadrics in PP^7
│ │ │ │       of discriminant 31 = det| 8 1 |
│ │ │ │                               | 1 4 |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Cubic__Fourfold_rp.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  -- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
│ │ │    -- top 1, degrees: 1^4 2^1 
│ │ │    -- top 2, degrees: 1^3 2^2 
│ │ │    -- top 3, degrees: 1^3 2^1 3^1 
│ │ │    -- top 4, degrees: 1^3 2^1 4^1 
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │  interpolation
│ │ │ │    -- top 1, degrees: 1^4 2^1
│ │ │ │    -- top 2, degrees: 1^3 2^2
│ │ │ │    -- top 3, degrees: 1^3 2^1 3^1
│ │ │ │    -- top 4, degrees: 1^3 2^1 4^1
│ │ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 2 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Special cubic fourfold of discriminant 14
│ │ │ │       containing a rational surface of degree 4 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degree 2
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │    -- top 3, degrees: 1^1 2^4 3^2 
│ │ │    -- top 4, degrees: 1^1 2^4 4^2 
│ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │  -- computing normalization of the surface image
│ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ + ✦ associated K3 successfully completed in 5 seconds (cpu: 7 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -84,15 +84,15 @@
│ │ │ │    -- top 3, degrees: 1^1 2^4 3^2
│ │ │ │    -- top 4, degrees: 1^1 2^4 4^2
│ │ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │ │  -- computing normalization of the surface image
│ │ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ │ + ✦ associated K3 successfully completed in 5 seconds (cpu: 7 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees 1^5 2^1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
      
│ │ │              
      
│ │ │                
      i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ + -- used 3.71194s (cpu); 2.26721s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
      
│ │ │              
      
│ │ │                
      i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ + -- used 0.605192s (cpu); 0.403906s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
      
│ │ │              
      
│ │ │                
      i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ │ + -- used 3.71194s (cpu); 2.26721s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the cubic map and passing through a
│ │ │ │  general point: 8
│ │ │ │  number 2-secant lines = 7
│ │ │ │  number 5-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │ │  
│ │ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ │ - -- used 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ │ + -- used 0.605192s (cpu); 0.403906s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree
│ │ │ │  (C * surface X) == 5 and isSubset(p, C))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_c_t_C_o_n_g_r_u_e_n_c_e_(_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_,_Z_Z_) -- detect and return a
│ │ │ │        congruence of (2e-1)-secant curves of degree e inside a del Pezzo
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │              
      
│ │ │                
      i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ + -- used 23.3875s (cpu); 9.42841s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
      
│ │ │              
      
│ │ │                
      i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ + -- used 0.909716s (cpu); 0.503881s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
      
│ │ │              
      
│ │ │                
      i7 : S = surface X;
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 20
│ │ │ │       containing a surface of degree 9 and sectional genus 2
│ │ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ │ + -- used 23.3875s (cpu); 9.42841s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the quadratic map and passing through a
│ │ │ │  general point: 7
│ │ │ │  number 1-secant lines = 6
│ │ │ │  number 3-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ │ @@ -52,15 +52,15 @@
│ │ │ │  i5 : p := point Y -- random point on Y
│ │ │ │  
│ │ │ │  o5 = point of coordinates [14360, -1933, -494, -6471, -10457, -2246, -11879, -
│ │ │ │  12725, 1]
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ │ - -- used 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ │ + -- used 0.909716s (cpu); 0.503881s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │ │  i7 : S = surface X;
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ │ │  i8 : assert(dim C == 1 and degree C == 2 and dim(C*S) == 0 and degree(C*S) == 3
│ │ │ │  and isSubset(p,C) and isSubset(C,Y))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ + -- used 0.619581s (cpu); 0.384104s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  i1 : K = ZZ/33331; S = PP_K^(1,5);
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ │ - -- used 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ │ + -- used 0.619581s (cpu); 0.384104s (thread); 0s (gc)
│ │ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3
│ │ │ │  hypersurfaces of degree 2
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 32
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Cubic__Fourfold_rp.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ + -- used 1.0497s (cpu); 0.693887s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = cubicFourfold V;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ │ + -- used 1.0497s (cpu); 0.693887s (thread); 0s (gc)
│ │ │ │  S: Veronese surface in PP^5
│ │ │ │  X: smooth cubic hypersurface in PP^5
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 27
│ │ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ + -- used 4.72968s (cpu); 3.42974s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │ │  i3 : X = gushelMukaiFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ │ + -- used 4.72968s (cpu); 3.42974s (thread); 0s (gc)
│ │ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │ │  X: GM fourfold containing S
│ │ │ │  Y: del Pezzo fivefold containing X
│ │ │ │  h^1(N_{S,Y}) = 0
│ │ │ │  h^0(N_{S,Y}) = 11
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
      
│ │ │              
      
│ │ │                
      i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ + -- used 1.1664s (cpu); 0.779962s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^9
│ │ │ │  i3 : ? X
│ │ │ │  
│ │ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │ │       1^2 2^5
│ │ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ │ - -- used 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ │ + -- used 1.1664s (cpu); 0.779962s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7
│ │ │ │  hypersurfaces of degrees 1^2 2^5
│ │ │ │       dominance: true
│ │ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html
│ │ │ @@ -176,15 +176,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- surface was already irreducible
│ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 15 seconds (cpu: 21 seconds)
│ │ │  
│ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -215,16 +215,16 @@
│ │ │  -- computing p2^*(H_PP^2)
│ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │    -- computing image on K3 surface...
│ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │  -- constructing lattice polarization...
│ │ │  -- verifying self-intersection of the curve...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 6 seconds)
│ │ │ +-- total time (K3 surface + polarization): 18 seconds (cpu: 27 seconds)
│ │ │  
│ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice intersection matrix on Ũ: | 14 7 |
│ │ │ @@ -327,15 +327,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- removing 1 components of degrees {3}
│ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 6 seconds (cpu: 10 seconds)
│ │ │  
│ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -351,15 +351,15 @@
│ │ │  -- available strategies: "SpecialCurve", "MapFromW", "MapFromU", "MapFromW-Virtual", "MapFromU-Virtual"
│ │ │  -- special curves already detected on U
│ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1 
│ │ │  -- constructing lattice polarization...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ +-- total time (K3 surface + polarization): 6 seconds (cpu: 10 seconds)
│ │ │  
│ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice intersection matrix on Ũ: | 6 5  |
│ │ │ ├── html2text {}
│ │ │ │ @@ -119,15 +119,15 @@
│ │ │ │  -- analyzing the base locus of the inverse map...
│ │ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │ │  -- projecting to PP^3 for surface decomposition
│ │ │ │    -- surface was already irreducible
│ │ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ │ + ✦ underlying K3 successfully completed in 15 seconds (cpu: 21 seconds)
│ │ │ │  
│ │ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       No exceptional curves
│ │ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by
│ │ │ │ @@ -160,16 +160,16 @@
│ │ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of
│ │ │ │  degrees 1^3 2^8
│ │ │ │    -- computing image on K3 surface...
│ │ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8
│ │ │ │  -- constructing lattice polarization...
│ │ │ │  -- verifying self-intersection of the curve...
│ │ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 6 seconds)
│ │ │ │ +-- total time (K3 surface + polarization): 18 seconds (cpu: 27 seconds)
│ │ │ │  
│ │ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       No exceptional curves
│ │ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by
│ │ │ │ @@ -267,15 +267,15 @@
│ │ │ │  -- analyzing the base locus of the inverse map...
│ │ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │ │  -- projecting to PP^3 for surface decomposition
│ │ │ │    -- removing 1 components of degrees {3}
│ │ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1
│ │ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ │ + ✦ underlying K3 successfully completed in 6 seconds (cpu: 10 seconds)
│ │ │ │  
│ │ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │ │        Mirror fourfold: PP^4
│ │ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2
│ │ │ │  hypersurfaces of degrees 2^1 3^1
│ │ │ │        No exceptional curves
│ │ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by
│ │ │ │ @@ -292,15 +292,15 @@
│ │ │ │  Virtual", "MapFromU-Virtual"
│ │ │ │  -- special curves already detected on U
│ │ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1
│ │ │ │  -- constructing lattice polarization...
│ │ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ │ +-- total time (K3 surface + polarization): 6 seconds (cpu: 10 seconds)
│ │ │ │  
│ │ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │ │        Mirror fourfold: PP^4
│ │ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2
│ │ │ │  hypersurfaces of degrees 2^1 3^1
│ │ │ │        No exceptional curves
│ │ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i3 : time toGrass X
│ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ + -- used 5.99311s (cpu); 3.58623s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
      
│ │ │              
      
│ │ │                
      i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ + -- used 6.50661s (cpu); 3.80633s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
      
│ │ │              
      
│ │ │                
      i4 : time f = unirationalParametrization X;
│ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ + -- used 1.29151s (cpu); 0.798655s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
      
│ │ │              
      
│ │ │                
      i5 : degreeSequence f
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = cubicFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time f = unirationalParametrization X;
│ │ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ │ + -- used 1.29151s (cpu); 0.798655s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/SpectralSequences/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  U3BlY3RyYWxTZXF1ZW5jZSBeIEluZmluaXRlTnVtYmVy
│ │ │  #:len=967
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGluZmluaXR5IHBhZ2Ugb2YgYSBz
│ │ │  cGVjdHJhbCBzZXF1ZW5jZSIsICJsaW5lbnVtIiA9PiAzNDExLCBJbnB1dHMgPT4ge1NQQU57VFR7
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  aXNMb29wbGVzcyhEaWdyYXBoKQ==
│ │ │  #:len=247
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0xvb3BsZXNzLERpZ3JhcGgpLCJpc0xvb3BsZXNz
│ │ ├── ./usr/share/doc/Macaulay2/StatePolytope/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  aW5pdGlhbElkZWFscw==
│ │ │  #:len=950
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2FsbHMgZ2ZhbiBhbmQgcmV0dXJucyB0
│ │ │  aGUgbGlzdCBvZiBpbml0aWFsIGlkZWFscyIsICJsaW5lbnVtIiA9PiAxNDAsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/StronglyStableIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  bWFjYXVsYXlEZWNvbXBvc2l0aW9uKFJpbmdFbGVtZW50KQ==
│ │ │  #:len=329
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODUxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWNhdWxheURlY29tcG9zaXRpb24sUmluZ0VsZW1l
│ │ ├── ./usr/share/doc/Macaulay2/Style/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=39
│ │ │  Z2VuZXJhdGVHcmFtbWFyKFN0cmluZyxTdHJpbmcsRnVuY3Rpb24p
│ │ │  #:len=282
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmF0ZUdyYW1tYXIsU3RyaW5nLFN0cmluZyxG
│ │ ├── ./usr/share/doc/Macaulay2/Style/example-output/_generate__Grammar.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 3455701143666534588
│ │ │  
│ │ │  i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10829-0/0
│ │ │ +o1 = /tmp/M2-10929-0/0
│ │ │  
│ │ │  i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10829-0/0.in
│ │ │ +o2 = /tmp/M2-10929-0/0.in
│ │ │  
│ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │  
│ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │  
│ │ │  i5 : template << endl;
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │        String regex: @M2STRINGS@
│ │ │        List of keywords: {
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │  
│ │ │  
│ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10829-0/0
│ │ │ + -- generating /tmp/M2-10929-0/0
│ │ │  
│ │ │  i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.26.06. Do not modify this file manually.
│ │ │  
│ │ │        This is an example file for the generateGrammar method!
│ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ ├── ./usr/share/doc/Macaulay2/Style/html/_generate__Grammar.html
│ │ │ @@ -87,22 +87,22 @@
│ │ │            
      The function demarkf indicates how the elements of each of the lists will be demarked in the resulting file.   The file outfile will then be generated, replacing each of these strings as indicated above.
      
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -148,15 +148,15 @@
│ │ │            @M2KEYWORDS@
│ │ │        }
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10829-0/0
      
│ │ │ +o1 = /tmp/M2-10929-0/0
│ │ │              
      
│ │ │                
      i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10829-0/0.in
      
│ │ │ +o2 = /tmp/M2-10929-0/0.in
│ │ │              
      
│ │ │                
      i3 : template << "@M2BANNER@" << endl << endl;
      
│ │ │              
      
│ │ │                
      i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10829-0/0
      
│ │ │ + -- generating /tmp/M2-10929-0/0
│ │ │              
      
│ │ │                
      i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.26.06. Do not modify this file manually.
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,18 +26,18 @@
│ │ │ │      * @M2CONSTANTS@, for a list of Macaulay2 symbols and packages.
│ │ │ │      * @M2STRINGS@, for a regular expression that matches Macaulay2 strings.
│ │ │ │  The function demarkf indicates how the elements of each of the lists will be
│ │ │ │  demarked in the resulting file. The file outfile will then be generated,
│ │ │ │  replacing each of these strings as indicated above.
│ │ │ │  i1 : outfile = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10829-0/0
│ │ │ │ +o1 = /tmp/M2-10929-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10829-0/0.in
│ │ │ │ +o2 = /tmp/M2-10929-0/0.in
│ │ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │ │  i5 : template << endl;
│ │ │ │  i6 : template << "String regex: @M2STRINGS@" << endl;
│ │ │ │  i7 : template << "List of keywords: {" << endl;
│ │ │ │  i8 : template << "    @M2KEYWORDS@" << endl;
│ │ │ │  i9 : template << "}" << endl << close;
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: @M2STRINGS@
│ │ │ │        List of keywords: {
│ │ │ │            @M2KEYWORDS@
│ │ │ │        }
│ │ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ │ - -- generating /tmp/M2-10829-0/0
│ │ │ │ + -- generating /tmp/M2-10929-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.26.06. Do not modify this file manually.
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ │ │        List of keywords: {
│ │ ├── ./usr/share/doc/Macaulay2/SubalgebraBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  c2FnYmkoLi4uLFN0cmF0ZWd5PT4uLi4p
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU3OSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2FnYmksU3RyYXRlZ3ldLCJzYWdiaSguLi4sU3Ry
│ │ ├── ./usr/share/doc/Macaulay2/SumsOfSquares/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  bG93ZXJCb3VuZCguLi4sVmVyYm9zaXR5PT4uLi4p
│ │ │  #:len=302
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tsb3dlckJvdW5kLFZlcmJvc2l0eV0sImxvd2VyQm91
│ │ ├── ./usr/share/doc/Macaulay2/SuperLinearAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  cGFyaXR5
│ │ │  #:len=1709
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicGFyaXR5IG9mIGFuIGVsZW1lbnQgb2Yg
│ │ │  YSBzdXBlciByaW5nLiIsICJsaW5lbnVtIiA9PiA2NDcsIElucHV0cyA9PiB7U1BBTntUVHsiZiJ9
│ │ ├── ./usr/share/doc/Macaulay2/SwitchingFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  ZmllbGRCYXNlQ2hhbmdlKFJpbmcsR2Fsb2lzRmllbGQp
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmaWVsZEJhc2VDaGFuZ2UsUmluZyxHYWxvaXNGaWVs
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  bm9QYWNrZWRBbGxTdWJz
│ │ │  #:len=1151
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZHMgYWxsIHN1YnN0aXR1dGlvbnMg
│ │ │  b2YgdmFyaWFibGVzIGJ5IDEgYW5kL29yIDAgZm9yIHdoaWNoIGlkZWFsIGlzIG5vdCBLb25pZy4i
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │  o5 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i6 : isHomogeneous P
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.441308s (cpu); 0.240277s (thread); 0s (gc)
│ │ │ + -- used 0.628112s (cpu); 0.283214s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │  
│ │ │               2         3         2     2
│ │ │  o8 = ideal (y  - x*z, x  - y*z, x y - z )
│ │ │ @@ -47,12 +47,12 @@
│ │ │  o8 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i9 : isHomogeneous Q
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.115068s (cpu); 0.0574305s (thread); 0s (gc)
│ │ │ + -- used 0.146378s (cpu); 0.0653208s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │  
│ │ │  o6 = false
      
│ │ │              
      
│ │ │                
      i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.441308s (cpu); 0.240277s (thread); 0s (gc)
│ │ │ + -- used 0.628112s (cpu); 0.283214s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
      
│ │ │              
      
│ │ │                
      i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o9 = true
      
│ │ │              
      
│ │ │                
      i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.115068s (cpu); 0.0574305s (thread); 0s (gc)
│ │ │ + -- used 0.146378s (cpu); 0.0653208s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
      
│ │ │              
      
│ │ │        
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,28 +59,28 @@
│ │ │ │  o5 = ideal (y  - x*z, x y - z , x  - y*z)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x..z]
│ │ │ │  i6 : isHomogeneous P
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : time symbolicPower(P,4);
│ │ │ │ - -- used 0.441308s (cpu); 0.240277s (thread); 0s (gc)
│ │ │ │ + -- used 0.628112s (cpu); 0.283214s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of QQ[x..z]
│ │ │ │  i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ │  
│ │ │ │               2         3         2     2
│ │ │ │  o8 = ideal (y  - x*z, x  - y*z, x y - z )
│ │ │ │  
│ │ │ │  o8 : Ideal of QQ[x..z]
│ │ │ │  i9 : isHomogeneous Q
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time symbolicPower(Q,4);
│ │ │ │ - -- used 0.115068s (cpu); 0.0574305s (thread); 0s (gc)
│ │ │ │ + -- used 0.146378s (cpu); 0.0653208s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of QQ[x..z]
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_y_m_b_P_o_w_e_r_P_r_i_m_e_P_o_s_C_h_a_r
│ │ │ │  ********** WWaayyss ttoo uussee ssyymmbboolliiccPPoowweerr:: **********
│ │ │ │      * symbolicPower(Ideal,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/SymmetricPolynomials/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  YnVpbGRTeW1tZXRyaWNHQihQb2x5bm9taWFsUmluZyk=
│ │ │  #:len=1047
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiR3JvZWJuZXIgYmFzaXMgb2YgZWxlbWVu
│ │ │  dGFyeSBzeW1tZXRyaWMgcG9seW5vbWlhbHMgYWxnZWJyYSIsICJsaW5lbnVtIiA9PiAxNzksIElu
│ │ ├── ./usr/share/doc/Macaulay2/TSpreadIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  Y291bnRUTGV4TW9uKC4uLixGaXhlZE1heD0+Li4uKQ==
│ │ │  #:len=268
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTQwMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY291bnRUTGV4TW9uLEZpeGVkTWF4XSwiY291bnRU
│ │ ├── ./usr/share/doc/Macaulay2/Tableaux/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  aXNDb3JuZXI=
│ │ │  #:len=1388
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIGlmIGEgYm94IGlzIGEgY29y
│ │ │  bmVyIG9mIGEgdGFibGVhdSIsICJsaW5lbnVtIiA9PiA2MTEsIElucHV0cyA9PiB7U1BBTntUVHsi
│ │ ├── ./usr/share/doc/Macaulay2/TangentCone/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  VGFuZ2VudENvbmU=
│ │ │  #:len=312
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFuZ2VudCBjb25lcyIsIERlc2NyaXB0
│ │ │  aW9uID0+IDE6KCJUaGlzIHBhY2thZ2UgcHJvdmlkZXMgYSBzaW5nbGUgZnVuY3Rpb24gdGhhdCBj
│ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=52
│ │ │  cHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyguLi4sQ29lZmZpY2llbnRGaWVsZD0+Li4uKQ==
│ │ │  #:len=347
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDM3Mywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbcHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyxDb2Vm
│ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  o3 = 0  <-- E  <-- 0
│ │ │                      
│ │ │       -1     0      1
│ │ │  
│ │ │  o3 : Complex
│ │ │  
│ │ │  i4 : time T=tateExtension W;
│ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc)
│ │ │ + -- used 1.18184s (cpu); 0.788632s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │       | 2h  h   0 1  2  3  4  |
│ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : Complex
      
│ │ │              
      
│ │ │                
      i4 : time T=tateExtension W;
│ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc)
      
│ │ │ + -- used 1.18184s (cpu); 0.788632s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │               1
│ │ │ │  o3 = 0  <-- E  <-- 0
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc)
│ │ │ │ + -- used 1.18184s (cpu); 0.788632s (thread); 0s (gc)
│ │ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │ │  
│ │ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │ │       | 2h  h   0 1  2  3  4  |
│ │ │ │       | 0   0   0 0  0  0  0  |
│ │ ├── ./usr/share/doc/Macaulay2/TensorComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  bWlub3JzTWFwKE1hdHJpeCxMYWJlbGVkTW9kdWxlKQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkzOCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWlub3JzTWFwLE1hdHJpeCxMYWJlbGVkTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/TerraciniLoci/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  dGVycmFjaW5pTG9jdXMoWlosUmluZ01hcCk=
│ │ │  #:len=278
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjM1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXJyYWNpbmlMb2N1cyxaWixSaW5nTWFwKSwidGVy
│ │ ├── ./usr/share/doc/Macaulay2/TestAudit/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  dGVzdEF1ZGl0KFN0cmluZyk=
│ │ │  #:len=238
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXN0QXVkaXQsU3RyaW5nKSwidGVzdEF1ZGl0KFN0
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  ZnJvYmVuaXVzUHJlaW1hZ2U=
│ │ │  #:len=934
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZHMgdGhlIGlkZWFsIG9mIGVsZW1l
│ │ │  bnRzIG1hcHBlZCBpbnRvIGEgZ2l2ZW4gaWRlYWwsIHVuZGVyIGFsbCAkcF57LWV9JC1saW5lYXIg
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out
│ │ │ @@ -63,20 +63,20 @@
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ + -- used 1.38459s (cpu); 0.889233s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ + -- used 3.25963s (cpu); 2.65285s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J1 == J2
│ │ │  
│ │ │  o19 = true
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │  
│ │ │  o3 : RingMap T <-- S
│ │ │  
│ │ │  i4 : R = S/(ker g);
│ │ │  
│ │ │  i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ + -- used 0.00331694s (cpu); 0.00331467s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ + -- used 0.00527997s (cpu); 0.00528991s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │  
│ │ │  i8 : isCohenMacaulay(R)
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out
│ │ │ @@ -60,49 +60,49 @@
│ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : time isFInjective(R)
│ │ │ - -- used 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ + -- used 0.0371275s (cpu); 0.0371297s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ + -- used 2.10131s (cpu); 1.55278s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │  
│ │ │  i23 : time isFInjective(R)
│ │ │ - -- used 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ + -- used 0.0813009s (cpu); 0.081308s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ + -- used 0.0829198s (cpu); 0.0829307s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │  
│ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │  
│ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │  
│ │ │  o27 : RingMap EP1 <-- S
│ │ │  
│ │ │  i28 : R = S/(ker f);
│ │ │  
│ │ │  i29 : time isFInjective(R)
│ │ │ - -- used 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ + -- used 0.855865s (cpu); 0.763457s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ + -- used 0.375159s (cpu); 0.283699s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : debugLevel = 1;
│ │ │  
│ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ + -- used 0.220881s (cpu); 0.108099s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ + -- used 0.196206s (cpu); 0.196215s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : debugLevel = 0;
│ │ │  
│ │ │  i30 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out
│ │ │ @@ -81,21 +81,21 @@
│ │ │  i22 : testIdeal({3/4, 2/3, 3/5}, L)
│ │ │  
│ │ │  o22 = ideal (y, x)
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ + -- used 0.309238s (cpu); 0.226464s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ + -- used 0.36985s (cpu); 0.279184s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -231,23 +231,23 @@
│ │ │  
│ │ │  o16 : Ideal of R
      
│ │ │              
      
│ │ │                
      i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ + -- used 1.38459s (cpu); 0.889233s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
      
│ │ │              
      
│ │ │                
      i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ + -- used 3.25963s (cpu); 2.65285s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
      
│ │ │              
      
│ │ │                
      i19 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  i15 : I2 = ideal(x^20*y^100, x + z^100);
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ │ + -- used 1.38459s (cpu); 0.889233s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ │ + -- used 3.25963s (cpu); 2.65285s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J1 == J2
│ │ │ │  
│ │ │ │  o19 = true
│ │ │ │  For legacy reasons, the last ideal in the list can be specified separately,
│ │ │ │  using frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}, I). The last
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html
│ │ │ @@ -101,23 +101,23 @@
│ │ │              
      		
│ │ │                
      i4 : R = S/(ker g);
      
│ │ │              
      
│ │ │                
      i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ + -- used 0.00331694s (cpu); 0.00331467s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
      
│ │ │              
      
│ │ │                
      i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ + -- used 0.00527997s (cpu); 0.00528991s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
      
│ │ │              
      
│ │ │          
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,19 +23,19 @@
│ │ │ │  i1 : T = ZZ/5[x,y];
│ │ │ │  i2 : S = ZZ/5[a,b,c,d];
│ │ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │ │  
│ │ │ │  o3 : RingMap T <-- S
│ │ │ │  i4 : R = S/(ker g);
│ │ │ │  i5 : time isCohenMacaulay(R)
│ │ │ │ - -- used 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ │ + -- used 0.00331694s (cpu); 0.00331467s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ │ + -- used 0.00527997s (cpu); 0.00528991s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │ │  i8 : isCohenMacaulay(R)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  The function isCohenMacaulay considers $R$ as a quotient of a polynomial ring,
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html
│ │ │ @@ -219,23 +219,23 @@
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i20 : time isFInjective(R)
│ │ │ - -- used 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ + -- used 0.0371275s (cpu); 0.0371297s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
      
│ │ │              
      
│ │ │                
      i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ + -- used 2.10131s (cpu); 1.55278s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
      
│ │ │              
      
│ │ │          
      
│ │ │            
      If the option AtOrigin (default value false) is set to true, isFInjective will only check $F$-injectivity at the origin. Otherwise, it will check $F$-injectivity globally. Note that checking $F$-injectivity at the origin can be slower than checking it globally. Consider the following example of a non-$F$-injective ring.
      
│ │ │ @@ -245,23 +245,23 @@
│ │ │              
      		
│ │ │                
      i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
      
│ │ │              
      
│ │ │                
      i23 : time isFInjective(R)
│ │ │ - -- used 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ + -- used 0.0813009s (cpu); 0.081308s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
      
│ │ │              
      
│ │ │                
      i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ + -- used 0.0829198s (cpu); 0.0829307s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
      
│ │ │              
      
│ │ │          
      
│ │ │            
      If the option AssumeCM (default value false) is set to true, then isFInjective only checks the Frobenius action on top cohomology (which is typically much faster). Note that it can give an incorrect answer if the non-injective Frobenius occurs in a lower degree.  Consider the example of the cone over a supersingular elliptic curve times $\mathbb{P}^1$.
      
│ │ │ @@ -288,23 +288,23 @@
│ │ │              
│ │ │                
      i28 : R = S/(ker f);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i29 : time isFInjective(R)
│ │ │ - -- used 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ + -- used 0.855865s (cpu); 0.763457s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ + -- used 0.375159s (cpu); 0.283699s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      If the option AssumedReduced is set to true (its default behavior), then the bottom local cohomology is avoided (this means the Frobenius action on the top potentially nonzero Ext is not computed).
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,52 +81,52 @@
│ │ │ │  much faster.
│ │ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : time isFInjective(R)
│ │ │ │ - -- used 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ │ + -- used 0.0371275s (cpu); 0.0371297s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ │ + -- used 2.10131s (cpu); 1.55278s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  If the option AtOrigin (default value false) is set to true, isFInjective will
│ │ │ │  only check $F$-injectivity at the origin. Otherwise, it will check $F$-
│ │ │ │  injectivity globally. Note that checking $F$-injectivity at the origin can be
│ │ │ │  slower than checking it globally. Consider the following example of a non-$F$-
│ │ │ │  injective ring.
│ │ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │ │  i23 : time isFInjective(R)
│ │ │ │ - -- used 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ │ + -- used 0.0813009s (cpu); 0.081308s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ │ + -- used 0.0829198s (cpu); 0.0829307s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  If the option AssumeCM (default value false) is set to true, then isFInjective
│ │ │ │  only checks the Frobenius action on top cohomology (which is typically much
│ │ │ │  faster). Note that it can give an incorrect answer if the non-injective
│ │ │ │  Frobenius occurs in a lower degree. Consider the example of the cone over a
│ │ │ │  supersingular elliptic curve times $\mathbb{P}^1$.
│ │ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │ │  
│ │ │ │  o27 : RingMap EP1 <-- S
│ │ │ │  i28 : R = S/(ker f);
│ │ │ │  i29 : time isFInjective(R)
│ │ │ │ - -- used 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ │ + -- used 0.855865s (cpu); 0.763457s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ │ + -- used 0.375159s (cpu); 0.283699s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  If the option AssumedReduced is set to true (its default behavior), then the
│ │ │ │  bottom local cohomology is avoided (this means the Frobenius action on the top
│ │ │ │  potentially nonzero Ext is not computed).
│ │ │ │  If the option AssumeNormal (default value false) is set to true, then the
│ │ │ │  bottom two local cohomology modules (or, rather, their duals) need not be
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html
│ │ │ @@ -278,23 +278,23 @@
│ │ │                
      i26 : debugLevel = 1;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ + -- used 0.220881s (cpu); 0.108099s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ + -- used 0.196206s (cpu); 0.196215s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i29 : debugLevel = 0;
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  i25 : I = minors(2, matrix {{x, y, z}, {u, v, w}});
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : debugLevel = 1;
│ │ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing
│ │ │ │  DepthOfSearch will let the function search more deeply.
│ │ │ │ - -- used 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ │ + -- used 0.220881s (cpu); 0.108099s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ │ + -- used 0.196206s (cpu); 0.196215s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : debugLevel = 0;
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_e_s_t_I_d_e_a_l -- compute a test ideal in a Q-Gorenstein ring
│ │ │ │      * _i_s_F_R_a_t_i_o_n_a_l -- whether a ring is F-rational
│ │ │ │  ********** WWaayyss ttoo uussee iissFFRReegguullaarr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html
│ │ │ @@ -260,25 +260,25 @@
│ │ │          
      
│ │ │            
      It is often more efficient to pass a list, as opposed to finding a common denominator and passing a single element, since testIdeal can do things in a more intelligent way for such a list.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ + -- used 0.309238s (cpu); 0.226464s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
      
│ │ │              
      
│ │ │                
      i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ + -- used 0.36985s (cpu); 0.279184s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
      
│ │ │              
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -100,21 +100,21 @@
│ │ │ │  o22 = ideal (y, x)
│ │ │ │  
│ │ │ │  o22 : Ideal of R
│ │ │ │  It is often more efficient to pass a list, as opposed to finding a common
│ │ │ │  denominator and passing a single element, since testIdeal can do things in a
│ │ │ │  more intelligent way for such a list.
│ │ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ │ - -- used 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ │ + -- used 0.309238s (cpu); 0.226464s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ │ + -- used 0.36985s (cpu); 0.279184s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = ideal (y, x)
│ │ │ │  
│ │ │ │  o24 : Ideal of R
│ │ │ │  The option AssumeDomain (default value false) is used when finding a test
│ │ │ │  element. The option FrobeniusRootStrategy (default value Substitution) is
│ │ │ │  passed to internal _f_r_o_b_e_n_i_u_s_R_o_o_t calls.
│ │ ├── ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  bmV3IFRPSCBmcm9tIFRoaW5n
│ │ │  #:len=219
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODQwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFRPSCxUaGluZyksIm5ldyBU
│ │ ├── ./usr/share/doc/Macaulay2/ThinSincereQuivers/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  Y2hhaW5RdWl2ZXIoLi4uLEhlaWdodD0+Li4uKQ==
│ │ │  #:len=283
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI5MSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2hhaW5RdWl2ZXIsSGVpZ2h0XSwiY2hhaW5RdWl2
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  dGdiKExpc3Qp
│ │ │  #:len=213
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0Z2IsTGlzdCksInRnYihMaXN0KSIsIlRocmVhZGVk
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out
│ │ │ @@ -2,19 +2,16 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o3 = LineageTable{((1, 2), 0) => c }
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out
│ │ │ @@ -2,39 +2,33 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                   3
│ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │ -                  ((1, 2), 0) => -c
│ │ │ +o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │ -                               2
│ │ │ -                  (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o4 = LineageTable{((1, 2), 0) => c }
│ │ │                    (0, 1) => null
│ │ │ -                  (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o4 : LineageTable
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out
│ │ │ @@ -2,18 +2,16 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -24,16 +22,15 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out
│ │ │ @@ -6,34 +6,50 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ -                                   4 4      2 6
│ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ -                                    2 6      2 5
│ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ -                                           4 4     2 4
│ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ +                                         2 4
│ │ │ +o4 = LineageTable{(((0, 2), 3), 3) => 37y z            }
│ │ │                                      3 4      2 4
│ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ -                                    3 4      2 5
│ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ -                                    2 5
│ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ +                  ((0, 1), 2) => 34y z  + 14y z
│ │ │ +                                    4 8      3 7
│ │ │ +                  ((0, 2), 1) => 14y z  + 20y z
│ │ │ +                                     4 5      3 7
│ │ │ +                  ((0, 2), 3) => - 9y z  + 27y z
│ │ │ +                                           3 7      2 7
│ │ │ +                  ((1, 2), (0, 2)) => - 39y z  - 22y z
│ │ │ +                                      3 18     3 17
│ │ │ +                  ((1, 2), 1) => - 24y z   + 7y z
│ │ │ +                                    3 7      3 6
│ │ │ +                  ((1, 2), 2) => 16y z  + 29y z
│ │ │                                     2 4
│ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ +                  ((1, 3), 1) => 4y z
│ │ │ +                                           2 5      2 4
│ │ │ +                  ((2, 3), (0, 2)) => - 40y z  - 22y z
│ │ │ +                                        4 4      2 4
│ │ │ +                  ((2, 3), (1, 2)) => 8y z  + 10y z
│ │ │ +                                   4 4      2 7
│ │ │ +                  ((2, 3), 2) => 9y z  + 50y z
│ │ │ +                                     4 4      3 4
│ │ │ +                  ((2, 3), 3) => - 9y z  - 34y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │                                 5       3 4
│ │ │                    (0, 3) => 28y z - 24y z
│ │ │ +                               5 6      4 5
│ │ │ +                  (1, 2) => 19y z  - 45y z
│ │ │ +                                 3 9      3 6
│ │ │ +                  (1, 3) => - 31y z  - 34y z
│ │ │ +                               5 2     2 4
│ │ │ +                  (2, 3) => 28y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html
│ │ │ @@ -91,19 +91,16 @@
│ │ │                
      i2 : allowableThreads= 2;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o3 = LineageTable{((1, 2), 0) => c }
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,19 +19,16 @@
│ │ │ │  This simple function just returns the Gröbner basis computed with threaded
│ │ │ │  Gröbner basis function _t_g_b in the expected Macaulay2 format, so that further
│ │ │ │  computation are one step easier to set up.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ -                  ((0, 1), 1) => null
│ │ │ │                                    2
│ │ │ │ -                  ((1, 2), 0) => c
│ │ │ │ +o3 = LineageTable{((1, 2), 0) => c }
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html
│ │ │ @@ -87,22 +87,18 @@
│ │ │                
      i2 : allowableThreads= 2;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -                                   3
│ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │ -                  ((1, 2), 0) => -c
│ │ │ +o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │ -                               2
│ │ │ -                  (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │                            3       2
│ │ │                    1 => - b c + a*b  + a*c
│ │ │                          2
│ │ │                    2 => b
│ │ │ @@ -110,19 +106,17 @@
│ │ │  o3 : LineageTable
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : minimize T
│ │ │  
│ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o4 = LineageTable{((1, 2), 0) => c }
│ │ │                    (0, 1) => null
│ │ │ -                  (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o4 : LineageTable
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,38 +19,32 @@
│ │ │ │  minimal generators of the ideal generated by the leading terms of the values of
│ │ │ │  H. If the values of H constitute a Gröbner basis of the ideal they generate,
│ │ │ │  this method returns a minimal Gröbner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -                                   3
│ │ │ │ -o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │ -                  ((1, 2), 0) => -c
│ │ │ │ +o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │ -                               2
│ │ │ │ -                  (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │                                  2
│ │ │ │                    0 => a*b*c + c
│ │ │ │                            3       2
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : minimize T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │ -                  ((1, 2), 0) => c
│ │ │ │ +o4 = LineageTable{((1, 2), 0) => c }
│ │ │ │                    (0, 1) => null
│ │ │ │ -                  (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o4 : LineageTable
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -87,18 +87,16 @@
│ │ │                
      i2 : allowableThreads= 2;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -112,16 +110,15 @@
│ │ │  o3 : LineageTable
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,18 +20,16 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gröbner basis of the ideal they generate, this method
│ │ │ │  returns a reduced Gröbner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ -                                     2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +                                   3
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -41,16 +39,15 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html
│ │ │ @@ -100,34 +100,50 @@
│ │ │                
      i3 : allowableThreads  = 4;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : H = tgb I
│ │ │  
│ │ │ -                                   4 4      2 6
│ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ -                                    2 6      2 5
│ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ -                                           4 4     2 4
│ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ +                                         2 4
│ │ │ +o4 = LineageTable{(((0, 2), 3), 3) => 37y z            }
│ │ │                                      3 4      2 4
│ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ -                                    3 4      2 5
│ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ -                                    2 5
│ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ +                  ((0, 1), 2) => 34y z  + 14y z
│ │ │ +                                    4 8      3 7
│ │ │ +                  ((0, 2), 1) => 14y z  + 20y z
│ │ │ +                                     4 5      3 7
│ │ │ +                  ((0, 2), 3) => - 9y z  + 27y z
│ │ │ +                                           3 7      2 7
│ │ │ +                  ((1, 2), (0, 2)) => - 39y z  - 22y z
│ │ │ +                                      3 18     3 17
│ │ │ +                  ((1, 2), 1) => - 24y z   + 7y z
│ │ │ +                                    3 7      3 6
│ │ │ +                  ((1, 2), 2) => 16y z  + 29y z
│ │ │                                     2 4
│ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ +                  ((1, 3), 1) => 4y z
│ │ │ +                                           2 5      2 4
│ │ │ +                  ((2, 3), (0, 2)) => - 40y z  - 22y z
│ │ │ +                                        4 4      2 4
│ │ │ +                  ((2, 3), (1, 2)) => 8y z  + 10y z
│ │ │ +                                   4 4      2 7
│ │ │ +                  ((2, 3), 2) => 9y z  + 50y z
│ │ │ +                                     4 4      3 4
│ │ │ +                  ((2, 3), 3) => - 9y z  - 34y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │                                 5       3 4
│ │ │                    (0, 3) => 28y z - 24y z
│ │ │ +                               5 6      4 5
│ │ │ +                  (1, 2) => 19y z  - 45y z
│ │ │ +                                 3 9      3 6
│ │ │ +                  (1, 3) => - 31y z  - 34y z
│ │ │ +                               5 2     2 4
│ │ │ +                  (2, 3) => 28y z  - 9y z
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,34 +26,50 @@
│ │ │ │  i2 : I = ideal {2*x + 10*y^2*z, 8*x^2*y + 10*x*y*z^3, 5*x*y^3*z^2 + 9*x*z^3,
│ │ │ │  9*x*y^3*z + 10*x*y^3};
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : allowableThreads  = 4;
│ │ │ │  i4 : H = tgb I
│ │ │ │  
│ │ │ │ -                                   4 4      2 6
│ │ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ │ -                                    2 6      2 5
│ │ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ │ -                                           4 4     2 4
│ │ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ │ +                                         2 4
│ │ │ │ +o4 = LineageTable{(((0, 2), 3), 3) => 37y z            }
│ │ │ │                                      3 4      2 4
│ │ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ │ -                                    3 4      2 5
│ │ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ │ -                                    2 5
│ │ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ │ +                  ((0, 1), 2) => 34y z  + 14y z
│ │ │ │ +                                    4 8      3 7
│ │ │ │ +                  ((0, 2), 1) => 14y z  + 20y z
│ │ │ │ +                                     4 5      3 7
│ │ │ │ +                  ((0, 2), 3) => - 9y z  + 27y z
│ │ │ │ +                                           3 7      2 7
│ │ │ │ +                  ((1, 2), (0, 2)) => - 39y z  - 22y z
│ │ │ │ +                                      3 18     3 17
│ │ │ │ +                  ((1, 2), 1) => - 24y z   + 7y z
│ │ │ │ +                                    3 7      3 6
│ │ │ │ +                  ((1, 2), 2) => 16y z  + 29y z
│ │ │ │                                     2 4
│ │ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ │ +                  ((1, 3), 1) => 4y z
│ │ │ │ +                                           2 5      2 4
│ │ │ │ +                  ((2, 3), (0, 2)) => - 40y z  - 22y z
│ │ │ │ +                                        4 4      2 4
│ │ │ │ +                  ((2, 3), (1, 2)) => 8y z  + 10y z
│ │ │ │ +                                   4 4      2 7
│ │ │ │ +                  ((2, 3), 2) => 9y z  + 50y z
│ │ │ │ +                                     4 4      3 4
│ │ │ │ +                  ((2, 3), 3) => - 9y z  - 34y z
│ │ │ │                                   5 2      3 4
│ │ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ │ -                                 3 5     2 4
│ │ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ │ +                              5 3     2 4
│ │ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ │                                 5       3 4
│ │ │ │                    (0, 3) => 28y z - 24y z
│ │ │ │ +                               5 6      4 5
│ │ │ │ +                  (1, 2) => 19y z  - 45y z
│ │ │ │ +                                 3 9      3 6
│ │ │ │ +                  (1, 3) => - 31y z  - 34y z
│ │ │ │ +                               5 2     2 4
│ │ │ │ +                  (2, 3) => 28y z  - 9y z
│ │ │ │                                 2
│ │ │ │                    0 => 2x + 10y z
│ │ │ │                           2           3
│ │ │ │                    1 => 8x y + 10x*y*z
│ │ │ │                             3 2       3
│ │ │ │                    2 => 5x*y z  + 9x*z
│ │ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Y2hpcm90b3BlU3RyaW5n
│ │ │  #:len=253
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzUzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJjaGlyb3RvcGVTdHJpbmciLCJjaGlyb3RvcGVTdHJp
│ │ ├── ./usr/share/doc/Macaulay2/TorAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  aXNHb3JlbnN0ZWlu
│ │ │  #:len=1402
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAid2hldGhlciB0aGUgcmluZyBpcyBHb3Jl
│ │ │  bnN0ZWluIiwgImxpbmVudW0iID0+IDEyNDMsIElucHV0cyA9PiB7U1BBTntUVHsiUiJ9LCIsICIs
│ │ ├── ./usr/share/doc/Macaulay2/ToricHigherDirectImages/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  VG9yaWNIaWdoZXJEaXJlY3RJbWFnZXM=
│ │ │  #:len=3556
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0YXRpb25zIGludm9sdmluZyBw
│ │ │  dXNoZm9yd2FyZHMgYW5kIGhpZ2hlciBkaXJlY3QgaW1hZ2VzIG9mIHRvcmljIG1hcHMiLCAibGlu
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  ZWREZWc=
│ │ │  #:len=2501
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIChnZW5lcmljKSBF
│ │ │  dWNsaWRlYW4gZGlzdGFuY2UgZGVncmVlIG9mIGEgcHJvamVjdGl2ZSB0b3JpYyB2YXJpZXR5Iiwg
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ + -- used 1.51164s (cpu); 1.01247s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
│ │ │  
│ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │  
│ │ │ @@ -56,14 +56,14 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ + -- used 5.77572s (cpu); 3.68672s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ + -- used 1.51164s (cpu); 1.01247s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
      
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -155,15 +155,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ + -- used 5.77572s (cpu); 3.68672s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
      
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,30 +66,30 @@
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │ - -- used 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ │ + -- used 1.51164s (cpu); 1.01247s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 252
│ │ │ │  
│ │ │ │  o4 : QQ
│ │ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ │  
│ │ │ │  The toric variety has degree = 28
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │ - -- used 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ │ + -- used 5.77572s (cpu); 3.68672s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 252
│ │ │ │  
│ │ │ │  o5 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee eeddDDeegg:: **********
│ │ │ │      * edDeg(Matrix)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  YmV0dGkoTm9ybWFsVG9yaWNWYXJpZXR5KQ==
│ │ │  #:len=272
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDc0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiZXR0aSxOb3JtYWxUb3JpY1ZhcmlldHkpLCJiZXR0
│ │ ├── ./usr/share/doc/Macaulay2/ToricVectorBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  ZHVhbChUb3JpY1ZlY3RvckJ1bmRsZSk=
│ │ │  #:len=1504
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiIHRoZSBkdWFsIGJ1bmRsZSBvZiBhIHRv
│ │ │  cmljIHZlY3RvciBidW5kbGUiLCAibGluZW51bSIgPT4gMjgwOCwgSW5wdXRzID0+IHtTUEFOe1RU
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  Y2hlY2tJbnRlcmZhY2U=
│ │ │  #:len=792
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAid2hldGhlciB0aGUgTWFwbGUgaW50ZXJm
│ │ │  YWNlIGlzIHdvcmtpbmcgKGZvciBkZXZlbG9wZXJzKSIsICJsaW5lbnVtIiA9PiAzMzUsIElucHV0
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=50
│ │ │  cmVndWxhclRyaWFuZ3VsYXRpb25XZWlnaHRzKC4uLixEZWdyZWVNYXRyaXg9Pi4uLik=
│ │ │  #:len=347
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgzNCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbcmVndWxhclRyaWFuZ3VsYXRpb25XZWlnaHRzLERl
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │       | -1 1  2  -1 -1 1 -1 1 0 0 |
│ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │  
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .116686s elapsed
│ │ │ + -- .0992883s elapsed
│ │ │  
│ │ │  i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -50,14 +50,14 @@
│ │ │  i7 : T = regularFineTriangulation A
│ │ │  
│ │ │  o7 = triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, 1, 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}}
│ │ │  
│ │ │  o7 : Triangulation
│ │ │  
│ │ │  i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 2.08042s elapsed
│ │ │ + -- 1.51096s elapsed
│ │ │  
│ │ │  i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html
│ │ │ @@ -168,15 +168,15 @@
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .116686s elapsed
      
│ │ │ + -- .0992883s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │ @@ -216,15 +216,15 @@
│ │ │  
│ │ │  o7 : Triangulation
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 2.08042s elapsed
      
│ │ │ + -- 1.51096s elapsed
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │       | 0  0  0  1  0  0 -1 0 0 0 |
│ │ │ │       | -1 1  2  -1 -1 1 -1 1 0 0 |
│ │ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │ │  
│ │ │ │                4       10
│ │ │ │  o2 : Matrix ZZ  <-- ZZ
│ │ │ │  i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ │ - -- .116686s elapsed
│ │ │ │ + -- .0992883s elapsed
│ │ │ │  i4 : select(Ts, T -> isStar T)
│ │ │ │  
│ │ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       5, 7, 9}, {0, 3, 5, 8, 9}, {0, 4, 6, 8, 9}, {0, 5, 6, 7, 9}, {0, 5, 6,
│ │ │ │ @@ -120,15 +120,15 @@
│ │ │ │  6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8},
│ │ │ │  {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6,
│ │ │ │  7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2,
│ │ │ │  3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}}
│ │ │ │  
│ │ │ │  o7 : Triangulation
│ │ │ │  i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ │ - -- 2.08042s elapsed
│ │ │ │ + -- 1.51096s elapsed
│ │ │ │  i9 : #Ts2 == #Ts
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _P_o_l_y_h_e_d_r_a -- convex polyhedra
│ │ │ │      * _T_o_p_c_o_m -- interface to the topcom software package which in particular
│ │ │ │        computes triangulations
│ │ ├── ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=7
│ │ │  cm90Rm9ydw==
│ │ │  #:len=229
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODM2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJyb3RGb3J3Iiwicm90Rm9ydyIsIlRyaXBsZXRzIn0s
│ │ ├── ./usr/share/doc/Macaulay2/Tropical/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  aXNCYWxhbmNlZA==
│ │ │  #:len=1097
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIHdoZXRoZXIgYSB0cm9waWNh
│ │ │  bCBjeWNsZSBpcyBiYWxhbmNlZCIsICJsaW5lbnVtIiA9PiA5NTEsIElucHV0cyA9PiB7U1BBTntU
│ │ ├── ./usr/share/doc/Macaulay2/TropicalToric/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=43
│ │ │  Y2xhc3NGcm9tVHJvcGljYWwoTm9ybWFsVG9yaWNWYXJpZXR5LElkZWFsKQ==
│ │ │  #:len=335
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjbGFzc0Zyb21Ucm9waWNhbCxOb3JtYWxUb3JpY1Zh
│ │ ├── ./usr/share/doc/Macaulay2/Truncations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  dHJ1bmNhdGUoSW5maW5pdGVOdW1iZXIsVGhpbmcp
│ │ │  #:len=222
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAsICJ1bmRvY3VtZW50ZWQiID0+IHRy
│ │ │  dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KHRydW5jYXRl
│ │ ├── ./usr/share/doc/Macaulay2/Units/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  VW5pdHM=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidW5pdHMgY29udmVyc2lvbiBhbmQgcGh5
│ │ │  c2ljYWwgY29uc3RhbnRzIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7VEVYeyJUaGlzIHBh
│ │ ├── ./usr/share/doc/Macaulay2/VNumber/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  c3RhYmxlTWF4
│ │ │  #:len=1353
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSB0aGUgc2V0IG9mIHN0YWJs
│ │ │  ZSBwcmltZXMgb2YgYSBtb25vbWlhbCBpZGVhbCB0aGF0IGFyZSBtYXhpbWFsIHdpdGggcmVzcGVj
│ │ ├── ./usr/share/doc/Macaulay2/Valuations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  VmFsdWF0aW9u
│ │ │  #:len=1226
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIHR5cGUgb2YgYWxsIHZhbHVhdGlv
│ │ │  bnMiLCAibGluZW51bSIgPT4gNzk4LCBTZWVBbHNvID0+IERJVntIRUFERVIyeyJTZWUgYWxzbyJ9
│ │ ├── ./usr/share/doc/Macaulay2/Varieties/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  PiBJbmZpbml0ZU51bWJlcg==
│ │ │  #:len=253
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgPixJbmZpbml0ZU51bWJlciksIj4gSW5m
│ │ ├── ./usr/share/doc/Macaulay2/VectorFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  Y29tbXV0YXRvcg==
│ │ │  #:len=3282
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNvbW11dGF0b3Igb2YgYSBjb2xs
│ │ │  ZWN0aW9uIG9mIHZlY3RvciBmaWVsZHMiLCAibGluZW51bSIgPT4gMjE1NSwgSW5wdXRzID0+IHtT
│ │ ├── ./usr/share/doc/Macaulay2/VectorGraphics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  dGV4KFNWRyk=
│ │ │  #:len=181
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg4OCwgInVuZG9jdW1lbnRlZCIgPT4g
│ │ │  dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4LFNW
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  bGlmdERlZm9ybWF0aW9uKC4uLixWZXJib3NlPT4uLi4p
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA1MCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbbGlmdERlZm9ybWF0aW9uLFZlcmJvc2VdLCJsaWZ0
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out
│ │ │ @@ -6,30 +6,30 @@
│ │ │  
│ │ │  o2 = | xz yz z2 x3 |
│ │ │  
│ │ │               1      4
│ │ │  o2 : Matrix S  <-- S
│ │ │  
│ │ │  i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.688076s (cpu); 0.53591s (thread); 0s (gc)
│ │ │ + -- used 0.965087s (cpu); 0.797017s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : T=ring first G;
│ │ │  
│ │ │  i5 : sum G
│ │ │  
│ │ │  o5 = | t_1t_16             |
│ │ │       | t_9t_16             |
│ │ │       | -t_4t_16            |
│ │ │       | -2t_14t_16+t_15t_16 |
│ │ │  
│ │ │               4      1
│ │ │  o5 : Matrix T  <-- T
│ │ │  
│ │ │  i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.537824s (cpu); 0.386837s (thread); 0s (gc)
│ │ │ + -- used 0.692149s (cpu); 0.510755s (thread); 0s (gc)
│ │ │  
│ │ │  i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16                                                             
│ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
│ │ │          
│ │ │          
      With the default setting SmartLift=>true we get very nice equations for the base space:
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -103,15 +103,15 @@
│ │ │            
│ │ │          
      
│ │ │                
      i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.688076s (cpu); 0.53591s (thread); 0s (gc)
      
│ │ │ + -- used 0.965087s (cpu); 0.797017s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i4 : T=ring first G;
      
│ │ │              
      
│ │ │          
      With the setting SmartLift=>false the calculation is faster, but the equations are no longer homogeneous:
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  o2 : List
│ │ │ │  i3 : range1=makeRange(L,{4,6})
│ │ │ │  
│ │ │ │  o3 = {4, 6, 8, 12}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1)
│ │ │ │ - -- .850657s elapsed
│ │ │ │ + -- .627901s elapsed
│ │ │ │  
│ │ │ │                      3           2 4      8     2    2    2 6   3    2
│ │ │ │  o4 = (true, ideal (x  - x x  - x z  - x z , x x  - x  - x z , x  - x x  +
│ │ │ │                      0    1 3    0      0     0 1    3    0     1    0 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            4        6      8
│ │ │ │       x x z  - x x z  + x z ), {0})
│ │ │ │ @@ -46,23 +46,23 @@
│ │ │ │  i5 : range2=drop(makeRange(L,{1}),9)
│ │ │ │  
│ │ │ │  o5 = {10, 11, 12, 13, 14, 15}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1)
│ │ │ │  time to decompose J1 :
│ │ │ │ - -- .000769529s elapsed
│ │ │ │ + -- .00152006s elapsed
│ │ │ │  
│ │ │ │  component number = 0
│ │ │ │  deformation weights = {{10}}
│ │ │ │  semigroup = {4, 5, 7}
│ │ │ │   smoothing components numbers = {0}
│ │ │ │ - -- .00146584s elapsed
│ │ │ │ + -- .00159642s elapsed
│ │ │ │  flat = true
│ │ │ │ - -- .720102s elapsed
│ │ │ │ + -- .544999s elapsed
│ │ │ │  
│ │ │ │                      3            2    2      10   3    2        10
│ │ │ │  o6 = (true, ideal (x  - x x , x x  - x  - x z  , x  - x x  - x z  ), {0})
│ │ │ │                      0    1 3   0 1    3    0      1    0 3    1
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_prune__Family.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.537824s (cpu); 0.386837s (thread); 0s (gc)
      
│ │ │ + -- used 0.692149s (cpu); 0.510755s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,29 +18,29 @@
│ │ │ │  o2 = | xz yz z2 x3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix S  <-- S
│ │ │ │  With the default setting SmartLift=>true we get very nice equations for the
│ │ │ │  base space:
│ │ │ │  i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ │ - -- used 0.688076s (cpu); 0.53591s (thread); 0s (gc)
│ │ │ │ + -- used 0.965087s (cpu); 0.797017s (thread); 0s (gc)
│ │ │ │  i4 : T=ring first G;
│ │ │ │  i5 : sum G
│ │ │ │  
│ │ │ │  o5 = | t_1t_16             |
│ │ │ │       | t_9t_16             |
│ │ │ │       | -t_4t_16            |
│ │ │ │       | -2t_14t_16+t_15t_16 |
│ │ │ │  
│ │ │ │               4      1
│ │ │ │  o5 : Matrix T  <-- T
│ │ │ │  With the setting SmartLift=>false the calculation is faster, but the equations
│ │ │ │  are no longer homogeneous:
│ │ │ │  i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ │ - -- used 0.537824s (cpu); 0.386837s (thread); 0s (gc)
│ │ │ │ + -- used 0.692149s (cpu); 0.510755s (thread); 0s (gc)
│ │ │ │  i7 : sum G
│ │ │ │  
│ │ │ │  o7 = | t_1t_16
│ │ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/VirtualResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=42
│ │ │  bXVsdGlncmFkZWRSZWd1bGFyaXR5KC4uLixMb3dlckxpbWl0PT4uLi4p
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjAzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttdWx0aWdyYWRlZFJlZ3VsYXJpdHksTG93ZXJMaW1p
│ │ ├── ./usr/share/doc/Macaulay2/Visualize/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  dmlzdWFsaXplKEdyYXBoLFZlcmJvc2U9Pi4uLik=
│ │ │  #:len=1329
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gdG8gdmlldyBjb21tdW5pY2F0aW9u
│ │ │  IGJldHdlZW4gTWFjYXVsYXkyIHNlcnZlciBhbmQgYnJvd3NlciAiLCAibGluZW51bSIgPT4gMjEw
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  Z2V0UmFuZ2VPZk9uZVBhcmFtZXRlckZhbWlseShJZGVhbCk=
│ │ │  #:len=349
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjMwMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2V0UmFuZ2VPZk9uZVBhcmFtZXRlckZhbWlseSxJ
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_append__Family.out
│ │ │ @@ -21,15 +21,15 @@
│ │ │  i5 : setRandomSeed("always successful");
│ │ │   -- setting random seed to 11500776554390917551191162798934277
│ │ │  
│ │ │  i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1)
│ │ │  number of components = 1, codimension of components = {0}
│ │ │  semigroup = {5, 7, 9}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .817637s elapsed
│ │ │ + -- .808135s elapsed
│ │ │  
│ │ │                      2             7      9     14   5      2    2 7    2   8
│ │ │  o6 = (true, ideal (x  - x x  - x z  - x z  - 2z  , x  - x x  - x z  - x x z 
│ │ │                      2    0 4    2      0            0    2 4    4      0 2  
│ │ │       ------------------------------------------------------------------------
│ │ │              9    3 10        11        13    2 15      16      18      20  
│ │ │       - x x z  - x z   - x x z   - x x z   - x z   - x z   - x z   - x z   -
│ │ │ @@ -43,15 +43,15 @@
│ │ │       x z   - x z   - z  ))
│ │ │        0       0
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1)
│ │ │  semigroup = {5, 7, 9}
│ │ │ - -- .0189065s elapsed
│ │ │ + -- .0244749s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : assert(flatten drop(degrees ring J,-1)==L)
│ │ │  
│ │ │  i9 : "appendFamily(L,J,X,Xdbm)";
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Smoothing__Family__With__Versal__Deformation.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  Calculating first order relations
│ │ │  Calculating standard expressions for obstructions
│ │ │  Starting lifting
│ │ │  Order 2
│ │ │  Order 3
│ │ │  Order 4
│ │ │  Solution is polynomial
│ │ │ - -- .253446s elapsed
│ │ │ + -- .193445s elapsed
│ │ │  
│ │ │                      3    2      2   4        5    3 6    2 7        9      10
│ │ │  o2 = (true, ideal (x  - x x  - x x z  - x x z  - x z  - x z  + x x z  - x z  
│ │ │                      2    0 1    0 2      0 1      0      2      0 2      1   
│ │ │       ------------------------------------------------------------------------
│ │ │          2 11      14       16     21   3      2        4    2   5        6  
│ │ │       - x z   + x z   - 3x z   - 2z  , x x  - x  - x x z  - x x z  - x x z  -
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : range1=makeRange(L,{4,6})
│ │ │  
│ │ │  o3 = {4, 6, 8, 12}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1)
│ │ │ - -- .850657s elapsed
│ │ │ + -- .627901s elapsed
│ │ │  
│ │ │                      3           2 4      8     2    2    2 6   3    2    
│ │ │  o4 = (true, ideal (x  - x x  - x z  - x z , x x  - x  - x z , x  - x x  +
│ │ │                      0    1 3    0      0     0 1    3    0     1    0 3  
│ │ │       ------------------------------------------------------------------------
│ │ │            4        6      8
│ │ │       x x z  - x x z  + x z ), {0})
│ │ │ @@ -35,23 +35,23 @@
│ │ │  
│ │ │  o5 = {10, 11, 12, 13, 14, 15}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1)
│ │ │  time to decompose J1 : 
│ │ │ - -- .000769529s elapsed
│ │ │ + -- .00152006s elapsed
│ │ │  
│ │ │  component number = 0
│ │ │  deformation weights = {{10}}
│ │ │  semigroup = {4, 5, 7}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .00146584s elapsed
│ │ │ + -- .00159642s elapsed
│ │ │  flat = true
│ │ │ - -- .720102s elapsed
│ │ │ + -- .544999s elapsed
│ │ │  
│ │ │                      3            2    2      10   3    2        10
│ │ │  o6 = (true, ideal (x  - x x , x x  - x  - x z  , x  - x x  - x z  ), {0})
│ │ │                      0    1 3   0 1    3    0      1    0 3    1
│ │ │  
│ │ │  o6 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_prune__Family.out
│ │ │ @@ -23,15 +23,15 @@
│ │ │  i6 : as=apply(numgens I,i->a=drop(support unfolding_{i},#L));
│ │ │  
│ │ │  i7 : rL=apply(#as,i->select(as_i,m->(degree m)_0> b));
│ │ │  
│ │ │  i8 : restrictionList=apply(flatten rL,m->sub(m,A));
│ │ │  
│ │ │  i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ - -- .325212s elapsed
│ │ │ + -- .272793s elapsed
│ │ │  
│ │ │  i10 : leadTerm J
│ │ │  
│ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │                {1, 4} {2, 6}   {2, 4} {1, 4}   {2, 3} {1, 4}   {2, 4} {1, 3} 
│ │ │        -----------------------------------------------------------------------
│ │ │        a      a      , a      a      )
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_smoothness__With__Reductions.out
│ │ │ @@ -27,12 +27,12 @@
│ │ │          0      3       1       0      0    1 3    0 3      3
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2)
│ │ │  semigroup = {5, 6, 8}
│ │ │  dim and degree singF = (0, 4)
│ │ │ - -- .0188212s elapsed
│ │ │ + -- .0321305s elapsed
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_append__Family.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │            
      
│ │ │                
      i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1)
│ │ │  number of components = 1, codimension of components = {0}
│ │ │  semigroup = {5, 7, 9}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .817637s elapsed
│ │ │ + -- .808135s elapsed
│ │ │  
│ │ │                      2             7      9     14   5      2    2 7    2   8
│ │ │  o6 = (true, ideal (x  - x x  - x z  - x z  - 2z  , x  - x x  - x z  - x x z 
│ │ │                      2    0 4    2      0            0    2 4    4      0 2  
│ │ │       ------------------------------------------------------------------------
│ │ │              9    3 10        11        13    2 15      16      18      20  
│ │ │       - x x z  - x z   - x x z   - x x z   - x z   - x z   - x z   - x z   -
│ │ │ @@ -138,15 +138,15 @@
│ │ │  o6 : Sequence
      
│ │ │              
      
│ │ │                
      i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1)
│ │ │  semigroup = {5, 7, 9}
│ │ │ - -- .0189065s elapsed
│ │ │ + -- .0244749s elapsed
│ │ │  
│ │ │  o7 = true
      
│ │ │              
      
│ │ │                
      i8 : assert(flatten drop(degrees ring J,-1)==L)
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  o4 = fam8.dbm
│ │ │ │  i5 : setRandomSeed("always successful");
│ │ │ │   -- setting random seed to 11500776554390917551191162798934277
│ │ │ │  i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1)
│ │ │ │  number of components = 1, codimension of components = {0}
│ │ │ │  semigroup = {5, 7, 9}
│ │ │ │   smoothing components numbers = {0}
│ │ │ │ - -- .817637s elapsed
│ │ │ │ + -- .808135s elapsed
│ │ │ │  
│ │ │ │                      2             7      9     14   5      2    2 7    2   8
│ │ │ │  o6 = (true, ideal (x  - x x  - x z  - x z  - 2z  , x  - x x  - x z  - x x z
│ │ │ │                      2    0 4    2      0            0    2 4    4      0 2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              9    3 10        11        13    2 15      16      18      20
│ │ │ │       - x x z  - x z   - x x z   - x x z   - x z   - x z   - x z   - x z   -
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │        2 17      22    27
│ │ │ │       x z   - x z   - z  ))
│ │ │ │        0       0
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1)
│ │ │ │  semigroup = {5, 7, 9}
│ │ │ │ - -- .0189065s elapsed
│ │ │ │ + -- .0244749s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : assert(flatten drop(degrees ring J,-1)==L)
│ │ │ │  i9 : "appendFamily(L,J,X,Xdbm)";
│ │ │ │  Reading and writing to the disk does not work in the documentation. Hence we
│ │ │ │  give the command in quotes.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Smoothing__Family__With__Versal__Deformation.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │  Calculating first order relations
│ │ │  Calculating standard expressions for obstructions
│ │ │  Starting lifting
│ │ │  Order 2
│ │ │  Order 3
│ │ │  Order 4
│ │ │  Solution is polynomial
│ │ │ - -- .253446s elapsed
│ │ │ + -- .193445s elapsed
│ │ │  
│ │ │                      3    2      2   4        5    3 6    2 7        9      10
│ │ │  o2 = (true, ideal (x  - x x  - x x z  - x x z  - x z  - x z  + x x z  - x z  
│ │ │                      2    0 1    0 2      0 1      0      2      0 2      1   
│ │ │       ------------------------------------------------------------------------
│ │ │          2 11      14       16     21   3      2        4    2   5        6  
│ │ │       - x z   + x z   - 3x z   - 2z  , x x  - x  - x x z  - x x z  - x x z  -
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │  Calculating first order relations
│ │ │ │  Calculating standard expressions for obstructions
│ │ │ │  Starting lifting
│ │ │ │  Order 2
│ │ │ │  Order 3
│ │ │ │  Order 4
│ │ │ │  Solution is polynomial
│ │ │ │ - -- .253446s elapsed
│ │ │ │ + -- .193445s elapsed
│ │ │ │  
│ │ │ │                      3    2      2   4        5    3 6    2 7        9      10
│ │ │ │  o2 = (true, ideal (x  - x x  - x x z  - x x z  - x z  - x z  + x x z  - x z
│ │ │ │                      2    0 1    0 2      0 1      0      2      0 2      1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2 11      14       16     21   3      2        4    2   5        6
│ │ │ │       - x z   + x z   - 3x z   - 2z  , x x  - x  - x x z  - x x z  - x x z  -
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Range.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
      
│ │ │                
      i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1)
│ │ │ - -- .850657s elapsed
│ │ │ + -- .627901s elapsed
│ │ │  
│ │ │                      3           2 4      8     2    2    2 6   3    2    
│ │ │  o4 = (true, ideal (x  - x x  - x z  - x z , x x  - x  - x z , x  - x x  +
│ │ │                      0    1 3    0      0     0 1    3    0     1    0 3  
│ │ │       ------------------------------------------------------------------------
│ │ │            4        6      8
│ │ │       x x z  - x x z  + x z ), {0})
│ │ │ @@ -137,23 +137,23 @@
│ │ │  o5 : List
      
│ │ │              
      
│ │ │                
      i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1)
│ │ │  time to decompose J1 : 
│ │ │ - -- .000769529s elapsed
│ │ │ + -- .00152006s elapsed
│ │ │  
│ │ │  component number = 0
│ │ │  deformation weights = {{10}}
│ │ │  semigroup = {4, 5, 7}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .00146584s elapsed
│ │ │ + -- .00159642s elapsed
│ │ │  flat = true
│ │ │ - -- .720102s elapsed
│ │ │ + -- .544999s elapsed
│ │ │  
│ │ │                      3            2    2      10   3    2        10
│ │ │  o6 = (true, ideal (x  - x x , x x  - x  - x z  , x  - x x  - x z  ), {0})
│ │ │                      0    1 3   0 1    3    0      1    0 3    1
│ │ │  
│ │ │  o6 : Sequence
      
│ │ │                    		
│ │ │                
      i8 : restrictionList=apply(flatten rL,m->sub(m,A));
      
│ │ │              
      
│ │ │                
      i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ - -- .325212s elapsed
      
│ │ │ + -- .272793s elapsed
│ │ │              
      
│ │ │                
      i10 : leadTerm J
│ │ │  
│ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │                10007  0   1   3   4
│ │ │ │  i4 : (A,unfolding)=makeUnfolding I;
│ │ │ │  i5 : b = genus L-1;
│ │ │ │  i6 : as=apply(numgens I,i->a=drop(support unfolding_{i},#L));
│ │ │ │  i7 : rL=apply(#as,i->select(as_i,m->(degree m)_0> b));
│ │ │ │  i8 : restrictionList=apply(flatten rL,m->sub(m,A));
│ │ │ │  i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ │ - -- .325212s elapsed
│ │ │ │ + -- .272793s elapsed
│ │ │ │  i10 : leadTerm J
│ │ │ │  
│ │ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │ │                {1, 4} {2, 6}   {2, 4} {1, 4}   {2, 3} {1, 4}   {2, 4} {1, 3}
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        a      a      , a      a      )
│ │ │ │         {2, 3} {1, 3}   {1, 2} {2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_smoothness__With__Reductions.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │              
      
│ │ │                
      i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2)
│ │ │  semigroup = {5, 6, 8}
│ │ │  dim and degree singF = (0, 4)
│ │ │ - -- .0188212s elapsed
│ │ │ + -- .0321305s elapsed
│ │ │  
│ │ │  o5 = true
      
│ │ │              
      
│ │ │          
      
│ │ │            
      The intermediate output dim and degree singF = (0, 4) says that after computing some minors of the jacobian matrix, we detect that the curve is smooth away from the zero dimensional scheme defined by singF of degree 4.
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │       - x z  - x z   - x z   + x z  , x  - x x  + x x z  + x z   - z  )
│ │ │ │          0      3       1       0      0    1 3    0 3      3
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2)
│ │ │ │  semigroup = {5, 6, 8}
│ │ │ │  dim and degree singF = (0, 4)
│ │ │ │ - -- .0188212s elapsed
│ │ │ │ + -- .0321305s elapsed
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  The intermediate output dim and degree singF = (0, 4) says that after computing
│ │ │ │  some minors of the jacobian matrix, we detect that the curve is smooth away
│ │ │ │  from the zero dimensional scheme defined by singF of degree 4.
│ │ │ │  The function checkSmoothness takes longer, some times much longer.
│ │ │ │  ********** WWaayyss ttoo uussee ssmmooootthhnneessssWWiitthhRReedduuccttiioonnss:: **********
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  Zmxvb3IoUldlaWxEaXZpc29yKQ==
│ │ │  #:len=275
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU1Niwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxvb3IsUldlaWxEaXZpc29yKSwiZmxvb3IoUldl
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Basic__Divisor_sp_pl_sp__Basic__Divisor.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 11085051886200177329
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │  
│ │ │ -o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │ +o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : D1 + D2
│ │ │  
│ │ │ -o4 = 6*Div(y) + -4*Div(x)
│ │ │ +o4 = -4*Div(x) + 6*Div(y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : D1 - D2
│ │ │  
│ │ │ -o5 = 4*Div(z) + 6*Div(x)
│ │ │ +o5 = 6*Div(x) + 4*Div(z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : R = QQ[x,y];
│ │ │  
│ │ │  i7 : D1 = divisor({3, 1}, {ideal(x), ideal(y)})
│ │ │  
│ │ │ @@ -64,30 +64,30 @@
│ │ │  
│ │ │  o12 : RWeilDivisor on R
│ │ │  
│ │ │  i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z);
│ │ │  
│ │ │  i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)})
│ │ │  
│ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z)
│ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : -D
│ │ │  
│ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z)
│ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z)
│ │ │  
│ │ │  o15 : WeilDivisor on R
│ │ │  
│ │ │  i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z)
│ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z)
│ │ │  
│ │ │  o16 : WeilDivisor on R
│ │ │  
│ │ │  i17 : -E
│ │ │  
│ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z)
│ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z)
│ │ │  
│ │ │  o17 : WeilDivisor on R
│ │ │  
│ │ │  i18 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Number_sp_st_sp__Basic__Divisor.out
│ │ │ @@ -16,21 +16,21 @@
│ │ │  
│ │ │  o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : 8*D
│ │ │  
│ │ │ -o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │ +o5 = 8*Div(y) + 16*Div(x) + -8*Div(x+y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : (-2/3)*D
│ │ │  
│ │ │ -o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │ +o6 = -2/3*Div(y) + -4/3*Div(x) + 2/3*Div(x+y)
│ │ │  
│ │ │  o6 : QWeilDivisor on R
│ │ │  
│ │ │  i7 : 0.0*D
│ │ │  
│ │ │  o7 = 0, the zero divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___O__O_sp__R__Weil__Divisor.out
│ │ │ @@ -90,15 +90,15 @@
│ │ │                                2
│ │ │  o15 : R-module, submodule of R
│ │ │  
│ │ │  i16 : R = ZZ/11[x,y];
│ │ │  
│ │ │  i17 : D = divisor(x*y/(x+y))
│ │ │  
│ │ │ -o17 = -Div(x+y) + Div(x) + Div(y)
│ │ │ +o17 = Div(x) + Div(y) + -Div(x+y)
│ │ │  
│ │ │  o17 : WeilDivisor on R
│ │ │  
│ │ │  i18 : divisor(OO(D))
│ │ │  
│ │ │  o18 = 0, the zero divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_apply__To__Coefficients.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 14937934652040812889
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D = divisor(x*y^2/z)
│ │ │  
│ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y)
│ │ │ +o2 = Div(x) + 2*Div(y) + -Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : applyToCoefficients(D, u->5*u)
│ │ │  
│ │ │  o3 = 5*Div(x) + 10*Div(y) + -5*Div(z)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ceiling_lp__R__Weil__Divisor_rp.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 992133077988949640
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
│ │ │  
│ │ │  i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
│ │ │  
│ │ │  i3 : ceiling( D )
│ │ │  
│ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : floor( D )
│ │ │  
│ │ │  o4 = Div(y, z)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │  
│ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 : ceiling( E )
│ │ │  
│ │ │  o6 = Div(x, z)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out
│ │ │ @@ -1,38 +1,38 @@
│ │ │  -- -*- M2-comint -*- hash: 16935688116980988371
│ │ │  
│ │ │  i1 : R = QQ[x,y,z];
│ │ │  
│ │ │  i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : E = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │  
│ │ │ -o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │  
│ │ │ -o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │  
│ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │ +o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │  
│ │ │  o6 : WeilDivisor on R
│ │ │  
│ │ │  i7 : R = QQ[x,y,z]/ideal(x^2-y*z);
│ │ │  
│ │ │  i8 : D = divisor({2}, {ideal(x,y)})
│ │ │  
│ │ │ @@ -60,29 +60,29 @@
│ │ │  
│ │ │  o14 = 3*Div(xz2, xyz, xy2, x2z, x2y, x3)
│ │ │  
│ │ │  o14 : WeilDivisor on A
│ │ │  
│ │ │  i15 : E = divisor(y2z)
│ │ │  
│ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │  
│ │ │  o15 : WeilDivisor on A
│ │ │  
│ │ │  i16 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o17 : QWeilDivisor on R
│ │ │  
│ │ │  i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │  
│ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o18 : QWeilDivisor on R
│ │ │  
│ │ │  i19 : R = ZZ/11[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i20 : D = divisor({1.1, -3.14159}, {ideal(x,u), ideal(x, v)}, CoefficientType=>RR)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out
│ │ │ @@ -44,51 +44,51 @@
│ │ │  i10 : J = m^9;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : M = J*R^1;
│ │ │  
│ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0706325s (cpu); 0.0706342s (thread); 0s (gc)
│ │ │ + -- used 0.0555153s (cpu); 0.0555098s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc)
│ │ │ + -- used 0.488536s (cpu); 0.488295s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc)
│ │ │ + -- used 0.699966s (cpu); 0.596083s (thread); 0s (gc)
│ │ │  
│ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc)
│ │ │ + -- used 0.003408s (cpu); 0.00341477s (thread); 0s (gc)
│ │ │  
│ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc)
│ │ │ + -- used 0.00312895s (cpu); 0.00313705s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i18 : I = ideal(x,u);
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J = I^15;
│ │ │  
│ │ │  o19 : Ideal of R
│ │ │  
│ │ │  i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0637156s (cpu); 0.0637222s (thread); 0s (gc)
│ │ │ + -- used 0.103835s (cpu); 0.103795s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
│ │ │  
│ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc)
│ │ │ + -- used 0.00859937s (cpu); 0.00854392s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i23 : J = ideal(x,y);
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Divisors.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 6816075660530398278
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)})
│ │ │  
│ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z)
│ │ │ +o2 = 2*Div(y) + 0*Div(x^2+z) + -8*Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : getPrimeDivisors( D )
│ │ │  
│ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)}
│ │ │ +o3 = {Div(y), Div(x^2+z), Div(x)}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : getPrimeDivisors( cleanSupport D )
│ │ │  
│ │ │ -o4 = {Div(x), Div(y)}
│ │ │ +o4 = {Div(y), Div(x)}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Cartier.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │  i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o5 = 2*Div(y, z) + Div(x, z)
│ │ │ +o5 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 : isCartier( D )
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │ @@ -48,15 +48,15 @@
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : isCartier(D, IsGraded => true)
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Linear__Equivalent.out
│ │ │ @@ -1,38 +1,38 @@
│ │ │  -- -*- M2-comint -*- hash: 6019119347082811396
│ │ │  
│ │ │  i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │  
│ │ │  i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : isLinearEquivalent(D1, D2)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │  
│ │ │  i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │  
│ │ │  o6 : WeilDivisor on R
│ │ │  
│ │ │  i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o7 : WeilDivisor on R
│ │ │  
│ │ │  i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │  
│ │ │  o8 = false
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Q__Cartier.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 13719144060491348416
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ +o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : isQCartier(10, D1)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │ @@ -44,21 +44,21 @@
│ │ │  
│ │ │  o10 = 0
│ │ │  
│ │ │  i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
│ │ │  
│ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o13 : QWeilDivisor on R
│ │ │  
│ │ │  i14 : isQCartier(10, D1, IsGraded => true)
│ │ │  
│ │ │  o14 = 1
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Q__Linear__Equivalent.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 13920959388108803216
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
│ │ │  
│ │ │  i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : isQLinearEquivalent(10, D, E)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │ @@ -36,21 +36,21 @@
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │  
│ │ │  i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │  
│ │ │  o11 : QWeilDivisor on R
│ │ │  
│ │ │  i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │  
│ │ │  o12 : QWeilDivisor on R
│ │ │  
│ │ │  i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │  
│ │ │  o13 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__S__N__C.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 2360371518304120718
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : isSNC( D )
│ │ │  
│ │ │  o3 = false
│ │ │  
│ │ │ @@ -48,27 +48,27 @@
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 : R = QQ[x, y];
│ │ │  
│ │ │  i14 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o14 : WeilDivisor on R
│ │ │  
│ │ │  i15 : isSNC( D, IsGraded => true )
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : R = QQ[x,y,z];
│ │ │  
│ │ │  i17 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o17 : WeilDivisor on R
│ │ │  
│ │ │  i18 : isSNC( D, IsGraded => true)
│ │ │  
│ │ │  o18 = false
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Weil__Divisor.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 9088327857146195664
│ │ │  
│ │ │  i1 : R = QQ[x, y, z];
│ │ │  
│ │ │  i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o2 = Div(y) + -2*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + Div(y) + -2*Div(z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
│ │ │  
│ │ │  i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x)
│ │ │ +o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : isWeilDivisor( D1 )
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out
│ │ │ @@ -36,18 +36,18 @@
│ │ │  
│ │ │  i10 : D = divisor(x*y*(x+y));
│ │ │  
│ │ │  o10 : WeilDivisor on R
│ │ │  
│ │ │  i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ +o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │  
│ │ │  o11 : WeilDivisor on S
│ │ │  
│ │ │  i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ +o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │  
│ │ │  o12 : WeilDivisor on S
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out
│ │ │ @@ -103,104 +103,104 @@
│ │ │  o21 : Ideal of R
│ │ │  
│ │ │  i22 : J = I^21;
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time reflexify(J);
│ │ │ - -- used 0.338433s (cpu); 0.253168s (thread); 0s (gc)
│ │ │ + -- used 0.322708s (cpu); 0.232027s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc)
│ │ │ + -- used 0.558297s (cpu); 0.448756s (thread); 0s (gc)
│ │ │  
│ │ │  i25 : R = ZZ/13[x,y,z]/ideal(x^3 + y^3-z^11*x*y);
│ │ │  
│ │ │  i26 : I = ideal(x-4*y, z);
│ │ │  
│ │ │  o26 : Ideal of R
│ │ │  
│ │ │  i27 : J = I^20;
│ │ │  
│ │ │  o27 : Ideal of R
│ │ │  
│ │ │  i28 : M = J*R^1;
│ │ │  
│ │ │  i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.168829s (cpu); 0.168835s (thread); 0s (gc)
│ │ │ + -- used 0.156379s (cpu); 0.156381s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc)
│ │ │ + -- used 7.88167s (cpu); 5.94876s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
│ │ │  
│ │ │  i31 : J1 == J2
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc)
│ │ │ + -- used 7.80245s (cpu); 5.81405s (thread); 0s (gc)
│ │ │  
│ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc)
│ │ │ + -- used 0.713827s (cpu); 0.477249s (thread); 0s (gc)
│ │ │  
│ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │  
│ │ │  i35 : I = ideal(x,u);
│ │ │  
│ │ │  o35 : Ideal of R
│ │ │  
│ │ │  i36 : J = I^20;
│ │ │  
│ │ │  o36 : Ideal of R
│ │ │  
│ │ │  i37 : M = I^20*R^1;
│ │ │  
│ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc)
│ │ │ + -- used 0.567643s (cpu); 0.33001s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o38 : Ideal of R
│ │ │  
│ │ │  i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.0135561s (cpu); 0.0135572s (thread); 0s (gc)
│ │ │ + -- used 0.0163127s (cpu); 0.0163172s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │         19    20
│ │ │        x  u, x  )
│ │ │  
│ │ │  o39 : Ideal of R
│ │ │  
│ │ │  i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.0397151s (cpu); 0.0397234s (thread); 0s (gc)
│ │ │ + -- used 0.055363s (cpu); 0.0553724s (thread); 0s (gc)
│ │ │  
│ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc)
│ │ │ + -- used 0.00812396s (cpu); 0.00813234s (thread); 0s (gc)
│ │ │  
│ │ │  i42 : R = QQ[x,y]/ideal(x*y);
│ │ │  
│ │ │  i43 : I = ideal(x,y);
│ │ │  
│ │ │  o43 : Ideal of R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out
│ │ │ @@ -23,44 +23,44 @@
│ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0292441s (cpu); 0.0292465s (thread); 0s (gc)
│ │ │ + -- used 0.0442378s (cpu); 0.0442393s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc)
│ │ │ + -- used 0.161455s (cpu); 0.161455s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : J20a == J20b
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │  
│ │ │  i12 : I = ideal(x-z,y-2*z);
│ │ │  
│ │ │  o12 : Ideal of R
│ │ │  
│ │ │  i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0299195s (cpu); 0.0299242s (thread); 0s (gc)
│ │ │ + -- used 0.0408921s (cpu); 0.0406934s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
│ │ │  
│ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc)
│ │ │ + -- used 0.0734406s (cpu); 0.0734493s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : J1 == J2
│ │ │  
│ │ │  o15 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ring_lp__Basic__Divisor_rp.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 5006859181202351713
│ │ │  
│ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │  
│ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : ring( D )
│ │ │  
│ │ │  o3 = R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__Q__Weil__Divisor.out
│ │ │ @@ -16,12 +16,12 @@
│ │ │  
│ │ │  o4 = Div(x)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
│ │ │  
│ │ │  i5 : F = divisor({3, 0, -2}, {ideal(x), ideal(y), ideal(x+y)})
│ │ │  
│ │ │ -o5 = -2*Div(x+y) + 3*Div(x) + 0*Div(y)
│ │ │ +o5 = 3*Div(x) + 0*Div(y) + -2*Div(x+y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__R__Weil__Divisor.out
│ │ │ @@ -1,32 +1,32 @@
│ │ │  -- -*- M2-comint -*- hash: 12819564349892123361
│ │ │  
│ │ │  i1 : R = ZZ/5[x,y];
│ │ │  
│ │ │  i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │  
│ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : E = (1/2)*D
│ │ │  
│ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
│ │ │  
│ │ │  i4 : F = toRWeilDivisor(D)
│ │ │  
│ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
│ │ │  
│ │ │  i5 : G = toRWeilDivisor(E)
│ │ │  
│ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
│ │ │  
│ │ │  i6 : F == 2*G
│ │ │  
│ │ │  o6 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_trim_lp__Basic__Divisor_rp.out
│ │ │ @@ -4,21 +4,21 @@
│ │ │  
│ │ │  i2 : D = divisor({1,0,-2}, {ideal(x, z), ideal(x-z,y-z), ideal(y+z, z)});
│ │ │  
│ │ │  o2 : WeilDivisor on R
│ │ │  
│ │ │  i3 : cleanSupport(D)
│ │ │  
│ │ │ -o3 = -2*Div(y+z, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + -2*Div(y+z, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
│ │ │  
│ │ │  i4 : trim(D)
│ │ │  
│ │ │ -o4 = -2*Div(z, y) + Div(z, x)
│ │ │ +o4 = Div(z, x) + -2*Div(z, y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
│ │ │  
│ │ │  i5 : D == trim(D)
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Basic__Divisor_sp_pl_sp__Basic__Divisor.html
│ │ │ @@ -85,42 +85,42 @@
│ │ │                
      i1 : R = QQ[x, y, z];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │  
│ │ │ -o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │ +o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │  
│ │ │  o3 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : D1 + D2
│ │ │  
│ │ │ -o4 = 6*Div(y) + -4*Div(x)
│ │ │ +o4 = -4*Div(x) + 6*Div(y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : D1 - D2
│ │ │  
│ │ │ -o5 = 4*Div(z) + 6*Div(x)
│ │ │ +o5 = 6*Div(x) + 4*Div(z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      We can also add or subtract divisors with different coefficients.
      
│ │ │ @@ -195,42 +195,42 @@
│ │ │                
      i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)})
│ │ │  
│ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z)
│ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i15 : -D
│ │ │  
│ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z)
│ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z)
│ │ │  
│ │ │  o15 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z)
│ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z)
│ │ │  
│ │ │  o16 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i17 : -E
│ │ │  
│ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z)
│ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z)
│ │ │  
│ │ │  o17 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,30 +16,30 @@
│ │ │ │      * Outputs:
│ │ │ │            o an instance of the type _B_a_s_i_c_D_i_v_i_s_o_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  We can add or subtract two divisors:
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │ │  
│ │ │ │ -o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │ │ +o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │ │  
│ │ │ │ -o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │ │ +o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : D1 + D2
│ │ │ │  
│ │ │ │ -o4 = 6*Div(y) + -4*Div(x)
│ │ │ │ +o4 = -4*Div(x) + 6*Div(y)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : D1 - D2
│ │ │ │  
│ │ │ │ -o5 = 4*Div(z) + 6*Div(x)
│ │ │ │ +o5 = 6*Div(x) + 4*Div(z)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  We can also add or subtract divisors with different coefficients.
│ │ │ │  i6 : R = QQ[x,y];
│ │ │ │  i7 : D1 = divisor({3, 1}, {ideal(x), ideal(y)})
│ │ │ │  
│ │ │ │  o7 = 3*Div(x) + Div(y)
│ │ │ │ @@ -70,30 +70,30 @@
│ │ │ │  o12 = -Div(y) + 2.75*Div(x)
│ │ │ │  
│ │ │ │  o12 : RWeilDivisor on R
│ │ │ │  Finally, we can negate a divisor.
│ │ │ │  i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z);
│ │ │ │  i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)})
│ │ │ │  
│ │ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z)
│ │ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : -D
│ │ │ │  
│ │ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z)
│ │ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z)
│ │ │ │  
│ │ │ │  o15 : WeilDivisor on R
│ │ │ │  i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z)
│ │ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z)
│ │ │ │  
│ │ │ │  o16 : WeilDivisor on R
│ │ │ │  i17 : -E
│ │ │ │  
│ │ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z)
│ │ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z)
│ │ │ │  
│ │ │ │  o17 : WeilDivisor on R
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _B_a_s_i_c_D_i_v_i_s_o_r_ _+_ _B_a_s_i_c_D_i_v_i_s_o_r -- add or subtract two divisors, or negate a
│ │ │ │        divisor
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Number_sp_st_sp__Basic__Divisor.html
│ │ │ @@ -108,24 +108,24 @@
│ │ │  o4 : RWeilDivisor on R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : 8*D
│ │ │  
│ │ │ -o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │ +o5 = 8*Div(y) + 16*Div(x) + -8*Div(x+y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i6 : (-2/3)*D
│ │ │  
│ │ │ -o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │ +o6 = -2/3*Div(y) + -4/3*Div(x) + 2/3*Div(x+y)
│ │ │  
│ │ │  o6 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i7 : 0.0*D
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,20 +27,20 @@
│ │ │ │  CoefficientType=>RR)
│ │ │ │  
│ │ │ │  o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2)
│ │ │ │  
│ │ │ │  o4 : RWeilDivisor on R
│ │ │ │  i5 : 8*D
│ │ │ │  
│ │ │ │ -o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x)
│ │ │ │ +o5 = 8*Div(y) + 16*Div(x) + -8*Div(x+y)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : (-2/3)*D
│ │ │ │  
│ │ │ │ -o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │ │ +o6 = -2/3*Div(y) + -4/3*Div(x) + 2/3*Div(x+y)
│ │ │ │  
│ │ │ │  o6 : QWeilDivisor on R
│ │ │ │  i7 : 0.0*D
│ │ │ │  
│ │ │ │  o7 = 0, the zero divisor
│ │ │ │  
│ │ │ │  o7 : RWeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___O__O_sp__R__Weil__Divisor.html
│ │ │ @@ -232,15 +232,15 @@
│ │ │                
      i16 : R = ZZ/11[x,y];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i17 : D = divisor(x*y/(x+y))
│ │ │  
│ │ │ -o17 = -Div(x+y) + Div(x) + Div(y)
│ │ │ +o17 = Div(x) + Div(y) + -Div(x+y)
│ │ │  
│ │ │  o17 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i18 : divisor(OO(D))
│ │ │ ├── html2text {}
│ │ │ │ @@ -100,15 +100,15 @@
│ │ │ │  Note that you can call the divisor constructor on the module you construct, but
│ │ │ │  it will only produce a divisor up to linear equivalence (which can mean
│ │ │ │  different things depending on whether or not you are keeping track of the
│ │ │ │  grading).
│ │ │ │  i16 : R = ZZ/11[x,y];
│ │ │ │  i17 : D = divisor(x*y/(x+y))
│ │ │ │  
│ │ │ │ -o17 = -Div(x+y) + Div(x) + Div(y)
│ │ │ │ +o17 = Div(x) + Div(y) + -Div(x+y)
│ │ │ │  
│ │ │ │  o17 : WeilDivisor on R
│ │ │ │  i18 : divisor(OO(D))
│ │ │ │  
│ │ │ │  o18 = 0, the zero divisor
│ │ │ │  
│ │ │ │  o18 : WeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_apply__To__Coefficients.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │                
      i1 : R = QQ[x, y, z];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D = divisor(x*y^2/z)
│ │ │  
│ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y)
│ │ │ +o2 = Div(x) + 2*Div(y) + -Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : applyToCoefficients(D, u->5*u)
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  the output D is the same as the class of the input D1 (WeilDivisor,
│ │ │ │  QWeilDivisor, RWeilDivisor, BasicDivisor). If Safe is set to true (the default
│ │ │ │  is false), then the function will check to make sure the output is a valid
│ │ │ │  divisor.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D = divisor(x*y^2/z)
│ │ │ │  
│ │ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y)
│ │ │ │ +o2 = Div(x) + 2*Div(y) + -Div(z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : applyToCoefficients(D, u->5*u)
│ │ │ │  
│ │ │ │  o3 = 5*Div(x) + 10*Div(y) + -5*Div(z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_ceiling_lp__R__Weil__Divisor_rp.html
│ │ │ @@ -83,24 +83,24 @@
│ │ │                
      i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : ceiling( D )
│ │ │  
│ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : floor( D )
│ │ │ @@ -110,15 +110,15 @@
│ │ │  o4 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │  
│ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i6 : ceiling( E )
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,30 +15,30 @@
│ │ │ │            o an instance of the type _W_e_i_l_D_i_v_i_s_o_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Start with a rational or real Weil divisor. We form a new divisor whose
│ │ │ │  coefficients are obtained by applying the ceiling or floor function to them.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x *y - z^2);
│ │ │ │  i2 : D = divisor({1/2, 4/3}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │ │  
│ │ │ │ -o2 = 4/3*Div(y, z) + 1/2*Div(x, z)
│ │ │ │ +o2 = 1/2*Div(x, z) + 4/3*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : ceiling( D )
│ │ │ │  
│ │ │ │ -o3 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o3 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : floor( D )
│ │ │ │  
│ │ │ │  o4 = Div(y, z)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : E = divisor({0.3, -0.7}, {ideal(x, z), ideal(y,z)}, CoefficientType => RR)
│ │ │ │  
│ │ │ │ -o5 = -.7*Div(y, z) + .3*Div(x, z)
│ │ │ │ +o5 = .3*Div(x, z) + -.7*Div(y, z)
│ │ │ │  
│ │ │ │  o5 : RWeilDivisor on R
│ │ │ │  i6 : ceiling( E )
│ │ │ │  
│ │ │ │  o6 = Div(x, z)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_divisor.html
│ │ │ @@ -100,51 +100,51 @@
│ │ │                
      i1 : R = QQ[x,y,z];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : E = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │  
│ │ │ -o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │  
│ │ │ -o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │  
│ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │ +o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │  
│ │ │  o6 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      Next we construct the same divisor in two different ways.  We are working on the quadric cone, and we are working with a divisor of a ruling of the cone.  This divisor is not Cartier, but 2 times it is.
      
│ │ │ @@ -209,15 +209,15 @@
│ │ │  o14 : WeilDivisor on A
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i15 : E = divisor(y2z)
│ │ │  
│ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │  
│ │ │  o15 : WeilDivisor on A
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      We can construct a Q-divisor as well.  Here are two ways to do it (we work in $A^2$ this time).
      
│ │ │ @@ -228,24 +228,24 @@
│ │ │                
      i16 : R = ZZ/7[x,y];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o17 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │  
│ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │  
│ │ │  o18 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      Or an R-divisor.  This time we work in the cone over $P^1 \times P^1$.
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,35 +43,35 @@
│ │ │ │  call it. In our first example, we construct divisors on $A^3$ (which can also
│ │ │ │  be viewed as divisors on $P^2$ since the ideals are homogeneous). The following
│ │ │ │  creates the same Weil divisor with coefficients 1, 2 and 3 in five different
│ │ │ │  ways.
│ │ │ │  i1 : R = QQ[x,y,z];
│ │ │ │  i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : E = divisor(x*y^2*z^3)
│ │ │ │  
│ │ │ │ -o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │ +o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │ │  
│ │ │ │ -o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │ +o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │ │  
│ │ │ │ -o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ │ +o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │ │  
│ │ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │ │ +o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ │ │  Next we construct the same divisor in two different ways. We are working on the
│ │ │ │  quadric cone, and we are working with a divisor of a ruling of the cone. This
│ │ │ │  divisor is not Cartier, but 2 times it is.
│ │ │ │  i7 : R = QQ[x,y,z]/ideal(x^2-y*z);
│ │ │ │  i8 : D = divisor({2}, {ideal(x,y)})
│ │ │ │ @@ -95,28 +95,28 @@
│ │ │ │  i14 : D = divisor(x3)
│ │ │ │  
│ │ │ │  o14 = 3*Div(xz2, xyz, xy2, x2z, x2y, x3)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on A
│ │ │ │  i15 : E = divisor(y2z)
│ │ │ │  
│ │ │ │ -o15 = 2*Div(yz2, y2z, y3, xyz, xy2, x2y) + Div(z3, yz2, y2z, xz2, xyz, x2z)
│ │ │ │ +o15 = Div(z3, yz2, y2z, xz2, xyz, x2z) + 2*Div(yz2, y2z, y3, xyz, xy2, x2y)
│ │ │ │  
│ │ │ │  o15 : WeilDivisor on A
│ │ │ │  We can construct a Q-divisor as well. Here are two ways to do it (we work in
│ │ │ │  $A^2$ this time).
│ │ │ │  i16 : R = ZZ/7[x,y];
│ │ │ │  i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o17 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ │ +o17 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ │  
│ │ │ │  o17 : QWeilDivisor on R
│ │ │ │  i18 : D = (-1/2)*divisor(y^2-x^3) + (2/1)*divisor(x)
│ │ │ │  
│ │ │ │ -o18 = -1/2*Div(-x^3+y^2) + 2*Div(x)
│ │ │ │ +o18 = 2*Div(x) + -1/2*Div(-x^3+y^2)
│ │ │ │  
│ │ │ │  o18 : QWeilDivisor on R
│ │ │ │  Or an R-divisor. This time we work in the cone over $P^1 \times P^1$.
│ │ │ │  i19 : R = ZZ/11[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i20 : D = divisor({1.1, -3.14159}, {ideal(x,u), ideal(x, v)},
│ │ │ │  CoefficientType=>RR)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html
│ │ │ @@ -168,43 +168,43 @@
│ │ │              
│ │ │                
      i11 : M = J*R^1;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0706325s (cpu); 0.0706342s (thread); 0s (gc)
│ │ │ + -- used 0.0555153s (cpu); 0.0555098s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc)
│ │ │ + -- used 0.488536s (cpu); 0.488295s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc)
      
│ │ │ + -- used 0.699966s (cpu); 0.596083s (thread); 0s (gc)
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc)
      
│ │ │ + -- used 0.003408s (cpu); 0.00341477s (thread); 0s (gc)
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc)
│ │ │ + -- used 0.00312895s (cpu); 0.00313705s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      For monomial ideals in toric rings, frequently ModuleStrategy appears faster.
      
│ │ │ @@ -228,23 +228,23 @@
│ │ │  
│ │ │  o19 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0637156s (cpu); 0.0637222s (thread); 0s (gc)
│ │ │ + -- used 0.103835s (cpu); 0.103795s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc)
│ │ │ + -- used 0.00859937s (cpu); 0.00854392s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      KnownDomain is an option for dualize.  If it is false (default is true), then the computer will first check whether the ring is a domain, if it is not then it will revert to ModuleStrategy.  If KnownDomain is set to true for a non-domain, then the function can return an incorrect answer.
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -60,43 +60,43 @@
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : J = m^9;
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : M = J*R^1;
│ │ │ │  i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.0706325s (cpu); 0.0706342s (thread); 0s (gc)
│ │ │ │ + -- used 0.0555153s (cpu); 0.0555098s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc)
│ │ │ │ + -- used 0.488536s (cpu); 0.488295s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc)
│ │ │ │ + -- used 0.699966s (cpu); 0.596083s (thread); 0s (gc)
│ │ │ │  i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc)
│ │ │ │ + -- used 0.003408s (cpu); 0.00341477s (thread); 0s (gc)
│ │ │ │  i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc)
│ │ │ │ + -- used 0.00312895s (cpu); 0.00313705s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  For monomial ideals in toric rings, frequently ModuleStrategy appears faster.
│ │ │ │  i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i18 : I = ideal(x,u);
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J = I^15;
│ │ │ │  
│ │ │ │  o19 : Ideal of R
│ │ │ │  i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.0637156s (cpu); 0.0637222s (thread); 0s (gc)
│ │ │ │ + -- used 0.103835s (cpu); 0.103795s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : Ideal of R
│ │ │ │  i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc)
│ │ │ │ + -- used 0.00859937s (cpu); 0.00854392s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 : Ideal of R
│ │ │ │  KnownDomain is an option for dualize. If it is false (default is true), then
│ │ │ │  the computer will first check whether the ring is a domain, if it is not then
│ │ │ │  it will revert to ModuleStrategy. If KnownDomain is set to true for a non-
│ │ │ │  domain, then the function can return an incorrect answer.
│ │ │ │  i22 : R = QQ[x,y]/ideal(x*y);
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_get__Prime__Divisors.html
│ │ │ @@ -81,33 +81,33 @@
│ │ │                
      i1 : R = QQ[x, y, z];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)})
│ │ │  
│ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z)
│ │ │ +o2 = 2*Div(y) + 0*Div(x^2+z) + -8*Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : getPrimeDivisors( D )
│ │ │  
│ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)}
│ │ │ +o3 = {Div(y), Div(x^2+z), Div(x)}
│ │ │  
│ │ │  o3 : List
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : getPrimeDivisors( cleanSupport D )
│ │ │  
│ │ │ -o4 = {Div(x), Div(y)}
│ │ │ +o4 = {Div(y), Div(x)}
│ │ │  
│ │ │  o4 : List
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,25 +14,25 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the list of prime divisors of a given divisor. The prime
│ │ │ │  divisors are all of the class WeilDivisor. If you do not call cleanSupport, you
│ │ │ │  may obtain divisors with zero coefficients.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)})
│ │ │ │  
│ │ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z)
│ │ │ │ +o2 = 2*Div(y) + 0*Div(x^2+z) + -8*Div(x)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : getPrimeDivisors( D )
│ │ │ │  
│ │ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)}
│ │ │ │ +o3 = {Div(y), Div(x^2+z), Div(x)}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : getPrimeDivisors( cleanSupport D )
│ │ │ │  
│ │ │ │ -o4 = {Div(x), Div(y)}
│ │ │ │ +o4 = {Div(y), Div(x)}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** WWaayyss ttoo uussee ggeettPPrriimmeeDDiivviissoorrss:: **********
│ │ │ │      * getPrimeDivisors(BasicDivisor)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _g_e_t_P_r_i_m_e_D_i_v_i_s_o_r_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Cartier.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │                
      i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o5 = 2*Div(y, z) + Div(x, z)
│ │ │ +o5 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i6 : isCartier( D )
│ │ │ @@ -186,15 +186,15 @@
│ │ │                
      i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i15 : isCartier(D, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  i3 : isCartier( D )
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  Neither is this divisor.
│ │ │ │  i4 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o5 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o5 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : isCartier( D )
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  Of course the next divisor is Cartier.
│ │ │ │  i7 : R = QQ[x, y, z];
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │  o11 : WeilDivisor on R
│ │ │ │  i12 : isCartier(D, IsGraded => true)
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │ │  i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o14 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o14 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isCartier(D, IsGraded => true)
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  The output value of this function is stored in the divisor's cache with the
│ │ │ │  value of the last IsGraded option. If you change the IsGraded option, the value
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Linear__Equivalent.html
│ │ │ @@ -86,24 +86,24 @@
│ │ │                
      i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : isLinearEquivalent(D1, D2)
│ │ │ @@ -121,24 +121,24 @@
│ │ │                
      i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │  
│ │ │  o6 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │  
│ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o7 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,36 +17,36 @@
│ │ │ │            o flag, a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Given two Weil divisors, this method checks whether they are linearly
│ │ │ │  equivalent.
│ │ │ │  i1 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │ │  i2 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ │ +o2 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │ │  
│ │ │ │ -o3 = 8*Div(y, z) + Div(x, z)
│ │ │ │ +o3 = Div(x, z) + 8*Div(y, z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : isLinearEquivalent(D1, D2)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  If IsGraded is set to true (by default it is false), then it treats the
│ │ │ │  divisors as divisors on the $Proj$ of their ambient ring.
│ │ │ │  i5 : R = QQ[x, y, z]/ ideal(x * y - z^2);
│ │ │ │  i6 : D1 = divisor({3, 8}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o6 = 3*Div(x, z) + 8*Div(y, z)
│ │ │ │ +o6 = 8*Div(y, z) + 3*Div(x, z)
│ │ │ │  
│ │ │ │  o6 : WeilDivisor on R
│ │ │ │  i7 : D2 = divisor({8, 1}, {ideal(y, z), ideal(x, z)})
│ │ │ │  
│ │ │ │ -o7 = Div(x, z) + 8*Div(y, z)
│ │ │ │ +o7 = 8*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o7 : WeilDivisor on R
│ │ │ │  i8 : isLinearEquivalent(D1, D2, IsGraded => true)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _O_O_ _R_W_e_i_l_D_i_v_i_s_o_r
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Q__Cartier.html
│ │ │ @@ -88,24 +88,24 @@
│ │ │                
      i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ +o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : isQCartier(10, D1)
│ │ │ @@ -169,24 +169,24 @@
│ │ │                
      i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │  
│ │ │  o12 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │  
│ │ │  o13 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i14 : isQCartier(10, D1, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,21 +21,21 @@
│ │ │ │  Check whether $m$ times a Weil or Q-divisor $D$ is Cartier for each $m$ from 1
│ │ │ │  to a fixed positive integer {\tt n1} (if the divisor is a QWeilDivisor, it can
│ │ │ │  search slightly higher than n1). If m * D1 is Cartier, it returns m. If it
│ │ │ │  fails to find an m, it returns 0.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o3 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ │ +o3 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : isQCartier(10, D1)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  i5 : isQCartier(10, D2)
│ │ │ │  
│ │ │ │ @@ -59,21 +59,21 @@
│ │ │ │  
│ │ │ │  o10 = 0
│ │ │ │  If the option IsGraded is set to true (by default it is false), then it treats
│ │ │ │  the divisor as a divisor on the $Proj$ of their ambient ring.
│ │ │ │  i11 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i12 : D1 = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o12 = 2*Div(y, z) + Div(x, z)
│ │ │ │ +o12 = Div(x, z) + 2*Div(y, z)
│ │ │ │  
│ │ │ │  o12 : WeilDivisor on R
│ │ │ │  i13 : D2 = divisor({1/2, 3/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o13 = 1/2*Div(y, z) + 3/4*Div(x, z)
│ │ │ │ +o13 = 3/4*Div(x, z) + 1/2*Div(y, z)
│ │ │ │  
│ │ │ │  o13 : QWeilDivisor on R
│ │ │ │  i14 : isQCartier(10, D1, IsGraded => true)
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : isQCartier(10, D2, IsGraded => true)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Q__Linear__Equivalent.html
│ │ │ @@ -87,24 +87,24 @@
│ │ │                
      i1 : R = QQ[x, y, z] / ideal(x * y - z^2);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : isQLinearEquivalent(10, D, E)
│ │ │ @@ -160,24 +160,24 @@
│ │ │                
      i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │  
│ │ │  o11 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │  
│ │ │  o12 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,20 +19,20 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Given two rational divisors, this method returns true if they linearly
│ │ │ │  equivalent after clearing denominators or if some further multiple up to n
│ │ │ │  makes them linearly equivalent. Otherwise it returns false.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │ │  i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │ │  
│ │ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │ │  
│ │ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : isQLinearEquivalent(10, D, E)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  In the above ring, every pair of divisors is Q-linearly equivalent because the
│ │ │ │  Weil divisor class group is isomorphic to Z/2. However, if we don't set n high
│ │ │ │ @@ -52,21 +52,21 @@
│ │ │ │  o9 = true
│ │ │ │  If IsGraded=>true (the default is false), then it treats the divisors as if
│ │ │ │  they are divisors on the $Proj$ of their ambient ring.
│ │ │ │  i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │ │  i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o11 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ │ +o11 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ │  
│ │ │ │  o11 : QWeilDivisor on R
│ │ │ │  i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType =>
│ │ │ │  QQ)
│ │ │ │  
│ │ │ │ -o12 = 3/2*Div(y, z) + -1/4*Div(x, z)
│ │ │ │ +o12 = -1/4*Div(x, z) + 3/2*Div(y, z)
│ │ │ │  
│ │ │ │  o12 : QWeilDivisor on R
│ │ │ │  i13 : isQLinearEquivalent(10, D, E, IsGraded => true)
│ │ │ │  
│ │ │ │  o13 = true
│ │ │ │  i14 : isQLinearEquivalent(10, 3*D, E, IsGraded => true)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__S__N__C.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │                
      i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : isSNC( D )
│ │ │ @@ -180,15 +180,15 @@
│ │ │                
      i13 : R = QQ[x, y];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i14 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o14 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i15 : isSNC( D, IsGraded => true )
│ │ │ @@ -203,15 +203,15 @@
│ │ │                
      i16 : R = QQ[x,y,z];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i17 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o17 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i18 : isSNC( D, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns true if the divisor is simple normal crossings, this
│ │ │ │  includes checking that the ambient ring is regular.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x, z) + -2*Div(y, z)
│ │ │ │ +o2 = -2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : isSNC( D )
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : R = QQ[x, y];
│ │ │ │  i5 : D = divisor(x*y*(x+y))
│ │ │ │ @@ -54,24 +54,24 @@
│ │ │ │  o11 : WeilDivisor on R
│ │ │ │  i12 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : R = QQ[x, y];
│ │ │ │  i14 : D = divisor(x*y*(x+y))
│ │ │ │  
│ │ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : R = QQ[x,y,z];
│ │ │ │  i17 : D = divisor(x*y*(x+y))
│ │ │ │  
│ │ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ │ +o17 = Div(x+y) + Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o17 : WeilDivisor on R
│ │ │ │  i18 : isSNC( D, IsGraded => true)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  The output value of this function is stored in the divisor's cache with the
│ │ │ │  value of the last IsGraded option. If you change the IsGraded option, the value
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Weil__Divisor.html
│ │ │ @@ -81,24 +81,24 @@
│ │ │                
      i1 : R = QQ[x, y, z];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o2 = Div(y) + -2*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + Div(y) + -2*Div(z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x)
│ │ │ +o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : isWeilDivisor( D1 )
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,21 +13,21 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Check if a rational/real divisor is a Weil divisor
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)},
│ │ │ │  CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o2 = Div(y) + -2*Div(z) + Div(x)
│ │ │ │ +o2 = Div(x) + Div(y) + -2*Div(z)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)},
│ │ │ │  CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x)
│ │ │ │ +o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : isWeilDivisor( D1 )
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : isWeilDivisor( D2 )
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html
│ │ │ @@ -155,24 +155,24 @@
│ │ │  o10 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ +o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │  
│ │ │  o11 : WeilDivisor on S
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ +o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │  
│ │ │  o12 : WeilDivisor on S
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      As illustrated by the previous example, the same functionality can also be accomplished by f^* (which creates a function which sends a divisor $D$ to $f^* D$).
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,20 +53,20 @@
│ │ │ │  
│ │ │ │  o9 : RingMap S <-- R
│ │ │ │  i10 : D = divisor(x*y*(x+y));
│ │ │ │  
│ │ │ │  o10 : WeilDivisor on R
│ │ │ │  i11 : D1 = pullback(f, D)
│ │ │ │  
│ │ │ │ -o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ │ +o11 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ │  
│ │ │ │  o11 : WeilDivisor on S
│ │ │ │  i12 : f^* D
│ │ │ │  
│ │ │ │ -o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ │ +o12 = Div(a+1) + 3*Div(b) + Div(a)
│ │ │ │  
│ │ │ │  o12 : WeilDivisor on S
│ │ │ │  As illustrated by the previous example, the same functionality can also be
│ │ │ │  accomplished by f^* (which creates a function which sends a divisor $D$ to $f^*
│ │ │ │  D$).
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _P_r_i_m_e_s -- a value for the option Strategy for the pullback method
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html
│ │ │ @@ -272,23 +272,23 @@
│ │ │  
│ │ │  o22 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i23 : time reflexify(J);
│ │ │ - -- used 0.338433s (cpu); 0.253168s (thread); 0s (gc)
│ │ │ + -- used 0.322708s (cpu); 0.232027s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc)
      
│ │ │ + -- used 0.558297s (cpu); 0.448756s (thread); 0s (gc)
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).
      
│ │ │          
      
│ │ │          
      
│ │ │ @@ -324,26 +324,26 @@
│ │ │              
│ │ │                
      i28 : M = J*R^1;
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.168829s (cpu); 0.168835s (thread); 0s (gc)
│ │ │ + -- used 0.156379s (cpu); 0.156381s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc)
│ │ │ + -- used 7.88167s (cpu); 5.94876s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │ @@ -353,21 +353,21 @@
│ │ │  
│ │ │  o31 = true
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc)
      
│ │ │ + -- used 7.80245s (cpu); 5.81405s (thread); 0s (gc)
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc)
      
│ │ │ + -- used 0.713827s (cpu); 0.477249s (thread); 0s (gc)
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
      
│ │ │          
      
│ │ │          
│ │ │ @@ -394,15 +394,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i37 : M = I^20*R^1;
      
│ │ │              
      
│ │ │                
      i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc)
│ │ │ + -- used 0.567643s (cpu); 0.33001s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -411,15 +411,15 @@
│ │ │  
│ │ │  o38 : Ideal of R
      
│ │ │              
      
│ │ │                
      i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.0135561s (cpu); 0.0135572s (thread); 0s (gc)
│ │ │ + -- used 0.0163127s (cpu); 0.0163172s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -428,21 +428,21 @@
│ │ │  
│ │ │  o39 : Ideal of R
      
│ │ │              
      
│ │ │                
      i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.0397151s (cpu); 0.0397234s (thread); 0s (gc)
      
│ │ │ + -- used 0.055363s (cpu); 0.0553724s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc)
      
│ │ │ + -- used 0.00812396s (cpu); 0.00813234s (thread); 0s (gc)
│ │ │              
      
│ │ │          
      
│ │ │            
      For ideals, if KnownDomain is false (default value is true), then the function will check whether it is a domain.  If it is a domain (or assumed to be a domain), it will reflexify using a strategy which can speed up computation, if not it will compute using a sometimes slower method which is essentially reflexifying it as a module.
      
│ │ │          
      
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  i21 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o21 : Ideal of R
│ │ │ │  i22 : J = I^21;
│ │ │ │  
│ │ │ │  o22 : Ideal of R
│ │ │ │  i23 : time reflexify(J);
│ │ │ │ - -- used 0.338433s (cpu); 0.253168s (thread); 0s (gc)
│ │ │ │ + -- used 0.322708s (cpu); 0.232027s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time reflexify(J*R^1);
│ │ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc)
│ │ │ │ + -- used 0.558297s (cpu); 0.448756s (thread); 0s (gc)
│ │ │ │  Because of this, there are two strategies for computing a reflexification (at
│ │ │ │  least if the module embeds as an ideal).
│ │ │ │  IdealStrategy. In the case that $R$ is a domain, and our module is isomorphic
│ │ │ │  to an ideal $I$, then one can compute the reflexification by computing colons.
│ │ │ │  ModuleStrategy. This computes the reflexification simply by computing $Hom$
│ │ │ │  twice.
│ │ │ │  ModuleStrategy is the default strategy for modules, IdealStrategy is the
│ │ │ │ @@ -139,73 +139,73 @@
│ │ │ │  
│ │ │ │  o26 : Ideal of R
│ │ │ │  i27 : J = I^20;
│ │ │ │  
│ │ │ │  o27 : Ideal of R
│ │ │ │  i28 : M = J*R^1;
│ │ │ │  i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ │ - -- used 0.168829s (cpu); 0.168835s (thread); 0s (gc)
│ │ │ │ + -- used 0.156379s (cpu); 0.156381s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc)
│ │ │ │ + -- used 7.88167s (cpu); 5.94876s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o30 : Ideal of R
│ │ │ │  i31 : J1 == J2
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc)
│ │ │ │ + -- used 7.80245s (cpu); 5.81405s (thread); 0s (gc)
│ │ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc)
│ │ │ │ + -- used 0.713827s (cpu); 0.477249s (thread); 0s (gc)
│ │ │ │  However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
│ │ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i35 : I = ideal(x,u);
│ │ │ │  
│ │ │ │  o35 : Ideal of R
│ │ │ │  i36 : J = I^20;
│ │ │ │  
│ │ │ │  o36 : Ideal of R
│ │ │ │  i37 : M = I^20*R^1;
│ │ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc)
│ │ │ │ + -- used 0.567643s (cpu); 0.33001s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12
│ │ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2
│ │ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         19    20
│ │ │ │        x  u, x  )
│ │ │ │  
│ │ │ │  o38 : Ideal of R
│ │ │ │  i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ │ - -- used 0.0135561s (cpu); 0.0135572s (thread); 0s (gc)
│ │ │ │ + -- used 0.0163127s (cpu); 0.0163172s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12
│ │ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2
│ │ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         19    20
│ │ │ │        x  u, x  )
│ │ │ │  
│ │ │ │  o39 : Ideal of R
│ │ │ │  i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ │ - -- used 0.0397151s (cpu); 0.0397234s (thread); 0s (gc)
│ │ │ │ + -- used 0.055363s (cpu); 0.0553724s (thread); 0s (gc)
│ │ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc)
│ │ │ │ + -- used 0.00812396s (cpu); 0.00813234s (thread); 0s (gc)
│ │ │ │  For ideals, if KnownDomain is false (default value is true), then the function
│ │ │ │  will check whether it is a domain. If it is a domain (or assumed to be a
│ │ │ │  domain), it will reflexify using a strategy which can speed up computation, if
│ │ │ │  not it will compute using a sometimes slower method which is essentially
│ │ │ │  reflexifying it as a module.
│ │ │ │  Consider the following example showing the importance of making the correct
│ │ │ │  assumption about the ring being a domain.
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html
│ │ │ @@ -129,30 +129,30 @@
│ │ │  
│ │ │  o6 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0292441s (cpu); 0.0292465s (thread); 0s (gc)
│ │ │ + -- used 0.0442378s (cpu); 0.0442393s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc)
│ │ │ + -- used 0.161455s (cpu); 0.161455s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i10 : J20a == J20b
│ │ │ @@ -176,23 +176,23 @@
│ │ │  
│ │ │  o12 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0299195s (cpu); 0.0299242s (thread); 0s (gc)
│ │ │ + -- used 0.0408921s (cpu); 0.0406934s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc)
│ │ │ + -- used 0.0734406s (cpu); 0.0734493s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i15 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,39 +40,39 @@
│ │ │ │  of the generators of $I$. Consider the example of a cone over a point on an
│ │ │ │  elliptic curve.
│ │ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ │ - -- used 0.0292441s (cpu); 0.0292465s (thread); 0s (gc)
│ │ │ │ + -- used 0.0442378s (cpu); 0.0442393s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc)
│ │ │ │ + -- used 0.161455s (cpu); 0.161455s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : J20a == J20b
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  This passes the Strategy option to a reflexify call. Valid options are
│ │ │ │  IdealStrategy and ModuleStrategy.
│ │ │ │  i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │ │  i12 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.0299195s (cpu); 0.0299242s (thread); 0s (gc)
│ │ │ │ + -- used 0.0408921s (cpu); 0.0406934s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc)
│ │ │ │ + -- used 0.0734406s (cpu); 0.0734493s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : J1 == J2
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_f_l_e_x_i_f_y -- calculate the double dual of an ideal or module Hom(Hom(M,
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_ring_lp__Basic__Divisor_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │                
      i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : ring( D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _r_i_n_g,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the ambient ring of a divisor.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2 );
│ │ │ │  i2 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
│ │ │ │  
│ │ │ │ -o2 = Div(x, z) + 2*Div(y, z)
│ │ │ │ +o2 = 2*Div(y, z) + Div(x, z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : ring( D )
│ │ │ │  
│ │ │ │  o3 = R
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__Q__Weil__Divisor.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │  o4 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : F = divisor({3, 0, -2}, {ideal(x), ideal(y), ideal(x+y)})
│ │ │  
│ │ │ -o5 = -2*Div(x+y) + 3*Div(x) + 0*Div(y)
│ │ │ +o5 = 3*Div(x) + 0*Div(y) + -2*Div(x+y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  i4 : toQWeilDivisor(E)
│ │ │ │  
│ │ │ │  o4 = Div(x)
│ │ │ │  
│ │ │ │  o4 : QWeilDivisor on R
│ │ │ │  i5 : F = divisor({3, 0, -2}, {ideal(x), ideal(y), ideal(x+y)})
│ │ │ │  
│ │ │ │ -o5 = -2*Div(x+y) + 3*Div(x) + 0*Div(y)
│ │ │ │ +o5 = 3*Div(x) + 0*Div(y) + -2*Div(x+y)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_o_W_e_i_l_D_i_v_i_s_o_r -- create a Weil divisor from a Q or R-divisor
│ │ │ │      * _t_o_R_W_e_i_l_D_i_v_i_s_o_r -- create a R-divisor from a Q or Weil divisor
│ │ │ │  ********** WWaayyss ttoo uussee ttooQQWWeeiillDDiivviissoorr:: **********
│ │ │ │      * toQWeilDivisor(QWeilDivisor)
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_to__R__Weil__Divisor.html
│ │ │ @@ -84,42 +84,42 @@
│ │ │                
      i1 : R = ZZ/5[x,y];
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │  
│ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : E = (1/2)*D
│ │ │  
│ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : F = toRWeilDivisor(D)
│ │ │  
│ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : G = toRWeilDivisor(E)
│ │ │  
│ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i6 : F == 2*G
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,30 +15,30 @@
│ │ │ │            o an instance of the type _R_W_e_i_l_D_i_v_i_s_o_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Turn a Weil divisor or a Q-divisor into a R-divisor (or do nothing to a R-
│ │ │ │  divisor).
│ │ │ │  i1 : R = ZZ/5[x,y];
│ │ │ │  i2 : D = divisor({2, 0, -4}, {ideal(x), ideal(y), ideal(x-y)})
│ │ │ │  
│ │ │ │ -o2 = -4*Div(x-y) + 2*Div(x) + 0*Div(y)
│ │ │ │ +o2 = 2*Div(x) + 0*Div(y) + -4*Div(x-y)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : E = (1/2)*D
│ │ │ │  
│ │ │ │ -o3 = -2*Div(x-y) + Div(x)
│ │ │ │ +o3 = Div(x) + -2*Div(x-y)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : F = toRWeilDivisor(D)
│ │ │ │  
│ │ │ │ -o4 = -4*Div(x-y) + 2*Div(x)
│ │ │ │ +o4 = 2*Div(x) + -4*Div(x-y)
│ │ │ │  
│ │ │ │  o4 : RWeilDivisor on R
│ │ │ │  i5 : G = toRWeilDivisor(E)
│ │ │ │  
│ │ │ │ -o5 = -2*Div(x-y) + Div(x)
│ │ │ │ +o5 = Div(x) + -2*Div(x-y)
│ │ │ │  
│ │ │ │  o5 : RWeilDivisor on R
│ │ │ │  i6 : F == 2*G
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_o_W_e_i_l_D_i_v_i_s_o_r -- create a Weil divisor from a Q or R-divisor
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_trim_lp__Basic__Divisor_rp.html
│ │ │ @@ -93,24 +93,24 @@
│ │ │  o2 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i3 : cleanSupport(D)
│ │ │  
│ │ │ -o3 = -2*Div(y+z, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + -2*Div(y+z, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : trim(D)
│ │ │  
│ │ │ -o4 = -2*Div(z, y) + Div(z, x)
│ │ │ +o4 = Div(z, x) + -2*Div(z, y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : D == trim(D)
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,20 +19,20 @@
│ │ │ │  removed and where the ideals displayed to the user are trimmed.
│ │ │ │  i1 : R = QQ[x,y,z]/ideal(x*y-z^2);
│ │ │ │  i2 : D = divisor({1,0,-2}, {ideal(x, z), ideal(x-z,y-z), ideal(y+z, z)});
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : cleanSupport(D)
│ │ │ │  
│ │ │ │ -o3 = -2*Div(y+z, z) + Div(x, z)
│ │ │ │ +o3 = Div(x, z) + -2*Div(y+z, z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : trim(D)
│ │ │ │  
│ │ │ │ -o4 = -2*Div(z, y) + Div(z, x)
│ │ │ │ +o4 = Div(z, x) + -2*Div(z, y)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : D == trim(D)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _t_r_i_m_(_B_a_s_i_c_D_i_v_i_s_o_r_) -- trims the ideals displayed to the user and removes
│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  ZXh0cmFjdFZhcnNBbGdlYnJh
│ │ │  #:len=1106
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidW5kZXJseWluZyBwb2x5bm9taWFsIHJp
│ │ │  bmcgaW4gdGhlIG9yZGluYXJ5IHZhcmlhYmxlcyBvZiBhIFdleWwgYWxnZWJyYSIsICJsaW5lbnVt
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  V2V5bEdyb3VwRWxlbWVudCAqIFJvb3Q=
│ │ │  #:len=994
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgYW4gZWxlbWVudCBvZiBhIFdl
│ │ │  eWwgZ3JvdXAgdG8gYSByb290IiwgImxpbmVudW0iID0+IDI5NjMsIElucHV0cyA9PiB7U1BBTntU
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_above__Bruhat_lp__Basic__List_rp.out
│ │ │ @@ -36,34 +36,34 @@
│ │ │                                             |  2 |
│ │ │                                             | -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : aboveBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ -                                             |  3 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, |  1
│ │ │ +                                             |  1 |        |  2 |       |  1
│ │ │ +                                             |  2 |        | -1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ -     |                                            |  1 |        |  1 |      
│ │ │ -     |                                            | -2 |        |  1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            |  3 |        | -1 |      
│ │ │ +     |                                            | -1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |},
│ │ │ -     |  2 |                                            | -2 |        | -1 |  
│ │ │ -     | -1 |                                            |  1 |        |  2 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |},
│ │ │ +     | -1 |                                            | -2 |        |  1 |  
│ │ │ +     |  0 |                                            |  3 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ -         |  1 |                                            |  2 |        | -1
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1
│ │ │ +         | -1 |                                            |  1 |        |  1
│ │ │ +         |  0 |                                            | -2 |        |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0,
│ │ │ -     |       |  1 |                                            |  3 |       
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ +     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1,
│ │ │ +     |       |  2 |                                            | -2 |       
│ │ │ +     |       | -1 |                                            |  1 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, |  2 |}}}}
│ │ │ -     | -1 |       | -1 |
│ │ │ -     |  2 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_hasse__Diagram__To__Graph_lp__Hasse__Diagram_rp.out
│ │ │ @@ -20,26 +20,26 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o3 : WeylGroupElement
│ │ │  
│ │ │  i4 : myInterval=intervalBruhat(w1,w2)
│ │ │  
│ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  2 |       |  1 |       | 0 |       |  1 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        |  1 |       | -1 |       | -1 |                                              | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | -1 |       | -1 |       | 1 |       |  1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  1 |       |  0 |       |  2 |                                              |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | 1 |}, {2, | -1 |}, {3, | -1 |}, {4, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | -1 |}, {1, |  0 |}, {4, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {2, |  0 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  0 |}, {3, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, |  0 |}, {3, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{2, |  0 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        | 0 |       |  1 |       |  2 |       |  1 |                                            |  2 |        |  2 |       | -1 |       | 0 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | -1 |       | 0 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | 1 |       |  1 |       | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |       | 1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        |  2 |       | 1 |                                            | -2 |        |  1 |       | -1 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                              |  2 |
│ │ │  
│ │ │  o4 : HasseDiagram
│ │ │  
│ │ │  i5 : hasseDiagramToGraph(myInterval)
│ │ │  
│ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 1}, {, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 4}}}, {, {{, 0}, {, 2}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, {{, 0}, {, 2}}}, {, {{, 0}, {, 1}}}, {, {{, 1}, {, 3}}}, {, {{, 2}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │  
│ │ │  o5 : HasseGraph
│ │ │  
│ │ │  i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │  
│ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232, 0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{12321, 1}, {232, 2}, {2, 3}, {121, 4}}}, {1232, {{2, 0}, {3, 1}, {12321, 4}}}, {1213, {{1, 0}, {3, 2}, {232, 3}}}}, {{123, {{3, 0}, {12321, 3}}}, {132, {{232, 0}, {121, 2}}}, {121, {{1, 0}, {2, 1}}}, {213, {{3, 1}, {1, 3}}}, {232, {{3, 2}, {2, 3}}}}, {{12, {{121, 0}}}, {21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}}, {{2, {}}}}
│ │ │  
│ │ │  o6 : HasseGraph
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.out
│ │ │ @@ -26,30 +26,30 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        |  1 |       | -1 |                                            |  3 |        |  0 |       | -1 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            | -2 |        |  1 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1
│ │ │                                               |  3 |        |  1 |       |  1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ +                                             | -2 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1,
│ │ │ +     |                                            | -1 |        | -1 |      
│ │ │ +     |                                            |  3 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ +     | -1 |}}}}
│ │ │ +     |  2 |
│ │ │       | -1 |
│ │ │ -     |  0 |
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Right__Coset_cm__Weyl__Group__Right__Coset_rp.out
│ │ │ @@ -26,30 +26,30 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            |  1 |        | 0 |       | -1 |                                              | -2 |        | -1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            |  1 |        | 1 |       |  2 |                                              |  3 |        |  0 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, {1, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  2 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | 0 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        | 1 |       |  2 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  0 |       |  2 |                                              |  1 |        |  2 |                                            |  2 |        | -1 |                                            |  3 |        |  0 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1
│ │ │                                               | -3 |        |  1 |       |  2
│ │ │                                               |  1 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1,
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1,
│ │ │       |                                            | -1 |        | -1 |      
│ │ │ -     |                                            |  2 |        |  2 |      
│ │ │ +     |                                            |  2 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, |  2 |}}}}
│ │ │ -     |  1 |       | -1 |
│ │ │ -     |  1 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_rp.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330744940387
│ │ │  
│ │ │  i1 : positiveRoots(rootSystemA(3))
│ │ │  
│ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ +o1 = set {|  2 |, |  0 |, | -1 |, | -1 |, | 1 |, |  1 |}
│ │ │ +          | -1 |  | -1 |  |  1 |  |  2 |  | 0 |  |  1 |
│ │ │ +          |  0 |  |  2 |  |  1 |  | -1 |  | 1 |  | -1 |
│ │ │  
│ │ │  o1 : Set
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_under__Bruhat_lp__Basic__List_rp.out
│ │ │ @@ -36,34 +36,34 @@
│ │ │                                             | -2 |
│ │ │                                             | -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : underBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             |  2 |        |  1 |       | -1
│ │ │ -                                             | -1 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |}, {2, | -1
│ │ │ +                                             |  2 |        | -1 |       |  1
│ │ │ +                                             | -3 |        |  2 |       |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  2 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            | -3 |        | -1 |      
│ │ │ +     |                                            |  1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |},
│ │ │ -     |  1 |                                            |  2 |        | -1 |  
│ │ │ -     | -1 |                                            | -3 |        |  2 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |},
│ │ │ +     | -1 |                                            | -1 |        |  1 |  
│ │ │ +     |  0 |                                            | -2 |        |  1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0
│ │ │ -         |  1 |                                            | -3 |        | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ +     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1
│ │ │ +         |  2 |                                            |  2 |        |  1
│ │ │ +         | -1 |                                            | -1 |        | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ +     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0,
│ │ │       |       | -1 |                                            | -1 |       
│ │ │ -     |       |  0 |                                            | -2 |       
│ │ │ +     |       |  0 |                                            |  2 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ -     |  1 |       |  2 |
│ │ │ -     |  1 |       | -1 |
│ │ │ +     | -1 |}, {1, |  1 |}}}}
│ │ │ +     |  2 |       |  1 |
│ │ │ +     | -1 |       | -1 |
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_above__Bruhat_lp__Basic__List_rp.html
│ │ │ @@ -121,37 +121,37 @@
│ │ │  o3 : List
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : aboveBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ -                                             |  3 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, |  1
│ │ │ +                                             |  1 |        |  2 |       |  1
│ │ │ +                                             |  2 |        | -1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ -     |                                            |  1 |        |  1 |      
│ │ │ -     |                                            | -2 |        |  1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            |  3 |        | -1 |      
│ │ │ +     |                                            | -1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |},
│ │ │ -     |  2 |                                            | -2 |        | -1 |  
│ │ │ -     | -1 |                                            |  1 |        |  2 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |},
│ │ │ +     | -1 |                                            | -2 |        |  1 |  
│ │ │ +     |  0 |                                            |  3 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ -         |  1 |                                            |  2 |        | -1
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1
│ │ │ +         | -1 |                                            |  1 |        |  1
│ │ │ +         |  0 |                                            | -2 |        |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0,
│ │ │ -     |       |  1 |                                            |  3 |       
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ +     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1,
│ │ │ +     |       |  2 |                                            | -2 |       
│ │ │ +     |       | -1 |                                            |  1 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, |  2 |}}}}
│ │ │ -     | -1 |       | -1 |
│ │ │ -     |  2 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,37 +48,37 @@
│ │ │ │       WeylGroupElement{RootSystem{...8...}, |  1 |}}
│ │ │ │                                             |  2 |
│ │ │ │                                             | -1 |
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : aboveBruhat(L1)
│ │ │ │  
│ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |}, {1, |  2
│ │ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ │ -                                             |  3 |        | -1 |       |  0
│ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, |  1
│ │ │ │ +                                             |  1 |        |  2 |       |  1
│ │ │ │ +                                             |  2 |        | -1 |       | -1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ │ -     |                                            |  1 |        |  1 |
│ │ │ │ -     |                                            | -2 |        |  1 |
│ │ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {2,
│ │ │ │ +     |                                            |  3 |        | -1 |
│ │ │ │ +     |                                            | -1 |        |  2 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |},
│ │ │ │ -     |  2 |                                            | -2 |        | -1 |
│ │ │ │ -     | -1 |                                            |  1 |        |  2 |
│ │ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |},
│ │ │ │ +     | -1 |                                            | -2 |        |  1 |
│ │ │ │ +     |  0 |                                            |  3 |        | -1 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ │ -         |  1 |                                            |  2 |        | -1
│ │ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1
│ │ │ │ +         | -1 |                                            |  1 |        |  1
│ │ │ │ +         |  0 |                                            | -2 |        |  1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0,
│ │ │ │ -     |       |  1 |                                            |  3 |
│ │ │ │ -     |       | -1 |                                            | -1 |
│ │ │ │ +     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1,
│ │ │ │ +     |       |  2 |                                            | -2 |
│ │ │ │ +     |       | -1 |                                            |  1 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |  0 |}, {2, |  2 |}}}}
│ │ │ │ -     | -1 |       | -1 |
│ │ │ │ -     |  2 |       |  0 |
│ │ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ │ +     | -1 |       |  1 |
│ │ │ │ +     |  2 |       |  1 |
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _a_b_o_v_e_B_r_u_h_a_t_(_B_a_s_i_c_L_i_s_t_) -- The Weyl group elements just under the ones in
│ │ │ │        the list for the Bruhat order
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_hasse__Diagram__To__Graph_lp__Hasse__Diagram_rp.html
│ │ │ @@ -112,40 +112,40 @@
│ │ │  o3 : WeylGroupElement
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : myInterval=intervalBruhat(w1,w2)
│ │ │  
│ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  2 |       |  1 |       | 0 |       |  1 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        |  1 |       | -1 |       | -1 |                                              | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | -1 |       | -1 |       | 1 |       |  1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  1 |       |  0 |       |  2 |                                              |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | 1 |}, {2, | -1 |}, {3, | -1 |}, {4, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | -1 |}, {1, |  0 |}, {4, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {2, |  0 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  0 |}, {3, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, |  0 |}, {3, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{2, |  0 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        | 0 |       |  1 |       |  2 |       |  1 |                                            |  2 |        |  2 |       | -1 |       | 0 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | -1 |       | 0 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | 1 |       |  1 |       | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |       | 1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        |  2 |       | 1 |                                            | -2 |        |  1 |       | -1 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                              |  2 |
│ │ │  
│ │ │  o4 : HasseDiagram
      
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : hasseDiagramToGraph(myInterval)
│ │ │  
│ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 1}, {, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 4}}}, {, {{, 0}, {, 2}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, {{, 0}, {, 2}}}, {, {{, 0}, {, 1}}}, {, {{, 1}, {, 3}}}, {, {{, 2}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │  
│ │ │  o5 : HasseGraph
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      It is also possible to ask for reduced decompositions as labels by changing the option "labels" as below.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │  
│ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232, 0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{12321, 1}, {232, 2}, {2, 3}, {121, 4}}}, {1232, {{2, 0}, {3, 1}, {12321, 4}}}, {1213, {{1, 0}, {3, 2}, {232, 3}}}}, {{123, {{3, 0}, {12321, 3}}}, {132, {{232, 0}, {121, 2}}}, {121, {{1, 0}, {2, 1}}}, {213, {{3, 1}, {1, 3}}}, {232, {{3, 2}, {2, 3}}}}, {{12, {{121, 0}}}, {21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}}, {{2, {}}}}
│ │ │  
│ │ │  o6 : HasseGraph
      
│ │ │              
      
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,72 +35,73 @@
│ │ │ │                                             |  1 |
│ │ │ │  
│ │ │ │  o3 : WeylGroupElement
│ │ │ │  i4 : myInterval=intervalBruhat(w1,w2)
│ │ │ │  
│ │ │ │  o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0
│ │ │ │  |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1
│ │ │ │ -|}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement
│ │ │ │ -{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}},
│ │ │ │ -{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4,
│ │ │ │ -|  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2,
│ │ │ │ -|  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, |
│ │ │ │ --1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0
│ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -
│ │ │ │ -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2
│ │ │ │ -|}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}},
│ │ │ │ +|}, {{1, | 1 |}, {2, | -1 |}, {3, | -1 |}, {4, |  1 |}}}, {WeylGroupElement
│ │ │ │ +{RootSystem{...8...}, | -3 |}, {{0, | -1 |}, {1, |  0 |}, {4, | 1 |}}},
│ │ │ │ +{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {2, |  0 |}, {3,
│ │ │ │ +| -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  0 |}, {3,
│ │ │ │ +| 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |
│ │ │ │ +1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -
│ │ │ │ +1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, |  0 |}, {3, |  2
│ │ │ │ +|}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{2, |  0 |}, {3, | -
│ │ │ │ +1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}},
│ │ │ │ +{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}},
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}},
│ │ │ │ -{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}},
│ │ │ │ -{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {
│ │ │ │ +{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ │                                                            | -2 |        | -1 |
│ │ │ │  |  1 |       |  2 |                                              | -3 |
│ │ │ │ -|  2 |       |  1 |       | 0 |       |  1 |
│ │ │ │ -|  2 |        | 0 |       |  2 |       | -1 |
│ │ │ │ -| -1 |        |  1 |       | -1 |       | -1 |
│ │ │ │ -| -3 |        | -1 |       | -1 |                                            |
│ │ │ │ --1 |        | -1 |       |  2 |                                            |  1
│ │ │ │ -|        | 0 |       | -1 |                                            |  3 |
│ │ │ │ -|  1 |       |  1 |                                            | -1 |        |
│ │ │ │ -2 |       | -1 |                                              | -2 |        | -
│ │ │ │ +| 0 |       |  1 |       |  2 |       |  1 |
│ │ │ │ +|  2 |        |  2 |       | -1 |       | 0 |
│ │ │ │ +| -1 |        | -1 |       | -1 |       |  1 |
│ │ │ │ +|  1 |        | -1 |       | 0 |                                            |
│ │ │ │ +3 |        |  1 |       |  1 |                                            | -
│ │ │ │ +1 |        | -1 |       |  2 |                                            | -
│ │ │ │ +3 |        | -1 |       | -1 |                                            | -
│ │ │ │ +1 |        | -1 |       |  2 |                                              |
│ │ │ │ +1 |        |  1 |                                            | -2 |        | -
│ │ │ │  1 |                                            |  1 |        |  1 |
│ │ │ │ -| -2 |        | -1 |                                            |  1 |        |
│ │ │ │ -1 |                                              | -1 |
│ │ │ │ +| -2 |        | -1 |                                              | -1 |
│ │ │ │                                                            |  1 |        |  2 |
│ │ │ │  | -1 |       | -1 |                                              |  1 |
│ │ │ │ -| -1 |       | -1 |       | 1 |       |  1 |
│ │ │ │ -| -1 |        | 1 |       | -1 |       |  2 |
│ │ │ │ -|  2 |        |  1 |       |  0 |       |  2 |
│ │ │ │ -|  2 |        |  2 |       |  0 |                                            |
│ │ │ │ --1 |        |  2 |       | -1 |                                            |  1
│ │ │ │ -|        | 1 |       |  2 |                                            | -2 |
│ │ │ │ -| -1 |       |  1 |                                            |  3 |        |
│ │ │ │ --1 |       |  0 |                                              |  3 |        |
│ │ │ │ -0 |                                            | -2 |        |  1 |
│ │ │ │ -|  1 |        |  2 |                                            |  2 |        |
│ │ │ │ --1 |                                              |  2 |
│ │ │ │ +| 1 |       |  1 |       | -1 |       | -1 |
│ │ │ │ +| -1 |        | -1 |       |  2 |       | 1 |
│ │ │ │ +|  2 |        |  0 |       |  2 |       |  1 |
│ │ │ │ +|  1 |        |  2 |       | 1 |                                            | -
│ │ │ │ +2 |        |  1 |       | -1 |                                            |  3
│ │ │ │ +|        |  0 |       | -1 |                                            |  2 |
│ │ │ │ +|  2 |       |  0 |                                            | -1 |        |
│ │ │ │ +2 |       | -1 |                                              |  2 |        | -
│ │ │ │ +1 |                                            |  3 |        |  0 |
│ │ │ │ +| -2 |        |  1 |                                            |  1 |        |
│ │ │ │ +2 |                                              |  2 |
│ │ │ │  
│ │ │ │  o4 : HasseDiagram
│ │ │ │  i5 : hasseDiagramToGraph(myInterval)
│ │ │ │  
│ │ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}},
│ │ │ │ -{, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {,
│ │ │ │ -{{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {
│ │ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 1}, {, 2}, {, 3}, {, 4}}},
│ │ │ │ +{, {{, 0}, {, 1}, {, 4}}}, {, {{, 0}, {, 2}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {,
│ │ │ │ +{{, 0}, {, 2}}}, {, {{, 0}, {, 1}}}, {, {{, 1}, {, 3}}}, {, {{, 2}, {, 3}}}}, {
│ │ │ │  {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ │  
│ │ │ │  o5 : HasseGraph
│ │ │ │  It is also possible to ask for reduced decompositions as labels by changing the
│ │ │ │  option "labels" as below.
│ │ │ │  i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │ │  
│ │ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121,
│ │ │ │ -1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232,
│ │ │ │ -0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123,
│ │ │ │ -{{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {
│ │ │ │ -{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{12321, 1},
│ │ │ │ +{232, 2}, {2, 3}, {121, 4}}}, {1232, {{2, 0}, {3, 1}, {12321, 4}}}, {1213, {{1,
│ │ │ │ +0}, {3, 2}, {232, 3}}}}, {{123, {{3, 0}, {12321, 3}}}, {132, {{232, 0}, {121,
│ │ │ │ +2}}}, {121, {{1, 0}, {2, 1}}}, {213, {{3, 1}, {1, 3}}}, {232, {{3, 2}, {2,
│ │ │ │ +3}}}}, {{12, {{121, 0}}}, {21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}}, {
│ │ │ │ +{2, {}}}}
│ │ │ │  
│ │ │ │  o6 : HasseGraph
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _h_a_s_s_e_D_i_a_g_r_a_m_T_o_G_r_a_p_h_(_H_a_s_s_e_D_i_a_g_r_a_m_) -- turning a hasse diagram into a graph
│ │ │ │        (intended for graphic representation)
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.html
│ │ │ @@ -115,41 +115,41 @@
│ │ │  o4 : WeylGroupElement
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        | -1 |       |  2 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        |  1 |       | -1 |                                            |  3 |        |  0 |       | -1 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            | -2 |        |  1 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      Each row of the Hasse diagram contains the elements of a certain length together with their links to the next row.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1
│ │ │                                               |  3 |        |  1 |       |  1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ +                                             | -2 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1,
│ │ │ +     |                                            | -1 |        | -1 |      
│ │ │ +     |                                            |  3 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ +     | -1 |}}}}
│ │ │ +     |  2 |
│ │ │       | -1 |
│ │ │ -     |  0 |
│ │ │  
│ │ │  o6 : List
      
│ │ │              
      
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,49 +41,49 @@
│ │ │ │                                             | -2 |
│ │ │ │                                             |  1 |
│ │ │ │  
│ │ │ │  o4 : WeylGroupElement
│ │ │ │  i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │ │  
│ │ │ │  o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |},
│ │ │ │ -{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |},
│ │ │ │ -{2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |},
│ │ │ │ -{2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2
│ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}},
│ │ │ │ -{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {
│ │ │ │ +{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |},
│ │ │ │ +{2, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |},
│ │ │ │ +{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1
│ │ │ │ +|}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}},
│ │ │ │ +{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ │                                                            | -3 |        | 0 |
│ │ │ │  |  1 |                                              |  3 |        |  1 |
│ │ │ │ -|  1 |                                            | -1 |        |  2 |       |
│ │ │ │ --1 |                                              | -2 |        | -1 |
│ │ │ │ -|  1 |        |  1 |                                            |  1 |        |
│ │ │ │ +|  1 |                                            | -1 |        | -1 |       |
│ │ │ │ +2 |                                              |  1 |        |  1 |
│ │ │ │ +| -2 |        | -1 |                                            |  1 |        |
│ │ │ │  1 |                                              | -1 |
│ │ │ │                                                            |  1 |        | 1 |
│ │ │ │ -|  1 |                                              | -2 |        | -1 |
│ │ │ │ -|  1 |                                            |  3 |        | -1 |       |
│ │ │ │ -0 |                                              |  3 |        |  0 |
│ │ │ │ -| -2 |        |  1 |                                            |  2 |        |
│ │ │ │ --1 |                                              |  2 |
│ │ │ │ +|  1 |                                              | -2 |        |  1 |
│ │ │ │ +| -1 |                                            |  3 |        |  0 |       |
│ │ │ │ +-1 |                                              |  2 |        | -1 |
│ │ │ │ +|  3 |        |  0 |                                            | -2 |        |
│ │ │ │ +1 |                                              |  2 |
│ │ │ │  
│ │ │ │  o5 : HasseDiagram
│ │ │ │  Each row of the Hasse diagram contains the elements of a certain length
│ │ │ │  together with their links to the next row.
│ │ │ │  i6 : myInterval#1
│ │ │ │  
│ │ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  1
│ │ │ │                                               |  3 |        |  1 |       |  1
│ │ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ │ +                                             | -2 |        |  1 |       | -1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ │ -     |                                            | -1 |        |  2 |
│ │ │ │ -     |                                            |  3 |        | -1 |
│ │ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1,
│ │ │ │ +     |                                            | -1 |        | -1 |
│ │ │ │ +     |                                            |  3 |        |  0 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |  2 |}}}}
│ │ │ │ +     | -1 |}}}}
│ │ │ │ +     |  2 |
│ │ │ │       | -1 |
│ │ │ │ -     |  0 |
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _i_n_t_e_r_v_a_l_B_r_u_h_a_t_(_W_e_y_l_G_r_o_u_p_L_e_f_t_C_o_s_e_t_,_W_e_y_l_G_r_o_u_p_L_e_f_t_C_o_s_e_t_) -- elements between
│ │ │ │        two given ones for the Bruhat order on a quotient of a Weyl group
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_interval__Bruhat_lp__Weyl__Group__Right__Coset_cm__Weyl__Group__Right__Coset_rp.html
│ │ │ @@ -115,41 +115,41 @@
│ │ │  o4 : WeylGroupElement
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            |  1 |        | 0 |       | -1 |                                              | -2 |        | -1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            |  1 |        | 1 |       |  2 |                                              |  3 |        |  0 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, {1, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  2 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  2 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | 0 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        | 1 |       |  2 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  0 |       |  2 |                                              |  1 |        |  2 |                                            |  2 |        | -1 |                                            |  3 |        |  0 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
      
│ │ │              
│ │ │            
│ │ │          
│ │ │          
      
│ │ │            
      Each row of the Hasse diagram contains the elements of a certain length together with their links to the next row.
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1
│ │ │                                               | -3 |        |  1 |       |  2
│ │ │                                               |  1 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1,
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1,
│ │ │       |                                            | -1 |        | -1 |      
│ │ │ -     |                                            |  2 |        |  2 |      
│ │ │ +     |                                            |  2 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, |  2 |}}}}
│ │ │ -     |  1 |       | -1 |
│ │ │ -     |  1 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o6 : List
      
│ │ │              
      
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,58 +42,58 @@
│ │ │ │                                             | -2 |
│ │ │ │                                             |  1 |
│ │ │ │  
│ │ │ │  o4 : WeylGroupElement
│ │ │ │  i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │ │  
│ │ │ │  o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0
│ │ │ │ -|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -
│ │ │ │ -1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0
│ │ │ │ -|}, {1, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -
│ │ │ │ -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2
│ │ │ │ -|}, {{0, |  0 |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -
│ │ │ │ -3 |}, {{1, | 1 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1
│ │ │ │ -|}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0
│ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {
│ │ │ │ +|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -
│ │ │ │ +1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2
│ │ │ │ +|}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -
│ │ │ │ +3 |}, {{0, | 1 |}, {1, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -
│ │ │ │ +1 |}, {{1, |  2 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2
│ │ │ │ +|}, {{0, |  2 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  3
│ │ │ │ +|}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1
│ │ │ │ +|}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ │                                                            | -2 |        | -1 |
│ │ │ │  |  2 |                                              | -3 |        |  1 |
│ │ │ │  |  2 |                                            | -1 |        | -1 |       |
│ │ │ │ -1 |       | -1 |                                              | -1 |        |
│ │ │ │ -2 |       | -1 |                                            | -3 |        | -
│ │ │ │ -1 |       | -1 |                                            |  1 |        | 0 |
│ │ │ │ -| -1 |                                              | -2 |        | -1 |
│ │ │ │ -| -2 |        | -1 |                                            |  1 |        |
│ │ │ │ -1 |                                              | -1 |
│ │ │ │ +-1 |       |  1 |                                              |  1 |        |
│ │ │ │ +0 |       | -1 |                                            | -1 |        | -
│ │ │ │ +1 |       |  2 |                                            | -3 |        | -
│ │ │ │ +1 |       | -1 |                                              | -2 |        | -
│ │ │ │ +1 |                                            |  1 |        |  1 |
│ │ │ │ +| -2 |        | -1 |                                              | -1 |
│ │ │ │                                                            |  1 |        |  2 |
│ │ │ │  | -1 |                                              |  1 |        |  1 |
│ │ │ │ -| -1 |                                            |  2 |        |  2 |       |
│ │ │ │ -1 |       |  0 |                                              |  3 |        | -
│ │ │ │ -1 |       |  0 |                                            |  2 |        |  2
│ │ │ │ -|       |  0 |                                            |  1 |        | 1 |
│ │ │ │ -|  2 |                                              |  3 |        |  0 |
│ │ │ │ -|  1 |        |  2 |                                            |  2 |        |
│ │ │ │ --1 |                                              |  2 |
│ │ │ │ +| -1 |                                            |  2 |        |  0 |       |
│ │ │ │ +2 |       |  1 |                                              |  1 |        | 1
│ │ │ │ +|       |  2 |                                            |  3 |        |  0 |
│ │ │ │ +| -1 |                                            |  2 |        |  0 |       |
│ │ │ │ +2 |                                              |  1 |        |  2 |
│ │ │ │ +|  2 |        | -1 |                                            |  3 |        |
│ │ │ │ +0 |                                              |  2 |
│ │ │ │  
│ │ │ │  o5 : HasseDiagram
│ │ │ │  Each row of the Hasse diagram contains the elements of a certain length
│ │ │ │  together with their links to the next row.
│ │ │ │  i6 : myInterval#1
│ │ │ │  
│ │ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1
│ │ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1
│ │ │ │                                               | -3 |        |  1 |       |  2
│ │ │ │                                               |  1 |        |  1 |       | -1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1,
│ │ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1,
│ │ │ │       |                                            | -1 |        | -1 |
│ │ │ │ -     |                                            |  2 |        |  2 |
│ │ │ │ +     |                                            |  2 |        |  0 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     | -1 |}, {2, |  2 |}}}}
│ │ │ │ -     |  1 |       | -1 |
│ │ │ │ -     |  1 |       |  0 |
│ │ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ │ +     | -1 |       |  1 |
│ │ │ │ +     |  2 |       |  1 |
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _i_n_t_e_r_v_a_l_B_r_u_h_a_t_(_W_e_y_l_G_r_o_u_p_R_i_g_h_t_C_o_s_e_t_,_W_e_y_l_G_r_o_u_p_R_i_g_h_t_C_o_s_e_t_) -- elements
│ │ │ │        between two given ones for the Bruhat order on a quotient of a Weyl group
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_positive__Roots_lp__Root__System_rp.html
│ │ │ @@ -74,17 +74,17 @@
│ │ │        
      
│ │ │          
      Description
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : positiveRoots(rootSystemA(3))
│ │ │  
│ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ +o1 = set {|  2 |, |  0 |, | -1 |, | -1 |, | 1 |, |  1 |}
│ │ │ +          | -1 |  | -1 |  |  1 |  |  2 |  | 0 |  |  1 |
│ │ │ +          |  0 |  |  2 |  |  1 |  | -1 |  | 1 |  | -1 |
│ │ │  
│ │ │  o1 : Set
      
│ │ │              
      
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,17 +10,17 @@
│ │ │ │      * Inputs:
│ │ │ │            o R, an instance of the type _R_o_o_t_S_y_s_t_e_m,
│ │ │ │      * Outputs:
│ │ │ │            o a _s_e_t, of all positive roots of R
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : positiveRoots(rootSystemA(3))
│ │ │ │  
│ │ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ │ +o1 = set {|  2 |, |  0 |, | -1 |, | -1 |, | 1 |, |  1 |}
│ │ │ │ +          | -1 |  | -1 |  |  1 |  |  2 |  | 0 |  |  1 |
│ │ │ │ +          |  0 |  |  2 |  |  1 |  | -1 |  | 1 |  | -1 |
│ │ │ │  
│ │ │ │  o1 : Set
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _p_o_s_i_t_i_v_e_R_o_o_t_s_(_R_o_o_t_S_y_s_t_e_m_) -- the set of all positive roots
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/WeylGroups.m2:2604:0.
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_under__Bruhat_lp__Basic__List_rp.html
│ │ │ @@ -121,37 +121,37 @@
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
      i4 : underBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             |  2 |        |  1 |       | -1
│ │ │ -                                             | -1 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |}, {2, | -1
│ │ │ +                                             |  2 |        | -1 |       |  1
│ │ │ +                                             | -3 |        |  2 |       |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  2 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            | -3 |        | -1 |      
│ │ │ +     |                                            |  1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |},
│ │ │ -     |  1 |                                            |  2 |        | -1 |  
│ │ │ -     | -1 |                                            | -3 |        |  2 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |},
│ │ │ +     | -1 |                                            | -1 |        |  1 |  
│ │ │ +     |  0 |                                            | -2 |        |  1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0
│ │ │ -         |  1 |                                            | -3 |        | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ +     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1
│ │ │ +         |  2 |                                            |  2 |        |  1
│ │ │ +         | -1 |                                            | -1 |        | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ +     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0,
│ │ │       |       | -1 |                                            | -1 |       
│ │ │ -     |       |  0 |                                            | -2 |       
│ │ │ +     |       |  0 |                                            |  2 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ -     |  1 |       |  2 |
│ │ │ -     |  1 |       | -1 |
│ │ │ +     | -1 |}, {1, |  1 |}}}}
│ │ │ +     |  2 |       |  1 |
│ │ │ +     | -1 |       | -1 |
│ │ │  
│ │ │  o4 : List
      
│ │ │              
│ │ │            
│ │ │          
│ │ │        
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,37 +48,37 @@
│ │ │ │       WeylGroupElement{RootSystem{...8...}, |  1 |}}
│ │ │ │                                             | -2 |
│ │ │ │                                             | -1 |
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : underBruhat(L1)
│ │ │ │  
│ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |}, {1, |  2
│ │ │ │ -                                             |  2 |        |  1 |       | -1
│ │ │ │ -                                             | -1 |        | -1 |       |  0
│ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |}, {2, | -1
│ │ │ │ +                                             |  2 |        | -1 |       |  1
│ │ │ │ +                                             | -3 |        |  2 |       |  1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ │ -     |                                            | -1 |        |  2 |
│ │ │ │ -     |                                            |  2 |        | -1 |
│ │ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2,
│ │ │ │ +     |                                            | -3 |        | -1 |
│ │ │ │ +     |                                            |  1 |        |  2 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |},
│ │ │ │ -     |  1 |                                            |  2 |        | -1 |
│ │ │ │ -     | -1 |                                            | -3 |        |  2 |
│ │ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |},
│ │ │ │ +     | -1 |                                            | -1 |        |  1 |
│ │ │ │ +     |  0 |                                            | -2 |        |  1 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0
│ │ │ │ -         |  1 |                                            | -3 |        | -1
│ │ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ │ +     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1
│ │ │ │ +         |  2 |                                            |  2 |        |  1
│ │ │ │ +         | -1 |                                            | -1 |        | -1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ │ +     |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0,
│ │ │ │       |       | -1 |                                            | -1 |
│ │ │ │ -     |       |  0 |                                            | -2 |
│ │ │ │ +     |       |  0 |                                            |  2 |
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ │ -     |  1 |       |  2 |
│ │ │ │ -     |  1 |       | -1 |
│ │ │ │ +     | -1 |}, {1, |  1 |}}}}
│ │ │ │ +     |  2 |       |  1 |
│ │ │ │ +     | -1 |       | -1 |
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _u_n_d_e_r_B_r_u_h_a_t_(_B_a_s_i_c_L_i_s_t_) -- Weyl group elements just under the ones in the
│ │ │ │        list for the Bruhat order
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  Y29ub3JtYWw=
│ │ │  #:len=1038
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNvbm9ybWFsIHZh
│ │ │  cmlldHkiLCAibGluZW51bSIgPT4gMTY2NCwgSW5wdXRzID0+IHtTUEFOe1RUeyJJIn0sIiwgIixT
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out
│ │ │ @@ -122,15 +122,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc)
│ │ │ + -- used 2.78156s (cpu); 1.51876s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : peek last ms
│ │ │  
│ │ │ @@ -142,15 +142,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 4.97937s (cpu); 2.77946s (thread); 0s (gc)
│ │ │ + -- used 7.89343s (cpu); 3.58368s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
│ │ │  
│ │ │  i26 : peek last ms
│ │ │  
│ │ │ @@ -162,15 +162,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 5.88833s (cpu); 3.18288s (thread); 0s (gc)
│ │ │ + -- used 9.68655s (cpu); 4.12208s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : peek last ms
│ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html
│ │ │ @@ -297,15 +297,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc)
│ │ │ + -- used 2.78156s (cpu); 1.51876s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o23 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -323,15 +323,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 4.97937s (cpu); 2.77946s (thread); 0s (gc)
│ │ │ + -- used 7.89343s (cpu); 3.58368s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o25 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -349,15 +349,15 @@
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  for j2....
│ │ │  loop over components of JY=ideal 1
│ │ │  loop over components of JY=ideal 1
│ │ │ - -- used 5.88833s (cpu); 3.18288s (thread); 0s (gc)
│ │ │ + -- used 9.68655s (cpu); 4.12208s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │  
│ │ │  o27 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -184,15 +184,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc)
│ │ │ │ + -- used 2.78156s (cpu); 1.51876s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : peek last ms
│ │ │ │  
│ │ │ │  o24 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │ │ @@ -202,15 +202,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 4.97937s (cpu); 2.77946s (thread); 0s (gc)
│ │ │ │ + -- used 7.89343s (cpu); 3.58368s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o25 : List
│ │ │ │  i26 : peek last ms
│ │ │ │  
│ │ │ │  o26 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ │ │ @@ -220,15 +220,15 @@
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  for j2....
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │  loop over components of JY=ideal 1
│ │ │ │ - -- used 5.88833s (cpu); 3.18288s (thread); 0s (gc)
│ │ │ │ + -- used 9.68655s (cpu); 4.12208s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : peek last ms
│ │ │ │  
│ │ │ │  o28 = MutableHashTable{0 => {ideal (P, M1)}                    }
│ │ ├── ./usr/share/doc/Macaulay2/WittVectors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=50
│ │ │  ZmluZEZyb2Jlbml1c0xpZnRDb25zdHJhaW50cyguLi4sSG9tb2dlbmVvdXM9Pi4uLik=
│ │ │  #:len=354
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEyOCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEZyb2Jlbml1c0xpZnRDb25zdHJhaW50cyxI
│ │ ├── ./usr/share/doc/Macaulay2/XML/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  Z2V0QXR0cmlidXRlcyhMaWJ4bWxOb2RlKQ==
│ │ │  #:len=465
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2V0IHRoZSBsaXN0IG9mIGF0dHJpYnV0
│ │ │  ZXMgb2YgYW4gWE1MIG5vZGUiLCBEZXNjcmlwdGlvbiA9PiAxOihUQUJMRXsiY2xhc3MiID0+ICJl
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  Z2ZhblBvbHlub21pYWxTZXRVbmlvbihMaXN0LExpc3Qp
│ │ │  #:len=309
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzY0Mywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2ZhblBvbHlub21pYWxTZXRVbmlvbixMaXN0LExp
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/example-output/___Installation_spand_sp__Configuration_spof_spgfan__Interface.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  i4 : prefixDirectory | currentLayout#"programs"
│ │ │  
│ │ │  o4 = /usr/x86_64-Linux-
│ │ │       Debian-forky/libexec/Macaulay2/bin/
│ │ │  
│ │ │  i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16342-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20465-0/172
│ │ │  This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │ @@ -38,16 +38,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174
│ │ │ +using temporary file /tmp/M2-20465-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20465-0/174
│ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read.
│ │ │  
│ │ │  -r:
│ │ │   Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used.
│ │ │ @@ -56,69 +56,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16342-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176
│ │ │ +using temporary file /tmp/M2-20465-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20465-0/176
│ │ │  This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false.
│ │ │  Options:
│ │ │  --remainder:
│ │ │   Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
│ │ │  --multiplier:
│ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.
│ │ │ -using temporary file /tmp/M2-16342-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178
│ │ │ +using temporary file /tmp/M2-20465-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20465-0/178
│ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │  --stable:
│ │ │   Compute the stable intersection.
│ │ │ -using temporary file /tmp/M2-16342-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180
│ │ │ +using temporary file /tmp/M2-20465-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20465-0/180
│ │ │  This program takes a polyhedral fan and a vector and computes the link of the polyhedral fan around that vertex. The link will have lineality space dimension equal to the dimension of the relative open polyhedral cone of the original fan containing the vector.
│ │ │  Options:
│ │ │  -i value:
│ │ │   Specify the name of the input file.
│ │ │  --symmetry:
│ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input.
│ │ │  
│ │ │  --star:
│ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector.
│ │ │ -using temporary file /tmp/M2-16342-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182
│ │ │ +using temporary file /tmp/M2-20465-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20465-0/182
│ │ │  This program takes two polyhedral fans and computes their product.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │ -using temporary file /tmp/M2-16342-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184
│ │ │ +using temporary file /tmp/M2-20465-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20465-0/184
│ │ │  This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option --pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant.
│ │ │  --pair:
│ │ │   The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual.
│ │ │  --asfan:
│ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
│ │ │  --vectorinput:
│ │ │   Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations.
│ │ │ -using temporary file /tmp/M2-16342-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186
│ │ │ +using temporary file /tmp/M2-20465-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20465-0/186
│ │ │  This program computes the homogeneity space of a list of polynomials - as a cone. Thus generators for the homogeneity space are found in the section LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first compute a set of homogeneous generators and call the program on these. A reduced Groebner basis will always suffice for this purpose.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188
│ │ │ +using temporary file /tmp/M2-20465-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20465-0/188
│ │ │  This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -126,30 +126,30 @@
│ │ │  -i:
│ │ │   Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
│ │ │  -w:
│ │ │   Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-16342-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190
│ │ │ +using temporary file /tmp/M2-20465-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20465-0/190
│ │ │  This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
│ │ │  Options:
│ │ │  --ideal:
│ │ │   Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program.
│ │ │  
│ │ │  --pair:
│ │ │   Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal.
│ │ │  
│ │ │  --mark:
│ │ │   If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector.
│ │ │  --list:
│ │ │   Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.
│ │ │ -using temporary file /tmp/M2-16342-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192
│ │ │ +using temporary file /tmp/M2-20465-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20465-0/192
│ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -164,57 +164,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-16342-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194
│ │ │ +using temporary file /tmp/M2-20465-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20465-0/194
│ │ │  This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196
│ │ │ +using temporary file /tmp/M2-20465-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20465-0/196
│ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ -using temporary file /tmp/M2-16342-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198
│ │ │ +using temporary file /tmp/M2-20465-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20465-0/198
│ │ │  This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice.
│ │ │  Options:
│ │ │  -t:
│ │ │   Compute the toric ideal of the matrix whose rows are given on the input instead.
│ │ │  --convert:
│ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ -using temporary file /tmp/M2-16342-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200
│ │ │ +using temporary file /tmp/M2-20465-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20465-0/200
│ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │  Options:
│ │ │  -m:
│ │ │   Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202
│ │ │ +using temporary file /tmp/M2-20465-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20465-0/202
│ │ │  This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204
│ │ │ +using temporary file /tmp/M2-20465-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20465-0/204
│ │ │  This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │  
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --nocones:
│ │ │   Tell the program to not list cones in the output.
│ │ │ -using temporary file /tmp/M2-16342-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206
│ │ │ +using temporary file /tmp/M2-20465-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20465-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -229,16 +229,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-16342-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208
│ │ │ +using temporary file /tmp/M2-20465-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20465-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -251,47 +251,47 @@
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │  --resume:
│ │ │   Continue enumeration from saved state...
│ │ │ -using temporary file /tmp/M2-16342-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210
│ │ │ +using temporary file /tmp/M2-20465-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20465-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │  --twosetswithrings:
│ │ │   Switch to mode where input is: ring polynomialset (same)ring polynomialset.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212
│ │ │ +using temporary file /tmp/M2-20465-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20465-0/212
│ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables.
│ │ │  Options:
│ │ │  -L:
│ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │  --shiftVariables value:
│ │ │   Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0.
│ │ │ -using temporary file /tmp/M2-16342-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214
│ │ │ +using temporary file /tmp/M2-20465-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20465-0/214
│ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables.
│ │ │  Options:
│ │ │  -m:
│ │ │   Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn.
│ │ │  
│ │ │  -d value:
│ │ │   Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8.
│ │ │  
│ │ │  -w value:
│ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216
│ │ │ +using temporary file /tmp/M2-20465-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20465-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -304,25 +304,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218
│ │ │ +using temporary file /tmp/M2-20465-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20465-0/218
│ │ │  This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │  Options:
│ │ │  -h:
│ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous generators).
│ │ │  
│ │ │  --noideal:
│ │ │   Do not treat input as an ideal but just factor out common monomial factors of the input polynomials.
│ │ │ -using temporary file /tmp/M2-16342-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220
│ │ │ +using temporary file /tmp/M2-20465-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20465-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -331,72 +331,72 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222
│ │ │ +using temporary file /tmp/M2-20465-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20465-0/222
│ │ │  This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224
│ │ │ +using temporary file /tmp/M2-20465-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20465-0/224
│ │ │  This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │  Output:
│ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226
│ │ │ +using temporary file /tmp/M2-20465-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20465-0/226
│ │ │  This program converts ASCII math to TeX math. The data-type is specified by the options.
│ │ │  Options:
│ │ │  -h:
│ │ │   Add a header to the output. Using this option the output will be LaTeXable right away.
│ │ │  --polynomialset_:
│ │ │   The data to be converted is a list of polynomials.
│ │ │  --polynomialsetlist_:
│ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ -using temporary file /tmp/M2-16342-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228
│ │ │ +using temporary file /tmp/M2-20465-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20465-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --generatedcones:
│ │ │   Instead of a list of reduced Groebner bases, read in a list of closed generated cones given in modern format (like _cone). In this case --restrict is ignored.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230
│ │ │ +using temporary file /tmp/M2-20465-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20465-0/230
│ │ │  This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used.
│ │ │  Options:
│ │ │  -h:
│ │ │   Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous.
│ │ │ -using temporary file /tmp/M2-16342-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232
│ │ │ +using temporary file /tmp/M2-20465-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20465-0/232
│ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234
│ │ │ +using temporary file /tmp/M2-20465-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20465-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236
│ │ │ +using temporary file /tmp/M2-20465-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20465-0/236
│ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options:
│ │ │  --exponents:
│ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ -using temporary file /tmp/M2-16342-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238
│ │ │ +using temporary file /tmp/M2-20465-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20465-0/238
│ │ │  This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options:
│ │ │ -using temporary file /tmp/M2-16342-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240
│ │ │ +using temporary file /tmp/M2-20465-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20465-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -408,16 +408,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16342-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242
│ │ │ +using temporary file /tmp/M2-20465-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20465-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -442,48 +442,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-16342-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244
│ │ │ +using temporary file /tmp/M2-20465-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20465-0/244
│ │ │  This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input.
│ │ │  Options:
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │   Specify n.
│ │ │  --trees:
│ │ │   list the boundary trees (assumes d=3)
│ │ │ -using temporary file /tmp/M2-16342-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246
│ │ │ +using temporary file /tmp/M2-20465-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20465-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248
│ │ │ +using temporary file /tmp/M2-20465-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20465-0/248
│ │ │  This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM.
│ │ │  Options:
│ │ │  --kapranov:
│ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │  --determinant:
│ │ │   Compute the tropical determinant instead.
│ │ │ -using temporary file /tmp/M2-16342-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250
│ │ │ +using temporary file /tmp/M2-20465-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20465-0/250
│ │ │  This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │  -d:
│ │ │   Output dimension information to standard error.
│ │ │  --stable:
│ │ │   Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │ -using temporary file /tmp/M2-16342-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252
│ │ │ +using temporary file /tmp/M2-20465-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20465-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -491,24 +491,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254
│ │ │ +using temporary file /tmp/M2-20465-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20465-0/254
│ │ │  This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │  -i2 value:
│ │ │   Specify the name of the Polymake input file containing the piecewise linear function.
│ │ │ -using temporary file /tmp/M2-16342-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256
│ │ │ +using temporary file /tmp/M2-20465-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20465-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -528,21 +528,21 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/256
│ │ │ +using temporary file /tmp/M2-20465-0/256
│ │ │  
│ │ │  i6 : QQ[x,y];
│ │ │  
│ │ │  i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20465-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16342-0/258
│ │ │ +using temporary file /tmp/M2-20465-0/258
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │            
      
│ │ │          
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16342-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20465-0/172
│ │ │  This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │ @@ -133,16 +133,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174
│ │ │ +using temporary file /tmp/M2-20465-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20465-0/174
│ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read.
│ │ │  
│ │ │  -r:
│ │ │   Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used.
│ │ │ @@ -151,69 +151,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16342-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176
│ │ │ +using temporary file /tmp/M2-20465-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20465-0/176
│ │ │  This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false.
│ │ │  Options:
│ │ │  --remainder:
│ │ │   Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
│ │ │  --multiplier:
│ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.
│ │ │ -using temporary file /tmp/M2-16342-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178
│ │ │ +using temporary file /tmp/M2-20465-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20465-0/178
│ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │  --stable:
│ │ │   Compute the stable intersection.
│ │ │ -using temporary file /tmp/M2-16342-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180
│ │ │ +using temporary file /tmp/M2-20465-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20465-0/180
│ │ │  This program takes a polyhedral fan and a vector and computes the link of the polyhedral fan around that vertex. The link will have lineality space dimension equal to the dimension of the relative open polyhedral cone of the original fan containing the vector.
│ │ │  Options:
│ │ │  -i value:
│ │ │   Specify the name of the input file.
│ │ │  --symmetry:
│ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input.
│ │ │  
│ │ │  --star:
│ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector.
│ │ │ -using temporary file /tmp/M2-16342-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182
│ │ │ +using temporary file /tmp/M2-20465-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20465-0/182
│ │ │  This program takes two polyhedral fans and computes their product.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │ -using temporary file /tmp/M2-16342-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184
│ │ │ +using temporary file /tmp/M2-20465-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20465-0/184
│ │ │  This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option --pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant.
│ │ │  --pair:
│ │ │   The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual.
│ │ │  --asfan:
│ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
│ │ │  --vectorinput:
│ │ │   Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations.
│ │ │ -using temporary file /tmp/M2-16342-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186
│ │ │ +using temporary file /tmp/M2-20465-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20465-0/186
│ │ │  This program computes the homogeneity space of a list of polynomials - as a cone. Thus generators for the homogeneity space are found in the section LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first compute a set of homogeneous generators and call the program on these. A reduced Groebner basis will always suffice for this purpose.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188
│ │ │ +using temporary file /tmp/M2-20465-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20465-0/188
│ │ │  This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -221,30 +221,30 @@
│ │ │  -i:
│ │ │   Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
│ │ │  -w:
│ │ │   Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-16342-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190
│ │ │ +using temporary file /tmp/M2-20465-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20465-0/190
│ │ │  This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
│ │ │  Options:
│ │ │  --ideal:
│ │ │   Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program.
│ │ │  
│ │ │  --pair:
│ │ │   Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal.
│ │ │  
│ │ │  --mark:
│ │ │   If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector.
│ │ │  --list:
│ │ │   Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.
│ │ │ -using temporary file /tmp/M2-16342-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192
│ │ │ +using temporary file /tmp/M2-20465-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20465-0/192
│ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -259,57 +259,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-16342-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194
│ │ │ +using temporary file /tmp/M2-20465-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20465-0/194
│ │ │  This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196
│ │ │ +using temporary file /tmp/M2-20465-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20465-0/196
│ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ -using temporary file /tmp/M2-16342-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198
│ │ │ +using temporary file /tmp/M2-20465-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20465-0/198
│ │ │  This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice.
│ │ │  Options:
│ │ │  -t:
│ │ │   Compute the toric ideal of the matrix whose rows are given on the input instead.
│ │ │  --convert:
│ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ -using temporary file /tmp/M2-16342-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200
│ │ │ +using temporary file /tmp/M2-20465-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20465-0/200
│ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │  Options:
│ │ │  -m:
│ │ │   Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202
│ │ │ +using temporary file /tmp/M2-20465-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20465-0/202
│ │ │  This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204
│ │ │ +using temporary file /tmp/M2-20465-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20465-0/204
│ │ │  This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │  
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --nocones:
│ │ │   Tell the program to not list cones in the output.
│ │ │ -using temporary file /tmp/M2-16342-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206
│ │ │ +using temporary file /tmp/M2-20465-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20465-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -324,16 +324,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-16342-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208
│ │ │ +using temporary file /tmp/M2-20465-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20465-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -346,47 +346,47 @@
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │  --resume:
│ │ │   Continue enumeration from saved state...
│ │ │ -using temporary file /tmp/M2-16342-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210
│ │ │ +using temporary file /tmp/M2-20465-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20465-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │  --twosetswithrings:
│ │ │   Switch to mode where input is: ring polynomialset (same)ring polynomialset.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212
│ │ │ +using temporary file /tmp/M2-20465-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20465-0/212
│ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables.
│ │ │  Options:
│ │ │  -L:
│ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │  --shiftVariables value:
│ │ │   Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0.
│ │ │ -using temporary file /tmp/M2-16342-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214
│ │ │ +using temporary file /tmp/M2-20465-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20465-0/214
│ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables.
│ │ │  Options:
│ │ │  -m:
│ │ │   Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn.
│ │ │  
│ │ │  -d value:
│ │ │   Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8.
│ │ │  
│ │ │  -w value:
│ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216
│ │ │ +using temporary file /tmp/M2-20465-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20465-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -399,25 +399,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218
│ │ │ +using temporary file /tmp/M2-20465-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20465-0/218
│ │ │  This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │  Options:
│ │ │  -h:
│ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous generators).
│ │ │  
│ │ │  --noideal:
│ │ │   Do not treat input as an ideal but just factor out common monomial factors of the input polynomials.
│ │ │ -using temporary file /tmp/M2-16342-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220
│ │ │ +using temporary file /tmp/M2-20465-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20465-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -426,72 +426,72 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222
│ │ │ +using temporary file /tmp/M2-20465-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20465-0/222
│ │ │  This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224
│ │ │ +using temporary file /tmp/M2-20465-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20465-0/224
│ │ │  This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │  Output:
│ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226
│ │ │ +using temporary file /tmp/M2-20465-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20465-0/226
│ │ │  This program converts ASCII math to TeX math. The data-type is specified by the options.
│ │ │  Options:
│ │ │  -h:
│ │ │   Add a header to the output. Using this option the output will be LaTeXable right away.
│ │ │  --polynomialset_:
│ │ │   The data to be converted is a list of polynomials.
│ │ │  --polynomialsetlist_:
│ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ -using temporary file /tmp/M2-16342-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228
│ │ │ +using temporary file /tmp/M2-20465-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20465-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --generatedcones:
│ │ │   Instead of a list of reduced Groebner bases, read in a list of closed generated cones given in modern format (like _cone). In this case --restrict is ignored.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230
│ │ │ +using temporary file /tmp/M2-20465-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20465-0/230
│ │ │  This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used.
│ │ │  Options:
│ │ │  -h:
│ │ │   Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous.
│ │ │ -using temporary file /tmp/M2-16342-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232
│ │ │ +using temporary file /tmp/M2-20465-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20465-0/232
│ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234
│ │ │ +using temporary file /tmp/M2-20465-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20465-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236
│ │ │ +using temporary file /tmp/M2-20465-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20465-0/236
│ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options:
│ │ │  --exponents:
│ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ -using temporary file /tmp/M2-16342-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238
│ │ │ +using temporary file /tmp/M2-20465-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20465-0/238
│ │ │  This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options:
│ │ │ -using temporary file /tmp/M2-16342-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240
│ │ │ +using temporary file /tmp/M2-20465-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20465-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -503,16 +503,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16342-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242
│ │ │ +using temporary file /tmp/M2-20465-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20465-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -537,48 +537,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-16342-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244
│ │ │ +using temporary file /tmp/M2-20465-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20465-0/244
│ │ │  This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input.
│ │ │  Options:
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │   Specify n.
│ │ │  --trees:
│ │ │   list the boundary trees (assumes d=3)
│ │ │ -using temporary file /tmp/M2-16342-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246
│ │ │ +using temporary file /tmp/M2-20465-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20465-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248
│ │ │ +using temporary file /tmp/M2-20465-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20465-0/248
│ │ │  This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM.
│ │ │  Options:
│ │ │  --kapranov:
│ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │  --determinant:
│ │ │   Compute the tropical determinant instead.
│ │ │ -using temporary file /tmp/M2-16342-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250
│ │ │ +using temporary file /tmp/M2-20465-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20465-0/250
│ │ │  This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │  -d:
│ │ │   Output dimension information to standard error.
│ │ │  --stable:
│ │ │   Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │ -using temporary file /tmp/M2-16342-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252
│ │ │ +using temporary file /tmp/M2-20465-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20465-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -586,24 +586,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254
│ │ │ +using temporary file /tmp/M2-20465-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20465-0/254
│ │ │  This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │  -i2 value:
│ │ │   Specify the name of the Polymake input file containing the piecewise linear function.
│ │ │ -using temporary file /tmp/M2-16342-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256
│ │ │ +using temporary file /tmp/M2-20465-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20465-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -623,32 +623,32 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/256
      
│ │ │ +using temporary file /tmp/M2-20465-0/256
│ │ │              
      
│ │ │                
      i6 : QQ[x,y];
      
│ │ │              
      
│ │ │                
      i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20465-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16342-0/258
      
│ │ │ +using temporary file /tmp/M2-20465-0/258
│ │ │              
      
│ │ │          
      
│ │ │            
      Finally, if you want to be able to render Groebner fans and monomial staircases to .png files, you should install fig2dev.  If it is installed in a non-standard location, then you may specify its path using programPaths.
      
│ │ │          
      
│ │ │        
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  If you would like to see the input and output files used to communicate with
│ │ │ │  gfan you can set the "keepfiles" configuration option to true. If "verbose" is
│ │ │ │  set to true, gfanInterface will output the names of the temporary files used.
│ │ │ │  i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true,
│ │ │ │  "verbose" => true}, Reload => true);
│ │ │ │   -- warning: reloading gfanInterface; recreate instances of types from this
│ │ │ │  package
│ │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16342-0/172
│ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20465-0/172
│ │ │ │  This is a program for computing all reduced Groebner bases of a polynomial
│ │ │ │  ideal. It takes the ring and a generating set for the ideal as input. By
│ │ │ │  default the enumeration is done by an almost memoryless reverse search. If the
│ │ │ │  ideal is symmetric the symmetry option is useful and enumeration will be done
│ │ │ │  up to symmetry using a breadth first search. The program needs a starting
│ │ │ │  Groebner basis to do its computations. If the -g option is not specified it
│ │ │ │  will compute one using Buchberger's algorithm.
│ │ │ │ @@ -81,16 +81,16 @@
│ │ │ │   With this option you can specify how many variables to treat as parameters
│ │ │ │  instead of variables. This makes it possible to do computations where the
│ │ │ │  coefficient field is the field of rational functions in the parameters.
│ │ │ │  --interrupt value:
│ │ │ │   Interrupt the enumeration after a specified number of facets have been
│ │ │ │  computed (works for usual symmetric traversals, but may not work in general for
│ │ │ │  non-symmetric traversals or for traversals restricted to fans).
│ │ │ │ -using temporary file /tmp/M2-16342-0/172
│ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174
│ │ │ │ +using temporary file /tmp/M2-20465-0/172
│ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20465-0/174
│ │ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial
│ │ │ │  ideal given as input. The default behavior is to use Buchberger's algorithm.
│ │ │ │  The ordering of the variables is $a>b>c...$ (assuming that the ring is Q
│ │ │ │  [a,b,c,...]).
│ │ │ │  Options:
│ │ │ │  -w:
│ │ │ │   Compute a Groebner basis with respect to a degree lexicographic order with
│ │ │ │ @@ -111,63 +111,63 @@
│ │ │ │  minimal Groebner basis with respect to the reverse lexicographic term order.
│ │ │ │  The target term order is always lexicographic. The -W option must be used.
│ │ │ │  
│ │ │ │  --parameters value:
│ │ │ │   With this option you can specify how many variables to treat as parameters
│ │ │ │  instead of variables. This makes it possible to do computations where the
│ │ │ │  coefficient field is the field of rational functions in the parameters.
│ │ │ │ -using temporary file /tmp/M2-16342-0/174
│ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176
│ │ │ │ +using temporary file /tmp/M2-20465-0/174
│ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20465-0/176
│ │ │ │  This program takes a marked Groebner basis of an ideal I and a set of
│ │ │ │  polynomials on its input and tests if the polynomial set is contained in I by
│ │ │ │  applying the division algorithm for each element. The output is 1 for true and
│ │ │ │  0 for false.
│ │ │ │  Options:
│ │ │ │  --remainder:
│ │ │ │   Tell the program to output the remainders of the divisions rather than
│ │ │ │  outputting 0 or 1.
│ │ │ │  --multiplier:
│ │ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided
│ │ │ │  before doing the division.
│ │ │ │ -using temporary file /tmp/M2-16342-0/176
│ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178
│ │ │ │ +using temporary file /tmp/M2-20465-0/176
│ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20465-0/178
│ │ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the first input file.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the second input file.
│ │ │ │  --stable:
│ │ │ │   Compute the stable intersection.
│ │ │ │ -using temporary file /tmp/M2-16342-0/178
│ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180
│ │ │ │ +using temporary file /tmp/M2-20465-0/178
│ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20465-0/180
│ │ │ │  This program takes a polyhedral fan and a vector and computes the link of the
│ │ │ │  polyhedral fan around that vertex. The link will have lineality space dimension
│ │ │ │  equal to the dimension of the relative open polyhedral cone of the original fan
│ │ │ │  containing the vector.
│ │ │ │  Options:
│ │ │ │  -i value:
│ │ │ │   Specify the name of the input file.
│ │ │ │  --symmetry:
│ │ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must
│ │ │ │  be given on the standard input.
│ │ │ │  
│ │ │ │  --star:
│ │ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan
│ │ │ │  containing all cones of the original fan containing the vector.
│ │ │ │ -using temporary file /tmp/M2-16342-0/180
│ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182
│ │ │ │ +using temporary file /tmp/M2-20465-0/180
│ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20465-0/182
│ │ │ │  This program takes two polyhedral fans and computes their product.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the first input file.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the second input file.
│ │ │ │ -using temporary file /tmp/M2-16342-0/182
│ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184
│ │ │ │ +using temporary file /tmp/M2-20465-0/182
│ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20465-0/184
│ │ │ │  This program computes a Groebner cone. Three different cases are handled. The
│ │ │ │  input may be a marked reduced Groebner basis in which case its Groebner cone is
│ │ │ │  computed. The input may be just a marked minimal basis in which case the cone
│ │ │ │  computed is not a Groebner cone in the usual sense but smaller. (These cones
│ │ │ │  are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case
│ │ │ │  is that the Groebner cone is possibly lower dimensional and given by a pair of
│ │ │ │  Groebner bases as it is useful to do for tropical varieties, see option --pair.
│ │ │ │ @@ -184,24 +184,24 @@
│ │ │ │  --asfan:
│ │ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way
│ │ │ │  the extreme rays of the cone are also computed.
│ │ │ │  --vectorinput:
│ │ │ │   Compute a cone given list of inequalities rather than a Groebner cone. The
│ │ │ │  input is an integer which specifies the dimension of the ambient space, a list
│ │ │ │  of inequalities given as vectors and a list of equations.
│ │ │ │ -using temporary file /tmp/M2-16342-0/184
│ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186
│ │ │ │ +using temporary file /tmp/M2-20465-0/184
│ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20465-0/186
│ │ │ │  This program computes the homogeneity space of a list of polynomials - as a
│ │ │ │  cone. Thus generators for the homogeneity space are found in the section
│ │ │ │  LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first
│ │ │ │  compute a set of homogeneous generators and call the program on these. A
│ │ │ │  reduced Groebner basis will always suffice for this purpose.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/186
│ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188
│ │ │ │ +using temporary file /tmp/M2-20465-0/186
│ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20465-0/188
│ │ │ │  This program homogenises a list of polynomials by introducing an extra
│ │ │ │  variable. The name of the variable to be introduced is read from the input
│ │ │ │  after the list of polynomials. Without the -w option the homogenisation is done
│ │ │ │  with respect to total degree.
│ │ │ │  Example:
│ │ │ │  Input:
│ │ │ │  Q[x,y]{y-1}
│ │ │ │ @@ -217,16 +217,16 @@
│ │ │ │   Specify a homogenisation vector. The length of the vector must be the same as
│ │ │ │  the number of variables in the ring. The vector is read from the input after
│ │ │ │  the list of polynomials.
│ │ │ │  
│ │ │ │  -H:
│ │ │ │   Let the name of the new variable be H rather than reading in a name from the
│ │ │ │  input.
│ │ │ │ -using temporary file /tmp/M2-16342-0/188
│ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190
│ │ │ │ +using temporary file /tmp/M2-20465-0/188
│ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20465-0/190
│ │ │ │  This program converts a list of polynomials to a list of their initial forms
│ │ │ │  with respect to the vector given after the list.
│ │ │ │  Options:
│ │ │ │  --ideal:
│ │ │ │   Treat input as an ideal. This will make the program compute the initial ideal
│ │ │ │  of the ideal generated by the input polynomials. The computation is done by
│ │ │ │  computing a Groebner basis with respect to the given vector. The vector must be
│ │ │ │ @@ -242,16 +242,16 @@
│ │ │ │  --mark:
│ │ │ │   If the --pair option is and the --ideal option is not used this option will
│ │ │ │  still make sure that the second output basis is marked consistently with the
│ │ │ │  vector.
│ │ │ │  --list:
│ │ │ │   Read in a list of vectors instead of a single vector and produce a list of
│ │ │ │  polynomial sets as output.
│ │ │ │ -using temporary file /tmp/M2-16342-0/190
│ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192
│ │ │ │ +using temporary file /tmp/M2-20465-0/190
│ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20465-0/192
│ │ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal.
│ │ │ │  The input is a Groebner basis defining the starting Groebner cone of the walk.
│ │ │ │  The program will list all flippable facets of the Groebner cone and ask the
│ │ │ │  user to choose one. The user types in the index (number) of the facet in the
│ │ │ │  list. The program will walk through the selected facet and display the new
│ │ │ │  Groebner basis and a list of new facet normals for the user to choose from.
│ │ │ │  Since the program reads the user's choices through the the standard input it is
│ │ │ │ @@ -281,54 +281,54 @@
│ │ │ │  -W:
│ │ │ │   Print weight vector. This will make the program print an interior vector of
│ │ │ │  the current Groebner cone and a relative interior point for each flippable
│ │ │ │  facet of the current Groebner cone.
│ │ │ │  
│ │ │ │  --tropical:
│ │ │ │   Traverse a tropical variety interactively.
│ │ │ │ -using temporary file /tmp/M2-16342-0/192
│ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194
│ │ │ │ +using temporary file /tmp/M2-20465-0/192
│ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20465-0/194
│ │ │ │  This program checks if a set of marked polynomials is a Groebner basis with
│ │ │ │  respect to its marking. First it is checked if the markings are consistent with
│ │ │ │  respect to a positive vector. Then Buchberger's S-criterion is checked. The
│ │ │ │  output is boolean value.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/194
│ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196
│ │ │ │ +using temporary file /tmp/M2-20465-0/194
│ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20465-0/196
│ │ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the
│ │ │ │  polynomial ring. This is done by first computing a Groebner basis.
│ │ │ │  Options:
│ │ │ │  -g:
│ │ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ │ -using temporary file /tmp/M2-16342-0/196
│ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198
│ │ │ │ +using temporary file /tmp/M2-20465-0/196
│ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20465-0/198
│ │ │ │  This program computes the lattice ideal of a lattice. The input is a list of
│ │ │ │  generators for the lattice.
│ │ │ │  Options:
│ │ │ │  -t:
│ │ │ │   Compute the toric ideal of the matrix whose rows are given on the input
│ │ │ │  instead.
│ │ │ │  --convert:
│ │ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ │ -using temporary file /tmp/M2-16342-0/198
│ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200
│ │ │ │ +using temporary file /tmp/M2-20465-0/198
│ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20465-0/200
│ │ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │ │  Options:
│ │ │ │  -m:
│ │ │ │   Do the same thing for a list of polynomial sets. That is, output the set of
│ │ │ │  sets of leading terms.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16342-0/200
│ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202
│ │ │ │ +using temporary file /tmp/M2-20465-0/200
│ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20465-0/202
│ │ │ │  This program marks a set of polynomials with respect to the vector given at the
│ │ │ │  end of the input, meaning that the largest terms are moved to the front. In
│ │ │ │  case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/202
│ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204
│ │ │ │ +using temporary file /tmp/M2-20465-0/202
│ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20465-0/204
│ │ │ │  This is a program for computing the normal fan of the Minkowski sum of the
│ │ │ │  Newton polytopes of a list of polynomials.
│ │ │ │  Options:
│ │ │ │  --symmetry:
│ │ │ │   Tells the program to read in generators for a group of symmetries (subgroup of
│ │ │ │  $S_n$) after having read in the ideal. The program checks that the ideal stays
│ │ │ │  fixed when permuting the variables with respect to elements in the group. The
│ │ │ │ @@ -338,16 +338,16 @@
│ │ │ │  --disableSymmetryTest:
│ │ │ │   When using --symmetry this option will disable the check that the group read
│ │ │ │  off from the input actually is a symmetry group with respect to the input
│ │ │ │  ideal.
│ │ │ │  
│ │ │ │  --nocones:
│ │ │ │   Tell the program to not list cones in the output.
│ │ │ │ -using temporary file /tmp/M2-16342-0/204
│ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206
│ │ │ │ +using temporary file /tmp/M2-20465-0/204
│ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20465-0/206
│ │ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │ │  Options:
│ │ │ │  -r value:
│ │ │ │   Specify r.
│ │ │ │  -d value:
│ │ │ │   Specify d.
│ │ │ │  -n value:
│ │ │ │ @@ -365,16 +365,16 @@
│ │ │ │   Produces a list of generators for the group of symmetries keeping the set of
│ │ │ │  minors fixed. (Only without --names).
│ │ │ │  --parametrize:
│ │ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal
│ │ │ │  to r-1 by a list of tropical polynomials.
│ │ │ │  --ultrametric:
│ │ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ │ -using temporary file /tmp/M2-16342-0/206
│ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208
│ │ │ │ +using temporary file /tmp/M2-20465-0/206
│ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20465-0/208
│ │ │ │  This program computes the mixed volume of the Newton polytopes of a list of
│ │ │ │  polynomials. The ring is specified on the input. After this follows the list of
│ │ │ │  polynomials.
│ │ │ │  Options:
│ │ │ │  --vectorinput:
│ │ │ │   Read in a list of point configurations instead of a polynomial ring and a list
│ │ │ │  of polynomials.
│ │ │ │ @@ -390,28 +390,28 @@
│ │ │ │   Use Gaukwa-n example instead of reading input.
│ │ │ │  --eco value:
│ │ │ │   Use Eco-n example instead of reading input.
│ │ │ │  -j value:
│ │ │ │   Number of threads
│ │ │ │  --resume:
│ │ │ │   Continue enumeration from saved state...
│ │ │ │ -using temporary file /tmp/M2-16342-0/208
│ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210
│ │ │ │ +using temporary file /tmp/M2-20465-0/208
│ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20465-0/210
│ │ │ │  This program computes the union of a list of polynomial sets given as input.
│ │ │ │  The polynomials must all belong to the same ring. The ring is specified on the
│ │ │ │  input. After this follows the list of polynomial sets.
│ │ │ │  Options:
│ │ │ │  -s:
│ │ │ │   Sort output by degree.
│ │ │ │  
│ │ │ │  --twosetswithrings:
│ │ │ │   Switch to mode where input is: ring polynomialset (same)ring polynomialset.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16342-0/210
│ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212
│ │ │ │ +using temporary file /tmp/M2-20465-0/210
│ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20465-0/212
│ │ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the
│ │ │ │  input is the list of all reduced Groebner bases of an ideal. The output is a
│ │ │ │  drawing of the Groebner fan intersected with a triangle. The corners of the
│ │ │ │  triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top.
│ │ │ │  If there are more than three variables in the ring these coordinates are
│ │ │ │  extended with zeros. It is possible to shift the 1 entry cyclic with the option
│ │ │ │  --shiftVariables.
│ │ │ │ @@ -419,16 +419,16 @@
│ │ │ │  -L:
│ │ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │ │  --shiftVariables value:
│ │ │ │   Shift the positions of the variables in the drawing. For example with the
│ │ │ │  value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...)
│ │ │ │  and top: (0,0,0,1,...). The shifting is done modulo the number of variables in
│ │ │ │  the polynomial ring. The default value is 0.
│ │ │ │ -using temporary file /tmp/M2-16342-0/212
│ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214
│ │ │ │ +using temporary file /tmp/M2-20465-0/212
│ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20465-0/214
│ │ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig
│ │ │ │  file. The input is a Groebner basis of a (not necessarily monomial) polynomial
│ │ │ │  ideal. The initial ideal is given by the leading terms in the Groebner basis.
│ │ │ │  Using the -m option it is possible to render more than one staircase diagram.
│ │ │ │  The program only works for ideals in a polynomial ring with three variables.
│ │ │ │  Options:
│ │ │ │  -m:
│ │ │ │ @@ -441,16 +441,16 @@
│ │ │ │  number is large enough to give a correct picture of the standard monomials. The
│ │ │ │  default value is 8.
│ │ │ │  
│ │ │ │  -w value:
│ │ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file.
│ │ │ │  The default value is 5.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16342-0/214
│ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216
│ │ │ │ +using temporary file /tmp/M2-20465-0/214
│ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20465-0/216
│ │ │ │  This program computes the resultant fan as defined in "Computing Tropical
│ │ │ │  Resultants" by Jensen and Yu. The input is a polynomial ring followed by
│ │ │ │  polynomials, whose coefficients are ignored. The output is the fan of
│ │ │ │  coefficients such that the input system has a tropical solution.
│ │ │ │  Options:
│ │ │ │  --codimension:
│ │ │ │   Compute only the codimension of the resultant fan and return.
│ │ │ │ @@ -478,28 +478,28 @@
│ │ │ │  of polynomials.
│ │ │ │  
│ │ │ │  --projection:
│ │ │ │   Use the projection method to compute the resultant fan. This works only if the
│ │ │ │  resultant fan is a hypersurface. If this option is combined with --special,
│ │ │ │  then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16342-0/216
│ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218
│ │ │ │ +using temporary file /tmp/M2-20465-0/216
│ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20465-0/218
│ │ │ │  This program computes the saturation of the input ideal with the product of the
│ │ │ │  variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous
│ │ │ │  generators).
│ │ │ │  
│ │ │ │  --noideal:
│ │ │ │   Do not treat input as an ideal but just factor out common monomial factors of
│ │ │ │  the input polynomials.
│ │ │ │ -using temporary file /tmp/M2-16342-0/218
│ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220
│ │ │ │ +using temporary file /tmp/M2-20465-0/218
│ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20465-0/220
│ │ │ │  This program computes the secondary fan of a vector configuration. The
│ │ │ │  configuration is given as an ordered list of vectors. In order to compute the
│ │ │ │  secondary fan of a point configuration an additional coordinate of ones must be
│ │ │ │  added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │ │  Options:
│ │ │ │  --unimodular:
│ │ │ │   Use heuristics to search for unimodular triangulation rather than computing
│ │ │ │ @@ -528,49 +528,49 @@
│ │ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but
│ │ │ │  still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is
│ │ │ │  used.
│ │ │ │  --interrupt value:
│ │ │ │   Interrupt the enumeration after a specified number of facets have been
│ │ │ │  computed (works for usual symmetric traversals, but may not work in general for
│ │ │ │  non-symmetric traversals or for traversals restricted to fans).
│ │ │ │ -using temporary file /tmp/M2-16342-0/220
│ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222
│ │ │ │ +using temporary file /tmp/M2-20465-0/220
│ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20465-0/222
│ │ │ │  This program takes a list of reduced Groebner bases for the same ideal and
│ │ │ │  computes various statistics. The following information is listed: the number of
│ │ │ │  bases in the input, the number of variables, the dimension of the homogeneity
│ │ │ │  space, the maximal total degree of any polynomial in the input and the minimal
│ │ │ │  total degree of any basis in the input, the maximal number of polynomials and
│ │ │ │  terms in a basis in the input.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/222
│ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224
│ │ │ │ +using temporary file /tmp/M2-20465-0/222
│ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20465-0/224
│ │ │ │  This program changes the variable names of a polynomial ring. The input is a
│ │ │ │  polynomial ring, a polynomial set in the ring and a new polynomial ring with
│ │ │ │  the same coefficient field but different variable names. The output is the
│ │ │ │  polynomial set written with the variable names of the second polynomial ring.
│ │ │ │  Example:
│ │ │ │  Input:
│ │ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │ │  Output:
│ │ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/224
│ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226
│ │ │ │ +using temporary file /tmp/M2-20465-0/224
│ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20465-0/226
│ │ │ │  This program converts ASCII math to TeX math. The data-type is specified by the
│ │ │ │  options.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Add a header to the output. Using this option the output will be LaTeXable
│ │ │ │  right away.
│ │ │ │  --polynomialset_:
│ │ │ │   The data to be converted is a list of polynomials.
│ │ │ │  --polynomialsetlist_:
│ │ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ │ -using temporary file /tmp/M2-16342-0/226
│ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228
│ │ │ │ +using temporary file /tmp/M2-20465-0/226
│ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20465-0/228
│ │ │ │  This program takes a list of reduced Groebner bases and produces the fan of all
│ │ │ │  faces of these. In this way by giving the complete list of reduced Groebner
│ │ │ │  bases, the Groebner fan can be computed as a polyhedral complex. The option --
│ │ │ │  restrict lets the user choose between computing the Groebner fan or the
│ │ │ │  restricted Groebner fan.
│ │ │ │  Options:
│ │ │ │  --restrict:
│ │ │ │ @@ -581,54 +581,54 @@
│ │ │ │  generated cones given in modern format (like _cone). In this case --restrict is
│ │ │ │  ignored.
│ │ │ │  --symmetry:
│ │ │ │   Tell the program to read in generators for a group of symmetries (subgroup of
│ │ │ │  $S_n$) after having read in the ring. The output is grouped according to these
│ │ │ │  symmetries. Only one representative for each orbit is needed on the input.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16342-0/228
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230
│ │ │ │ +using temporary file /tmp/M2-20465-0/228
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20465-0/230
│ │ │ │  This program computes a tropical basis for an ideal defining a tropical curve.
│ │ │ │  Defining a tropical curve means that the Krull dimension of R/I is at most 1 +
│ │ │ │  the dimension of the homogeneity space of I where R is the polynomial ring. The
│ │ │ │  input is a generating set for the ideal. If the input is not homogeneous option
│ │ │ │  -h must be used.
│ │ │ │  Options:
│ │ │ │  -h:
│ │ │ │   Homogenise the input before computing a tropical basis and dehomogenise the
│ │ │ │  output. This is needed if the input generators are not already homogeneous.
│ │ │ │ -using temporary file /tmp/M2-16342-0/230
│ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232
│ │ │ │ +using temporary file /tmp/M2-20465-0/230
│ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20465-0/232
│ │ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and
│ │ │ │  computes the tropical variety of the ideal as a subfan of the Groebner fan. The
│ │ │ │  program is slow but works for any homogeneous ideal. If you know that your
│ │ │ │  ideal is prime over the complex numbers or you simply know that its tropical
│ │ │ │  variety is pure and connected in codimension one then use
│ │ │ │  gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/232
│ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234
│ │ │ │ +using temporary file /tmp/M2-20465-0/232
│ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20465-0/234
│ │ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/234
│ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236
│ │ │ │ +using temporary file /tmp/M2-20465-0/234
│ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20465-0/236
│ │ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise
│ │ │ │  linear function represented by a fan whose cones are the linear regions. Each
│ │ │ │  ray of the fan gets the value of the tropical function assigned to it. In other
│ │ │ │  words this program computes the normal fan of the Newton polytope of the input
│ │ │ │  polynomial with additional information.Options:
│ │ │ │  --exponents:
│ │ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ │ -using temporary file /tmp/M2-16342-0/236
│ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238
│ │ │ │ +using temporary file /tmp/M2-20465-0/236
│ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20465-0/238
│ │ │ │  This program computes the tropical hypersurface defined by a principal ideal.
│ │ │ │  The input is the polynomial ring followed by a set containing just a generator
│ │ │ │  of the ideal.Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/238
│ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240
│ │ │ │ +using temporary file /tmp/M2-20465-0/238
│ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20465-0/240
│ │ │ │  This program computes the set theoretical intersection of a set of tropical
│ │ │ │  hypersurfaces (or to be precise, their common refinement as a fan). The input
│ │ │ │  is a list of polynomials with each polynomial defining a hypersurface.
│ │ │ │  Considering tropical hypersurfaces as fans, the intersection can be computed as
│ │ │ │  the common refinement of these. Thus the output is a fan whose support is the
│ │ │ │  intersection of the tropical hypersurfaces.
│ │ │ │  Options:
│ │ │ │ @@ -665,16 +665,16 @@
│ │ │ │  --stable:
│ │ │ │   Find the stable intersection of the input polynomials using tropical
│ │ │ │  intersection theory. This can be slow. Most other options are ignored.
│ │ │ │  --parameters value:
│ │ │ │   With this option you can specify how many variables to treat as parameters
│ │ │ │  instead of variables. This makes it possible to do computations where the
│ │ │ │  coefficient field is the field of rational functions in the parameters.
│ │ │ │ -using temporary file /tmp/M2-16342-0/240
│ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242
│ │ │ │ +using temporary file /tmp/M2-20465-0/240
│ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20465-0/242
│ │ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and
│ │ │ │  Singular. The Singular part of the implementation can be found in:
│ │ │ │  
│ │ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry,
│ │ │ │  2007
│ │ │ │  
│ │ │ │ @@ -702,54 +702,54 @@
│ │ │ │  Options:
│ │ │ │  --noMult:
│ │ │ │   Disable the multiplicity computation.
│ │ │ │  -n value:
│ │ │ │   Number of variables that should have negative weight.
│ │ │ │  -c:
│ │ │ │   Only output a list of vectors being the possible choices.
│ │ │ │ -using temporary file /tmp/M2-16342-0/242
│ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244
│ │ │ │ +using temporary file /tmp/M2-20465-0/242
│ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20465-0/244
│ │ │ │  This program generates tropical equations for a tropical linear space in the
│ │ │ │  Speyer sense given the tropical Pluecker coordinates as input.
│ │ │ │  Options:
│ │ │ │  -d value:
│ │ │ │   Specify d.
│ │ │ │  -n value:
│ │ │ │   Specify n.
│ │ │ │  --trees:
│ │ │ │   list the boundary trees (assumes d=3)
│ │ │ │ -using temporary file /tmp/M2-16342-0/244
│ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246
│ │ │ │ +using temporary file /tmp/M2-20465-0/244
│ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20465-0/246
│ │ │ │  This program computes the multiplicity of a tropical cone given a marked
│ │ │ │  reduced Groebner basis for its initial ideal.
│ │ │ │  Options:
│ │ │ │ -using temporary file /tmp/M2-16342-0/246
│ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248
│ │ │ │ +using temporary file /tmp/M2-20465-0/246
│ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20465-0/248
│ │ │ │  This program will compute the tropical rank of matrix given as input. Tropical
│ │ │ │  addition is MAXIMUM.
│ │ │ │  Options:
│ │ │ │  --kapranov:
│ │ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │ │  --determinant:
│ │ │ │   Compute the tropical determinant instead.
│ │ │ │ -using temporary file /tmp/M2-16342-0/248
│ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250
│ │ │ │ +using temporary file /tmp/M2-20465-0/248
│ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20465-0/250
│ │ │ │  This program computes a starting pair of marked reduced Groebner bases to be
│ │ │ │  used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose
│ │ │ │  tropical variety is a pure d-dimensional polyhedral complex.
│ │ │ │  Options:
│ │ │ │  -g:
│ │ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ │  -d:
│ │ │ │   Output dimension information to standard error.
│ │ │ │  --stable:
│ │ │ │   Find starting cone in the stable intersection or, equivalently, pretend that
│ │ │ │  the coefficients are genereric.
│ │ │ │ -using temporary file /tmp/M2-16342-0/250
│ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252
│ │ │ │ +using temporary file /tmp/M2-20465-0/250
│ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20465-0/252
│ │ │ │  This program computes a polyhedral fan representation of the tropical variety
│ │ │ │  of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let
│ │ │ │  $\omega$ be a relative interior point of $d$-dimensional Groebner cone
│ │ │ │  contained in the tropical variety. The input for this program is a pair of
│ │ │ │  marked reduced Groebner bases with respect to the term order represented by
│ │ │ │  $\omega$, tie-broken in some way. The first one is for the initial ideal
│ │ │ │  $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point
│ │ │ │ @@ -779,27 +779,27 @@
│ │ │ │  --stable:
│ │ │ │   Traverse the stable intersection or, equivalently, pretend that the
│ │ │ │  coefficients are genereric.
│ │ │ │  --interrupt value:
│ │ │ │   Interrupt the enumeration after a specified number of facets have been
│ │ │ │  computed (works for usual symmetric traversals, but may not work in general for
│ │ │ │  non-symmetric traversals or for traversals restricted to fans).
│ │ │ │ -using temporary file /tmp/M2-16342-0/252
│ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254
│ │ │ │ +using temporary file /tmp/M2-20465-0/252
│ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20465-0/254
│ │ │ │  This program computes the tropical Weil divisor of piecewise linear (or
│ │ │ │  tropical rational) function on a tropical k-cycle. See the Gfan manual for more
│ │ │ │  information.
│ │ │ │  Options:
│ │ │ │  -i1 value:
│ │ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │ │  -i2 value:
│ │ │ │   Specify the name of the Polymake input file containing the piecewise linear
│ │ │ │  function.
│ │ │ │ -using temporary file /tmp/M2-16342-0/254
│ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256
│ │ │ │ +using temporary file /tmp/M2-20465-0/254
│ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20465-0/256
│ │ │ │  This program is an experimental implementation of Groebner bases for ideals in
│ │ │ │  Z[x_1,...,x_n].
│ │ │ │  Several operations are supported by specifying the appropriate option:
│ │ │ │   (1) computation of the reduced Groebner basis with respect to a given vector
│ │ │ │  (tiebroken lexicographically),
│ │ │ │   (2) computation of an initial ideal,
│ │ │ │   (3) computation of the Groebner fan,
│ │ │ │ @@ -834,23 +834,23 @@
│ │ │ │   For the operations taking a vector as input, read in a list of vectors
│ │ │ │  instead, and perform the operation for each vector in the list.
│ │ │ │  -g:
│ │ │ │   Tells the program that the input is already a Groebner basis (with the initial
│ │ │ │  term of each polynomial being the first ones listed). Use this option if the
│ │ │ │  usual --groebnerFan is too slow.
│ │ │ │  
│ │ │ │ -using temporary file /tmp/M2-16342-0/256
│ │ │ │ +using temporary file /tmp/M2-20465-0/256
│ │ │ │  i6 : QQ[x,y];
│ │ │ │  i7 : gfan {x,y};
│ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258
│ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20465-0/258
│ │ │ │  Q[x1,x2]
│ │ │ │  {{
│ │ │ │  x2,
│ │ │ │  x1}
│ │ │ │  }
│ │ │ │ -using temporary file /tmp/M2-16342-0/258
│ │ │ │ +using temporary file /tmp/M2-20465-0/258
│ │ │ │  Finally, if you want to be able to render Groebner fans and monomial staircases
│ │ │ │  to .png files, you should install fig2dev. If it is installed in a non-standard
│ │ │ │  location, then you may specify its path using _p_r_o_g_r_a_m_P_a_t_h_s.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/gfanInterface.m2:2631:0.
│ │ ├── ./usr/share/info/AInfinity.info.gz
│ │ │ ├── AInfinity.info
│ │ │ │ @@ -6133,16 +6133,16 @@
│ │ │ │  00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017f70: 2d2d 2d2d 2b0a 7c69 3320 3a20 656c 6170  ----+.|i3 : elap
│ │ │ │  00017f80: 7365 6454 696d 6520 6275 726b 6552 6573  sedTime burkeRes
│ │ │ │  00017f90: 6f6c 7574 696f 6e28 4d2c 2037 2c20 4368  olution(M, 7, Ch
│ │ │ │  00017fa0: 6563 6b20 3d3e 2066 616c 7365 2920 2020  eck => false)   
│ │ │ │ -00017fb0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2032          |.| -- 2
│ │ │ │ -00017fc0: 2e34 3430 3938 7320 656c 6170 7365 6420  .44098s elapsed 
│ │ │ │ +00017fb0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2031          |.| -- 1
│ │ │ │ +00017fc0: 2e37 3630 3233 7320 656c 6170 7365 6420  .76023s elapsed 
│ │ │ │  00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -6176,15 +6176,15 @@
│ │ │ │  000181f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00018220: 3420 3a20 656c 6170 7365 6454 696d 6520  4 : elapsedTime 
│ │ │ │  00018230: 6275 726b 6552 6573 6f6c 7574 696f 6e28  burkeResolution(
│ │ │ │  00018240: 4d2c 2037 2c20 4368 6563 6b20 3d3e 2074  M, 7, Check => t
│ │ │ │  00018250: 7275 6529 2020 2020 2020 2020 2020 2020  rue)            
│ │ │ │ -00018260: 7c0a 7c20 2d2d 2032 2e33 3931 3435 7320  |.| -- 2.39145s 
│ │ │ │ +00018260: 7c0a 7c20 2d2d 2032 2e32 3133 3339 7320  |.| -- 2.21339s 
│ │ │ │  00018270: 656c 6170 7365 6420 2020 2020 2020 2020  elapsed         
│ │ │ │  00018280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000182b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182d0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/AdjunctionForSurfaces.info.gz
│ │ │ ├── AdjunctionForSurfaces.info
│ │ │ │ @@ -741,16 +741,16 @@
│ │ │ │  00002e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00002e70: 7c69 3130 203a 2065 6c61 7073 6564 5469  |i10 : elapsedTi
│ │ │ │  00002e80: 6d65 2066 493d 7265 7320 4920 2020 2020  me fI=res I     
│ │ │ │  00002e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3235 3738     |.| -- .02578
│ │ │ │ -00002ec0: 3338 7320 656c 6170 7365 6420 2020 2020  38s elapsed     
│ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3337 3838     |.| -- .03788
│ │ │ │ +00002ec0: 3639 7320 656c 6170 7365 6420 2020 2020  69s elapsed     
│ │ │ │  00002ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00002f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ @@ -1596,15 +1596,15 @@
│ │ │ │  000063b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000063c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000063d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000063e0: 7c69 3135 203a 2065 6c61 7073 6564 5469  |i15 : elapsedTi
│ │ │ │  000063f0: 6d65 2062 6574 7469 2849 273d 7472 696d  me betti(I'=trim
│ │ │ │  00006400: 206b 6572 2070 6869 2920 2020 2020 2020   ker phi)       
│ │ │ │  00006410: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00006420: 2e36 3039 3138 3573 2065 6c61 7073 6564  .609185s elapsed
│ │ │ │ +00006420: 2e37 3431 3133 3173 2065 6c61 7073 6564  .741131s elapsed
│ │ │ │  00006430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006450: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00006460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00006490: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ @@ -1651,16 +1651,16 @@
│ │ │ │  00006720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006750: 2d2d 2d2b 0a7c 6931 3720 3a20 656c 6170  ---+.|i17 : elap
│ │ │ │  00006760: 7365 6454 696d 6520 6261 7365 5074 733d  sedTime basePts=
│ │ │ │  00006770: 7072 696d 6172 7944 6563 6f6d 706f 7369  primaryDecomposi
│ │ │ │  00006780: 7469 6f6e 2069 6465 616c 2048 3b20 7c0a  tion ideal H; |.
│ │ │ │ -00006790: 7c20 2d2d 2036 2e39 3831 3434 7320 656c  | -- 6.98144s el
│ │ │ │ -000067a0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │ +00006790: 7c20 2d2d 2036 2e33 3930 3873 2065 6c61  | -- 6.3908s ela
│ │ │ │ +000067a0: 7073 6564 2020 2020 2020 2020 2020 2020  psed            
│ │ │ │  000067b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000067c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  000067d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000067e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000067f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006800: 2d2d 2d2d 2b0a 7c69 3138 203a 2074 616c  ----+.|i18 : tal
│ │ │ │  00006810: 6c79 2061 7070 6c79 2862 6173 6550 7473  ly apply(basePts
│ │ │ │ @@ -2608,15 +2608,15 @@
│ │ │ │  0000a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0000a320: 3134 203a 2065 6c61 7073 6564 5469 6d65  14 : elapsedTime
│ │ │ │  0000a330: 2073 7562 2849 2c48 2920 2020 2020 2020   sub(I,H)       
│ │ │ │  0000a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a350: 2020 2020 2020 207c 0a7c 202d 2d20 2e30         |.| -- .0
│ │ │ │ -0000a360: 3230 3035 3236 7320 656c 6170 7365 6420  200526s elapsed 
│ │ │ │ +0000a360: 3232 3832 3234 7320 656c 6170 7365 6420  228224s elapsed 
│ │ │ │  0000a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a390: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a3c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0000a3d0: 6f31 3420 3d20 6964 6561 6c20 2830 2c20  o14 = ideal (0, 
│ │ │ │ @@ -2648,15 +2648,15 @@
│ │ │ │  0000a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a5a0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 656c  -----+.|i16 : el
│ │ │ │  0000a5b0: 6170 7365 6454 696d 6520 6265 7474 6928  apsedTime betti(
│ │ │ │  0000a5c0: 4927 3d74 7269 6d20 6b65 7220 7068 6929  I'=trim ker phi)
│ │ │ │  0000a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0000a5e0: 7c0a 7c20 2d2d 202e 3035 3635 3536 3673  |.| -- .0565566s
│ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3037 3932 3634 3673  |.| -- .0792646s
│ │ │ │  0000a5f0: 2065 6c61 7073 6564 2020 2020 2020 2020   elapsed        
│ │ │ │  0000a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a610: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a650: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ @@ -2700,16 +2700,16 @@
│ │ │ │  0000a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000a8e0: 0a7c 6931 3820 3a20 656c 6170 7365 6454  .|i18 : elapsedT
│ │ │ │  0000a8f0: 696d 6520 6261 7365 5074 733d 7072 696d  ime basePts=prim
│ │ │ │  0000a900: 6172 7944 6563 6f6d 706f 7369 7469 6f6e  aryDecomposition
│ │ │ │  0000a910: 2069 6465 616c 2048 3b20 7c0a 7c20 2d2d   ideal H; |.| --
│ │ │ │ -0000a920: 2031 2e39 3630 3431 7320 656c 6170 7365   1.96041s elapse
│ │ │ │ -0000a930: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ +0000a920: 2031 2e38 3436 3573 2065 6c61 7073 6564   1.8465s elapsed
│ │ │ │ +0000a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a950: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0000a960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a990: 2b0a 7c69 3139 203a 2074 616c 6c79 2061  +.|i19 : tally a
│ │ │ │  0000a9a0: 7070 6c79 2862 6173 6550 7473 2c63 2d3e  pply(basePts,c->
│ │ ├── ./usr/share/info/BGG.info.gz
│ │ │ ├── BGG.info
│ │ │ │ @@ -4338,16 +4338,16 @@
│ │ │ │  00010f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00010f40: 3134 203a 2074 696d 6520 6265 7474 6920  14 : time betti 
│ │ │ │  00010f50: 2846 203d 2070 7572 6552 6573 6f6c 7574  (F = pureResolut
│ │ │ │  00010f60: 696f 6e28 4d2c 7b30 2c32 2c34 7d29 2920  ion(M,{0,2,4})) 
│ │ │ │  00010f70: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ -00010f80: 302e 3435 3030 3137 7320 2863 7075 293b  0.450017s (cpu);
│ │ │ │ -00010f90: 2030 2e33 3736 3836 3373 2028 7468 7265   0.376863s (thre
│ │ │ │ +00010f80: 302e 3830 3534 3534 7320 2863 7075 293b  0.805454s (cpu);
│ │ │ │ +00010f90: 2030 2e36 3038 3535 3773 2028 7468 7265   0.608557s (thre
│ │ │ │  00010fa0: 6164 293b 2030 7320 2867 6329 7c0a 7c20  ad); 0s (gc)|.| 
│ │ │ │  00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00010ff0: 2020 2020 3020 3120 3220 2020 2020 2020      0 1 2       
│ │ │ │  00011000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4403,16 +4403,16 @@
│ │ │ │  00011320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00011350: 0a7c 6931 3520 3a20 7469 6d65 2062 6574  .|i15 : time bet
│ │ │ │  00011360: 7469 2028 4620 3d20 7075 7265 5265 736f  ti (F = pureReso
│ │ │ │  00011370: 6c75 7469 6f6e 2831 312c 342c 7b30 2c32  lution(11,4,{0,2
│ │ │ │  00011380: 2c34 7d29 2920 207c 0a7c 202d 2d20 7573  ,4}))  |.| -- us
│ │ │ │ -00011390: 6564 2030 2e34 3638 3335 3173 2028 6370  ed 0.468351s (cp
│ │ │ │ -000113a0: 7529 3b20 302e 3430 3337 3332 7320 2874  u); 0.403732s (t
│ │ │ │ +00011390: 6564 2030 2e36 3639 3131 3573 2028 6370  ed 0.669115s (cp
│ │ │ │ +000113a0: 7529 3b20 302e 3536 3334 3632 7320 2874  u); 0.563462s (t
│ │ │ │  000113b0: 6872 6561 6429 3b20 3073 2028 6763 297c  hread); 0s (gc)|
│ │ │ │  000113c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000113d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00011400: 2020 2020 2020 2030 2031 2032 2020 2020         0 1 2    
│ │ │ │  00011410: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/Benchmark.info.gz
│ │ │ ├── Benchmark.info
│ │ │ │ @@ -200,46 +200,46 @@
│ │ │ │  00000c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00000c80: 3120 3a20 7275 6e42 656e 6368 6d61 726b  1 : runBenchmark
│ │ │ │  00000c90: 7320 2272 6573 3339 2220 2020 2020 2020  s "res39"       
│ │ │ │  00000ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  00000cd0: 2d20 6265 6769 6e6e 696e 6720 636f 6d70  - beginning comp
│ │ │ │ -00000ce0: 7574 6174 696f 6e20 5361 7420 4f63 7420  utation Sat Oct 
│ │ │ │ -00000cf0: 3234 2030 333a 3230 3a30 3620 5554 4320  24 03:20:06 UTC 
│ │ │ │ +00000ce0: 7574 6174 696f 6e20 4d6f 6e20 4f63 7420  utation Mon Oct 
│ │ │ │ +00000cf0: 3236 2031 303a 3532 3a34 3520 5554 4320  26 10:52:45 UTC 
│ │ │ │  00000d00: 3230 3236 2020 2020 2020 2020 2020 2020  2026            
│ │ │ │  00000d10: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  00000d20: 2d20 4c69 6e75 7820 7362 7569 6c64 2036  - Linux sbuild 6
│ │ │ │ -00000d30: 2e31 322e 3131 332b 6465 6231 332d 616d  .12.113+deb13-am
│ │ │ │ -00000d40: 6436 3420 2331 2053 4d50 2050 5245 454d  d64 #1 SMP PREEM
│ │ │ │ -00000d50: 5054 5f44 594e 414d 4943 2044 6562 6961  PT_DYNAMIC Debia
│ │ │ │ -00000d60: 6e20 362e 3132 2e31 3133 2d31 7c0a 7c2d  n 6.12.113-1|.|-
│ │ │ │ -00000d70: 2d20 414d 4420 4550 5943 2037 3730 3250  - AMD EPYC 7702P
│ │ │ │ -00000d80: 2036 342d 436f 7265 2050 726f 6365 7373   64-Core Process
│ │ │ │ -00000d90: 6f72 2020 4175 7468 656e 7469 6341 4d44  or  AuthenticAMD
│ │ │ │ -00000da0: 2020 6370 7520 4d48 7a20 3139 3936 2e32    cpu MHz 1996.2
│ │ │ │ -00000db0: 3530 2020 2020 2020 2020 2020 7c0a 7c2d  50          |.|-
│ │ │ │ +00000d30: 2e31 322e 3130 302b 6465 6231 332d 636c  .12.100+deb13-cl
│ │ │ │ +00000d40: 6f75 642d 616d 6436 3420 2331 2053 4d50  oud-amd64 #1 SMP
│ │ │ │ +00000d50: 2050 5245 454d 5054 5f44 594e 414d 4943   PREEMPT_DYNAMIC
│ │ │ │ +00000d60: 2044 6562 6961 6e20 2020 2020 7c0a 7c2d   Debian     |.|-
│ │ │ │ +00000d70: 2d20 496e 7465 6c20 5865 6f6e 2050 726f  - Intel Xeon Pro
│ │ │ │ +00000d80: 6365 7373 6f72 2028 536b 796c 616b 652c  cessor (Skylake,
│ │ │ │ +00000d90: 2049 4252 5329 2020 4765 6e75 696e 6549   IBRS)  GenuineI
│ │ │ │ +00000da0: 6e74 656c 2020 6370 7520 4d48 7a20 3230  ntel  cpu MHz 20
│ │ │ │ +00000db0: 3939 2e39 3938 2020 2020 2020 7c0a 7c2d  99.998      |.|-
│ │ │ │  00000dc0: 2d20 4d61 6361 756c 6179 3220 312e 3236  - Macaulay2 1.26
│ │ │ │  00000dd0: 2e30 362c 2063 6f6d 7069 6c65 6420 7769  .06, compiled wi
│ │ │ │  00000de0: 7468 2067 6363 2031 362e 322e 3020 2020  th gcc 16.2.0   
│ │ │ │  00000df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000e00: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  00000e10: 2d20 7265 7333 393a 2072 6573 206f 6620  - res39: res of 
│ │ │ │  00000e20: 6120 6765 6e65 7269 6320 3320 6279 2039  a generic 3 by 9
│ │ │ │  00000e30: 206d 6174 7269 7820 6f76 6572 205a 5a2f   matrix over ZZ/
│ │ │ │ -00000e40: 3130 313a 202e 3330 3331 3520 7365 636f  101: .30315 seco
│ │ │ │ -00000e50: 6e64 7320 2020 2020 2020 2020 7c0a 7c2d  nds         |.|-
│ │ │ │ +00000e40: 3130 313a 202e 3139 3536 3533 2073 6563  101: .195653 sec
│ │ │ │ +00000e50: 6f6e 6473 2020 2020 2020 2020 7c0a 7c2d  onds        |.|-
│ │ │ │  00000e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00000ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c28  ------------|.|(
│ │ │ │ -00000eb0: 3230 3236 2d31 302d 3135 2920 7838 365f  2026-10-15) x86_
│ │ │ │ -00000ec0: 3634 2047 4e55 2f4c 696e 7578 2020 2020  64 GNU/Linux    
│ │ │ │ -00000ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00000ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c36  ------------|.|6
│ │ │ │ +00000eb0: 2e31 322e 3130 302d 3120 2832 3032 362d  .12.100-1 (2026-
│ │ │ │ +00000ec0: 3037 2d33 3029 2078 3836 5f36 3420 474e  07-30) x86_64 GN
│ │ │ │ +00000ed0: 552f 4c69 6e75 7820 2020 2020 2020 2020  U/Linux         
│ │ │ │  00000ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000ef0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00000f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46  ------------+..F
│ │ ├── ./usr/share/info/Bertini.info.gz
│ │ │ ├── Bertini.info
│ │ │ │ @@ -2253,15 +2253,15 @@
│ │ │ │  00008cc0: 616c 206e 756d 6265 720a 2020 2020 2020  al number.      
│ │ │ │  00008cd0: 2020 6f72 2072 616e 646f 6d20 636f 6d70    or random comp
│ │ │ │  00008ce0: 6c65 7820 6e75 6d62 6572 0a20 2020 2020  lex number.     
│ │ │ │  00008cf0: 202a 202a 6e6f 7465 2054 6f70 4469 7265   * *note TopDire
│ │ │ │  00008d00: 6374 6f72 793a 2054 6f70 4469 7265 6374  ctory: TopDirect
│ │ │ │  00008d10: 6f72 792c 203d 3e20 2e2e 2e2c 2064 6566  ory, => ..., def
│ │ │ │  00008d20: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -00008d30: 2020 2022 2f74 6d70 2f4d 322d 3233 3636     "/tmp/M2-2366
│ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3330 3130     "/tmp/M2-3010
│ │ │ │  00008d40: 392d 302f 3022 2c20 4f70 7469 6f6e 2074  9-0/0", Option t
│ │ │ │  00008d50: 6f20 6368 616e 6765 2064 6972 6563 746f  o change directo
│ │ │ │  00008d60: 7279 2066 6f72 2066 696c 6520 7374 6f72  ry for file stor
│ │ │ │  00008d70: 6167 652e 0a20 2020 2020 202a 202a 6e6f  age..      * *no
│ │ │ │  00008d80: 7465 2056 6572 626f 7365 3a20 6265 7274  te Verbose: bert
│ │ │ │  00008d90: 696e 6954 7261 636b 486f 6d6f 746f 7079  iniTrackHomotopy
│ │ │ │  00008da0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56  _lp_pd_pd_pd_cmV
│ │ │ │ @@ -4971,15 +4971,15 @@
│ │ │ │  000136a0: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │  000136b0: 2064 6566 6175 6c74 2076 616c 7565 207b   default value {
│ │ │ │  000136c0: 7d2c 200a 2020 2020 2020 2a20 2a6e 6f74  }, .      * *not
│ │ │ │  000136d0: 6520 546f 7044 6972 6563 746f 7279 3a20  e TopDirectory: 
│ │ │ │  000136e0: 546f 7044 6972 6563 746f 7279 2c20 3d3e  TopDirectory, =>
│ │ │ │  000136f0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │  00013700: 6c75 650a 2020 2020 2020 2020 222f 746d  lue.        "/tm
│ │ │ │ -00013710: 702f 4d32 2d32 3336 3639 2d30 2f30 222c  p/M2-23669-0/0",
│ │ │ │ +00013710: 702f 4d32 2d33 3031 3039 2d30 2f30 222c  p/M2-30109-0/0",
│ │ │ │  00013720: 204f 7074 696f 6e20 746f 2063 6861 6e67   Option to chang
│ │ │ │  00013730: 6520 6469 7265 6374 6f72 7920 666f 7220  e directory for 
│ │ │ │  00013740: 6669 6c65 2073 746f 7261 6765 2e0a 2020  file storage..  
│ │ │ │  00013750: 2020 2020 2a20 2a6e 6f74 6520 5665 7262      * *note Verb
│ │ │ │  00013760: 6f73 653a 2062 6572 7469 6e69 5472 6163  ose: bertiniTrac
│ │ │ │  00013770: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f  kHomotopy_lp_pd_
│ │ │ │  00013780: 7064 5f70 645f 636d 5665 7262 6f73 653d  pd_pd_cmVerbose=
│ │ │ │ @@ -5472,16 +5472,16 @@
│ │ │ │  000155f0: 6561 6c20 6e75 6d62 6572 0a20 2020 2020  eal number.     
│ │ │ │  00015600: 2020 206f 7220 7261 6e64 6f6d 2063 6f6d     or random com
│ │ │ │  00015610: 706c 6578 206e 756d 6265 720a 2020 2020  plex number.    
│ │ │ │  00015620: 2020 2a20 2a6e 6f74 6520 546f 7044 6972    * *note TopDir
│ │ │ │  00015630: 6563 746f 7279 3a20 546f 7044 6972 6563  ectory: TopDirec
│ │ │ │  00015640: 746f 7279 2c20 3d3e 202e 2e2e 2c20 6465  tory, => ..., de
│ │ │ │  00015650: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ -00015660: 2020 2020 222f 746d 702f 4d32 2d32 3336      "/tmp/M2-236
│ │ │ │ -00015670: 3639 2d30 2f30 222c 204f 7074 696f 6e20  69-0/0", Option 
│ │ │ │ +00015660: 2020 2020 222f 746d 702f 4d32 2d33 3031      "/tmp/M2-301
│ │ │ │ +00015670: 3039 2d30 2f30 222c 204f 7074 696f 6e20  09-0/0", Option 
│ │ │ │  00015680: 746f 2063 6861 6e67 6520 6469 7265 6374  to change direct
│ │ │ │  00015690: 6f72 7920 666f 7220 6669 6c65 2073 746f  ory for file sto
│ │ │ │  000156a0: 7261 6765 2e0a 2020 2020 2020 2a20 5573  rage..      * Us
│ │ │ │  000156b0: 6552 6567 656e 6572 6174 696f 6e20 286d  eRegeneration (m
│ │ │ │  000156c0: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │  000156d0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465  tion) => ..., de
│ │ │ │  000156e0: 6661 756c 7420 7661 6c75 6520 2d31 2c20  fault value -1,
│ │ ├── ./usr/share/info/BettiCharacters.info.gz
│ │ │ ├── BettiCharacters.info
│ │ │ │ @@ -16843,15 +16843,15 @@
│ │ │ │  00041ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00041cc0: 3920 3a20 656c 6170 7365 6454 696d 6520  9 : elapsedTime 
│ │ │ │  00041cd0: 6320 3d20 6368 6172 6163 7465 7220 4120  c = character A 
│ │ │ │  00041ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00041d10: 2d2d 202e 3434 3931 3832 7320 656c 6170  -- .449182s elap
│ │ │ │ +00041d10: 2d2d 202e 3430 3537 3837 7320 656c 6170  -- .405787s elap
│ │ │ │  00041d20: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  00041d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00041d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -18058,16 +18058,16 @@
│ │ │ │  00046890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000468a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000468b0: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 656c  ------+.|i7 : el
│ │ │ │  000468c0: 6170 7365 6454 696d 6520 633d 6368 6172  apsedTime c=char
│ │ │ │  000468d0: 6163 7465 7220 4120 2020 2020 2020 2020  acter A         
│ │ │ │  000468e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000468f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046900: 2020 2020 2020 7c0a 7c20 2d2d 2031 2e30        |.| -- 1.0
│ │ │ │ -00046910: 3632 3635 7320 656c 6170 7365 6420 2020  6265s elapsed   
│ │ │ │ +00046900: 2020 2020 2020 7c0a 7c20 2d2d 202e 3634        |.| -- .64
│ │ │ │ +00046910: 3132 3433 7320 656c 6170 7365 6420 2020  1243s elapsed   
│ │ │ │  00046920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046950: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00046960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -19501,15 +19501,15 @@
│ │ │ │  0004c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c2f0: 2b0a 7c69 3230 203a 2065 6c61 7073 6564  +.|i20 : elapsed
│ │ │ │  0004c300: 5469 6d65 2061 3120 3d20 6368 6172 6163  Time a1 = charac
│ │ │ │  0004c310: 7465 7220 4131 2020 2020 2020 2020 2020  ter A1          
│ │ │ │  0004c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c330: 2020 7c0a 7c20 2d2d 2031 2e30 3637 3132    |.| -- 1.06712
│ │ │ │ +0004c330: 2020 7c0a 7c20 2d2d 202e 3739 3039 3335    |.| -- .790935
│ │ │ │  0004c340: 7320 656c 6170 7365 6420 2020 2020 2020  s elapsed       
│ │ │ │  0004c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c370: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0004c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -19555,15 +19555,15 @@
│ │ │ │  0004c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231  ----------+.|i21
│ │ │ │  0004c650: 203a 2065 6c61 7073 6564 5469 6d65 2061   : elapsedTime a
│ │ │ │  0004c660: 3220 3d20 6368 6172 6163 7465 7220 4132  2 = character A2
│ │ │ │  0004c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0004c690: 2d2d 2033 362e 3335 3339 7320 656c 6170  -- 36.3539s elap
│ │ │ │ +0004c690: 2d2d 2032 372e 3931 3636 7320 656c 6170  -- 27.9166s elap
│ │ │ │  0004c6a0: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  0004c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0004c6d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0004c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -20066,15 +20066,15 @@
│ │ │ │  0004e610: 4f6e 4772 6164 6564 4d6f 6475 6c65 2020  OnGradedModule  
│ │ │ │  0004e620: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0004e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004e650: 2d2b 0a7c 6933 3220 3a20 656c 6170 7365  -+.|i32 : elapse
│ │ │ │  0004e660: 6454 696d 6520 6220 3d20 6368 6172 6163  dTime b = charac
│ │ │ │  0004e670: 7465 7228 422c 3231 297c 0a7c 202d 2d20  ter(B,21)|.| -- 
│ │ │ │ -0004e680: 3136 2e34 3237 3773 2065 6c61 7073 6564  16.4277s elapsed
│ │ │ │ +0004e680: 3132 2e39 3430 3973 2065 6c61 7073 6564  12.9409s elapsed
│ │ │ │  0004e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0004e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e6c0: 2020 2020 2020 2020 207c 0a7c 6f33 3220           |.|o32 
│ │ │ │  0004e6d0: 3d20 4368 6172 6163 7465 7220 6f76 6572  = Character over
│ │ │ │  0004e6e0: 206b 6b20 2020 2020 2020 2020 2020 2020   kk             
│ │ │ │  0004e6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|
│ │ ├── ./usr/share/info/Bruns.info.gz
│ │ │ ├── Bruns.info
│ │ │ │ @@ -1095,17 +1095,17 @@
│ │ │ │  00004460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  00004480: 6932 3320 3a20 7469 6d65 206a 3d62 7275  i23 : time j=bru
│ │ │ │  00004490: 6e73 2046 2e64 645f 333b 2020 2020 2020  ns F.dd_3;      
│ │ │ │  000044a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000044b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000044c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000044d0: 202d 2d20 7573 6564 2030 2e34 3032 3236   -- used 0.40226
│ │ │ │ -000044e0: 3873 2028 6370 7529 3b20 302e 3234 3632  8s (cpu); 0.2462
│ │ │ │ -000044f0: 3439 7320 2874 6872 6561 6429 3b20 3073  49s (thread); 0s
│ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e35 3632 3636   -- used 0.56266
│ │ │ │ +000044e0: 3673 2028 6370 7529 3b20 302e 3337 3234  6s (cpu); 0.3724
│ │ │ │ +000044f0: 3939 7320 2874 6872 6561 6429 3b20 3073  99s (thread); 0s
│ │ │ │  00004500: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  00004510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00004520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004560: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ ├── ./usr/share/info/CellularResolutions.info.gz
│ │ │ ├── CellularResolutions.info
│ │ │ │ @@ -3003,63 +3003,63 @@
│ │ │ │  0000bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0000bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbf0: 2020 2020 2020 7c0a 7c6f 3820 3d20 5265        |.|o8 = Re
│ │ │ │  0000bc00: 6c61 7469 6f6e 204d 6174 7269 783a 207c  lation Matrix: |
│ │ │ │ -0000bc10: 2031 2030 2030 2030 2030 2030 2031 2031   1 0 0 0 0 0 1 1
│ │ │ │ -0000bc20: 2031 2030 2030 2030 2031 2031 2031 2031   1 0 0 0 1 1 1 1
│ │ │ │ +0000bc10: 2031 2030 2030 2030 2030 2030 2030 2030   1 0 0 0 0 0 0 0
│ │ │ │ +0000bc20: 2031 2030 2031 2031 2031 2031 2031 2031   1 0 1 1 1 1 1 1
│ │ │ │  0000bc30: 2031 207c 7c0a 7c20 2020 2020 2020 2020   1 ||.|         
│ │ │ │  0000bc40: 2020 2020 2020 2020 2020 2020 207c 2030               | 0
│ │ │ │ -0000bc50: 2031 2030 2030 2030 2030 2030 2030 2030   1 0 0 0 0 0 0 0
│ │ │ │ -0000bc60: 2031 2031 2030 2031 2031 2031 2030 2030   1 1 0 1 1 1 0 0
│ │ │ │ +0000bc50: 2031 2030 2030 2030 2031 2031 2030 2031   1 0 0 0 1 1 0 1
│ │ │ │ +0000bc60: 2030 2030 2030 2030 2031 2031 2030 2030   0 0 0 0 1 1 0 0
│ │ │ │  0000bc70: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │  0000bc80: 2020 2020 2020 2020 2020 207c 2030 2030             | 0 0
│ │ │ │ -0000bc90: 2031 2030 2030 2030 2031 2030 2030 2031   1 0 0 0 1 0 0 1
│ │ │ │ -0000bca0: 2030 2031 2030 2031 2030 2031 2030 207c   0 1 0 1 0 1 0 |
│ │ │ │ +0000bc90: 2031 2030 2030 2031 2030 2031 2030 2030   1 0 0 1 0 1 0 0
│ │ │ │ +0000bca0: 2031 2030 2030 2031 2030 2031 2030 207c   1 0 0 1 0 1 0 |
│ │ │ │  0000bcb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000bcc0: 2020 2020 2020 2020 207c 2030 2030 2030           | 0 0 0
│ │ │ │ -0000bcd0: 2031 2030 2031 2030 2031 2030 2030 2031   1 0 1 0 1 0 0 1
│ │ │ │ -0000bce0: 2030 2030 2030 2031 2030 2031 207c 7c0a   0 0 0 1 0 1 ||.
│ │ │ │ +0000bcd0: 2031 2030 2030 2031 2030 2030 2031 2030   1 0 0 1 0 0 1 0
│ │ │ │ +0000bce0: 2031 2030 2030 2031 2030 2031 207c 7c0a   1 0 0 1 0 1 ||.
│ │ │ │  0000bcf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0000bd00: 2020 2020 2020 207c 2030 2030 2030 2030         | 0 0 0 0
│ │ │ │ -0000bd10: 2031 2031 2030 2030 2031 2030 2030 2031   1 1 0 0 1 0 0 1
│ │ │ │ -0000bd20: 2030 2030 2030 2031 2031 207c 7c0a 7c20   0 0 0 1 1 ||.| 
│ │ │ │ +0000bd10: 2031 2030 2030 2031 2030 2031 2030 2030   1 0 0 1 0 1 0 0
│ │ │ │ +0000bd20: 2031 2030 2030 2031 2031 207c 7c0a 7c20   1 0 0 1 1 ||.| 
│ │ │ │  0000bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd40: 2020 2020 207c 2030 2030 2030 2030 2030       | 0 0 0 0 0
│ │ │ │  0000bd50: 2031 2030 2030 2030 2030 2030 2030 2030   1 0 0 0 0 0 0 0
│ │ │ │ -0000bd60: 2030 2030 2030 2031 207c 7c0a 7c20 2020   0 0 0 1 ||.|   
│ │ │ │ +0000bd60: 2031 2030 2030 2030 207c 7c0a 7c20 2020   1 0 0 0 ||.|   
│ │ │ │  0000bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd80: 2020 207c 2030 2030 2030 2030 2030 2030     | 0 0 0 0 0 0
│ │ │ │ -0000bd90: 2031 2030 2030 2030 2030 2030 2030 2031   1 0 0 0 0 0 0 1
│ │ │ │ -0000bda0: 2030 2031 2030 207c 7c0a 7c20 2020 2020   0 1 0 ||.|     
│ │ │ │ +0000bd90: 2031 2030 2030 2030 2030 2030 2030 2030   1 0 0 0 0 0 0 0
│ │ │ │ +0000bda0: 2031 2030 2030 207c 7c0a 7c20 2020 2020   1 0 0 ||.|     
│ │ │ │  0000bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bdc0: 207c 2030 2030 2030 2030 2030 2030 2030   | 0 0 0 0 0 0 0
│ │ │ │ -0000bdd0: 2031 2030 2030 2030 2030 2030 2030 2031   1 0 0 0 0 0 0 1
│ │ │ │ -0000bde0: 2030 2031 207c 7c0a 7c20 2020 2020 2020   0 1 ||.|       
│ │ │ │ +0000bdd0: 2031 2030 2030 2030 2030 2030 2030 2030   1 0 0 0 0 0 0 0
│ │ │ │ +0000bde0: 2031 2030 207c 7c0a 7c20 2020 2020 2020   1 0 ||.|       
│ │ │ │  0000bdf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000be00: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │ -0000be10: 2031 2030 2030 2030 2030 2030 2030 2031   1 0 0 0 0 0 0 1
│ │ │ │ -0000be20: 2031 207c 7c0a 7c20 2020 2020 2020 2020   1 ||.|         
│ │ │ │ +0000be10: 2031 2030 2030 2030 2030 2031 2031 2030   1 0 0 0 0 1 1 0
│ │ │ │ +0000be20: 2030 207c 7c0a 7c20 2020 2020 2020 2020   0 ||.|         
│ │ │ │  0000be30: 2020 2020 2020 2020 2020 2020 207c 2030               | 0
│ │ │ │  0000be40: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │ -0000be50: 2031 2030 2030 2030 2031 2030 2030 2030   1 0 0 0 1 0 0 0
│ │ │ │ +0000be50: 2031 2030 2030 2030 2030 2030 2030 2031   1 0 0 0 0 0 0 1
│ │ │ │  0000be60: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │  0000be70: 2020 2020 2020 2020 2020 207c 2030 2030             | 0 0
│ │ │ │  0000be80: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │ -0000be90: 2031 2030 2030 2030 2031 2030 2030 207c   1 0 0 0 1 0 0 |
│ │ │ │ +0000be90: 2031 2030 2030 2031 2030 2031 2030 207c   1 0 0 1 0 1 0 |
│ │ │ │  0000bea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0000beb0: 2020 2020 2020 2020 207c 2030 2030 2030           | 0 0 0
│ │ │ │  0000bec0: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │ -0000bed0: 2031 2030 2030 2030 2031 2030 207c 7c0a   1 0 0 0 1 0 ||.
│ │ │ │ +0000bed0: 2031 2030 2030 2031 2030 2031 207c 7c0a   1 0 0 1 0 1 ||.
│ │ │ │  0000bee0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0000bef0: 2020 2020 2020 207c 2030 2030 2030 2030         | 0 0 0 0
│ │ │ │  0000bf00: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │ -0000bf10: 2031 2031 2031 2030 2030 207c 7c0a 7c20   1 1 1 0 0 ||.| 
│ │ │ │ +0000bf10: 2031 2030 2030 2031 2031 207c 7c0a 7c20   1 0 0 1 1 ||.| 
│ │ │ │  0000bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bf30: 2020 2020 207c 2030 2030 2030 2030 2030       | 0 0 0 0 0
│ │ │ │  0000bf40: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │  0000bf50: 2031 2030 2030 2030 207c 7c0a 7c20 2020   1 0 0 0 ||.|   
│ │ │ │  0000bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bf70: 2020 207c 2030 2030 2030 2030 2030 2030     | 0 0 0 0 0 0
│ │ │ │  0000bf80: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │ @@ -3679,23 +3679,23 @@
│ │ │ │  0000e5e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000e5f0: 0a7c 6f38 203d 2048 6173 6854 6162 6c65  .|o8 = HashTable
│ │ │ │  0000e600: 7b30 203d 3e20 7b78 202c 2078 2079 2c20  {0 => {x , x y, 
│ │ │ │  0000e610: 7820 7920 2c20 7820 7920 2c20 782a 7920  x y , x y , x*y 
│ │ │ │  0000e620: 2c20 7820 7d20 2020 2020 2020 2020 2020  , x }           
│ │ │ │  0000e630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000e640: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0000e650: 2020 2020 2020 2020 3520 3320 2020 3520          5 3   5 
│ │ │ │ -0000e660: 3420 2020 3420 3220 2020 3420 3420 2020  4   4 2   4 4   
│ │ │ │ -0000e670: 3520 2020 2033 2033 2020 2035 2032 2020  5    3 3   5 2  
│ │ │ │ -0000e680: 2032 2034 2020 2035 2033 2020 2020 207c   2 4   5 3     |
│ │ │ │ +0000e650: 2020 2020 2020 2020 3420 3420 2020 3520          4 4   5 
│ │ │ │ +0000e660: 2020 2033 2033 2020 2035 2032 2020 2032     3 3   5 2   2
│ │ │ │ +0000e670: 2034 2020 2035 2033 2020 2035 2034 2020   4   5 3   5 4  
│ │ │ │ +0000e680: 2035 2020 2020 3520 3220 2020 2020 207c   5    5 2      |
│ │ │ │  0000e690: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000e6a0: 2031 203d 3e20 7b78 2079 202c 2078 2079   1 => {x y , x y
│ │ │ │ -0000e6b0: 202c 2078 2079 202c 2078 2079 202c 2078   , x y , x y , x
│ │ │ │ -0000e6c0: 2079 2c20 7820 7920 2c20 7820 7920 2c20   y, x y , x y , 
│ │ │ │ -0000e6d0: 7820 7920 2c20 7820 7920 2c20 2020 207c  x y , x y ,    |
│ │ │ │ +0000e6b0: 2c20 7820 7920 2c20 7820 7920 2c20 7820  , x y , x y , x 
│ │ │ │ +0000e6c0: 7920 2c20 7820 7920 2c20 7820 7920 2c20  y , x y , x y , 
│ │ │ │ +0000e6d0: 7820 792c 2078 2079 202c 2020 2020 207c  x y, x y ,     |
│ │ │ │  0000e6e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000e6f0: 2020 2020 2020 2020 3520 3220 2020 3520          5 2   5 
│ │ │ │  0000e700: 3420 2020 3520 3320 2020 3520 3420 2020  4   5 3   5 4   
│ │ │ │  0000e710: 3520 3220 2020 3520 3420 2020 3520 3320  5 2   5 4   5 3 
│ │ │ │  0000e720: 2020 3520 3420 2020 2020 2020 2020 207c    5 4          |
│ │ │ │  0000e730: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000e740: 2032 203d 3e20 7b78 2079 202c 2078 2079   2 => {x y , x y
│ │ │ │ @@ -3714,25 +3714,25 @@
│ │ │ │  0000e810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000e820: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │  0000e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │  0000e870: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0000e880: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │ +0000e880: 2020 207d 2020 2020 2020 2020 2020 2020     }            
│ │ │ │  0000e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e8b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000e8c0: 0a7c 2035 2034 2020 2035 2020 2020 3520  .| 5 4   5    5 
│ │ │ │ -0000e8d0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0000e8c0: 0a7c 2035 2033 2020 2035 2034 2020 2034  .| 5 3   5 4   4
│ │ │ │ +0000e8d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0000e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000e910: 0a7c 7820 7920 2c20 7820 792c 2078 2079  .|x y , x y, x y
│ │ │ │ -0000e920: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +0000e910: 0a7c 7820 7920 2c20 7820 7920 2c20 7820  .|x y , x y , x 
│ │ │ │ +0000e920: 7920 7d20 2020 2020 2020 2020 2020 2020  y }             
│ │ │ │  0000e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000e950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000e960: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0000e970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -4695,15 +4695,15 @@
│ │ │ │  00012560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000125a0: 2020 207c 0a7c 6f37 203d 2048 6173 6854     |.|o7 = HashT
│ │ │ │  000125b0: 6162 6c65 7b30 203d 3e20 7b43 656c 6c20  able{0 => {Cell 
│ │ │ │  000125c0: 6f66 2064 696d 656e 7369 6f6e 2030 2077  of dimension 0 w
│ │ │ │ -000125d0: 6974 6820 6c61 6265 6c20 792c 2043 656c  ith label y, Cel
│ │ │ │ +000125d0: 6974 6820 6c61 6265 6c20 782c 2043 656c  ith label x, Cel
│ │ │ │  000125e0: 6c20 6f66 2064 696d 656e 7369 6f6e 2030  l of dimension 0
│ │ │ │  000125f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00012600: 2020 2020 2031 203d 3e20 7b43 656c 6c20       1 => {Cell 
│ │ │ │  00012610: 6f66 2064 696d 656e 7369 6f6e 2031 2077  of dimension 1 w
│ │ │ │  00012620: 6974 6820 6c61 6265 6c20 782a 797d 2020  ith label x*y}  
│ │ │ │  00012630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012640: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ @@ -4720,15 +4720,15 @@
│ │ │ │  000126f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012730: 2d2d 2d7c 0a7c 7769 7468 206c 6162 656c  ---|.|with label
│ │ │ │  00012740: 207a 2c20 4365 6c6c 206f 6620 6469 6d65   z, Cell of dime
│ │ │ │  00012750: 6e73 696f 6e20 3020 7769 7468 206c 6162  nsion 0 with lab
│ │ │ │ -00012760: 656c 2078 7d7d 2020 2020 2020 2020 2020  el x}}          
│ │ │ │ +00012760: 656c 2079 7d7d 2020 2020 2020 2020 2020  el y}}          
│ │ │ │  00012770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012780: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00012790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000127a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000127b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000127c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000127d0: 2d2d 2d2b 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ---+.+----------
│ │ │ │ @@ -7252,22 +7252,22 @@
│ │ │ │  0001c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5a0: 7c0a 7c20 2020 2020 2020 3520 3420 2020  |.|       5 4   
│ │ │ │ -0001c5b0: 3320 3520 2020 2034 2035 2020 2032 2020  3 5    4 5   2  
│ │ │ │ -0001c5c0: 2020 2020 2020 3220 2020 2034 2034 2020        2    4 4  
│ │ │ │ -0001c5d0: 2020 3520 3320 2020 2020 2020 2020 2020    5 3           
│ │ │ │ +0001c5a0: 7c0a 7c20 2020 2020 2020 2020 3220 2020  |.|         2   
│ │ │ │ +0001c5b0: 2034 2034 2020 2020 3520 3320 2020 2035   4 4    5 3    5
│ │ │ │ +0001c5c0: 2034 2020 2033 2035 2020 2020 3420 3520   4   3 5    4 5 
│ │ │ │ +0001c5d0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0001c5e0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -0001c5f0: 203d 207b 7820 7920 2c20 7820 7920 7a2c   = {x y , x y z,
│ │ │ │ -0001c600: 2078 2079 202c 2078 2079 2a7a 2c20 782a   x y , x y*z, x*
│ │ │ │ -0001c610: 7920 7a2c 2078 2079 207a 2c20 7820 7920  y z, x y z, x y 
│ │ │ │ +0001c5f0: 203d 207b 782a 7920 7a2c 2078 2079 207a   = {x*y z, x y z
│ │ │ │ +0001c600: 2c20 7820 7920 7a2c 2078 2079 202c 2078  , x y z, x y , x
│ │ │ │ +0001c610: 2079 207a 2c20 7820 7920 2c20 7820 792a   y z, x y , x y*
│ │ │ │  0001c620: 7a7d 2020 2020 2020 2020 2020 2020 2020  z}              
│ │ │ │  0001c630: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c680: 207c 0a7c 6f35 203a 204c 6973 7420 2020   |.|o5 : List   
│ │ │ │ @@ -7285,16 +7285,16 @@
│ │ │ │  0001c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c760: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001c7b0: 2020 2020 2020 2035 2034 2020 2020 3420         5 4    4 
│ │ │ │ -0001c7c0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ +0001c7b0: 2020 2020 2020 2034 2035 2020 2020 3520         4 5    5 
│ │ │ │ +0001c7c0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7f0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │  0001c800: 7b78 2079 207a 2c20 7820 7920 7a7d 2020  {x y z, x y z}  
│ │ │ │  0001c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -9367,21 +9367,21 @@
│ │ │ │  00024960: 6162 656c 2863 2920 2020 2020 2020 2020  abel(c)         
│ │ │ │  00024970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024980: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000249e0: 3220 2020 2020 2032 2020 2032 2020 2020  2      2   2    
│ │ │ │ +000249d0: 7c0a 7c20 2020 2020 2020 2032 2020 2020  |.|        2    
│ │ │ │ +000249e0: 2020 3220 2020 3220 2020 2020 2020 2020    2   2         
│ │ │ │  000249f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a20: 7c0a 7c6f 3133 203d 207b 612a 632c 2061  |.|o13 = {a*c, a
│ │ │ │ -00024a30: 2062 2c20 622a 6320 2c20 6220 7d20 2020   b, b*c , b }   
│ │ │ │ +00024a20: 7c0a 7c6f 3133 203d 207b 6120 622c 2062  |.|o13 = {a b, b
│ │ │ │ +00024a30: 2a63 202c 2062 202c 2061 2a63 7d20 2020  *c , b , a*c}   
│ │ │ │  00024a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -9402,22 +9402,22 @@
│ │ │ │  00024b90: 6162 656c 2863 2920 2020 2020 2020 2020  abel(c)         
│ │ │ │  00024ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024bb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c00: 7c0a 7c20 2020 2020 2020 2020 2032 2020  |.|          2  
│ │ │ │ -00024c10: 2020 3220 2020 3220 2020 3220 2020 2020    2   2   2     
│ │ │ │ -00024c20: 2032 2032 2020 2020 2020 2032 2020 2020   2 2       2    
│ │ │ │ +00024c00: 7c0a 7c20 2020 2020 2020 2032 2032 2020  |.|        2 2  
│ │ │ │ +00024c10: 2032 2020 2032 2020 2020 2032 2020 2020   2   2     2    
│ │ │ │ +00024c20: 3220 3220 2020 2020 2020 3220 2020 2020  2 2       2     
│ │ │ │  00024c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c50: 7c0a 7c6f 3134 203d 207b 612a 6220 632c  |.|o14 = {a*b c,
│ │ │ │ -00024c60: 2061 2062 2a63 202c 2061 2062 2a63 2c20   a b*c , a b*c, 
│ │ │ │ -00024c70: 6120 6220 2c20 612a 622a 6320 7d20 2020  a b , a*b*c }   
│ │ │ │ +00024c50: 7c0a 7c6f 3134 203d 207b 6220 6320 2c20  |.|o14 = {b c , 
│ │ │ │ +00024c60: 6120 622a 6320 2c20 612a 6220 632c 2061  a b*c , a*b c, a
│ │ │ │ +00024c70: 2062 202c 2061 2a62 2a63 207d 2020 2020   b , a*b*c }    
│ │ │ │  00024c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ca0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00024cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -10856,38 +10856,38 @@
│ │ │ │  0002a670: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0002a680: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0002a690: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │  0002a6a0: 2020 7c0a 7c20 2020 2020 3020 3a20 5320    |.|     0 : S 
│ │ │ │  0002a6b0: 203c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   <--------------
│ │ │ │  0002a6c0: 2d2d 2d2d 2d2d 2d2d 2d20 5320 203a 2031  --------- S  : 1
│ │ │ │  0002a6d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002a6e0: 2020 7b32 7d20 7c20 2d79 3220 3020 2020    {2} | -y2 0   
│ │ │ │ -0002a6f0: 2d7a 3220 7c20 2020 2020 2020 2020 7c0a  -z2 |         |.
│ │ │ │ +0002a6e0: 2020 7b32 7d20 7c20 2d7a 3220 2d79 3220    {2} | -z2 -y2 
│ │ │ │ +0002a6f0: 3020 2020 7c20 2020 2020 2020 2020 7c0a  0   |         |.
│ │ │ │  0002a700: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002a710: 7b32 7d20 7c20 7832 2020 2d7a 3220 3020  {2} | x2  -z2 0 
│ │ │ │ -0002a720: 2020 7c20 2020 2020 2020 2020 7c0a 7c20    |         |.| 
│ │ │ │ +0002a710: 7b32 7d20 7c20 3020 2020 7832 2020 2d7a  {2} | 0   x2  -z
│ │ │ │ +0002a720: 3220 7c20 2020 2020 2020 2020 7c0a 7c20  2 |         |.| 
│ │ │ │  0002a730: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -0002a740: 7d20 7c20 3020 2020 7932 2020 7832 2020  } | 0   y2  x2  
│ │ │ │ +0002a740: 7d20 7c20 7832 2020 3020 2020 7932 2020  } | x2  0   y2  
│ │ │ │  0002a750: 7c20 2020 2020 2020 2020 7c0a 7c20 2020  |         |.|   
│ │ │ │  0002a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a780: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0002a790: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │  0002a7a0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │  0002a7b0: 2020 2020 2020 7c0a 7c20 2020 2020 3120        |.|     1 
│ │ │ │  0002a7c0: 3a20 5320 203c 2d2d 2d2d 2d2d 2d2d 2d2d  : S  <----------
│ │ │ │  0002a7d0: 2d2d 2d2d 2d20 5320 203a 2032 2020 2020  ----- S  : 2    
│ │ │ │  0002a7e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002a7f0: 2020 2020 2020 7b34 7d20 7c20 2d7a 3220        {4} | -z2 
│ │ │ │ +0002a7f0: 2020 2020 2020 7b34 7d20 7c20 7932 2020        {4} | y2  
│ │ │ │  0002a800: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002a810: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002a820: 2020 2020 7b34 7d20 7c20 2d78 3220 7c20      {4} | -x2 | 
│ │ │ │ +0002a820: 2020 2020 7b34 7d20 7c20 2d7a 3220 7c20      {4} | -z2 | 
│ │ │ │  0002a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002a850: 2020 7b34 7d20 7c20 7932 2020 7c20 2020    {4} | y2  |   
│ │ │ │ +0002a850: 2020 7b34 7d20 7c20 2d78 3220 7c20 2020    {4} | -x2 |   
│ │ │ │  0002a860: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  0002a870: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0002a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a890: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │  0002a8a0: 3520 3a20 436f 6d70 6c65 784d 6170 2020  5 : ComplexMap  
│ │ │ │  0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ ├── ./usr/share/info/ChainComplexExtras.info.gz
│ │ │ ├── ChainComplexExtras.info
│ │ │ │ @@ -6049,17 +6049,17 @@
│ │ │ │  00017a00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00017a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017a30: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2074  ------+.|i13 : t
│ │ │ │  00017a40: 696d 6520 6d20 3d20 6d69 6e69 6d69 7a65  ime m = minimize
│ │ │ │  00017a50: 2028 455b 315d 293b 2020 2020 2020 2020   (E[1]);        
│ │ │ │  00017a60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00017a70: 7c20 2d2d 2075 7365 6420 302e 3239 3635  | -- used 0.2965
│ │ │ │ -00017a80: 3733 7320 2863 7075 293b 2030 2e32 3137  73s (cpu); 0.217
│ │ │ │ -00017a90: 3036 3473 2028 7468 7265 6164 293b 2030  064s (thread); 0
│ │ │ │ +00017a70: 7c20 2d2d 2075 7365 6420 302e 3434 3333  | -- used 0.4433
│ │ │ │ +00017a80: 3434 7320 2863 7075 293b 2030 2e33 3231  44s (cpu); 0.321
│ │ │ │ +00017a90: 3030 3273 2028 7468 7265 6164 293b 2030  002s (thread); 0
│ │ │ │  00017aa0: 7320 2867 6329 7c0a 2b2d 2d2d 2d2d 2d2d  s (gc)|.+-------
│ │ │ │  00017ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00017ae0: 7c69 3134 203a 2069 7351 7561 7369 4973  |i14 : isQuasiIs
│ │ │ │  00017af0: 6f6d 6f72 7068 6973 6d20 6d20 2020 2020  omorphism m     
│ │ │ │  00017b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -7810,32 +7810,32 @@
│ │ │ │  0001e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e820: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074  -------+.|i8 : t
│ │ │ │  0001e830: 696d 6520 6d20 3d20 7265 736f 6c75 7469  ime m = resoluti
│ │ │ │  0001e840: 6f6e 4f66 4368 6169 6e43 6f6d 706c 6578  onOfChainComplex
│ │ │ │  0001e850: 2043 3b20 2020 2020 2020 2020 2020 2020   C;             
│ │ │ │  0001e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e870: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -0001e880: 6564 2030 2e30 3939 3530 3833 7320 2863  ed 0.0995083s (c
│ │ │ │ -0001e890: 7075 293b 2030 2e30 3938 3736 3537 7320  pu); 0.0987657s 
│ │ │ │ -0001e8a0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -0001e8b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001e880: 6564 2030 2e31 3331 3939 3373 2028 6370  ed 0.131993s (cp
│ │ │ │ +0001e890: 7529 3b20 302e 3132 3932 3533 7320 2874  u); 0.129253s (t
│ │ │ │ +0001e8a0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0001e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e910: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074  -------+.|i9 : t
│ │ │ │  0001e920: 696d 6520 6e20 3d20 6361 7274 616e 4569  ime n = cartanEi
│ │ │ │  0001e930: 6c65 6e62 6572 6752 6573 6f6c 7574 696f  lenbergResolutio
│ │ │ │  0001e940: 6e20 433b 2020 2020 2020 2020 2020 2020  n C;            
│ │ │ │  0001e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e960: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -0001e970: 6564 2030 2e31 3133 3236 3173 2028 6370  ed 0.113261s (cp
│ │ │ │ -0001e980: 7529 3b20 302e 3131 3439 7320 2874 6872  u); 0.1149s (thr
│ │ │ │ -0001e990: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +0001e970: 6564 2030 2e31 3631 3632 3873 2028 6370  ed 0.161628s (cp
│ │ │ │ +0001e980: 7529 3b20 302e 3136 3237 3635 7320 2874  u); 0.162765s (t
│ │ │ │ +0001e990: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  0001e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e9b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0001e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ea00: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 :
│ │ ├── ./usr/share/info/CharacteristicClasses.info.gz
│ │ │ ├── CharacteristicClasses.info
│ │ │ │ @@ -1215,16 +1215,16 @@
│ │ │ │  00004be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004bf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │  00004c00: 2074 696d 6520 4353 4d20 5520 2020 2020   time CSM U     
│ │ │ │  00004c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004c40: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -00004c50: 7573 6564 2030 2e32 3732 3834 3173 2028  used 0.272841s (
│ │ │ │ -00004c60: 6370 7529 3b20 302e 3137 3635 3436 7320  cpu); 0.176546s 
│ │ │ │ +00004c50: 7573 6564 2030 2e33 3339 3434 3873 2028  used 0.339448s (
│ │ │ │ +00004c60: 6370 7529 3b20 302e 3233 3430 3033 7320  cpu); 0.234003s 
│ │ │ │  00004c70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  00004c80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00004c90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1300,18 +1300,18 @@
│ │ │ │  00005130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00005140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │  00005150: 2074 696d 6520 4353 4d28 552c 4368 6563   time CSM(U,Chec
│ │ │ │  00005160: 6b53 6d6f 6f74 683d 3e66 616c 7365 2920  kSmooth=>false) 
│ │ │ │  00005170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005190: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000051a0: 7573 6564 2030 2e34 3134 3636 3273 2028  used 0.414662s (
│ │ │ │ -000051b0: 6370 7529 3b20 302e 3332 3231 3836 7320  cpu); 0.322186s 
│ │ │ │ -000051c0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -000051d0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000051a0: 7573 6564 2030 2e34 3937 3973 2028 6370  used 0.4979s (cp
│ │ │ │ +000051b0: 7529 3b20 302e 3338 3836 3438 7320 2874  u); 0.388648s (t
│ │ │ │ +000051c0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +000051d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000051e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000051f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005230: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00005240: 2020 2037 2020 2020 2020 3620 2020 2020     7      6     
│ │ │ │ @@ -4341,18 +4341,18 @@
│ │ │ │  00010f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │  00010f70: 2074 696d 6520 4353 4d28 492c 436f 6d70   time CSM(I,Comp
│ │ │ │  00010f80: 4d65 7468 6f64 3d3e 5072 6f6a 6563 7469  Method=>Projecti
│ │ │ │  00010f90: 7665 4465 6772 6565 2920 2020 2020 2020  veDegree)       
│ │ │ │  00010fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00010fb0: 2d2d 2075 7365 6420 302e 3535 3336 3233  -- used 0.553623
│ │ │ │ -00010fc0: 7320 2863 7075 293b 2030 2e33 3833 3036  s (cpu); 0.38306
│ │ │ │ -00010fd0: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ -00010fe0: 2867 6329 2020 2020 2020 2020 2020 207c  (gc)           |
│ │ │ │ +00010fb0: 2d2d 2075 7365 6420 312e 3132 3434 3673  -- used 1.12446s
│ │ │ │ +00010fc0: 2028 6370 7529 3b20 302e 3439 3539 3538   (cpu); 0.495958
│ │ │ │ +00010fd0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00010fe0: 6763 2920 2020 2020 2020 2020 2020 207c  gc)            |
│ │ │ │  00010ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00011000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011030: 2020 7c0a 7c20 2020 2020 2020 3520 2020    |.|       5   
│ │ │ │  00011040: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │  00011050: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ @@ -4400,16 +4400,16 @@
│ │ │ │  000112f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011310: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520  ---+.|i6 : time 
│ │ │ │  00011320: 4353 4d28 492c 436f 6d70 4d65 7468 6f64  CSM(I,CompMethod
│ │ │ │  00011330: 3d3e 506e 5265 7369 6475 616c 2920 2020  =>PnResidual)   
│ │ │ │  00011340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011350: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00011360: 6420 322e 3331 3331 3373 2028 6370 7529  d 2.31313s (cpu)
│ │ │ │ -00011370: 3b20 312e 3835 3435 3173 2028 7468 7265  ; 1.85451s (thre
│ │ │ │ +00011360: 6420 322e 3536 3033 3373 2028 6370 7529  d 2.56033s (cpu)
│ │ │ │ +00011370: 3b20 322e 3234 3234 3673 2028 7468 7265  ; 2.24246s (thre
│ │ │ │  00011380: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00011390: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000113a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000113d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000113e0: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ @@ -4488,16 +4488,16 @@
│ │ │ │  00011870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011890: 2d2d 2b0a 7c69 3130 203a 2074 696d 6520  --+.|i10 : time 
│ │ │ │  000118a0: 4353 4d28 4b2c 436f 6d70 4d65 7468 6f64  CSM(K,CompMethod
│ │ │ │  000118b0: 3d3e 5072 6f6a 6563 7469 7665 4465 6772  =>ProjectiveDegr
│ │ │ │  000118c0: 6565 2920 2020 2020 2020 2020 2020 2020  ee)             
│ │ │ │  000118d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -000118e0: 2030 2e33 3032 3131 3273 2028 6370 7529   0.302112s (cpu)
│ │ │ │ -000118f0: 3b20 302e 3230 3531 3735 7320 2874 6872  ; 0.205175s (thr
│ │ │ │ +000118e0: 2030 2e33 3736 3137 3173 2028 6370 7529   0.376171s (cpu)
│ │ │ │ +000118f0: 3b20 302e 3237 3137 3136 7320 2874 6872  ; 0.271716s (thr
│ │ │ │  00011900: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00011910: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00011920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00011960: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ @@ -4546,18 +4546,18 @@
│ │ │ │  00011c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00011c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  00011c40: 3131 203a 2074 696d 6520 4353 4d28 4b2c  11 : time CSM(K,
│ │ │ │  00011c50: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265  CompMethod=>PnRe
│ │ │ │  00011c60: 7369 6475 616c 2920 2020 2020 2020 2020  sidual)         
│ │ │ │  00011c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00011c80: 0a7c 202d 2d20 7573 6564 2030 2e30 3837  .| -- used 0.087
│ │ │ │ -00011c90: 3934 3638 7320 2863 7075 293b 2030 2e30  9468s (cpu); 0.0
│ │ │ │ -00011ca0: 3837 3935 3573 2028 7468 7265 6164 293b  87955s (thread);
│ │ │ │ -00011cb0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │ +00011c80: 0a7c 202d 2d20 7573 6564 2030 2e31 3237  .| -- used 0.127
│ │ │ │ +00011c90: 3336 3473 2028 6370 7529 3b20 302e 3132  364s (cpu); 0.12
│ │ │ │ +00011ca0: 3733 3732 7320 2874 6872 6561 6429 3b20  7372s (thread); 
│ │ │ │ +00011cb0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  00011cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00011d10: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │  00011d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5446,6790 +5446,6790 @@
│ │ │ │  00015450: 2072 6574 7572 6e65 6420 696e 2074 6865   returned in the
│ │ │ │  00015460: 2073 616d 6520 7269 6e67 2e20 5765 206d   same ring. We m
│ │ │ │  00015470: 6179 2061 6c73 6f20 7265 7475 726e 2061  ay also return a
│ │ │ │  00015480: 0a4d 7574 6162 6c65 4861 7368 5461 626c  .MutableHashTabl
│ │ │ │  00015490: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e...+-----------
│ │ │ │  000154a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000154b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000154c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -000154d0: 203a 2052 3d4d 756c 7469 5072 6f6a 436f   : R=MultiProjCo
│ │ │ │ -000154e0: 6f72 6452 696e 6728 7b32 2c32 7d29 2020  ordRing({2,2})  
│ │ │ │ +000154c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +000154d0: 3a20 523d 4d75 6c74 6950 726f 6a43 6f6f  : R=MultiProjCoo
│ │ │ │ +000154e0: 7264 5269 6e67 287b 322c 327d 2920 2020  rdRing({2,2})   
│ │ │ │  000154f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015500: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015500: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015530: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -00015540: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ +00015530: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ +00015540: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  00015550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015570: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015560: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015570: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00015580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000155a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -000155b0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -000155c0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000155e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000155a0: 2020 2020 207c 0a7c 6f31 3120 3a20 506f       |.|o11 : Po
│ │ │ │ +000155b0: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +000155c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000155e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000155f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ -00015620: 203a 2041 3d43 686f 7752 696e 6728 5229   : A=ChowRing(R)
│ │ │ │ +00015610: 2d2d 2d2b 0a7c 6931 3220 3a20 413d 4368  ---+.|i12 : A=Ch
│ │ │ │ +00015620: 6f77 5269 6e67 2852 2920 2020 2020 2020  owRing(R)       
│ │ │ │  00015630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015650: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015640: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015680: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -00015690: 203d 2041 2020 2020 2020 2020 2020 2020   = A            
│ │ │ │ +00015680: 207c 0a7c 6f31 3220 3d20 4120 2020 2020   |.|o12 = A     
│ │ │ │ +00015690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000156a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000156b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000156c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000156d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000156f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ -00015700: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ +000156e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000156f0: 0a7c 6f31 3220 3a20 5175 6f74 6965 6e74  .|o12 : Quotient
│ │ │ │ +00015700: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  00015710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015730: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00015720: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00015730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015760: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ -00015770: 203a 2072 3d67 656e 7320 5220 2020 2020   : r=gens R     
│ │ │ │ +00015750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00015760: 6931 3320 3a20 723d 6765 6e73 2052 2020  i13 : r=gens R  
│ │ │ │ +00015770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015790: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000157a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000157b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3133            |.|o13
│ │ │ │ -000157e0: 203d 207b 7820 2c20 7820 2c20 7820 2c20   = {x , x , x , 
│ │ │ │ -000157f0: 7820 2c20 7820 2c20 7820 7d20 2020 2020  x , x , x }     
│ │ │ │ -00015800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015810: 2020 7c0a 7c20 2020 2020 2020 2030 2020    |.|        0  
│ │ │ │ -00015820: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00015830: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -00015840: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000157c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +000157d0: 3320 3d20 7b78 202c 2078 202c 2078 202c  3 = {x , x , x ,
│ │ │ │ +000157e0: 2078 202c 2078 202c 2078 207d 2020 2020   x , x , x }    
│ │ │ │ +000157f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015800: 2020 7c0a 7c20 2020 2020 2020 2030 2020    |.|        0  
│ │ │ │ +00015810: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ +00015820: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +00015830: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00015840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015880: 2020 7c0a 7c6f 3133 203a 204c 6973 7420    |.|o13 : List 
│ │ │ │ +00015870: 7c0a 7c6f 3133 203a 204c 6973 7420 2020  |.|o13 : List   
│ │ │ │ +00015880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000158a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000158b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000158c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000158f0: 2d2d 2b0a 7c69 3134 203a 204b 3d69 6465  --+.|i14 : K=ide
│ │ │ │ -00015900: 616c 2872 5f30 5e32 2a72 5f33 2d72 5f34  al(r_0^2*r_3-r_4
│ │ │ │ -00015910: 2a72 5f31 2a72 5f32 2c72 5f32 5e32 2a72  *r_1*r_2,r_2^2*r
│ │ │ │ -00015920: 5f35 2920 2020 2020 2020 7c0a 7c20 2020  _5)       |.|   
│ │ │ │ +000158d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000158e0: 7c69 3134 203a 204b 3d69 6465 616c 2872  |i14 : K=ideal(r
│ │ │ │ +000158f0: 5f30 5e32 2a72 5f33 2d72 5f34 2a72 5f31  _0^2*r_3-r_4*r_1
│ │ │ │ +00015900: 2a72 5f32 2c72 5f32 5e32 2a72 5f35 2920  *r_2,r_2^2*r_5) 
│ │ │ │ +00015910: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00015920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015960: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015970: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00015980: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00015990: 2020 2020 2020 2020 2020 7c0a 7c6f 3134            |.|o14
│ │ │ │ -000159a0: 203d 2069 6465 616c 2028 7820 7820 202d   = ideal (x x  -
│ │ │ │ -000159b0: 2078 2078 2078 202c 2078 2078 2029 2020   x x x , x x )  
│ │ │ │ -000159c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000159e0: 2020 2030 2033 2020 2020 3120 3220 3420     0 3    1 2 4 
│ │ │ │ -000159f0: 2020 3220 3520 2020 2020 2020 2020 2020    2 5           
│ │ │ │ -00015a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00015950: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00015960: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015980: 2020 207c 0a7c 6f31 3420 3d20 6964 6561     |.|o14 = idea
│ │ │ │ +00015990: 6c20 2878 2078 2020 2d20 7820 7820 7820  l (x x  - x x x 
│ │ │ │ +000159a0: 2c20 7820 7820 2920 2020 2020 2020 2020  , x x )         
│ │ │ │ +000159b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000159c0: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ +000159d0: 2020 3120 3220 3420 2020 3220 3520 2020    1 2 4   2 5   
│ │ │ │ +000159e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000159f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a40: 2020 7c0a 7c6f 3134 203a 2049 6465 616c    |.|o14 : Ideal
│ │ │ │ -00015a50: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ -00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00015a20: 2020 2020 2020 2020 7c0a 7c6f 3134 203a          |.|o14 :
│ │ │ │ +00015a30: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ +00015a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015a60: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ab0: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520  --+.|i15 : time 
│ │ │ │ -00015ac0: 6373 6d4b 3d43 534d 2841 2c4b 2920 2020  csmK=CSM(A,K)   
│ │ │ │ -00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -00015af0: 2075 7365 6420 302e 3339 3939 3339 7320   used 0.399939s 
│ │ │ │ -00015b00: 2863 7075 293b 2030 2e33 3130 3437 3373  (cpu); 0.310473s
│ │ │ │ -00015b10: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00015b20: 6329 7c0a 7c20 2020 2020 2020 2020 2020  c)|.|           
│ │ │ │ -00015b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00015b60: 2020 2020 2032 2032 2020 2020 2032 2020       2 2     2  
│ │ │ │ -00015b70: 2020 2020 2020 2032 2020 2020 3220 2020         2    2   
│ │ │ │ -00015b80: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00015b90: 2020 7c0a 7c6f 3135 203d 2037 6820 6820    |.|o15 = 7h h 
│ │ │ │ -00015ba0: 202b 2035 6820 6820 202b 2034 6820 6820   + 5h h  + 4h h 
│ │ │ │ -00015bb0: 202b 2068 2020 2b20 3368 2068 2020 2b20   + h  + 3h h  + 
│ │ │ │ -00015bc0: 6820 2020 2020 2020 2020 7c0a 7c20 2020  h         |.|   
│ │ │ │ -00015bd0: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ -00015be0: 2020 2020 2031 2032 2020 2020 3120 2020       1 2    1   
│ │ │ │ -00015bf0: 2020 3120 3220 2020 2032 2020 2020 2020    1 2    2      
│ │ │ │ -00015c00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c30: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -00015c40: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │ -00015c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c70: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136  ----------+.|i16
│ │ │ │ -00015cb0: 203a 2063 736d 4b48 6173 683d 2043 534d   : csmKHash= CSM
│ │ │ │ -00015cc0: 2841 2c4b 2c4f 7574 7075 743d 3e48 6173  (A,K,Output=>Has
│ │ │ │ -00015cd0: 6846 6f72 6d29 2020 2020 2020 2020 2020  hForm)          
│ │ │ │ -00015ce0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d10: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ -00015d20: 203d 204d 7574 6162 6c65 4861 7368 5461   = MutableHashTa
│ │ │ │ -00015d30: 626c 657b 2e2e 2e34 2e2e 2e7d 2020 2020  ble{...4...}    
│ │ │ │ +00015a90: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2074  ------+.|i15 : t
│ │ │ │ +00015aa0: 696d 6520 6373 6d4b 3d43 534d 2841 2c4b  ime csmK=CSM(A,K
│ │ │ │ +00015ab0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00015ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015ad0: 202d 2d20 7573 6564 2031 2e30 3738 3432   -- used 1.07842
│ │ │ │ +00015ae0: 7320 2863 7075 293b 2030 2e34 3330 3334  s (cpu); 0.43034
│ │ │ │ +00015af0: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │ +00015b00: 2867 6329 7c0a 7c20 2020 2020 2020 2020  (gc)|.|         
│ │ │ │ +00015b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00015b40: 2020 2020 2020 3220 3220 2020 2020 3220        2 2     2 
│ │ │ │ +00015b50: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +00015b60: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00015b70: 2020 7c0a 7c6f 3135 203d 2037 6820 6820    |.|o15 = 7h h 
│ │ │ │ +00015b80: 202b 2035 6820 6820 202b 2034 6820 6820   + 5h h  + 4h h 
│ │ │ │ +00015b90: 202b 2068 2020 2b20 3368 2068 2020 2b20   + h  + 3h h  + 
│ │ │ │ +00015ba0: 6820 2020 2020 2020 207c 0a7c 2020 2020  h        |.|    
│ │ │ │ +00015bb0: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ +00015bc0: 2020 2020 3120 3220 2020 2031 2020 2020      1 2    1    
│ │ │ │ +00015bd0: 2031 2032 2020 2020 3220 2020 2020 2020   1 2    2       
│ │ │ │ +00015be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c10: 2020 2020 2020 207c 0a7c 6f31 3520 3a20         |.|o15 : 
│ │ │ │ +00015c20: 4120 2020 2020 2020 2020 2020 2020 2020  A               
│ │ │ │ +00015c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015c50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00015c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015c80: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 6373  -----+.|i16 : cs
│ │ │ │ +00015c90: 6d4b 4861 7368 3d20 4353 4d28 412c 4b2c  mKHash= CSM(A,K,
│ │ │ │ +00015ca0: 4f75 7470 7574 3d3e 4861 7368 466f 726d  Output=>HashForm
│ │ │ │ +00015cb0: 2920 2020 2020 2020 2020 2020 7c0a 7c20  )           |.| 
│ │ │ │ +00015cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015cf0: 2020 207c 0a7c 6f31 3620 3d20 4d75 7461     |.|o16 = Muta
│ │ │ │ +00015d00: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +00015d10: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
│ │ │ │ +00015d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d80: 2020 2020 2020 2020 2020 7c0a 7c6f 3136            |.|o16
│ │ │ │ -00015d90: 203a 204d 7574 6162 6c65 4861 7368 5461   : MutableHashTa
│ │ │ │ -00015da0: 626c 6520 2020 2020 2020 2020 2020 2020  ble             
│ │ │ │ -00015db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015dc0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137  ----------+.|i17
│ │ │ │ -00015e00: 203a 2063 736d 4b3d 3d63 736d 4b48 6173   : csmK==csmKHas
│ │ │ │ -00015e10: 6823 2243 534d 2220 2020 2020 2020 2020  h#"CSM"         
│ │ │ │ +00015d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d60: 207c 0a7c 6f31 3620 3a20 4d75 7461 626c   |.|o16 : Mutabl
│ │ │ │ +00015d70: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │ +00015d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00015da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00015dd0: 0a7c 6931 3720 3a20 6373 6d4b 3d3d 6373  .|i17 : csmK==cs
│ │ │ │ +00015de0: 6d4b 4861 7368 2322 4353 4d22 2020 2020  mKHash#"CSM"    
│ │ │ │ +00015df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00015e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00015e40: 6f31 3720 3d20 7472 7565 2020 2020 2020  o17 = true      
│ │ │ │  00015e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e60: 2020 2020 2020 2020 2020 7c0a 7c6f 3137            |.|o17
│ │ │ │ -00015e70: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ -00015e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ -00015ee0: 203a 2043 534d 2841 2c69 6465 616c 284b   : CSM(A,ideal(K
│ │ │ │ -00015ef0: 5f30 2929 3d3d 6373 6d4b 4861 7368 237b  _0))==csmKHash#{
│ │ │ │ -00015f00: 307d 2020 2020 2020 2020 2020 2020 2020  0}              
│ │ │ │ -00015f10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00015f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00015e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00015eb0: 3820 3a20 4353 4d28 412c 6964 6561 6c28  8 : CSM(A,ideal(
│ │ │ │ +00015ec0: 4b5f 3029 293d 3d63 736d 4b48 6173 6823  K_0))==csmKHash#
│ │ │ │ +00015ed0: 7b30 7d20 2020 2020 2020 2020 2020 2020  {0}             
│ │ │ │ +00015ee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00015ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f10: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +00015f20: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  00015f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f40: 2020 2020 2020 2020 2020 7c0a 7c6f 3138            |.|o18
│ │ │ │ -00015f50: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ -00015f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00015f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 7570  ----------+..Sup
│ │ │ │ -00015fc0: 706f 7365 2077 6520 6861 7665 2061 6c72  pose we have alr
│ │ │ │ -00015fd0: 6561 6479 2063 6f6d 7075 7465 6420 736f  eady computed so
│ │ │ │ -00015fe0: 6d65 206f 6620 4353 4d20 636c 6173 7365  me of CSM classe
│ │ │ │ -00015ff0: 7320 6f66 2068 7970 6572 7375 7266 6163  s of hypersurfac
│ │ │ │ -00016000: 6573 2069 6e76 6f6c 7665 640a 696e 2074  es involved.in t
│ │ │ │ -00016010: 6865 2069 6e63 6c75 7369 6f6e 2d65 7863  he inclusion-exc
│ │ │ │ -00016020: 6c75 7369 6f6e 2070 726f 6365 6475 7265  lusion procedure
│ │ │ │ -00016030: 2c20 7468 656e 2077 6520 6d61 7920 696e  , then we may in
│ │ │ │ -00016040: 7075 7420 7468 6573 6520 746f 2062 6520  put these to be 
│ │ │ │ -00016050: 7573 6564 2062 7920 7468 650a 4353 4d20  used by the.CSM 
│ │ │ │ -00016060: 6675 6e63 7469 6f6e 2e20 496e 2074 6865  function. In the
│ │ │ │ -00016070: 2065 7861 6d70 6c65 2062 656c 6f77 2077   example below w
│ │ │ │ -00016080: 6520 696e 7075 7420 7468 6520 4353 4d20  e input the CSM 
│ │ │ │ -00016090: 636c 6173 7320 6f66 2056 284b 5f30 2920  class of V(K_0) 
│ │ │ │ -000160a0: 2874 6861 7420 6973 206f 660a 7468 6520  (that is of.the 
│ │ │ │ -000160b0: 6879 7065 7273 7572 6661 6365 2064 6566  hypersurface def
│ │ │ │ -000160c0: 696e 6564 2062 7920 7468 6520 6669 7273  ined by the firs
│ │ │ │ -000160d0: 7420 706f 6c79 6e6f 6d69 616c 2067 656e  t polynomial gen
│ │ │ │ -000160e0: 6572 6174 696e 6720 4b29 2061 6e64 2074  erating K) and t
│ │ │ │ -000160f0: 6865 2043 534d 0a63 6c61 7373 206f 6620  he CSM.class of 
│ │ │ │ -00016100: 7468 6520 6879 7065 7273 7572 6661 6365  the hypersurface
│ │ │ │ -00016110: 2064 6566 696e 6564 2062 7920 7468 6520   defined by the 
│ │ │ │ -00016120: 7072 6f64 7563 7420 6f66 2074 6865 2067  product of the g
│ │ │ │ -00016130: 656e 6572 6174 6f72 7320 6f66 204b 2e0a  enerators of K..
│ │ │ │ -00016140: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00016150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00016180: 6931 3920 3a20 6d3d 6e65 7720 4d75 7461  i19 : m=new Muta
│ │ │ │ -00016190: 626c 6548 6173 6854 6162 6c65 3b20 2020  bleHashTable;   
│ │ │ │ -000161a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000161b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ -00016200: 3a20 6d23 7b30 7d3d 6373 6d4b 4861 7368  : m#{0}=csmKHash
│ │ │ │ -00016210: 237b 307d 2020 2020 2020 2020 2020 2020  #{0}            
│ │ │ │ +00015f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00015f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015f80: 2d2d 2d2d 2d2d 2d2b 0a0a 5375 7070 6f73  -------+..Suppos
│ │ │ │ +00015f90: 6520 7765 2068 6176 6520 616c 7265 6164  e we have alread
│ │ │ │ +00015fa0: 7920 636f 6d70 7574 6564 2073 6f6d 6520  y computed some 
│ │ │ │ +00015fb0: 6f66 2043 534d 2063 6c61 7373 6573 206f  of CSM classes o
│ │ │ │ +00015fc0: 6620 6879 7065 7273 7572 6661 6365 7320  f hypersurfaces 
│ │ │ │ +00015fd0: 696e 766f 6c76 6564 0a69 6e20 7468 6520  involved.in the 
│ │ │ │ +00015fe0: 696e 636c 7573 696f 6e2d 6578 636c 7573  inclusion-exclus
│ │ │ │ +00015ff0: 696f 6e20 7072 6f63 6564 7572 652c 2074  ion procedure, t
│ │ │ │ +00016000: 6865 6e20 7765 206d 6179 2069 6e70 7574  hen we may input
│ │ │ │ +00016010: 2074 6865 7365 2074 6f20 6265 2075 7365   these to be use
│ │ │ │ +00016020: 6420 6279 2074 6865 0a43 534d 2066 756e  d by the.CSM fun
│ │ │ │ +00016030: 6374 696f 6e2e 2049 6e20 7468 6520 6578  ction. In the ex
│ │ │ │ +00016040: 616d 706c 6520 6265 6c6f 7720 7765 2069  ample below we i
│ │ │ │ +00016050: 6e70 7574 2074 6865 2043 534d 2063 6c61  nput the CSM cla
│ │ │ │ +00016060: 7373 206f 6620 5628 4b5f 3029 2028 7468  ss of V(K_0) (th
│ │ │ │ +00016070: 6174 2069 7320 6f66 0a74 6865 2068 7970  at is of.the hyp
│ │ │ │ +00016080: 6572 7375 7266 6163 6520 6465 6669 6e65  ersurface define
│ │ │ │ +00016090: 6420 6279 2074 6865 2066 6972 7374 2070  d by the first p
│ │ │ │ +000160a0: 6f6c 796e 6f6d 6961 6c20 6765 6e65 7261  olynomial genera
│ │ │ │ +000160b0: 7469 6e67 204b 2920 616e 6420 7468 6520  ting K) and the 
│ │ │ │ +000160c0: 4353 4d0a 636c 6173 7320 6f66 2074 6865  CSM.class of the
│ │ │ │ +000160d0: 2068 7970 6572 7375 7266 6163 6520 6465   hypersurface de
│ │ │ │ +000160e0: 6669 6e65 6420 6279 2074 6865 2070 726f  fined by the pro
│ │ │ │ +000160f0: 6475 6374 206f 6620 7468 6520 6765 6e65  duct of the gene
│ │ │ │ +00016100: 7261 746f 7273 206f 6620 4b2e 0a0a 2b2d  rators of K...+-
│ │ │ │ +00016110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016140: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139  ----------+.|i19
│ │ │ │ +00016150: 203a 206d 3d6e 6577 204d 7574 6162 6c65   : m=new Mutable
│ │ │ │ +00016160: 4861 7368 5461 626c 653b 2020 2020 2020  HashTable;      
│ │ │ │ +00016170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016180: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161c0: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 206d  ------+.|i20 : m
│ │ │ │ +000161d0: 237b 307d 3d63 736d 4b48 6173 6823 7b30  #{0}=csmKHash#{0
│ │ │ │ +000161e0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000161f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016200: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00016240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016280: 3220 3220 2020 2020 3220 2020 2020 2020  2 2     2       
│ │ │ │ -00016290: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ -000162a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000162b0: 2020 207c 0a7c 6f32 3020 3d20 3868 2068     |.|o20 = 8h h
│ │ │ │ -000162c0: 2020 2b20 3768 2068 2020 2b20 3668 2068    + 7h h  + 6h h
│ │ │ │ -000162d0: 2020 2b20 3268 2020 2b20 3568 2068 2020    + 2h  + 5h h  
│ │ │ │ -000162e0: 2b20 3268 2020 2b20 3268 2020 2b20 6820  + 2h  + 2h  + h 
│ │ │ │ -000162f0: 207c 0a7c 2020 2020 2020 2020 3120 3220   |.|        1 2 
│ │ │ │ -00016300: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ -00016310: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -00016320: 2020 3220 2020 2020 3120 2020 2032 207c    2     1    2 |
│ │ │ │ -00016330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016240: 2020 7c0a 7c20 2020 2020 2020 2032 2032    |.|        2 2
│ │ │ │ +00016250: 2020 2020 2032 2020 2020 2020 2020 2032       2         2
│ │ │ │ +00016260: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00016270: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00016280: 7c0a 7c6f 3230 203d 2038 6820 6820 202b  |.|o20 = 8h h  +
│ │ │ │ +00016290: 2037 6820 6820 202b 2036 6820 6820 202b   7h h  + 6h h  +
│ │ │ │ +000162a0: 2032 6820 202b 2035 6820 6820 202b 2032   2h  + 5h h  + 2
│ │ │ │ +000162b0: 6820 202b 2032 6820 202b 2068 2020 7c0a  h  + 2h  + h  |.
│ │ │ │ +000162c0: 7c20 2020 2020 2020 2031 2032 2020 2020  |        1 2    
│ │ │ │ +000162d0: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +000162e0: 2031 2020 2020 2031 2032 2020 2020 2032   1     1 2     2
│ │ │ │ +000162f0: 2020 2020 2031 2020 2020 3220 7c0a 7c20       1    2 |.| 
│ │ │ │ +00016300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016330: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ +00016340: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │  00016350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016370: 6f32 3020 3a20 4120 2020 2020 2020 2020  o20 : A         
│ │ │ │ -00016380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000163a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000163b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000163e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120  ---------+.|i21 
│ │ │ │ -000163f0: 3a20 6d23 7b30 2c31 7d3d 6373 6d4b 4861  : m#{0,1}=csmKHa
│ │ │ │ -00016400: 7368 237b 302c 317d 2020 2020 2020 2020  sh#{0,1}        
│ │ │ │ +00016360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000163a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000163b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 206d  ------+.|i21 : m
│ │ │ │ +000163c0: 237b 302c 317d 3d63 736d 4b48 6173 6823  #{0,1}=csmKHash#
│ │ │ │ +000163d0: 7b30 2c31 7d20 2020 2020 2020 2020 2020  {0,1}           
│ │ │ │ +000163e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000163f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016420: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00016430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016470: 3220 3220 2020 2020 3220 2020 2020 2020  2 2     2       
│ │ │ │ -00016480: 2020 3220 2020 2020 3220 2020 2020 2020    2     2       
│ │ │ │ -00016490: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000164a0: 2020 207c 0a7c 6f32 3120 3d20 3968 2068     |.|o21 = 9h h
│ │ │ │ -000164b0: 2020 2b20 3968 2068 2020 2b20 3968 2068    + 9h h  + 9h h
│ │ │ │ -000164c0: 2020 2b20 3368 2020 2b20 3768 2068 2020    + 3h  + 7h h  
│ │ │ │ -000164d0: 2b20 3368 2020 2b20 3368 2020 2b20 3268  + 3h  + 3h  + 2h
│ │ │ │ -000164e0: 207c 0a7c 2020 2020 2020 2020 3120 3220   |.|        1 2 
│ │ │ │ -000164f0: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ -00016500: 2020 2020 3120 2020 2020 3120 3220 2020      1     1 2   
│ │ │ │ -00016510: 2020 3220 2020 2020 3120 2020 2020 327c    2     1     2|
│ │ │ │ -00016520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016430: 2020 7c0a 7c20 2020 2020 2020 2032 2032    |.|        2 2
│ │ │ │ +00016440: 2020 2020 2032 2020 2020 2020 2020 2032       2         2
│ │ │ │ +00016450: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00016460: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00016470: 7c0a 7c6f 3231 203d 2039 6820 6820 202b  |.|o21 = 9h h  +
│ │ │ │ +00016480: 2039 6820 6820 202b 2039 6820 6820 202b   9h h  + 9h h  +
│ │ │ │ +00016490: 2033 6820 202b 2037 6820 6820 202b 2033   3h  + 7h h  + 3
│ │ │ │ +000164a0: 6820 202b 2033 6820 202b 2032 6820 7c0a  h  + 3h  + 2h |.
│ │ │ │ +000164b0: 7c20 2020 2020 2020 2031 2032 2020 2020  |        1 2    
│ │ │ │ +000164c0: 2031 2032 2020 2020 2031 2032 2020 2020   1 2     1 2    
│ │ │ │ +000164d0: 2031 2020 2020 2031 2032 2020 2020 2032   1     1 2     2
│ │ │ │ +000164e0: 2020 2020 2031 2020 2020 2032 7c0a 7c20       1     2|.| 
│ │ │ │ +000164f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016520: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ +00016530: 203a 2041 2020 2020 2020 2020 2020 2020   : A            
│ │ │ │  00016540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016560: 6f32 3120 3a20 4120 2020 2020 2020 2020  o21 : A         
│ │ │ │ -00016570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016590: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000165a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000165b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000165c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000165d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220  ---------+.|i22 
│ │ │ │ -000165e0: 3a20 7469 6d65 2043 534d 2841 2c4b 2c6d  : time CSM(A,K,m
│ │ │ │ -000165f0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00016600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016610: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00016620: 6564 2030 2e31 3832 3638 3673 2028 6370  ed 0.182686s (cp
│ │ │ │ -00016630: 7529 3b20 302e 3130 3736 3831 7320 2874  u); 0.107681s (t
│ │ │ │ -00016640: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -00016650: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016690: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ -000166a0: 3220 2020 2020 3220 2020 2020 2020 2020  2     2         
│ │ │ │ -000166b0: 3220 2020 2032 2020 2020 2020 2020 2020  2    2          
│ │ │ │ -000166c0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000166d0: 207c 0a7c 6f32 3220 3d20 3768 2068 2020   |.|o22 = 7h h  
│ │ │ │ -000166e0: 2b20 3568 2068 2020 2b20 3468 2068 2020  + 5h h  + 4h h  
│ │ │ │ -000166f0: 2b20 6820 202b 2033 6820 6820 202b 2068  + h  + 3h h  + h
│ │ │ │ -00016700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016710: 0a7c 2020 2020 2020 2020 3120 3220 2020  .|        1 2   
│ │ │ │ -00016720: 2020 3120 3220 2020 2020 3120 3220 2020    1 2     1 2   
│ │ │ │ -00016730: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ -00016740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00016750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016560: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00016570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000165a0: 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a 2074  ------+.|i22 : t
│ │ │ │ +000165b0: 696d 6520 4353 4d28 412c 4b2c 6d29 2020  ime CSM(A,K,m)  
│ │ │ │ +000165c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000165d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000165e0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420      |.| -- used 
│ │ │ │ +000165f0: 302e 3235 3939 3738 7320 2863 7075 293b  0.259978s (cpu);
│ │ │ │ +00016600: 2030 2e31 3231 3033 3273 2028 7468 7265   0.121032s (thre
│ │ │ │ +00016610: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +00016620: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016660: 7c0a 7c20 2020 2020 2020 2032 2032 2020  |.|        2 2  
│ │ │ │ +00016670: 2020 2032 2020 2020 2020 2020 2032 2020     2         2  
│ │ │ │ +00016680: 2020 3220 2020 2020 2020 2020 2020 2032    2            2
│ │ │ │ +00016690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000166a0: 7c6f 3232 203d 2037 6820 6820 202b 2035  |o22 = 7h h  + 5
│ │ │ │ +000166b0: 6820 6820 202b 2034 6820 6820 202b 2068  h h  + 4h h  + h
│ │ │ │ +000166c0: 2020 2b20 3368 2068 2020 2b20 6820 2020    + 3h h  + h   
│ │ │ │ +000166d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000166e0: 2020 2020 2020 2031 2032 2020 2020 2031         1 2     1
│ │ │ │ +000166f0: 2032 2020 2020 2031 2032 2020 2020 3120   2     1 2    1 
│ │ │ │ +00016700: 2020 2020 3120 3220 2020 2032 2020 2020      1 2    2    
│ │ │ │ +00016710: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00016720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016750: 2020 2020 2020 2020 7c0a 7c6f 3232 203a          |.|o22 :
│ │ │ │ +00016760: 2041 2020 2020 2020 2020 2020 2020 2020   A              
│ │ │ │  00016770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016780: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00016790: 3220 3a20 4120 2020 2020 2020 2020 2020  2 : A           
│ │ │ │ -000167a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000167d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000167e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000167f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016800: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6865  -------+..In the
│ │ │ │ -00016810: 2063 6173 6520 7768 6572 6520 7468 6520   case where the 
│ │ │ │ -00016820: 616d 6269 656e 7420 7370 6163 6520 6973  ambient space is
│ │ │ │ -00016830: 2061 2074 6f72 6963 2076 6172 6965 7479   a toric variety
│ │ │ │ -00016840: 2077 6869 6368 2069 7320 6e6f 7420 6120   which is not a 
│ │ │ │ -00016850: 7072 6f64 7563 740a 6f66 2070 726f 6a65  product.of proje
│ │ │ │ -00016860: 6374 6976 6520 7370 6163 6573 2077 6520  ctive spaces we 
│ │ │ │ -00016870: 6d75 7374 206c 6f61 6420 7468 6520 4e6f  must load the No
│ │ │ │ -00016880: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ -00016890: 6573 2070 6163 6b61 6765 2061 6e64 206d  es package and m
│ │ │ │ -000168a0: 7573 740a 616c 736f 2069 6e70 7574 2074  ust.also input t
│ │ │ │ -000168b0: 6865 2074 6f72 6963 2076 6172 6965 7479  he toric variety
│ │ │ │ -000168c0: 2e20 4966 2074 6865 2074 6f72 6963 2076  . If the toric v
│ │ │ │ -000168d0: 6172 6965 7479 2069 7320 6120 7072 6f64  ariety is a prod
│ │ │ │ -000168e0: 7563 7420 6f66 2070 726f 6a65 6374 6976  uct of projectiv
│ │ │ │ -000168f0: 650a 7370 6163 6520 6974 2069 7320 7265  e.space it is re
│ │ │ │ -00016900: 636f 6d6d 656e 6420 746f 2075 7365 2074  commend to use t
│ │ │ │ -00016910: 6865 2066 6f72 6d20 6162 6f76 6520 7261  he form above ra
│ │ │ │ -00016920: 7468 6572 2074 6861 6e20 696e 7075 7474  ther than inputt
│ │ │ │ -00016930: 696e 6720 7468 6520 746f 7269 630a 7661  ing the toric.va
│ │ │ │ -00016940: 7269 6574 7920 666f 7220 6566 6669 6369  riety for effici
│ │ │ │ -00016950: 656e 6379 2072 6561 736f 6e73 2e0a 0a2b  ency reasons...+
│ │ │ │ +00016780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016790: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000167a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000167d0: 2d2d 2d2d 2b0a 0a49 6e20 7468 6520 6361  ----+..In the ca
│ │ │ │ +000167e0: 7365 2077 6865 7265 2074 6865 2061 6d62  se where the amb
│ │ │ │ +000167f0: 6965 6e74 2073 7061 6365 2069 7320 6120  ient space is a 
│ │ │ │ +00016800: 746f 7269 6320 7661 7269 6574 7920 7768  toric variety wh
│ │ │ │ +00016810: 6963 6820 6973 206e 6f74 2061 2070 726f  ich is not a pro
│ │ │ │ +00016820: 6475 6374 0a6f 6620 7072 6f6a 6563 7469  duct.of projecti
│ │ │ │ +00016830: 7665 2073 7061 6365 7320 7765 206d 7573  ve spaces we mus
│ │ │ │ +00016840: 7420 6c6f 6164 2074 6865 204e 6f72 6d61  t load the Norma
│ │ │ │ +00016850: 6c54 6f72 6963 5661 7269 6574 6965 7320  lToricVarieties 
│ │ │ │ +00016860: 7061 636b 6167 6520 616e 6420 6d75 7374  package and must
│ │ │ │ +00016870: 0a61 6c73 6f20 696e 7075 7420 7468 6520  .also input the 
│ │ │ │ +00016880: 746f 7269 6320 7661 7269 6574 792e 2049  toric variety. I
│ │ │ │ +00016890: 6620 7468 6520 746f 7269 6320 7661 7269  f the toric vari
│ │ │ │ +000168a0: 6574 7920 6973 2061 2070 726f 6475 6374  ety is a product
│ │ │ │ +000168b0: 206f 6620 7072 6f6a 6563 7469 7665 0a73   of projective.s
│ │ │ │ +000168c0: 7061 6365 2069 7420 6973 2072 6563 6f6d  pace it is recom
│ │ │ │ +000168d0: 6d65 6e64 2074 6f20 7573 6520 7468 6520  mend to use the 
│ │ │ │ +000168e0: 666f 726d 2061 626f 7665 2072 6174 6865  form above rathe
│ │ │ │ +000168f0: 7220 7468 616e 2069 6e70 7574 7469 6e67  r than inputting
│ │ │ │ +00016900: 2074 6865 2074 6f72 6963 0a76 6172 6965   the toric.varie
│ │ │ │ +00016910: 7479 2066 6f72 2065 6666 6963 6965 6e63  ty for efficienc
│ │ │ │ +00016920: 7920 7265 6173 6f6e 732e 0a0a 2b2d 2d2d  y reasons...+---
│ │ │ │ +00016930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000169a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a  --------+.|i23 :
│ │ │ │ -000169b0: 206e 6565 6473 5061 636b 6167 6520 224e   needsPackage "N
│ │ │ │ -000169c0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -000169d0: 6965 7322 2020 2020 2020 2020 2020 2020  ies"            
│ │ │ │ +00016970: 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20 6e65  -----+.|i23 : ne
│ │ │ │ +00016980: 6564 7350 6163 6b61 6765 2022 4e6f 726d  edsPackage "Norm
│ │ │ │ +00016990: 616c 546f 7269 6356 6172 6965 7469 6573  alToricVarieties
│ │ │ │ +000169a0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +000169b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000169c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000169d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000169e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000169f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00016a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00016a40: 7c6f 3233 203d 204e 6f72 6d61 6c54 6f72  |o23 = NormalTor
│ │ │ │ -00016a50: 6963 5661 7269 6574 6965 7320 2020 2020  icVarieties     
│ │ │ │ +000169f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a00: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00016a10: 3320 3d20 4e6f 726d 616c 546f 7269 6356  3 = NormalToricV
│ │ │ │ +00016a20: 6172 6965 7469 6573 2020 2020 2020 2020  arieties        
│ │ │ │ +00016a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00016a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00016a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016aa0: 207c 0a7c 6f32 3320 3a20 5061 636b 6167   |.|o23 : Packag
│ │ │ │ +00016ab0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │  00016ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ad0: 2020 2020 7c0a 7c6f 3233 203a 2050 6163      |.|o23 : Pac
│ │ │ │ -00016ae0: 6b61 6765 2020 2020 2020 2020 2020 2020  kage            
│ │ │ │ -00016af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016b20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00016b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234  ----------+.|i24
│ │ │ │ -00016b70: 203a 2052 686f 203d 207b 7b31 2c30 2c30   : Rho = {{1,0,0
│ │ │ │ -00016b80: 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 2c31  },{0,1,0},{0,0,1
│ │ │ │ -00016b90: 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 2c30  },{-1,-1,0},{0,0
│ │ │ │ -00016ba0: 2c2d 317d 7d20 2020 2020 2020 2020 2020  ,-1}}           
│ │ │ │ -00016bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00016bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c00: 7c0a 7c6f 3234 203d 207b 7b31 2c20 302c  |.|o24 = {{1, 0,
│ │ │ │ -00016c10: 2030 7d2c 207b 302c 2031 2c20 307d 2c20   0}, {0, 1, 0}, 
│ │ │ │ -00016c20: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20  {0, 0, 1}, {-1, 
│ │ │ │ -00016c30: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d  -1, 0}, {0, 0, -
│ │ │ │ -00016c40: 317d 7d20 2020 2020 2020 207c 0a7c 2020  1}}        |.|  
│ │ │ │ +00016ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ae0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00016af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016b30: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20  -------+.|i24 : 
│ │ │ │ +00016b40: 5268 6f20 3d20 7b7b 312c 302c 307d 2c7b  Rho = {{1,0,0},{
│ │ │ │ +00016b50: 302c 312c 307d 2c7b 302c 302c 317d 2c7b  0,1,0},{0,0,1},{
│ │ │ │ +00016b60: 2d31 2c2d 312c 307d 2c7b 302c 302c 2d31  -1,-1,0},{0,0,-1
│ │ │ │ +00016b70: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00016b80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00016bd0: 6f32 3420 3d20 7b7b 312c 2030 2c20 307d  o24 = {{1, 0, 0}
│ │ │ │ +00016be0: 2c20 7b30 2c20 312c 2030 7d2c 207b 302c  , {0, 1, 0}, {0,
│ │ │ │ +00016bf0: 2030 2c20 317d 2c20 7b2d 312c 202d 312c   0, 1}, {-1, -1,
│ │ │ │ +00016c00: 2030 7d2c 207b 302c 2030 2c20 2d31 7d7d   0}, {0, 0, -1}}
│ │ │ │ +00016c10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c60: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374     |.|o24 : List
│ │ │ │  00016c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c90: 2020 2020 2020 7c0a 7c6f 3234 203a 204c        |.|o24 : L
│ │ │ │ -00016ca0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ce0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00016d30: 3235 203a 2053 6967 6d61 203d 207b 7b30  25 : Sigma = {{0
│ │ │ │ -00016d40: 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c 7b30  ,1,2},{1,2,3},{0
│ │ │ │ -00016d50: 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c 7b31  ,2,3},{0,1,4},{1
│ │ │ │ -00016d60: 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d 2020  ,3,4},{0,3,4}}  
│ │ │ │ -00016d70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00016d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016dc0: 2020 7c0a 7c6f 3235 203d 207b 7b30 2c20    |.|o25 = {{0, 
│ │ │ │ -00016dd0: 312c 2032 7d2c 207b 312c 2032 2c20 337d  1, 2}, {1, 2, 3}
│ │ │ │ -00016de0: 2c20 7b30 2c20 322c 2033 7d2c 207b 302c  , {0, 2, 3}, {0,
│ │ │ │ -00016df0: 2031 2c20 347d 2c20 7b31 2c20 332c 2034   1, 4}, {1, 3, 4
│ │ │ │ -00016e00: 7d2c 207b 302c 2033 2c20 347d 7d7c 0a7c  }, {0, 3, 4}}|.|
│ │ │ │ +00016c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016cb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00016cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3520  ---------+.|i25 
│ │ │ │ +00016d00: 3a20 5369 676d 6120 3d20 7b7b 302c 312c  : Sigma = {{0,1,
│ │ │ │ +00016d10: 327d 2c7b 312c 322c 337d 2c7b 302c 322c  2},{1,2,3},{0,2,
│ │ │ │ +00016d20: 337d 2c7b 302c 312c 347d 2c7b 312c 332c  3},{0,1,4},{1,3,
│ │ │ │ +00016d30: 347d 2c7b 302c 332c 347d 7d20 2020 2020  4},{0,3,4}}     
│ │ │ │ +00016d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00016d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00016d90: 0a7c 6f32 3520 3d20 7b7b 302c 2031 2c20  .|o25 = {{0, 1, 
│ │ │ │ +00016da0: 327d 2c20 7b31 2c20 322c 2033 7d2c 207b  2}, {1, 2, 3}, {
│ │ │ │ +00016db0: 302c 2032 2c20 337d 2c20 7b30 2c20 312c  0, 2, 3}, {0, 1,
│ │ │ │ +00016dc0: 2034 7d2c 207b 312c 2033 2c20 347d 2c20   4}, {1, 3, 4}, 
│ │ │ │ +00016dd0: 7b30 2c20 332c 2034 7d7d 7c0a 7c20 2020  {0, 3, 4}}|.|   
│ │ │ │ +00016de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e20: 2020 2020 207c 0a7c 6f32 3520 3a20 4c69       |.|o25 : Li
│ │ │ │ +00016e30: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │  00016e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e50: 2020 2020 2020 2020 7c0a 7c6f 3235 203a          |.|o25 :
│ │ │ │ -00016e60: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ -00016e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ea0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00016ef0: 7c69 3236 203a 2058 203d 206e 6f72 6d61  |i26 : X = norma
│ │ │ │ -00016f00: 6c54 6f72 6963 5661 7269 6574 7928 5268  lToricVariety(Rh
│ │ │ │ -00016f10: 6f2c 5369 676d 612c 436f 6566 6669 6369  o,Sigma,Coeffici
│ │ │ │ -00016f20: 656e 7452 696e 6720 3d3e 5a5a 2f33 3237  entRing =>ZZ/327
│ │ │ │ -00016f30: 3439 2920 2020 2020 207c 0a7c 2020 2020  49)      |.|    
│ │ │ │ +00016e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00016e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00016ec0: 3620 3a20 5820 3d20 6e6f 726d 616c 546f  6 : X = normalTo
│ │ │ │ +00016ed0: 7269 6356 6172 6965 7479 2852 686f 2c53  ricVariety(Rho,S
│ │ │ │ +00016ee0: 6967 6d61 2c43 6f65 6666 6963 6965 6e74  igma,Coefficient
│ │ │ │ +00016ef0: 5269 6e67 203d 3e5a 5a2f 3332 3734 3929  Ring =>ZZ/32749)
│ │ │ │ +00016f00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f50: 207c 0a7c 6f32 3620 3d20 5820 2020 2020   |.|o26 = X     
│ │ │ │  00016f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f80: 2020 2020 7c0a 7c6f 3236 203d 2058 2020      |.|o26 = X  
│ │ │ │ -00016f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017010: 2020 2020 2020 2020 2020 7c0a 7c6f 3236            |.|o26
│ │ │ │ -00017020: 203a 204e 6f72 6d61 6c54 6f72 6963 5661   : NormalToricVa
│ │ │ │ -00017030: 7269 6574 7920 2020 2020 2020 2020 2020  riety           
│ │ │ │ -00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017060: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000170a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000170b0: 2b0a 7c69 3237 203a 2063 736d 583d 4353  +.|i27 : csmX=CS
│ │ │ │ -000170c0: 4d20 5820 2020 2020 2020 2020 2020 2020  M X             
│ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fe0: 2020 2020 2020 207c 0a7c 6f32 3620 3a20         |.|o26 : 
│ │ │ │ +00016ff0: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ +00017000: 7479 2020 2020 2020 2020 2020 2020 2020  ty              
│ │ │ │ +00017010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017030: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00017040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017080: 6932 3720 3a20 6373 6d58 3d43 534d 2058  i27 : csmX=CSM X
│ │ │ │ +00017090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000170e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017110: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ +00017120: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017150: 2032 2020 2020 2020 2032 2020 2020 2020   2       2      
│ │ │ │ -00017160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017190: 207c 0a7c 6f32 3720 3d20 3678 2078 2020   |.|o27 = 6x x  
│ │ │ │ -000171a0: 2b20 3378 2020 2b20 3678 2078 2020 2b20  + 3x  + 6x x  + 
│ │ │ │ -000171b0: 3378 2020 2b20 3278 2020 2b20 3120 2020  3x  + 2x  + 1   
│ │ │ │ -000171c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000171e0: 2020 2020 2020 2033 2034 2020 2020 2033         3 4     3
│ │ │ │ -000171f0: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
│ │ │ │ -00017200: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017160: 7c6f 3237 203d 2036 7820 7820 202b 2033  |o27 = 6x x  + 3
│ │ │ │ +00017170: 7820 202b 2036 7820 7820 202b 2033 7820  x  + 6x x  + 3x 
│ │ │ │ +00017180: 202b 2032 7820 202b 2031 2020 2020 2020   + 2x  + 1      
│ │ │ │ +00017190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000171b0: 2020 2020 3320 3420 2020 2020 3320 2020      3 4     3   
│ │ │ │ +000171c0: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +000171d0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +000171e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00017200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00017230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017270: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00017280: 2020 2020 2020 2020 2020 205a 5a5b 7820             ZZ[x 
│ │ │ │ -00017290: 2e2e 7820 5d20 2020 2020 2020 2020 2020  ..x ]           
│ │ │ │ -000172a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00017240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00017250: 2020 2020 2020 2020 5a5a 5b78 202e 2e78          ZZ[x ..x
│ │ │ │ +00017260: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +00017270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017280: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00017290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172a0: 2020 2020 2020 2030 2020 2034 2020 2020         0   4    
│ │ │ │ +000172b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172d0: 2020 2020 2020 2020 2020 3020 2020 3420            0   4 
│ │ │ │ -000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017300: 2020 2020 2020 2020 7c0a 7c6f 3237 203a          |.|o27 :
│ │ │ │ -00017310: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ -00017320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017330: 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 2020  ----------      
│ │ │ │ -00017340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017350: 2020 207c 0a7c 2020 2020 2020 2878 2078     |.|      (x x
│ │ │ │ -00017360: 202c 2078 2078 2078 202c 2078 2020 2d20   , x x x , x  - 
│ │ │ │ -00017370: 7820 2c20 7820 202d 2078 202c 2078 2020  x , x  - x , x  
│ │ │ │ -00017380: 2d20 7820 2920 2020 2020 2020 2020 2020  - x )           
│ │ │ │ -00017390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000173a0: 7c20 2020 2020 2020 2032 2034 2020 2030  |        2 4   0
│ │ │ │ -000173b0: 2031 2033 2020 2030 2020 2020 3320 2020   1 3   0    3   
│ │ │ │ -000173c0: 3120 2020 2033 2020 2032 2020 2020 3420  1    3   2    4 
│ │ │ │ -000173d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000173e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000172d0: 2020 2020 207c 0a7c 6f32 3720 3a20 2d2d       |.|o27 : --
│ │ │ │ +000172e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000172f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017300: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +00017310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017320: 7c0a 7c20 2020 2020 2028 7820 7820 2c20  |.|      (x x , 
│ │ │ │ +00017330: 7820 7820 7820 2c20 7820 202d 2078 202c  x x x , x  - x ,
│ │ │ │ +00017340: 2078 2020 2d20 7820 2c20 7820 202d 2078   x  - x , x  - x
│ │ │ │ +00017350: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │ +00017360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00017370: 2020 2020 2020 3220 3420 2020 3020 3120        2 4   0 1 
│ │ │ │ +00017380: 3320 2020 3020 2020 2033 2020 2031 2020  3   0    3   1  
│ │ │ │ +00017390: 2020 3320 2020 3220 2020 2034 2020 2020    3   2    4    
│ │ │ │ +000173a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000173b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000173c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000173e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000173f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017430: 2d2d 2d2d 2b0a 7c69 3238 203a 2043 683d  ----+.|i28 : Ch=
│ │ │ │ -00017440: 7269 6e67 2063 736d 5820 2020 2020 2020  ring csmX       
│ │ │ │ +00017400: 2d2b 0a7c 6932 3820 3a20 4368 3d72 696e  -+.|i28 : Ch=rin
│ │ │ │ +00017410: 6720 6373 6d58 2020 2020 2020 2020 2020  g csmX          
│ │ │ │ +00017420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00017490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017490: 2020 2020 2020 207c 0a7c 6f32 3820 3d20         |.|o28 = 
│ │ │ │ +000174a0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  000174b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3238            |.|o28
│ │ │ │ -000174d0: 203d 2043 6820 2020 2020 2020 2020 2020   = Ch           
│ │ │ │ -000174e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000174f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017510: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00017520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017530: 6f32 3820 3a20 5175 6f74 6965 6e74 5269  o28 : QuotientRi
│ │ │ │ +00017540: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │  00017550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017560: 7c0a 7c6f 3238 203a 2051 756f 7469 656e  |.|o28 : Quotien
│ │ │ │ -00017570: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ -00017580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000175a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00017560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017570: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00017580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000175a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000175b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175f0: 2d2d 2d2d 2d2d 2b0a 7c69 3239 203a 2043  ------+.|i29 : C
│ │ │ │ -00017600: 6865 636b 546f 7269 6356 6172 6965 7479  heckToricVariety
│ │ │ │ -00017610: 5661 6c69 6428 5829 2020 2020 2020 2020  Valid(X)        
│ │ │ │ +000175c0: 2d2d 2d2b 0a7c 6932 3920 3a20 4368 6563  ---+.|i29 : Chec
│ │ │ │ +000175d0: 6b54 6f72 6963 5661 7269 6574 7956 616c  kToricVarietyVal
│ │ │ │ +000175e0: 6964 2858 2920 2020 2020 2020 2020 2020  id(X)           
│ │ │ │ +000175f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017610: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00017620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00017650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017650: 2020 2020 2020 2020 207c 0a7c 6f32 3920           |.|o29 
│ │ │ │ +00017660: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │  00017670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017680: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00017690: 3239 203d 2074 7275 6520 2020 2020 2020  29 = true       
│ │ │ │ -000176a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000176f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017720: 2d2d 2b0a 7c69 3330 203a 2052 3d72 696e  --+.|i30 : R=rin
│ │ │ │ -00017730: 6728 5829 2020 2020 2020 2020 2020 2020  g(X)            
│ │ │ │ +00017680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000176a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000176b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000176f0: 0a7c 6933 3020 3a20 523d 7269 6e67 2858  .|i30 : R=ring(X
│ │ │ │ +00017700: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00017710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017730: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00017740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00017760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017780: 2020 2020 207c 0a7c 6f33 3020 3d20 5220       |.|o30 = R 
│ │ │ │  00017790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000177b0: 2020 2020 2020 2020 7c0a 7c6f 3330 203d          |.|o30 =
│ │ │ │ -000177c0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -000177d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000177b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000177c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000177d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000177e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017800: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00017810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00017850: 7c6f 3330 203a 2050 6f6c 796e 6f6d 6961  |o30 : Polynomia
│ │ │ │ -00017860: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ -00017870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017890: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00017800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017810: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00017820: 3020 3a20 506f 6c79 6e6f 6d69 616c 5269  0 : PolynomialRi
│ │ │ │ +00017830: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00017840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017860: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00017870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000178a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000178e0: 2d2d 2d2d 2b0a 7c69 3331 203a 2049 3d69  ----+.|i31 : I=i
│ │ │ │ -000178f0: 6465 616c 2852 5f30 5e34 2a52 5f31 2c52  deal(R_0^4*R_1,R
│ │ │ │ -00017900: 5f30 2a52 5f33 2a52 5f34 2a52 5f32 2d52  _0*R_3*R_4*R_2-R
│ │ │ │ -00017910: 5f32 5e32 2a52 5f30 5e32 2920 2020 2020  _2^2*R_0^2)     
│ │ │ │ -00017920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00017940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017970: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00017980: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ -00017990: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ -000179a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179c0: 2020 2020 207c 0a7c 6f33 3120 3d20 6964       |.|o31 = id
│ │ │ │ -000179d0: 6561 6c20 2878 2078 202c 202d 2078 2078  eal (x x , - x x
│ │ │ │ -000179e0: 2020 2b20 7820 7820 7820 7820 2920 2020    + x x x x )   
│ │ │ │ -000179f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00017a20: 2030 2031 2020 2020 2030 2032 2020 2020   0 1     0 2    
│ │ │ │ -00017a30: 3020 3220 3320 3420 2020 2020 2020 2020  0 2 3 4         
│ │ │ │ +000178b0: 2d2b 0a7c 6933 3120 3a20 493d 6964 6561  -+.|i31 : I=idea
│ │ │ │ +000178c0: 6c28 525f 305e 342a 525f 312c 525f 302a  l(R_0^4*R_1,R_0*
│ │ │ │ +000178d0: 525f 332a 525f 342a 525f 322d 525f 325e  R_3*R_4*R_2-R_2^
│ │ │ │ +000178e0: 322a 525f 305e 3229 2020 2020 2020 2020  2*R_0^2)        
│ │ │ │ +000178f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017940: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00017950: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ +00017960: 3220 3220 2020 2020 2020 2020 2020 2020  2 2             
│ │ │ │ +00017970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017990: 2020 7c0a 7c6f 3331 203d 2069 6465 616c    |.|o31 = ideal
│ │ │ │ +000179a0: 2028 7820 7820 2c20 2d20 7820 7820 202b   (x x , - x x  +
│ │ │ │ +000179b0: 2078 2078 2078 2078 2029 2020 2020 2020   x x x x )      
│ │ │ │ +000179c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000179d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000179e0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +000179f0: 3120 2020 2020 3020 3220 2020 2030 2032  1     0 2    0 2
│ │ │ │ +00017a00: 2033 2034 2020 2020 2020 2020 2020 2020   3 4            
│ │ │ │ +00017a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017a20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00017a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017a70: 2020 207c 0a7c 6f33 3120 3a20 4964 6561     |.|o31 : Idea
│ │ │ │ +00017a80: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │  00017a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017aa0: 2020 2020 2020 7c0a 7c6f 3331 203a 2049        |.|o31 : I
│ │ │ │ -00017ab0: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ -00017ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017af0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00017b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00017b40: 3332 203a 2043 534d 2858 2c49 2920 2020  32 : CSM(X,I)   
│ │ │ │ -00017b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017ac0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00017ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017b00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3220  ---------+.|i32 
│ │ │ │ +00017b10: 3a20 4353 4d28 582c 4929 2020 2020 2020  : CSM(X,I)      
│ │ │ │ +00017b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00017b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00017ba0: 0a7c 2020 2020 2020 2020 3220 2020 2020  .|        2     
│ │ │ │ +00017bb0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00017bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017bd0: 2020 7c0a 7c20 2020 2020 2020 2032 2020    |.|        2  
│ │ │ │ -00017be0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00017bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00017c20: 6f33 3220 3d20 3578 2078 2020 2b20 3378  o32 = 5x x  + 3x
│ │ │ │ -00017c30: 2020 2b20 3478 2078 2020 2b20 7820 2020    + 4x x  + x   
│ │ │ │ -00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00017c70: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ -00017c80: 2033 2034 2020 2020 3320 2020 2020 2020   3 4    3       
│ │ │ │ +00017bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017be0: 2020 2020 2020 2020 2020 7c0a 7c6f 3332            |.|o32
│ │ │ │ +00017bf0: 203d 2035 7820 7820 202b 2033 7820 202b   = 5x x  + 3x  +
│ │ │ │ +00017c00: 2034 7820 7820 202b 2078 2020 2020 2020   4x x  + x      
│ │ │ │ +00017c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00017c40: 3320 3420 2020 2020 3320 2020 2020 3320  3 4     3     3 
│ │ │ │ +00017c50: 3420 2020 2033 2020 2020 2020 2020 2020  4    3          
│ │ │ │ +00017c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00017c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00017cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017cc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00017cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00017d00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00017d10: 2020 2020 2020 205a 5a5b 7820 2e2e 7820         ZZ[x ..x 
│ │ │ │ -00017d20: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -00017d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017ce0: 2020 2020 5a5a 5b78 202e 2e78 205d 2020      ZZ[x ..x ]  
│ │ │ │ +00017cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017d30: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │ +00017d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d60: 2020 2020 2020 3020 2020 3420 2020 2020        0   4     
│ │ │ │ -00017d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d90: 2020 2020 7c0a 7c6f 3332 203a 202d 2d2d      |.|o32 : ---
│ │ │ │ -00017da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017dc0: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ -00017dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017de0: 0a7c 2020 2020 2020 2878 2078 202c 2078  .|      (x x , x
│ │ │ │ -00017df0: 2078 2078 202c 2078 2020 2d20 7820 2c20   x x , x  - x , 
│ │ │ │ -00017e00: 7820 202d 2078 202c 2078 2020 2d20 7820  x  - x , x  - x 
│ │ │ │ -00017e10: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00017e20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00017e30: 2020 2020 2032 2034 2020 2030 2031 2033       2 4   0 1 3
│ │ │ │ -00017e40: 2020 2030 2020 2020 3320 2020 3120 2020     0    3   1   
│ │ │ │ -00017e50: 2033 2020 2032 2020 2020 3420 2020 2020   3   2    4     
│ │ │ │ -00017e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017e70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ec0: 2b0a 7c69 3333 203a 2043 534d 2843 682c  +.|i33 : CSM(Ch,
│ │ │ │ -00017ed0: 582c 4929 2020 2020 2020 2020 2020 2020  X,I)            
│ │ │ │ +00017d60: 207c 0a7c 6f33 3220 3a20 2d2d 2d2d 2d2d   |.|o32 : ------
│ │ │ │ +00017d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017d90: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │ +00017da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017db0: 2020 2020 2028 7820 7820 2c20 7820 7820       (x x , x x 
│ │ │ │ +00017dc0: 7820 2c20 7820 202d 2078 202c 2078 2020  x , x  - x , x  
│ │ │ │ +00017dd0: 2d20 7820 2c20 7820 202d 2078 2029 2020  - x , x  - x )  
│ │ │ │ +00017de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017df0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00017e00: 2020 3220 3420 2020 3020 3120 3320 2020    2 4   0 1 3   
│ │ │ │ +00017e10: 3020 2020 2033 2020 2031 2020 2020 3320  0    3   1    3 
│ │ │ │ +00017e20: 2020 3220 2020 2034 2020 2020 2020 2020    2    4        
│ │ │ │ +00017e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017e40: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00017e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00017e90: 6933 3320 3a20 4353 4d28 4368 2c58 2c49  i33 : CSM(Ch,X,I
│ │ │ │ +00017ea0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00017eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00017f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f20: 2020 207c 0a7c 2020 2020 2020 2020 3220     |.|        2 
│ │ │ │ +00017f30: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00017f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017f60: 2032 2020 2020 2020 2032 2020 2020 2020   2       2      
│ │ │ │ -00017f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017f70: 7c6f 3333 203d 2038 7820 7820 202b 2033  |o33 = 8x x  + 3
│ │ │ │ +00017f80: 7820 202b 2035 7820 7820 202b 2078 2020  x  + 5x x  + x  
│ │ │ │  00017f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fa0: 207c 0a7c 6f33 3320 3d20 3878 2078 2020   |.|o33 = 8x x  
│ │ │ │ -00017fb0: 2b20 3378 2020 2b20 3578 2078 2020 2b20  + 3x  + 5x x  + 
│ │ │ │ -00017fc0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ -00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00017ff0: 2020 2020 2020 2033 2034 2020 2020 2033         3 4     3
│ │ │ │ -00018000: 2020 2020 2033 2034 2020 2020 3320 2020       3 4    3   
│ │ │ │ +00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017fb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00017fc0: 2020 2020 3320 3420 2020 2020 3320 2020      3 4     3   
│ │ │ │ +00017fd0: 2020 3320 3420 2020 2033 2020 2020 2020    3 4    3      
│ │ │ │ +00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018000: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00018040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018050: 0a7c 6f33 3320 3a20 4368 2020 2020 2020  .|o33 : Ch      
│ │ │ │  00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018080: 2020 7c0a 7c6f 3333 203a 2043 6820 2020    |.|o33 : Ch   
│ │ │ │ -00018090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00018080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018090: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000180a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000180c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000180d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000180e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000180f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018110: 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6869 7320  --------+..This 
│ │ │ │ -00018120: 6675 6e63 7469 6f6e 206d 6179 2061 6c73  function may als
│ │ │ │ -00018130: 6f20 636f 6d70 7574 6520 7468 6520 4353  o compute the CS
│ │ │ │ -00018140: 4d20 636c 6173 7320 6f66 2061 206e 6f72  M class of a nor
│ │ │ │ -00018150: 6d61 6c20 746f 7269 6320 7661 7269 6574  mal toric variet
│ │ │ │ -00018160: 7920 6465 6669 6e65 640a 6279 2061 2066  y defined.by a f
│ │ │ │ -00018170: 616e 2e20 496e 2074 6869 7320 6361 7365  an. In this case
│ │ │ │ -00018180: 2061 2063 6f6d 6269 6e61 746f 7269 616c   a combinatorial
│ │ │ │ -00018190: 206d 6574 686f 6420 6973 2075 7365 642e   method is used.
│ │ │ │ -000181a0: 2054 6869 7320 6d65 7468 6f64 2069 7320   This method is 
│ │ │ │ -000181b0: 6163 6365 7373 6564 0a77 6974 6820 7468  accessed.with th
│ │ │ │ -000181c0: 6520 7573 7561 6c20 4353 4d20 636f 6d6d  e usual CSM comm
│ │ │ │ -000181d0: 616e 6420 7769 7468 2065 6974 6865 7220  and with either 
│ │ │ │ -000181e0: 6f6e 6c79 2061 2074 6f72 6963 2076 6172  only a toric var
│ │ │ │ -000181f0: 6965 7479 206f 7220 6120 746f 7269 6320  iety or a toric 
│ │ │ │ -00018200: 7661 7269 6574 790a 616e 6420 6120 4368  variety.and a Ch
│ │ │ │ -00018210: 6f77 2072 696e 6720 6173 2069 6e70 7574  ow ring as input
│ │ │ │ -00018220: 2e20 496e 2074 6869 7320 6361 7365 2077  . In this case w
│ │ │ │ -00018230: 6520 6f6e 6c79 2072 6571 7569 7265 2074  e only require t
│ │ │ │ -00018240: 6861 7420 7468 6520 696e 7075 7420 746f  hat the input to
│ │ │ │ -00018250: 7269 630a 7661 7269 6574 7920 6973 2063  ric.variety is c
│ │ │ │ -00018260: 6f6d 706c 6574 6520 616e 6420 7369 6d70  omplete and simp
│ │ │ │ -00018270: 6c69 6369 616c 2028 696e 2070 6172 7469  licial (in parti
│ │ │ │ -00018280: 6375 6c61 7220 7765 2064 6f20 6e6f 7420  cular we do not 
│ │ │ │ -00018290: 6e65 6564 2069 7420 746f 2062 650a 736d  need it to be.sm
│ │ │ │ -000182a0: 6f6f 7468 292e 0a0a 2b2d 2d2d 2d2d 2d2d  ooth)...+-------
│ │ │ │ -000182b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182d0: 2d2d 2d2b 0a7c 6933 3420 3a20 6e65 6564  ---+.|i34 : need
│ │ │ │ -000182e0: 7350 6163 6b61 6765 2022 4e6f 726d 616c  sPackage "Normal
│ │ │ │ -000182f0: 546f 7269 6356 6172 6965 7469 6573 2220  ToricVarieties" 
│ │ │ │ -00018300: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00018330: 6f33 3420 3d20 4e6f 726d 616c 546f 7269  o34 = NormalTori
│ │ │ │ -00018340: 6356 6172 6965 7469 6573 2020 2020 2020  cVarieties      
│ │ │ │ -00018350: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00018360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000180e0: 2d2d 2d2d 2d2b 0a0a 5468 6973 2066 756e  -----+..This fun
│ │ │ │ +000180f0: 6374 696f 6e20 6d61 7920 616c 736f 2063  ction may also c
│ │ │ │ +00018100: 6f6d 7075 7465 2074 6865 2043 534d 2063  ompute the CSM c
│ │ │ │ +00018110: 6c61 7373 206f 6620 6120 6e6f 726d 616c  lass of a normal
│ │ │ │ +00018120: 2074 6f72 6963 2076 6172 6965 7479 2064   toric variety d
│ │ │ │ +00018130: 6566 696e 6564 0a62 7920 6120 6661 6e2e  efined.by a fan.
│ │ │ │ +00018140: 2049 6e20 7468 6973 2063 6173 6520 6120   In this case a 
│ │ │ │ +00018150: 636f 6d62 696e 6174 6f72 6961 6c20 6d65  combinatorial me
│ │ │ │ +00018160: 7468 6f64 2069 7320 7573 6564 2e20 5468  thod is used. Th
│ │ │ │ +00018170: 6973 206d 6574 686f 6420 6973 2061 6363  is method is acc
│ │ │ │ +00018180: 6573 7365 640a 7769 7468 2074 6865 2075  essed.with the u
│ │ │ │ +00018190: 7375 616c 2043 534d 2063 6f6d 6d61 6e64  sual CSM command
│ │ │ │ +000181a0: 2077 6974 6820 6569 7468 6572 206f 6e6c   with either onl
│ │ │ │ +000181b0: 7920 6120 746f 7269 6320 7661 7269 6574  y a toric variet
│ │ │ │ +000181c0: 7920 6f72 2061 2074 6f72 6963 2076 6172  y or a toric var
│ │ │ │ +000181d0: 6965 7479 0a61 6e64 2061 2043 686f 7720  iety.and a Chow 
│ │ │ │ +000181e0: 7269 6e67 2061 7320 696e 7075 742e 2049  ring as input. I
│ │ │ │ +000181f0: 6e20 7468 6973 2063 6173 6520 7765 206f  n this case we o
│ │ │ │ +00018200: 6e6c 7920 7265 7175 6972 6520 7468 6174  nly require that
│ │ │ │ +00018210: 2074 6865 2069 6e70 7574 2074 6f72 6963   the input toric
│ │ │ │ +00018220: 0a76 6172 6965 7479 2069 7320 636f 6d70  .variety is comp
│ │ │ │ +00018230: 6c65 7465 2061 6e64 2073 696d 706c 6963  lete and simplic
│ │ │ │ +00018240: 6961 6c20 2869 6e20 7061 7274 6963 756c  ial (in particul
│ │ │ │ +00018250: 6172 2077 6520 646f 206e 6f74 206e 6565  ar we do not nee
│ │ │ │ +00018260: 6420 6974 2074 6f20 6265 0a73 6d6f 6f74  d it to be.smoot
│ │ │ │ +00018270: 6829 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  h)...+----------
│ │ │ │ +00018280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000182a0: 2b0a 7c69 3334 203a 206e 6565 6473 5061  +.|i34 : needsPa
│ │ │ │ +000182b0: 636b 6167 6520 224e 6f72 6d61 6c54 6f72  ckage "NormalTor
│ │ │ │ +000182c0: 6963 5661 7269 6574 6965 7322 207c 0a7c  icVarieties" |.|
│ │ │ │ +000182d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3334            |.|o34
│ │ │ │ +00018300: 203d 204e 6f72 6d61 6c54 6f72 6963 5661   = NormalToricVa
│ │ │ │ +00018310: 7269 6574 6965 7320 2020 2020 2020 2020  rieties         
│ │ │ │ +00018320: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00018330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018350: 2020 2020 7c0a 7c6f 3334 203a 2050 6163      |.|o34 : Pac
│ │ │ │ +00018360: 6b61 6765 2020 2020 2020 2020 2020 2020  kage            
│ │ │ │  00018370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018380: 2020 2020 2020 207c 0a7c 6f33 3420 3a20         |.|o34 : 
│ │ │ │ -00018390: 5061 636b 6167 6520 2020 2020 2020 2020  Package         
│ │ │ │ -000183a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000183c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000183d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000183e0: 2d2b 0a7c 6933 3520 3a20 5520 3d20 6869  -+.|i35 : U = hi
│ │ │ │ -000183f0: 727a 6562 7275 6368 5375 7266 6163 6520  rzebruchSurface 
│ │ │ │ -00018400: 3720 2020 2020 2020 2020 2020 2020 7c0a  7             |.
│ │ │ │ -00018410: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00018380: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00018390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000183a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000183b0: 7c69 3335 203a 2055 203d 2068 6972 7a65  |i35 : U = hirze
│ │ │ │ +000183c0: 6272 7563 6853 7572 6661 6365 2037 2020  bruchSurface 7  
│ │ │ │ +000183d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000183e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000183f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018400: 2020 2020 2020 2020 7c0a 7c6f 3335 203d          |.|o35 =
│ │ │ │ +00018410: 2055 2020 2020 2020 2020 2020 2020 2020   U              
│ │ │ │  00018420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018430: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ -00018440: 3520 3d20 5520 2020 2020 2020 2020 2020  5 = U           
│ │ │ │ +00018430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00018470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018490: 2020 2020 207c 0a7c 6f33 3520 3a20 4e6f       |.|o35 : No
│ │ │ │ -000184a0: 726d 616c 546f 7269 6356 6172 6965 7479  rmalToricVariety
│ │ │ │ -000184b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -000184d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000184e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000184f0: 0a7c 6933 3620 3a20 4368 3d54 6f72 6963  .|i36 : Ch=Toric
│ │ │ │ -00018500: 4368 6f77 5269 6e67 2855 2920 2020 2020  ChowRing(U)     
│ │ │ │ -00018510: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018460: 2020 7c0a 7c6f 3335 203a 204e 6f72 6d61    |.|o35 : Norma
│ │ │ │ +00018470: 6c54 6f72 6963 5661 7269 6574 7920 2020  lToricVariety   
│ │ │ │ +00018480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018490: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000184a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000184b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000184c0: 3336 203a 2043 683d 546f 7269 6343 686f  36 : Ch=ToricCho
│ │ │ │ +000184d0: 7752 696e 6728 5529 2020 2020 2020 2020  wRing(U)        
│ │ │ │ +000184e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000184f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018510: 2020 2020 2020 7c0a 7c6f 3336 203d 2043        |.|o36 = C
│ │ │ │ +00018520: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │  00018530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018540: 2020 2020 2020 2020 207c 0a7c 6f33 3620           |.|o36 
│ │ │ │ -00018550: 3d20 4368 2020 2020 2020 2020 2020 2020  = Ch            
│ │ │ │ +00018540: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00018550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018570: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00018580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185a0: 2020 207c 0a7c 6f33 3620 3a20 5175 6f74     |.|o36 : Quot
│ │ │ │ -000185b0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00018600: 6933 3720 3a20 4353 4d20 5520 2020 2020  i37 : CSM U     
│ │ │ │ +00018570: 7c0a 7c6f 3336 203a 2051 756f 7469 656e  |.|o36 : Quotien
│ │ │ │ +00018580: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ +00018590: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000185a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000185b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000185c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3337  ----------+.|i37
│ │ │ │ +000185d0: 203a 2043 534d 2055 2020 2020 2020 2020   : CSM U        
│ │ │ │ +000185e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000185f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00018600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018620: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00018630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00018630: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00018640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00018660: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00018670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018680: 2020 2020 7c0a 7c6f 3337 203d 202d 2033      |.|o37 = - 3
│ │ │ │ -00018690: 7820 7820 202b 2078 2020 2d20 3578 2020  x x  + x  - 5x  
│ │ │ │ -000186a0: 2b20 3278 2020 2b20 3120 2020 2020 2020  + 2x  + 1       
│ │ │ │ -000186b0: 207c 0a7c 2020 2020 2020 2020 2020 3220   |.|          2 
│ │ │ │ -000186c0: 3320 2020 2033 2020 2020 2032 2020 2020  3    3     2    
│ │ │ │ -000186d0: 2033 2020 2020 2020 2020 2020 2020 7c0a   3            |.
│ │ │ │ -000186e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000186f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018700: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00018710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018720: 205a 5a5b 7820 2e2e 7820 5d20 2020 2020   ZZ[x ..x ]     
│ │ │ │ -00018730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00018740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018750: 2020 3020 2020 3320 2020 2020 2020 2020    0   3         
│ │ │ │ -00018760: 2020 2020 207c 0a7c 6f33 3720 3a20 2d2d       |.|o37 : --
│ │ │ │ -00018770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018790: 2d2d 7c0a 7c20 2020 2020 2028 7820 7820  --|.|      (x x 
│ │ │ │ -000187a0: 2c20 7820 7820 2c20 7820 202d 2078 202c  , x x , x  - x ,
│ │ │ │ -000187b0: 2078 2020 2b20 3778 2020 2d20 7820 297c   x  + 7x  - x )|
│ │ │ │ -000187c0: 0a7c 2020 2020 2020 2020 3020 3220 2020  .|        0 2   
│ │ │ │ -000187d0: 3120 3320 2020 3020 2020 2032 2020 2031  1 3   0    2   1
│ │ │ │ -000187e0: 2020 2020 2032 2020 2020 3320 7c0a 2b2d       2    3 |.+-
│ │ │ │ -000187f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018810: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3820  ---------+.|i38 
│ │ │ │ -00018820: 3a20 6373 6d31 3d43 534d 2843 682c 5529  : csm1=CSM(Ch,U)
│ │ │ │ +00018650: 207c 0a7c 6f33 3720 3d20 2d20 3378 2078   |.|o37 = - 3x x
│ │ │ │ +00018660: 2020 2b20 7820 202d 2035 7820 202b 2032    + x  - 5x  + 2
│ │ │ │ +00018670: 7820 202b 2031 2020 2020 2020 2020 7c0a  x  + 1        |.
│ │ │ │ +00018680: 7c20 2020 2020 2020 2020 2032 2033 2020  |          2 3  
│ │ │ │ +00018690: 2020 3320 2020 2020 3220 2020 2020 3320    3     2     3 
│ │ │ │ +000186a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000186b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000186c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000186d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000186e0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +000186f0: 5b78 202e 2e78 205d 2020 2020 2020 2020  [x ..x ]        
│ │ │ │ +00018700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00018710: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00018720: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00018730: 2020 7c0a 7c6f 3337 203a 202d 2d2d 2d2d    |.|o37 : -----
│ │ │ │ +00018740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00018760: 0a7c 2020 2020 2020 2878 2078 202c 2078  .|      (x x , x
│ │ │ │ +00018770: 2078 202c 2078 2020 2d20 7820 2c20 7820   x , x  - x , x 
│ │ │ │ +00018780: 202b 2037 7820 202d 2078 2029 7c0a 7c20   + 7x  - x )|.| 
│ │ │ │ +00018790: 2020 2020 2020 2030 2032 2020 2031 2033         0 2   1 3
│ │ │ │ +000187a0: 2020 2030 2020 2020 3220 2020 3120 2020     0    2   1   
│ │ │ │ +000187b0: 2020 3220 2020 2033 207c 0a2b 2d2d 2d2d    2    3 |.+----
│ │ │ │ +000187c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000187d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000187e0: 2d2d 2d2d 2d2d 2b0a 7c69 3338 203a 2063  ------+.|i38 : c
│ │ │ │ +000187f0: 736d 313d 4353 4d28 4368 2c55 2920 2020  sm1=CSM(Ch,U)   
│ │ │ │ +00018800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018810: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00018820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018840: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00018850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018870: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00018880: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188a0: 7c0a 7c6f 3338 203d 202d 2033 7820 7820  |.|o38 = - 3x x 
│ │ │ │ -000188b0: 202b 2078 2020 2d20 3578 2020 2b20 3278   + x  - 5x  + 2x
│ │ │ │ -000188c0: 2020 2b20 3120 2020 2020 2020 207c 0a7c    + 1        |.|
│ │ │ │ -000188d0: 2020 2020 2020 2020 2020 3220 3320 2020            2 3   
│ │ │ │ -000188e0: 2033 2020 2020 2032 2020 2020 2033 2020   3     2     3  
│ │ │ │ -000188f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00018840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00018850: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00018860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00018870: 6f33 3820 3d20 2d20 3378 2078 2020 2b20  o38 = - 3x x  + 
│ │ │ │ +00018880: 7820 202d 2035 7820 202b 2032 7820 202b  x  - 5x  + 2x  +
│ │ │ │ +00018890: 2031 2020 2020 2020 2020 7c0a 7c20 2020   1        |.|   
│ │ │ │ +000188a0: 2020 2020 2020 2032 2033 2020 2020 3320         2 3    3 
│ │ │ │ +000188b0: 2020 2020 3220 2020 2020 3320 2020 2020      2     3     
│ │ │ │ +000188c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000188d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000188e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000188f0: 2020 2020 7c0a 7c6f 3338 203a 2043 6820      |.|o38 : Ch 
│ │ │ │  00018900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018920: 2020 2020 2020 207c 0a7c 6f33 3820 3a20         |.|o38 : 
│ │ │ │ -00018930: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │ -00018940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018950: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00018960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018980: 2d2b 0a0a 416c 6c20 7468 6520 6578 616d  -+..All the exam
│ │ │ │ -00018990: 706c 6573 2077 6572 6520 646f 6e65 2075  ples were done u
│ │ │ │ -000189a0: 7369 6e67 2073 796d 626f 6c69 6320 636f  sing symbolic co
│ │ │ │ -000189b0: 6d70 7574 6174 696f 6e73 2077 6974 6820  mputations with 
│ │ │ │ -000189c0: 4772 5c22 6f62 6e65 7220 6261 7365 732e  Gr\"obner bases.
│ │ │ │ -000189d0: 0a43 6861 6e67 696e 6720 7468 6520 6f70  .Changing the op
│ │ │ │ -000189e0: 7469 6f6e 202a 6e6f 7465 2043 6f6d 704d  tion *note CompM
│ │ │ │ -000189f0: 6574 686f 643a 2043 6f6d 704d 6574 686f  ethod: CompMetho
│ │ │ │ -00018a00: 642c 2074 6f20 6265 7274 696e 6920 7769  d, to bertini wi
│ │ │ │ -00018a10: 6c6c 2064 6f20 7468 6520 6d61 696e 0a63  ll do the main.c
│ │ │ │ -00018a20: 6f6d 7075 7461 7469 6f6e 7320 6e75 6d65  omputations nume
│ │ │ │ -00018a30: 7269 6361 6c6c 792c 2070 726f 7669 6465  rically, provide
│ │ │ │ -00018a40: 6420 4265 7274 696e 6920 6973 202a 6e6f  d Bertini is *no
│ │ │ │ -00018a50: 7465 2069 6e73 7461 6c6c 6564 2061 6e64  te installed and
│ │ │ │ -00018a60: 2063 6f6e 6669 6775 7265 643a 0a63 6f6e   configured:.con
│ │ │ │ -00018a70: 6669 6775 7269 6e67 2042 6572 7469 6e69  figuring Bertini
│ │ │ │ -00018a80: 2c2e 204e 6f74 6520 7468 6174 2074 6865  ,. Note that the
│ │ │ │ -00018a90: 2062 6572 7469 6e69 2061 6e64 2050 6e52   bertini and PnR
│ │ │ │ -00018aa0: 6573 6964 7561 6c20 6f70 7469 6f6e 7320  esidual options 
│ │ │ │ -00018ab0: 6d61 7920 6f6e 6c79 2062 650a 7573 6564  may only be.used
│ │ │ │ -00018ac0: 2066 6f72 2073 7562 7363 6865 6d65 7320   for subschemes 
│ │ │ │ -00018ad0: 6f66 205c 5050 5e6e 2e0a 0a4f 6273 6572  of \PP^n...Obser
│ │ │ │ -00018ae0: 7665 2074 6861 7420 7468 6520 616c 676f  ve that the algo
│ │ │ │ -00018af0: 7269 7468 6d20 6973 2061 2070 726f 6261  rithm is a proba
│ │ │ │ -00018b00: 6269 6c69 7374 6963 2061 6c67 6f72 6974  bilistic algorit
│ │ │ │ -00018b10: 686d 2061 6e64 206d 6179 2067 6976 6520  hm and may give 
│ │ │ │ -00018b20: 6120 7772 6f6e 670a 616e 7377 6572 2077  a wrong.answer w
│ │ │ │ -00018b30: 6974 6820 6120 736d 616c 6c20 6275 7420  ith a small but 
│ │ │ │ -00018b40: 6e6f 6e7a 6572 6f20 7072 6f62 6162 696c  nonzero probabil
│ │ │ │ -00018b50: 6974 792e 2052 6561 6420 6d6f 7265 2075  ity. Read more u
│ │ │ │ -00018b60: 6e64 6572 202a 6e6f 7465 0a70 726f 6261  nder *note.proba
│ │ │ │ -00018b70: 6269 6c69 7374 6963 2061 6c67 6f72 6974  bilistic algorit
│ │ │ │ -00018b80: 686d 3a20 7072 6f62 6162 696c 6973 7469  hm: probabilisti
│ │ │ │ -00018b90: 6320 616c 676f 7269 7468 6d2c 2e0a 0a0a  c algorithm,....
│ │ │ │ -00018ba0: 0a57 6179 7320 746f 2075 7365 2043 534d  .Ways to use CSM
│ │ │ │ -00018bb0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -00018bc0: 3d3d 0a0a 2020 2a20 2243 534d 2849 6465  ==..  * "CSM(Ide
│ │ │ │ -00018bd0: 616c 2922 0a20 202a 2022 4353 4d28 4964  al)".  * "CSM(Id
│ │ │ │ -00018be0: 6561 6c2c 5379 6d62 6f6c 2922 0a20 202a  eal,Symbol)".  *
│ │ │ │ -00018bf0: 2022 4353 4d28 5175 6f74 6965 6e74 5269   "CSM(QuotientRi
│ │ │ │ -00018c00: 6e67 2c49 6465 616c 2922 0a20 202a 2022  ng,Ideal)".  * "
│ │ │ │ -00018c10: 4353 4d28 5175 6f74 6965 6e74 5269 6e67  CSM(QuotientRing
│ │ │ │ -00018c20: 2c49 6465 616c 2c4d 7574 6162 6c65 4861  ,Ideal,MutableHa
│ │ │ │ -00018c30: 7368 5461 626c 6529 220a 0a46 6f72 2074  shTable)"..For t
│ │ │ │ -00018c40: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ -00018c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00018c60: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -00018c70: 7465 2043 534d 3a20 4353 4d2c 2069 7320  te CSM: CSM, is 
│ │ │ │ -00018c80: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ -00018c90: 756e 6374 696f 6e20 7769 7468 206f 7074  unction with opt
│ │ │ │ -00018ca0: 696f 6e73 3a0a 284d 6163 6175 6c61 7932  ions:.(Macaulay2
│ │ │ │ -00018cb0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -00018cc0: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a  onWithOptions,..
│ │ │ │ -00018cd0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ -00018ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d20: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -00018d30: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ -00018d40: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ -00018d50: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -00018d60: 6175 6c61 7932 2d31 2e32 362e 3036 2b64  aulay2-1.26.06+d
│ │ │ │ -00018d70: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -00018d80: 6163 6b61 6765 732f 0a43 6861 7261 6374  ackages/.Charact
│ │ │ │ -00018d90: 6572 6973 7469 6343 6c61 7373 6573 2e6d  eristicClasses.m
│ │ │ │ -00018da0: 323a 3232 3231 3a30 2e0a 1f0a 4669 6c65  2:2221:0....File
│ │ │ │ -00018db0: 3a20 4368 6172 6163 7465 7269 7374 6963  : Characteristic
│ │ │ │ -00018dc0: 436c 6173 7365 732e 696e 666f 2c20 4e6f  Classes.info, No
│ │ │ │ -00018dd0: 6465 3a20 4575 6c65 722c 204e 6578 743a  de: Euler, Next:
│ │ │ │ -00018de0: 2045 756c 6572 4166 6669 6e65 2c20 5072   EulerAffine, Pr
│ │ │ │ -00018df0: 6576 3a20 4353 4d2c 2055 703a 2054 6f70  ev: CSM, Up: Top
│ │ │ │ -00018e00: 0a0a 4575 6c65 7220 2d2d 2054 6865 2045  ..Euler -- The E
│ │ │ │ -00018e10: 756c 6572 2043 6861 7261 6374 6572 6973  uler Characteris
│ │ │ │ -00018e20: 7469 630a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  tic.************
│ │ │ │ -00018e30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00018e40: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -00018e50: 3a20 0a20 2020 2020 2020 2045 756c 6572  : .        Euler
│ │ │ │ -00018e60: 2049 0a20 2020 2020 2020 2045 756c 6572   I.        Euler
│ │ │ │ -00018e70: 2858 2c4a 290a 2020 2020 2020 2020 4575  (X,J).        Eu
│ │ │ │ -00018e80: 6c65 7220 6373 6d0a 2020 2a20 496e 7075  ler csm.  * Inpu
│ │ │ │ -00018e90: 7473 3a0a 2020 2020 2020 2a20 492c 2061  ts:.      * I, a
│ │ │ │ -00018ea0: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028  n *note ideal: (
│ │ │ │ -00018eb0: 4d61 6361 756c 6179 3244 6f63 2949 6465  Macaulay2Doc)Ide
│ │ │ │ -00018ec0: 616c 2c2c 2061 206d 756c 7469 2d68 6f6d  al,, a multi-hom
│ │ │ │ -00018ed0: 6f67 656e 656f 7573 2069 6465 616c 2069  ogeneous ideal i
│ │ │ │ -00018ee0: 6e20 610a 2020 2020 2020 2020 6772 6164  n a.        grad
│ │ │ │ -00018ef0: 6564 2070 6f6c 796e 6f6d 6961 6c20 7269  ed polynomial ri
│ │ │ │ -00018f00: 6e67 206f 7665 7220 6120 6669 656c 6420  ng over a field 
│ │ │ │ -00018f10: 6465 6669 6e69 6e67 2061 2063 6c6f 7365  defining a close
│ │ │ │ -00018f20: 6420 7375 6273 6368 656d 6520 5620 6f66  d subscheme V of
│ │ │ │ -00018f30: 0a20 2020 2020 2020 205c 5050 5e7b 6e5f  .        \PP^{n_
│ │ │ │ -00018f40: 317d 782e 2e2e 785c 5050 5e7b 6e5f 6d7d  1}x...x\PP^{n_m}
│ │ │ │ -00018f50: 0a20 2020 2020 202a 204a 2c20 616e 202a  .      * J, an *
│ │ │ │ -00018f60: 6e6f 7465 2069 6465 616c 3a20 284d 6163  note ideal: (Mac
│ │ │ │ -00018f70: 6175 6c61 7932 446f 6329 4964 6561 6c2c  aulay2Doc)Ideal,
│ │ │ │ -00018f80: 2c20 616e 2069 6465 616c 2069 6e20 7468  , an ideal in th
│ │ │ │ -00018f90: 6520 6772 6164 6564 0a20 2020 2020 2020  e graded.       
│ │ │ │ -00018fa0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67   polynomial ring
│ │ │ │ -00018fb0: 2077 6869 6368 2069 7320 636f 6f72 6469   which is coordi
│ │ │ │ -00018fc0: 6e61 7465 2072 696e 6720 6f66 2074 6865  nate ring of the
│ │ │ │ -00018fd0: 204e 6f72 6d61 6c20 546f 7269 6320 5661   Normal Toric Va
│ │ │ │ -00018fe0: 7269 6574 7920 580a 2020 2020 2020 2a20  riety X.      * 
│ │ │ │ -00018ff0: 582c 2061 202a 6e6f 7465 206e 6f72 6d61  X, a *note norma
│ │ │ │ -00019000: 6c20 746f 7269 6320 7661 7269 6574 793a  l toric variety:
│ │ │ │ -00019010: 0a20 2020 2020 2020 2028 4e6f 726d 616c  .        (Normal
│ │ │ │ -00019020: 546f 7269 6356 6172 6965 7469 6573 294e  ToricVarieties)N
│ │ │ │ -00019030: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -00019040: 792c 2c20 6120 6e6f 726d 616c 2074 6f72  y,, a normal tor
│ │ │ │ -00019050: 6963 2076 6172 6965 7479 2077 6869 6368  ic variety which
│ │ │ │ -00019060: 0a20 2020 2020 2020 2069 7320 7468 6520  .        is the 
│ │ │ │ -00019070: 616d 6269 656e 7420 7370 6163 6520 7468  ambient space th
│ │ │ │ -00019080: 6174 2077 6520 6172 6520 776f 726b 696e  at we are workin
│ │ │ │ -00019090: 6720 696e 0a20 2020 2020 202a 2063 736d  g in.      * csm
│ │ │ │ -000190a0: 2c20 6120 2a6e 6f74 6520 7269 6e67 2065  , a *note ring e
│ │ │ │ -000190b0: 6c65 6d65 6e74 3a20 284d 6163 6175 6c61  lement: (Macaula
│ │ │ │ -000190c0: 7932 446f 6329 5269 6e67 456c 656d 656e  y2Doc)RingElemen
│ │ │ │ -000190d0: 742c 2c20 7468 6520 4353 4d20 636c 6173  t,, the CSM clas
│ │ │ │ -000190e0: 7320 6f66 0a20 2020 2020 2020 2073 6f6d  s of.        som
│ │ │ │ -000190f0: 6520 7661 7269 6574 7920 560a 2020 2a20  e variety V.  * 
│ │ │ │ -00019100: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069  *note Optional i
│ │ │ │ -00019110: 6e70 7574 733a 2028 4d61 6361 756c 6179  nputs: (Macaulay
│ │ │ │ -00019120: 3244 6f63 2975 7369 6e67 2066 756e 6374  2Doc)using funct
│ │ │ │ -00019130: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ -00019140: 616c 2069 6e70 7574 732c 3a0a 2020 2020  al inputs,:.    
│ │ │ │ -00019150: 2020 2a20 436f 6d70 4d65 7468 6f64 2028    * CompMethod (
│ │ │ │ -00019160: 6d69 7373 696e 6720 646f 6375 6d65 6e74  missing document
│ │ │ │ -00019170: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064  ation) => ..., d
│ │ │ │ -00019180: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ -00019190: 2020 2020 2050 726f 6a65 6374 6976 6544       ProjectiveD
│ │ │ │ -000191a0: 6567 7265 652c 2050 726f 6a65 6374 6976  egree, Projectiv
│ │ │ │ -000191b0: 6544 6567 7265 652c 2061 7070 6c69 6361  eDegree, applica
│ │ │ │ -000191c0: 626c 6520 666f 7220 616c 6c20 6361 7365  ble for all case
│ │ │ │ -000191d0: 7320 7768 6572 6520 7468 650a 2020 2020  s where the.    
│ │ │ │ -000191e0: 2020 2020 6d65 7468 6f64 7320 696e 2074      methods in t
│ │ │ │ -000191f0: 6865 2070 6163 6b61 6765 206d 6179 2062  he package may b
│ │ │ │ -00019200: 6520 7573 6564 0a20 2020 2020 202a 2043  e used.      * C
│ │ │ │ -00019210: 6f6d 704d 6574 686f 6420 286d 6973 7369  ompMethod (missi
│ │ │ │ -00019220: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ -00019230: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c  ) => ..., defaul
│ │ │ │ -00019240: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ -00019250: 5072 6f6a 6563 7469 7665 4465 6772 6565  ProjectiveDegree
│ │ │ │ -00019260: 2c20 506e 5265 7369 6475 616c 2c20 7468  , PnResidual, th
│ │ │ │ -00019270: 6973 2061 6c67 6f72 6974 686d 206d 6179  is algorithm may
│ │ │ │ -00019280: 2062 6520 7573 6564 2066 6f72 2073 7562   be used for sub
│ │ │ │ -00019290: 7363 6865 6d65 730a 2020 2020 2020 2020  schemes.        
│ │ │ │ -000192a0: 6f66 205c 5050 5e6e 206f 6e6c 790a 2020  of \PP^n only.  
│ │ │ │ -000192b0: 2020 2020 2a20 4d65 7468 6f64 2028 6d69      * Method (mi
│ │ │ │ -000192c0: 7373 696e 6720 646f 6375 6d65 6e74 6174  ssing documentat
│ │ │ │ -000192d0: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566  ion) => ..., def
│ │ │ │ -000192e0: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -000192f0: 2020 2049 6e63 6c75 7369 6f6e 4578 636c     InclusionExcl
│ │ │ │ -00019300: 7573 696f 6e2c 2049 6e63 6c75 7369 6f6e  usion, Inclusion
│ │ │ │ -00019310: 4578 636c 7573 696f 6e2c 2061 7070 6c69  Exclusion, appli
│ │ │ │ -00019320: 6361 626c 6520 666f 7220 616c 6c20 696e  cable for all in
│ │ │ │ -00019330: 7075 7473 0a20 2020 2020 202a 204d 6574  puts.      * Met
│ │ │ │ -00019340: 686f 6420 286d 6973 7369 6e67 2064 6f63  hod (missing doc
│ │ │ │ -00019350: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ -00019360: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -00019370: 650a 2020 2020 2020 2020 496e 636c 7573  e.        Inclus
│ │ │ │ -00019380: 696f 6e45 7863 6c75 7369 6f6e 2c20 4469  ionExclusion, Di
│ │ │ │ -00019390: 7265 6374 436f 6d70 6c65 7465 496e 742c  rectCompleteInt,
│ │ │ │ -000193a0: 2074 6869 7320 6d65 7468 6f64 206d 6179   this method may
│ │ │ │ -000193b0: 2070 726f 7669 6465 2061 0a20 2020 2020   provide a.     
│ │ │ │ -000193c0: 2020 2070 6572 666f 726d 616e 6365 2069     performance i
│ │ │ │ -000193d0: 6d70 726f 7665 6d65 6e74 2077 6865 6e20  mprovement when 
│ │ │ │ -000193e0: 7468 6520 696e 7075 7420 6973 2061 2063  the input is a c
│ │ │ │ -000193f0: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ -00019400: 7469 6f6e 2c20 6966 0a20 2020 2020 2020  tion, if.       
│ │ │ │ -00019410: 2074 6865 2069 6e70 7574 2069 7320 6e6f   the input is no
│ │ │ │ -00019420: 7420 6120 636f 6d70 6c65 7465 2069 6e74  t a complete int
│ │ │ │ -00019430: 6572 7365 6374 696f 6e20 696e 636c 7573  ersection inclus
│ │ │ │ -00019440: 696f 6e2f 6578 636c 7573 696f 6e20 6974  ion/exclusion it
│ │ │ │ -00019450: 2077 696c 6c0a 2020 2020 2020 2020 7265   will.        re
│ │ │ │ -00019460: 7475 726e 2061 6e20 6572 726f 720a 2020  turn an error.  
│ │ │ │ -00019470: 2020 2020 2a20 496e 7075 7449 7353 6d6f      * InputIsSmo
│ │ │ │ -00019480: 6f74 6820 286d 6973 7369 6e67 2064 6f63  oth (missing doc
│ │ │ │ -00019490: 756d 656e 7461 7469 6f6e 2920 3d3e 202e  umentation) => .
│ │ │ │ -000194a0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ -000194b0: 6520 6661 6c73 652c 2074 6869 730a 2020  e false, this.  
│ │ │ │ -000194c0: 2020 2020 2020 6f70 7469 6f6e 2068 6173        option has
│ │ │ │ -000194d0: 2076 616c 7565 7320 7472 7565 2f66 616c   values true/fal
│ │ │ │ -000194e0: 7365 2061 6e64 2074 656c 6c73 2074 6865  se and tells the
│ │ │ │ -000194f0: 206d 6574 686f 6420 7768 6574 6865 7220   method whether 
│ │ │ │ -00019500: 746f 2061 7373 756d 6520 7468 650a 2020  to assume the.  
│ │ │ │ -00019510: 2020 2020 2020 696e 7075 7420 6964 6561        input idea
│ │ │ │ -00019520: 6c20 6465 6669 6e65 7320 6120 736d 6f6f  l defines a smoo
│ │ │ │ -00019530: 7468 2073 6368 656d 652c 2061 6e64 2068  th scheme, and h
│ │ │ │ -00019540: 656e 6365 2074 6f20 6361 6c6c 2074 6865  ence to call the
│ │ │ │ -00019550: 206d 6574 686f 6420 4368 6572 6e0a 2020   method Chern.  
│ │ │ │ -00019560: 2020 2020 2020 696e 7374 6561 6420 666f        instead fo
│ │ │ │ -00019570: 7220 7265 6475 6365 6420 7275 6e20 7469  r reduced run ti
│ │ │ │ -00019580: 6d65 2c20 616c 7465 726e 6174 6976 656c  me, alternativel
│ │ │ │ -00019590: 7920 7468 6520 4368 6572 6e20 6675 6e63  y the Chern func
│ │ │ │ -000195a0: 7469 6f6e 2063 616e 2062 650a 2020 2020  tion can be.    
│ │ │ │ -000195b0: 2020 2020 7573 6564 2064 6972 6563 746c      used directl
│ │ │ │ -000195c0: 790a 2020 2020 2020 2a20 4f75 7470 7574  y.      * Output
│ │ │ │ -000195d0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -000195e0: 2076 616c 7565 2043 686f 7752 696e 6745   value ChowRingE
│ │ │ │ -000195f0: 6c65 6d65 6e74 2c20 7468 6520 7479 7065  lement, the type
│ │ │ │ -00019600: 206f 6620 6f75 7470 7574 2074 6f0a 2020   of output to.  
│ │ │ │ -00019610: 2020 2020 2020 7265 7475 726e 2074 6865        return the
│ │ │ │ -00019620: 2064 6566 6175 6c74 206f 7574 7075 7420   default output 
│ │ │ │ -00019630: 6973 2061 6e20 696e 7465 6765 720a 2020  is an integer.  
│ │ │ │ -00019640: 2020 2020 2a20 4f75 7470 7574 203d 3e20      * Output => 
│ │ │ │ -00019650: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -00019660: 7565 2043 686f 7752 696e 6745 6c65 6d65  ue ChowRingEleme
│ │ │ │ -00019670: 6e74 2c20 4861 7368 466f 726d 2c20 7468  nt, HashForm, th
│ │ │ │ -00019680: 6520 7479 7065 206f 660a 2020 2020 2020  e type of.      
│ │ │ │ -00019690: 2020 6f75 7470 7574 2074 6f20 7265 7475    output to retu
│ │ │ │ -000196a0: 726e 2c20 4861 7368 466f 726d 2072 6574  rn, HashForm ret
│ │ │ │ -000196b0: 7572 6e73 2061 204d 7574 6162 6c65 4861  urns a MutableHa
│ │ │ │ -000196c0: 7368 5461 626c 6520 636f 6e74 6169 6e69  shTable containi
│ │ │ │ -000196d0: 6e67 2074 6865 0a20 2020 2020 2020 206b  ng the.        k
│ │ │ │ -000196e0: 6579 2022 4353 4d22 2028 7468 6520 4353  ey "CSM" (the CS
│ │ │ │ -000196f0: 4d20 636c 6173 7329 2c20 616e 6420 6b65  M class), and ke
│ │ │ │ -00019700: 7973 206f 6620 7468 6520 666f 726d 0a20  ys of the form. 
│ │ │ │ -00019710: 2020 2020 2020 205c 7b30 5c7d 2c5c 7b31         \{0\},\{1
│ │ │ │ -00019720: 5c7d 2c5c 7b32 5c7d 2c2e 2e2e 2c5c 7b30  \},\{2\},...,\{0
│ │ │ │ -00019730: 2c31 5c7d 2c5c 7b30 2c32 5c7d 202e 2e2e  ,1\},\{0,2\} ...
│ │ │ │ -00019740: 2e5c 7b30 2c31 2c32 5c7d 2e2e 2e20 616e  .\{0,1,2\}... an
│ │ │ │ -00019750: 6420 736f 206f 6e20 7768 6963 680a 2020  d so on which.  
│ │ │ │ -00019760: 2020 2020 2020 636f 7272 6573 706f 6e64        correspond
│ │ │ │ -00019770: 2074 6f20 7468 6520 696e 6469 6365 7320   to the indices 
│ │ │ │ -00019780: 6f66 2074 6865 2070 6f73 7369 626c 6520  of the possible 
│ │ │ │ -00019790: 7375 6273 6574 7320 6f66 2074 6865 2067  subsets of the g
│ │ │ │ -000197a0: 656e 6572 6174 6f72 7320 6f66 0a20 2020  enerators of.   
│ │ │ │ -000197b0: 2020 2020 2074 6865 2069 6e70 7574 2069       the input i
│ │ │ │ -000197c0: 6465 616c 2c20 666f 7220 6561 6368 2073  deal, for each s
│ │ │ │ -000197d0: 6574 206f 6620 696e 6469 6365 7320 7468  et of indices th
│ │ │ │ -000197e0: 6520 4353 4d20 636c 6173 7320 6f66 2074  e CSM class of t
│ │ │ │ -000197f0: 6865 0a20 2020 2020 2020 2068 7970 6572  he.        hyper
│ │ │ │ -00019800: 7375 7266 6163 6520 6769 7665 6e20 6279  surface given by
│ │ │ │ -00019810: 2074 6865 2070 726f 6475 6374 206f 6620   the product of 
│ │ │ │ -00019820: 616c 6c20 706f 6c79 6e6f 6d69 616c 7320  all polynomials 
│ │ │ │ -00019830: 696e 2074 6865 0a20 2020 2020 2020 2063  in the.        c
│ │ │ │ -00019840: 6f72 7265 7370 6f6e 6469 6e67 2073 6574  orresponding set
│ │ │ │ -00019850: 206f 6620 6765 6e65 7261 746f 7273 2069   of generators i
│ │ │ │ -00019860: 7320 7374 6f72 6564 2c20 7468 6572 6520  s stored, there 
│ │ │ │ -00019870: 6973 206e 6f20 6578 7472 6120 636f 7374  is no extra cost
│ │ │ │ -00019880: 2074 6f0a 2020 2020 2020 2020 7573 696e   to.        usin
│ │ │ │ -00019890: 6720 7468 6973 206f 7074 696f 6e0a 2020  g this option.  
│ │ │ │ -000198a0: 2020 2020 2a20 496e 6473 4f66 536d 6f6f      * IndsOfSmoo
│ │ │ │ -000198b0: 7468 2028 6d69 7373 696e 6720 646f 6375  th (missing docu
│ │ │ │ -000198c0: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e  mentation) => ..
│ │ │ │ -000198d0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -000198e0: 207b 7d2c 2074 6869 730a 2020 2020 2020   {}, this.      
│ │ │ │ -000198f0: 2020 6f70 7469 6f6e 206d 6179 2073 7065    option may spe
│ │ │ │ -00019900: 6564 2075 7020 7468 6520 7275 6e20 7469  ed up the run ti
│ │ │ │ -00019910: 6d65 2077 6865 6e20 7573 696e 6720 7468  me when using th
│ │ │ │ -00019920: 6520 4469 7265 6374 436f 6d70 6c65 7465  e DirectComplete
│ │ │ │ -00019930: 496e 740a 2020 2020 2020 2020 4d65 7468  Int.        Meth
│ │ │ │ -00019940: 6f64 2069 6620 7468 6520 7573 6572 206b  od if the user k
│ │ │ │ -00019950: 6e6f 7773 2061 6464 6974 696f 6e61 6c20  nows additional 
│ │ │ │ -00019960: 696e 666f 726d 6174 696f 6e20 6162 6f75  information abou
│ │ │ │ -00019970: 7420 7468 6520 696e 7075 7420 6964 6561  t the input idea
│ │ │ │ -00019980: 6c2c 0a20 2020 2020 2020 2073 6565 202a  l,.        see *
│ │ │ │ -00019990: 6e6f 7465 2049 6e64 734f 6653 6d6f 6f74  note IndsOfSmoot
│ │ │ │ -000199a0: 683a 2049 6e64 734f 6653 6d6f 6f74 682c  h: IndsOfSmooth,
│ │ │ │ -000199b0: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -000199c0: 2020 2020 2a20 6120 2a6e 6f74 6520 7269      * a *note ri
│ │ │ │ -000199d0: 6e67 2065 6c65 6d65 6e74 3a20 284d 6163  ng element: (Mac
│ │ │ │ -000199e0: 6175 6c61 7932 446f 6329 5269 6e67 456c  aulay2Doc)RingEl
│ │ │ │ -000199f0: 656d 656e 742c 2c20 7468 6520 4575 6c65  ement,, the Eule
│ │ │ │ -00019a00: 720a 2020 2020 2020 2020 6368 6172 6163  r.        charac
│ │ │ │ -00019a10: 7465 7269 7374 6963 0a0a 4465 7363 7269  teristic..Descri
│ │ │ │ -00019a20: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -00019a30: 3d0a 0a46 6f72 2061 2073 7562 7363 6865  =..For a subsche
│ │ │ │ -00019a40: 6d65 2056 206f 6620 616e 2061 7070 6c69  me V of an appli
│ │ │ │ -00019a50: 6361 626c 6520 746f 7269 6320 7661 7269  cable toric vari
│ │ │ │ -00019a60: 6574 7920 582c 2074 6869 7320 636f 6d6d  ety X, this comm
│ │ │ │ -00019a70: 616e 6420 636f 6d70 7574 6573 2074 6865  and computes the
│ │ │ │ -00019a80: 0a45 756c 6572 2063 6861 7261 6374 6572  .Euler character
│ │ │ │ -00019a90: 6973 7469 630a 0a2b 2d2d 2d2d 2d2d 2d2d  istic..+--------
│ │ │ │ +00018920: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00018930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00018950: 0a41 6c6c 2074 6865 2065 7861 6d70 6c65  .All the example
│ │ │ │ +00018960: 7320 7765 7265 2064 6f6e 6520 7573 696e  s were done usin
│ │ │ │ +00018970: 6720 7379 6d62 6f6c 6963 2063 6f6d 7075  g symbolic compu
│ │ │ │ +00018980: 7461 7469 6f6e 7320 7769 7468 2047 725c  tations with Gr\
│ │ │ │ +00018990: 226f 626e 6572 2062 6173 6573 2e0a 4368  "obner bases..Ch
│ │ │ │ +000189a0: 616e 6769 6e67 2074 6865 206f 7074 696f  anging the optio
│ │ │ │ +000189b0: 6e20 2a6e 6f74 6520 436f 6d70 4d65 7468  n *note CompMeth
│ │ │ │ +000189c0: 6f64 3a20 436f 6d70 4d65 7468 6f64 2c20  od: CompMethod, 
│ │ │ │ +000189d0: 746f 2062 6572 7469 6e69 2077 696c 6c20  to bertini will 
│ │ │ │ +000189e0: 646f 2074 6865 206d 6169 6e0a 636f 6d70  do the main.comp
│ │ │ │ +000189f0: 7574 6174 696f 6e73 206e 756d 6572 6963  utations numeric
│ │ │ │ +00018a00: 616c 6c79 2c20 7072 6f76 6964 6564 2042  ally, provided B
│ │ │ │ +00018a10: 6572 7469 6e69 2069 7320 2a6e 6f74 6520  ertini is *note 
│ │ │ │ +00018a20: 696e 7374 616c 6c65 6420 616e 6420 636f  installed and co
│ │ │ │ +00018a30: 6e66 6967 7572 6564 3a0a 636f 6e66 6967  nfigured:.config
│ │ │ │ +00018a40: 7572 696e 6720 4265 7274 696e 692c 2e20  uring Bertini,. 
│ │ │ │ +00018a50: 4e6f 7465 2074 6861 7420 7468 6520 6265  Note that the be
│ │ │ │ +00018a60: 7274 696e 6920 616e 6420 506e 5265 7369  rtini and PnResi
│ │ │ │ +00018a70: 6475 616c 206f 7074 696f 6e73 206d 6179  dual options may
│ │ │ │ +00018a80: 206f 6e6c 7920 6265 0a75 7365 6420 666f   only be.used fo
│ │ │ │ +00018a90: 7220 7375 6273 6368 656d 6573 206f 6620  r subschemes of 
│ │ │ │ +00018aa0: 5c50 505e 6e2e 0a0a 4f62 7365 7276 6520  \PP^n...Observe 
│ │ │ │ +00018ab0: 7468 6174 2074 6865 2061 6c67 6f72 6974  that the algorit
│ │ │ │ +00018ac0: 686d 2069 7320 6120 7072 6f62 6162 696c  hm is a probabil
│ │ │ │ +00018ad0: 6973 7469 6320 616c 676f 7269 7468 6d20  istic algorithm 
│ │ │ │ +00018ae0: 616e 6420 6d61 7920 6769 7665 2061 2077  and may give a w
│ │ │ │ +00018af0: 726f 6e67 0a61 6e73 7765 7220 7769 7468  rong.answer with
│ │ │ │ +00018b00: 2061 2073 6d61 6c6c 2062 7574 206e 6f6e   a small but non
│ │ │ │ +00018b10: 7a65 726f 2070 726f 6261 6269 6c69 7479  zero probability
│ │ │ │ +00018b20: 2e20 5265 6164 206d 6f72 6520 756e 6465  . Read more unde
│ │ │ │ +00018b30: 7220 2a6e 6f74 650a 7072 6f62 6162 696c  r *note.probabil
│ │ │ │ +00018b40: 6973 7469 6320 616c 676f 7269 7468 6d3a  istic algorithm:
│ │ │ │ +00018b50: 2070 726f 6261 6269 6c69 7374 6963 2061   probabilistic a
│ │ │ │ +00018b60: 6c67 6f72 6974 686d 2c2e 0a0a 0a0a 5761  lgorithm,.....Wa
│ │ │ │ +00018b70: 7973 2074 6f20 7573 6520 4353 4d3a 0a3d  ys to use CSM:.=
│ │ │ │ +00018b80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00018b90: 0a20 202a 2022 4353 4d28 4964 6561 6c29  .  * "CSM(Ideal)
│ │ │ │ +00018ba0: 220a 2020 2a20 2243 534d 2849 6465 616c  ".  * "CSM(Ideal
│ │ │ │ +00018bb0: 2c53 796d 626f 6c29 220a 2020 2a20 2243  ,Symbol)".  * "C
│ │ │ │ +00018bc0: 534d 2851 756f 7469 656e 7452 696e 672c  SM(QuotientRing,
│ │ │ │ +00018bd0: 4964 6561 6c29 220a 2020 2a20 2243 534d  Ideal)".  * "CSM
│ │ │ │ +00018be0: 2851 756f 7469 656e 7452 696e 672c 4964  (QuotientRing,Id
│ │ │ │ +00018bf0: 6561 6c2c 4d75 7461 626c 6548 6173 6854  eal,MutableHashT
│ │ │ │ +00018c00: 6162 6c65 2922 0a0a 466f 7220 7468 6520  able)"..For the 
│ │ │ │ +00018c10: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00018c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00018c30: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +00018c40: 4353 4d3a 2043 534d 2c20 6973 2061 202a  CSM: CSM, is a *
│ │ │ │ +00018c50: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +00018c60: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ +00018c70: 733a 0a28 4d61 6361 756c 6179 3244 6f63  s:.(Macaulay2Doc
│ │ │ │ +00018c80: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ +00018c90: 6974 684f 7074 696f 6e73 2c2e 0a0a 2d2d  ithOptions,...--
│ │ │ │ +00018ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +00018cf0: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +00018d00: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +00018d10: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +00018d20: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +00018d30: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ +00018d40: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +00018d50: 6167 6573 2f0a 4368 6172 6163 7465 7269  ages/.Characteri
│ │ │ │ +00018d60: 7374 6963 436c 6173 7365 732e 6d32 3a32  sticClasses.m2:2
│ │ │ │ +00018d70: 3232 313a 302e 0a1f 0a46 696c 653a 2043  221:0....File: C
│ │ │ │ +00018d80: 6861 7261 6374 6572 6973 7469 6343 6c61  haracteristicCla
│ │ │ │ +00018d90: 7373 6573 2e69 6e66 6f2c 204e 6f64 653a  sses.info, Node:
│ │ │ │ +00018da0: 2045 756c 6572 2c20 4e65 7874 3a20 4575   Euler, Next: Eu
│ │ │ │ +00018db0: 6c65 7241 6666 696e 652c 2050 7265 763a  lerAffine, Prev:
│ │ │ │ +00018dc0: 2043 534d 2c20 5570 3a20 546f 700a 0a45   CSM, Up: Top..E
│ │ │ │ +00018dd0: 756c 6572 202d 2d20 5468 6520 4575 6c65  uler -- The Eule
│ │ │ │ +00018de0: 7220 4368 6172 6163 7465 7269 7374 6963  r Characteristic
│ │ │ │ +00018df0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +00018e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00018e10: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +00018e20: 2020 2020 2020 2020 4575 6c65 7220 490a          Euler I.
│ │ │ │ +00018e30: 2020 2020 2020 2020 4575 6c65 7228 582c          Euler(X,
│ │ │ │ +00018e40: 4a29 0a20 2020 2020 2020 2045 756c 6572  J).        Euler
│ │ │ │ +00018e50: 2063 736d 0a20 202a 2049 6e70 7574 733a   csm.  * Inputs:
│ │ │ │ +00018e60: 0a20 2020 2020 202a 2049 2c20 616e 202a  .      * I, an *
│ │ │ │ +00018e70: 6e6f 7465 2069 6465 616c 3a20 284d 6163  note ideal: (Mac
│ │ │ │ +00018e80: 6175 6c61 7932 446f 6329 4964 6561 6c2c  aulay2Doc)Ideal,
│ │ │ │ +00018e90: 2c20 6120 6d75 6c74 692d 686f 6d6f 6765  , a multi-homoge
│ │ │ │ +00018ea0: 6e65 6f75 7320 6964 6561 6c20 696e 2061  neous ideal in a
│ │ │ │ +00018eb0: 0a20 2020 2020 2020 2067 7261 6465 6420  .        graded 
│ │ │ │ +00018ec0: 706f 6c79 6e6f 6d69 616c 2072 696e 6720  polynomial ring 
│ │ │ │ +00018ed0: 6f76 6572 2061 2066 6965 6c64 2064 6566  over a field def
│ │ │ │ +00018ee0: 696e 696e 6720 6120 636c 6f73 6564 2073  ining a closed s
│ │ │ │ +00018ef0: 7562 7363 6865 6d65 2056 206f 660a 2020  ubscheme V of.  
│ │ │ │ +00018f00: 2020 2020 2020 5c50 505e 7b6e 5f31 7d78        \PP^{n_1}x
│ │ │ │ +00018f10: 2e2e 2e78 5c50 505e 7b6e 5f6d 7d0a 2020  ...x\PP^{n_m}.  
│ │ │ │ +00018f20: 2020 2020 2a20 4a2c 2061 6e20 2a6e 6f74      * J, an *not
│ │ │ │ +00018f30: 6520 6964 6561 6c3a 2028 4d61 6361 756c  e ideal: (Macaul
│ │ │ │ +00018f40: 6179 3244 6f63 2949 6465 616c 2c2c 2061  ay2Doc)Ideal,, a
│ │ │ │ +00018f50: 6e20 6964 6561 6c20 696e 2074 6865 2067  n ideal in the g
│ │ │ │ +00018f60: 7261 6465 640a 2020 2020 2020 2020 706f  raded.        po
│ │ │ │ +00018f70: 6c79 6e6f 6d69 616c 2072 696e 6720 7768  lynomial ring wh
│ │ │ │ +00018f80: 6963 6820 6973 2063 6f6f 7264 696e 6174  ich is coordinat
│ │ │ │ +00018f90: 6520 7269 6e67 206f 6620 7468 6520 4e6f  e ring of the No
│ │ │ │ +00018fa0: 726d 616c 2054 6f72 6963 2056 6172 6965  rmal Toric Varie
│ │ │ │ +00018fb0: 7479 2058 0a20 2020 2020 202a 2058 2c20  ty X.      * X, 
│ │ │ │ +00018fc0: 6120 2a6e 6f74 6520 6e6f 726d 616c 2074  a *note normal t
│ │ │ │ +00018fd0: 6f72 6963 2076 6172 6965 7479 3a0a 2020  oric variety:.  
│ │ │ │ +00018fe0: 2020 2020 2020 284e 6f72 6d61 6c54 6f72        (NormalTor
│ │ │ │ +00018ff0: 6963 5661 7269 6574 6965 7329 4e6f 726d  icVarieties)Norm
│ │ │ │ +00019000: 616c 546f 7269 6356 6172 6965 7479 2c2c  alToricVariety,,
│ │ │ │ +00019010: 2061 206e 6f72 6d61 6c20 746f 7269 6320   a normal toric 
│ │ │ │ +00019020: 7661 7269 6574 7920 7768 6963 680a 2020  variety which.  
│ │ │ │ +00019030: 2020 2020 2020 6973 2074 6865 2061 6d62        is the amb
│ │ │ │ +00019040: 6965 6e74 2073 7061 6365 2074 6861 7420  ient space that 
│ │ │ │ +00019050: 7765 2061 7265 2077 6f72 6b69 6e67 2069  we are working i
│ │ │ │ +00019060: 6e0a 2020 2020 2020 2a20 6373 6d2c 2061  n.      * csm, a
│ │ │ │ +00019070: 202a 6e6f 7465 2072 696e 6720 656c 656d   *note ring elem
│ │ │ │ +00019080: 656e 743a 2028 4d61 6361 756c 6179 3244  ent: (Macaulay2D
│ │ │ │ +00019090: 6f63 2952 696e 6745 6c65 6d65 6e74 2c2c  oc)RingElement,,
│ │ │ │ +000190a0: 2074 6865 2043 534d 2063 6c61 7373 206f   the CSM class o
│ │ │ │ +000190b0: 660a 2020 2020 2020 2020 736f 6d65 2076  f.        some v
│ │ │ │ +000190c0: 6172 6965 7479 2056 0a20 202a 202a 6e6f  ariety V.  * *no
│ │ │ │ +000190d0: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ +000190e0: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
│ │ │ │ +000190f0: 6329 7573 696e 6720 6675 6e63 7469 6f6e  c)using function
│ │ │ │ +00019100: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ +00019110: 696e 7075 7473 2c3a 0a20 2020 2020 202a  inputs,:.      *
│ │ │ │ +00019120: 2043 6f6d 704d 6574 686f 6420 286d 6973   CompMethod (mis
│ │ │ │ +00019130: 7369 6e67 2064 6f63 756d 656e 7461 7469  sing documentati
│ │ │ │ +00019140: 6f6e 2920 3d3e 202e 2e2e 2c20 6465 6661  on) => ..., defa
│ │ │ │ +00019150: 756c 7420 7661 6c75 650a 2020 2020 2020  ult value.      
│ │ │ │ +00019160: 2020 5072 6f6a 6563 7469 7665 4465 6772    ProjectiveDegr
│ │ │ │ +00019170: 6565 2c20 5072 6f6a 6563 7469 7665 4465  ee, ProjectiveDe
│ │ │ │ +00019180: 6772 6565 2c20 6170 706c 6963 6162 6c65  gree, applicable
│ │ │ │ +00019190: 2066 6f72 2061 6c6c 2063 6173 6573 2077   for all cases w
│ │ │ │ +000191a0: 6865 7265 2074 6865 0a20 2020 2020 2020  here the.       
│ │ │ │ +000191b0: 206d 6574 686f 6473 2069 6e20 7468 6520   methods in the 
│ │ │ │ +000191c0: 7061 636b 6167 6520 6d61 7920 6265 2075  package may be u
│ │ │ │ +000191d0: 7365 640a 2020 2020 2020 2a20 436f 6d70  sed.      * Comp
│ │ │ │ +000191e0: 4d65 7468 6f64 2028 6d69 7373 696e 6720  Method (missing 
│ │ │ │ +000191f0: 646f 6375 6d65 6e74 6174 696f 6e29 203d  documentation) =
│ │ │ │ +00019200: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +00019210: 616c 7565 0a20 2020 2020 2020 2050 726f  alue.        Pro
│ │ │ │ +00019220: 6a65 6374 6976 6544 6567 7265 652c 2050  jectiveDegree, P
│ │ │ │ +00019230: 6e52 6573 6964 7561 6c2c 2074 6869 7320  nResidual, this 
│ │ │ │ +00019240: 616c 676f 7269 7468 6d20 6d61 7920 6265  algorithm may be
│ │ │ │ +00019250: 2075 7365 6420 666f 7220 7375 6273 6368   used for subsch
│ │ │ │ +00019260: 656d 6573 0a20 2020 2020 2020 206f 6620  emes.        of 
│ │ │ │ +00019270: 5c50 505e 6e20 6f6e 6c79 0a20 2020 2020  \PP^n only.     
│ │ │ │ +00019280: 202a 204d 6574 686f 6420 286d 6973 7369   * Method (missi
│ │ │ │ +00019290: 6e67 2064 6f63 756d 656e 7461 7469 6f6e  ng documentation
│ │ │ │ +000192a0: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c  ) => ..., defaul
│ │ │ │ +000192b0: 7420 7661 6c75 650a 2020 2020 2020 2020  t value.        
│ │ │ │ +000192c0: 496e 636c 7573 696f 6e45 7863 6c75 7369  InclusionExclusi
│ │ │ │ +000192d0: 6f6e 2c20 496e 636c 7573 696f 6e45 7863  on, InclusionExc
│ │ │ │ +000192e0: 6c75 7369 6f6e 2c20 6170 706c 6963 6162  lusion, applicab
│ │ │ │ +000192f0: 6c65 2066 6f72 2061 6c6c 2069 6e70 7574  le for all input
│ │ │ │ +00019300: 730a 2020 2020 2020 2a20 4d65 7468 6f64  s.      * Method
│ │ │ │ +00019310: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ +00019320: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │ +00019330: 2064 6566 6175 6c74 2076 616c 7565 0a20   default value. 
│ │ │ │ +00019340: 2020 2020 2020 2049 6e63 6c75 7369 6f6e         Inclusion
│ │ │ │ +00019350: 4578 636c 7573 696f 6e2c 2044 6972 6563  Exclusion, Direc
│ │ │ │ +00019360: 7443 6f6d 706c 6574 6549 6e74 2c20 7468  tCompleteInt, th
│ │ │ │ +00019370: 6973 206d 6574 686f 6420 6d61 7920 7072  is method may pr
│ │ │ │ +00019380: 6f76 6964 6520 610a 2020 2020 2020 2020  ovide a.        
│ │ │ │ +00019390: 7065 7266 6f72 6d61 6e63 6520 696d 7072  performance impr
│ │ │ │ +000193a0: 6f76 656d 656e 7420 7768 656e 2074 6865  ovement when the
│ │ │ │ +000193b0: 2069 6e70 7574 2069 7320 6120 636f 6d70   input is a comp
│ │ │ │ +000193c0: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +000193d0: 6e2c 2069 660a 2020 2020 2020 2020 7468  n, if.        th
│ │ │ │ +000193e0: 6520 696e 7075 7420 6973 206e 6f74 2061  e input is not a
│ │ │ │ +000193f0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ +00019400: 6563 7469 6f6e 2069 6e63 6c75 7369 6f6e  ection inclusion
│ │ │ │ +00019410: 2f65 7863 6c75 7369 6f6e 2069 7420 7769  /exclusion it wi
│ │ │ │ +00019420: 6c6c 0a20 2020 2020 2020 2072 6574 7572  ll.        retur
│ │ │ │ +00019430: 6e20 616e 2065 7272 6f72 0a20 2020 2020  n an error.     
│ │ │ │ +00019440: 202a 2049 6e70 7574 4973 536d 6f6f 7468   * InputIsSmooth
│ │ │ │ +00019450: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ +00019460: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c  ntation) => ...,
│ │ │ │ +00019470: 2064 6566 6175 6c74 2076 616c 7565 2066   default value f
│ │ │ │ +00019480: 616c 7365 2c20 7468 6973 0a20 2020 2020  alse, this.     
│ │ │ │ +00019490: 2020 206f 7074 696f 6e20 6861 7320 7661     option has va
│ │ │ │ +000194a0: 6c75 6573 2074 7275 652f 6661 6c73 6520  lues true/false 
│ │ │ │ +000194b0: 616e 6420 7465 6c6c 7320 7468 6520 6d65  and tells the me
│ │ │ │ +000194c0: 7468 6f64 2077 6865 7468 6572 2074 6f20  thod whether to 
│ │ │ │ +000194d0: 6173 7375 6d65 2074 6865 0a20 2020 2020  assume the.     
│ │ │ │ +000194e0: 2020 2069 6e70 7574 2069 6465 616c 2064     input ideal d
│ │ │ │ +000194f0: 6566 696e 6573 2061 2073 6d6f 6f74 6820  efines a smooth 
│ │ │ │ +00019500: 7363 6865 6d65 2c20 616e 6420 6865 6e63  scheme, and henc
│ │ │ │ +00019510: 6520 746f 2063 616c 6c20 7468 6520 6d65  e to call the me
│ │ │ │ +00019520: 7468 6f64 2043 6865 726e 0a20 2020 2020  thod Chern.     
│ │ │ │ +00019530: 2020 2069 6e73 7465 6164 2066 6f72 2072     instead for r
│ │ │ │ +00019540: 6564 7563 6564 2072 756e 2074 696d 652c  educed run time,
│ │ │ │ +00019550: 2061 6c74 6572 6e61 7469 7665 6c79 2074   alternatively t
│ │ │ │ +00019560: 6865 2043 6865 726e 2066 756e 6374 696f  he Chern functio
│ │ │ │ +00019570: 6e20 6361 6e20 6265 0a20 2020 2020 2020  n can be.       
│ │ │ │ +00019580: 2075 7365 6420 6469 7265 6374 6c79 0a20   used directly. 
│ │ │ │ +00019590: 2020 2020 202a 204f 7574 7075 7420 3d3e       * Output =>
│ │ │ │ +000195a0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +000195b0: 6c75 6520 4368 6f77 5269 6e67 456c 656d  lue ChowRingElem
│ │ │ │ +000195c0: 656e 742c 2074 6865 2074 7970 6520 6f66  ent, the type of
│ │ │ │ +000195d0: 206f 7574 7075 7420 746f 0a20 2020 2020   output to.     
│ │ │ │ +000195e0: 2020 2072 6574 7572 6e20 7468 6520 6465     return the de
│ │ │ │ +000195f0: 6661 756c 7420 6f75 7470 7574 2069 7320  fault output is 
│ │ │ │ +00019600: 616e 2069 6e74 6567 6572 0a20 2020 2020  an integer.     
│ │ │ │ +00019610: 202a 204f 7574 7075 7420 3d3e 202e 2e2e   * Output => ...
│ │ │ │ +00019620: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00019630: 4368 6f77 5269 6e67 456c 656d 656e 742c  ChowRingElement,
│ │ │ │ +00019640: 2048 6173 6846 6f72 6d2c 2074 6865 2074   HashForm, the t
│ │ │ │ +00019650: 7970 6520 6f66 0a20 2020 2020 2020 206f  ype of.        o
│ │ │ │ +00019660: 7574 7075 7420 746f 2072 6574 7572 6e2c  utput to return,
│ │ │ │ +00019670: 2048 6173 6846 6f72 6d20 7265 7475 726e   HashForm return
│ │ │ │ +00019680: 7320 6120 4d75 7461 626c 6548 6173 6854  s a MutableHashT
│ │ │ │ +00019690: 6162 6c65 2063 6f6e 7461 696e 696e 6720  able containing 
│ │ │ │ +000196a0: 7468 650a 2020 2020 2020 2020 6b65 7920  the.        key 
│ │ │ │ +000196b0: 2243 534d 2220 2874 6865 2043 534d 2063  "CSM" (the CSM c
│ │ │ │ +000196c0: 6c61 7373 292c 2061 6e64 206b 6579 7320  lass), and keys 
│ │ │ │ +000196d0: 6f66 2074 6865 2066 6f72 6d0a 2020 2020  of the form.    
│ │ │ │ +000196e0: 2020 2020 5c7b 305c 7d2c 5c7b 315c 7d2c      \{0\},\{1\},
│ │ │ │ +000196f0: 5c7b 325c 7d2c 2e2e 2e2c 5c7b 302c 315c  \{2\},...,\{0,1\
│ │ │ │ +00019700: 7d2c 5c7b 302c 325c 7d20 2e2e 2e2e 5c7b  },\{0,2\} ....\{
│ │ │ │ +00019710: 302c 312c 325c 7d2e 2e2e 2061 6e64 2073  0,1,2\}... and s
│ │ │ │ +00019720: 6f20 6f6e 2077 6869 6368 0a20 2020 2020  o on which.     
│ │ │ │ +00019730: 2020 2063 6f72 7265 7370 6f6e 6420 746f     correspond to
│ │ │ │ +00019740: 2074 6865 2069 6e64 6963 6573 206f 6620   the indices of 
│ │ │ │ +00019750: 7468 6520 706f 7373 6962 6c65 2073 7562  the possible sub
│ │ │ │ +00019760: 7365 7473 206f 6620 7468 6520 6765 6e65  sets of the gene
│ │ │ │ +00019770: 7261 746f 7273 206f 660a 2020 2020 2020  rators of.      
│ │ │ │ +00019780: 2020 7468 6520 696e 7075 7420 6964 6561    the input idea
│ │ │ │ +00019790: 6c2c 2066 6f72 2065 6163 6820 7365 7420  l, for each set 
│ │ │ │ +000197a0: 6f66 2069 6e64 6963 6573 2074 6865 2043  of indices the C
│ │ │ │ +000197b0: 534d 2063 6c61 7373 206f 6620 7468 650a  SM class of the.
│ │ │ │ +000197c0: 2020 2020 2020 2020 6879 7065 7273 7572          hypersur
│ │ │ │ +000197d0: 6661 6365 2067 6976 656e 2062 7920 7468  face given by th
│ │ │ │ +000197e0: 6520 7072 6f64 7563 7420 6f66 2061 6c6c  e product of all
│ │ │ │ +000197f0: 2070 6f6c 796e 6f6d 6961 6c73 2069 6e20   polynomials in 
│ │ │ │ +00019800: 7468 650a 2020 2020 2020 2020 636f 7272  the.        corr
│ │ │ │ +00019810: 6573 706f 6e64 696e 6720 7365 7420 6f66  esponding set of
│ │ │ │ +00019820: 2067 656e 6572 6174 6f72 7320 6973 2073   generators is s
│ │ │ │ +00019830: 746f 7265 642c 2074 6865 7265 2069 7320  tored, there is 
│ │ │ │ +00019840: 6e6f 2065 7874 7261 2063 6f73 7420 746f  no extra cost to
│ │ │ │ +00019850: 0a20 2020 2020 2020 2075 7369 6e67 2074  .        using t
│ │ │ │ +00019860: 6869 7320 6f70 7469 6f6e 0a20 2020 2020  his option.     
│ │ │ │ +00019870: 202a 2049 6e64 734f 6653 6d6f 6f74 6820   * IndsOfSmooth 
│ │ │ │ +00019880: 286d 6973 7369 6e67 2064 6f63 756d 656e  (missing documen
│ │ │ │ +00019890: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20  tation) => ..., 
│ │ │ │ +000198a0: 6465 6661 756c 7420 7661 6c75 6520 7b7d  default value {}
│ │ │ │ +000198b0: 2c20 7468 6973 0a20 2020 2020 2020 206f  , this.        o
│ │ │ │ +000198c0: 7074 696f 6e20 6d61 7920 7370 6565 6420  ption may speed 
│ │ │ │ +000198d0: 7570 2074 6865 2072 756e 2074 696d 6520  up the run time 
│ │ │ │ +000198e0: 7768 656e 2075 7369 6e67 2074 6865 2044  when using the D
│ │ │ │ +000198f0: 6972 6563 7443 6f6d 706c 6574 6549 6e74  irectCompleteInt
│ │ │ │ +00019900: 0a20 2020 2020 2020 204d 6574 686f 6420  .        Method 
│ │ │ │ +00019910: 6966 2074 6865 2075 7365 7220 6b6e 6f77  if the user know
│ │ │ │ +00019920: 7320 6164 6469 7469 6f6e 616c 2069 6e66  s additional inf
│ │ │ │ +00019930: 6f72 6d61 7469 6f6e 2061 626f 7574 2074  ormation about t
│ │ │ │ +00019940: 6865 2069 6e70 7574 2069 6465 616c 2c0a  he input ideal,.
│ │ │ │ +00019950: 2020 2020 2020 2020 7365 6520 2a6e 6f74          see *not
│ │ │ │ +00019960: 6520 496e 6473 4f66 536d 6f6f 7468 3a20  e IndsOfSmooth: 
│ │ │ │ +00019970: 496e 6473 4f66 536d 6f6f 7468 2c0a 2020  IndsOfSmooth,.  
│ │ │ │ +00019980: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +00019990: 202a 2061 202a 6e6f 7465 2072 696e 6720   * a *note ring 
│ │ │ │ +000199a0: 656c 656d 656e 743a 2028 4d61 6361 756c  element: (Macaul
│ │ │ │ +000199b0: 6179 3244 6f63 2952 696e 6745 6c65 6d65  ay2Doc)RingEleme
│ │ │ │ +000199c0: 6e74 2c2c 2074 6865 2045 756c 6572 0a20  nt,, the Euler. 
│ │ │ │ +000199d0: 2020 2020 2020 2063 6861 7261 6374 6572         character
│ │ │ │ +000199e0: 6973 7469 630a 0a44 6573 6372 6970 7469  istic..Descripti
│ │ │ │ +000199f0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +00019a00: 466f 7220 6120 7375 6273 6368 656d 6520  For a subscheme 
│ │ │ │ +00019a10: 5620 6f66 2061 6e20 6170 706c 6963 6162  V of an applicab
│ │ │ │ +00019a20: 6c65 2074 6f72 6963 2076 6172 6965 7479  le toric variety
│ │ │ │ +00019a30: 2058 2c20 7468 6973 2063 6f6d 6d61 6e64   X, this command
│ │ │ │ +00019a40: 2063 6f6d 7075 7465 7320 7468 650a 4575   computes the.Eu
│ │ │ │ +00019a50: 6c65 7220 6368 6172 6163 7465 7269 7374  ler characterist
│ │ │ │ +00019a60: 6963 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  ic..+-----------
│ │ │ │ +00019a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ae0: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b3d  -----+.|i1 : kk=
│ │ │ │ -00019af0: 5a5a 2f33 3237 3439 3b20 2020 2020 2020  ZZ/32749;       
│ │ │ │ -00019b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00019ab0: 2d2d 2b0a 7c69 3120 3a20 6b6b 3d5a 5a2f  --+.|i1 : kk=ZZ/
│ │ │ │ +00019ac0: 3332 3734 393b 2020 2020 2020 2020 2020  32749;          
│ │ │ │ +00019ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019b00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00019b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019b80: 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 3d6b  -----+.|i2 : R=k
│ │ │ │ -00019b90: 6b5b 785f 302e 2e78 5f34 5d20 2020 2020  k[x_0..x_4]     
│ │ │ │ -00019ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019b50: 2d2d 2b0a 7c69 3220 3a20 523d 6b6b 5b78  --+.|i2 : R=kk[x
│ │ │ │ +00019b60: 5f30 2e2e 785f 345d 2020 2020 2020 2020  _0..x_4]        
│ │ │ │ +00019b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019ba0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00019bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00019bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019bf0: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020    |.|o2 = R     
│ │ │ │  00019c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c20: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │ +00019c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00019c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00019c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c90: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f    |.|o2 : Polyno
│ │ │ │ +00019ca0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │  00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cc0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ -00019cd0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ -00019ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019d10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00019cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019ce0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00019cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d60: 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 3d69  -----+.|i3 : I=i
│ │ │ │ -00019d70: 6465 616c 2872 616e 646f 6d28 312c 5229  deal(random(1,R)
│ │ │ │ -00019d80: 2c72 616e 646f 6d28 322c 5229 2920 2020  ,random(2,R))   
│ │ │ │ +00019d30: 2d2d 2b0a 7c69 3320 3a20 493d 6964 6561  --+.|i3 : I=idea
│ │ │ │ +00019d40: 6c28 7261 6e64 6f6d 2831 2c52 292c 7261  l(random(1,R),ra
│ │ │ │ +00019d50: 6e64 6f6d 2832 2c52 2929 2020 2020 2020  ndom(2,R))      
│ │ │ │ +00019d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019d80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00019d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00019db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019dd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00019de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00019e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e40: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00019e50: 2020 2020 207c 0a7c 6f33 203d 2069 6465       |.|o3 = ide
│ │ │ │ -00019e60: 616c 2028 3130 3778 2020 2b20 3433 3736  al (107x  + 4376
│ │ │ │ -00019e70: 7820 202d 2036 3331 3678 2020 2b20 3331  x  - 6316x  + 31
│ │ │ │ -00019e80: 3837 7820 202b 2033 3738 3378 202c 202d  87x  + 3783x , -
│ │ │ │ -00019e90: 2036 3035 3378 2020 2b20 3835 3730 7820   6053x  + 8570x 
│ │ │ │ -00019ea0: 7820 202b 207c 0a7c 2020 2020 2020 2020  x  + |.|        
│ │ │ │ -00019eb0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -00019ec0: 2031 2020 2020 2020 2020 3220 2020 2020   1        2     
│ │ │ │ -00019ed0: 2020 2033 2020 2020 2020 2020 3420 2020     3        4   
│ │ │ │ -00019ee0: 2020 2020 2020 3020 2020 2020 2020 2030        0        0
│ │ │ │ -00019ef0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d   1   |.|     ---
│ │ │ │ +00019e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019e10: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00019e20: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20    |.|o3 = ideal 
│ │ │ │ +00019e30: 2831 3037 7820 202b 2034 3337 3678 2020  (107x  + 4376x  
│ │ │ │ +00019e40: 2d20 3633 3136 7820 202b 2033 3138 3778  - 6316x  + 3187x
│ │ │ │ +00019e50: 2020 2b20 3337 3833 7820 2c20 2d20 3630    + 3783x , - 60
│ │ │ │ +00019e60: 3533 7820 202b 2038 3537 3078 2078 2020  53x  + 8570x x  
│ │ │ │ +00019e70: 2b20 7c0a 7c20 2020 2020 2020 2020 2020  + |.|           
│ │ │ │ +00019e80: 2020 2020 2030 2020 2020 2020 2020 3120       0        1 
│ │ │ │ +00019e90: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00019ea0: 3320 2020 2020 2020 2034 2020 2020 2020  3        4      
│ │ │ │ +00019eb0: 2020 2030 2020 2020 2020 2020 3020 3120     0        0 1 
│ │ │ │ +00019ec0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
│ │ │ │ +00019ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019f40: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -00019f50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00019f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f70: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00019f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f90: 2020 2020 207c 0a7c 2020 2020 2031 3033       |.|     103
│ │ │ │ -00019fa0: 3539 7820 202d 2031 3630 3930 7820 7820  59x  - 16090x x 
│ │ │ │ -00019fb0: 202d 2038 3231 3078 2078 2020 2b20 3530   - 8210x x  + 50
│ │ │ │ -00019fc0: 3731 7820 202b 2038 3434 3478 2078 2020  71x  + 8444x x  
│ │ │ │ -00019fd0: 2d20 3839 3937 7820 7820 202d 2036 3934  - 8997x x  - 694
│ │ │ │ -00019fe0: 3978 2078 207c 0a7c 2020 2020 2020 2020  9x x |.|        
│ │ │ │ -00019ff0: 2020 2031 2020 2020 2020 2020 2030 2032     1         0 2
│ │ │ │ -0001a000: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -0001a010: 2020 2032 2020 2020 2020 2020 3020 3320     2        0 3 
│ │ │ │ -0001a020: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ -0001a030: 2020 3220 337c 0a7c 2020 2020 202d 2d2d    2 3|.|     ---
│ │ │ │ +00019f10: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +00019f20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00019f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019f40: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00019f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019f60: 2020 7c0a 7c20 2020 2020 3130 3335 3978    |.|     10359x
│ │ │ │ +00019f70: 2020 2d20 3136 3039 3078 2078 2020 2d20    - 16090x x  - 
│ │ │ │ +00019f80: 3832 3130 7820 7820 202b 2035 3037 3178  8210x x  + 5071x
│ │ │ │ +00019f90: 2020 2b20 3834 3434 7820 7820 202d 2038    + 8444x x  - 8
│ │ │ │ +00019fa0: 3939 3778 2078 2020 2d20 3639 3439 7820  997x x  - 6949x 
│ │ │ │ +00019fb0: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +00019fc0: 3120 2020 2020 2020 2020 3020 3220 2020  1         0 2   
│ │ │ │ +00019fd0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +00019fe0: 3220 2020 2020 2020 2030 2033 2020 2020  2        0 3    
│ │ │ │ +00019ff0: 2020 2020 3120 3320 2020 2020 2020 2032      1 3        2
│ │ │ │ +0001a000: 2033 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   3|.|     ------
│ │ │ │ +0001a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a080: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0001a090: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001a0d0: 2020 2020 207c 0a7c 2020 2020 202d 2031       |.|     - 1
│ │ │ │ -0001a0e0: 3432 3534 7820 202d 2031 3132 3236 7820  4254x  - 11226x 
│ │ │ │ -0001a0f0: 7820 202b 2032 3635 3378 2078 2020 2b20  x  + 2653x x  + 
│ │ │ │ -0001a100: 3132 3336 3578 2078 2020 2d20 3130 3232  12365x x  - 1022
│ │ │ │ -0001a110: 3678 2078 2020 2d20 3132 3639 3678 2029  6x x  - 12696x )
│ │ │ │ -0001a120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a130: 2020 2020 2033 2020 2020 2020 2020 2030       3         0
│ │ │ │ -0001a140: 2034 2020 2020 2020 2020 3120 3420 2020   4        1 4   
│ │ │ │ -0001a150: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ -0001a160: 2020 3320 3420 2020 2020 2020 2020 3420    3 4         4 
│ │ │ │ -0001a170: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a050: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001a060: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a090: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001a0a0: 2020 7c0a 7c20 2020 2020 2d20 3134 3235    |.|     - 1425
│ │ │ │ +0001a0b0: 3478 2020 2d20 3131 3232 3678 2078 2020  4x  - 11226x x  
│ │ │ │ +0001a0c0: 2b20 3236 3533 7820 7820 202b 2031 3233  + 2653x x  + 123
│ │ │ │ +0001a0d0: 3635 7820 7820 202d 2031 3032 3236 7820  65x x  - 10226x 
│ │ │ │ +0001a0e0: 7820 202d 2031 3236 3936 7820 2920 2020  x  - 12696x )   
│ │ │ │ +0001a0f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a100: 2020 3320 2020 2020 2020 2020 3020 3420    3         0 4 
│ │ │ │ +0001a110: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ +0001a120: 2020 2032 2034 2020 2020 2020 2020 2033     2 4         3
│ │ │ │ +0001a130: 2034 2020 2020 2020 2020 2034 2020 2020   4         4    
│ │ │ │ +0001a140: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a190: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20    |.|o3 : Ideal 
│ │ │ │ +0001a1a0: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  0001a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1c0: 2020 2020 207c 0a7c 6f33 203a 2049 6465       |.|o3 : Ide
│ │ │ │ -0001a1d0: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ -0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a210: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a1e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a260: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 696d  -----+.|i4 : tim
│ │ │ │ -0001a270: 6520 4575 6c65 7228 492c 496e 7075 7449  e Euler(I,InputI
│ │ │ │ -0001a280: 7353 6d6f 6f74 683d 3e74 7275 6529 2020  sSmooth=>true)  
│ │ │ │ -0001a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a2b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0001a2c0: 2030 2e30 3337 3639 3636 7320 2863 7075   0.0376966s (cpu
│ │ │ │ -0001a2d0: 293b 2030 2e30 3336 3939 3531 7320 2874  ); 0.0369951s (t
│ │ │ │ -0001a2e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +0001a230: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2045  --+.|i4 : time E
│ │ │ │ +0001a240: 756c 6572 2849 2c49 6e70 7574 4973 536d  uler(I,InputIsSm
│ │ │ │ +0001a250: 6f6f 7468 3d3e 7472 7565 2920 2020 2020  ooth=>true)     
│ │ │ │ +0001a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a280: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +0001a290: 3038 3130 3232 3673 2028 6370 7529 3b20  0810226s (cpu); 
│ │ │ │ +0001a2a0: 302e 3036 3134 3133 3573 2028 7468 7265  0.0614135s (thre
│ │ │ │ +0001a2b0: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │ +0001a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a2d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a320: 2020 7c0a 7c6f 3420 3d20 3420 2020 2020    |.|o4 = 4     
│ │ │ │  0001a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a350: 2020 2020 207c 0a7c 6f34 203d 2034 2020       |.|o4 = 4  
│ │ │ │ +0001a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a3a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a370: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a3f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 696d  -----+.|i5 : tim
│ │ │ │ -0001a400: 6520 4575 6c65 7220 4920 2020 2020 2020  e Euler I       
│ │ │ │ -0001a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a440: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -0001a450: 2030 2e32 3537 3335 3473 2028 6370 7529   0.257354s (cpu)
│ │ │ │ -0001a460: 3b20 302e 3136 3434 3435 7320 2874 6872  ; 0.164445s (thr
│ │ │ │ -0001a470: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +0001a3c0: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2045  --+.|i5 : time E
│ │ │ │ +0001a3d0: 756c 6572 2049 2020 2020 2020 2020 2020  uler I          
│ │ │ │ +0001a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a410: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ +0001a420: 3337 3735 3933 7320 2863 7075 293b 2030  377593s (cpu); 0
│ │ │ │ +0001a430: 2e32 3138 3138 3573 2028 7468 7265 6164  .218185s (thread
│ │ │ │ +0001a440: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +0001a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a460: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a4b0: 2020 7c0a 7c6f 3520 3d20 3420 2020 2020    |.|o5 = 4     
│ │ │ │  0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4e0: 2020 2020 207c 0a7c 6f35 203d 2034 2020       |.|o5 = 4  
│ │ │ │ +0001a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a530: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a500: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a580: 2d2d 2d2d 2d2b 0a7c 6936 203a 2045 756c  -----+.|i6 : Eul
│ │ │ │ -0001a590: 6572 4948 6173 683d 4575 6c65 7228 492c  erIHash=Euler(I,
│ │ │ │ -0001a5a0: 4f75 7470 7574 3d3e 4861 7368 466f 726d  Output=>HashForm
│ │ │ │ -0001a5b0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -0001a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a5d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a550: 2d2d 2b0a 7c69 3620 3a20 4575 6c65 7249  --+.|i6 : EulerI
│ │ │ │ +0001a560: 4861 7368 3d45 756c 6572 2849 2c4f 7574  Hash=Euler(I,Out
│ │ │ │ +0001a570: 7075 743d 3e48 6173 6846 6f72 6d29 3b20  put=>HashForm); 
│ │ │ │ +0001a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a5a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a620: 2d2d 2d2d 2d2b 0a7c 6937 203a 2041 3d72  -----+.|i7 : A=r
│ │ │ │ -0001a630: 696e 6720 4575 6c65 7249 4861 7368 2322  ing EulerIHash#"
│ │ │ │ -0001a640: 4353 4d22 2020 2020 2020 2020 2020 2020  CSM"            
│ │ │ │ +0001a5f0: 2d2d 2b0a 7c69 3720 3a20 413d 7269 6e67  --+.|i7 : A=ring
│ │ │ │ +0001a600: 2045 756c 6572 4948 6173 6823 2243 534d   EulerIHash#"CSM
│ │ │ │ +0001a610: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +0001a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a640: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a690: 2020 7c0a 7c6f 3720 3d20 4120 2020 2020    |.|o7 = A     
│ │ │ │  0001a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6c0: 2020 2020 207c 0a7c 6f37 203d 2041 2020       |.|o7 = A  
│ │ │ │ +0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a6e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a730: 2020 7c0a 7c6f 3720 3a20 5175 6f74 6965    |.|o7 : Quotie
│ │ │ │ +0001a740: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │  0001a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a760: 2020 2020 207c 0a7c 6f37 203a 2051 756f       |.|o7 : Quo
│ │ │ │ -0001a770: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ -0001a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a800: 2d2d 2d2d 2d2b 0a7c 6938 203a 2045 756c  -----+.|i8 : Eul
│ │ │ │ -0001a810: 6572 4948 6173 6823 7b30 2c31 7d3d 3d43  erIHash#{0,1}==C
│ │ │ │ -0001a820: 534d 2841 2c69 6465 616c 2849 5f30 2a49  SM(A,ideal(I_0*I
│ │ │ │ -0001a830: 5f31 2929 2020 2020 2020 2020 2020 2020  _1))            
│ │ │ │ +0001a7d0: 2d2d 2b0a 7c69 3820 3a20 4575 6c65 7249  --+.|i8 : EulerI
│ │ │ │ +0001a7e0: 4861 7368 237b 302c 317d 3d3d 4353 4d28  Hash#{0,1}==CSM(
│ │ │ │ +0001a7f0: 412c 6964 6561 6c28 495f 302a 495f 3129  A,ideal(I_0*I_1)
│ │ │ │ +0001a800: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a820: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a870: 2020 7c0a 7c6f 3820 3d20 7472 7565 2020    |.|o8 = true  
│ │ │ │  0001a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8a0: 2020 2020 207c 0a7c 6f38 203d 2074 7275       |.|o8 = tru
│ │ │ │ -0001a8b0: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a940: 2d2d 2d2d 2d2b 0a7c 6939 203a 204a 3d49  -----+.|i9 : J=I
│ │ │ │ -0001a950: 2b69 6465 616c 2878 5f30 2a78 5f32 2d78  +ideal(x_0*x_2-x
│ │ │ │ -0001a960: 5f33 2a78 5f30 2920 2020 2020 2020 2020  _3*x_0)         
│ │ │ │ +0001a910: 2d2d 2b0a 7c69 3920 3a20 4a3d 492b 6964  --+.|i9 : J=I+id
│ │ │ │ +0001a920: 6561 6c28 785f 302a 785f 322d 785f 332a  eal(x_0*x_2-x_3*
│ │ │ │ +0001a930: 785f 3029 2020 2020 2020 2020 2020 2020  x_0)            
│ │ │ │ +0001a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a960: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa20: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0001aa30: 2020 2020 207c 0a7c 6f39 203d 2069 6465       |.|o9 = ide
│ │ │ │ -0001aa40: 616c 2028 3130 3778 2020 2b20 3433 3736  al (107x  + 4376
│ │ │ │ -0001aa50: 7820 202d 2036 3331 3678 2020 2b20 3331  x  - 6316x  + 31
│ │ │ │ -0001aa60: 3837 7820 202b 2033 3738 3378 202c 202d  87x  + 3783x , -
│ │ │ │ -0001aa70: 2036 3035 3378 2020 2b20 3835 3730 7820   6053x  + 8570x 
│ │ │ │ -0001aa80: 7820 202b 207c 0a7c 2020 2020 2020 2020  x  + |.|        
│ │ │ │ -0001aa90: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -0001aaa0: 2031 2020 2020 2020 2020 3220 2020 2020   1        2     
│ │ │ │ -0001aab0: 2020 2033 2020 2020 2020 2020 3420 2020     3        4   
│ │ │ │ -0001aac0: 2020 2020 2020 3020 2020 2020 2020 2030        0        0
│ │ │ │ -0001aad0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d   1   |.|     ---
│ │ │ │ +0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9f0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0001aa00: 2020 7c0a 7c6f 3920 3d20 6964 6561 6c20    |.|o9 = ideal 
│ │ │ │ +0001aa10: 2831 3037 7820 202b 2034 3337 3678 2020  (107x  + 4376x  
│ │ │ │ +0001aa20: 2d20 3633 3136 7820 202b 2033 3138 3778  - 6316x  + 3187x
│ │ │ │ +0001aa30: 2020 2b20 3337 3833 7820 2c20 2d20 3630    + 3783x , - 60
│ │ │ │ +0001aa40: 3533 7820 202b 2038 3537 3078 2078 2020  53x  + 8570x x  
│ │ │ │ +0001aa50: 2b20 7c0a 7c20 2020 2020 2020 2020 2020  + |.|           
│ │ │ │ +0001aa60: 2020 2020 2030 2020 2020 2020 2020 3120       0        1 
│ │ │ │ +0001aa70: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001aa80: 3320 2020 2020 2020 2034 2020 2020 2020  3        4      
│ │ │ │ +0001aa90: 2020 2030 2020 2020 2020 2020 3020 3120     0        0 1 
│ │ │ │ +0001aaa0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d    |.|     ------
│ │ │ │ +0001aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001aae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0001ab30: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0001ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0001ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab70: 2020 2020 207c 0a7c 2020 2020 2031 3033       |.|     103
│ │ │ │ -0001ab80: 3539 7820 202d 2031 3630 3930 7820 7820  59x  - 16090x x 
│ │ │ │ -0001ab90: 202d 2038 3231 3078 2078 2020 2b20 3530   - 8210x x  + 50
│ │ │ │ -0001aba0: 3731 7820 202b 2038 3434 3478 2078 2020  71x  + 8444x x  
│ │ │ │ -0001abb0: 2d20 3839 3937 7820 7820 202d 2036 3934  - 8997x x  - 694
│ │ │ │ -0001abc0: 3978 2078 207c 0a7c 2020 2020 2020 2020  9x x |.|        
│ │ │ │ -0001abd0: 2020 2031 2020 2020 2020 2020 2030 2032     1         0 2
│ │ │ │ -0001abe0: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -0001abf0: 2020 2032 2020 2020 2020 2020 3020 3320     2        0 3 
│ │ │ │ -0001ac00: 2020 2020 2020 2031 2033 2020 2020 2020         1 3      
│ │ │ │ -0001ac10: 2020 3220 337c 0a7c 2020 2020 202d 2d2d    2 3|.|     ---
│ │ │ │ +0001aaf0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001ab00: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab40: 2020 7c0a 7c20 2020 2020 3130 3335 3978    |.|     10359x
│ │ │ │ +0001ab50: 2020 2d20 3136 3039 3078 2078 2020 2d20    - 16090x x  - 
│ │ │ │ +0001ab60: 3832 3130 7820 7820 202b 2035 3037 3178  8210x x  + 5071x
│ │ │ │ +0001ab70: 2020 2b20 3834 3434 7820 7820 202d 2038    + 8444x x  - 8
│ │ │ │ +0001ab80: 3939 3778 2078 2020 2d20 3639 3439 7820  997x x  - 6949x 
│ │ │ │ +0001ab90: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +0001aba0: 3120 2020 2020 2020 2020 3020 3220 2020  1         0 2   
│ │ │ │ +0001abb0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +0001abc0: 3220 2020 2020 2020 2030 2033 2020 2020  2        0 3    
│ │ │ │ +0001abd0: 2020 2020 3120 3320 2020 2020 2020 2032      1 3        2
│ │ │ │ +0001abe0: 2033 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   3|.|     ------
│ │ │ │ +0001abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ac00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ac10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ac20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ac30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ac50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ac60: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │ -0001ac70: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aca0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0001acb0: 2020 2020 207c 0a7c 2020 2020 202d 2031       |.|     - 1
│ │ │ │ -0001acc0: 3432 3534 7820 202d 2031 3132 3236 7820  4254x  - 11226x 
│ │ │ │ -0001acd0: 7820 202b 2032 3635 3378 2078 2020 2b20  x  + 2653x x  + 
│ │ │ │ -0001ace0: 3132 3336 3578 2078 2020 2d20 3130 3232  12365x x  - 1022
│ │ │ │ -0001acf0: 3678 2078 2020 2d20 3132 3639 3678 202c  6x x  - 12696x ,
│ │ │ │ -0001ad00: 2078 2078 207c 0a7c 2020 2020 2020 2020   x x |.|        
│ │ │ │ -0001ad10: 2020 2020 2033 2020 2020 2020 2020 2030       3         0
│ │ │ │ -0001ad20: 2034 2020 2020 2020 2020 3120 3420 2020   4        1 4   
│ │ │ │ -0001ad30: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ -0001ad40: 2020 3320 3420 2020 2020 2020 2020 3420    3 4         4 
│ │ │ │ -0001ad50: 2020 3020 327c 0a7c 2020 2020 202d 2d2d    0 2|.|     ---
│ │ │ │ +0001ac30: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020  --|.|           
│ │ │ │ +0001ac40: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac70: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001ac80: 2020 7c0a 7c20 2020 2020 2d20 3134 3235    |.|     - 1425
│ │ │ │ +0001ac90: 3478 2020 2d20 3131 3232 3678 2078 2020  4x  - 11226x x  
│ │ │ │ +0001aca0: 2b20 3236 3533 7820 7820 202b 2031 3233  + 2653x x  + 123
│ │ │ │ +0001acb0: 3635 7820 7820 202d 2031 3032 3236 7820  65x x  - 10226x 
│ │ │ │ +0001acc0: 7820 202d 2031 3236 3936 7820 2c20 7820  x  - 12696x , x 
│ │ │ │ +0001acd0: 7820 7c0a 7c20 2020 2020 2020 2020 2020  x |.|           
│ │ │ │ +0001ace0: 2020 3320 2020 2020 2020 2020 3020 3420    3         0 4 
│ │ │ │ +0001acf0: 2020 2020 2020 2031 2034 2020 2020 2020         1 4      
│ │ │ │ +0001ad00: 2020 2032 2034 2020 2020 2020 2020 2033     2 4         3
│ │ │ │ +0001ad10: 2034 2020 2020 2020 2020 2034 2020 2030   4         4   0
│ │ │ │ +0001ad20: 2032 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d   2|.|     ------
│ │ │ │ +0001ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ada0: 2d2d 2d2d 2d7c 0a7c 2020 2020 202d 2078  -----|.|     - x
│ │ │ │ -0001adb0: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ -0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad70: 2d2d 7c0a 7c20 2020 2020 2d20 7820 7820  --|.|     - x x 
│ │ │ │ +0001ad80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001adc0: 2020 7c0a 7c20 2020 2020 2020 2030 2033    |.|        0 3
│ │ │ │  0001add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001adf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001ae00: 3020 3320 2020 2020 2020 2020 2020 2020  0 3             
│ │ │ │ -0001ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae60: 2020 7c0a 7c6f 3920 3a20 4964 6561 6c20    |.|o9 : Ideal 
│ │ │ │ +0001ae70: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  0001ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae90: 2020 2020 207c 0a7c 6f39 203a 2049 6465       |.|o9 : Ide
│ │ │ │ -0001aea0: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ -0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aee0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aeb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001af30: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2074 6861  -----+..Note tha
│ │ │ │ -0001af40: 7420 7468 6520 6964 6561 6c20 4a20 6162  t the ideal J ab
│ │ │ │ -0001af50: 6f76 6520 6973 2061 2063 6f6d 706c 6574  ove is a complet
│ │ │ │ -0001af60: 6520 696e 7465 7273 6563 7469 6f6e 2c20  e intersection, 
│ │ │ │ -0001af70: 7468 7573 2077 6520 6d61 7920 6368 616e  thus we may chan
│ │ │ │ -0001af80: 6765 2074 6865 0a6d 6574 686f 6420 6f70  ge the.method op
│ │ │ │ -0001af90: 7469 6f6e 2077 6869 6368 206d 6179 2073  tion which may s
│ │ │ │ -0001afa0: 7065 6564 2063 6f6d 7075 7461 7469 6f6e  peed computation
│ │ │ │ -0001afb0: 2069 6e20 736f 6d65 2063 6173 6573 2e20   in some cases. 
│ │ │ │ -0001afc0: 5765 206d 6179 2061 6c73 6f20 6e6f 7465  We may also note
│ │ │ │ -0001afd0: 2074 6861 740a 7468 6520 6964 6561 6c20   that.the ideal 
│ │ │ │ -0001afe0: 6765 6e65 7261 7465 6420 6279 2074 6865  generated by the
│ │ │ │ -0001aff0: 2066 6972 7374 2032 2067 656e 6572 6174   first 2 generat
│ │ │ │ -0001b000: 6f72 7320 6f66 2049 2064 6566 696e 6573  ors of I defines
│ │ │ │ -0001b010: 2061 2073 6d6f 6f74 6820 7363 6865 6d65   a smooth scheme
│ │ │ │ -0001b020: 2061 6e64 0a69 6e70 7574 2074 6869 7320   and.input this 
│ │ │ │ -0001b030: 696e 666f 726d 6174 696f 6e20 696e 746f  information into
│ │ │ │ -0001b040: 2074 6865 206d 6574 686f 642e 2054 6869   the method. Thi
│ │ │ │ -0001b050: 7320 6d61 7920 616c 736f 2069 6d70 726f  s may also impro
│ │ │ │ -0001b060: 7665 2063 6f6d 7075 7461 7469 6f6e 0a73  ve computation.s
│ │ │ │ -0001b070: 7065 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  peed...+--------
│ │ │ │ -0001b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b0b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ -0001b0c0: 3a20 7469 6d65 2045 756c 6572 284a 2c4d  : time Euler(J,M
│ │ │ │ -0001b0d0: 6574 686f 643d 3e44 6972 6563 7443 6f6d  ethod=>DirectCom
│ │ │ │ -0001b0e0: 706c 6574 6549 6e74 2920 2020 2020 2020  pleteInt)       
│ │ │ │ -0001b0f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b100: 202d 2d20 7573 6564 2030 2e30 3739 3233   -- used 0.07923
│ │ │ │ -0001b110: 3233 7320 2863 7075 293b 2030 2e30 3735  23s (cpu); 0.075
│ │ │ │ -0001b120: 3436 3438 7320 2874 6872 6561 6429 3b20  4648s (thread); 
│ │ │ │ -0001b130: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ -0001b140: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001af00: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2074  --+..Note that t
│ │ │ │ +0001af10: 6865 2069 6465 616c 204a 2061 626f 7665  he ideal J above
│ │ │ │ +0001af20: 2069 7320 6120 636f 6d70 6c65 7465 2069   is a complete i
│ │ │ │ +0001af30: 6e74 6572 7365 6374 696f 6e2c 2074 6875  ntersection, thu
│ │ │ │ +0001af40: 7320 7765 206d 6179 2063 6861 6e67 6520  s we may change 
│ │ │ │ +0001af50: 7468 650a 6d65 7468 6f64 206f 7074 696f  the.method optio
│ │ │ │ +0001af60: 6e20 7768 6963 6820 6d61 7920 7370 6565  n which may spee
│ │ │ │ +0001af70: 6420 636f 6d70 7574 6174 696f 6e20 696e  d computation in
│ │ │ │ +0001af80: 2073 6f6d 6520 6361 7365 732e 2057 6520   some cases. We 
│ │ │ │ +0001af90: 6d61 7920 616c 736f 206e 6f74 6520 7468  may also note th
│ │ │ │ +0001afa0: 6174 0a74 6865 2069 6465 616c 2067 656e  at.the ideal gen
│ │ │ │ +0001afb0: 6572 6174 6564 2062 7920 7468 6520 6669  erated by the fi
│ │ │ │ +0001afc0: 7273 7420 3220 6765 6e65 7261 746f 7273  rst 2 generators
│ │ │ │ +0001afd0: 206f 6620 4920 6465 6669 6e65 7320 6120   of I defines a 
│ │ │ │ +0001afe0: 736d 6f6f 7468 2073 6368 656d 6520 616e  smooth scheme an
│ │ │ │ +0001aff0: 640a 696e 7075 7420 7468 6973 2069 6e66  d.input this inf
│ │ │ │ +0001b000: 6f72 6d61 7469 6f6e 2069 6e74 6f20 7468  ormation into th
│ │ │ │ +0001b010: 6520 6d65 7468 6f64 2e20 5468 6973 206d  e method. This m
│ │ │ │ +0001b020: 6179 2061 6c73 6f20 696d 7072 6f76 6520  ay also improve 
│ │ │ │ +0001b030: 636f 6d70 7574 6174 696f 6e0a 7370 6565  computation.spee
│ │ │ │ +0001b040: 642e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  d...+-----------
│ │ │ │ +0001b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b080: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2074  ------+.|i10 : t
│ │ │ │ +0001b090: 696d 6520 4575 6c65 7228 4a2c 4d65 7468  ime Euler(J,Meth
│ │ │ │ +0001b0a0: 6f64 3d3e 4469 7265 6374 436f 6d70 6c65  od=>DirectComple
│ │ │ │ +0001b0b0: 7465 496e 7429 2020 2020 2020 2020 2020  teInt)          
│ │ │ │ +0001b0c0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ +0001b0d0: 2075 7365 6420 302e 3134 3934 3239 7320   used 0.149429s 
│ │ │ │ +0001b0e0: 2863 7075 293b 2030 2e30 3933 3330 3638  (cpu); 0.0933068
│ │ │ │ +0001b0f0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +0001b100: 6763 2920 2020 2020 2020 2020 2020 7c0a  gc)           |.
│ │ │ │ +0001b110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b150: 2020 7c0a 7c6f 3130 203d 2032 2020 2020    |.|o10 = 2    
│ │ │ │  0001b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b180: 2020 2020 207c 0a7c 6f31 3020 3d20 3220       |.|o10 = 2 
│ │ │ │ -0001b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0001b210: 6931 3120 3a20 7469 6d65 2045 756c 6572  i11 : time Euler
│ │ │ │ -0001b220: 284a 2c4d 6574 686f 643d 3e44 6972 6563  (J,Method=>Direc
│ │ │ │ -0001b230: 7443 6f6d 706c 6574 6549 6e74 2c49 6e64  tCompleteInt,Ind
│ │ │ │ -0001b240: 734f 6653 6d6f 6f74 683d 3e7b 302c 317d  sOfSmooth=>{0,1}
│ │ │ │ -0001b250: 297c 0a7c 202d 2d20 7573 6564 2030 2e31  )|.| -- used 0.1
│ │ │ │ -0001b260: 3935 3536 3273 2028 6370 7529 3b20 302e  95562s (cpu); 0.
│ │ │ │ -0001b270: 3130 3337 3035 7320 2874 6872 6561 6429  103705s (thread)
│ │ │ │ -0001b280: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ -0001b290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b190: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ +0001b1e0: 203a 2074 696d 6520 4575 6c65 7228 4a2c   : time Euler(J,
│ │ │ │ +0001b1f0: 4d65 7468 6f64 3d3e 4469 7265 6374 436f  Method=>DirectCo
│ │ │ │ +0001b200: 6d70 6c65 7465 496e 742c 496e 6473 4f66  mpleteInt,IndsOf
│ │ │ │ +0001b210: 536d 6f6f 7468 3d3e 7b30 2c31 7d29 7c0a  Smooth=>{0,1})|.
│ │ │ │ +0001b220: 7c20 2d2d 2075 7365 6420 302e 3330 3834  | -- used 0.3084
│ │ │ │ +0001b230: 3938 7320 2863 7075 293b 2030 2e31 3334  98s (cpu); 0.134
│ │ │ │ +0001b240: 3434 7320 2874 6872 6561 6429 3b20 3073  44s (thread); 0s
│ │ │ │ +0001b250: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +0001b260: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b2a0: 2020 2020 2020 7c0a 7c6f 3131 203d 2032        |.|o11 = 2
│ │ │ │  0001b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2d0: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ -0001b2e0: 3d20 3220 2020 2020 2020 2020 2020 2020  = 2             
│ │ │ │ -0001b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b360: 2d2b 0a0a 4e6f 7720 636f 6e73 6964 6572  -+..Now consider
│ │ │ │ -0001b370: 2061 6e20 6578 616d 706c 6520 696e 205c   an example in \
│ │ │ │ -0001b380: 5050 5e32 205c 7469 6d65 7320 5c50 505e  PP^2 \times \PP^
│ │ │ │ -0001b390: 322e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  2...+-----------
│ │ │ │ -0001b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b3c0: 2d2d 2d2b 0a7c 6931 3220 3a20 523d 4d75  ---+.|i12 : R=Mu
│ │ │ │ -0001b3d0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ -0001b3e0: 287b 322c 327d 2920 2020 2020 2020 2020  ({2,2})         
│ │ │ │ -0001b3f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b2e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0001b330: 0a4e 6f77 2063 6f6e 7369 6465 7220 616e  .Now consider an
│ │ │ │ +0001b340: 2065 7861 6d70 6c65 2069 6e20 5c50 505e   example in \PP^
│ │ │ │ +0001b350: 3220 5c74 696d 6573 205c 5050 5e32 2e0a  2 \times \PP^2..
│ │ │ │ +0001b360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b390: 2b0a 7c69 3132 203a 2052 3d4d 756c 7469  +.|i12 : R=Multi
│ │ │ │ +0001b3a0: 5072 6f6a 436f 6f72 6452 696e 6728 7b32  ProjCoordRing({2
│ │ │ │ +0001b3b0: 2c32 7d29 2020 2020 2020 2020 2020 2020  ,2})            
│ │ │ │ +0001b3c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b3f0: 2020 7c0a 7c6f 3132 203d 2052 2020 2020    |.|o12 = R    
│ │ │ │  0001b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b420: 2020 2020 207c 0a7c 6f31 3220 3d20 5220       |.|o12 = R 
│ │ │ │ +0001b420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b450: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b450: 2020 2020 7c0a 7c6f 3132 203a 2050 6f6c      |.|o12 : Pol
│ │ │ │ +0001b460: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │  0001b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b480: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ -0001b490: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ -0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b4b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001b4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b4e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ -0001b4f0: 3a20 723d 6765 6e73 2052 2020 2020 2020  : r=gens R      
│ │ │ │ +0001b480: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001b490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b4b0: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2072  ------+.|i13 : r
│ │ │ │ +0001b4c0: 3d67 656e 7320 5220 2020 2020 2020 2020  =gens R         
│ │ │ │ +0001b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b4e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b510: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b540: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001b550: 3320 3d20 7b78 202c 2078 202c 2078 202c  3 = {x , x , x ,
│ │ │ │ -0001b560: 2078 202c 2078 202c 2078 207d 2020 2020   x , x , x }    
│ │ │ │ -0001b570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001b580: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ -0001b590: 2020 2033 2020 2034 2020 2035 2020 2020     3   4   5    
│ │ │ │ -0001b5a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b510: 2020 2020 2020 2020 7c0a 7c6f 3133 203d          |.|o13 =
│ │ │ │ +0001b520: 207b 7820 2c20 7820 2c20 7820 2c20 7820   {x , x , x , x 
│ │ │ │ +0001b530: 2c20 7820 2c20 7820 7d20 2020 2020 2020  , x , x }       
│ │ │ │ +0001b540: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001b550: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0001b560: 3320 2020 3420 2020 3520 2020 2020 2020  3   4   5       
│ │ │ │ +0001b570: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b5a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001b5b0: 3320 3a20 4c69 7374 2020 2020 2020 2020  3 : List        
│ │ │ │  0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b5e0: 7c6f 3133 203a 204c 6973 7420 2020 2020  |o13 : List     
│ │ │ │ -0001b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001b610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0001b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b640: 2b0a 7c69 3134 203a 204b 3d69 6465 616c  +.|i14 : K=ideal
│ │ │ │ -0001b650: 2872 5f30 5e32 2a72 5f33 2d72 5f34 2a72  (r_0^2*r_3-r_4*r
│ │ │ │ -0001b660: 5f31 2a72 5f32 2c72 5f32 5e32 2a72 5f35  _1*r_2,r_2^2*r_5
│ │ │ │ -0001b670: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ -0001b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b5d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001b610: 6931 3420 3a20 4b3d 6964 6561 6c28 725f  i14 : K=ideal(r_
│ │ │ │ +0001b620: 305e 322a 725f 332d 725f 342a 725f 312a  0^2*r_3-r_4*r_1*
│ │ │ │ +0001b630: 725f 322c 725f 325e 322a 725f 3529 7c0a  r_2,r_2^2*r_5)|.
│ │ │ │ +0001b640: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001b680: 3220 2020 2020 2020 2020 2020 2020 2032  2              2
│ │ │ │  0001b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b6a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001b6b0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -0001b6c0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0001b6d0: 2020 207c 0a7c 6f31 3420 3d20 6964 6561     |.|o14 = idea
│ │ │ │ -0001b6e0: 6c20 2878 2078 2020 2d20 7820 7820 7820  l (x x  - x x x 
│ │ │ │ -0001b6f0: 2c20 7820 7820 2920 2020 2020 2020 2020  , x x )         
│ │ │ │ -0001b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001b710: 2020 2020 2030 2033 2020 2020 3120 3220       0 3    1 2 
│ │ │ │ -0001b720: 3420 2020 3220 3520 2020 2020 2020 2020  4   2 5         
│ │ │ │ -0001b730: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b6a0: 7c0a 7c6f 3134 203d 2069 6465 616c 2028  |.|o14 = ideal (
│ │ │ │ +0001b6b0: 7820 7820 202d 2078 2078 2078 202c 2078  x x  - x x x , x
│ │ │ │ +0001b6c0: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ +0001b6d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b6e0: 2020 3020 3320 2020 2031 2032 2034 2020    0 3    1 2 4  
│ │ │ │ +0001b6f0: 2032 2035 2020 2020 2020 2020 2020 2020   2 5            
│ │ │ │ +0001b700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b730: 2020 207c 0a7c 6f31 3420 3a20 4964 6561     |.|o14 : Idea
│ │ │ │ +0001b740: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │  0001b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b760: 2020 2020 2020 7c0a 7c6f 3134 203a 2049        |.|o14 : I
│ │ │ │ -0001b770: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ -0001b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b790: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a  --------+.|i15 :
│ │ │ │ -0001b7d0: 2045 756c 6572 4b3d 4575 6c65 7228 4b29   EulerK=Euler(K)
│ │ │ │ +0001b760: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b790: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 4575  -----+.|i15 : Eu
│ │ │ │ +0001b7a0: 6c65 724b 3d45 756c 6572 284b 2920 2020  lerK=Euler(K)   
│ │ │ │ +0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b7c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b7f0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20         |.|o15 = 
│ │ │ │ +0001b800: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │  0001b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b820: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -0001b830: 203d 2037 2020 2020 2020 2020 2020 2020   = 7            
│ │ │ │ -0001b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b850: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0001b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001b890: 3136 203a 2063 736d 4b3d 2043 534d 284b  16 : csmK= CSM(K
│ │ │ │ -0001b8a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b8f0: 7c20 2020 2020 2020 2032 2032 2020 2020  |        2 2    
│ │ │ │ -0001b900: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │ -0001b910: 3220 2020 2020 2020 2020 2020 2032 207c  2            2 |
│ │ │ │ -0001b920: 0a7c 6f31 3620 3d20 3768 2068 2020 2b20  .|o16 = 7h h  + 
│ │ │ │ -0001b930: 3568 2068 2020 2b20 3468 2068 2020 2b20  5h h  + 4h h  + 
│ │ │ │ -0001b940: 6820 202b 2033 6820 6820 202b 2068 2020  h  + 3h h  + h  
│ │ │ │ -0001b950: 7c0a 7c20 2020 2020 2020 2031 2032 2020  |.|        1 2  
│ │ │ │ -0001b960: 2020 2031 2032 2020 2020 2031 2032 2020     1 2     1 2  
│ │ │ │ -0001b970: 2020 3120 2020 2020 3120 3220 2020 2032    1     1 2    2
│ │ │ │ -0001b980: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b820: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b850: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620  ---------+.|i16 
│ │ │ │ +0001b860: 3a20 6373 6d4b 3d20 4353 4d28 4b29 2020  : csmK= CSM(K)  
│ │ │ │ +0001b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b880: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b8b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001b8c0: 2020 2020 2020 3220 3220 2020 2020 3220        2 2     2 
│ │ │ │ +0001b8d0: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +0001b8e0: 2020 2020 2020 2020 2020 3220 7c0a 7c6f            2 |.|o
│ │ │ │ +0001b8f0: 3136 203d 2037 6820 6820 202b 2035 6820  16 = 7h h  + 5h 
│ │ │ │ +0001b900: 6820 202b 2034 6820 6820 202b 2068 2020  h  + 4h h  + h  
│ │ │ │ +0001b910: 2b20 3368 2068 2020 2b20 6820 207c 0a7c  + 3h h  + h  |.|
│ │ │ │ +0001b920: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ +0001b930: 3120 3220 2020 2020 3120 3220 2020 2031  1 2     1 2    1
│ │ │ │ +0001b940: 2020 2020 2031 2032 2020 2020 3220 7c0a       1 2    2 |.
│ │ │ │ +0001b950: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b980: 0a7c 2020 2020 2020 5a5a 5b68 202e 2e68  .|      ZZ[h ..h
│ │ │ │ +0001b990: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │  0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9b0: 2020 7c0a 7c20 2020 2020 205a 5a5b 6820    |.|      ZZ[h 
│ │ │ │ -0001b9c0: 2e2e 6820 5d20 2020 2020 2020 2020 2020  ..h ]           
│ │ │ │ +0001b9b0: 7c0a 7c20 2020 2020 2020 2020 2031 2020  |.|          1  
│ │ │ │ +0001b9c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001b9f0: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │ +0001b9e0: 207c 0a7c 6f31 3620 3a20 2d2d 2d2d 2d2d   |.|o16 : ------
│ │ │ │ +0001b9f0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba10: 2020 2020 7c0a 7c6f 3136 203a 202d 2d2d      |.|o16 : ---
│ │ │ │ -0001ba20: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │ +0001ba10: 2020 7c0a 7c20 2020 2020 2020 2020 3320    |.|         3 
│ │ │ │ +0001ba20: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │  0001ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001ba50: 2033 2020 2033 2020 2020 2020 2020 2020   3   3          
│ │ │ │ +0001ba40: 2020 207c 0a7c 2020 2020 2020 2028 6820     |.|       (h 
│ │ │ │ +0001ba50: 2c20 6820 2920 2020 2020 2020 2020 2020  , h )           
│ │ │ │  0001ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ba80: 2868 202c 2068 2029 2020 2020 2020 2020  (h , h )        
│ │ │ │ +0001ba70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001ba80: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │  0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001baa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001bab0: 2020 2031 2020 2032 2020 2020 2020 2020     1   2        
│ │ │ │ -0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bad0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bb00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ -0001bb10: 3a20 4575 6c65 724b 3d3d 4575 6c65 7228  : EulerK==Euler(
│ │ │ │ -0001bb20: 6373 6d4b 2920 2020 2020 2020 2020 2020  csmK)           
│ │ │ │ -0001bb30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001baa0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bad0: 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a 2045  ------+.|i17 : E
│ │ │ │ +0001bae0: 756c 6572 4b3d 3d45 756c 6572 2863 736d  ulerK==Euler(csm
│ │ │ │ +0001baf0: 4b29 2020 2020 2020 2020 2020 2020 2020  K)              
│ │ │ │ +0001bb00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb30: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ +0001bb40: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │  0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb60: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001bb70: 3720 3d20 7472 7565 2020 2020 2020 2020  7 = true        
│ │ │ │ -0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0001bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -0001bbd0: 496e 2074 6865 2063 6173 6520 7768 6572  In the case wher
│ │ │ │ -0001bbe0: 6520 7468 6520 616d 6269 656e 7420 7370  e the ambient sp
│ │ │ │ -0001bbf0: 6163 6520 6973 2061 2074 6f72 6963 2076  ace is a toric v
│ │ │ │ -0001bc00: 6172 6965 7479 2077 6869 6368 2069 7320  ariety which is 
│ │ │ │ -0001bc10: 6e6f 7420 6120 7072 6f64 7563 740a 6f66  not a product.of
│ │ │ │ -0001bc20: 2070 726f 6a65 6374 6976 6520 7370 6163   projective spac
│ │ │ │ -0001bc30: 6573 2077 6520 6d75 7374 206c 6f61 6420  es we must load 
│ │ │ │ -0001bc40: 7468 6520 4e6f 726d 616c 546f 7269 6356  the NormalToricV
│ │ │ │ -0001bc50: 6172 6965 7469 6573 2070 6163 6b61 6765  arieties package
│ │ │ │ -0001bc60: 2061 6e64 206d 7573 740a 616c 736f 2069   and must.also i
│ │ │ │ -0001bc70: 6e70 7574 2074 6865 2074 6f72 6963 2076  nput the toric v
│ │ │ │ -0001bc80: 6172 6965 7479 2e20 4966 2074 6865 2074  ariety. If the t
│ │ │ │ -0001bc90: 6f72 6963 2076 6172 6965 7479 2069 7320  oric variety is 
│ │ │ │ -0001bca0: 6120 7072 6f64 7563 7420 6f66 2070 726f  a product of pro
│ │ │ │ -0001bcb0: 6a65 6374 6976 650a 7370 6163 6520 6974  jective.space it
│ │ │ │ -0001bcc0: 2069 7320 7265 636f 6d6d 656e 6465 6420   is recommended 
│ │ │ │ -0001bcd0: 746f 2075 7365 2074 6865 2066 6f72 6d20  to use the form 
│ │ │ │ -0001bce0: 6162 6f76 6520 7261 7468 6572 2074 6861  above rather tha
│ │ │ │ -0001bcf0: 6e20 696e 7075 7474 696e 6720 7468 6520  n inputting the 
│ │ │ │ -0001bd00: 746f 7269 630a 7661 7269 6574 7920 666f  toric.variety fo
│ │ │ │ -0001bd10: 7220 6566 6669 6369 656e 6379 2072 6561  r efficiency rea
│ │ │ │ -0001bd20: 736f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  sons...+--------
│ │ │ │ -0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd70: 2b0a 7c69 3138 203a 206e 6565 6473 5061  +.|i18 : needsPa
│ │ │ │ -0001bd80: 636b 6167 6520 224e 6f72 6d61 6c54 6f72  ckage "NormalTor
│ │ │ │ -0001bd90: 6963 5661 7269 6574 6965 7322 2020 2020  icVarieties"    
│ │ │ │ +0001bb60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20  ----------+..In 
│ │ │ │ +0001bba0: 7468 6520 6361 7365 2077 6865 7265 2074  the case where t
│ │ │ │ +0001bbb0: 6865 2061 6d62 6965 6e74 2073 7061 6365  he ambient space
│ │ │ │ +0001bbc0: 2069 7320 6120 746f 7269 6320 7661 7269   is a toric vari
│ │ │ │ +0001bbd0: 6574 7920 7768 6963 6820 6973 206e 6f74  ety which is not
│ │ │ │ +0001bbe0: 2061 2070 726f 6475 6374 0a6f 6620 7072   a product.of pr
│ │ │ │ +0001bbf0: 6f6a 6563 7469 7665 2073 7061 6365 7320  ojective spaces 
│ │ │ │ +0001bc00: 7765 206d 7573 7420 6c6f 6164 2074 6865  we must load the
│ │ │ │ +0001bc10: 204e 6f72 6d61 6c54 6f72 6963 5661 7269   NormalToricVari
│ │ │ │ +0001bc20: 6574 6965 7320 7061 636b 6167 6520 616e  eties package an
│ │ │ │ +0001bc30: 6420 6d75 7374 0a61 6c73 6f20 696e 7075  d must.also inpu
│ │ │ │ +0001bc40: 7420 7468 6520 746f 7269 6320 7661 7269  t the toric vari
│ │ │ │ +0001bc50: 6574 792e 2049 6620 7468 6520 746f 7269  ety. If the tori
│ │ │ │ +0001bc60: 6320 7661 7269 6574 7920 6973 2061 2070  c variety is a p
│ │ │ │ +0001bc70: 726f 6475 6374 206f 6620 7072 6f6a 6563  roduct of projec
│ │ │ │ +0001bc80: 7469 7665 0a73 7061 6365 2069 7420 6973  tive.space it is
│ │ │ │ +0001bc90: 2072 6563 6f6d 6d65 6e64 6564 2074 6f20   recommended to 
│ │ │ │ +0001bca0: 7573 6520 7468 6520 666f 726d 2061 626f  use the form abo
│ │ │ │ +0001bcb0: 7665 2072 6174 6865 7220 7468 616e 2069  ve rather than i
│ │ │ │ +0001bcc0: 6e70 7574 7469 6e67 2074 6865 2074 6f72  nputting the tor
│ │ │ │ +0001bcd0: 6963 0a76 6172 6965 7479 2066 6f72 2065  ic.variety for e
│ │ │ │ +0001bce0: 6666 6963 6965 6e63 7920 7265 6173 6f6e  fficiency reason
│ │ │ │ +0001bcf0: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  s...+-----------
│ │ │ │ +0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001bd40: 6931 3820 3a20 6e65 6564 7350 6163 6b61  i18 : needsPacka
│ │ │ │ +0001bd50: 6765 2022 4e6f 726d 616c 546f 7269 6356  ge "NormalToricV
│ │ │ │ +0001bd60: 6172 6965 7469 6573 2220 2020 2020 2020  arieties"       
│ │ │ │ +0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bd80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bdb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bdd0: 2020 207c 0a7c 6f31 3820 3d20 4e6f 726d     |.|o18 = Norm
│ │ │ │ +0001bde0: 616c 546f 7269 6356 6172 6965 7469 6573  alToricVarieties
│ │ │ │  0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be00: 2020 2020 2020 7c0a 7c6f 3138 203d 204e        |.|o18 = N
│ │ │ │ -0001be10: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ -0001be20: 6965 7320 2020 2020 2020 2020 2020 2020  ies             
│ │ │ │ +0001be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001be20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  0001be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001be60: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +0001be70: 3a20 5061 636b 6167 6520 2020 2020 2020  : Package       
│ │ │ │  0001be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001bea0: 3138 203a 2050 6163 6b61 6765 2020 2020  18 : Package    
│ │ │ │ -0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bee0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bf30: 2d2d 2b0a 7c69 3139 203a 2052 686f 203d  --+.|i19 : Rho =
│ │ │ │ -0001bf40: 207b 7b31 2c30 2c30 7d2c 7b30 2c31 2c30   {{1,0,0},{0,1,0
│ │ │ │ -0001bf50: 7d2c 7b30 2c30 2c31 7d2c 7b2d 312c 2d31  },{0,0,1},{-1,-1
│ │ │ │ -0001bf60: 2c30 7d2c 7b30 2c30 2c2d 317d 7d20 2020  ,0},{0,0,-1}}   
│ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001beb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001bf00: 0a7c 6931 3920 3a20 5268 6f20 3d20 7b7b  .|i19 : Rho = {{
│ │ │ │ +0001bf10: 312c 302c 307d 2c7b 302c 312c 307d 2c7b  1,0,0},{0,1,0},{
│ │ │ │ +0001bf20: 302c 302c 317d 2c7b 2d31 2c2d 312c 307d  0,0,1},{-1,-1,0}
│ │ │ │ +0001bf30: 2c7b 302c 302c 2d31 7d7d 2020 2020 2020  ,{0,0,-1}}      
│ │ │ │ +0001bf40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfc0: 2020 2020 2020 2020 7c0a 7c6f 3139 203d          |.|o19 =
│ │ │ │ -0001bfd0: 207b 7b31 2c20 302c 2030 7d2c 207b 302c   {{1, 0, 0}, {0,
│ │ │ │ -0001bfe0: 2031 2c20 307d 2c20 7b30 2c20 302c 2031   1, 0}, {0, 0, 1
│ │ │ │ -0001bff0: 7d2c 207b 2d31 2c20 2d31 2c20 307d 2c20  }, {-1, -1, 0}, 
│ │ │ │ -0001c000: 7b30 2c20 302c 202d 317d 7d20 2020 2020  {0, 0, -1}}     
│ │ │ │ -0001c010: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf90: 2020 2020 207c 0a7c 6f31 3920 3d20 7b7b       |.|o19 = {{
│ │ │ │ +0001bfa0: 312c 2030 2c20 307d 2c20 7b30 2c20 312c  1, 0, 0}, {0, 1,
│ │ │ │ +0001bfb0: 2030 7d2c 207b 302c 2030 2c20 317d 2c20   0}, {0, 0, 1}, 
│ │ │ │ +0001bfc0: 7b2d 312c 202d 312c 2030 7d2c 207b 302c  {-1, -1, 0}, {0,
│ │ │ │ +0001bfd0: 2030 2c20 2d31 7d7d 2020 2020 2020 2020   0, -1}}        
│ │ │ │ +0001bfe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c020: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001c030: 3920 3a20 4c69 7374 2020 2020 2020 2020  9 : List        
│ │ │ │  0001c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001c060: 7c6f 3139 203a 204c 6973 7420 2020 2020  |o19 : List     
│ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c070: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c0f0: 2d2d 2d2d 2b0a 7c69 3230 203a 2053 6967  ----+.|i20 : Sig
│ │ │ │ -0001c100: 6d61 203d 207b 7b30 2c31 2c32 7d2c 7b31  ma = {{0,1,2},{1
│ │ │ │ -0001c110: 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30  ,2,3},{0,2,3},{0
│ │ │ │ -0001c120: 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30  ,1,4},{1,3,4},{0
│ │ │ │ -0001c130: 2c33 2c34 7d7d 2020 2020 2020 2020 207c  ,3,4}}         |
│ │ │ │ -0001c140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c180: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ -0001c190: 203d 207b 7b30 2c20 312c 2032 7d2c 207b   = {{0, 1, 2}, {
│ │ │ │ -0001c1a0: 312c 2032 2c20 337d 2c20 7b30 2c20 322c  1, 2, 3}, {0, 2,
│ │ │ │ -0001c1b0: 2033 7d2c 207b 302c 2031 2c20 347d 2c20   3}, {0, 1, 4}, 
│ │ │ │ -0001c1c0: 7b31 2c20 332c 2034 7d2c 207b 302c 2033  {1, 3, 4}, {0, 3
│ │ │ │ -0001c1d0: 2c20 347d 7d7c 0a7c 2020 2020 2020 2020  , 4}}|.|        
│ │ │ │ -0001c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c0c0: 2d2b 0a7c 6932 3020 3a20 5369 676d 6120  -+.|i20 : Sigma 
│ │ │ │ +0001c0d0: 3d20 7b7b 302c 312c 327d 2c7b 312c 322c  = {{0,1,2},{1,2,
│ │ │ │ +0001c0e0: 337d 2c7b 302c 322c 337d 2c7b 302c 312c  3},{0,2,3},{0,1,
│ │ │ │ +0001c0f0: 347d 2c7b 312c 332c 347d 2c7b 302c 332c  4},{1,3,4},{0,3,
│ │ │ │ +0001c100: 347d 7d20 2020 2020 2020 2020 7c0a 7c20  4}}         |.| 
│ │ │ │ +0001c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c150: 2020 2020 2020 207c 0a7c 6f32 3020 3d20         |.|o20 = 
│ │ │ │ +0001c160: 7b7b 302c 2031 2c20 327d 2c20 7b31 2c20  {{0, 1, 2}, {1, 
│ │ │ │ +0001c170: 322c 2033 7d2c 207b 302c 2032 2c20 337d  2, 3}, {0, 2, 3}
│ │ │ │ +0001c180: 2c20 7b30 2c20 312c 2034 7d2c 207b 312c  , {0, 1, 4}, {1,
│ │ │ │ +0001c190: 2033 2c20 347d 2c20 7b30 2c20 332c 2034   3, 4}, {0, 3, 4
│ │ │ │ +0001c1a0: 7d7d 7c0a 7c20 2020 2020 2020 2020 2020  }}|.|           
│ │ │ │ +0001c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c1f0: 6f32 3020 3a20 4c69 7374 2020 2020 2020  o20 : List      
│ │ │ │  0001c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c220: 7c0a 7c6f 3230 203a 204c 6973 7420 2020  |.|o20 : List   
│ │ │ │ -0001c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c260: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c230: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c2b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 2058  ------+.|i21 : X
│ │ │ │ -0001c2c0: 203d 206e 6f72 6d61 6c54 6f72 6963 5661   = normalToricVa
│ │ │ │ -0001c2d0: 7269 6574 7928 5268 6f2c 5369 676d 612c  riety(Rho,Sigma,
│ │ │ │ -0001c2e0: 436f 6566 6669 6369 656e 7452 696e 6720  CoefficientRing 
│ │ │ │ -0001c2f0: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020  =>ZZ/32749)     
│ │ │ │ -0001c300: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c280: 2d2d 2d2b 0a7c 6932 3120 3a20 5820 3d20  ---+.|i21 : X = 
│ │ │ │ +0001c290: 6e6f 726d 616c 546f 7269 6356 6172 6965  normalToricVarie
│ │ │ │ +0001c2a0: 7479 2852 686f 2c53 6967 6d61 2c43 6f65  ty(Rho,Sigma,Coe
│ │ │ │ +0001c2b0: 6666 6963 6965 6e74 5269 6e67 203d 3e5a  fficientRing =>Z
│ │ │ │ +0001c2c0: 5a2f 3332 3734 3929 2020 2020 2020 7c0a  Z/32749)      |.
│ │ │ │ +0001c2d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c310: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +0001c320: 3d20 5820 2020 2020 2020 2020 2020 2020  = X             
│ │ │ │  0001c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c340: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001c350: 3231 203d 2058 2020 2020 2020 2020 2020  21 = X          
│ │ │ │ -0001c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0001c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c390: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c3a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c3b0: 0a7c 6f32 3120 3a20 4e6f 726d 616c 546f  .|o21 : NormalTo
│ │ │ │ +0001c3c0: 7269 6356 6172 6965 7479 2020 2020 2020  ricVariety      
│ │ │ │  0001c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3e0: 2020 7c0a 7c6f 3231 203a 204e 6f72 6d61    |.|o21 : Norma
│ │ │ │ -0001c3f0: 6c54 6f72 6963 5661 7269 6574 7920 2020  lToricVariety   
│ │ │ │ -0001c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c420: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c3f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a  --------+.|i22 :
│ │ │ │ -0001c480: 2043 6865 636b 546f 7269 6356 6172 6965   CheckToricVarie
│ │ │ │ -0001c490: 7479 5661 6c69 6428 5829 2020 2020 2020  tyValid(X)      
│ │ │ │ +0001c440: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 4368  -----+.|i22 : Ch
│ │ │ │ +0001c450: 6563 6b54 6f72 6963 5661 7269 6574 7956  eckToricVarietyV
│ │ │ │ +0001c460: 616c 6964 2858 2920 2020 2020 2020 2020  alid(X)         
│ │ │ │ +0001c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c490: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c4d0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001c4e0: 3220 3d20 7472 7565 2020 2020 2020 2020  2 = true        
│ │ │ │  0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c500: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001c510: 7c6f 3232 203d 2074 7275 6520 2020 2020  |o22 = true     
│ │ │ │ -0001c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c550: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c520: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c5a0: 2d2d 2d2d 2b0a 7c69 3233 203a 2052 3d72  ----+.|i23 : R=r
│ │ │ │ -0001c5b0: 696e 6728 5829 2020 2020 2020 2020 2020  ing(X)          
│ │ │ │ +0001c570: 2d2b 0a7c 6932 3320 3a20 523d 7269 6e67  -+.|i23 : R=ring
│ │ │ │ +0001c580: 2858 2920 2020 2020 2020 2020 2020 2020  (X)             
│ │ │ │ +0001c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001c5f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c600: 2020 2020 2020 207c 0a7c 6f32 3320 3d20         |.|o23 = 
│ │ │ │ +0001c610: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0001c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c630: 2020 2020 2020 2020 2020 7c0a 7c6f 3233            |.|o23
│ │ │ │ -0001c640: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ -0001c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c650: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c6a0: 6f32 3320 3a20 506f 6c79 6e6f 6d69 616c  o23 : Polynomial
│ │ │ │ +0001c6b0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │  0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6d0: 7c0a 7c6f 3233 203a 2050 6f6c 796e 6f6d  |.|o23 : Polynom
│ │ │ │ -0001c6e0: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ -0001c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c710: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c6e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c760: 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a 2049  ------+.|i24 : I
│ │ │ │ -0001c770: 3d69 6465 616c 2852 5f30 5e34 2a52 5f31  =ideal(R_0^4*R_1
│ │ │ │ -0001c780: 2c52 5f30 2a52 5f33 2a52 5f34 2a52 5f32  ,R_0*R_3*R_4*R_2
│ │ │ │ -0001c790: 2d52 5f32 5e32 2a52 5f30 5e32 2920 2020  -R_2^2*R_0^2)   
│ │ │ │ +0001c730: 2d2d 2d2b 0a7c 6932 3420 3a20 493d 6964  ---+.|i24 : I=id
│ │ │ │ +0001c740: 6561 6c28 525f 305e 342a 525f 312c 525f  eal(R_0^4*R_1,R_
│ │ │ │ +0001c750: 302a 525f 332a 525f 342a 525f 322d 525f  0*R_3*R_4*R_2-R_
│ │ │ │ +0001c760: 325e 322a 525f 305e 3229 2020 2020 2020  2^2*R_0^2)      
│ │ │ │ +0001c770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001c780: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001c800: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -0001c810: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ -0001c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c840: 2020 2020 2020 207c 0a7c 6f32 3420 3d20         |.|o24 = 
│ │ │ │ -0001c850: 6964 6561 6c20 2878 2078 202c 202d 2078  ideal (x x , - x
│ │ │ │ -0001c860: 2078 2020 2b20 7820 7820 7820 7820 2920   x  + x x x x ) 
│ │ │ │ -0001c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001c8a0: 2020 2030 2031 2020 2020 2030 2032 2020     0 1     0 2  
│ │ │ │ -0001c8b0: 2020 3020 3220 3320 3420 2020 2020 2020    0 2 3 4       
│ │ │ │ +0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c7c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001c7d0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0001c7e0: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +0001c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c810: 2020 2020 7c0a 7c6f 3234 203d 2069 6465      |.|o24 = ide
│ │ │ │ +0001c820: 616c 2028 7820 7820 2c20 2d20 7820 7820  al (x x , - x x 
│ │ │ │ +0001c830: 202b 2078 2078 2078 2078 2029 2020 2020   + x x x x )    
│ │ │ │ +0001c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001c860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001c870: 3020 3120 2020 2020 3020 3220 2020 2030  0 1     0 2    0
│ │ │ │ +0001c880: 2032 2033 2034 2020 2020 2020 2020 2020   2 3 4          
│ │ │ │ +0001c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c8a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c8f0: 2020 2020 207c 0a7c 6f32 3420 3a20 4964       |.|o24 : Id
│ │ │ │ +0001c900: 6561 6c20 6f66 2052 2020 2020 2020 2020  eal of R        
│ │ │ │  0001c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c920: 2020 2020 2020 2020 7c0a 7c6f 3234 203a          |.|o24 :
│ │ │ │ -0001c930: 2049 6465 616c 206f 6620 5220 2020 2020   Ideal of R     
│ │ │ │ -0001c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c970: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001c9c0: 7c69 3235 203a 2063 736d 493d 4353 4d28  |i25 : csmI=CSM(
│ │ │ │ -0001c9d0: 582c 4929 2020 2020 2020 2020 2020 2020  X,I)            
│ │ │ │ +0001c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c940: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001c990: 3520 3a20 6373 6d49 3d43 534d 2858 2c49  5 : csmI=CSM(X,I
│ │ │ │ +0001c9a0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c9d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca20: 207c 0a7c 2020 2020 2020 2020 3220 2020   |.|        2   
│ │ │ │ +0001ca30: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0001ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca50: 2020 2020 7c0a 7c20 2020 2020 2020 2032      |.|        2
│ │ │ │ -0001ca60: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0001ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001caa0: 0a7c 6f32 3520 3d20 3578 2078 2020 2b20  .|o25 = 5x x  + 
│ │ │ │ -0001cab0: 3378 2020 2b20 3478 2078 2020 2b20 7820  3x  + 4x x  + x 
│ │ │ │ -0001cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001caf0: 2020 2020 2033 2034 2020 2020 2033 2020       3 4     3  
│ │ │ │ -0001cb00: 2020 2033 2034 2020 2020 3320 2020 2020     3 4    3     
│ │ │ │ +0001ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ca60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001ca70: 3235 203d 2035 7820 7820 202b 2033 7820  25 = 5x x  + 3x 
│ │ │ │ +0001ca80: 202b 2034 7820 7820 202b 2078 2020 2020   + 4x x  + x    
│ │ │ │ +0001ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001cac0: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +0001cad0: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │ +0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0001cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb60: 2020 2020 2020 5a5a 5b78 202e 2e78 205d        ZZ[x ..x ]
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001cb90: 2020 2020 2020 2020 205a 5a5b 7820 2e2e           ZZ[x ..
│ │ │ │ -0001cba0: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │ -0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cbb0: 2020 2020 2030 2020 2034 2020 2020 2020       0   4      
│ │ │ │ +0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbe0: 2020 2020 2020 2020 3020 2020 3420 2020          0   4   
│ │ │ │ -0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc10: 2020 2020 2020 7c0a 7c6f 3235 203a 202d        |.|o25 : -
│ │ │ │ -0001cc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cc40: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc60: 207c 0a7c 2020 2020 2020 2878 2078 202c   |.|      (x x ,
│ │ │ │ -0001cc70: 2078 2078 2078 202c 2078 2020 2d20 7820   x x x , x  - x 
│ │ │ │ -0001cc80: 2c20 7820 202d 2078 202c 2078 2020 2d20  , x  - x , x  - 
│ │ │ │ -0001cc90: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │ -0001cca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001ccb0: 2020 2020 2020 2032 2034 2020 2030 2031         2 4   0 1
│ │ │ │ -0001ccc0: 2033 2020 2030 2020 2020 3320 2020 3120   3   0    3   1 
│ │ │ │ -0001ccd0: 2020 2033 2020 2032 2020 2020 3420 2020     3   2    4   
│ │ │ │ -0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ccf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cd40: 2d2d 2b0a 7c69 3236 203a 2045 756c 6572  --+.|i26 : Euler
│ │ │ │ -0001cd50: 493d 4575 6c65 7228 582c 4929 2020 2020  I=Euler(X,I)    
│ │ │ │ +0001cbe0: 2020 207c 0a7c 6f32 3520 3a20 2d2d 2d2d     |.|o25 : ----
│ │ │ │ +0001cbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cc10: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020  -----           
│ │ │ │ +0001cc20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001cc30: 7c20 2020 2020 2028 7820 7820 2c20 7820  |      (x x , x 
│ │ │ │ +0001cc40: 7820 7820 2c20 7820 202d 2078 202c 2078  x x , x  - x , x
│ │ │ │ +0001cc50: 2020 2d20 7820 2c20 7820 202d 2078 2029    - x , x  - x )
│ │ │ │ +0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cc70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001cc80: 2020 2020 3220 3420 2020 3020 3120 3320      2 4   0 1 3 
│ │ │ │ +0001cc90: 2020 3020 2020 2033 2020 2031 2020 2020    0    3   1    
│ │ │ │ +0001cca0: 3320 2020 3220 2020 2034 2020 2020 2020  3   2    4      
│ │ │ │ +0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ccc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001ccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001cd10: 0a7c 6932 3620 3a20 4575 6c65 7249 3d45  .|i26 : EulerI=E
│ │ │ │ +0001cd20: 756c 6572 2858 2c49 2920 2020 2020 2020  uler(X,I)       
│ │ │ │ +0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cda0: 2020 2020 207c 0a7c 6f32 3620 3d20 3520       |.|o26 = 5 
│ │ │ │  0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdd0: 2020 2020 2020 2020 7c0a 7c6f 3236 203d          |.|o26 =
│ │ │ │ -0001cde0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001ce70: 7c69 3237 203a 2045 756c 6572 2863 736d  |i27 : Euler(csm
│ │ │ │ -0001ce80: 4929 3d3d 4575 6c65 7249 2020 2020 2020  I)==EulerI      
│ │ │ │ +0001cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cdf0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0001ce40: 3720 3a20 4575 6c65 7228 6373 6d49 293d  7 : Euler(csmI)=
│ │ │ │ +0001ce50: 3d45 756c 6572 4920 2020 2020 2020 2020  =EulerI         
│ │ │ │ +0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0001ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ceb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ced0: 207c 0a7c 6f32 3720 3d20 7472 7565 2020   |.|o27 = true  
│ │ │ │  0001cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf00: 2020 2020 7c0a 7c6f 3237 203d 2074 7275      |.|o27 = tru
│ │ │ │ -0001cf10: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001cf50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0001cf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6c6c  ----------+..All
│ │ │ │ -0001cfa0: 2074 6865 2065 7861 6d70 6c65 7320 7765   the examples we
│ │ │ │ -0001cfb0: 7265 2064 6f6e 6520 7573 696e 6720 7379  re done using sy
│ │ │ │ -0001cfc0: 6d62 6f6c 6963 2063 6f6d 7075 7461 7469  mbolic computati
│ │ │ │ -0001cfd0: 6f6e 7320 7769 7468 2047 725c 226f 626e  ons with Gr\"obn
│ │ │ │ -0001cfe0: 6572 2062 6173 6573 2e0a 4368 616e 6769  er bases..Changi
│ │ │ │ -0001cff0: 6e67 2074 6865 206f 7074 696f 6e20 2a6e  ng the option *n
│ │ │ │ -0001d000: 6f74 6520 436f 6d70 4d65 7468 6f64 3a20  ote CompMethod: 
│ │ │ │ -0001d010: 436f 6d70 4d65 7468 6f64 2c20 746f 2062  CompMethod, to b
│ │ │ │ -0001d020: 6572 7469 6e69 2077 696c 6c20 646f 2074  ertini will do t
│ │ │ │ -0001d030: 6865 206d 6169 6e0a 636f 6d70 7574 6174  he main.computat
│ │ │ │ -0001d040: 696f 6e73 206e 756d 6572 6963 616c 6c79  ions numerically
│ │ │ │ -0001d050: 2c20 7072 6f76 6964 6564 2042 6572 7469  , provided Berti
│ │ │ │ -0001d060: 6e69 2069 7320 2a6e 6f74 6520 696e 7374  ni is *note inst
│ │ │ │ -0001d070: 616c 6c65 6420 616e 6420 636f 6e66 6967  alled and config
│ │ │ │ -0001d080: 7572 6564 3a0a 636f 6e66 6967 7572 696e  ured:.configurin
│ │ │ │ -0001d090: 6720 4265 7274 696e 692c 2e20 4e6f 7465  g Bertini,. Note
│ │ │ │ -0001d0a0: 2074 6861 7420 7468 6520 6265 7274 696e   that the bertin
│ │ │ │ -0001d0b0: 6920 616e 6420 506e 5265 7369 6475 616c  i and PnResidual
│ │ │ │ -0001d0c0: 206f 7074 696f 6e73 206d 6179 206f 6e6c   options may onl
│ │ │ │ -0001d0d0: 7920 6265 0a75 7365 6420 666f 7220 7375  y be.used for su
│ │ │ │ -0001d0e0: 6273 6368 656d 6573 206f 6620 5c50 505e  bschemes of \PP^
│ │ │ │ -0001d0f0: 6e2e 0a0a 4f62 7365 7276 6520 7468 6174  n...Observe that
│ │ │ │ -0001d100: 2074 6865 2061 6c67 6f72 6974 686d 2069   the algorithm i
│ │ │ │ -0001d110: 7320 6120 7072 6f62 6162 696c 6973 7469  s a probabilisti
│ │ │ │ -0001d120: 6320 616c 676f 7269 7468 6d20 616e 6420  c algorithm and 
│ │ │ │ -0001d130: 6d61 7920 6769 7665 2061 2077 726f 6e67  may give a wrong
│ │ │ │ -0001d140: 0a61 6e73 7765 7220 7769 7468 2061 2073  .answer with a s
│ │ │ │ -0001d150: 6d61 6c6c 2062 7574 206e 6f6e 7a65 726f  mall but nonzero
│ │ │ │ -0001d160: 2070 726f 6261 6269 6c69 7479 2e20 5265   probability. Re
│ │ │ │ -0001d170: 6164 206d 6f72 6520 756e 6465 7220 2a6e  ad more under *n
│ │ │ │ -0001d180: 6f74 650a 7072 6f62 6162 696c 6973 7469  ote.probabilisti
│ │ │ │ -0001d190: 6320 616c 676f 7269 7468 6d3a 2070 726f  c algorithm: pro
│ │ │ │ -0001d1a0: 6261 6269 6c69 7374 6963 2061 6c67 6f72  babilistic algor
│ │ │ │ -0001d1b0: 6974 686d 2c2e 0a0a 5761 7973 2074 6f20  ithm,...Ways to 
│ │ │ │ -0001d1c0: 7573 6520 4575 6c65 723a 0a3d 3d3d 3d3d  use Euler:.=====
│ │ │ │ -0001d1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -0001d1e0: 202a 2022 4575 6c65 7228 4964 6561 6c29   * "Euler(Ideal)
│ │ │ │ -0001d1f0: 220a 2020 2a20 2245 756c 6572 2852 696e  ".  * "Euler(Rin
│ │ │ │ -0001d200: 6745 6c65 6d65 6e74 2922 0a0a 466f 7220  gElement)"..For 
│ │ │ │ -0001d210: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -0001d220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d230: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0001d240: 6f74 6520 4575 6c65 723a 2045 756c 6572  ote Euler: Euler
│ │ │ │ -0001d250: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -0001d260: 686f 6420 6675 6e63 7469 6f6e 2077 6974  hod function wit
│ │ │ │ -0001d270: 6820 6f70 7469 6f6e 733a 0a28 4d61 6361  h options:.(Maca
│ │ │ │ -0001d280: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ -0001d290: 756e 6374 696f 6e57 6974 684f 7074 696f  unctionWithOptio
│ │ │ │ -0001d2a0: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  ns,...----------
│ │ │ │ +0001cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cf10: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001cf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cf60: 2d2d 2d2d 2d2d 2d2b 0a0a 416c 6c20 7468  -------+..All th
│ │ │ │ +0001cf70: 6520 6578 616d 706c 6573 2077 6572 6520  e examples were 
│ │ │ │ +0001cf80: 646f 6e65 2075 7369 6e67 2073 796d 626f  done using symbo
│ │ │ │ +0001cf90: 6c69 6320 636f 6d70 7574 6174 696f 6e73  lic computations
│ │ │ │ +0001cfa0: 2077 6974 6820 4772 5c22 6f62 6e65 7220   with Gr\"obner 
│ │ │ │ +0001cfb0: 6261 7365 732e 0a43 6861 6e67 696e 6720  bases..Changing 
│ │ │ │ +0001cfc0: 7468 6520 6f70 7469 6f6e 202a 6e6f 7465  the option *note
│ │ │ │ +0001cfd0: 2043 6f6d 704d 6574 686f 643a 2043 6f6d   CompMethod: Com
│ │ │ │ +0001cfe0: 704d 6574 686f 642c 2074 6f20 6265 7274  pMethod, to bert
│ │ │ │ +0001cff0: 696e 6920 7769 6c6c 2064 6f20 7468 6520  ini will do the 
│ │ │ │ +0001d000: 6d61 696e 0a63 6f6d 7075 7461 7469 6f6e  main.computation
│ │ │ │ +0001d010: 7320 6e75 6d65 7269 6361 6c6c 792c 2070  s numerically, p
│ │ │ │ +0001d020: 726f 7669 6465 6420 4265 7274 696e 6920  rovided Bertini 
│ │ │ │ +0001d030: 6973 202a 6e6f 7465 2069 6e73 7461 6c6c  is *note install
│ │ │ │ +0001d040: 6564 2061 6e64 2063 6f6e 6669 6775 7265  ed and configure
│ │ │ │ +0001d050: 643a 0a63 6f6e 6669 6775 7269 6e67 2042  d:.configuring B
│ │ │ │ +0001d060: 6572 7469 6e69 2c2e 204e 6f74 6520 7468  ertini,. Note th
│ │ │ │ +0001d070: 6174 2074 6865 2062 6572 7469 6e69 2061  at the bertini a
│ │ │ │ +0001d080: 6e64 2050 6e52 6573 6964 7561 6c20 6f70  nd PnResidual op
│ │ │ │ +0001d090: 7469 6f6e 7320 6d61 7920 6f6e 6c79 2062  tions may only b
│ │ │ │ +0001d0a0: 650a 7573 6564 2066 6f72 2073 7562 7363  e.used for subsc
│ │ │ │ +0001d0b0: 6865 6d65 7320 6f66 205c 5050 5e6e 2e0a  hemes of \PP^n..
│ │ │ │ +0001d0c0: 0a4f 6273 6572 7665 2074 6861 7420 7468  .Observe that th
│ │ │ │ +0001d0d0: 6520 616c 676f 7269 7468 6d20 6973 2061  e algorithm is a
│ │ │ │ +0001d0e0: 2070 726f 6261 6269 6c69 7374 6963 2061   probabilistic a
│ │ │ │ +0001d0f0: 6c67 6f72 6974 686d 2061 6e64 206d 6179  lgorithm and may
│ │ │ │ +0001d100: 2067 6976 6520 6120 7772 6f6e 670a 616e   give a wrong.an
│ │ │ │ +0001d110: 7377 6572 2077 6974 6820 6120 736d 616c  swer with a smal
│ │ │ │ +0001d120: 6c20 6275 7420 6e6f 6e7a 6572 6f20 7072  l but nonzero pr
│ │ │ │ +0001d130: 6f62 6162 696c 6974 792e 2052 6561 6420  obability. Read 
│ │ │ │ +0001d140: 6d6f 7265 2075 6e64 6572 202a 6e6f 7465  more under *note
│ │ │ │ +0001d150: 0a70 726f 6261 6269 6c69 7374 6963 2061  .probabilistic a
│ │ │ │ +0001d160: 6c67 6f72 6974 686d 3a20 7072 6f62 6162  lgorithm: probab
│ │ │ │ +0001d170: 696c 6973 7469 6320 616c 676f 7269 7468  ilistic algorith
│ │ │ │ +0001d180: 6d2c 2e0a 0a57 6179 7320 746f 2075 7365  m,...Ways to use
│ │ │ │ +0001d190: 2045 756c 6572 3a0a 3d3d 3d3d 3d3d 3d3d   Euler:.========
│ │ │ │ +0001d1a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +0001d1b0: 2245 756c 6572 2849 6465 616c 2922 0a20  "Euler(Ideal)". 
│ │ │ │ +0001d1c0: 202a 2022 4575 6c65 7228 5269 6e67 456c   * "Euler(RingEl
│ │ │ │ +0001d1d0: 656d 656e 7429 220a 0a46 6f72 2074 6865  ement)"..For the
│ │ │ │ +0001d1e0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +0001d1f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0001d200: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +0001d210: 2045 756c 6572 3a20 4575 6c65 722c 2069   Euler: Euler, i
│ │ │ │ +0001d220: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +0001d230: 2066 756e 6374 696f 6e20 7769 7468 206f   function with o
│ │ │ │ +0001d240: 7074 696f 6e73 3a0a 284d 6163 6175 6c61  ptions:.(Macaula
│ │ │ │ +0001d250: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ +0001d260: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c  tionWithOptions,
│ │ │ │ +0001d270: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0001d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d2f0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -0001d300: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -0001d310: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -0001d320: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -0001d330: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ -0001d340: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ -0001d350: 6179 322f 7061 636b 6167 6573 2f0a 4368  ay2/packages/.Ch
│ │ │ │ -0001d360: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ -0001d370: 7365 732e 6d32 3a32 3331 323a 302e 0a1f  ses.m2:2312:0...
│ │ │ │ -0001d380: 0a46 696c 653a 2043 6861 7261 6374 6572  .File: Character
│ │ │ │ -0001d390: 6973 7469 6343 6c61 7373 6573 2e69 6e66  isticClasses.inf
│ │ │ │ -0001d3a0: 6f2c 204e 6f64 653a 2045 756c 6572 4166  o, Node: EulerAf
│ │ │ │ -0001d3b0: 6669 6e65 2c20 4e65 7874 3a20 496e 6473  fine, Next: Inds
│ │ │ │ -0001d3c0: 4f66 536d 6f6f 7468 2c20 5072 6576 3a20  OfSmooth, Prev: 
│ │ │ │ -0001d3d0: 4575 6c65 722c 2055 703a 2054 6f70 0a0a  Euler, Up: Top..
│ │ │ │ -0001d3e0: 4575 6c65 7241 6666 696e 6520 2d2d 2054  EulerAffine -- T
│ │ │ │ -0001d3f0: 6865 2045 756c 6572 2043 6861 7261 6374  he Euler Charact
│ │ │ │ -0001d400: 6572 6973 7469 6320 6f66 2061 6e20 6166  eristic of an af
│ │ │ │ -0001d410: 6669 6e65 2076 6172 6965 7479 2e0a 2a2a  fine variety..**
│ │ │ │ -0001d420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -0001d460: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0001d470: 2045 756c 6572 4166 6669 6e65 2049 0a20   EulerAffine I. 
│ │ │ │ -0001d480: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ -0001d490: 202a 2049 2c20 616e 202a 6e6f 7465 2069   * I, an *note i
│ │ │ │ -0001d4a0: 6465 616c 3a20 284d 6163 6175 6c61 7932  deal: (Macaulay2
│ │ │ │ -0001d4b0: 446f 6329 4964 6561 6c2c 2c20 616e 2069  Doc)Ideal,, an i
│ │ │ │ -0001d4c0: 6465 616c 2069 6e20 6120 706f 6c79 6e6f  deal in a polyno
│ │ │ │ -0001d4d0: 6d69 616c 2072 696e 670a 2020 2020 2020  mial ring.      
│ │ │ │ -0001d4e0: 2020 6f76 6572 2061 2066 6965 6c64 2064    over a field d
│ │ │ │ -0001d4f0: 6566 696e 696e 6720 616e 2061 6666 696e  efining an affin
│ │ │ │ -0001d500: 6520 7661 7269 6574 792e 0a20 202a 204f  e variety..  * O
│ │ │ │ -0001d510: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -0001d520: 6120 2a6e 6f74 6520 7269 6e67 2065 6c65  a *note ring ele
│ │ │ │ -0001d530: 6d65 6e74 3a20 284d 6163 6175 6c61 7932  ment: (Macaulay2
│ │ │ │ -0001d540: 446f 6329 5269 6e67 456c 656d 656e 742c  Doc)RingElement,
│ │ │ │ -0001d550: 2c20 7468 6520 4575 6c65 720a 2020 2020  , the Euler.    
│ │ │ │ -0001d560: 2020 2020 6368 6172 6163 7465 7269 7374      characterist
│ │ │ │ -0001d570: 6963 0a0a 4465 7363 7269 7074 696f 6e0a  ic..Description.
│ │ │ │ -0001d580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ -0001d590: 7320 636f 6d6d 616e 6420 636f 6d70 7574  s command comput
│ │ │ │ -0001d5a0: 6573 2074 6865 2045 756c 6572 2063 6861  es the Euler cha
│ │ │ │ -0001d5b0: 7261 6374 6572 6973 7469 6320 6f66 2061  racteristic of a
│ │ │ │ -0001d5c0: 2063 6f6d 706c 6578 2061 6666 696e 6520   complex affine 
│ │ │ │ -0001d5d0: 7661 7269 6574 792e 0a0a 2b2d 2d2d 2d2d  variety...+-----
│ │ │ │ -0001d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d610: 2d2b 0a7c 6931 203a 206b 6b3d 5a5a 2f33  -+.|i1 : kk=ZZ/3
│ │ │ │ -0001d620: 3237 3439 3b20 2020 2020 2020 2020 2020  2749;           
│ │ │ │ -0001d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d640: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0001d650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d680: 2d2d 2d2b 0a7c 6932 203a 2052 3d6b 6b5b  ---+.|i2 : R=kk[
│ │ │ │ -0001d690: 785f 312e 2e78 5f33 5d20 2020 2020 2020  x_1..x_3]       
│ │ │ │ +0001d2c0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +0001d2d0: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +0001d2e0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +0001d2f0: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +0001d300: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ +0001d310: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +0001d320: 2f70 6163 6b61 6765 732f 0a43 6861 7261  /packages/.Chara
│ │ │ │ +0001d330: 6374 6572 6973 7469 6343 6c61 7373 6573  cteristicClasses
│ │ │ │ +0001d340: 2e6d 323a 3233 3132 3a30 2e0a 1f0a 4669  .m2:2312:0....Fi
│ │ │ │ +0001d350: 6c65 3a20 4368 6172 6163 7465 7269 7374  le: Characterist
│ │ │ │ +0001d360: 6963 436c 6173 7365 732e 696e 666f 2c20  icClasses.info, 
│ │ │ │ +0001d370: 4e6f 6465 3a20 4575 6c65 7241 6666 696e  Node: EulerAffin
│ │ │ │ +0001d380: 652c 204e 6578 743a 2049 6e64 734f 6653  e, Next: IndsOfS
│ │ │ │ +0001d390: 6d6f 6f74 682c 2050 7265 763a 2045 756c  mooth, Prev: Eul
│ │ │ │ +0001d3a0: 6572 2c20 5570 3a20 546f 700a 0a45 756c  er, Up: Top..Eul
│ │ │ │ +0001d3b0: 6572 4166 6669 6e65 202d 2d20 5468 6520  erAffine -- The 
│ │ │ │ +0001d3c0: 4575 6c65 7220 4368 6172 6163 7465 7269  Euler Characteri
│ │ │ │ +0001d3d0: 7374 6963 206f 6620 616e 2061 6666 696e  stic of an affin
│ │ │ │ +0001d3e0: 6520 7661 7269 6574 792e 0a2a 2a2a 2a2a  e variety..*****
│ │ │ │ +0001d3f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d420: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +0001d430: 6167 653a 200a 2020 2020 2020 2020 4575  age: .        Eu
│ │ │ │ +0001d440: 6c65 7241 6666 696e 6520 490a 2020 2a20  lerAffine I.  * 
│ │ │ │ +0001d450: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +0001d460: 492c 2061 6e20 2a6e 6f74 6520 6964 6561  I, an *note idea
│ │ │ │ +0001d470: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ +0001d480: 2949 6465 616c 2c2c 2061 6e20 6964 6561  )Ideal,, an idea
│ │ │ │ +0001d490: 6c20 696e 2061 2070 6f6c 796e 6f6d 6961  l in a polynomia
│ │ │ │ +0001d4a0: 6c20 7269 6e67 0a20 2020 2020 2020 206f  l ring.        o
│ │ │ │ +0001d4b0: 7665 7220 6120 6669 656c 6420 6465 6669  ver a field defi
│ │ │ │ +0001d4c0: 6e69 6e67 2061 6e20 6166 6669 6e65 2076  ning an affine v
│ │ │ │ +0001d4d0: 6172 6965 7479 2e0a 2020 2a20 4f75 7470  ariety..  * Outp
│ │ │ │ +0001d4e0: 7574 733a 0a20 2020 2020 202a 2061 202a  uts:.      * a *
│ │ │ │ +0001d4f0: 6e6f 7465 2072 696e 6720 656c 656d 656e  note ring elemen
│ │ │ │ +0001d500: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +0001d510: 2952 696e 6745 6c65 6d65 6e74 2c2c 2074  )RingElement,, t
│ │ │ │ +0001d520: 6865 2045 756c 6572 0a20 2020 2020 2020  he Euler.       
│ │ │ │ +0001d530: 2063 6861 7261 6374 6572 6973 7469 630a   characteristic.
│ │ │ │ +0001d540: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0001d550: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2063  ========..This c
│ │ │ │ +0001d560: 6f6d 6d61 6e64 2063 6f6d 7075 7465 7320  ommand computes 
│ │ │ │ +0001d570: 7468 6520 4575 6c65 7220 6368 6172 6163  the Euler charac
│ │ │ │ +0001d580: 7465 7269 7374 6963 206f 6620 6120 636f  teristic of a co
│ │ │ │ +0001d590: 6d70 6c65 7820 6166 6669 6e65 2076 6172  mplex affine var
│ │ │ │ +0001d5a0: 6965 7479 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  iety...+--------
│ │ │ │ +0001d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001d5e0: 0a7c 6931 203a 206b 6b3d 5a5a 2f33 3237  .|i1 : kk=ZZ/327
│ │ │ │ +0001d5f0: 3439 3b20 2020 2020 2020 2020 2020 2020  49;             
│ │ │ │ +0001d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d610: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d650: 2d2d 2d2b 0a7c 6932 203a 2052 3d6b 6b5b  ---+.|i2 : R=kk[
│ │ │ │ +0001d660: 785f 312e 2e78 5f33 5d20 2020 2020 2020  x_1..x_3]       
│ │ │ │ +0001d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d6c0: 2020 2020 2020 207c 0a7c 6f32 203d 2052         |.|o2 = R
│ │ │ │  0001d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6f0: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │ -0001d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d700: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001d730: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0001d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d760: 2020 2020 2020 207c 0a7c 6f32 203a 2050         |.|o2 : P
│ │ │ │ -0001d770: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ -0001d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d7a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0001d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d7d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0001d7e0: 2049 3d69 6465 616c 2878 5f31 5e32 2b78   I=ideal(x_1^2+x
│ │ │ │ -0001d7f0: 5f32 5e32 2b78 5f33 5e32 2d31 2920 2020  _2^2+x_3^2-1)   
│ │ │ │ +0001d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d730: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001d740: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +0001d750: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0001d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d770: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001d7b0: 0a7c 6933 203a 2049 3d69 6465 616c 2878  .|i3 : I=ideal(x
│ │ │ │ +0001d7c0: 5f31 5e32 2b78 5f32 5e32 2b78 5f33 5e32  _1^2+x_2^2+x_3^2
│ │ │ │ +0001d7d0: 2d31 2920 2020 2020 2020 2020 2020 2020  -1)             
│ │ │ │ +0001d7e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d810: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001d850: 2020 2020 2020 2020 2020 3220 2020 2032            2    2
│ │ │ │ -0001d860: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0001d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d880: 2020 2020 7c0a 7c6f 3320 3d20 6964 6561      |.|o3 = idea
│ │ │ │ -0001d890: 6c28 7820 202b 2078 2020 2b20 7820 202d  l(x  + x  + x  -
│ │ │ │ -0001d8a0: 2031 2920 2020 2020 2020 2020 2020 2020   1)             
│ │ │ │ -0001d8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001d8c0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -0001d8d0: 2032 2020 2020 3320 2020 2020 2020 2020   2    3         
│ │ │ │ +0001d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d820: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001d830: 2020 3220 2020 2032 2020 2020 3220 2020    2    2    2   
│ │ │ │ +0001d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001d860: 6f33 203d 2069 6465 616c 2878 2020 2b20  o3 = ideal(x  + 
│ │ │ │ +0001d870: 7820 202b 2078 2020 2d20 3129 2020 2020  x  + x  - 1)    
│ │ │ │ +0001d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d890: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001d8a0: 2020 2020 2020 3120 2020 2032 2020 2020        1    2    
│ │ │ │ +0001d8b0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0001d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d8f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001d930: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ -0001d940: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ -0001d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d960: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d9a0: 2d2b 0a7c 6934 203a 2074 696d 6520 4575  -+.|i4 : time Eu
│ │ │ │ -0001d9b0: 6c65 7241 6666 696e 6520 4920 2020 2020  lerAffine I     
│ │ │ │ -0001d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -0001d9e0: 2075 7365 6420 302e 3034 3731 3038 7320   used 0.047108s 
│ │ │ │ -0001d9f0: 2863 7075 293b 2030 2e30 3436 3733 3439  (cpu); 0.0467349
│ │ │ │ -0001da00: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -0001da10: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │ -0001da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001da50: 3420 3d20 3220 2020 2020 2020 2020 2020  4 = 2           
│ │ │ │ -0001da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d900: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +0001d910: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │ +0001d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d940: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001d980: 0a7c 6934 203a 2074 696d 6520 4575 6c65  .|i4 : time Eule
│ │ │ │ +0001d990: 7241 6666 696e 6520 4920 2020 2020 2020  rAffine I       
│ │ │ │ +0001d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d9b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +0001d9c0: 7573 6564 2030 2e30 3932 3434 3531 7320  used 0.0924451s 
│ │ │ │ +0001d9d0: 2863 7075 293b 2030 2e30 3734 3934 3832  (cpu); 0.0749482
│ │ │ │ +0001d9e0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +0001d9f0: 6763 297c 0a7c 2020 2020 2020 2020 2020  gc)|.|          
│ │ │ │ +0001da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001da30: 6f34 203d 2032 2020 2020 2020 2020 2020  o4 = 2          
│ │ │ │ +0001da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001da60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001dac0: 0a4f 6273 6572 7665 2074 6861 7420 7468  .Observe that th
│ │ │ │ -0001dad0: 6520 616c 676f 7269 7468 6d20 6973 2061  e algorithm is a
│ │ │ │ -0001dae0: 2070 726f 6261 6269 6c69 7374 6963 2061   probabilistic a
│ │ │ │ -0001daf0: 6c67 6f72 6974 686d 2061 6e64 206d 6179  lgorithm and may
│ │ │ │ -0001db00: 2067 6976 6520 6120 7772 6f6e 670a 616e   give a wrong.an
│ │ │ │ -0001db10: 7377 6572 2077 6974 6820 6120 736d 616c  swer with a smal
│ │ │ │ -0001db20: 6c20 6275 7420 6e6f 6e7a 6572 6f20 7072  l but nonzero pr
│ │ │ │ -0001db30: 6f62 6162 696c 6974 792e 2052 6561 6420  obability. Read 
│ │ │ │ -0001db40: 6d6f 7265 2075 6e64 6572 202a 6e6f 7465  more under *note
│ │ │ │ -0001db50: 0a70 726f 6261 6269 6c69 7374 6963 2061  .probabilistic a
│ │ │ │ -0001db60: 6c67 6f72 6974 686d 3a20 7072 6f62 6162  lgorithm: probab
│ │ │ │ -0001db70: 696c 6973 7469 6320 616c 676f 7269 7468  ilistic algorith
│ │ │ │ -0001db80: 6d2c 2e0a 0a57 6179 7320 746f 2075 7365  m,...Ways to use
│ │ │ │ -0001db90: 2045 756c 6572 4166 6669 6e65 3a0a 3d3d   EulerAffine:.==
│ │ │ │ -0001dba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001dbb0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 756c  ======..  * "Eul
│ │ │ │ -0001dbc0: 6572 4166 6669 6e65 2849 6465 616c 2922  erAffine(Ideal)"
│ │ │ │ -0001dbd0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0001dbe0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0001dbf0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0001dc00: 6563 7420 2a6e 6f74 6520 4575 6c65 7241  ect *note EulerA
│ │ │ │ -0001dc10: 6666 696e 653a 2045 756c 6572 4166 6669  ffine: EulerAffi
│ │ │ │ -0001dc20: 6e65 2c20 6973 2061 202a 6e6f 7465 206d  ne, is a *note m
│ │ │ │ -0001dc30: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a  ethod function:.
│ │ │ │ -0001dc40: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -0001dc50: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a  thodFunction,...
│ │ │ │ +0001daa0: 2d2b 0a0a 4f62 7365 7276 6520 7468 6174  -+..Observe that
│ │ │ │ +0001dab0: 2074 6865 2061 6c67 6f72 6974 686d 2069   the algorithm i
│ │ │ │ +0001dac0: 7320 6120 7072 6f62 6162 696c 6973 7469  s a probabilisti
│ │ │ │ +0001dad0: 6320 616c 676f 7269 7468 6d20 616e 6420  c algorithm and 
│ │ │ │ +0001dae0: 6d61 7920 6769 7665 2061 2077 726f 6e67  may give a wrong
│ │ │ │ +0001daf0: 0a61 6e73 7765 7220 7769 7468 2061 2073  .answer with a s
│ │ │ │ +0001db00: 6d61 6c6c 2062 7574 206e 6f6e 7a65 726f  mall but nonzero
│ │ │ │ +0001db10: 2070 726f 6261 6269 6c69 7479 2e20 5265   probability. Re
│ │ │ │ +0001db20: 6164 206d 6f72 6520 756e 6465 7220 2a6e  ad more under *n
│ │ │ │ +0001db30: 6f74 650a 7072 6f62 6162 696c 6973 7469  ote.probabilisti
│ │ │ │ +0001db40: 6320 616c 676f 7269 7468 6d3a 2070 726f  c algorithm: pro
│ │ │ │ +0001db50: 6261 6269 6c69 7374 6963 2061 6c67 6f72  babilistic algor
│ │ │ │ +0001db60: 6974 686d 2c2e 0a0a 5761 7973 2074 6f20  ithm,...Ways to 
│ │ │ │ +0001db70: 7573 6520 4575 6c65 7241 6666 696e 653a  use EulerAffine:
│ │ │ │ +0001db80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0001db90: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +0001dba0: 4575 6c65 7241 6666 696e 6528 4964 6561  EulerAffine(Idea
│ │ │ │ +0001dbb0: 6c29 220a 0a46 6f72 2074 6865 2070 726f  l)"..For the pro
│ │ │ │ +0001dbc0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +0001dbd0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +0001dbe0: 6f62 6a65 6374 202a 6e6f 7465 2045 756c  object *note Eul
│ │ │ │ +0001dbf0: 6572 4166 6669 6e65 3a20 4575 6c65 7241  erAffine: EulerA
│ │ │ │ +0001dc00: 6666 696e 652c 2069 7320 6120 2a6e 6f74  ffine, is a *not
│ │ │ │ +0001dc10: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ +0001dc20: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ +0001dc30: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +0001dc40: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0001dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -0001dcb0: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -0001dcc0: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ -0001dcd0: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ -0001dce0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -0001dcf0: 756c 6179 322d 312e 3236 2e30 362b 6473  ulay2-1.26.06+ds
│ │ │ │ -0001dd00: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -0001dd10: 636b 6167 6573 2f0a 4368 6172 6163 7465  ckages/.Characte
│ │ │ │ -0001dd20: 7269 7374 6963 436c 6173 7365 732e 6d32  risticClasses.m2
│ │ │ │ -0001dd30: 3a32 3534 313a 302e 0a1f 0a46 696c 653a  :2541:0....File:
│ │ │ │ -0001dd40: 2043 6861 7261 6374 6572 6973 7469 6343   CharacteristicC
│ │ │ │ -0001dd50: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64  lasses.info, Nod
│ │ │ │ -0001dd60: 653a 2049 6e64 734f 6653 6d6f 6f74 682c  e: IndsOfSmooth,
│ │ │ │ -0001dd70: 204e 6578 743a 2049 6e70 7574 4973 536d   Next: InputIsSm
│ │ │ │ -0001dd80: 6f6f 7468 2c20 5072 6576 3a20 4575 6c65  ooth, Prev: Eule
│ │ │ │ -0001dd90: 7241 6666 696e 652c 2055 703a 2054 6f70  rAffine, Up: Top
│ │ │ │ -0001dda0: 0a0a 496e 6473 4f66 536d 6f6f 7468 0a2a  ..IndsOfSmooth.*
│ │ │ │ -0001ddb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573  ***********..Des
│ │ │ │ -0001ddc0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0001ddd0: 3d3d 3d3d 0a0a 5468 6520 6f70 7469 6f6e  ====..The option
│ │ │ │ -0001dde0: 2049 6e64 734f 6653 6d6f 6f74 6820 6973   IndsOfSmooth is
│ │ │ │ -0001ddf0: 206f 6e6c 7920 7573 6564 2062 7920 7468   only used by th
│ │ │ │ -0001de00: 6520 636f 6d6d 616e 6473 202a 6e6f 7465  e commands *note
│ │ │ │ -0001de10: 2043 534d 3a20 4353 4d2c 2c20 616e 6420   CSM: CSM,, and 
│ │ │ │ -0001de20: 2a6e 6f74 650a 4575 6c65 723a 2045 756c  *note.Euler: Eul
│ │ │ │ -0001de30: 6572 2c20 696e 2063 6f6d 6269 6e61 7469  er, in combinati
│ │ │ │ -0001de40: 6f6e 2077 6974 6820 7468 6520 6f70 7469  on with the opti
│ │ │ │ -0001de50: 6f6e 204d 6574 686f 643d 3e44 6972 6563  on Method=>Direc
│ │ │ │ -0001de60: 7443 6f6d 706c 6574 496e 742e 2057 6865  tCompletInt. Whe
│ │ │ │ -0001de70: 6e0a 7573 6564 2074 6869 7320 6f70 7469  n.used this opti
│ │ │ │ -0001de80: 6f6e 206d 6179 2061 6c6c 6f77 2074 6865  on may allow the
│ │ │ │ -0001de90: 2075 7365 7220 746f 2073 7065 6564 2075   user to speed u
│ │ │ │ -0001dea0: 7020 7468 6520 636f 6d70 7574 6174 696f  p the computatio
│ │ │ │ -0001deb0: 6e20 6279 2074 656c 6c69 6e67 0a67 6976  n by telling.giv
│ │ │ │ -0001dec0: 696e 6720 7468 6520 6d65 7468 6f64 2061  ing the method a
│ │ │ │ -0001ded0: 206c 6973 7420 6f66 2069 6e64 6963 6573   list of indices
│ │ │ │ -0001dee0: 2066 6f72 2074 6865 2067 656e 6572 6174   for the generat
│ │ │ │ -0001def0: 6f72 7320 6f66 2074 6865 2069 6e70 7574  ors of the input
│ │ │ │ -0001df00: 2069 6465 616c 2074 6861 742c 0a77 6865   ideal that,.whe
│ │ │ │ -0001df10: 6e20 7461 6b65 6e20 746f 6765 7468 6572  n taken together
│ │ │ │ -0001df20: 2c20 6465 6669 6e65 2061 2073 6d6f 6f74  , define a smoot
│ │ │ │ -0001df30: 6820 7375 6273 6368 656d 6520 6f66 2074  h subscheme of t
│ │ │ │ -0001df40: 6865 2061 6d62 6965 6e74 2073 7061 6365  he ambient space
│ │ │ │ -0001df50: 2e20 5468 6973 0a6f 7074 696f 6e20 7769  . This.option wi
│ │ │ │ -0001df60: 6c6c 2062 6520 6967 6e6f 7265 6420 6f74  ll be ignored ot
│ │ │ │ -0001df70: 6865 7277 6973 652e 0a0a 2b2d 2d2d 2d2d  herwise...+-----
│ │ │ │ +0001dc90: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +0001dca0: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +0001dcb0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +0001dcc0: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +0001dcd0: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ +0001dce0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +0001dcf0: 2f70 6163 6b61 6765 732f 0a43 6861 7261  /packages/.Chara
│ │ │ │ +0001dd00: 6374 6572 6973 7469 6343 6c61 7373 6573  cteristicClasses
│ │ │ │ +0001dd10: 2e6d 323a 3235 3431 3a30 2e0a 1f0a 4669  .m2:2541:0....Fi
│ │ │ │ +0001dd20: 6c65 3a20 4368 6172 6163 7465 7269 7374  le: Characterist
│ │ │ │ +0001dd30: 6963 436c 6173 7365 732e 696e 666f 2c20  icClasses.info, 
│ │ │ │ +0001dd40: 4e6f 6465 3a20 496e 6473 4f66 536d 6f6f  Node: IndsOfSmoo
│ │ │ │ +0001dd50: 7468 2c20 4e65 7874 3a20 496e 7075 7449  th, Next: InputI
│ │ │ │ +0001dd60: 7353 6d6f 6f74 682c 2050 7265 763a 2045  sSmooth, Prev: E
│ │ │ │ +0001dd70: 756c 6572 4166 6669 6e65 2c20 5570 3a20  ulerAffine, Up: 
│ │ │ │ +0001dd80: 546f 700a 0a49 6e64 734f 6653 6d6f 6f74  Top..IndsOfSmoot
│ │ │ │ +0001dd90: 680a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  h.************..
│ │ │ │ +0001dda0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ +0001ddb0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 7074  =======..The opt
│ │ │ │ +0001ddc0: 696f 6e20 496e 6473 4f66 536d 6f6f 7468  ion IndsOfSmooth
│ │ │ │ +0001ddd0: 2069 7320 6f6e 6c79 2075 7365 6420 6279   is only used by
│ │ │ │ +0001dde0: 2074 6865 2063 6f6d 6d61 6e64 7320 2a6e   the commands *n
│ │ │ │ +0001ddf0: 6f74 6520 4353 4d3a 2043 534d 2c2c 2061  ote CSM: CSM,, a
│ │ │ │ +0001de00: 6e64 202a 6e6f 7465 0a45 756c 6572 3a20  nd *note.Euler: 
│ │ │ │ +0001de10: 4575 6c65 722c 2069 6e20 636f 6d62 696e  Euler, in combin
│ │ │ │ +0001de20: 6174 696f 6e20 7769 7468 2074 6865 206f  ation with the o
│ │ │ │ +0001de30: 7074 696f 6e20 4d65 7468 6f64 3d3e 4469  ption Method=>Di
│ │ │ │ +0001de40: 7265 6374 436f 6d70 6c65 7449 6e74 2e20  rectCompletInt. 
│ │ │ │ +0001de50: 5768 656e 0a75 7365 6420 7468 6973 206f  When.used this o
│ │ │ │ +0001de60: 7074 696f 6e20 6d61 7920 616c 6c6f 7720  ption may allow 
│ │ │ │ +0001de70: 7468 6520 7573 6572 2074 6f20 7370 6565  the user to spee
│ │ │ │ +0001de80: 6420 7570 2074 6865 2063 6f6d 7075 7461  d up the computa
│ │ │ │ +0001de90: 7469 6f6e 2062 7920 7465 6c6c 696e 670a  tion by telling.
│ │ │ │ +0001dea0: 6769 7669 6e67 2074 6865 206d 6574 686f  giving the metho
│ │ │ │ +0001deb0: 6420 6120 6c69 7374 206f 6620 696e 6469  d a list of indi
│ │ │ │ +0001dec0: 6365 7320 666f 7220 7468 6520 6765 6e65  ces for the gene
│ │ │ │ +0001ded0: 7261 746f 7273 206f 6620 7468 6520 696e  rators of the in
│ │ │ │ +0001dee0: 7075 7420 6964 6561 6c20 7468 6174 2c0a  put ideal that,.
│ │ │ │ +0001def0: 7768 656e 2074 616b 656e 2074 6f67 6574  when taken toget
│ │ │ │ +0001df00: 6865 722c 2064 6566 696e 6520 6120 736d  her, define a sm
│ │ │ │ +0001df10: 6f6f 7468 2073 7562 7363 6865 6d65 206f  ooth subscheme o
│ │ │ │ +0001df20: 6620 7468 6520 616d 6269 656e 7420 7370  f the ambient sp
│ │ │ │ +0001df30: 6163 652e 2054 6869 730a 6f70 7469 6f6e  ace. This.option
│ │ │ │ +0001df40: 2077 696c 6c20 6265 2069 676e 6f72 6564   will be ignored
│ │ │ │ +0001df50: 206f 7468 6572 7769 7365 2e0a 0a2b 2d2d   otherwise...+--
│ │ │ │ +0001df60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001dfc0: 0a7c 6931 203a 2052 203d 204d 756c 7469  .|i1 : R = Multi
│ │ │ │ -0001dfd0: 5072 6f6a 436f 6f72 6452 696e 6728 7b32  ProjCoordRing({2
│ │ │ │ -0001dfe0: 2c32 7d29 2020 2020 2020 2020 2020 2020  ,2})            
│ │ │ │ +0001dfa0: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 4d75  --+.|i1 : R = Mu
│ │ │ │ +0001dfb0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ +0001dfc0: 287b 322c 327d 2920 2020 2020 2020 2020  ({2,2})         
│ │ │ │ +0001dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001dfe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e000: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e040: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001e050: 6f31 203d 2052 2020 2020 2020 2020 2020  o1 = R          
│ │ │ │ +0001e030: 7c0a 7c6f 3120 3d20 5220 2020 2020 2020  |.|o1 = R       
│ │ │ │ +0001e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e070: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e090: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e0d0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0001e0e0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -0001e0f0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -0001e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e120: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001e0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e0c0: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ +0001e0d0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +0001e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e100: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e160: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -0001e170: 2049 3d69 6465 616c 2852 5f30 2a52 5f31   I=ideal(R_0*R_1
│ │ │ │ -0001e180: 2a52 5f33 2d52 5f30 5e32 2a52 5f33 2c72  *R_3-R_0^2*R_3,r
│ │ │ │ -0001e190: 616e 646f 6d28 7b30 2c31 7d2c 5229 2c72  andom({0,1},R),r
│ │ │ │ -0001e1a0: 616e 646f 6d28 7b31 2c32 7d2c 5229 293b  andom({1,2},R));
│ │ │ │ -0001e1b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001e150: 3220 3a20 493d 6964 6561 6c28 525f 302a  2 : I=ideal(R_0*
│ │ │ │ +0001e160: 525f 312a 525f 332d 525f 305e 322a 525f  R_1*R_3-R_0^2*R_
│ │ │ │ +0001e170: 332c 7261 6e64 6f6d 287b 302c 317d 2c52  3,random({0,1},R
│ │ │ │ +0001e180: 292c 7261 6e64 6f6d 287b 312c 327d 2c52  ),random({1,2},R
│ │ │ │ +0001e190: 2929 3b7c 0a7c 2020 2020 2020 2020 2020  ));|.|          
│ │ │ │ +0001e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1f0: 2020 2020 2020 207c 0a7c 6f32 203a 2049         |.|o2 : I
│ │ │ │ -0001e200: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ +0001e1d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0001e1e0: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │ +0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e240: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001e220: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e280: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d  -----+.|i3 : tim
│ │ │ │ -0001e290: 6520 4353 4d28 492c 4d65 7468 6f64 3d3e  e CSM(I,Method=>
│ │ │ │ -0001e2a0: 4469 7265 6374 436f 6d70 6c65 7449 6e74  DirectCompletInt
│ │ │ │ -0001e2b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001e2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001e2d0: 2d2d 2075 7365 6420 312e 3732 3139 3673  -- used 1.72196s
│ │ │ │ -0001e2e0: 2028 6370 7529 3b20 312e 3136 3739 3273   (cpu); 1.16792s
│ │ │ │ -0001e2f0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -0001e300: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │ -0001e310: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001e260: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0001e270: 7469 6d65 2043 534d 2849 2c4d 6574 686f  time CSM(I,Metho
│ │ │ │ +0001e280: 643d 3e44 6972 6563 7443 6f6d 706c 6574  d=>DirectComplet
│ │ │ │ +0001e290: 496e 7429 2020 2020 2020 2020 2020 2020  Int)            
│ │ │ │ +0001e2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e2b0: 0a7c 202d 2d20 7573 6564 2035 2e32 3938  .| -- used 5.298
│ │ │ │ +0001e2c0: 3939 7320 2863 7075 293b 2031 2e34 3939  99s (cpu); 1.499
│ │ │ │ +0001e2d0: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +0001e2e0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +0001e2f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e350: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001e360: 2020 2020 3220 3220 2020 2020 3220 2020      2 2     2   
│ │ │ │ -0001e370: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0001e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3a0: 207c 0a7c 6f33 203d 2032 6820 6820 202b   |.|o3 = 2h h  +
│ │ │ │ -0001e3b0: 2032 6820 6820 202b 2035 6820 6820 2020   2h h  + 5h h   
│ │ │ │ -0001e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001e3f0: 2020 3120 3220 2020 2020 3120 3220 2020    1 2     1 2   
│ │ │ │ -0001e400: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ -0001e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e430: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001e330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e340: 2020 2020 2020 2032 2032 2020 2020 2032         2 2     2
│ │ │ │ +0001e350: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0001e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e380: 2020 2020 7c0a 7c6f 3320 3d20 3268 2068      |.|o3 = 2h h
│ │ │ │ +0001e390: 2020 2b20 3268 2068 2020 2b20 3568 2068    + 2h h  + 5h h
│ │ │ │ +0001e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e3c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001e3d0: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ +0001e3e0: 2020 2020 2031 2032 2020 2020 2020 2020       1 2        
│ │ │ │ +0001e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e410: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e470: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a        |.|     ZZ
│ │ │ │ -0001e480: 5b68 202e 2e68 205d 2020 2020 2020 2020  [h ..h ]        
│ │ │ │ +0001e450: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001e460: 205a 5a5b 6820 2e2e 6820 5d20 2020 2020   ZZ[h ..h ]     
│ │ │ │ +0001e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001e4c0: 2020 2020 2020 2020 2031 2020 2032 2020           1   2  
│ │ │ │ +0001e4a0: 7c0a 7c20 2020 2020 2020 2020 3120 2020  |.|         1   
│ │ │ │ +0001e4b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e500: 2020 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d      |.|o3 : ----
│ │ │ │ -0001e510: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ -0001e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e540: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001e550: 2020 2020 2020 3320 2020 3320 2020 2020        3   3     
│ │ │ │ +0001e4e0: 2020 2020 2020 207c 0a7c 6f33 203a 202d         |.|o3 : -
│ │ │ │ +0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020 2020  ---------       
│ │ │ │ +0001e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e530: 7c20 2020 2020 2020 2033 2020 2033 2020  |        3   3  
│ │ │ │ +0001e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e590: 2020 7c0a 7c20 2020 2020 2028 6820 2c20    |.|      (h , 
│ │ │ │ -0001e5a0: 6820 2920 2020 2020 2020 2020 2020 2020  h )             
│ │ │ │ -0001e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001e5e0: 2020 2020 3120 2020 3220 2020 2020 2020      1   2       
│ │ │ │ +0001e570: 2020 2020 207c 0a7c 2020 2020 2020 2868       |.|      (h
│ │ │ │ +0001e580: 202c 2068 2029 2020 2020 2020 2020 2020   , h )          
│ │ │ │ +0001e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001e5c0: 2020 2020 2020 2031 2020 2032 2020 2020         1   2    
│ │ │ │ +0001e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e620: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001e600: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e660: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074  -------+.|i4 : t
│ │ │ │ -0001e670: 696d 6520 4353 4d28 492c 4d65 7468 6f64  ime CSM(I,Method
│ │ │ │ -0001e680: 3d3e 4469 7265 6374 436f 6d70 6c65 7449  =>DirectCompletI
│ │ │ │ -0001e690: 6e74 2c49 6e64 734f 6653 6d6f 6f74 683d  nt,IndsOfSmooth=
│ │ │ │ -0001e6a0: 3e7b 312c 327d 2920 2020 2020 2020 7c0a  >{1,2})       |.
│ │ │ │ -0001e6b0: 7c20 2d2d 2075 7365 6420 312e 3430 3632  | -- used 1.4062
│ │ │ │ -0001e6c0: 3273 2028 6370 7529 3b20 312e 3035 3636  2s (cpu); 1.0566
│ │ │ │ -0001e6d0: 3773 2028 7468 7265 6164 293b 2030 7320  7s (thread); 0s 
│ │ │ │ -0001e6e0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -0001e6f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +0001e650: 3a20 7469 6d65 2043 534d 2849 2c4d 6574  : time CSM(I,Met
│ │ │ │ +0001e660: 686f 643d 3e44 6972 6563 7443 6f6d 706c  hod=>DirectCompl
│ │ │ │ +0001e670: 6574 496e 742c 496e 6473 4f66 536d 6f6f  etInt,IndsOfSmoo
│ │ │ │ +0001e680: 7468 3d3e 7b31 2c32 7d29 2020 2020 2020  th=>{1,2})      
│ │ │ │ +0001e690: 207c 0a7c 202d 2d20 7573 6564 2035 2e30   |.| -- used 5.0
│ │ │ │ +0001e6a0: 3538 3533 7320 2863 7075 293b 2031 2e34  5853s (cpu); 1.4
│ │ │ │ +0001e6b0: 3631 3238 7320 2874 6872 6561 6429 3b20  6128s (thread); 
│ │ │ │ +0001e6c0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │ +0001e6d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001e740: 2020 2020 2020 3220 3220 2020 2020 3220        2 2     2 
│ │ │ │ -0001e750: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -0001e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e780: 2020 207c 0a7c 6f34 203d 2032 6820 6820     |.|o4 = 2h h 
│ │ │ │ -0001e790: 202b 2032 6820 6820 202b 2035 6820 6820   + 2h h  + 5h h 
│ │ │ │ -0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001e7d0: 2020 2020 3120 3220 2020 2020 3120 3220      1 2     1 2 
│ │ │ │ -0001e7e0: 2020 2020 3120 3220 2020 2020 2020 2020      1 2         
│ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e720: 0a7c 2020 2020 2020 2032 2032 2020 2020  .|       2 2    
│ │ │ │ +0001e730: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │ +0001e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e760: 2020 2020 2020 7c0a 7c6f 3420 3d20 3268        |.|o4 = 2h
│ │ │ │ +0001e770: 2068 2020 2b20 3268 2068 2020 2b20 3568   h  + 2h h  + 5h
│ │ │ │ +0001e780: 2068 2020 2020 2020 2020 2020 2020 2020   h              
│ │ │ │ +0001e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e7b0: 2020 2020 2020 2031 2032 2020 2020 2031         1 2     1
│ │ │ │ +0001e7c0: 2032 2020 2020 2031 2032 2020 2020 2020   2     1 2      
│ │ │ │ +0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0001e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e810: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e850: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001e860: 5a5a 5b68 202e 2e68 205d 2020 2020 2020  ZZ[h ..h ]      
│ │ │ │ +0001e830: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001e840: 2020 205a 5a5b 6820 2e2e 6820 5d20 2020     ZZ[h ..h ]   
│ │ │ │ +0001e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001e8a0: 0a7c 2020 2020 2020 2020 2031 2020 2032  .|         1   2
│ │ │ │ +0001e880: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ +0001e890: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8e0: 2020 2020 2020 7c0a 7c6f 3420 3a20 2d2d        |.|o4 : --
│ │ │ │ -0001e8f0: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │ +0001e8c0: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +0001e8d0: 202d 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020   ----------     
│ │ │ │ +0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e920: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001e930: 2020 2020 2020 2020 3320 2020 3320 2020          3   3   
│ │ │ │ +0001e910: 7c0a 7c20 2020 2020 2020 2033 2020 2033  |.|        3   3
│ │ │ │ +0001e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e970: 2020 2020 7c0a 7c20 2020 2020 2028 6820      |.|      (h 
│ │ │ │ -0001e980: 2c20 6820 2920 2020 2020 2020 2020 2020  , h )           
│ │ │ │ -0001e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001e9c0: 2020 2020 2020 3120 2020 3220 2020 2020        1   2     
│ │ │ │ +0001e950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001e960: 2868 202c 2068 2029 2020 2020 2020 2020  (h , h )        
│ │ │ │ +0001e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001e9a0: 7c20 2020 2020 2020 2031 2020 2032 2020  |        1   2  
│ │ │ │ +0001e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001e9e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675 6e63  ---------+..Func
│ │ │ │ -0001ea50: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ -0001ea60: 6e61 6c20 6172 6775 6d65 6e74 206e 616d  nal argument nam
│ │ │ │ -0001ea70: 6564 2049 6e64 734f 6653 6d6f 6f74 683a  ed IndsOfSmooth:
│ │ │ │ -0001ea80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0001ea90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001eaa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001eab0: 3d3d 3d3d 3d0a 0a20 202a 2022 4353 4d28  =====..  * "CSM(
│ │ │ │ -0001eac0: 2e2e 2e2c 496e 6473 4f66 536d 6f6f 7468  ...,IndsOfSmooth
│ │ │ │ -0001ead0: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ -0001eae0: 6e6f 7465 2043 534d 3a20 4353 4d2c 202d  note CSM: CSM, -
│ │ │ │ -0001eaf0: 2d20 5468 650a 2020 2020 4368 6572 6e2d  - The.    Chern-
│ │ │ │ -0001eb00: 5363 6877 6172 747a 2d4d 6163 5068 6572  Schwartz-MacPher
│ │ │ │ -0001eb10: 736f 6e20 636c 6173 730a 2020 2a20 4575  son class.  * Eu
│ │ │ │ -0001eb20: 6c65 7228 2e2e 2e2c 496e 6473 4f66 536d  ler(...,IndsOfSm
│ │ │ │ -0001eb30: 6f6f 7468 3d3e 2e2e 2e29 2028 6d69 7373  ooth=>...) (miss
│ │ │ │ -0001eb40: 696e 6720 646f 6375 6d65 6e74 6174 696f  ing documentatio
│ │ │ │ -0001eb50: 6e29 0a0a 466f 7220 7468 6520 7072 6f67  n)..For the prog
│ │ │ │ -0001eb60: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -0001eb70: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -0001eb80: 626a 6563 7420 2a6e 6f74 6520 496e 6473  bject *note Inds
│ │ │ │ -0001eb90: 4f66 536d 6f6f 7468 3a20 496e 6473 4f66  OfSmooth: IndsOf
│ │ │ │ -0001eba0: 536d 6f6f 7468 2c20 6973 2061 202a 6e6f  Smooth, is a *no
│ │ │ │ -0001ebb0: 7465 2073 796d 626f 6c3a 0a28 4d61 6361  te symbol:.(Maca
│ │ │ │ -0001ebc0: 756c 6179 3244 6f63 2953 796d 626f 6c2c  ulay2Doc)Symbol,
│ │ │ │ -0001ebd0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46  ------------+..F
│ │ │ │ +0001ea30: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ +0001ea40: 7469 6f6e 616c 2061 7267 756d 656e 7420  tional argument 
│ │ │ │ +0001ea50: 6e61 6d65 6420 496e 6473 4f66 536d 6f6f  named IndsOfSmoo
│ │ │ │ +0001ea60: 7468 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  th:.============
│ │ │ │ +0001ea70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001ea80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001ea90: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2243  ========..  * "C
│ │ │ │ +0001eaa0: 534d 282e 2e2e 2c49 6e64 734f 6653 6d6f  SM(...,IndsOfSmo
│ │ │ │ +0001eab0: 6f74 683d 3e2e 2e2e 2922 202d 2d20 7365  oth=>...)" -- se
│ │ │ │ +0001eac0: 6520 2a6e 6f74 6520 4353 4d3a 2043 534d  e *note CSM: CSM
│ │ │ │ +0001ead0: 2c20 2d2d 2054 6865 0a20 2020 2043 6865  , -- The.    Che
│ │ │ │ +0001eae0: 726e 2d53 6368 7761 7274 7a2d 4d61 6350  rn-Schwartz-MacP
│ │ │ │ +0001eaf0: 6865 7273 6f6e 2063 6c61 7373 0a20 202a  herson class.  *
│ │ │ │ +0001eb00: 2045 756c 6572 282e 2e2e 2c49 6e64 734f   Euler(...,IndsO
│ │ │ │ +0001eb10: 6653 6d6f 6f74 683d 3e2e 2e2e 2920 286d  fSmooth=>...) (m
│ │ │ │ +0001eb20: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ +0001eb30: 7469 6f6e 290a 0a46 6f72 2074 6865 2070  tion)..For the p
│ │ │ │ +0001eb40: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +0001eb50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +0001eb60: 6520 6f62 6a65 6374 202a 6e6f 7465 2049  e object *note I
│ │ │ │ +0001eb70: 6e64 734f 6653 6d6f 6f74 683a 2049 6e64  ndsOfSmooth: Ind
│ │ │ │ +0001eb80: 734f 6653 6d6f 6f74 682c 2069 7320 6120  sOfSmooth, is a 
│ │ │ │ +0001eb90: 2a6e 6f74 6520 7379 6d62 6f6c 3a0a 284d  *note symbol:.(M
│ │ │ │ +0001eba0: 6163 6175 6c61 7932 446f 6329 5379 6d62  acaulay2Doc)Symb
│ │ │ │ +0001ebb0: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  ol,...----------
│ │ │ │ +0001ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ebd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ebe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec20: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -0001ec30: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -0001ec40: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -0001ec50: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -0001ec60: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ -0001ec70: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -0001ec80: 2f70 6163 6b61 6765 732f 0a43 6861 7261  /packages/.Chara
│ │ │ │ -0001ec90: 6374 6572 6973 7469 6343 6c61 7373 6573  cteristicClasses
│ │ │ │ -0001eca0: 2e6d 323a 3234 3832 3a30 2e0a 1f0a 4669  .m2:2482:0....Fi
│ │ │ │ -0001ecb0: 6c65 3a20 4368 6172 6163 7465 7269 7374  le: Characterist
│ │ │ │ -0001ecc0: 6963 436c 6173 7365 732e 696e 666f 2c20  icClasses.info, 
│ │ │ │ -0001ecd0: 4e6f 6465 3a20 496e 7075 7449 7353 6d6f  Node: InputIsSmo
│ │ │ │ -0001ece0: 6f74 682c 204e 6578 743a 2069 734d 756c  oth, Next: isMul
│ │ │ │ -0001ecf0: 7469 486f 6d6f 6765 6e65 6f75 732c 2050  tiHomogeneous, P
│ │ │ │ -0001ed00: 7265 763a 2049 6e64 734f 6653 6d6f 6f74  rev: IndsOfSmoot
│ │ │ │ -0001ed10: 682c 2055 703a 2054 6f70 0a0a 496e 7075  h, Up: Top..Inpu
│ │ │ │ -0001ed20: 7449 7353 6d6f 6f74 680a 2a2a 2a2a 2a2a  tIsSmooth.******
│ │ │ │ -0001ed30: 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 6970  *******..Descrip
│ │ │ │ -0001ed40: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -0001ed50: 0a0a 5468 6520 6f70 7469 6f6e 2049 6e70  ..The option Inp
│ │ │ │ -0001ed60: 7574 4973 536d 6f6f 7468 2069 7320 6f6e  utIsSmooth is on
│ │ │ │ -0001ed70: 6c79 2075 7365 6420 6279 2074 6865 2063  ly used by the c
│ │ │ │ -0001ed80: 6f6d 6d61 6e64 7320 2a6e 6f74 6520 4353  ommands *note CS
│ │ │ │ -0001ed90: 4d3a 2043 534d 2c2c 2061 6e64 0a2a 6e6f  M: CSM,, and.*no
│ │ │ │ -0001eda0: 7465 2045 756c 6572 3a20 4575 6c65 722c  te Euler: Euler,
│ │ │ │ -0001edb0: 2e20 4966 2074 6865 2069 6e70 7574 2069  . If the input i
│ │ │ │ -0001edc0: 6465 616c 2069 7320 6b6e 6f77 6e20 746f  deal is known to
│ │ │ │ -0001edd0: 2064 6566 696e 6520 6120 736d 6f6f 7468   define a smooth
│ │ │ │ -0001ede0: 2073 7562 7363 6865 6d65 0a73 6574 7469   subscheme.setti
│ │ │ │ -0001edf0: 6e67 2074 6869 7320 6f70 7469 6f6e 2074  ng this option t
│ │ │ │ -0001ee00: 6f20 7472 7565 2077 696c 6c20 7370 6565  o true will spee
│ │ │ │ -0001ee10: 6420 7570 2063 6f6d 7075 7461 7469 6f6e  d up computation
│ │ │ │ -0001ee20: 7320 2869 7420 6973 2073 6574 2074 6f20  s (it is set to 
│ │ │ │ -0001ee30: 6661 6c73 6520 6279 0a64 6566 6175 6c74  false by.default
│ │ │ │ -0001ee40: 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  )...+-----------
│ │ │ │ -0001ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ee60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0001ee80: 203a 2052 203d 205a 5a2f 3332 3734 395b   : R = ZZ/32749[
│ │ │ │ -0001ee90: 785f 302e 2e78 5f34 5d3b 2020 2020 2020  x_0..x_4];      
│ │ │ │ -0001eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eeb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001ec00: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +0001ec10: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +0001ec20: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +0001ec30: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +0001ec40: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ +0001ec50: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ +0001ec60: 6179 322f 7061 636b 6167 6573 2f0a 4368  ay2/packages/.Ch
│ │ │ │ +0001ec70: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ +0001ec80: 7365 732e 6d32 3a32 3438 323a 302e 0a1f  ses.m2:2482:0...
│ │ │ │ +0001ec90: 0a46 696c 653a 2043 6861 7261 6374 6572  .File: Character
│ │ │ │ +0001eca0: 6973 7469 6343 6c61 7373 6573 2e69 6e66  isticClasses.inf
│ │ │ │ +0001ecb0: 6f2c 204e 6f64 653a 2049 6e70 7574 4973  o, Node: InputIs
│ │ │ │ +0001ecc0: 536d 6f6f 7468 2c20 4e65 7874 3a20 6973  Smooth, Next: is
│ │ │ │ +0001ecd0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573  MultiHomogeneous
│ │ │ │ +0001ece0: 2c20 5072 6576 3a20 496e 6473 4f66 536d  , Prev: IndsOfSm
│ │ │ │ +0001ecf0: 6f6f 7468 2c20 5570 3a20 546f 700a 0a49  ooth, Up: Top..I
│ │ │ │ +0001ed00: 6e70 7574 4973 536d 6f6f 7468 0a2a 2a2a  nputIsSmooth.***
│ │ │ │ +0001ed10: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363  **********..Desc
│ │ │ │ +0001ed20: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0001ed30: 3d3d 3d0a 0a54 6865 206f 7074 696f 6e20  ===..The option 
│ │ │ │ +0001ed40: 496e 7075 7449 7353 6d6f 6f74 6820 6973  InputIsSmooth is
│ │ │ │ +0001ed50: 206f 6e6c 7920 7573 6564 2062 7920 7468   only used by th
│ │ │ │ +0001ed60: 6520 636f 6d6d 616e 6473 202a 6e6f 7465  e commands *note
│ │ │ │ +0001ed70: 2043 534d 3a20 4353 4d2c 2c20 616e 640a   CSM: CSM,, and.
│ │ │ │ +0001ed80: 2a6e 6f74 6520 4575 6c65 723a 2045 756c  *note Euler: Eul
│ │ │ │ +0001ed90: 6572 2c2e 2049 6620 7468 6520 696e 7075  er,. If the inpu
│ │ │ │ +0001eda0: 7420 6964 6561 6c20 6973 206b 6e6f 776e  t ideal is known
│ │ │ │ +0001edb0: 2074 6f20 6465 6669 6e65 2061 2073 6d6f   to define a smo
│ │ │ │ +0001edc0: 6f74 6820 7375 6273 6368 656d 650a 7365  oth subscheme.se
│ │ │ │ +0001edd0: 7474 696e 6720 7468 6973 206f 7074 696f  tting this optio
│ │ │ │ +0001ede0: 6e20 746f 2074 7275 6520 7769 6c6c 2073  n to true will s
│ │ │ │ +0001edf0: 7065 6564 2075 7020 636f 6d70 7574 6174  peed up computat
│ │ │ │ +0001ee00: 696f 6e73 2028 6974 2069 7320 7365 7420  ions (it is set 
│ │ │ │ +0001ee10: 746f 2066 616c 7365 2062 790a 6465 6661  to false by.defa
│ │ │ │ +0001ee20: 756c 7429 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ult)...+--------
│ │ │ │ +0001ee30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ee40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001ee60: 0a7c 6931 203a 2052 203d 205a 5a2f 3332  .|i1 : R = ZZ/32
│ │ │ │ +0001ee70: 3734 395b 785f 302e 2e78 5f34 5d3b 2020  749[x_0..x_4];  
│ │ │ │ +0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0001eef0: 6932 203a 2049 3d69 6465 616c 2872 616e  i2 : I=ideal(ran
│ │ │ │ -0001ef00: 646f 6d28 322c 5229 2c72 616e 646f 6d28  dom(2,R),random(
│ │ │ │ -0001ef10: 322c 5229 2c72 616e 646f 6d28 312c 5229  2,R),random(1,R)
│ │ │ │ -0001ef20: 293b 2020 2020 7c0a 7c20 2020 2020 2020  );    |.|       
│ │ │ │ +0001eed0: 2d2d 2d2b 0a7c 6932 203a 2049 3d69 6465  ---+.|i2 : I=ide
│ │ │ │ +0001eee0: 616c 2872 616e 646f 6d28 322c 5229 2c72  al(random(2,R),r
│ │ │ │ +0001eef0: 616e 646f 6d28 322c 5229 2c72 616e 646f  andom(2,R),rando
│ │ │ │ +0001ef00: 6d28 312c 5229 293b 2020 2020 207c 0a7c  m(1,R));     |.|
│ │ │ │ +0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001ef60: 0a7c 6f32 203a 2049 6465 616c 206f 6620  .|o2 : Ideal of 
│ │ │ │ -0001ef70: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ -0001ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001ef40: 2020 2020 2020 207c 0a7c 6f32 203a 2049         |.|o2 : I
│ │ │ │ +0001ef50: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ +0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001ef90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001efa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001efb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001efc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001efd0: 2d2b 0a7c 6933 203a 2074 696d 6520 4353  -+.|i3 : time CS
│ │ │ │ -0001efe0: 4d20 4920 2020 2020 2020 2020 2020 2020  M I             
│ │ │ │ -0001eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f000: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -0001f010: 2075 7365 6420 302e 3631 3837 3534 7320   used 0.618754s 
│ │ │ │ -0001f020: 2863 7075 293b 2030 2e34 3636 3337 3673  (cpu); 0.466376s
│ │ │ │ -0001f030: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -0001f040: 6329 207c 0a7c 2020 2020 2020 2020 2020  c) |.|          
│ │ │ │ +0001efb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +0001efc0: 203a 2074 696d 6520 4353 4d20 4920 2020   : time CSM I   
│ │ │ │ +0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eff0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +0001f000: 2031 2e31 3531 3733 7320 2863 7075 293b   1.15173s (cpu);
│ │ │ │ +0001f010: 2030 2e36 3038 3333 3373 2028 7468 7265   0.608333s (thre
│ │ │ │ +0001f020: 6164 293b 2030 7320 2867 6329 2020 207c  ad); 0s (gc)   |
│ │ │ │ +0001f030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f070: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f080: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0001f060: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f070: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +0001f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0b0: 2020 2020 207c 0a7c 6f33 203d 2034 6820       |.|o3 = 4h 
│ │ │ │ +0001f0a0: 2020 207c 0a7c 6f33 203d 2034 6820 2020     |.|o3 = 4h   
│ │ │ │ +0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f0f0: 7c20 2020 2020 2020 3120 2020 2020 2020  |       1       
│ │ │ │ +0001f0d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f0e0: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +0001f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f120: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f110: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f160: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020  |.|     ZZ[h ]  
│ │ │ │ +0001f150: 207c 0a7c 2020 2020 205a 5a5b 6820 5d20   |.|     ZZ[h ] 
│ │ │ │ +0001f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f190: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001f1a0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +0001f180: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f190: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ +0001f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1d0: 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d 2d2d    |.|o3 : ------
│ │ │ │ +0001f1c0: 2020 2020 207c 0a7c 6f33 203a 202d 2d2d       |.|o3 : ---
│ │ │ │ +0001f1d0: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │  0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f200: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001f210: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │ +0001f1f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f200: 0a7c 2020 2020 2020 2020 3520 2020 2020  .|        5     
│ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f240: 2020 2020 7c0a 7c20 2020 2020 2020 6820      |.|       h 
│ │ │ │ +0001f230: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f240: 2020 2068 2020 2020 2020 2020 2020 2020     h            
│ │ │ │  0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0001f280: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0001f270: 2020 207c 0a7c 2020 2020 2020 2020 3120     |.|        1 
│ │ │ │ +0001f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001f2a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001f2f0: 0a7c 6934 203a 2074 696d 6520 4353 4d28  .|i4 : time CSM(
│ │ │ │ -0001f300: 492c 496e 7075 7449 7353 6d6f 6f74 683d  I,InputIsSmooth=
│ │ │ │ -0001f310: 3e74 7275 6529 2020 2020 2020 2020 2020  >true)          
│ │ │ │ -0001f320: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -0001f330: 7365 6420 302e 3033 3339 3837 7320 2863  sed 0.033987s (c
│ │ │ │ -0001f340: 7075 293b 2030 2e30 3333 3432 3133 7320  pu); 0.0334213s 
│ │ │ │ -0001f350: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -0001f360: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +0001f2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074  -------+.|i4 : t
│ │ │ │ +0001f2f0: 696d 6520 4353 4d28 492c 496e 7075 7449  ime CSM(I,InputI
│ │ │ │ +0001f300: 7353 6d6f 6f74 683d 3e74 7275 6529 2020  sSmooth=>true)  
│ │ │ │ +0001f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f320: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ +0001f330: 3932 3033 3935 7320 2863 7075 293b 2030  920395s (cpu); 0
│ │ │ │ +0001f340: 2e30 3533 3630 3131 7320 2874 6872 6561  .0536011s (threa
│ │ │ │ +0001f350: 6429 3b20 3073 2028 6763 297c 0a7c 2020  d); 0s (gc)|.|  
│ │ │ │ +0001f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f390: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f3a0: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +0001f390: 2020 2020 207c 0a7c 2020 2020 2020 2033       |.|       3
│ │ │ │ +0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3d0: 2020 207c 0a7c 6f34 203d 2034 6820 2020     |.|o4 = 4h   
│ │ │ │ +0001f3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f3d0: 0a7c 6f34 203d 2034 6820 2020 2020 2020  .|o4 = 4h       
│ │ │ │  0001f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f410: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +0001f400: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f410: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  0001f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f440: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f480: 7c20 2020 2020 5a5a 5b68 205d 2020 2020  |     ZZ[h ]    
│ │ │ │ +0001f470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f480: 2020 2020 205a 5a5b 6820 5d20 2020 2020       ZZ[h ]     
│ │ │ │  0001f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001f4c0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4f0: 7c0a 7c6f 3420 3a20 2d2d 2d2d 2d2d 2020  |.|o4 : ------  
│ │ │ │ +0001f4f0: 207c 0a7c 6f34 203a 202d 2d2d 2d2d 2d20   |.|o4 : ------ 
│ │ │ │  0001f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f520: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001f530: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +0001f520: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f530: 2020 2020 2020 3520 2020 2020 2020 2020        5         
│ │ │ │  0001f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f560: 2020 7c0a 7c20 2020 2020 2020 6820 2020    |.|       h   
│ │ │ │ +0001f560: 2020 2020 207c 0a7c 2020 2020 2020 2068       |.|       h
│ │ │ │  0001f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0001f5a0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ +0001f590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f5a0: 0a7c 2020 2020 2020 2020 3120 2020 2020  .|        1     
│ │ │ │  0001f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001f5d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │  0001f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -0001f610: 4e6f 7465 2074 6861 7420 6f6e 6520 636f  Note that one co
│ │ │ │ -0001f620: 756c 642c 2065 7175 6976 616c 656e 746c  uld, equivalentl
│ │ │ │ -0001f630: 792c 2075 7365 2074 6865 2063 6f6d 6d61  y, use the comma
│ │ │ │ -0001f640: 6e64 202a 6e6f 7465 2043 6865 726e 3a20  nd *note Chern: 
│ │ │ │ -0001f650: 4368 6572 6e2c 2069 6e73 7465 6164 0a69  Chern, instead.i
│ │ │ │ -0001f660: 6e20 7468 6973 2063 6173 652e 0a0a 2b2d  n this case...+-
│ │ │ │ -0001f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f610: 2d2d 2d2b 0a0a 4e6f 7465 2074 6861 7420  ---+..Note that 
│ │ │ │ +0001f620: 6f6e 6520 636f 756c 642c 2065 7175 6976  one could, equiv
│ │ │ │ +0001f630: 616c 656e 746c 792c 2075 7365 2074 6865  alently, use the
│ │ │ │ +0001f640: 2063 6f6d 6d61 6e64 202a 6e6f 7465 2043   command *note C
│ │ │ │ +0001f650: 6865 726e 3a20 4368 6572 6e2c 2069 6e73  hern: Chern, ins
│ │ │ │ +0001f660: 7465 6164 0a69 6e20 7468 6973 2063 6173  tead.in this cas
│ │ │ │ +0001f670: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  e...+-----------
│ │ │ │  0001f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6a0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7469  ------+.|i5 : ti
│ │ │ │ -0001f6b0: 6d65 2043 6865 726e 2049 2020 2020 2020  me Chern I      
│ │ │ │ +0001f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001f6b0: 3520 3a20 7469 6d65 2043 6865 726e 2049  5 : time Chern I
│ │ │ │  0001f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6e0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3033  |.| -- used 0.03
│ │ │ │ -0001f6f0: 3232 3236 3873 2028 6370 7529 3b20 302e  22268s (cpu); 0.
│ │ │ │ -0001f700: 3033 3137 3730 3973 2028 7468 7265 6164  0317709s (thread
│ │ │ │ -0001f710: 293b 2030 7320 2867 6329 7c0a 7c20 2020  ); 0s (gc)|.|   
│ │ │ │ -0001f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6e0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ +0001f6f0: 6420 302e 3039 3031 3435 3273 2028 6370  d 0.0901452s (cp
│ │ │ │ +0001f700: 7529 3b20 302e 3035 3233 3832 3973 2028  u); 0.0523829s (
│ │ │ │ +0001f710: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +0001f720: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f750: 2020 2020 7c0a 7c20 2020 2020 2020 3320      |.|       3 
│ │ │ │ -0001f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f750: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f760: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │  0001f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f780: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f790: 7c6f 3520 3d20 3468 2020 2020 2020 2020  |o5 = 4h        
│ │ │ │ +0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f790: 2020 2020 7c0a 7c6f 3520 3d20 3468 2020      |.|o5 = 4h  
│ │ │ │  0001f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f7d0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +0001f7c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f7d0: 7c20 2020 2020 2020 3120 2020 2020 2020  |       1       
│ │ │ │  0001f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f800: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001f800: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0001f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f830: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f840: 2020 2020 5a5a 5b68 205d 2020 2020 2020      ZZ[h ]      
│ │ │ │ +0001f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f840: 2020 7c0a 7c20 2020 2020 5a5a 5b68 205d    |.|     ZZ[h ]
│ │ │ │  0001f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f870: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f880: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +0001f870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f880: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │  0001f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8b0: 7c0a 7c6f 3520 3a20 2d2d 2d2d 2d2d 2020  |.|o5 : ------  
│ │ │ │ -0001f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f8b0: 2020 2020 2020 7c0a 7c6f 3520 3a20 2d2d        |.|o5 : --
│ │ │ │ +0001f8c0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  0001f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f8f0: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ +0001f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f8f0: 7c0a 7c20 2020 2020 2020 2035 2020 2020  |.|        5    
│ │ │ │  0001f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f920: 2020 2020 7c0a 7c20 2020 2020 2020 6820      |.|       h 
│ │ │ │ -0001f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f920: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f930: 2020 2020 6820 2020 2020 2020 2020 2020      h           
│ │ │ │  0001f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f950: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f960: 7c20 2020 2020 2020 2031 2020 2020 2020  |        1      
│ │ │ │ +0001f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f960: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │  0001f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f990: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0001f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f990: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0001f9a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  0001f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f9d0: 2d2d 2b0a 0a46 756e 6374 696f 6e73 2077  --+..Functions w
│ │ │ │ -0001f9e0: 6974 6820 6f70 7469 6f6e 616c 2061 7267  ith optional arg
│ │ │ │ -0001f9f0: 756d 656e 7420 6e61 6d65 6420 496e 7075  ument named Inpu
│ │ │ │ -0001fa00: 7449 7353 6d6f 6f74 683a 0a3d 3d3d 3d3d  tIsSmooth:.=====
│ │ │ │ -0001fa10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001f9d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 756e 6374  --------+..Funct
│ │ │ │ +0001f9e0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +0001f9f0: 616c 2061 7267 756d 656e 7420 6e61 6d65  al argument name
│ │ │ │ +0001fa00: 6420 496e 7075 7449 7353 6d6f 6f74 683a  d InputIsSmooth:
│ │ │ │ +0001fa10: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │  0001fa20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  0001fa30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001fa40: 0a0a 2020 2a20 2243 534d 282e 2e2e 2c49  ..  * "CSM(...,I
│ │ │ │ -0001fa50: 6e70 7574 4973 536d 6f6f 7468 3d3e 2e2e  nputIsSmooth=>..
│ │ │ │ -0001fa60: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ -0001fa70: 2043 534d 3a20 4353 4d2c 202d 2d20 5468   CSM: CSM, -- Th
│ │ │ │ -0001fa80: 650a 2020 2020 4368 6572 6e2d 5363 6877  e.    Chern-Schw
│ │ │ │ -0001fa90: 6172 747a 2d4d 6163 5068 6572 736f 6e20  artz-MacPherson 
│ │ │ │ -0001faa0: 636c 6173 730a 2020 2a20 4575 6c65 7228  class.  * Euler(
│ │ │ │ -0001fab0: 2e2e 2e2c 496e 7075 7449 7353 6d6f 6f74  ...,InputIsSmoot
│ │ │ │ -0001fac0: 683d 3e2e 2e2e 2920 286d 6973 7369 6e67  h=>...) (missing
│ │ │ │ -0001fad0: 2064 6f63 756d 656e 7461 7469 6f6e 290a   documentation).
│ │ │ │ -0001fae0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ -0001faf0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -0001fb00: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ -0001fb10: 6374 202a 6e6f 7465 2049 6e70 7574 4973  ct *note InputIs
│ │ │ │ -0001fb20: 536d 6f6f 7468 3a20 496e 7075 7449 7353  Smooth: InputIsS
│ │ │ │ -0001fb30: 6d6f 6f74 682c 2069 7320 6120 2a6e 6f74  mooth, is a *not
│ │ │ │ -0001fb40: 6520 7379 6d62 6f6c 3a0a 284d 6163 6175  e symbol:.(Macau
│ │ │ │ -0001fb50: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e  lay2Doc)Symbol,.
│ │ │ │ -0001fb60: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │ +0001fa40: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2243 534d  ======..  * "CSM
│ │ │ │ +0001fa50: 282e 2e2e 2c49 6e70 7574 4973 536d 6f6f  (...,InputIsSmoo
│ │ │ │ +0001fa60: 7468 3d3e 2e2e 2e29 2220 2d2d 2073 6565  th=>...)" -- see
│ │ │ │ +0001fa70: 202a 6e6f 7465 2043 534d 3a20 4353 4d2c   *note CSM: CSM,
│ │ │ │ +0001fa80: 202d 2d20 5468 650a 2020 2020 4368 6572   -- The.    Cher
│ │ │ │ +0001fa90: 6e2d 5363 6877 6172 747a 2d4d 6163 5068  n-Schwartz-MacPh
│ │ │ │ +0001faa0: 6572 736f 6e20 636c 6173 730a 2020 2a20  erson class.  * 
│ │ │ │ +0001fab0: 4575 6c65 7228 2e2e 2e2c 496e 7075 7449  Euler(...,InputI
│ │ │ │ +0001fac0: 7353 6d6f 6f74 683d 3e2e 2e2e 2920 286d  sSmooth=>...) (m
│ │ │ │ +0001fad0: 6973 7369 6e67 2064 6f63 756d 656e 7461  issing documenta
│ │ │ │ +0001fae0: 7469 6f6e 290a 0a46 6f72 2074 6865 2070  tion)..For the p
│ │ │ │ +0001faf0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +0001fb00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +0001fb10: 6520 6f62 6a65 6374 202a 6e6f 7465 2049  e object *note I
│ │ │ │ +0001fb20: 6e70 7574 4973 536d 6f6f 7468 3a20 496e  nputIsSmooth: In
│ │ │ │ +0001fb30: 7075 7449 7353 6d6f 6f74 682c 2069 7320  putIsSmooth, is 
│ │ │ │ +0001fb40: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a0a  a *note symbol:.
│ │ │ │ +0001fb50: 284d 6163 6175 6c61 7932 446f 6329 5379  (Macaulay2Doc)Sy
│ │ │ │ +0001fb60: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  mbol,...--------
│ │ │ │  0001fb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fbb0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ -0001fbc0: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ -0001fbd0: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ -0001fbe0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ -0001fbf0: 6361 756c 6179 322d 312e 3236 2e30 362b  caulay2-1.26.06+
│ │ │ │ -0001fc00: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ -0001fc10: 7061 636b 6167 6573 2f0a 4368 6172 6163  packages/.Charac
│ │ │ │ -0001fc20: 7465 7269 7374 6963 436c 6173 7365 732e  teristicClasses.
│ │ │ │ -0001fc30: 6d32 3a32 3530 303a 302e 0a1f 0a46 696c  m2:2500:0....Fil
│ │ │ │ -0001fc40: 653a 2043 6861 7261 6374 6572 6973 7469  e: Characteristi
│ │ │ │ -0001fc50: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e  cClasses.info, N
│ │ │ │ -0001fc60: 6f64 653a 2069 734d 756c 7469 486f 6d6f  ode: isMultiHomo
│ │ │ │ -0001fc70: 6765 6e65 6f75 732c 204e 6578 743a 204d  geneous, Next: M
│ │ │ │ -0001fc80: 6574 686f 642c 2050 7265 763a 2049 6e70  ethod, Prev: Inp
│ │ │ │ -0001fc90: 7574 4973 536d 6f6f 7468 2c20 5570 3a20  utIsSmooth, Up: 
│ │ │ │ -0001fca0: 546f 700a 0a69 734d 756c 7469 486f 6d6f  Top..isMultiHomo
│ │ │ │ -0001fcb0: 6765 6e65 6f75 7320 2d2d 2043 6865 636b  geneous -- Check
│ │ │ │ -0001fcc0: 7320 6966 2061 6e20 6964 6561 6c20 6973  s if an ideal is
│ │ │ │ -0001fcd0: 2068 6f6d 6f67 656e 656f 7573 2077 6974   homogeneous wit
│ │ │ │ -0001fce0: 6820 7265 7370 6563 7420 746f 2074 6865  h respect to the
│ │ │ │ -0001fcf0: 2067 7261 6469 6e67 206f 6e20 6974 7320   grading on its 
│ │ │ │ -0001fd00: 7269 6e67 2028 692e 652e 206d 756c 7469  ring (i.e. multi
│ │ │ │ -0001fd10: 2d68 6f6d 6f67 656e 656f 7573 2069 6e20  -homogeneous in 
│ │ │ │ -0001fd20: 7468 6520 6d75 6c74 692d 6772 6164 6564  the multi-graded
│ │ │ │ -0001fd30: 2063 6173 6529 0a2a 2a2a 2a2a 2a2a 2a2a   case).*********
│ │ │ │ +0001fbb0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ +0001fbc0: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ +0001fbd0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ +0001fbe0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ +0001fbf0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ +0001fc00: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ +0001fc10: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ +0001fc20: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ +0001fc30: 6173 7365 732e 6d32 3a32 3530 303a 302e  asses.m2:2500:0.
│ │ │ │ +0001fc40: 0a1f 0a46 696c 653a 2043 6861 7261 6374  ...File: Charact
│ │ │ │ +0001fc50: 6572 6973 7469 6343 6c61 7373 6573 2e69  eristicClasses.i
│ │ │ │ +0001fc60: 6e66 6f2c 204e 6f64 653a 2069 734d 756c  nfo, Node: isMul
│ │ │ │ +0001fc70: 7469 486f 6d6f 6765 6e65 6f75 732c 204e  tiHomogeneous, N
│ │ │ │ +0001fc80: 6578 743a 204d 6574 686f 642c 2050 7265  ext: Method, Pre
│ │ │ │ +0001fc90: 763a 2049 6e70 7574 4973 536d 6f6f 7468  v: InputIsSmooth
│ │ │ │ +0001fca0: 2c20 5570 3a20 546f 700a 0a69 734d 756c  , Up: Top..isMul
│ │ │ │ +0001fcb0: 7469 486f 6d6f 6765 6e65 6f75 7320 2d2d  tiHomogeneous --
│ │ │ │ +0001fcc0: 2043 6865 636b 7320 6966 2061 6e20 6964   Checks if an id
│ │ │ │ +0001fcd0: 6561 6c20 6973 2068 6f6d 6f67 656e 656f  eal is homogeneo
│ │ │ │ +0001fce0: 7573 2077 6974 6820 7265 7370 6563 7420  us with respect 
│ │ │ │ +0001fcf0: 746f 2074 6865 2067 7261 6469 6e67 206f  to the grading o
│ │ │ │ +0001fd00: 6e20 6974 7320 7269 6e67 2028 692e 652e  n its ring (i.e.
│ │ │ │ +0001fd10: 206d 756c 7469 2d68 6f6d 6f67 656e 656f   multi-homogeneo
│ │ │ │ +0001fd20: 7573 2069 6e20 7468 6520 6d75 6c74 692d  us in the multi-
│ │ │ │ +0001fd30: 6772 6164 6564 2063 6173 6529 0a2a 2a2a  graded case).***
│ │ │ │  0001fd40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fd50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fd60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fd70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fd80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fd90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001fdb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001fdc0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ -0001fdd0: 6167 653a 200a 2020 2020 2020 2020 6973  age: .        is
│ │ │ │ -0001fde0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573  MultiHomogeneous
│ │ │ │ -0001fdf0: 2049 0a20 2020 2020 2020 2069 734d 756c   I.        isMul
│ │ │ │ -0001fe00: 7469 486f 6d6f 6765 6e65 6f75 7320 660a  tiHomogeneous f.
│ │ │ │ -0001fe10: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -0001fe20: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520    * I, an *note 
│ │ │ │ -0001fe30: 6964 6561 6c3a 2028 4d61 6361 756c 6179  ideal: (Macaulay
│ │ │ │ -0001fe40: 3244 6f63 2949 6465 616c 2c2c 2061 6e20  2Doc)Ideal,, an 
│ │ │ │ -0001fe50: 6964 6561 6c20 696e 2061 2067 7261 6465  ideal in a grade
│ │ │ │ -0001fe60: 6420 6f72 0a20 2020 2020 2020 206d 756c  d or.        mul
│ │ │ │ -0001fe70: 7469 2d67 7261 6465 6420 7269 6e67 0a20  ti-graded ring. 
│ │ │ │ -0001fe80: 2020 2020 202a 2066 2c20 6120 2a6e 6f74       * f, a *not
│ │ │ │ -0001fe90: 6520 7269 6e67 2065 6c65 6d65 6e74 3a20  e ring element: 
│ │ │ │ -0001fea0: 284d 6163 6175 6c61 7932 446f 6329 5269  (Macaulay2Doc)Ri
│ │ │ │ -0001feb0: 6e67 456c 656d 656e 742c 2c20 6120 656c  ngElement,, a el
│ │ │ │ -0001fec0: 656d 656e 7420 696e 2061 0a20 2020 2020  ement in a.     
│ │ │ │ -0001fed0: 2020 2067 7261 6465 6420 6f72 206d 756c     graded or mul
│ │ │ │ -0001fee0: 7469 2d67 7261 6465 6420 7269 6e67 0a20  ti-graded ring. 
│ │ │ │ -0001fef0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -0001ff00: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c    * a *note Bool
│ │ │ │ -0001ff10: 6561 6e20 7661 6c75 653a 2028 4d61 6361  ean value: (Maca
│ │ │ │ -0001ff20: 756c 6179 3244 6f63 2942 6f6f 6c65 616e  ulay2Doc)Boolean
│ │ │ │ -0001ff30: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e  ,, ..Description
│ │ │ │ -0001ff40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5465  .===========..Te
│ │ │ │ -0001ff50: 7374 7320 6966 2074 6865 2069 6e70 7574  sts if the input
│ │ │ │ -0001ff60: 2049 6465 616c 206f 7220 5269 6e67 456c   Ideal or RingEl
│ │ │ │ -0001ff70: 656d 656e 7420 6973 2048 6f6d 6f67 656e  ement is Homogen
│ │ │ │ -0001ff80: 656f 7573 2077 6974 6820 7265 7370 6563  eous with respec
│ │ │ │ -0001ff90: 7420 746f 2074 6865 0a67 7261 6469 6e67  t to the.grading
│ │ │ │ -0001ffa0: 206f 6e20 7468 6520 7269 6e67 2e20 486f   on the ring. Ho
│ │ │ │ -0001ffb0: 6d6f 6765 6e65 6f75 7320 696e 7075 7420  mogeneous input 
│ │ │ │ -0001ffc0: 6973 2072 6571 7569 7265 6420 666f 7220  is required for 
│ │ │ │ -0001ffd0: 616c 6c20 6d65 7468 6f64 7320 746f 2063  all methods to c
│ │ │ │ -0001ffe0: 6f6d 7075 7465 0a63 6861 7261 6374 6572  ompute.character
│ │ │ │ -0001fff0: 6973 7469 6320 636c 6173 7365 732e 0a0a  istic classes...
│ │ │ │ -00020000: 5468 6973 206d 6574 686f 6420 776f 726b  This method work
│ │ │ │ -00020010: 7320 666f 7220 6964 6561 6c73 2069 6e20  s for ideals in 
│ │ │ │ -00020020: 7468 6520 6772 6164 6564 2063 6f6f 7264  the graded coord
│ │ │ │ -00020030: 696e 6174 6520 7269 6e67 7320 6f66 2074  inate rings of t
│ │ │ │ -00020040: 6f72 6963 2076 6172 6965 7469 6573 2c0a  oric varieties,.
│ │ │ │ -00020050: 616e 6420 6865 6e63 6520 666f 7220 7072  and hence for pr
│ │ │ │ -00020060: 6f64 7563 7473 206f 6620 7072 6f6a 6563  oducts of projec
│ │ │ │ -00020070: 7469 7665 2073 7061 6365 732e 2054 6865  tive spaces. The
│ │ │ │ -00020080: 7365 2063 616e 2062 6520 6372 6561 7465  se can be create
│ │ │ │ -00020090: 6420 6469 7265 6374 6c79 2c20 6f72 0a75  d directly, or.u
│ │ │ │ -000200a0: 7369 6e67 206d 6574 686f 6473 2074 6865  sing methods the
│ │ │ │ -000200b0: 202a 6e6f 7465 204d 756c 7469 5072 6f6a   *note MultiProj
│ │ │ │ -000200c0: 436f 6f72 6452 696e 673a 204d 756c 7469  CoordRing: Multi
│ │ │ │ -000200d0: 5072 6f6a 436f 6f72 6452 696e 672c 206d  ProjCoordRing, m
│ │ │ │ -000200e0: 6574 686f 6420 6f66 2074 6869 730a 7061  ethod of this.pa
│ │ │ │ -000200f0: 636b 6167 652c 206f 7220 7769 7468 206d  ckage, or with m
│ │ │ │ -00020100: 6574 686f 6473 2066 726f 6d20 7468 6520  ethods from the 
│ │ │ │ -00020110: 4e6f 726d 616c 546f 7269 6356 6172 6965  NormalToricVarie
│ │ │ │ -00020120: 7469 6573 2050 6163 6b61 6765 2e0a 0a2b  ties Package...+
│ │ │ │ -00020130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001fdc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ +0001fdd0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +0001fde0: 2020 2020 6973 4d75 6c74 6948 6f6d 6f67      isMultiHomog
│ │ │ │ +0001fdf0: 656e 656f 7573 2049 0a20 2020 2020 2020  eneous I.       
│ │ │ │ +0001fe00: 2069 734d 756c 7469 486f 6d6f 6765 6e65   isMultiHomogene
│ │ │ │ +0001fe10: 6f75 7320 660a 2020 2a20 496e 7075 7473  ous f.  * Inputs
│ │ │ │ +0001fe20: 3a0a 2020 2020 2020 2a20 492c 2061 6e20  :.      * I, an 
│ │ │ │ +0001fe30: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ +0001fe40: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ +0001fe50: 2c2c 2061 6e20 6964 6561 6c20 696e 2061  ,, an ideal in a
│ │ │ │ +0001fe60: 2067 7261 6465 6420 6f72 0a20 2020 2020   graded or.     
│ │ │ │ +0001fe70: 2020 206d 756c 7469 2d67 7261 6465 6420     multi-graded 
│ │ │ │ +0001fe80: 7269 6e67 0a20 2020 2020 202a 2066 2c20  ring.      * f, 
│ │ │ │ +0001fe90: 6120 2a6e 6f74 6520 7269 6e67 2065 6c65  a *note ring ele
│ │ │ │ +0001fea0: 6d65 6e74 3a20 284d 6163 6175 6c61 7932  ment: (Macaulay2
│ │ │ │ +0001feb0: 446f 6329 5269 6e67 456c 656d 656e 742c  Doc)RingElement,
│ │ │ │ +0001fec0: 2c20 6120 656c 656d 656e 7420 696e 2061  , a element in a
│ │ │ │ +0001fed0: 0a20 2020 2020 2020 2067 7261 6465 6420  .        graded 
│ │ │ │ +0001fee0: 6f72 206d 756c 7469 2d67 7261 6465 6420  or multi-graded 
│ │ │ │ +0001fef0: 7269 6e67 0a20 202a 204f 7574 7075 7473  ring.  * Outputs
│ │ │ │ +0001ff00: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74  :.      * a *not
│ │ │ │ +0001ff10: 6520 426f 6f6c 6561 6e20 7661 6c75 653a  e Boolean value:
│ │ │ │ +0001ff20: 2028 4d61 6361 756c 6179 3244 6f63 2942   (Macaulay2Doc)B
│ │ │ │ +0001ff30: 6f6f 6c65 616e 2c2c 200a 0a44 6573 6372  oolean,, ..Descr
│ │ │ │ +0001ff40: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0001ff50: 3d3d 0a0a 5465 7374 7320 6966 2074 6865  ==..Tests if the
│ │ │ │ +0001ff60: 2069 6e70 7574 2049 6465 616c 206f 7220   input Ideal or 
│ │ │ │ +0001ff70: 5269 6e67 456c 656d 656e 7420 6973 2048  RingElement is H
│ │ │ │ +0001ff80: 6f6d 6f67 656e 656f 7573 2077 6974 6820  omogeneous with 
│ │ │ │ +0001ff90: 7265 7370 6563 7420 746f 2074 6865 0a67  respect to the.g
│ │ │ │ +0001ffa0: 7261 6469 6e67 206f 6e20 7468 6520 7269  rading on the ri
│ │ │ │ +0001ffb0: 6e67 2e20 486f 6d6f 6765 6e65 6f75 7320  ng. Homogeneous 
│ │ │ │ +0001ffc0: 696e 7075 7420 6973 2072 6571 7569 7265  input is require
│ │ │ │ +0001ffd0: 6420 666f 7220 616c 6c20 6d65 7468 6f64  d for all method
│ │ │ │ +0001ffe0: 7320 746f 2063 6f6d 7075 7465 0a63 6861  s to compute.cha
│ │ │ │ +0001fff0: 7261 6374 6572 6973 7469 6320 636c 6173  racteristic clas
│ │ │ │ +00020000: 7365 732e 0a0a 5468 6973 206d 6574 686f  ses...This metho
│ │ │ │ +00020010: 6420 776f 726b 7320 666f 7220 6964 6561  d works for idea
│ │ │ │ +00020020: 6c73 2069 6e20 7468 6520 6772 6164 6564  ls in the graded
│ │ │ │ +00020030: 2063 6f6f 7264 696e 6174 6520 7269 6e67   coordinate ring
│ │ │ │ +00020040: 7320 6f66 2074 6f72 6963 2076 6172 6965  s of toric varie
│ │ │ │ +00020050: 7469 6573 2c0a 616e 6420 6865 6e63 6520  ties,.and hence 
│ │ │ │ +00020060: 666f 7220 7072 6f64 7563 7473 206f 6620  for products of 
│ │ │ │ +00020070: 7072 6f6a 6563 7469 7665 2073 7061 6365  projective space
│ │ │ │ +00020080: 732e 2054 6865 7365 2063 616e 2062 6520  s. These can be 
│ │ │ │ +00020090: 6372 6561 7465 6420 6469 7265 6374 6c79  created directly
│ │ │ │ +000200a0: 2c20 6f72 0a75 7369 6e67 206d 6574 686f  , or.using metho
│ │ │ │ +000200b0: 6473 2074 6865 202a 6e6f 7465 204d 756c  ds the *note Mul
│ │ │ │ +000200c0: 7469 5072 6f6a 436f 6f72 6452 696e 673a  tiProjCoordRing:
│ │ │ │ +000200d0: 204d 756c 7469 5072 6f6a 436f 6f72 6452   MultiProjCoordR
│ │ │ │ +000200e0: 696e 672c 206d 6574 686f 6420 6f66 2074  ing, method of t
│ │ │ │ +000200f0: 6869 730a 7061 636b 6167 652c 206f 7220  his.package, or 
│ │ │ │ +00020100: 7769 7468 206d 6574 686f 6473 2066 726f  with methods fro
│ │ │ │ +00020110: 6d20 7468 6520 4e6f 726d 616c 546f 7269  m the NormalTori
│ │ │ │ +00020120: 6356 6172 6965 7469 6573 2050 6163 6b61  cVarieties Packa
│ │ │ │ +00020130: 6765 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ge...+----------
│ │ │ │  00020140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00020170: 0a7c 6931 203a 2052 3d4d 756c 7469 5072  .|i1 : R=MultiPr
│ │ │ │ -00020180: 6f6a 436f 6f72 6452 696e 6728 7b31 2c32  ojCoordRing({1,2
│ │ │ │ -00020190: 2c31 7d29 2020 2020 2020 2020 2020 2020  ,1})            
│ │ │ │ +00020160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020170: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 3d4d  -----+.|i1 : R=M
│ │ │ │ +00020180: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ +00020190: 6728 7b31 2c32 2c31 7d29 2020 2020 2020  g({1,2,1})      
│ │ │ │  000201a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000201b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000201c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201f0: 2020 207c 0a7c 6f31 203d 2052 2020 2020     |.|o1 = R    
│ │ │ │ -00020200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000201f0: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ +00020200: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  00020210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020230: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020230: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00020240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020270: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -00020280: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ -00020290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00020280: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ +00020290: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │  000202a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000202b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000202b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000202c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00020300: 203a 2078 3d67 656e 7328 5229 2020 2020   : x=gens(R)    
│ │ │ │ -00020310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020300: 2d2b 0a7c 6932 203a 2078 3d67 656e 7328  -+.|i2 : x=gens(
│ │ │ │ +00020310: 5229 2020 2020 2020 2020 2020 2020 2020  R)              
│ │ │ │  00020320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020340: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00020350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020380: 0a7c 6f32 203d 207b 7820 2c20 7820 2c20  .|o2 = {x , x , 
│ │ │ │ -00020390: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -000203a0: 7820 7d20 2020 2020 2020 2020 2020 2020  x }             
│ │ │ │ +00020370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020380: 2020 2020 207c 0a7c 6f32 203d 207b 7820       |.|o2 = {x 
│ │ │ │ +00020390: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +000203a0: 2c20 7820 2c20 7820 7d20 2020 2020 2020  , x , x }       
│ │ │ │  000203b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203c0: 207c 0a7c 2020 2020 2020 2030 2020 2031   |.|       0   1
│ │ │ │ -000203d0: 2020 2032 2020 2033 2020 2034 2020 2035     2   3   4   5
│ │ │ │ -000203e0: 2020 2036 2020 2020 2020 2020 2020 2020     6            
│ │ │ │ +000203c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000203d0: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ +000203e0: 2034 2020 2035 2020 2036 2020 2020 2020   4   5   6      
│ │ │ │  000203f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020400: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00020400: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00020410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020440: 2020 2020 207c 0a7c 6f32 203a 204c 6973       |.|o2 : Lis
│ │ │ │ -00020450: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00020440: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00020450: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │  00020460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020480: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00020480: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000204a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000204b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000204c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -000204d0: 2049 3d69 6465 616c 2878 5f30 5e32 2a78   I=ideal(x_0^2*x
│ │ │ │ -000204e0: 5f33 2d78 5f31 2a78 5f30 2a78 5f34 2c78  _3-x_1*x_0*x_4,x
│ │ │ │ -000204f0: 5f36 5e33 2920 2020 2020 2020 2020 2020  _6^3)           
│ │ │ │ -00020500: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00020510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000204c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000204d0: 0a7c 6933 203a 2049 3d69 6465 616c 2878  .|i3 : I=ideal(x
│ │ │ │ +000204e0: 5f30 5e32 2a78 5f33 2d78 5f31 2a78 5f30  _0^2*x_3-x_1*x_0
│ │ │ │ +000204f0: 2a78 5f34 2c78 5f36 5e33 2920 2020 2020  *x_4,x_6^3)     
│ │ │ │ +00020500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020510: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00020520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020550: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00020560: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -00020570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020590: 0a7c 6f33 203d 2069 6465 616c 2028 7820  .|o3 = ideal (x 
│ │ │ │ -000205a0: 7820 202d 2078 2078 2078 202c 2078 2029  x  - x x x , x )
│ │ │ │ -000205b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020550: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00020560: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00020570: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +00020580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020590: 2020 2020 207c 0a7c 6f33 203d 2069 6465       |.|o3 = ide
│ │ │ │ +000205a0: 616c 2028 7820 7820 202d 2078 2078 2078  al (x x  - x x x
│ │ │ │ +000205b0: 202c 2078 2029 2020 2020 2020 2020 2020   , x )          
│ │ │ │  000205c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000205d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000205e0: 2030 2033 2020 2020 3020 3120 3420 2020   0 3    0 1 4   
│ │ │ │ -000205f0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +000205d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000205e0: 2020 2020 2020 2030 2033 2020 2020 3020         0 3    0 
│ │ │ │ +000205f0: 3120 3420 2020 3620 2020 2020 2020 2020  1 4   6         
│ │ │ │  00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020610: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00020610: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00020620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020650: 2020 2020 207c 0a7c 6f33 203a 2049 6465       |.|o3 : Ide
│ │ │ │ -00020660: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ +00020650: 2020 2020 2020 2020 2020 207c 0a7c 6f33             |.|o3
│ │ │ │ +00020660: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │  00020670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020690: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00020690: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  000206a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000206b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000206c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000206d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -000206e0: 2069 734d 756c 7469 486f 6d6f 6765 6e65   isMultiHomogene
│ │ │ │ -000206f0: 6f75 7320 4920 2020 2020 2020 2020 2020  ous I           
│ │ │ │ +000206d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000206e0: 0a7c 6934 203a 2069 734d 756c 7469 486f  .|i4 : isMultiHo
│ │ │ │ +000206f0: 6d6f 6765 6e65 6f75 7320 4920 2020 2020  mogeneous I     
│ │ │ │  00020700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020710: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00020720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020720: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00020730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020760: 6f34 203d 2074 7275 6520 2020 2020 2020  o4 = true       
│ │ │ │ +00020750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020760: 2020 207c 0a7c 6f34 203d 2074 7275 6520     |.|o4 = true 
│ │ │ │  00020770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000207a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00020790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000207a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000207d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000207e0: 2d2b 0a7c 6935 203a 2069 734d 756c 7469  -+.|i5 : isMulti
│ │ │ │ -000207f0: 486f 6d6f 6765 6e65 6f75 7320 6964 6561  Homogeneous idea
│ │ │ │ -00020800: 6c28 785f 302a 785f 332d 785f 312a 785f  l(x_0*x_3-x_1*x_
│ │ │ │ -00020810: 302a 785f 342c 785f 365e 3329 2020 2020  0*x_4,x_6^3)    
│ │ │ │ -00020820: 2020 207c 0a7c 496e 7075 7420 7465 726d     |.|Input term
│ │ │ │ -00020830: 2062 656c 6f77 2069 7320 6e6f 7420 686f   below is not ho
│ │ │ │ -00020840: 6d6f 6765 6e65 6f75 7320 7769 7468 2072  mogeneous with r
│ │ │ │ -00020850: 6573 7065 6374 2074 6f20 7468 6520 6772  espect to the gr
│ │ │ │ -00020860: 6164 696e 677c 0a7c 2d20 7820 7820 7820  ading|.|- x x x 
│ │ │ │ -00020870: 202b 2078 2078 2020 2020 2020 2020 2020   + x x          
│ │ │ │ +000207e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2069  -------+.|i5 : i
│ │ │ │ +000207f0: 734d 756c 7469 486f 6d6f 6765 6e65 6f75  sMultiHomogeneou
│ │ │ │ +00020800: 7320 6964 6561 6c28 785f 302a 785f 332d  s ideal(x_0*x_3-
│ │ │ │ +00020810: 785f 312a 785f 302a 785f 342c 785f 365e  x_1*x_0*x_4,x_6^
│ │ │ │ +00020820: 3329 2020 2020 2020 207c 0a7c 496e 7075  3)       |.|Inpu
│ │ │ │ +00020830: 7420 7465 726d 2062 656c 6f77 2069 7320  t term below is 
│ │ │ │ +00020840: 6e6f 7420 686f 6d6f 6765 6e65 6f75 7320  not homogeneous 
│ │ │ │ +00020850: 7769 7468 2072 6573 7065 6374 2074 6f20  with respect to 
│ │ │ │ +00020860: 7468 6520 6772 6164 696e 677c 0a7c 2d20  the grading|.|- 
│ │ │ │ +00020870: 7820 7820 7820 202b 2078 2078 2020 2020  x x x  + x x    
│ │ │ │  00020880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208a0: 2020 2020 2020 207c 0a7c 2020 2030 2031         |.|   0 1
│ │ │ │ -000208b0: 2034 2020 2020 3020 3320 2020 2020 2020   4    0 3       
│ │ │ │ +000208a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000208b0: 2020 2030 2031 2034 2020 2020 3020 3320     0 1 4    0 3 
│ │ │ │  000208c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000208d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000208f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000208e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000208f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00020900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020920: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -00020930: 203d 2066 616c 7365 2020 2020 2020 2020   = false        
│ │ │ │ +00020920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020930: 207c 0a7c 6f35 203d 2066 616c 7365 2020   |.|o5 = false  
│ │ │ │  00020940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020960: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00020970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020970: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00020980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000209b0: 0a0a 4e6f 7465 2074 6861 7420 666f 7220  ..Note that for 
│ │ │ │ -000209c0: 616e 2069 6465 616c 2074 6f20 6265 206d  an ideal to be m
│ │ │ │ -000209d0: 756c 7469 2d68 6f6d 6f67 656e 656f 7573  ulti-homogeneous
│ │ │ │ -000209e0: 2074 6865 2064 6567 7265 6520 7665 6374   the degree vect
│ │ │ │ -000209f0: 6f72 206f 6620 616c 6c0a 6d6f 6e6f 6d69  or of all.monomi
│ │ │ │ -00020a00: 616c 7320 696e 2061 2067 6976 656e 2067  als in a given g
│ │ │ │ -00020a10: 656e 6572 6174 6f72 206d 7573 7420 6265  enerator must be
│ │ │ │ -00020a20: 2074 6865 2073 616d 652e 0a0a 5761 7973   the same...Ways
│ │ │ │ -00020a30: 2074 6f20 7573 6520 6973 4d75 6c74 6948   to use isMultiH
│ │ │ │ -00020a40: 6f6d 6f67 656e 656f 7573 3a0a 3d3d 3d3d  omogeneous:.====
│ │ │ │ -00020a50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00020a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00020a70: 2022 6973 4d75 6c74 6948 6f6d 6f67 656e   "isMultiHomogen
│ │ │ │ -00020a80: 656f 7573 2849 6465 616c 2922 0a20 202a  eous(Ideal)".  *
│ │ │ │ -00020a90: 2022 6973 4d75 6c74 6948 6f6d 6f67 656e   "isMultiHomogen
│ │ │ │ -00020aa0: 656f 7573 2852 696e 6745 6c65 6d65 6e74  eous(RingElement
│ │ │ │ -00020ab0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ -00020ac0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -00020ad0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -00020ae0: 626a 6563 7420 2a6e 6f74 6520 6973 4d75  bject *note isMu
│ │ │ │ -00020af0: 6c74 6948 6f6d 6f67 656e 656f 7573 3a20  ltiHomogeneous: 
│ │ │ │ -00020b00: 6973 4d75 6c74 6948 6f6d 6f67 656e 656f  isMultiHomogeneo
│ │ │ │ -00020b10: 7573 2c20 6973 2061 202a 6e6f 7465 206d  us, is a *note m
│ │ │ │ -00020b20: 6574 686f 640a 6675 6e63 7469 6f6e 3a20  ethod.function: 
│ │ │ │ -00020b30: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -00020b40: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a  thodFunction,...
│ │ │ │ -00020b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000209b0: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2074 6861  -----+..Note tha
│ │ │ │ +000209c0: 7420 666f 7220 616e 2069 6465 616c 2074  t for an ideal t
│ │ │ │ +000209d0: 6f20 6265 206d 756c 7469 2d68 6f6d 6f67  o be multi-homog
│ │ │ │ +000209e0: 656e 656f 7573 2074 6865 2064 6567 7265  eneous the degre
│ │ │ │ +000209f0: 6520 7665 6374 6f72 206f 6620 616c 6c0a  e vector of all.
│ │ │ │ +00020a00: 6d6f 6e6f 6d69 616c 7320 696e 2061 2067  monomials in a g
│ │ │ │ +00020a10: 6976 656e 2067 656e 6572 6174 6f72 206d  iven generator m
│ │ │ │ +00020a20: 7573 7420 6265 2074 6865 2073 616d 652e  ust be the same.
│ │ │ │ +00020a30: 0a0a 5761 7973 2074 6f20 7573 6520 6973  ..Ways to use is
│ │ │ │ +00020a40: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573  MultiHomogeneous
│ │ │ │ +00020a50: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00020a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00020a70: 3d0a 0a20 202a 2022 6973 4d75 6c74 6948  =..  * "isMultiH
│ │ │ │ +00020a80: 6f6d 6f67 656e 656f 7573 2849 6465 616c  omogeneous(Ideal
│ │ │ │ +00020a90: 2922 0a20 202a 2022 6973 4d75 6c74 6948  )".  * "isMultiH
│ │ │ │ +00020aa0: 6f6d 6f67 656e 656f 7573 2852 696e 6745  omogeneous(RingE
│ │ │ │ +00020ab0: 6c65 6d65 6e74 2922 0a0a 466f 7220 7468  lement)"..For th
│ │ │ │ +00020ac0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +00020ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00020ae0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +00020af0: 6520 6973 4d75 6c74 6948 6f6d 6f67 656e  e isMultiHomogen
│ │ │ │ +00020b00: 656f 7573 3a20 6973 4d75 6c74 6948 6f6d  eous: isMultiHom
│ │ │ │ +00020b10: 6f67 656e 656f 7573 2c20 6973 2061 202a  ogeneous, is a *
│ │ │ │ +00020b20: 6e6f 7465 206d 6574 686f 640a 6675 6e63  note method.func
│ │ │ │ +00020b30: 7469 6f6e 3a20 284d 6163 6175 6c61 7932  tion: (Macaulay2
│ │ │ │ +00020b40: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +00020b50: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │  00020b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -00020ba0: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -00020bb0: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ -00020bc0: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ -00020bd0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -00020be0: 756c 6179 322d 312e 3236 2e30 362b 6473  ulay2-1.26.06+ds
│ │ │ │ -00020bf0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -00020c00: 636b 6167 6573 2f0a 4368 6172 6163 7465  ckages/.Characte
│ │ │ │ -00020c10: 7269 7374 6963 436c 6173 7365 732e 6d32  risticClasses.m2
│ │ │ │ -00020c20: 3a32 3031 323a 302e 0a1f 0a46 696c 653a  :2012:0....File:
│ │ │ │ -00020c30: 2043 6861 7261 6374 6572 6973 7469 6343   CharacteristicC
│ │ │ │ -00020c40: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64  lasses.info, Nod
│ │ │ │ -00020c50: 653a 204d 6574 686f 642c 204e 6578 743a  e: Method, Next:
│ │ │ │ -00020c60: 204d 756c 7469 5072 6f6a 436f 6f72 6452   MultiProjCoordR
│ │ │ │ -00020c70: 696e 672c 2050 7265 763a 2069 734d 756c  ing, Prev: isMul
│ │ │ │ -00020c80: 7469 486f 6d6f 6765 6e65 6f75 732c 2055  tiHomogeneous, U
│ │ │ │ -00020c90: 703a 2054 6f70 0a0a 4d65 7468 6f64 0a2a  p: Top..Method.*
│ │ │ │ -00020ca0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469  *****..Descripti
│ │ │ │ -00020cb0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00020cc0: 5468 6520 6f70 7469 6f6e 204d 6574 686f  The option Metho
│ │ │ │ -00020cd0: 6420 6973 206f 6e6c 7920 7573 6564 2062  d is only used b
│ │ │ │ -00020ce0: 7920 7468 6520 636f 6d6d 616e 6473 202a  y the commands *
│ │ │ │ -00020cf0: 6e6f 7465 2043 534d 3a20 4353 4d2c 2061  note CSM: CSM, a
│ │ │ │ -00020d00: 6e64 202a 6e6f 7465 2045 756c 6572 3a0a  nd *note Euler:.
│ │ │ │ -00020d10: 4575 6c65 722c 2061 6e64 206f 6e6c 7920  Euler, and only 
│ │ │ │ -00020d20: 696e 2063 6f6d 6269 6e61 7469 6f6e 2077  in combination w
│ │ │ │ -00020d30: 6974 6820 2a6e 6f74 6520 436f 6d70 4d65  ith *note CompMe
│ │ │ │ -00020d40: 7468 6f64 3a0a 436f 6d70 4d65 7468 6f64  thod:.CompMethod
│ │ │ │ -00020d50: 2c3d 3e50 726f 6a65 6374 6976 6544 6567  ,=>ProjectiveDeg
│ │ │ │ -00020d60: 7265 652e 2054 6865 204d 6574 686f 6420  ree. The Method 
│ │ │ │ -00020d70: 496e 636c 7573 696f 6e45 7863 6c75 7369  InclusionExclusi
│ │ │ │ -00020d80: 6f6e 2077 696c 6c20 616c 7761 7973 2062  on will always b
│ │ │ │ -00020d90: 650a 7573 6564 2077 6974 6820 2a6e 6f74  e.used with *not
│ │ │ │ -00020da0: 6520 436f 6d70 4d65 7468 6f64 3a20 436f  e CompMethod: Co
│ │ │ │ -00020db0: 6d70 4d65 7468 6f64 2c20 506e 5265 7369  mpMethod, PnResi
│ │ │ │ -00020dc0: 6475 616c 206f 7220 6265 7274 696e 692e  dual or bertini.
│ │ │ │ -00020dd0: 2057 6865 6e20 7468 6520 696e 7075 740a   When the input.
│ │ │ │ -00020de0: 6964 6561 6c20 6973 2061 2063 6f6d 706c  ideal is a compl
│ │ │ │ -00020df0: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ -00020e00: 206f 6e65 206d 6179 2c20 706f 7465 6e74   one may, potent
│ │ │ │ -00020e10: 6961 6c6c 792c 2073 7065 6564 2075 7020  ially, speed up 
│ │ │ │ -00020e20: 7468 6520 636f 6d70 7574 6174 696f 6e0a  the computation.
│ │ │ │ -00020e30: 6279 2073 6574 7469 6e67 204d 6574 686f  by setting Metho
│ │ │ │ -00020e40: 643d 3e20 4469 7265 6374 436f 6d70 6c65  d=> DirectComple
│ │ │ │ -00020e50: 7465 496e 742e 2054 6865 206f 7074 696f  teInt. The optio
│ │ │ │ -00020e60: 6e20 4d65 7468 6f64 2069 7320 6f6e 6c79  n Method is only
│ │ │ │ -00020e70: 2075 7365 6420 6279 2074 6865 0a63 6f6d   used by the.com
│ │ │ │ -00020e80: 6d61 6e64 7320 2a6e 6f74 6520 4353 4d3a  mands *note CSM:
│ │ │ │ -00020e90: 2043 534d 2c20 616e 6420 2a6e 6f74 6520   CSM, and *note 
│ │ │ │ -00020ea0: 4575 6c65 723a 2045 756c 6572 2c20 616e  Euler: Euler, an
│ │ │ │ -00020eb0: 6420 6f6e 6c79 2069 6e20 636f 6d62 696e  d only in combin
│ │ │ │ -00020ec0: 6174 696f 6e20 7769 7468 0a2a 6e6f 7465  ation with.*note
│ │ │ │ -00020ed0: 2043 6f6d 704d 6574 686f 643a 2043 6f6d   CompMethod: Com
│ │ │ │ -00020ee0: 704d 6574 686f 642c 3d3e 5072 6f6a 6563  pMethod,=>Projec
│ │ │ │ -00020ef0: 7469 7665 4465 6772 6565 2e20 5468 6520  tiveDegree. The 
│ │ │ │ -00020f00: 4d65 7468 6f64 2049 6e63 6c75 7369 6f6e  Method Inclusion
│ │ │ │ -00020f10: 4578 636c 7573 696f 6e0a 7769 6c6c 2061  Exclusion.will a
│ │ │ │ -00020f20: 6c77 6179 7320 6265 2075 7365 6420 7769  lways be used wi
│ │ │ │ -00020f30: 7468 202a 6e6f 7465 2043 6f6d 704d 6574  th *note CompMet
│ │ │ │ -00020f40: 686f 643a 2043 6f6d 704d 6574 686f 642c  hod: CompMethod,
│ │ │ │ -00020f50: 2050 6e52 6573 6964 7561 6c20 6f72 2062   PnResidual or b
│ │ │ │ -00020f60: 6572 7469 6e69 2e0a 0a2b 2d2d 2d2d 2d2d  ertini...+------
│ │ │ │ +00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ba0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +00020bb0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +00020bc0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +00020bd0: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +00020be0: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ +00020bf0: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ +00020c00: 6179 322f 7061 636b 6167 6573 2f0a 4368  ay2/packages/.Ch
│ │ │ │ +00020c10: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ +00020c20: 7365 732e 6d32 3a32 3031 323a 302e 0a1f  ses.m2:2012:0...
│ │ │ │ +00020c30: 0a46 696c 653a 2043 6861 7261 6374 6572  .File: Character
│ │ │ │ +00020c40: 6973 7469 6343 6c61 7373 6573 2e69 6e66  isticClasses.inf
│ │ │ │ +00020c50: 6f2c 204e 6f64 653a 204d 6574 686f 642c  o, Node: Method,
│ │ │ │ +00020c60: 204e 6578 743a 204d 756c 7469 5072 6f6a   Next: MultiProj
│ │ │ │ +00020c70: 436f 6f72 6452 696e 672c 2050 7265 763a  CoordRing, Prev:
│ │ │ │ +00020c80: 2069 734d 756c 7469 486f 6d6f 6765 6e65   isMultiHomogene
│ │ │ │ +00020c90: 6f75 732c 2055 703a 2054 6f70 0a0a 4d65  ous, Up: Top..Me
│ │ │ │ +00020ca0: 7468 6f64 0a2a 2a2a 2a2a 2a0a 0a44 6573  thod.******..Des
│ │ │ │ +00020cb0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00020cc0: 3d3d 3d3d 0a0a 5468 6520 6f70 7469 6f6e  ====..The option
│ │ │ │ +00020cd0: 204d 6574 686f 6420 6973 206f 6e6c 7920   Method is only 
│ │ │ │ +00020ce0: 7573 6564 2062 7920 7468 6520 636f 6d6d  used by the comm
│ │ │ │ +00020cf0: 616e 6473 202a 6e6f 7465 2043 534d 3a20  ands *note CSM: 
│ │ │ │ +00020d00: 4353 4d2c 2061 6e64 202a 6e6f 7465 2045  CSM, and *note E
│ │ │ │ +00020d10: 756c 6572 3a0a 4575 6c65 722c 2061 6e64  uler:.Euler, and
│ │ │ │ +00020d20: 206f 6e6c 7920 696e 2063 6f6d 6269 6e61   only in combina
│ │ │ │ +00020d30: 7469 6f6e 2077 6974 6820 2a6e 6f74 6520  tion with *note 
│ │ │ │ +00020d40: 436f 6d70 4d65 7468 6f64 3a0a 436f 6d70  CompMethod:.Comp
│ │ │ │ +00020d50: 4d65 7468 6f64 2c3d 3e50 726f 6a65 6374  Method,=>Project
│ │ │ │ +00020d60: 6976 6544 6567 7265 652e 2054 6865 204d  iveDegree. The M
│ │ │ │ +00020d70: 6574 686f 6420 496e 636c 7573 696f 6e45  ethod InclusionE
│ │ │ │ +00020d80: 7863 6c75 7369 6f6e 2077 696c 6c20 616c  xclusion will al
│ │ │ │ +00020d90: 7761 7973 2062 650a 7573 6564 2077 6974  ways be.used wit
│ │ │ │ +00020da0: 6820 2a6e 6f74 6520 436f 6d70 4d65 7468  h *note CompMeth
│ │ │ │ +00020db0: 6f64 3a20 436f 6d70 4d65 7468 6f64 2c20  od: CompMethod, 
│ │ │ │ +00020dc0: 506e 5265 7369 6475 616c 206f 7220 6265  PnResidual or be
│ │ │ │ +00020dd0: 7274 696e 692e 2057 6865 6e20 7468 6520  rtini. When the 
│ │ │ │ +00020de0: 696e 7075 740a 6964 6561 6c20 6973 2061  input.ideal is a
│ │ │ │ +00020df0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ +00020e00: 6563 7469 6f6e 206f 6e65 206d 6179 2c20  ection one may, 
│ │ │ │ +00020e10: 706f 7465 6e74 6961 6c6c 792c 2073 7065  potentially, spe
│ │ │ │ +00020e20: 6564 2075 7020 7468 6520 636f 6d70 7574  ed up the comput
│ │ │ │ +00020e30: 6174 696f 6e0a 6279 2073 6574 7469 6e67  ation.by setting
│ │ │ │ +00020e40: 204d 6574 686f 643d 3e20 4469 7265 6374   Method=> Direct
│ │ │ │ +00020e50: 436f 6d70 6c65 7465 496e 742e 2054 6865  CompleteInt. The
│ │ │ │ +00020e60: 206f 7074 696f 6e20 4d65 7468 6f64 2069   option Method i
│ │ │ │ +00020e70: 7320 6f6e 6c79 2075 7365 6420 6279 2074  s only used by t
│ │ │ │ +00020e80: 6865 0a63 6f6d 6d61 6e64 7320 2a6e 6f74  he.commands *not
│ │ │ │ +00020e90: 6520 4353 4d3a 2043 534d 2c20 616e 6420  e CSM: CSM, and 
│ │ │ │ +00020ea0: 2a6e 6f74 6520 4575 6c65 723a 2045 756c  *note Euler: Eul
│ │ │ │ +00020eb0: 6572 2c20 616e 6420 6f6e 6c79 2069 6e20  er, and only in 
│ │ │ │ +00020ec0: 636f 6d62 696e 6174 696f 6e20 7769 7468  combination with
│ │ │ │ +00020ed0: 0a2a 6e6f 7465 2043 6f6d 704d 6574 686f  .*note CompMetho
│ │ │ │ +00020ee0: 643a 2043 6f6d 704d 6574 686f 642c 3d3e  d: CompMethod,=>
│ │ │ │ +00020ef0: 5072 6f6a 6563 7469 7665 4465 6772 6565  ProjectiveDegree
│ │ │ │ +00020f00: 2e20 5468 6520 4d65 7468 6f64 2049 6e63  . The Method Inc
│ │ │ │ +00020f10: 6c75 7369 6f6e 4578 636c 7573 696f 6e0a  lusionExclusion.
│ │ │ │ +00020f20: 7769 6c6c 2061 6c77 6179 7320 6265 2075  will always be u
│ │ │ │ +00020f30: 7365 6420 7769 7468 202a 6e6f 7465 2043  sed with *note C
│ │ │ │ +00020f40: 6f6d 704d 6574 686f 643a 2043 6f6d 704d  ompMethod: CompM
│ │ │ │ +00020f50: 6574 686f 642c 2050 6e52 6573 6964 7561  ethod, PnResidua
│ │ │ │ +00020f60: 6c20 6f72 2062 6572 7469 6e69 2e0a 0a2b  l or bertini...+
│ │ │ │  00020f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020fa0: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 5a5a  --+.|i1 : R = ZZ
│ │ │ │ -00020fb0: 2f33 3237 3439 5b78 5f30 2e2e 785f 365d  /32749[x_0..x_6]
│ │ │ │ -00020fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020fa0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00020fb0: 5220 3d20 5a5a 2f33 3237 3439 5b78 5f30  R = ZZ/32749[x_0
│ │ │ │ +00020fc0: 2e2e 785f 365d 2020 2020 2020 2020 2020  ..x_6]          
│ │ │ │ +00020fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020fe0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00020ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021010: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -00021020: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00021010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021020: 7c6f 3120 3d20 5220 2020 2020 2020 2020  |o1 = R         
│ │ │ │  00021030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021050: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00021050: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00021060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021080: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021090: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c  |o1 : Polynomial
│ │ │ │ -000210a0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +00021080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021090: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ +000210a0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │  000210b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000210d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000210c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000210d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000210e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021100: 2d2d 2d2d 2b0a 7c69 3220 3a20 493d 6964  ----+.|i2 : I=id
│ │ │ │ -00021110: 6561 6c28 7261 6e64 6f6d 2832 2c52 292c  eal(random(2,R),
│ │ │ │ -00021120: 7261 6e64 6f6d 2831 2c52 292c 525f 302a  random(1,R),R_0*
│ │ │ │ -00021130: 525f 312a 525f 362d 525f 305e 3329 3b7c  R_1*R_6-R_0^3);|
│ │ │ │ -00021140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021100: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00021110: 3a20 493d 6964 6561 6c28 7261 6e64 6f6d  : I=ideal(random
│ │ │ │ +00021120: 2832 2c52 292c 7261 6e64 6f6d 2831 2c52  (2,R),random(1,R
│ │ │ │ +00021130: 292c 525f 302a 525f 312a 525f 362d 525f  ),R_0*R_1*R_6-R_
│ │ │ │ +00021140: 305e 3329 3b7c 0a7c 2020 2020 2020 2020  0^3);|.|        
│ │ │ │  00021150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021170: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ -00021180: 3a20 4964 6561 6c20 6f66 2052 2020 2020  : Ideal of R    
│ │ │ │ -00021190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021180: 7c0a 7c6f 3220 3a20 4964 6561 6c20 6f66  |.|o2 : Ideal of
│ │ │ │ +00021190: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │  000211a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000211b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  000211c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000211d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000211e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211f0: 2b0a 7c69 3320 3a20 7469 6d65 2043 534d  +.|i3 : time CSM
│ │ │ │ -00021200: 2049 2020 2020 2020 2020 2020 2020 2020   I              
│ │ │ │ +000211f0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469  ------+.|i3 : ti
│ │ │ │ +00021200: 6d65 2043 534d 2049 2020 2020 2020 2020  me CSM I        
│ │ │ │  00021210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021220: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00021230: 2d20 7573 6564 2031 2e31 3939 3131 7320  - used 1.19911s 
│ │ │ │ -00021240: 2863 7075 293b 2030 2e38 3737 3230 3973  (cpu); 0.877209s
│ │ │ │ -00021250: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ -00021260: 6329 2020 2020 7c0a 7c20 2020 2020 2020  c)    |.|       
│ │ │ │ +00021220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021230: 207c 0a7c 202d 2d20 7573 6564 2033 2e31   |.| -- used 3.1
│ │ │ │ +00021240: 3839 3532 7320 2863 7075 293b 2031 2e32  8952s (cpu); 1.2
│ │ │ │ +00021250: 3537 3334 7320 2874 6872 6561 6429 3b20  5734s (thread); 
│ │ │ │ +00021260: 3073 2028 6763 2920 2020 2020 7c0a 7c20  0s (gc)     |.| 
│ │ │ │  00021270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212a0: 207c 0a7c 2020 2020 2020 2020 3520 2020   |.|        5   
│ │ │ │ -000212b0: 2020 2034 2020 2020 2033 2020 2020 2020     4     3      
│ │ │ │ +000212a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000212b0: 2020 3520 2020 2020 2034 2020 2020 2033    5      4     3
│ │ │ │  000212c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000212e0: 3320 3d20 3132 6820 202b 2031 3068 2020  3 = 12h  + 10h  
│ │ │ │ -000212f0: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │ +000212d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000212e0: 2020 7c0a 7c6f 3320 3d20 3132 6820 202b    |.|o3 = 12h  +
│ │ │ │ +000212f0: 2031 3068 2020 2b20 3668 2020 2020 2020   10h  + 6h      
│ │ │ │  00021300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021310: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00021320: 2020 3120 2020 2020 2031 2020 2020 2031    1      1     1
│ │ │ │ -00021330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021320: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ +00021330: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │  00021340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021350: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00021350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00021360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021380: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021390: 2020 2020 205a 5a5b 6820 5d20 2020 2020       ZZ[h ]     
│ │ │ │ -000213a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021390: 2020 207c 0a7c 2020 2020 205a 5a5b 6820     |.|     ZZ[h 
│ │ │ │ +000213a0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  000213b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000213c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000213d0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +000213c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000213d0: 7c20 2020 2020 2020 2020 3120 2020 2020  |         1     
│ │ │ │  000213e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000213f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021400: 2020 207c 0a7c 6f33 203a 202d 2d2d 2d2d     |.|o3 : -----
│ │ │ │ -00021410: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +00021400: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +00021410: 202d 2d2d 2d2d 2d20 2020 2020 2020 2020   ------         
│ │ │ │  00021420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021440: 7c20 2020 2020 2020 2037 2020 2020 2020  |        7      
│ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021440: 2020 2020 7c0a 7c20 2020 2020 2020 2037      |.|        7
│ │ │ │  00021450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021470: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00021480: 2020 2068 2020 2020 2020 2020 2020 2020     h            
│ │ │ │ +00021470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021480: 0a7c 2020 2020 2020 2068 2020 2020 2020  .|       h      
│ │ │ │  00021490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214b0: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │ -000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000214b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000214c0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │  000214d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000214f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000214e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000214f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00021500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021520: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ -00021530: 3a20 7469 6d65 2043 534d 2849 2c4d 6574  : time CSM(I,Met
│ │ │ │ -00021540: 686f 643d 3e44 6972 6563 7443 6f6d 706c  hod=>DirectCompl
│ │ │ │ -00021550: 6574 6549 6e74 2920 2020 2020 2020 2020  eteInt)         
│ │ │ │ -00021560: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00021570: 2030 2e33 3130 3536 3273 2028 6370 7529   0.310562s (cpu)
│ │ │ │ -00021580: 3b20 302e 3233 3333 3431 7320 2874 6872  ; 0.233341s (thr
│ │ │ │ -00021590: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -000215a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00021520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021530: 2b0a 7c69 3420 3a20 7469 6d65 2043 534d  +.|i4 : time CSM
│ │ │ │ +00021540: 2849 2c4d 6574 686f 643d 3e44 6972 6563  (I,Method=>Direc
│ │ │ │ +00021550: 7443 6f6d 706c 6574 6549 6e74 2920 2020  tCompleteInt)   
│ │ │ │ +00021560: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00021570: 2d20 7573 6564 2030 2e36 3839 3732 3273  - used 0.689722s
│ │ │ │ +00021580: 2028 6370 7529 3b20 302e 3238 3439 3232   (cpu); 0.284922
│ │ │ │ +00021590: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +000215a0: 6763 2920 2020 7c0a 7c20 2020 2020 2020  gc)   |.|       
│ │ │ │  000215b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000215e0: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -000215f0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +000215d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000215e0: 207c 0a7c 2020 2020 2020 2020 3520 2020   |.|        5   
│ │ │ │ +000215f0: 2020 2034 2020 2020 2033 2020 2020 2020     4     3      
│ │ │ │  00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021610: 2020 2020 2020 7c0a 7c6f 3420 3d20 3132        |.|o4 = 12
│ │ │ │ -00021620: 6820 202b 2031 3068 2020 2b20 3668 2020  h  + 10h  + 6h  
│ │ │ │ -00021630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00021620: 3420 3d20 3132 6820 202b 2031 3068 2020  4 = 12h  + 10h  
│ │ │ │ +00021630: 2b20 3668 2020 2020 2020 2020 2020 2020  + 6h            
│ │ │ │  00021640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021650: 207c 0a7c 2020 2020 2020 2020 3120 2020   |.|        1   
│ │ │ │ -00021660: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ +00021650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00021660: 2020 3120 2020 2020 2031 2020 2020 2031    1      1     1
│ │ │ │  00021670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00021690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021690: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000216a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216c0: 2020 2020 2020 207c 0a7c 2020 2020 205a         |.|     Z
│ │ │ │ -000216d0: 5a5b 6820 5d20 2020 2020 2020 2020 2020  Z[h ]           
│ │ │ │ +000216c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000216d0: 2020 2020 205a 5a5b 6820 5d20 2020 2020       ZZ[h ]     
│ │ │ │  000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021700: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ -00021710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021710: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  00021720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021740: 6f34 203a 202d 2d2d 2d2d 2d20 2020 2020  o4 : ------     
│ │ │ │ -00021750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021740: 2020 207c 0a7c 6f34 203a 202d 2d2d 2d2d     |.|o4 : -----
│ │ │ │ +00021750: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │  00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021770: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00021780: 2020 2037 2020 2020 2020 2020 2020 2020     7            
│ │ │ │ +00021770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021780: 7c20 2020 2020 2020 2037 2020 2020 2020  |        7      
│ │ │ │  00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000217a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217b0: 2020 207c 0a7c 2020 2020 2020 2068 2020     |.|       h  
│ │ │ │ -000217c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000217c0: 2020 2068 2020 2020 2020 2020 2020 2020     h            
│ │ │ │  000217d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000217e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000217f0: 7c20 2020 2020 2020 2031 2020 2020 2020  |        1      
│ │ │ │ +000217e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217f0: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │  00021800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021820: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00021830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021830: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00021840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021860: 2d2d 2d2d 2b0a 0a57 6865 6e20 7573 696e  ----+..When usin
│ │ │ │ -00021870: 6720 7468 6520 4469 7265 6374 436f 6d70  g the DirectComp
│ │ │ │ -00021880: 6c65 7465 496e 7420 6d65 7468 6f64 206f  leteInt method o
│ │ │ │ -00021890: 6e65 206d 6179 2070 6f74 656e 7469 616c  ne may potential
│ │ │ │ -000218a0: 6c79 2066 7572 7468 6572 2073 7065 6564  ly further speed
│ │ │ │ -000218b0: 2075 700a 636f 6d70 7574 6174 696f 6e20   up.computation 
│ │ │ │ -000218c0: 7469 6d65 2062 7920 7370 6563 6966 7969  time by specifyi
│ │ │ │ -000218d0: 6e67 2077 6861 7420 7375 6273 6574 206f  ng what subset o
│ │ │ │ -000218e0: 6620 7468 6520 6765 6e65 7261 746f 7273  f the generators
│ │ │ │ -000218f0: 206f 6620 7468 6520 696e 7075 7420 6964   of the input id
│ │ │ │ -00021900: 6561 6c0a 6465 6669 6e65 2061 2073 6d6f  eal.define a smo
│ │ │ │ -00021910: 6f74 6820 7375 6273 6368 656d 6520 2869  oth subscheme (i
│ │ │ │ -00021920: 6620 7468 6973 2069 7320 6b6e 6f77 6e29  f this is known)
│ │ │ │ -00021930: 2c20 7365 6520 2a6e 6f74 6520 496e 6473  , see *note Inds
│ │ │ │ -00021940: 4f66 536d 6f6f 7468 3a0a 496e 6473 4f66  OfSmooth:.IndsOf
│ │ │ │ -00021950: 536d 6f6f 7468 2c2e 0a0a 4675 6e63 7469  Smooth,...Functi
│ │ │ │ -00021960: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ -00021970: 6c20 6172 6775 6d65 6e74 206e 616d 6564  l argument named
│ │ │ │ -00021980: 204d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d   Method:.=======
│ │ │ │ +00021860: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6865  ----------+..Whe
│ │ │ │ +00021870: 6e20 7573 696e 6720 7468 6520 4469 7265  n using the Dire
│ │ │ │ +00021880: 6374 436f 6d70 6c65 7465 496e 7420 6d65  ctCompleteInt me
│ │ │ │ +00021890: 7468 6f64 206f 6e65 206d 6179 2070 6f74  thod one may pot
│ │ │ │ +000218a0: 656e 7469 616c 6c79 2066 7572 7468 6572  entially further
│ │ │ │ +000218b0: 2073 7065 6564 2075 700a 636f 6d70 7574   speed up.comput
│ │ │ │ +000218c0: 6174 696f 6e20 7469 6d65 2062 7920 7370  ation time by sp
│ │ │ │ +000218d0: 6563 6966 7969 6e67 2077 6861 7420 7375  ecifying what su
│ │ │ │ +000218e0: 6273 6574 206f 6620 7468 6520 6765 6e65  bset of the gene
│ │ │ │ +000218f0: 7261 746f 7273 206f 6620 7468 6520 696e  rators of the in
│ │ │ │ +00021900: 7075 7420 6964 6561 6c0a 6465 6669 6e65  put ideal.define
│ │ │ │ +00021910: 2061 2073 6d6f 6f74 6820 7375 6273 6368   a smooth subsch
│ │ │ │ +00021920: 656d 6520 2869 6620 7468 6973 2069 7320  eme (if this is 
│ │ │ │ +00021930: 6b6e 6f77 6e29 2c20 7365 6520 2a6e 6f74  known), see *not
│ │ │ │ +00021940: 6520 496e 6473 4f66 536d 6f6f 7468 3a0a  e IndsOfSmooth:.
│ │ │ │ +00021950: 496e 6473 4f66 536d 6f6f 7468 2c2e 0a0a  IndsOfSmooth,...
│ │ │ │ +00021960: 4675 6e63 7469 6f6e 7320 7769 7468 206f  Functions with o
│ │ │ │ +00021970: 7074 696f 6e61 6c20 6172 6775 6d65 6e74  ptional argument
│ │ │ │ +00021980: 206e 616d 6564 204d 6574 686f 643a 0a3d   named Method:.=
│ │ │ │  00021990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  000219a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000219b0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4353  =======..  * "CS
│ │ │ │ -000219c0: 4d28 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e  M(...,Method=>..
│ │ │ │ -000219d0: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ -000219e0: 2043 534d 3a20 4353 4d2c 202d 2d20 5468   CSM: CSM, -- Th
│ │ │ │ -000219f0: 650a 2020 2020 4368 6572 6e2d 5363 6877  e.    Chern-Schw
│ │ │ │ -00021a00: 6172 747a 2d4d 6163 5068 6572 736f 6e20  artz-MacPherson 
│ │ │ │ -00021a10: 636c 6173 730a 2020 2a20 4575 6c65 7228  class.  * Euler(
│ │ │ │ -00021a20: 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e 2e29  ...,Method=>...)
│ │ │ │ -00021a30: 2028 6d69 7373 696e 6720 646f 6375 6d65   (missing docume
│ │ │ │ -00021a40: 6e74 6174 696f 6e29 0a0a 466f 7220 7468  ntation)..For th
│ │ │ │ -00021a50: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00021a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00021a70: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00021a80: 6520 4d65 7468 6f64 3a20 4d65 7468 6f64  e Method: Method
│ │ │ │ -00021a90: 2c20 6973 2061 202a 6e6f 7465 2073 796d  , is a *note sym
│ │ │ │ -00021aa0: 626f 6c3a 2028 4d61 6361 756c 6179 3244  bol: (Macaulay2D
│ │ │ │ -00021ab0: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d  oc)Symbol,...---
│ │ │ │ -00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000219b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +000219c0: 202a 2022 4353 4d28 2e2e 2e2c 4d65 7468   * "CSM(...,Meth
│ │ │ │ +000219d0: 6f64 3d3e 2e2e 2e29 2220 2d2d 2073 6565  od=>...)" -- see
│ │ │ │ +000219e0: 202a 6e6f 7465 2043 534d 3a20 4353 4d2c   *note CSM: CSM,
│ │ │ │ +000219f0: 202d 2d20 5468 650a 2020 2020 4368 6572   -- The.    Cher
│ │ │ │ +00021a00: 6e2d 5363 6877 6172 747a 2d4d 6163 5068  n-Schwartz-MacPh
│ │ │ │ +00021a10: 6572 736f 6e20 636c 6173 730a 2020 2a20  erson class.  * 
│ │ │ │ +00021a20: 4575 6c65 7228 2e2e 2e2c 4d65 7468 6f64  Euler(...,Method
│ │ │ │ +00021a30: 3d3e 2e2e 2e29 2028 6d69 7373 696e 6720  =>...) (missing 
│ │ │ │ +00021a40: 646f 6375 6d65 6e74 6174 696f 6e29 0a0a  documentation)..
│ │ │ │ +00021a50: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +00021a60: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +00021a70: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +00021a80: 7420 2a6e 6f74 6520 4d65 7468 6f64 3a20  t *note Method: 
│ │ │ │ +00021a90: 4d65 7468 6f64 2c20 6973 2061 202a 6e6f  Method, is a *no
│ │ │ │ +00021aa0: 7465 2073 796d 626f 6c3a 2028 4d61 6361  te symbol: (Maca
│ │ │ │ +00021ab0: 756c 6179 3244 6f63 2953 796d 626f 6c2c  ulay2Doc)Symbol,
│ │ │ │ +00021ac0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │  00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ -00021b10: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ -00021b20: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ -00021b30: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ -00021b40: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ -00021b50: 7932 2d31 2e32 362e 3036 2b64 732f 4d32  y2-1.26.06+ds/M2
│ │ │ │ -00021b60: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ -00021b70: 6765 732f 0a43 6861 7261 6374 6572 6973  ges/.Characteris
│ │ │ │ -00021b80: 7469 6343 6c61 7373 6573 2e6d 323a 3234  ticClasses.m2:24
│ │ │ │ -00021b90: 3332 3a30 2e0a 1f0a 4669 6c65 3a20 4368  32:0....File: Ch
│ │ │ │ -00021ba0: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ -00021bb0: 7365 732e 696e 666f 2c20 4e6f 6465 3a20  ses.info, Node: 
│ │ │ │ -00021bc0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269  MultiProjCoordRi
│ │ │ │ -00021bd0: 6e67 2c20 4e65 7874 3a20 4f75 7470 7574  ng, Next: Output
│ │ │ │ -00021be0: 2c20 5072 6576 3a20 4d65 7468 6f64 2c20  , Prev: Method, 
│ │ │ │ -00021bf0: 5570 3a20 546f 700a 0a4d 756c 7469 5072  Up: Top..MultiPr
│ │ │ │ -00021c00: 6f6a 436f 6f72 6452 696e 6720 2d2d 2041  ojCoordRing -- A
│ │ │ │ -00021c10: 2071 7569 636b 2077 6179 2074 6f20 6275   quick way to bu
│ │ │ │ -00021c20: 696c 6420 7468 6520 636f 6f72 6469 6e61  ild the coordina
│ │ │ │ -00021c30: 7465 2072 696e 6720 6f66 2061 2070 726f  te ring of a pro
│ │ │ │ -00021c40: 6475 6374 206f 6620 7072 6f6a 6563 7469  duct of projecti
│ │ │ │ -00021c50: 7665 2073 7061 6365 730a 2a2a 2a2a 2a2a  ve spaces.******
│ │ │ │ +00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021b10: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +00021b20: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +00021b30: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +00021b40: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +00021b50: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ +00021b60: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +00021b70: 2f70 6163 6b61 6765 732f 0a43 6861 7261  /packages/.Chara
│ │ │ │ +00021b80: 6374 6572 6973 7469 6343 6c61 7373 6573  cteristicClasses
│ │ │ │ +00021b90: 2e6d 323a 3234 3332 3a30 2e0a 1f0a 4669  .m2:2432:0....Fi
│ │ │ │ +00021ba0: 6c65 3a20 4368 6172 6163 7465 7269 7374  le: Characterist
│ │ │ │ +00021bb0: 6963 436c 6173 7365 732e 696e 666f 2c20  icClasses.info, 
│ │ │ │ +00021bc0: 4e6f 6465 3a20 4d75 6c74 6950 726f 6a43  Node: MultiProjC
│ │ │ │ +00021bd0: 6f6f 7264 5269 6e67 2c20 4e65 7874 3a20  oordRing, Next: 
│ │ │ │ +00021be0: 4f75 7470 7574 2c20 5072 6576 3a20 4d65  Output, Prev: Me
│ │ │ │ +00021bf0: 7468 6f64 2c20 5570 3a20 546f 700a 0a4d  thod, Up: Top..M
│ │ │ │ +00021c00: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ +00021c10: 6720 2d2d 2041 2071 7569 636b 2077 6179  g -- A quick way
│ │ │ │ +00021c20: 2074 6f20 6275 696c 6420 7468 6520 636f   to build the co
│ │ │ │ +00021c30: 6f72 6469 6e61 7465 2072 696e 6720 6f66  ordinate ring of
│ │ │ │ +00021c40: 2061 2070 726f 6475 6374 206f 6620 7072   a product of pr
│ │ │ │ +00021c50: 6f6a 6563 7469 7665 2073 7061 6365 730a  ojective spaces.
│ │ │ │  00021c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00021cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -00021cc0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00021cd0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269  MultiProjCoordRi
│ │ │ │ -00021ce0: 6e67 2044 696d 730a 2020 2020 2020 2020  ng Dims.        
│ │ │ │ -00021cf0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269  MultiProjCoordRi
│ │ │ │ -00021d00: 6e67 2028 436f 6566 6652 696e 672c 4469  ng (CoeffRing,Di
│ │ │ │ -00021d10: 6d73 290a 2020 2020 2020 2020 4d75 6c74  ms).        Mult
│ │ │ │ -00021d20: 6950 726f 6a43 6f6f 7264 5269 6e67 2028  iProjCoordRing (
│ │ │ │ -00021d30: 7661 722c 4469 6d73 290a 2020 2020 2020  var,Dims).      
│ │ │ │ -00021d40: 2020 4d75 6c74 6950 726f 6a43 6f6f 7264    MultiProjCoord
│ │ │ │ -00021d50: 5269 6e67 2028 436f 6566 6652 696e 672c  Ring (CoeffRing,
│ │ │ │ -00021d60: 7661 722c 4469 6d73 290a 2020 2a20 496e  var,Dims).  * In
│ │ │ │ -00021d70: 7075 7473 3a0a 2020 2020 2020 2a20 4469  puts:.      * Di
│ │ │ │ -00021d80: 6d73 2c20 6120 2a6e 6f74 6520 6c69 7374  ms, a *note list
│ │ │ │ -00021d90: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00021da0: 4c69 7374 2c2c 2072 6570 7265 7365 6e74  List,, represent
│ │ │ │ -00021db0: 696e 6720 7468 6520 6469 6d65 6e73 696f  ing the dimensio
│ │ │ │ -00021dc0: 6e73 206f 660a 2020 2020 2020 2020 7468  ns of.        th
│ │ │ │ -00021dd0: 6520 7072 6f6a 6563 7469 7665 2073 7061  e projective spa
│ │ │ │ -00021de0: 6365 732c 2069 2e65 2e20 7b6e 5f31 2c2e  ces, i.e. {n_1,.
│ │ │ │ -00021df0: 2e2e 2c6e 5f6d 7d20 636f 7272 6573 706f  ..,n_m} correspo
│ │ │ │ -00021e00: 6e64 7320 746f 205c 5050 5e7b 6e5f 317d  nds to \PP^{n_1}
│ │ │ │ -00021e10: 0a20 2020 2020 2020 2078 2e2e 2e2e 2078  .        x.... x
│ │ │ │ -00021e20: 205c 5050 5e7b 6e5f 6d7d 0a20 2020 2020   \PP^{n_m}.     
│ │ │ │ -00021e30: 202a 2043 6f65 6666 5269 6e67 2c20 6120   * CoeffRing, a 
│ │ │ │ -00021e40: 2a6e 6f74 6520 7269 6e67 3a20 284d 6163  *note ring: (Mac
│ │ │ │ -00021e50: 6175 6c61 7932 446f 6329 5269 6e67 2c2c  aulay2Doc)Ring,,
│ │ │ │ -00021e60: 2074 6865 2063 6f65 6666 6963 6965 6e74   the coefficient
│ │ │ │ -00021e70: 2072 696e 6720 6f66 0a20 2020 2020 2020   ring of.       
│ │ │ │ -00021e80: 2074 6865 2067 7261 6465 6420 706f 6c79   the graded poly
│ │ │ │ -00021e90: 6e6f 6d69 616c 2072 696e 6720 746f 2062  nomial ring to b
│ │ │ │ -00021ea0: 6520 6275 696c 7420 6279 2074 6865 206d  e built by the m
│ │ │ │ -00021eb0: 6574 686f 642c 2062 7920 6465 6661 756c  ethod, by defaul
│ │ │ │ -00021ec0: 7420 7468 6973 0a20 2020 2020 2020 2069  t this.        i
│ │ │ │ -00021ed0: 7320 5c5a 5a2f 3332 3734 390a 2020 2020  s \ZZ/32749.    
│ │ │ │ -00021ee0: 2020 2a20 7661 722c 2061 202a 6e6f 7465    * var, a *note
│ │ │ │ -00021ef0: 2073 796d 626f 6c3a 2028 4d61 6361 756c   symbol: (Macaul
│ │ │ │ -00021f00: 6179 3244 6f63 2953 796d 626f 6c2c 2c20  ay2Doc)Symbol,, 
│ │ │ │ -00021f10: 746f 2062 6520 7573 6564 2066 6f72 2074  to be used for t
│ │ │ │ -00021f20: 6865 0a20 2020 2020 2020 2069 6e74 6572  he.        inter
│ │ │ │ -00021f30: 6d65 6469 6174 6573 206f 6620 7468 6520  mediates of the 
│ │ │ │ -00021f40: 6772 6164 6564 2070 6f6c 796e 6f6d 6961  graded polynomia
│ │ │ │ -00021f50: 6c20 7269 6e67 2074 6f20 6265 2062 7569  l ring to be bui
│ │ │ │ -00021f60: 6c74 2062 7920 7468 6520 6d65 7468 6f64  lt by the method
│ │ │ │ -00021f70: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00021f80: 2020 2020 2a20 6120 2a6e 6f74 6520 7269      * a *note ri
│ │ │ │ -00021f90: 6e67 3a20 284d 6163 6175 6c61 7932 446f  ng: (Macaulay2Do
│ │ │ │ -00021fa0: 6329 5269 6e67 2c2c 2074 6865 2067 7261  c)Ring,, the gra
│ │ │ │ -00021fb0: 6465 6420 636f 6f72 6469 6e61 7465 2072  ded coordinate r
│ │ │ │ -00021fc0: 696e 6720 6f66 2074 6865 0a20 2020 2020  ing of the.     
│ │ │ │ -00021fd0: 2020 205c 5050 5e7b 6e5f 317d 2078 2e2e     \PP^{n_1} x..
│ │ │ │ -00021fe0: 2e2e 2078 205c 5050 5e7b 6e5f 6d7d 2077  .. x \PP^{n_m} w
│ │ │ │ -00021ff0: 6865 7265 207b 6e5f 312c 2e2e 2e2c 6e5f  here {n_1,...,n_
│ │ │ │ -00022000: 6d7d 2069 7320 7468 6520 696e 7075 7420  m} is the input 
│ │ │ │ -00022010: 6c69 7374 206f 660a 2020 2020 2020 2020  list of.        
│ │ │ │ -00022020: 6469 6d65 6e73 696f 6e73 0a0a 4465 7363  dimensions..Desc
│ │ │ │ -00022030: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -00022040: 3d3d 3d0a 0a43 6f6d 7075 7465 7320 7468  ===..Computes th
│ │ │ │ -00022050: 6520 6772 6164 6564 2063 6f6f 7264 696e  e graded coordin
│ │ │ │ -00022060: 6174 6520 7269 6e67 206f 6620 7468 6520  ate ring of the 
│ │ │ │ -00022070: 5c50 505e 7b6e 5f31 7d20 782e 2e2e 2e20  \PP^{n_1} x.... 
│ │ │ │ -00022080: 7820 5c50 505e 7b6e 5f6d 7d20 7768 6572  x \PP^{n_m} wher
│ │ │ │ -00022090: 650a 7b6e 5f31 2c2e 2e2e 2c6e 5f6d 7d20  e.{n_1,...,n_m} 
│ │ │ │ -000220a0: 6973 2074 6865 2069 6e70 7574 206c 6973  is the input lis
│ │ │ │ -000220b0: 7420 6f66 2064 696d 656e 7369 6f6e 732e  t of dimensions.
│ │ │ │ -000220c0: 2054 6869 7320 6d65 7468 6f64 2069 7320   This method is 
│ │ │ │ -000220d0: 7573 6564 2074 6f20 7175 6963 6b6c 790a  used to quickly.
│ │ │ │ -000220e0: 6275 696c 6420 7468 6520 636f 6f72 6469  build the coordi
│ │ │ │ -000220f0: 6e61 7465 2072 696e 6720 6f66 2061 2070  nate ring of a p
│ │ │ │ -00022100: 726f 6475 6374 206f 6620 7072 6f6a 6563  roduct of projec
│ │ │ │ -00022110: 7469 7665 2073 7061 6365 7320 666f 7220  tive spaces for 
│ │ │ │ -00022120: 7573 6520 696e 0a63 6f6d 7075 7461 7469  use in.computati
│ │ │ │ -00022130: 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ons...+---------
│ │ │ │ +00021cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00021cc0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ +00021cd0: 2020 2020 2020 4d75 6c74 6950 726f 6a43        MultiProjC
│ │ │ │ +00021ce0: 6f6f 7264 5269 6e67 2044 696d 730a 2020  oordRing Dims.  
│ │ │ │ +00021cf0: 2020 2020 2020 4d75 6c74 6950 726f 6a43        MultiProjC
│ │ │ │ +00021d00: 6f6f 7264 5269 6e67 2028 436f 6566 6652  oordRing (CoeffR
│ │ │ │ +00021d10: 696e 672c 4469 6d73 290a 2020 2020 2020  ing,Dims).      
│ │ │ │ +00021d20: 2020 4d75 6c74 6950 726f 6a43 6f6f 7264    MultiProjCoord
│ │ │ │ +00021d30: 5269 6e67 2028 7661 722c 4469 6d73 290a  Ring (var,Dims).
│ │ │ │ +00021d40: 2020 2020 2020 2020 4d75 6c74 6950 726f          MultiPro
│ │ │ │ +00021d50: 6a43 6f6f 7264 5269 6e67 2028 436f 6566  jCoordRing (Coef
│ │ │ │ +00021d60: 6652 696e 672c 7661 722c 4469 6d73 290a  fRing,var,Dims).
│ │ │ │ +00021d70: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00021d80: 2020 2a20 4469 6d73 2c20 6120 2a6e 6f74    * Dims, a *not
│ │ │ │ +00021d90: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +00021da0: 7932 446f 6329 4c69 7374 2c2c 2072 6570  y2Doc)List,, rep
│ │ │ │ +00021db0: 7265 7365 6e74 696e 6720 7468 6520 6469  resenting the di
│ │ │ │ +00021dc0: 6d65 6e73 696f 6e73 206f 660a 2020 2020  mensions of.    
│ │ │ │ +00021dd0: 2020 2020 7468 6520 7072 6f6a 6563 7469      the projecti
│ │ │ │ +00021de0: 7665 2073 7061 6365 732c 2069 2e65 2e20  ve spaces, i.e. 
│ │ │ │ +00021df0: 7b6e 5f31 2c2e 2e2e 2c6e 5f6d 7d20 636f  {n_1,...,n_m} co
│ │ │ │ +00021e00: 7272 6573 706f 6e64 7320 746f 205c 5050  rresponds to \PP
│ │ │ │ +00021e10: 5e7b 6e5f 317d 0a20 2020 2020 2020 2078  ^{n_1}.        x
│ │ │ │ +00021e20: 2e2e 2e2e 2078 205c 5050 5e7b 6e5f 6d7d  .... x \PP^{n_m}
│ │ │ │ +00021e30: 0a20 2020 2020 202a 2043 6f65 6666 5269  .      * CoeffRi
│ │ │ │ +00021e40: 6e67 2c20 6120 2a6e 6f74 6520 7269 6e67  ng, a *note ring
│ │ │ │ +00021e50: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00021e60: 5269 6e67 2c2c 2074 6865 2063 6f65 6666  Ring,, the coeff
│ │ │ │ +00021e70: 6963 6965 6e74 2072 696e 6720 6f66 0a20  icient ring of. 
│ │ │ │ +00021e80: 2020 2020 2020 2074 6865 2067 7261 6465         the grade
│ │ │ │ +00021e90: 6420 706f 6c79 6e6f 6d69 616c 2072 696e  d polynomial rin
│ │ │ │ +00021ea0: 6720 746f 2062 6520 6275 696c 7420 6279  g to be built by
│ │ │ │ +00021eb0: 2074 6865 206d 6574 686f 642c 2062 7920   the method, by 
│ │ │ │ +00021ec0: 6465 6661 756c 7420 7468 6973 0a20 2020  default this.   
│ │ │ │ +00021ed0: 2020 2020 2069 7320 5c5a 5a2f 3332 3734       is \ZZ/3274
│ │ │ │ +00021ee0: 390a 2020 2020 2020 2a20 7661 722c 2061  9.      * var, a
│ │ │ │ +00021ef0: 202a 6e6f 7465 2073 796d 626f 6c3a 2028   *note symbol: (
│ │ │ │ +00021f00: 4d61 6361 756c 6179 3244 6f63 2953 796d  Macaulay2Doc)Sym
│ │ │ │ +00021f10: 626f 6c2c 2c20 746f 2062 6520 7573 6564  bol,, to be used
│ │ │ │ +00021f20: 2066 6f72 2074 6865 0a20 2020 2020 2020   for the.       
│ │ │ │ +00021f30: 2069 6e74 6572 6d65 6469 6174 6573 206f   intermediates o
│ │ │ │ +00021f40: 6620 7468 6520 6772 6164 6564 2070 6f6c  f the graded pol
│ │ │ │ +00021f50: 796e 6f6d 6961 6c20 7269 6e67 2074 6f20  ynomial ring to 
│ │ │ │ +00021f60: 6265 2062 7569 6c74 2062 7920 7468 6520  be built by the 
│ │ │ │ +00021f70: 6d65 7468 6f64 0a20 202a 204f 7574 7075  method.  * Outpu
│ │ │ │ +00021f80: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e  ts:.      * a *n
│ │ │ │ +00021f90: 6f74 6520 7269 6e67 3a20 284d 6163 6175  ote ring: (Macau
│ │ │ │ +00021fa0: 6c61 7932 446f 6329 5269 6e67 2c2c 2074  lay2Doc)Ring,, t
│ │ │ │ +00021fb0: 6865 2067 7261 6465 6420 636f 6f72 6469  he graded coordi
│ │ │ │ +00021fc0: 6e61 7465 2072 696e 6720 6f66 2074 6865  nate ring of the
│ │ │ │ +00021fd0: 0a20 2020 2020 2020 205c 5050 5e7b 6e5f  .        \PP^{n_
│ │ │ │ +00021fe0: 317d 2078 2e2e 2e2e 2078 205c 5050 5e7b  1} x.... x \PP^{
│ │ │ │ +00021ff0: 6e5f 6d7d 2077 6865 7265 207b 6e5f 312c  n_m} where {n_1,
│ │ │ │ +00022000: 2e2e 2e2c 6e5f 6d7d 2069 7320 7468 6520  ...,n_m} is the 
│ │ │ │ +00022010: 696e 7075 7420 6c69 7374 206f 660a 2020  input list of.  
│ │ │ │ +00022020: 2020 2020 2020 6469 6d65 6e73 696f 6e73        dimensions
│ │ │ │ +00022030: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00022040: 3d3d 3d3d 3d3d 3d3d 3d0a 0a43 6f6d 7075  =========..Compu
│ │ │ │ +00022050: 7465 7320 7468 6520 6772 6164 6564 2063  tes the graded c
│ │ │ │ +00022060: 6f6f 7264 696e 6174 6520 7269 6e67 206f  oordinate ring o
│ │ │ │ +00022070: 6620 7468 6520 5c50 505e 7b6e 5f31 7d20  f the \PP^{n_1} 
│ │ │ │ +00022080: 782e 2e2e 2e20 7820 5c50 505e 7b6e 5f6d  x.... x \PP^{n_m
│ │ │ │ +00022090: 7d20 7768 6572 650a 7b6e 5f31 2c2e 2e2e  } where.{n_1,...
│ │ │ │ +000220a0: 2c6e 5f6d 7d20 6973 2074 6865 2069 6e70  ,n_m} is the inp
│ │ │ │ +000220b0: 7574 206c 6973 7420 6f66 2064 696d 656e  ut list of dimen
│ │ │ │ +000220c0: 7369 6f6e 732e 2054 6869 7320 6d65 7468  sions. This meth
│ │ │ │ +000220d0: 6f64 2069 7320 7573 6564 2074 6f20 7175  od is used to qu
│ │ │ │ +000220e0: 6963 6b6c 790a 6275 696c 6420 7468 6520  ickly.build the 
│ │ │ │ +000220f0: 636f 6f72 6469 6e61 7465 2072 696e 6720  coordinate ring 
│ │ │ │ +00022100: 6f66 2061 2070 726f 6475 6374 206f 6620  of a product of 
│ │ │ │ +00022110: 7072 6f6a 6563 7469 7665 2073 7061 6365  projective space
│ │ │ │ +00022120: 7320 666f 7220 7573 6520 696e 0a63 6f6d  s for use in.com
│ │ │ │ +00022130: 7075 7461 7469 6f6e 732e 0a0a 2b2d 2d2d  putations...+---
│ │ │ │  00022140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022180: 2d2d 2d2d 2b0a 7c69 3120 3a20 533d 4d75  ----+.|i1 : S=Mu
│ │ │ │ -00022190: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ -000221a0: 2851 512c 7379 6d62 6f6c 207a 2c7b 312c  (QQ,symbol z,{1,
│ │ │ │ -000221b0: 332c 337d 2920 2020 2020 2020 2020 2020  3,3})           
│ │ │ │ +00022180: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00022190: 3a20 533d 4d75 6c74 6950 726f 6a43 6f6f  : S=MultiProjCoo
│ │ │ │ +000221a0: 7264 5269 6e67 2851 512c 7379 6d62 6f6c  rdRing(QQ,symbol
│ │ │ │ +000221b0: 207a 2c7b 312c 332c 337d 2920 2020 2020   z,{1,3,3})     
│ │ │ │  000221c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000221d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000221d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000221e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000221f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022220: 2020 2020 7c0a 7c6f 3120 3d20 5320 2020      |.|o1 = S   
│ │ │ │ -00022230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022220: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00022230: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │  00022240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022270: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022270: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222c0: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ -000222d0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +000222c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +000222d0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │  000222e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000222f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022310: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00022310: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022360: 2d2d 2d2d 2b0a 7c69 3220 3a20 6465 6772  ----+.|i2 : degr
│ │ │ │ -00022370: 6565 7320 5320 2020 2020 2020 2020 2020  ees S           
│ │ │ │ +00022360: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00022370: 3a20 6465 6772 6565 7320 5320 2020 2020  : degrees S     
│ │ │ │  00022380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000223b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000223c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000223f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022400: 2020 2020 7c0a 7c6f 3220 3d20 7b7b 312c      |.|o2 = {{1,
│ │ │ │ -00022410: 2030 2c20 307d 2c20 7b31 2c20 302c 2030   0, 0}, {1, 0, 0
│ │ │ │ -00022420: 7d2c 207b 302c 2031 2c20 307d 2c20 7b30  }, {0, 1, 0}, {0
│ │ │ │ -00022430: 2c20 312c 2030 7d2c 207b 302c 2031 2c20  , 1, 0}, {0, 1, 
│ │ │ │ -00022440: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b  0}, {0, 1, 0}, {
│ │ │ │ -00022450: 302c 2020 7c0a 7c20 2020 2020 2d2d 2d2d  0,  |.|     ----
│ │ │ │ -00022460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022400: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00022410: 3d20 7b7b 312c 2030 2c20 307d 2c20 7b31  = {{1, 0, 0}, {1
│ │ │ │ +00022420: 2c20 302c 2030 7d2c 207b 302c 2031 2c20  , 0, 0}, {0, 1, 
│ │ │ │ +00022430: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b  0}, {0, 1, 0}, {
│ │ │ │ +00022440: 302c 2031 2c20 307d 2c20 7b30 2c20 312c  0, 1, 0}, {0, 1,
│ │ │ │ +00022450: 2030 7d2c 207b 302c 2020 7c0a 7c20 2020   0}, {0,  |.|   
│ │ │ │ +00022460: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d    --------------
│ │ │ │  00022470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000224a0: 2d2d 2d2d 7c0a 7c20 2020 2020 302c 2031  ----|.|     0, 1
│ │ │ │ -000224b0: 7d2c 207b 302c 2030 2c20 317d 2c20 7b30  }, {0, 0, 1}, {0
│ │ │ │ -000224c0: 2c20 302c 2031 7d2c 207b 302c 2030 2c20  , 0, 1}, {0, 0, 
│ │ │ │ -000224d0: 317d 7d20 2020 2020 2020 2020 2020 2020  1}}             
│ │ │ │ +000224a0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020  ----------|.|   
│ │ │ │ +000224b0: 2020 302c 2031 7d2c 207b 302c 2030 2c20    0, 1}, {0, 0, 
│ │ │ │ +000224c0: 317d 2c20 7b30 2c20 302c 2031 7d2c 207b  1}, {0, 0, 1}, {
│ │ │ │ +000224d0: 302c 2030 2c20 317d 7d20 2020 2020 2020  0, 0, 1}}       
│ │ │ │  000224e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000224f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022540: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374      |.|o2 : List
│ │ │ │ -00022550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022540: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00022550: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │  00022560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022590: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00022590: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  000225a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000225b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000225c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000225d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000225e0: 2d2d 2d2d 2b0a 7c69 3320 3a20 523d 4d75  ----+.|i3 : R=Mu
│ │ │ │ -000225f0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67  ltiProjCoordRing
│ │ │ │ -00022600: 207b 322c 337d 2020 2020 2020 2020 2020   {2,3}          
│ │ │ │ +000225e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ +000225f0: 3a20 523d 4d75 6c74 6950 726f 6a43 6f6f  : R=MultiProjCoo
│ │ │ │ +00022600: 7264 5269 6e67 207b 322c 337d 2020 2020  rdRing {2,3}    
│ │ │ │  00022610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022630: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022630: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022680: 2020 2020 7c0a 7c6f 3320 3d20 5220 2020      |.|o3 = R   
│ │ │ │ -00022690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022680: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +00022690: 3d20 5220 2020 2020 2020 2020 2020 2020  = R             
│ │ │ │  000226a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000226d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000226e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000226f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022720: 2020 2020 7c0a 7c6f 3320 3a20 506f 6c79      |.|o3 : Poly
│ │ │ │ -00022730: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +00022720: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ +00022730: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │  00022740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022770: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00022770: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00022780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000227a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000227b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000227c0: 2d2d 2d2d 2b0a 7c69 3420 3a20 636f 6566  ----+.|i4 : coef
│ │ │ │ -000227d0: 6669 6369 656e 7452 696e 6720 5220 2020  ficientRing R   
│ │ │ │ -000227e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000227c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +000227d0: 3a20 636f 6566 6669 6369 656e 7452 696e  : coefficientRin
│ │ │ │ +000227e0: 6720 5220 2020 2020 2020 2020 2020 2020  g R             
│ │ │ │  000227f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022810: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022810: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022860: 2020 2020 7c0a 7c20 2020 2020 2020 5a5a      |.|       ZZ
│ │ │ │ -00022870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022860: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022870: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │  00022880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228b0: 2020 2020 7c0a 7c6f 3420 3d20 2d2d 2d2d      |.|o4 = ----
│ │ │ │ -000228c0: 2d20 2020 2020 2020 2020 2020 2020 2020  -               
│ │ │ │ +000228b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +000228c0: 3d20 2d2d 2d2d 2d20 2020 2020 2020 2020  = -----         
│ │ │ │  000228d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022900: 2020 2020 7c0a 7c20 2020 2020 3332 3734      |.|     3274
│ │ │ │ -00022910: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ +00022900: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022910: 2020 3332 3734 3920 2020 2020 2020 2020    32749         
│ │ │ │  00022920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022950: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022950: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229a0: 2020 2020 7c0a 7c6f 3420 3a20 5175 6f74      |.|o4 : Quot
│ │ │ │ -000229b0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +000229a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +000229b0: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │  000229c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000229f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00022a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a40: 2d2d 2d2d 2b0a 7c69 3520 3a20 6465 7363  ----+.|i5 : desc
│ │ │ │ -00022a50: 7269 6265 2052 2020 2020 2020 2020 2020  ribe R          
│ │ │ │ +00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +00022a50: 3a20 6465 7363 7269 6265 2052 2020 2020  : describe R    
│ │ │ │  00022a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022a90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ae0: 2020 2020 7c0a 7c20 2020 2020 2020 5a5a      |.|       ZZ
│ │ │ │ -00022af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022af0: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │  00022b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b30: 2020 2020 7c0a 7c6f 3520 3d20 2d2d 2d2d      |.|o5 = ----
│ │ │ │ -00022b40: 2d5b 7820 2e2e 7820 2c20 4465 6772 6565  -[x ..x , Degree
│ │ │ │ -00022b50: 7320 3d3e 207b 333a 7b31 7d2c 2034 3a7b  s => {3:{1}, 4:{
│ │ │ │ -00022b60: 307d 7d2c 2048 6566 7420 3d3e 207b 323a  0}}, Heft => {2:
│ │ │ │ -00022b70: 317d 5d20 2020 2020 2020 2020 2020 2020  1}]             
│ │ │ │ -00022b80: 2020 2020 7c0a 7c20 2020 2020 3332 3734      |.|     3274
│ │ │ │ -00022b90: 3920 2030 2020 2036 2020 2020 2020 2020  9  0   6        
│ │ │ │ -00022ba0: 2020 2020 2020 2020 7b30 7d20 2020 207b          {0}    {
│ │ │ │ -00022bb0: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │ +00022b30: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00022b40: 3d20 2d2d 2d2d 2d5b 7820 2e2e 7820 2c20  = -----[x ..x , 
│ │ │ │ +00022b50: 4465 6772 6565 7320 3d3e 207b 333a 7b31  Degrees => {3:{1
│ │ │ │ +00022b60: 7d2c 2034 3a7b 307d 7d2c 2048 6566 7420  }, 4:{0}}, Heft 
│ │ │ │ +00022b70: 3d3e 207b 323a 317d 5d20 2020 2020 2020  => {2:1}]       
│ │ │ │ +00022b80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022b90: 2020 3332 3734 3920 2030 2020 2036 2020    32749  0   6  
│ │ │ │ +00022ba0: 2020 2020 2020 2020 2020 2020 2020 7b30                {0
│ │ │ │ +00022bb0: 7d20 2020 207b 317d 2020 2020 2020 2020  }    {1}        
│ │ │ │  00022bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022bd0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00022bd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00022be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022c20: 2d2d 2d2d 2b0a 7c69 3620 3a20 413d 4368  ----+.|i6 : A=Ch
│ │ │ │ -00022c30: 6f77 5269 6e67 2052 2020 2020 2020 2020  owRing R        
│ │ │ │ +00022c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ +00022c30: 3a20 413d 4368 6f77 5269 6e67 2052 2020  : A=ChowRing R  
│ │ │ │  00022c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022c70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cc0: 2020 2020 7c0a 7c6f 3620 3d20 4120 2020      |.|o6 = A   
│ │ │ │ -00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +00022cd0: 3d20 4120 2020 2020 2020 2020 2020 2020  = A             
│ │ │ │  00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022d10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d60: 2020 2020 7c0a 7c6f 3620 3a20 5175 6f74      |.|o6 : Quot
│ │ │ │ -00022d70: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +00022d60: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +00022d70: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │  00022d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022db0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00022db0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00022dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022e00: 2d2d 2d2d 2b0a 7c69 3720 3a20 6465 7363  ----+.|i7 : desc
│ │ │ │ -00022e10: 7269 6265 2041 2020 2020 2020 2020 2020  ribe A          
│ │ │ │ +00022e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +00022e10: 3a20 6465 7363 7269 6265 2041 2020 2020  : describe A    
│ │ │ │  00022e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022e50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ea0: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b68      |.|     ZZ[h
│ │ │ │ -00022eb0: 202e 2e68 205d 2020 2020 2020 2020 2020   ..h ]          
│ │ │ │ +00022ea0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022eb0: 2020 5a5a 5b68 202e 2e68 205d 2020 2020    ZZ[h ..h ]    
│ │ │ │  00022ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ef0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00022f00: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │ +00022ef0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022f00: 2020 2020 2020 3120 2020 3220 2020 2020        1   2     
│ │ │ │  00022f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f40: 2020 2020 7c0a 7c6f 3720 3d20 2d2d 2d2d      |.|o7 = ----
│ │ │ │ -00022f50: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ +00022f40: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +00022f50: 3d20 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020  = ----------    
│ │ │ │  00022f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f90: 2020 2020 7c0a 7c20 2020 2020 2020 2033      |.|        3
│ │ │ │ -00022fa0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00022f90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022fa0: 2020 2020 2033 2020 2034 2020 2020 2020       3   4      
│ │ │ │  00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fe0: 2020 2020 7c0a 7c20 2020 2020 2028 6820      |.|      (h 
│ │ │ │ -00022ff0: 2c20 6820 2920 2020 2020 2020 2020 2020  , h )           
│ │ │ │ +00022fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022ff0: 2020 2028 6820 2c20 6820 2920 2020 2020     (h , h )     
│ │ │ │  00023000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023030: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │ -00023040: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00023030: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023040: 2020 2020 2031 2020 2032 2020 2020 2020       1   2      
│ │ │ │  00023050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023080: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023080: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00023090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000230a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000230b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000230c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000230d0: 2d2d 2d2d 2b0a 7c69 3820 3a20 5365 6772  ----+.|i8 : Segr
│ │ │ │ -000230e0: 6528 412c 6964 6561 6c20 7261 6e64 6f6d  e(A,ideal random
│ │ │ │ -000230f0: 287b 312c 317d 2c52 2929 2020 2020 2020  ({1,1},R))      
│ │ │ │ +000230d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +000230e0: 3a20 5365 6772 6528 412c 6964 6561 6c20  : Segre(A,ideal 
│ │ │ │ +000230f0: 7261 6e64 6f6d 287b 312c 317d 2c52 2929  random({1,1},R))
│ │ │ │  00023100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023120: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023120: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023170: 2020 2020 7c0a 7c20 2020 2020 2020 2032      |.|        2
│ │ │ │ -00023180: 2033 2020 2020 2032 2032 2020 2020 2020   3     2 2      
│ │ │ │ -00023190: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │ -000231a0: 2032 2020 2020 3320 2020 2032 2020 2020   2    3    2    
│ │ │ │ -000231b0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -000231c0: 2020 2020 7c0a 7c6f 3820 3d20 3130 6820      |.|o8 = 10h 
│ │ │ │ -000231d0: 6820 202d 2036 6820 6820 202d 2034 6820  h  - 6h h  - 4h 
│ │ │ │ -000231e0: 6820 202b 2033 6820 6820 202b 2033 6820  h  + 3h h  + 3h 
│ │ │ │ -000231f0: 6820 202b 2068 2020 2d20 6820 202d 2032  h  + h  - h  - 2
│ │ │ │ -00023200: 6820 6820 202d 2068 2020 2b20 6820 202b  h h  - h  + h  +
│ │ │ │ -00023210: 2068 2020 7c0a 7c20 2020 2020 2020 2031   h  |.|        1
│ │ │ │ -00023220: 2032 2020 2020 2031 2032 2020 2020 2031   2     1 2     1
│ │ │ │ -00023230: 2032 2020 2020 2031 2032 2020 2020 2031   2     1 2     1
│ │ │ │ -00023240: 2032 2020 2020 3220 2020 2031 2020 2020   2    2    1    
│ │ │ │ -00023250: 2031 2032 2020 2020 3220 2020 2031 2020   1 2    2    1  
│ │ │ │ -00023260: 2020 3220 7c0a 7c20 2020 2020 2020 2020    2 |.|         
│ │ │ │ +00023170: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023180: 2020 2020 2032 2033 2020 2020 2032 2032       2 3     2 2
│ │ │ │ +00023190: 2020 2020 2020 2033 2020 2020 2032 2020         3     2  
│ │ │ │ +000231a0: 2020 2020 2020 2032 2020 2020 3320 2020         2    3   
│ │ │ │ +000231b0: 2032 2020 2020 2020 2020 2020 2020 3220   2            2 
│ │ │ │ +000231c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +000231d0: 3d20 3130 6820 6820 202d 2036 6820 6820  = 10h h  - 6h h 
│ │ │ │ +000231e0: 202d 2034 6820 6820 202b 2033 6820 6820   - 4h h  + 3h h 
│ │ │ │ +000231f0: 202b 2033 6820 6820 202b 2068 2020 2d20   + 3h h  + h  - 
│ │ │ │ +00023200: 6820 202d 2032 6820 6820 202d 2068 2020  h  - 2h h  - h  
│ │ │ │ +00023210: 2b20 6820 202b 2068 2020 7c0a 7c20 2020  + h  + h  |.|   
│ │ │ │ +00023220: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ +00023230: 2020 2020 2031 2032 2020 2020 2031 2032       1 2     1 2
│ │ │ │ +00023240: 2020 2020 2031 2032 2020 2020 3220 2020       1 2    2   
│ │ │ │ +00023250: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ +00023260: 2020 2031 2020 2020 3220 7c0a 7c20 2020     1    2 |.|   
│ │ │ │  00023270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232b0: 2020 2020 7c0a 7c6f 3820 3a20 4120 2020      |.|o8 : A   
│ │ │ │ -000232c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000232b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +000232c0: 3a20 4120 2020 2020 2020 2020 2020 2020  : A             
│ │ │ │  000232d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023300: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023300: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00023310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023350: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
│ │ │ │ -00023360: 7365 204d 756c 7469 5072 6f6a 436f 6f72  se MultiProjCoor
│ │ │ │ -00023370: 6452 696e 673a 0a3d 3d3d 3d3d 3d3d 3d3d  dRing:.=========
│ │ │ │ +00023350: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179  ----------+..Way
│ │ │ │ +00023360: 7320 746f 2075 7365 204d 756c 7469 5072  s to use MultiPr
│ │ │ │ +00023370: 6f6a 436f 6f72 6452 696e 673a 0a3d 3d3d  ojCoordRing:.===
│ │ │ │  00023380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00023390: 3d3d 3d3d 3d3d 0a0a 2020 2a20 224d 756c  ======..  * "Mul
│ │ │ │ -000233a0: 7469 5072 6f6a 436f 6f72 6452 696e 6728  tiProjCoordRing(
│ │ │ │ -000233b0: 4c69 7374 2922 0a20 202a 2022 4d75 6c74  List)".  * "Mult
│ │ │ │ -000233c0: 6950 726f 6a43 6f6f 7264 5269 6e67 2852  iProjCoordRing(R
│ │ │ │ -000233d0: 696e 672c 4c69 7374 2922 0a20 202a 2022  ing,List)".  * "
│ │ │ │ -000233e0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269  MultiProjCoordRi
│ │ │ │ -000233f0: 6e67 2852 696e 672c 5379 6d62 6f6c 2c4c  ng(Ring,Symbol,L
│ │ │ │ -00023400: 6973 7429 220a 2020 2a20 224d 756c 7469  ist)".  * "Multi
│ │ │ │ -00023410: 5072 6f6a 436f 6f72 6452 696e 6728 5379  ProjCoordRing(Sy
│ │ │ │ -00023420: 6d62 6f6c 2c4c 6973 7429 220a 0a46 6f72  mbol,List)"..For
│ │ │ │ -00023430: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ -00023440: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00023450: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ -00023460: 6e6f 7465 204d 756c 7469 5072 6f6a 436f  note MultiProjCo
│ │ │ │ -00023470: 6f72 6452 696e 673a 204d 756c 7469 5072  ordRing: MultiPr
│ │ │ │ -00023480: 6f6a 436f 6f72 6452 696e 672c 2069 7320  ojCoordRing, is 
│ │ │ │ -00023490: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66  a *note method.f
│ │ │ │ -000234a0: 756e 6374 696f 6e3a 2028 4d61 6361 756c  unction: (Macaul
│ │ │ │ -000234b0: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -000234c0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +00023390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +000233a0: 2a20 224d 756c 7469 5072 6f6a 436f 6f72  * "MultiProjCoor
│ │ │ │ +000233b0: 6452 696e 6728 4c69 7374 2922 0a20 202a  dRing(List)".  *
│ │ │ │ +000233c0: 2022 4d75 6c74 6950 726f 6a43 6f6f 7264   "MultiProjCoord
│ │ │ │ +000233d0: 5269 6e67 2852 696e 672c 4c69 7374 2922  Ring(Ring,List)"
│ │ │ │ +000233e0: 0a20 202a 2022 4d75 6c74 6950 726f 6a43  .  * "MultiProjC
│ │ │ │ +000233f0: 6f6f 7264 5269 6e67 2852 696e 672c 5379  oordRing(Ring,Sy
│ │ │ │ +00023400: 6d62 6f6c 2c4c 6973 7429 220a 2020 2a20  mbol,List)".  * 
│ │ │ │ +00023410: 224d 756c 7469 5072 6f6a 436f 6f72 6452  "MultiProjCoordR
│ │ │ │ +00023420: 696e 6728 5379 6d62 6f6c 2c4c 6973 7429  ing(Symbol,List)
│ │ │ │ +00023430: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +00023440: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +00023450: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +00023460: 6a65 6374 202a 6e6f 7465 204d 756c 7469  ject *note Multi
│ │ │ │ +00023470: 5072 6f6a 436f 6f72 6452 696e 673a 204d  ProjCoordRing: M
│ │ │ │ +00023480: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ +00023490: 672c 2069 7320 6120 2a6e 6f74 6520 6d65  g, is a *note me
│ │ │ │ +000234a0: 7468 6f64 0a66 756e 6374 696f 6e3a 2028  thod.function: (
│ │ │ │ +000234b0: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +000234c0: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │  000234d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000234e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000234f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023510: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -00023520: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -00023530: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -00023540: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -00023550: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -00023560: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ -00023570: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -00023580: 0a43 6861 7261 6374 6572 6973 7469 6343  .CharacteristicC
│ │ │ │ -00023590: 6c61 7373 6573 2e6d 323a 3230 3530 3a30  lasses.m2:2050:0
│ │ │ │ -000235a0: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163  ....File: Charac
│ │ │ │ -000235b0: 7465 7269 7374 6963 436c 6173 7365 732e  teristicClasses.
│ │ │ │ -000235c0: 696e 666f 2c20 4e6f 6465 3a20 4f75 7470  info, Node: Outp
│ │ │ │ -000235d0: 7574 2c20 4e65 7874 3a20 7072 6f62 6162  ut, Next: probab
│ │ │ │ -000235e0: 696c 6973 7469 6320 616c 676f 7269 7468  ilistic algorith
│ │ │ │ -000235f0: 6d2c 2050 7265 763a 204d 756c 7469 5072  m, Prev: MultiPr
│ │ │ │ -00023600: 6f6a 436f 6f72 6452 696e 672c 2055 703a  ojCoordRing, Up:
│ │ │ │ -00023610: 2054 6f70 0a0a 4f75 7470 7574 0a2a 2a2a   Top..Output.***
│ │ │ │ -00023620: 2a2a 2a0a 0a44 6573 6372 6970 7469 6f6e  ***..Description
│ │ │ │ -00023630: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ -00023640: 6520 6f70 7469 6f6e 204f 7574 7075 7420  e option Output 
│ │ │ │ -00023650: 6973 206f 6e6c 7920 7573 6564 2062 7920  is only used by 
│ │ │ │ -00023660: 7468 6520 636f 6d6d 616e 6473 202a 6e6f  the commands *no
│ │ │ │ -00023670: 7465 2043 534d 3a20 4353 4d2c 2c20 2a6e  te CSM: CSM,, *n
│ │ │ │ -00023680: 6f74 6520 5365 6772 653a 0a53 6567 7265  ote Segre:.Segre
│ │ │ │ -00023690: 2c2c 202a 6e6f 7465 2043 6865 726e 3a20  ,, *note Chern: 
│ │ │ │ -000236a0: 4368 6572 6e2c 2061 6e64 202a 6e6f 7465  Chern, and *note
│ │ │ │ -000236b0: 2045 756c 6572 3a20 4575 6c65 722c 2074   Euler: Euler, t
│ │ │ │ -000236c0: 6f20 7370 6563 6966 7920 7468 6520 7479  o specify the ty
│ │ │ │ -000236d0: 7065 206f 660a 6f75 7470 7574 2074 6f20  pe of.output to 
│ │ │ │ -000236e0: 6265 2072 6574 7572 6e65 6420 746f 2074  be returned to t
│ │ │ │ -000236f0: 6865 2075 7365 642e 2054 6869 7320 6f70  he used. This op
│ │ │ │ -00023700: 7469 6f6e 2077 696c 6c20 6265 2069 676e  tion will be ign
│ │ │ │ -00023710: 6f72 6564 2077 6865 6e20 7573 6564 2077  ored when used w
│ │ │ │ -00023720: 6974 680a 2a6e 6f74 6520 436f 6d70 4d65  ith.*note CompMe
│ │ │ │ -00023730: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64  thod: CompMethod
│ │ │ │ -00023740: 2c20 506e 5265 7369 6475 616c 206f 7220  , PnResidual or 
│ │ │ │ -00023750: 6265 7274 696e 692e 2054 6865 206f 7074  bertini. The opt
│ │ │ │ -00023760: 696f 6e20 7769 6c6c 2061 6c73 6f20 6265  ion will also be
│ │ │ │ -00023770: 0a69 676e 6f72 6520 7768 656e 202a 6e6f  .ignore when *no
│ │ │ │ -00023780: 7465 204d 6574 686f 643a 204d 6574 686f  te Method: Metho
│ │ │ │ -00023790: 642c 3d3e 4469 7265 6374 436f 6d70 6c65  d,=>DirectComple
│ │ │ │ -000237a0: 7465 496e 7420 6973 2075 7365 642e 2054  teInt is used. T
│ │ │ │ -000237b0: 6865 2064 6566 6175 6c74 0a6f 7574 7075  he default.outpu
│ │ │ │ -000237c0: 7420 666f 7220 616c 6c20 7468 6573 6520  t for all these 
│ │ │ │ -000237d0: 6d65 7468 6f64 7320 6973 2043 686f 7752  methods is ChowR
│ │ │ │ -000237e0: 696e 6745 6c65 6c6d 656e 7420 7768 6963  ingElelment whic
│ │ │ │ -000237f0: 6820 7769 6c6c 2072 6574 7572 6e20 616e  h will return an
│ │ │ │ -00023800: 2065 6c65 6d65 6e74 0a6f 6620 7468 6520   element.of the 
│ │ │ │ -00023810: 6170 7072 6f70 7269 6174 6520 4368 6f77  appropriate Chow
│ │ │ │ -00023820: 2072 696e 672e 2041 6c6c 206d 6574 686f   ring. All metho
│ │ │ │ -00023830: 6473 2061 6c73 6f20 6861 7665 2061 6e20  ds also have an 
│ │ │ │ -00023840: 6f70 7469 6f6e 2048 6173 6846 6f72 6d20  option HashForm 
│ │ │ │ -00023850: 7768 6963 680a 7265 7475 726e 7320 6164  which.returns ad
│ │ │ │ -00023860: 6469 7469 6f6e 616c 2069 6e66 6f72 6d61  ditional informa
│ │ │ │ -00023870: 7469 6f6e 2063 6f6d 7075 7465 6420 6279  tion computed by
│ │ │ │ -00023880: 2074 6865 206d 6574 686f 6473 2064 7572   the methods dur
│ │ │ │ -00023890: 696e 6720 7468 6569 7220 7374 616e 6461  ing their standa
│ │ │ │ -000238a0: 7264 0a6f 7065 7261 7469 6f6e 2e0a 0a2b  rd.operation...+
│ │ │ │ -000238b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +00023520: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +00023530: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +00023540: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +00023550: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +00023560: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ +00023570: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +00023580: 6b61 6765 732f 0a43 6861 7261 6374 6572  kages/.Character
│ │ │ │ +00023590: 6973 7469 6343 6c61 7373 6573 2e6d 323a  isticClasses.m2:
│ │ │ │ +000235a0: 3230 3530 3a30 2e0a 1f0a 4669 6c65 3a20  2050:0....File: 
│ │ │ │ +000235b0: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ +000235c0: 6173 7365 732e 696e 666f 2c20 4e6f 6465  asses.info, Node
│ │ │ │ +000235d0: 3a20 4f75 7470 7574 2c20 4e65 7874 3a20  : Output, Next: 
│ │ │ │ +000235e0: 7072 6f62 6162 696c 6973 7469 6320 616c  probabilistic al
│ │ │ │ +000235f0: 676f 7269 7468 6d2c 2050 7265 763a 204d  gorithm, Prev: M
│ │ │ │ +00023600: 756c 7469 5072 6f6a 436f 6f72 6452 696e  ultiProjCoordRin
│ │ │ │ +00023610: 672c 2055 703a 2054 6f70 0a0a 4f75 7470  g, Up: Top..Outp
│ │ │ │ +00023620: 7574 0a2a 2a2a 2a2a 2a0a 0a44 6573 6372  ut.******..Descr
│ │ │ │ +00023630: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +00023640: 3d3d 0a0a 5468 6520 6f70 7469 6f6e 204f  ==..The option O
│ │ │ │ +00023650: 7574 7075 7420 6973 206f 6e6c 7920 7573  utput is only us
│ │ │ │ +00023660: 6564 2062 7920 7468 6520 636f 6d6d 616e  ed by the comman
│ │ │ │ +00023670: 6473 202a 6e6f 7465 2043 534d 3a20 4353  ds *note CSM: CS
│ │ │ │ +00023680: 4d2c 2c20 2a6e 6f74 6520 5365 6772 653a  M,, *note Segre:
│ │ │ │ +00023690: 0a53 6567 7265 2c2c 202a 6e6f 7465 2043  .Segre,, *note C
│ │ │ │ +000236a0: 6865 726e 3a20 4368 6572 6e2c 2061 6e64  hern: Chern, and
│ │ │ │ +000236b0: 202a 6e6f 7465 2045 756c 6572 3a20 4575   *note Euler: Eu
│ │ │ │ +000236c0: 6c65 722c 2074 6f20 7370 6563 6966 7920  ler, to specify 
│ │ │ │ +000236d0: 7468 6520 7479 7065 206f 660a 6f75 7470  the type of.outp
│ │ │ │ +000236e0: 7574 2074 6f20 6265 2072 6574 7572 6e65  ut to be returne
│ │ │ │ +000236f0: 6420 746f 2074 6865 2075 7365 642e 2054  d to the used. T
│ │ │ │ +00023700: 6869 7320 6f70 7469 6f6e 2077 696c 6c20  his option will 
│ │ │ │ +00023710: 6265 2069 676e 6f72 6564 2077 6865 6e20  be ignored when 
│ │ │ │ +00023720: 7573 6564 2077 6974 680a 2a6e 6f74 6520  used with.*note 
│ │ │ │ +00023730: 436f 6d70 4d65 7468 6f64 3a20 436f 6d70  CompMethod: Comp
│ │ │ │ +00023740: 4d65 7468 6f64 2c20 506e 5265 7369 6475  Method, PnResidu
│ │ │ │ +00023750: 616c 206f 7220 6265 7274 696e 692e 2054  al or bertini. T
│ │ │ │ +00023760: 6865 206f 7074 696f 6e20 7769 6c6c 2061  he option will a
│ │ │ │ +00023770: 6c73 6f20 6265 0a69 676e 6f72 6520 7768  lso be.ignore wh
│ │ │ │ +00023780: 656e 202a 6e6f 7465 204d 6574 686f 643a  en *note Method:
│ │ │ │ +00023790: 204d 6574 686f 642c 3d3e 4469 7265 6374   Method,=>Direct
│ │ │ │ +000237a0: 436f 6d70 6c65 7465 496e 7420 6973 2075  CompleteInt is u
│ │ │ │ +000237b0: 7365 642e 2054 6865 2064 6566 6175 6c74  sed. The default
│ │ │ │ +000237c0: 0a6f 7574 7075 7420 666f 7220 616c 6c20  .output for all 
│ │ │ │ +000237d0: 7468 6573 6520 6d65 7468 6f64 7320 6973  these methods is
│ │ │ │ +000237e0: 2043 686f 7752 696e 6745 6c65 6c6d 656e   ChowRingElelmen
│ │ │ │ +000237f0: 7420 7768 6963 6820 7769 6c6c 2072 6574  t which will ret
│ │ │ │ +00023800: 7572 6e20 616e 2065 6c65 6d65 6e74 0a6f  urn an element.o
│ │ │ │ +00023810: 6620 7468 6520 6170 7072 6f70 7269 6174  f the appropriat
│ │ │ │ +00023820: 6520 4368 6f77 2072 696e 672e 2041 6c6c  e Chow ring. All
│ │ │ │ +00023830: 206d 6574 686f 6473 2061 6c73 6f20 6861   methods also ha
│ │ │ │ +00023840: 7665 2061 6e20 6f70 7469 6f6e 2048 6173  ve an option Has
│ │ │ │ +00023850: 6846 6f72 6d20 7768 6963 680a 7265 7475  hForm which.retu
│ │ │ │ +00023860: 726e 7320 6164 6469 7469 6f6e 616c 2069  rns additional i
│ │ │ │ +00023870: 6e66 6f72 6d61 7469 6f6e 2063 6f6d 7075  nformation compu
│ │ │ │ +00023880: 7465 6420 6279 2074 6865 206d 6574 686f  ted by the metho
│ │ │ │ +00023890: 6473 2064 7572 696e 6720 7468 6569 7220  ds during their 
│ │ │ │ +000238a0: 7374 616e 6461 7264 0a6f 7065 7261 7469  standard.operati
│ │ │ │ +000238b0: 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  on...+----------
│ │ │ │  000238c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000238d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000238e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000238f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00023900: 6931 203a 2052 203d 205a 5a2f 3332 3734  i1 : R = ZZ/3274
│ │ │ │ -00023910: 395b 785f 302e 2e78 5f36 5d20 2020 2020  9[x_0..x_6]     
│ │ │ │ -00023920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000238f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023900: 2d2d 2d2b 0a7c 6931 203a 2052 203d 205a  ---+.|i1 : R = Z
│ │ │ │ +00023910: 5a2f 3332 3734 395b 785f 302e 2e78 5f36  Z/32749[x_0..x_6
│ │ │ │ +00023920: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  00023930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023950: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023990: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000239a0: 6f31 203d 2052 2020 2020 2020 2020 2020  o1 = R          
│ │ │ │ +00023990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000239a0: 2020 207c 0a7c 6f31 203d 2052 2020 2020     |.|o1 = R    
│ │ │ │  000239b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000239f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000239e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000239f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023a40: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ -00023a50: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00023a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a40: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e     |.|o1 : Polyn
│ │ │ │ +00023a50: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │  00023a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a80: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00023ae0: 6932 203a 2041 3d43 686f 7752 696e 6728  i2 : A=ChowRing(
│ │ │ │ -00023af0: 5229 2020 2020 2020 2020 2020 2020 2020  R)              
│ │ │ │ +00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023ae0: 2d2d 2d2b 0a7c 6932 203a 2041 3d43 686f  ---+.|i2 : A=Cho
│ │ │ │ +00023af0: 7752 696e 6728 5229 2020 2020 2020 2020  wRing(R)        
│ │ │ │  00023b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023b80: 6f32 203d 2041 2020 2020 2020 2020 2020  o2 = A          
│ │ │ │ +00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b80: 2020 207c 0a7c 6f32 203d 2041 2020 2020     |.|o2 = A    
│ │ │ │  00023b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023bd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023c20: 6f32 203a 2051 756f 7469 656e 7452 696e  o2 : QuotientRin
│ │ │ │ -00023c30: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00023c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c20: 2020 207c 0a7c 6f32 203a 2051 756f 7469     |.|o2 : Quoti
│ │ │ │ +00023c30: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │  00023c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c60: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00023c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c70: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00023c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00023cc0: 6933 203a 2049 3d69 6465 616c 2872 616e  i3 : I=ideal(ran
│ │ │ │ -00023cd0: 646f 6d28 322c 5229 2c52 5f30 2a52 5f31  dom(2,R),R_0*R_1
│ │ │ │ -00023ce0: 2a52 5f36 2d52 5f30 5e33 293b 2020 2020  *R_6-R_0^3);    
│ │ │ │ -00023cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023cc0: 2d2d 2d2b 0a7c 6933 203a 2049 3d69 6465  ---+.|i3 : I=ide
│ │ │ │ +00023cd0: 616c 2872 616e 646f 6d28 322c 5229 2c52  al(random(2,R),R
│ │ │ │ +00023ce0: 5f30 2a52 5f31 2a52 5f36 2d52 5f30 5e33  _0*R_1*R_6-R_0^3
│ │ │ │ +00023cf0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +00023d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023d60: 6f33 203a 2049 6465 616c 206f 6620 5220  o3 : Ideal of R 
│ │ │ │ -00023d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d60: 2020 207c 0a7c 6f33 203a 2049 6465 616c     |.|o3 : Ideal
│ │ │ │ +00023d70: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │  00023d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023da0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00023db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023db0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00023dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00023e00: 6934 203a 2063 736d 3d43 534d 2841 2c49  i4 : csm=CSM(A,I
│ │ │ │ -00023e10: 2c4f 7574 7075 743d 3e48 6173 6846 6f72  ,Output=>HashFor
│ │ │ │ -00023e20: 6d29 2020 2020 2020 2020 2020 2020 2020  m)              
│ │ │ │ +00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023e00: 2d2d 2d2b 0a7c 6934 203a 2063 736d 3d43  ---+.|i4 : csm=C
│ │ │ │ +00023e10: 534d 2841 2c49 2c4f 7574 7075 743d 3e48  SM(A,I,Output=>H
│ │ │ │ +00023e20: 6173 6846 6f72 6d29 2020 2020 2020 2020  ashForm)        
│ │ │ │  00023e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023ea0: 6f34 203d 204d 7574 6162 6c65 4861 7368  o4 = MutableHash
│ │ │ │ -00023eb0: 5461 626c 657b 2e2e 2e34 2e2e 2e7d 2020  Table{...4...}  
│ │ │ │ -00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ea0: 2020 207c 0a7c 6f34 203d 204d 7574 6162     |.|o4 = Mutab
│ │ │ │ +00023eb0: 6c65 4861 7368 5461 626c 657b 2e2e 2e34  leHashTable{...4
│ │ │ │ +00023ec0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │  00023ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ef0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00023f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023f40: 6f34 203a 204d 7574 6162 6c65 4861 7368  o4 : MutableHash
│ │ │ │ -00023f50: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │ +00023f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023f40: 2020 207c 0a7c 6f34 203a 204d 7574 6162     |.|o4 : Mutab
│ │ │ │ +00023f50: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │  00023f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f80: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00023f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023f90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00023fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00023fe0: 6935 203a 2070 6565 6b20 6373 6d20 2020  i5 : peek csm   
│ │ │ │ -00023ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023fe0: 2d2d 2d2b 0a7c 6935 203a 2070 6565 6b20  ---+.|i5 : peek 
│ │ │ │ +00023ff0: 6373 6d20 2020 2020 2020 2020 2020 2020  csm             
│ │ │ │  00024000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024030: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024080: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240a0: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ -000240b0: 3420 2020 2020 2033 2020 2020 2020 3220  4      3      2 
│ │ │ │ -000240c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000240d0: 6f35 203d 204d 7574 6162 6c65 4861 7368  o5 = MutableHash
│ │ │ │ -000240e0: 5461 626c 657b 7b30 2c20 317d 203d 3e20  Table{{0, 1} => 
│ │ │ │ -000240f0: 3268 2020 2b20 3233 6820 202b 2033 3268  2h  + 23h  + 32h
│ │ │ │ -00024100: 2020 2b20 3333 6820 202b 2031 3868 2020    + 33h  + 18h  
│ │ │ │ -00024110: 2b20 3568 207d 2020 2020 2020 207c 0a7c  + 5h }       |.|
│ │ │ │ -00024120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000240a0: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ +000240b0: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ +000240c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +000240d0: 2020 207c 0a7c 6f35 203d 204d 7574 6162     |.|o5 = Mutab
│ │ │ │ +000240e0: 6c65 4861 7368 5461 626c 657b 7b30 2c20  leHashTable{{0, 
│ │ │ │ +000240f0: 317d 203d 3e20 3268 2020 2b20 3233 6820  1} => 2h  + 23h 
│ │ │ │ +00024100: 202b 2033 3268 2020 2b20 3333 6820 202b   + 32h  + 33h  +
│ │ │ │ +00024110: 2031 3868 2020 2b20 3568 207d 2020 2020   18h  + 5h }    
│ │ │ │ +00024120: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024140: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00024150: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00024160: 2020 2020 3120 2020 2020 2020 207c 0a7c      1        |.|
│ │ │ │ -00024170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024140: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ +00024150: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024160: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +00024170: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024190: 3620 2020 2020 2035 2020 2020 2020 3420  6      5      4 
│ │ │ │ -000241a0: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ -000241b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000241c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000241d0: 2020 2020 2020 4353 4d20 3d3e 2031 3068        CSM => 10h
│ │ │ │ -000241e0: 2020 2b20 3132 6820 202b 2032 3268 2020    + 12h  + 22h  
│ │ │ │ -000241f0: 2b20 3136 6820 202b 2036 6820 2020 2020  + 16h  + 6h     
│ │ │ │ -00024200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024190: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +000241a0: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +000241b0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000241c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000241d0: 2020 2020 2020 2020 2020 2020 4353 4d20              CSM 
│ │ │ │ +000241e0: 3d3e 2031 3068 2020 2b20 3132 6820 202b  => 10h  + 12h  +
│ │ │ │ +000241f0: 2032 3268 2020 2b20 3136 6820 202b 2036   22h  + 16h  + 6
│ │ │ │ +00024200: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +00024210: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024230: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00024240: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00024250: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024270: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00024280: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -00024290: 2020 2020 3320 2020 2020 2032 2020 2020      3      2    
│ │ │ │ -000242a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000242b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000242c0: 2020 2020 2020 7b30 7d20 3d3e 2036 6820        {0} => 6h 
│ │ │ │ -000242d0: 202b 2031 3868 2020 2b20 3236 6820 202b   + 18h  + 26h  +
│ │ │ │ -000242e0: 2032 3268 2020 2b20 3130 6820 202b 2032   22h  + 10h  + 2
│ │ │ │ -000242f0: 6820 2020 2020 2020 2020 2020 207c 0a7c  h            |.|
│ │ │ │ -00024300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024310: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00024320: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024330: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00024340: 2031 2020 2020 2020 2020 2020 207c 0a7c   1           |.|
│ │ │ │ -00024350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024360: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00024370: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -00024380: 2020 2020 3320 2020 2020 2032 2020 2020      3      2    
│ │ │ │ -00024390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000243a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243b0: 2020 2020 2020 7b31 7d20 3d3e 2036 6820        {1} => 6h 
│ │ │ │ -000243c0: 202b 2031 3768 2020 2b20 3238 6820 202b   + 17h  + 28h  +
│ │ │ │ -000243d0: 2032 3768 2020 2b20 3134 6820 202b 2033   27h  + 14h  + 3
│ │ │ │ -000243e0: 6820 2020 2020 2020 2020 2020 207c 0a7c  h            |.|
│ │ │ │ -000243f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024400: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00024410: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024420: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00024430: 2031 2020 2020 2020 2020 2020 207c 0a2b   1           |.+
│ │ │ │ -00024440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024230: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024240: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00024250: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00024260: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024280: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00024290: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +000242a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000242b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000242c0: 2020 2020 2020 2020 2020 2020 7b30 7d20              {0} 
│ │ │ │ +000242d0: 3d3e 2036 6820 202b 2031 3868 2020 2b20  => 6h  + 18h  + 
│ │ │ │ +000242e0: 3236 6820 202b 2032 3268 2020 2b20 3130  26h  + 22h  + 10
│ │ │ │ +000242f0: 6820 202b 2032 6820 2020 2020 2020 2020  h  + 2h         
│ │ │ │ +00024300: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024320: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024330: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024340: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00024350: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024370: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00024380: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00024390: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000243a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000243b0: 2020 2020 2020 2020 2020 2020 7b31 7d20              {1} 
│ │ │ │ +000243c0: 3d3e 2036 6820 202b 2031 3768 2020 2b20  => 6h  + 17h  + 
│ │ │ │ +000243d0: 3238 6820 202b 2032 3768 2020 2b20 3134  28h  + 27h  + 14
│ │ │ │ +000243e0: 6820 202b 2033 6820 2020 2020 2020 2020  h  + 3h         
│ │ │ │ +000243f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024410: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024420: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024430: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00024440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00024450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00024490: 6936 203a 2043 534d 2841 2c69 6465 616c  i6 : CSM(A,ideal
│ │ │ │ -000244a0: 2049 5f30 293d 3d63 736d 237b 307d 2020   I_0)==csm#{0}  
│ │ │ │ -000244b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024490: 2d2d 2d2b 0a7c 6936 203a 2043 534d 2841  ---+.|i6 : CSM(A
│ │ │ │ +000244a0: 2c69 6465 616c 2049 5f30 293d 3d63 736d  ,ideal I_0)==csm
│ │ │ │ +000244b0: 237b 307d 2020 2020 2020 2020 2020 2020  #{0}            
│ │ │ │  000244c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000244d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000244e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000244f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024530: 6f36 203d 2074 7275 6520 2020 2020 2020  o6 = true       
│ │ │ │ +00024520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024530: 2020 207c 0a7c 6f36 203d 2074 7275 6520     |.|o6 = true 
│ │ │ │  00024540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024570: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00024580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024580: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00024590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000245a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000245b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000245c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000245d0: 6937 203a 2043 534d 2841 2c69 6465 616c  i7 : CSM(A,ideal
│ │ │ │ -000245e0: 2849 5f30 2a49 5f31 2929 3d3d 6373 6d23  (I_0*I_1))==csm#
│ │ │ │ -000245f0: 7b30 2c31 7d20 2020 2020 2020 2020 2020  {0,1}           
│ │ │ │ +000245c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000245d0: 2d2d 2d2b 0a7c 6937 203a 2043 534d 2841  ---+.|i7 : CSM(A
│ │ │ │ +000245e0: 2c69 6465 616c 2849 5f30 2a49 5f31 2929  ,ideal(I_0*I_1))
│ │ │ │ +000245f0: 3d3d 6373 6d23 7b30 2c31 7d20 2020 2020  ==csm#{0,1}     
│ │ │ │  00024600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024620: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024660: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024670: 6f37 203d 2074 7275 6520 2020 2020 2020  o7 = true       
│ │ │ │ +00024660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024670: 2020 207c 0a7c 6f37 203d 2074 7275 6520     |.|o7 = true 
│ │ │ │  00024680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000246c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000246b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000246c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  000246d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000246e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000246f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00024710: 6938 203a 2063 3d43 6865 726e 2820 492c  i8 : c=Chern( I,
│ │ │ │ -00024720: 204f 7574 7075 743d 3e48 6173 6846 6f72   Output=>HashFor
│ │ │ │ -00024730: 6d29 2020 2020 2020 2020 2020 2020 2020  m)              
│ │ │ │ +00024700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024710: 2d2d 2d2b 0a7c 6938 203a 2063 3d43 6865  ---+.|i8 : c=Che
│ │ │ │ +00024720: 726e 2820 492c 204f 7574 7075 743d 3e48  rn( I, Output=>H
│ │ │ │ +00024730: 6173 6846 6f72 6d29 2020 2020 2020 2020  ashForm)        
│ │ │ │  00024740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024760: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000247b0: 6f38 203d 204d 7574 6162 6c65 4861 7368  o8 = MutableHash
│ │ │ │ -000247c0: 5461 626c 657b 2e2e 2e36 2e2e 2e7d 2020  Table{...6...}  
│ │ │ │ -000247d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000247a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000247b0: 2020 207c 0a7c 6f38 203d 204d 7574 6162     |.|o8 = Mutab
│ │ │ │ +000247c0: 6c65 4861 7368 5461 626c 657b 2e2e 2e36  leHashTable{...6
│ │ │ │ +000247d0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │  000247e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000247f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024800: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024850: 6f38 203a 204d 7574 6162 6c65 4861 7368  o8 : MutableHash
│ │ │ │ -00024860: 5461 626c 6520 2020 2020 2020 2020 2020  Table           
│ │ │ │ +00024840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024850: 2020 207c 0a7c 6f38 203a 204d 7574 6162     |.|o8 : Mutab
│ │ │ │ +00024860: 6c65 4861 7368 5461 626c 6520 2020 2020  leHashTable     
│ │ │ │  00024870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024890: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000248a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000248a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  000248b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000248c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000248d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000248e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000248f0: 6939 203a 2070 6565 6b20 6320 2020 2020  i9 : peek c     
│ │ │ │ -00024900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000248e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000248f0: 2d2d 2d2b 0a7c 6939 203a 2070 6565 6b20  ---+.|i9 : peek 
│ │ │ │ +00024900: 6320 2020 2020 2020 2020 2020 2020 2020  c               
│ │ │ │  00024910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024990: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000249a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249b0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000249c0: 2020 2033 2020 2020 2020 3420 2020 2020     3      4     
│ │ │ │ -000249d0: 2020 3520 2020 2020 2020 3620 207c 0a7c    5       6  |.|
│ │ │ │ -000249e0: 6f39 203d 204d 7574 6162 6c65 4861 7368  o9 = MutableHash
│ │ │ │ -000249f0: 5461 626c 657b 5365 6772 654c 6973 7420  Table{SegreList 
│ │ │ │ -00024a00: 3d3e 207b 302c 2030 2c20 3668 202c 202d  => {0, 0, 6h , -
│ │ │ │ -00024a10: 3330 6820 2c20 3131 3468 202c 202d 3339  30h , 114h , -39
│ │ │ │ -00024a20: 3068 202c 2031 3236 3668 207d 7d7c 0a7c  0h , 1266h }}|.|
│ │ │ │ -00024a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000249b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000249c0: 2020 3220 2020 2020 2033 2020 2020 2020    2      3      
│ │ │ │ +000249d0: 3420 2020 2020 2020 3520 2020 2020 2020  4       5       
│ │ │ │ +000249e0: 3620 207c 0a7c 6f39 203d 204d 7574 6162  6  |.|o9 = Mutab
│ │ │ │ +000249f0: 6c65 4861 7368 5461 626c 657b 5365 6772  leHashTable{Segr
│ │ │ │ +00024a00: 654c 6973 7420 3d3e 207b 302c 2030 2c20  eList => {0, 0, 
│ │ │ │ +00024a10: 3668 202c 202d 3330 6820 2c20 3131 3468  6h , -30h , 114h
│ │ │ │ +00024a20: 202c 202d 3339 3068 202c 2031 3236 3668   , -390h , 1266h
│ │ │ │ +00024a30: 207d 7d7c 0a7c 2020 2020 2020 2020 2020   }}|.|          
│ │ │ │  00024a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a50: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -00024a60: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00024a70: 2020 3120 2020 2020 2020 3120 207c 0a7c    1       1  |.|
│ │ │ │ -00024a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a60: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00024a70: 3120 2020 2020 2020 3120 2020 2020 2020  1       1       
│ │ │ │ +00024a80: 3120 207c 0a7c 2020 2020 2020 2020 2020  1  |.|          
│ │ │ │  00024a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024aa0: 2020 2020 2020 2020 2020 3220 2020 2033            2    3
│ │ │ │ -00024ab0: 2020 2020 3420 2020 2035 2020 2020 3620      4    5    6 
│ │ │ │ -00024ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ae0: 2020 2020 2020 476c 6973 7420 3d3e 207b        Glist => {
│ │ │ │ -00024af0: 312c 2033 6820 2c20 3368 202c 2033 6820  1, 3h , 3h , 3h 
│ │ │ │ -00024b00: 2c20 3368 202c 2033 6820 2c20 3368 207d  , 3h , 3h , 3h }
│ │ │ │ -00024b10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ab0: 3220 2020 2033 2020 2020 3420 2020 2035  2    3    4    5
│ │ │ │ +00024ac0: 2020 2020 3620 2020 2020 2020 2020 2020      6           
│ │ │ │ +00024ad0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024ae0: 2020 2020 2020 2020 2020 2020 476c 6973              Glis
│ │ │ │ +00024af0: 7420 3d3e 207b 312c 2033 6820 2c20 3368  t => {1, 3h , 3h
│ │ │ │ +00024b00: 202c 2033 6820 2c20 3368 202c 2033 6820   , 3h , 3h , 3h 
│ │ │ │ +00024b10: 2c20 3368 207d 2020 2020 2020 2020 2020  , 3h }          
│ │ │ │ +00024b20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b40: 2020 2020 2031 2020 2020 3120 2020 2031       1    1    1
│ │ │ │ -00024b50: 2020 2020 3120 2020 2031 2020 2020 3120      1    1    1 
│ │ │ │ -00024b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b40: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +00024b50: 3120 2020 2031 2020 2020 3120 2020 2031  1    1    1    1
│ │ │ │ +00024b60: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00024b70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b90: 2020 2020 3620 2020 2020 2020 3520 2020      6       5   
│ │ │ │ -00024ba0: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ -00024bb0: 2032 2020 2020 2020 2020 2020 207c 0a7c   2           |.|
│ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bd0: 2020 2020 2020 5365 6772 6520 3d3e 2031        Segre => 1
│ │ │ │ -00024be0: 3236 3668 2020 2d20 3339 3068 2020 2b20  266h  - 390h  + 
│ │ │ │ -00024bf0: 3131 3468 2020 2d20 3330 6820 202b 2036  114h  - 30h  + 6
│ │ │ │ -00024c00: 6820 2020 2020 2020 2020 2020 207c 0a7c  h            |.|
│ │ │ │ -00024c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b90: 2020 2020 2020 2020 2020 3620 2020 2020            6     
│ │ │ │ +00024ba0: 2020 3520 2020 2020 2020 3420 2020 2020    5       4     
│ │ │ │ +00024bb0: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │ +00024bc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024bd0: 2020 2020 2020 2020 2020 2020 5365 6772              Segr
│ │ │ │ +00024be0: 6520 3d3e 2031 3236 3668 2020 2d20 3339  e => 1266h  - 39
│ │ │ │ +00024bf0: 3068 2020 2b20 3131 3468 2020 2d20 3330  0h  + 114h  - 30
│ │ │ │ +00024c00: 6820 202b 2036 6820 2020 2020 2020 2020  h  + 6h         
│ │ │ │ +00024c10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c30: 2020 2020 3120 2020 2020 2020 3120 2020      1       1   
│ │ │ │ -00024c40: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00024c50: 2031 2020 2020 2020 2020 2020 207c 0a7c   1           |.|
│ │ │ │ -00024c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c30: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +00024c40: 2020 3120 2020 2020 2020 3120 2020 2020    1       1     
│ │ │ │ +00024c50: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00024c60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c80: 2020 3620 2020 2020 2035 2020 2020 2020    6      5      
│ │ │ │ -00024c90: 3420 2020 2020 2033 2020 2020 2032 2020  4      3     2  
│ │ │ │ -00024ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024cc0: 2020 2020 2020 4368 6572 6e20 3d3e 2039        Chern => 9
│ │ │ │ -00024cd0: 3068 2020 2d20 3132 6820 202b 2033 3068  0h  - 12h  + 30h
│ │ │ │ -00024ce0: 2020 2b20 3132 6820 202b 2036 6820 2020    + 12h  + 6h   
│ │ │ │ -00024cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c80: 2020 2020 2020 2020 3620 2020 2020 2035          6      5
│ │ │ │ +00024c90: 2020 2020 2020 3420 2020 2020 2033 2020        4      3  
│ │ │ │ +00024ca0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00024cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024cc0: 2020 2020 2020 2020 2020 2020 4368 6572              Cher
│ │ │ │ +00024cd0: 6e20 3d3e 2039 3068 2020 2d20 3132 6820  n => 90h  - 12h 
│ │ │ │ +00024ce0: 202b 2033 3068 2020 2b20 3132 6820 202b   + 30h  + 12h  +
│ │ │ │ +00024cf0: 2036 6820 2020 2020 2020 2020 2020 2020   6h             
│ │ │ │ +00024d00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d20: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -00024d30: 3120 2020 2020 2031 2020 2020 2031 2020  1      1     1  
│ │ │ │ -00024d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d60: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00024d70: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -00024d80: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ -00024d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024db0: 2020 2020 2020 4346 203d 3e20 3930 6820        CF => 90h 
│ │ │ │ -00024dc0: 202d 2031 3268 2020 2b20 3330 6820 202b   - 12h  + 30h  +
│ │ │ │ -00024dd0: 2031 3268 2020 2b20 3668 2020 2020 2020   12h  + 6h      
│ │ │ │ -00024de0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e00: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00024e10: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00024e20: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00024e30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e50: 2020 2020 2020 2020 2020 2020 2036 2020               6  
│ │ │ │ -00024e60: 2020 2035 2020 2020 2034 2020 2020 2033     5     4     3
│ │ │ │ -00024e70: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00024e80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ea0: 2020 2020 2020 4720 3d3e 2033 6820 202b        G => 3h  +
│ │ │ │ +00024d20: 2020 2020 2020 2020 3120 2020 2020 2031          1      1
│ │ │ │ +00024d30: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00024d40: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00024d50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d70: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00024d80: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00024d90: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00024da0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024db0: 2020 2020 2020 2020 2020 2020 4346 203d              CF =
│ │ │ │ +00024dc0: 3e20 3930 6820 202d 2031 3268 2020 2b20  > 90h  - 12h  + 
│ │ │ │ +00024dd0: 3330 6820 202b 2031 3268 2020 2b20 3668  30h  + 12h  + 6h
│ │ │ │ +00024de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024df0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e10: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00024e20: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00024e30: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00024e40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e60: 2020 2036 2020 2020 2035 2020 2020 2034     6     5     4
│ │ │ │ +00024e70: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ +00024e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024ea0: 2020 2020 2020 2020 2020 2020 4720 3d3e              G =>
│ │ │ │  00024eb0: 2033 6820 202b 2033 6820 202b 2033 6820   3h  + 3h  + 3h 
│ │ │ │ -00024ec0: 202b 2033 6820 202b 2033 6820 202b 2031   + 3h  + 3h  + 1
│ │ │ │ -00024ed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ef0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00024ec0: 202b 2033 6820 202b 2033 6820 202b 2033   + 3h  + 3h  + 3
│ │ │ │ +00024ed0: 6820 202b 2031 2020 2020 2020 2020 2020  h  + 1          
│ │ │ │ +00024ee0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f00: 2020 2031 2020 2020 2031 2020 2020 2031     1     1     1
│ │ │ │  00024f10: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ -00024f20: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00024f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f20: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00024f30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00024f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00024f80: 6931 3020 3a20 7365 673d 5365 6772 6528  i10 : seg=Segre(
│ │ │ │ -00024f90: 2049 2c20 4f75 7470 7574 3d3e 4861 7368   I, Output=>Hash
│ │ │ │ -00024fa0: 466f 726d 2920 2020 2020 2020 2020 2020  Form)           
│ │ │ │ +00024f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f80: 2d2d 2d2b 0a7c 6931 3020 3a20 7365 673d  ---+.|i10 : seg=
│ │ │ │ +00024f90: 5365 6772 6528 2049 2c20 4f75 7470 7574  Segre( I, Output
│ │ │ │ +00024fa0: 3d3e 4861 7368 466f 726d 2920 2020 2020  =>HashForm)     
│ │ │ │  00024fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00024fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025020: 6f31 3020 3d20 4d75 7461 626c 6548 6173  o10 = MutableHas
│ │ │ │ -00025030: 6854 6162 6c65 7b2e 2e2e 342e 2e2e 7d20  hTable{...4...} 
│ │ │ │ -00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025020: 2020 207c 0a7c 6f31 3020 3d20 4d75 7461     |.|o10 = Muta
│ │ │ │ +00025030: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +00025040: 342e 2e2e 7d20 2020 2020 2020 2020 2020  4...}           
│ │ │ │  00025050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025070: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000250c0: 6f31 3020 3a20 4d75 7461 626c 6548 6173  o10 : MutableHas
│ │ │ │ -000250d0: 6854 6162 6c65 2020 2020 2020 2020 2020  hTable          
│ │ │ │ +000250b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000250c0: 2020 207c 0a7c 6f31 3020 3a20 4d75 7461     |.|o10 : Muta
│ │ │ │ +000250d0: 626c 6548 6173 6854 6162 6c65 2020 2020  bleHashTable    
│ │ │ │  000250e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025100: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00025110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025110: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00025120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00025160: 6931 3120 3a20 7065 656b 2073 6567 2020  i11 : peek seg  
│ │ │ │ -00025170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025160: 2d2d 2d2b 0a7c 6931 3120 3a20 7065 656b  ---+.|i11 : peek
│ │ │ │ +00025170: 2073 6567 2020 2020 2020 2020 2020 2020   seg            
│ │ │ │  00025180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000251a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000251b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000251a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000251b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000251c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000251e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000251f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000251f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025200: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025220: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00025230: 2020 2020 3320 2020 2020 2034 2020 2020      3      4    
│ │ │ │ -00025240: 2020 2035 2020 2020 2020 2020 207c 0a7c     5         |.|
│ │ │ │ -00025250: 6f31 3120 3d20 4d75 7461 626c 6548 6173  o11 = MutableHas
│ │ │ │ -00025260: 6854 6162 6c65 7b53 6567 7265 4c69 7374  hTable{SegreList
│ │ │ │ -00025270: 203d 3e20 7b30 2c20 302c 2036 6820 2c20   => {0, 0, 6h , 
│ │ │ │ -00025280: 2d33 3068 202c 2031 3134 6820 2c20 2d33  -30h , 114h , -3
│ │ │ │ -00025290: 3930 6820 2c20 2020 2020 2020 207c 0a7c  90h ,        |.|
│ │ │ │ -000252a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025230: 2020 2032 2020 2020 2020 3320 2020 2020     2      3     
│ │ │ │ +00025240: 2034 2020 2020 2020 2035 2020 2020 2020   4       5      
│ │ │ │ +00025250: 2020 207c 0a7c 6f31 3120 3d20 4d75 7461     |.|o11 = Muta
│ │ │ │ +00025260: 626c 6548 6173 6854 6162 6c65 7b53 6567  bleHashTable{Seg
│ │ │ │ +00025270: 7265 4c69 7374 203d 3e20 7b30 2c20 302c  reList => {0, 0,
│ │ │ │ +00025280: 2036 6820 2c20 2d33 3068 202c 2031 3134   6h , -30h , 114
│ │ │ │ +00025290: 6820 2c20 2d33 3930 6820 2c20 2020 2020  h , -390h ,     
│ │ │ │ +000252a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000252b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252c0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -000252d0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -000252e0: 2020 2031 2020 2020 2020 2020 207c 0a7c     1         |.|
│ │ │ │ -000252f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000252c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000252d0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +000252e0: 2031 2020 2020 2020 2031 2020 2020 2020   1       1      
│ │ │ │ +000252f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025310: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00025320: 3320 2020 2034 2020 2020 3520 2020 2036  3    4    5    6
│ │ │ │ -00025330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025350: 2020 2020 2020 2047 6c69 7374 203d 3e20         Glist => 
│ │ │ │ -00025360: 7b31 2c20 3368 202c 2033 6820 2c20 3368  {1, 3h , 3h , 3h
│ │ │ │ -00025370: 202c 2033 6820 2c20 3368 202c 2033 6820   , 3h , 3h , 3h 
│ │ │ │ -00025380: 7d20 2020 2020 2020 2020 2020 207c 0a7c  }            |.|
│ │ │ │ -00025390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025320: 2032 2020 2020 3320 2020 2034 2020 2020   2    3    4    
│ │ │ │ +00025330: 3520 2020 2036 2020 2020 2020 2020 2020  5    6          
│ │ │ │ +00025340: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025350: 2020 2020 2020 2020 2020 2020 2047 6c69               Gli
│ │ │ │ +00025360: 7374 203d 3e20 7b31 2c20 3368 202c 2033  st => {1, 3h , 3
│ │ │ │ +00025370: 6820 2c20 3368 202c 2033 6820 2c20 3368  h , 3h , 3h , 3h
│ │ │ │ +00025380: 202c 2033 6820 7d20 2020 2020 2020 2020   , 3h }         
│ │ │ │ +00025390: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000253a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253b0: 2020 2020 2020 3120 2020 2031 2020 2020        1    1    
│ │ │ │ -000253c0: 3120 2020 2031 2020 2020 3120 2020 2031  1    1    1    1
│ │ │ │ -000253d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000253e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000253b0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +000253c0: 2031 2020 2020 3120 2020 2031 2020 2020   1    1    1    
│ │ │ │ +000253d0: 3120 2020 2031 2020 2020 2020 2020 2020  1    1          
│ │ │ │ +000253e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000253f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025400: 2020 2020 2036 2020 2020 2020 2035 2020       6       5  
│ │ │ │ -00025410: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ -00025420: 2020 3220 2020 2020 2020 2020 207c 0a7c    2          |.|
│ │ │ │ -00025430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025440: 2020 2020 2020 2053 6567 7265 203d 3e20         Segre => 
│ │ │ │ -00025450: 3132 3636 6820 202d 2033 3930 6820 202b  1266h  - 390h  +
│ │ │ │ -00025460: 2031 3134 6820 202d 2033 3068 2020 2b20   114h  - 30h  + 
│ │ │ │ -00025470: 3668 2020 2020 2020 2020 2020 207c 0a7c  6h           |.|
│ │ │ │ -00025480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025400: 2020 2020 2020 2020 2020 2036 2020 2020             6    
│ │ │ │ +00025410: 2020 2035 2020 2020 2020 2034 2020 2020     5       4    
│ │ │ │ +00025420: 2020 3320 2020 2020 3220 2020 2020 2020    3     2       
│ │ │ │ +00025430: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025440: 2020 2020 2020 2020 2020 2020 2053 6567               Seg
│ │ │ │ +00025450: 7265 203d 3e20 3132 3636 6820 202d 2033  re => 1266h  - 3
│ │ │ │ +00025460: 3930 6820 202b 2031 3134 6820 202d 2033  90h  + 114h  - 3
│ │ │ │ +00025470: 3068 2020 2b20 3668 2020 2020 2020 2020  0h  + 6h        
│ │ │ │ +00025480: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254a0: 2020 2020 2031 2020 2020 2020 2031 2020       1       1  
│ │ │ │ -000254b0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -000254c0: 2020 3120 2020 2020 2020 2020 207c 0a7c    1          |.|
│ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254e0: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -000254f0: 2020 2020 3520 2020 2020 3420 2020 2020      5     4     
│ │ │ │ -00025500: 3320 2020 2020 3220 2020 2020 2020 2020  3     2         
│ │ │ │ -00025510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025530: 2020 2020 2020 2047 203d 3e20 3368 2020         G => 3h  
│ │ │ │ -00025540: 2b20 3368 2020 2b20 3368 2020 2b20 3368  + 3h  + 3h  + 3h
│ │ │ │ +000254a0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ +000254b0: 2020 2031 2020 2020 2020 2031 2020 2020     1       1    
│ │ │ │ +000254c0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +000254d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000254e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000254f0: 2020 2020 3620 2020 2020 3520 2020 2020      6     5     
│ │ │ │ +00025500: 3420 2020 2020 3320 2020 2020 3220 2020  4     3     2   
│ │ │ │ +00025510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025520: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025530: 2020 2020 2020 2020 2020 2020 2047 203d               G =
│ │ │ │ +00025540: 3e20 3368 2020 2b20 3368 2020 2b20 3368  > 3h  + 3h  + 3h
│ │ │ │  00025550: 2020 2b20 3368 2020 2b20 3368 2020 2b20    + 3h  + 3h  + 
│ │ │ │ -00025560: 3120 2020 2020 2020 2020 2020 207c 0a7c  1            |.|
│ │ │ │ -00025570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025580: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00025560: 3368 2020 2b20 3120 2020 2020 2020 2020  3h  + 1         
│ │ │ │ +00025570: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025590: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │  000255a0: 3120 2020 2020 3120 2020 2020 3120 2020  1     1     1   
│ │ │ │ -000255b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000255b0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +000255c0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │  000255d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000255e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000255f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -00025610: 2020 2020 2036 2020 2020 2020 2020 2020       6          
│ │ │ │ +00025600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025610: 2d2d 2d7c 0a7c 2020 2020 2036 2020 2020  ---|.|     6    
│ │ │ │  00025620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025650: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025660: 3132 3636 6820 7d7d 2020 2020 2020 2020  1266h }}        
│ │ │ │ +00025650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025660: 2020 207c 0a7c 3132 3636 6820 7d7d 2020     |.|1266h }}  
│ │ │ │  00025670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000256b0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +000256a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000256b0: 2020 207c 0a7c 2020 2020 2031 2020 2020     |.|     1    
│ │ │ │  000256c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000256e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000256f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00025700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000256f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025700: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00025710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00025750: 6931 3220 3a20 6575 3d45 756c 6572 2820  i12 : eu=Euler( 
│ │ │ │ -00025760: 492c 204f 7574 7075 743d 3e48 6173 6846  I, Output=>HashF
│ │ │ │ -00025770: 6f72 6d29 2020 2020 2020 2020 2020 2020  orm)            
│ │ │ │ +00025740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025750: 2d2d 2d2b 0a7c 6931 3220 3a20 6575 3d45  ---+.|i12 : eu=E
│ │ │ │ +00025760: 756c 6572 2820 492c 204f 7574 7075 743d  uler( I, Output=
│ │ │ │ +00025770: 3e48 6173 6846 6f72 6d29 2020 2020 2020  >HashForm)      
│ │ │ │  00025780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025790: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000257a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000257a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000257b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000257d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000257f0: 6f31 3220 3d20 4d75 7461 626c 6548 6173  o12 = MutableHas
│ │ │ │ -00025800: 6854 6162 6c65 7b2e 2e2e 352e 2e2e 7d20  hTable{...5...} 
│ │ │ │ -00025810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000257e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000257f0: 2020 207c 0a7c 6f31 3220 3d20 4d75 7461     |.|o12 = Muta
│ │ │ │ +00025800: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e  bleHashTable{...
│ │ │ │ +00025810: 352e 2e2e 7d20 2020 2020 2020 2020 2020  5...}           
│ │ │ │  00025820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025830: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025840: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025880: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025890: 6f31 3220 3a20 4d75 7461 626c 6548 6173  o12 : MutableHas
│ │ │ │ -000258a0: 6854 6162 6c65 2020 2020 2020 2020 2020  hTable          
│ │ │ │ +00025880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025890: 2020 207c 0a7c 6f31 3220 3a20 4d75 7461     |.|o12 : Muta
│ │ │ │ +000258a0: 626c 6548 6173 6854 6162 6c65 2020 2020  bleHashTable    
│ │ │ │  000258b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000258c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000258d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000258e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000258d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000258e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  000258f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00025930: 6931 3320 3a20 7065 656b 2065 7520 2020  i13 : peek eu   
│ │ │ │ -00025940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025930: 2d2d 2d2b 0a7c 6931 3320 3a20 7065 656b  ---+.|i13 : peek
│ │ │ │ +00025940: 2065 7520 2020 2020 2020 2020 2020 2020   eu             
│ │ │ │  00025950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025970: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000259d0: 6f31 3320 3d20 4d75 7461 626c 6548 6173  o13 = MutableHas
│ │ │ │ -000259e0: 6854 6162 6c65 7b45 756c 6572 203d 3e20  hTable{Euler => 
│ │ │ │ -000259f0: 3130 2020 2020 2020 2020 2020 2020 2020  10              
│ │ │ │ +000259c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000259d0: 2020 207c 0a7c 6f31 3320 3d20 4d75 7461     |.|o13 = Muta
│ │ │ │ +000259e0: 626c 6548 6173 6854 6162 6c65 7b45 756c  bleHashTable{Eul
│ │ │ │ +000259f0: 6572 203d 3e20 3130 2020 2020 2020 2020  er => 10        
│ │ │ │  00025a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a10: 2020 2020 2020 7d20 2020 2020 207c 0a7c        }      |.|
│ │ │ │ -00025a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a10: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ +00025a20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a40: 2020 2036 2020 2020 2020 3520 2020 2020     6      5     
│ │ │ │ -00025a50: 2034 2020 2020 2020 3320 2020 2020 2032   4      3      2
│ │ │ │ -00025a60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a80: 2020 2020 2020 207b 302c 2031 7d20 3d3e         {0, 1} =>
│ │ │ │ -00025a90: 2032 6820 202b 2032 3368 2020 2b20 3332   2h  + 23h  + 32
│ │ │ │ -00025aa0: 6820 202b 2033 3368 2020 2b20 3138 6820  h  + 33h  + 18h 
│ │ │ │ -00025ab0: 202b 2035 6820 2020 2020 2020 207c 0a7c   + 5h        |.|
│ │ │ │ -00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a40: 2020 2020 2020 2020 2036 2020 2020 2020           6      
│ │ │ │ +00025a50: 3520 2020 2020 2034 2020 2020 2020 3320  5      4      3 
│ │ │ │ +00025a60: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00025a70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025a80: 2020 2020 2020 2020 2020 2020 207b 302c               {0,
│ │ │ │ +00025a90: 2031 7d20 3d3e 2032 6820 202b 2032 3368   1} => 2h  + 23h
│ │ │ │ +00025aa0: 2020 2b20 3332 6820 202b 2033 3368 2020    + 32h  + 33h  
│ │ │ │ +00025ab0: 2b20 3138 6820 202b 2035 6820 2020 2020  + 18h  + 5h     
│ │ │ │ +00025ac0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ae0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00025af0: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ -00025b00: 2020 2020 2031 2020 2020 2020 207c 0a7c       1       |.|
│ │ │ │ -00025b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025ae0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +00025af0: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ +00025b00: 2020 2020 2031 2020 2020 2031 2020 2020       1     1    
│ │ │ │ +00025b10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b30: 2036 2020 2020 2020 3520 2020 2020 2034   6      5      4
│ │ │ │ -00025b40: 2020 2020 2020 3320 2020 2020 3220 2020        3     2   
│ │ │ │ -00025b50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b70: 2020 2020 2020 2043 534d 203d 3e20 3130         CSM => 10
│ │ │ │ -00025b80: 6820 202b 2031 3268 2020 2b20 3232 6820  h  + 12h  + 22h 
│ │ │ │ -00025b90: 202b 2031 3668 2020 2b20 3668 2020 2020   + 16h  + 6h    
│ │ │ │ -00025ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025b30: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00025b40: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ +00025b50: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00025b60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025b70: 2020 2020 2020 2020 2020 2020 2043 534d               CSM
│ │ │ │ +00025b80: 203d 3e20 3130 6820 202b 2031 3268 2020   => 10h  + 12h  
│ │ │ │ +00025b90: 2b20 3232 6820 202b 2031 3668 2020 2b20  + 22h  + 16h  + 
│ │ │ │ +00025ba0: 3668 2020 2020 2020 2020 2020 2020 2020  6h              
│ │ │ │ +00025bb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025bd0: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ -00025be0: 2020 2020 2020 3120 2020 2020 3120 2020        1     1   
│ │ │ │ -00025bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025bd0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00025be0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00025bf0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00025c00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c20: 3620 2020 2020 2035 2020 2020 2020 3420  6      5      4 
│ │ │ │ -00025c30: 2020 2020 2033 2020 2020 2020 3220 2020       3      2   
│ │ │ │ -00025c40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c60: 2020 2020 2020 207b 307d 203d 3e20 3668         {0} => 6h
│ │ │ │ -00025c70: 2020 2b20 3138 6820 202b 2032 3668 2020    + 18h  + 26h  
│ │ │ │ -00025c80: 2b20 3232 6820 202b 2031 3068 2020 2b20  + 22h  + 10h  + 
│ │ │ │ -00025c90: 3268 2020 2020 2020 2020 2020 207c 0a7c  2h           |.|
│ │ │ │ -00025ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025c20: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00025c30: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00025c40: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00025c50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025c60: 2020 2020 2020 2020 2020 2020 207b 307d               {0}
│ │ │ │ +00025c70: 203d 3e20 3668 2020 2b20 3138 6820 202b   => 6h  + 18h  +
│ │ │ │ +00025c80: 2032 3668 2020 2b20 3232 6820 202b 2031   26h  + 22h  + 1
│ │ │ │ +00025c90: 3068 2020 2b20 3268 2020 2020 2020 2020  0h  + 2h        
│ │ │ │ +00025ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025cc0: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00025cd0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00025ce0: 2020 3120 2020 2020 2020 2020 207c 0a7c    1          |.|
│ │ │ │ -00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025cc0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00025cd0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00025ce0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +00025cf0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d10: 3620 2020 2020 2035 2020 2020 2020 3420  6      5      4 
│ │ │ │ -00025d20: 2020 2020 2033 2020 2020 2020 3220 2020       3      2   
│ │ │ │ -00025d30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00025d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d50: 2020 2020 2020 207b 317d 203d 3e20 3668         {1} => 6h
│ │ │ │ -00025d60: 2020 2b20 3137 6820 202b 2032 3868 2020    + 17h  + 28h  
│ │ │ │ -00025d70: 2b20 3237 6820 202b 2031 3468 2020 2b20  + 27h  + 14h  + 
│ │ │ │ -00025d80: 3368 2020 2020 2020 2020 2020 207c 0a7c  3h           |.|
│ │ │ │ -00025d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025d10: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ +00025d20: 2020 2020 3420 2020 2020 2033 2020 2020      4      3    
│ │ │ │ +00025d30: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00025d40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00025d50: 2020 2020 2020 2020 2020 2020 207b 317d               {1}
│ │ │ │ +00025d60: 203d 3e20 3668 2020 2b20 3137 6820 202b   => 6h  + 17h  +
│ │ │ │ +00025d70: 2032 3868 2020 2b20 3237 6820 202b 2031   28h  + 27h  + 1
│ │ │ │ +00025d80: 3468 2020 2b20 3368 2020 2020 2020 2020  4h  + 3h        
│ │ │ │ +00025d90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00025da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025db0: 3120 2020 2020 2031 2020 2020 2020 3120  1      1      1 
│ │ │ │ -00025dc0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00025dd0: 2020 3120 2020 2020 2020 2020 207c 0a2b    1          |.+
│ │ │ │ -00025de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025db0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ +00025dc0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00025dd0: 2020 3120 2020 2020 3120 2020 2020 2020    1     1       
│ │ │ │ +00025de0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00025df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00025e30: 5468 6520 4d75 7461 626c 6548 6173 6854  The MutableHashT
│ │ │ │ -00025e40: 6162 6c65 2072 6574 7572 6e65 6420 7769  able returned wi
│ │ │ │ -00025e50: 7468 2074 6865 206f 7074 696f 6e20 4f75  th the option Ou
│ │ │ │ -00025e60: 7470 7574 3d3e 4861 7368 466f 726d 2063  tput=>HashForm c
│ │ │ │ -00025e70: 6f6e 7461 696e 730a 6469 6666 6572 656e  ontains.differen
│ │ │ │ -00025e80: 7420 696e 666f 726d 6174 696f 6e20 6465  t information de
│ │ │ │ -00025e90: 7065 6e64 696e 6720 6f6e 2074 6865 206d  pending on the m
│ │ │ │ -00025ea0: 6574 686f 6420 7769 7468 2077 6869 6368  ethod with which
│ │ │ │ -00025eb0: 2069 7420 6973 2075 7365 642e 0a41 6464   it is used..Add
│ │ │ │ -00025ec0: 6974 696f 6e61 6c6c 7920 6966 2074 6865  itionally if the
│ │ │ │ -00025ed0: 206f 7074 696f 6e20 2a6e 6f74 6520 496e   option *note In
│ │ │ │ -00025ee0: 7075 7449 7353 6d6f 6f74 683a 2049 6e70  putIsSmooth: Inp
│ │ │ │ -00025ef0: 7574 4973 536d 6f6f 7468 2c20 6973 2075  utIsSmooth, is u
│ │ │ │ -00025f00: 7365 6420 7468 656e 2074 6865 0a68 6173  sed then the.has
│ │ │ │ -00025f10: 6820 7461 626c 6520 7265 7475 726e 6564  h table returned
│ │ │ │ -00025f20: 2062 7920 7468 6520 6d65 7468 6f64 7320   by the methods 
│ │ │ │ -00025f30: 4575 6c65 7220 616e 6420 4353 4d20 7769  Euler and CSM wi
│ │ │ │ -00025f40: 6c6c 2062 6520 7468 6520 7361 6d65 2061  ll be the same a
│ │ │ │ -00025f50: 7320 7468 6174 0a72 6574 7572 6e65 6420  s that.returned 
│ │ │ │ -00025f60: 6279 2043 6865 726e 2e20 5768 656e 2075  by Chern. When u
│ │ │ │ -00025f70: 7369 6e67 2074 6865 202a 6e6f 7465 2043  sing the *note C
│ │ │ │ -00025f80: 534d 3a20 4353 4d2c 2020 636f 6d6d 616e  SM: CSM,  comman
│ │ │ │ -00025f90: 6420 696e 2074 6865 2064 6566 6175 6c74  d in the default
│ │ │ │ -00025fa0: 0a63 6f6e 6669 6775 7261 7469 6f6e 7320  .configurations 
│ │ │ │ -00025fb0: 2874 6861 7420 6973 202a 6e6f 7465 204d  (that is *note M
│ │ │ │ -00025fc0: 6574 686f 643a 204d 6574 686f 642c 3d3e  ethod: Method,=>
│ │ │ │ -00025fd0: 496e 636c 7573 696f 6e45 7863 6c75 7369  InclusionExclusi
│ │ │ │ -00025fe0: 6f6e 2c20 2a6e 6f74 650a 436f 6d70 4d65  on, *note.CompMe
│ │ │ │ -00025ff0: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64  thod: CompMethod
│ │ │ │ -00026000: 2c3d 3e50 726f 6a65 6374 6976 6544 6567  ,=>ProjectiveDeg
│ │ │ │ -00026010: 7265 6529 2074 6865 7265 2069 7320 7468  ree) there is th
│ │ │ │ -00026020: 6520 6164 6469 7469 6f6e 616c 206f 7074  e additional opt
│ │ │ │ -00026030: 696f 6e20 746f 0a73 6574 204f 7574 7075  ion to.set Outpu
│ │ │ │ -00026040: 743d 3e48 6173 6846 6f72 6d58 4c2e 2054  t=>HashFormXL. T
│ │ │ │ -00026050: 6869 7320 7265 7475 726e 7320 616c 6c20  his returns all 
│ │ │ │ -00026060: 7468 6520 7573 7561 6c20 696e 666f 726d  the usual inform
│ │ │ │ -00026070: 6174 696f 6e20 7468 6174 0a4f 7574 7075  ation that.Outpu
│ │ │ │ -00026080: 743d 3e48 6173 6846 6f72 6d20 776f 756c  t=>HashForm woul
│ │ │ │ -00026090: 6420 666f 7220 7468 6973 2063 6f6e 6669  d for this confi
│ │ │ │ -000260a0: 6775 7261 7469 6f6e 2077 6974 6820 7468  guration with th
│ │ │ │ -000260b0: 6520 6164 6469 7469 6f6e 206f 6620 7468  e addition of th
│ │ │ │ -000260c0: 650a 7072 6f6a 6563 7469 7665 2064 6567  e.projective deg
│ │ │ │ -000260d0: 7265 6573 2061 6e64 2053 6567 7265 2063  rees and Segre c
│ │ │ │ -000260e0: 6c61 7373 6573 206f 6620 7369 6e67 756c  lasses of singul
│ │ │ │ -000260f0: 6172 6974 7920 7375 6273 6368 656d 6573  arity subschemes
│ │ │ │ -00026100: 2067 656e 6572 6174 6564 2062 7920 7468   generated by th
│ │ │ │ -00026110: 650a 6879 7065 7273 7572 6661 6365 7320  e.hypersurfaces 
│ │ │ │ -00026120: 636f 6e73 6964 6572 6564 2069 6e20 7468  considered in th
│ │ │ │ -00026130: 6520 696e 636c 7573 696f 6e2f 6578 636c  e inclusion/excl
│ │ │ │ -00026140: 7573 696f 6e20 7072 6f63 6564 7572 652c  usion procedure,
│ │ │ │ -00026150: 2074 6861 7420 6973 2069 6e0a 6669 6e64   that is in.find
│ │ │ │ -00026160: 696e 6720 7468 6520 4353 4d20 636c 6173  ing the CSM clas
│ │ │ │ -00026170: 7320 6f66 2061 6c6c 2068 7970 6572 7375  s of all hypersu
│ │ │ │ -00026180: 7266 6163 6573 2067 656e 6572 6174 6564  rfaces generated
│ │ │ │ -00026190: 2062 7920 7461 6b69 6e67 2061 2070 726f   by taking a pro
│ │ │ │ -000261a0: 6475 6374 206f 660a 736f 6d65 2073 7562  duct of.some sub
│ │ │ │ -000261b0: 7365 7473 206f 6620 6765 6e65 7261 746f  sets of generato
│ │ │ │ -000261c0: 7273 206f 6620 7468 6520 696e 7075 7420  rs of the input 
│ │ │ │ -000261d0: 6964 6561 6c2e 204e 6f74 6520 7468 6174  ideal. Note that
│ │ │ │ -000261e0: 2c20 7369 6e63 6520 7468 6520 4353 4d20  , since the CSM 
│ │ │ │ -000261f0: 636c 6173 730a 6f66 2061 2073 7562 7363  class.of a subsc
│ │ │ │ -00026200: 6865 6d65 2065 7175 616c 7320 7468 6520  heme equals the 
│ │ │ │ -00026210: 4353 4d20 636c 6173 7320 6f66 2069 7473  CSM class of its
│ │ │ │ -00026220: 2072 6564 7563 6564 2073 6368 656d 652c   reduced scheme,
│ │ │ │ -00026230: 206f 7220 6571 7569 7661 6c65 6e74 6c79   or equivalently
│ │ │ │ -00026240: 2066 6f72 0a75 7320 7468 6520 4353 4d20   for.us the CSM 
│ │ │ │ -00026250: 636c 6173 7320 636f 7272 6573 706f 6e64  class correspond
│ │ │ │ -00026260: 696e 6720 746f 2061 6e20 6964 6561 6c20  ing to an ideal 
│ │ │ │ -00026270: 4920 6571 7561 6c73 2074 6865 2043 534d  I equals the CSM
│ │ │ │ -00026280: 2063 6c61 7373 206f 6620 7468 650a 7261   class of the.ra
│ │ │ │ -00026290: 6469 6361 6c20 6f66 2049 2c20 7468 656e  dical of I, then
│ │ │ │ -000262a0: 2069 6e74 6572 6e61 6c6c 7920 7765 2061   internally we a
│ │ │ │ -000262b0: 6c77 6179 7320 776f 726b 2077 6974 6820  lways work with 
│ │ │ │ -000262c0: 7261 6469 6361 6c20 6964 6561 6c73 2028  radical ideals (
│ │ │ │ -000262d0: 666f 720a 6566 6669 6369 656e 6379 2072  for.efficiency r
│ │ │ │ -000262e0: 6561 736f 6e73 292e 2048 656e 6365 2074  easons). Hence t
│ │ │ │ -000262f0: 6865 2070 726f 6a65 6374 6976 6520 6465  he projective de
│ │ │ │ -00026300: 6772 6565 7320 616e 6420 5365 6772 6520  grees and Segre 
│ │ │ │ -00026310: 636c 6173 7365 7320 636f 6d70 7574 6564  classes computed
│ │ │ │ -00026320: 0a69 6e74 6572 6e61 6c6c 7920 7769 6c6c  .internally will
│ │ │ │ -00026330: 2062 6520 7468 6f73 6520 6f66 2074 6865   be those of the
│ │ │ │ -00026340: 2072 6164 6963 616c 206f 6620 616e 2069   radical of an i
│ │ │ │ -00026350: 6465 616c 2064 6566 696e 6564 2062 7920  deal defined by 
│ │ │ │ -00026360: 6120 706f 6c79 6e6f 6d69 616c 0a77 6869  a polynomial.whi
│ │ │ │ -00026370: 6368 2069 7320 6120 7072 6f64 7563 7420  ch is a product 
│ │ │ │ -00026380: 6f66 2073 6f6d 6520 7375 6273 6574 206f  of some subset o
│ │ │ │ -00026390: 6620 7468 6520 6765 6e65 7261 746f 7273  f the generators
│ │ │ │ -000263a0: 2e20 5765 2069 6c6c 7573 7472 6174 6520  . We illustrate 
│ │ │ │ -000263b0: 7468 6973 2077 6974 6820 616e 0a65 7861  this with an.exa
│ │ │ │ -000263c0: 6d70 6c65 2062 656c 6f77 2e0a 0a2b 2d2d  mple below...+--
│ │ │ │ -000263d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025e30: 2d2d 2d2b 0a0a 5468 6520 4d75 7461 626c  ---+..The Mutabl
│ │ │ │ +00025e40: 6548 6173 6854 6162 6c65 2072 6574 7572  eHashTable retur
│ │ │ │ +00025e50: 6e65 6420 7769 7468 2074 6865 206f 7074  ned with the opt
│ │ │ │ +00025e60: 696f 6e20 4f75 7470 7574 3d3e 4861 7368  ion Output=>Hash
│ │ │ │ +00025e70: 466f 726d 2063 6f6e 7461 696e 730a 6469  Form contains.di
│ │ │ │ +00025e80: 6666 6572 656e 7420 696e 666f 726d 6174  fferent informat
│ │ │ │ +00025e90: 696f 6e20 6465 7065 6e64 696e 6720 6f6e  ion depending on
│ │ │ │ +00025ea0: 2074 6865 206d 6574 686f 6420 7769 7468   the method with
│ │ │ │ +00025eb0: 2077 6869 6368 2069 7420 6973 2075 7365   which it is use
│ │ │ │ +00025ec0: 642e 0a41 6464 6974 696f 6e61 6c6c 7920  d..Additionally 
│ │ │ │ +00025ed0: 6966 2074 6865 206f 7074 696f 6e20 2a6e  if the option *n
│ │ │ │ +00025ee0: 6f74 6520 496e 7075 7449 7353 6d6f 6f74  ote InputIsSmoot
│ │ │ │ +00025ef0: 683a 2049 6e70 7574 4973 536d 6f6f 7468  h: InputIsSmooth
│ │ │ │ +00025f00: 2c20 6973 2075 7365 6420 7468 656e 2074  , is used then t
│ │ │ │ +00025f10: 6865 0a68 6173 6820 7461 626c 6520 7265  he.hash table re
│ │ │ │ +00025f20: 7475 726e 6564 2062 7920 7468 6520 6d65  turned by the me
│ │ │ │ +00025f30: 7468 6f64 7320 4575 6c65 7220 616e 6420  thods Euler and 
│ │ │ │ +00025f40: 4353 4d20 7769 6c6c 2062 6520 7468 6520  CSM will be the 
│ │ │ │ +00025f50: 7361 6d65 2061 7320 7468 6174 0a72 6574  same as that.ret
│ │ │ │ +00025f60: 7572 6e65 6420 6279 2043 6865 726e 2e20  urned by Chern. 
│ │ │ │ +00025f70: 5768 656e 2075 7369 6e67 2074 6865 202a  When using the *
│ │ │ │ +00025f80: 6e6f 7465 2043 534d 3a20 4353 4d2c 2020  note CSM: CSM,  
│ │ │ │ +00025f90: 636f 6d6d 616e 6420 696e 2074 6865 2064  command in the d
│ │ │ │ +00025fa0: 6566 6175 6c74 0a63 6f6e 6669 6775 7261  efault.configura
│ │ │ │ +00025fb0: 7469 6f6e 7320 2874 6861 7420 6973 202a  tions (that is *
│ │ │ │ +00025fc0: 6e6f 7465 204d 6574 686f 643a 204d 6574  note Method: Met
│ │ │ │ +00025fd0: 686f 642c 3d3e 496e 636c 7573 696f 6e45  hod,=>InclusionE
│ │ │ │ +00025fe0: 7863 6c75 7369 6f6e 2c20 2a6e 6f74 650a  xclusion, *note.
│ │ │ │ +00025ff0: 436f 6d70 4d65 7468 6f64 3a20 436f 6d70  CompMethod: Comp
│ │ │ │ +00026000: 4d65 7468 6f64 2c3d 3e50 726f 6a65 6374  Method,=>Project
│ │ │ │ +00026010: 6976 6544 6567 7265 6529 2074 6865 7265  iveDegree) there
│ │ │ │ +00026020: 2069 7320 7468 6520 6164 6469 7469 6f6e   is the addition
│ │ │ │ +00026030: 616c 206f 7074 696f 6e20 746f 0a73 6574  al option to.set
│ │ │ │ +00026040: 204f 7574 7075 743d 3e48 6173 6846 6f72   Output=>HashFor
│ │ │ │ +00026050: 6d58 4c2e 2054 6869 7320 7265 7475 726e  mXL. This return
│ │ │ │ +00026060: 7320 616c 6c20 7468 6520 7573 7561 6c20  s all the usual 
│ │ │ │ +00026070: 696e 666f 726d 6174 696f 6e20 7468 6174  information that
│ │ │ │ +00026080: 0a4f 7574 7075 743d 3e48 6173 6846 6f72  .Output=>HashFor
│ │ │ │ +00026090: 6d20 776f 756c 6420 666f 7220 7468 6973  m would for this
│ │ │ │ +000260a0: 2063 6f6e 6669 6775 7261 7469 6f6e 2077   configuration w
│ │ │ │ +000260b0: 6974 6820 7468 6520 6164 6469 7469 6f6e  ith the addition
│ │ │ │ +000260c0: 206f 6620 7468 650a 7072 6f6a 6563 7469   of the.projecti
│ │ │ │ +000260d0: 7665 2064 6567 7265 6573 2061 6e64 2053  ve degrees and S
│ │ │ │ +000260e0: 6567 7265 2063 6c61 7373 6573 206f 6620  egre classes of 
│ │ │ │ +000260f0: 7369 6e67 756c 6172 6974 7920 7375 6273  singularity subs
│ │ │ │ +00026100: 6368 656d 6573 2067 656e 6572 6174 6564  chemes generated
│ │ │ │ +00026110: 2062 7920 7468 650a 6879 7065 7273 7572   by the.hypersur
│ │ │ │ +00026120: 6661 6365 7320 636f 6e73 6964 6572 6564  faces considered
│ │ │ │ +00026130: 2069 6e20 7468 6520 696e 636c 7573 696f   in the inclusio
│ │ │ │ +00026140: 6e2f 6578 636c 7573 696f 6e20 7072 6f63  n/exclusion proc
│ │ │ │ +00026150: 6564 7572 652c 2074 6861 7420 6973 2069  edure, that is i
│ │ │ │ +00026160: 6e0a 6669 6e64 696e 6720 7468 6520 4353  n.finding the CS
│ │ │ │ +00026170: 4d20 636c 6173 7320 6f66 2061 6c6c 2068  M class of all h
│ │ │ │ +00026180: 7970 6572 7375 7266 6163 6573 2067 656e  ypersurfaces gen
│ │ │ │ +00026190: 6572 6174 6564 2062 7920 7461 6b69 6e67  erated by taking
│ │ │ │ +000261a0: 2061 2070 726f 6475 6374 206f 660a 736f   a product of.so
│ │ │ │ +000261b0: 6d65 2073 7562 7365 7473 206f 6620 6765  me subsets of ge
│ │ │ │ +000261c0: 6e65 7261 746f 7273 206f 6620 7468 6520  nerators of the 
│ │ │ │ +000261d0: 696e 7075 7420 6964 6561 6c2e 204e 6f74  input ideal. Not
│ │ │ │ +000261e0: 6520 7468 6174 2c20 7369 6e63 6520 7468  e that, since th
│ │ │ │ +000261f0: 6520 4353 4d20 636c 6173 730a 6f66 2061  e CSM class.of a
│ │ │ │ +00026200: 2073 7562 7363 6865 6d65 2065 7175 616c   subscheme equal
│ │ │ │ +00026210: 7320 7468 6520 4353 4d20 636c 6173 7320  s the CSM class 
│ │ │ │ +00026220: 6f66 2069 7473 2072 6564 7563 6564 2073  of its reduced s
│ │ │ │ +00026230: 6368 656d 652c 206f 7220 6571 7569 7661  cheme, or equiva
│ │ │ │ +00026240: 6c65 6e74 6c79 2066 6f72 0a75 7320 7468  lently for.us th
│ │ │ │ +00026250: 6520 4353 4d20 636c 6173 7320 636f 7272  e CSM class corr
│ │ │ │ +00026260: 6573 706f 6e64 696e 6720 746f 2061 6e20  esponding to an 
│ │ │ │ +00026270: 6964 6561 6c20 4920 6571 7561 6c73 2074  ideal I equals t
│ │ │ │ +00026280: 6865 2043 534d 2063 6c61 7373 206f 6620  he CSM class of 
│ │ │ │ +00026290: 7468 650a 7261 6469 6361 6c20 6f66 2049  the.radical of I
│ │ │ │ +000262a0: 2c20 7468 656e 2069 6e74 6572 6e61 6c6c  , then internall
│ │ │ │ +000262b0: 7920 7765 2061 6c77 6179 7320 776f 726b  y we always work
│ │ │ │ +000262c0: 2077 6974 6820 7261 6469 6361 6c20 6964   with radical id
│ │ │ │ +000262d0: 6561 6c73 2028 666f 720a 6566 6669 6369  eals (for.effici
│ │ │ │ +000262e0: 656e 6379 2072 6561 736f 6e73 292e 2048  ency reasons). H
│ │ │ │ +000262f0: 656e 6365 2074 6865 2070 726f 6a65 6374  ence the project
│ │ │ │ +00026300: 6976 6520 6465 6772 6565 7320 616e 6420  ive degrees and 
│ │ │ │ +00026310: 5365 6772 6520 636c 6173 7365 7320 636f  Segre classes co
│ │ │ │ +00026320: 6d70 7574 6564 0a69 6e74 6572 6e61 6c6c  mputed.internall
│ │ │ │ +00026330: 7920 7769 6c6c 2062 6520 7468 6f73 6520  y will be those 
│ │ │ │ +00026340: 6f66 2074 6865 2072 6164 6963 616c 206f  of the radical o
│ │ │ │ +00026350: 6620 616e 2069 6465 616c 2064 6566 696e  f an ideal defin
│ │ │ │ +00026360: 6564 2062 7920 6120 706f 6c79 6e6f 6d69  ed by a polynomi
│ │ │ │ +00026370: 616c 0a77 6869 6368 2069 7320 6120 7072  al.which is a pr
│ │ │ │ +00026380: 6f64 7563 7420 6f66 2073 6f6d 6520 7375  oduct of some su
│ │ │ │ +00026390: 6273 6574 206f 6620 7468 6520 6765 6e65  bset of the gene
│ │ │ │ +000263a0: 7261 746f 7273 2e20 5765 2069 6c6c 7573  rators. We illus
│ │ │ │ +000263b0: 7472 6174 6520 7468 6973 2077 6974 6820  trate this with 
│ │ │ │ +000263c0: 616e 0a65 7861 6d70 6c65 2062 656c 6f77  an.example below
│ │ │ │ +000263d0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  000263e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000263f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00026420: 3420 3a20 6373 6d58 4c68 6173 683d 4353  4 : csmXLhash=CS
│ │ │ │ -00026430: 4d28 412c 492c 4f75 7470 7574 3d3e 4861  M(A,I,Output=>Ha
│ │ │ │ -00026440: 7368 466f 726d 584c 2920 2020 2020 2020  shFormXL)       
│ │ │ │ +00026410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026420: 2d2b 0a7c 6931 3420 3a20 6373 6d58 4c68  -+.|i14 : csmXLh
│ │ │ │ +00026430: 6173 683d 4353 4d28 412c 492c 4f75 7470  ash=CSM(A,I,Outp
│ │ │ │ +00026440: 7574 3d3e 4861 7368 466f 726d 584c 2920  ut=>HashFormXL) 
│ │ │ │  00026450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026460: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026470: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000264c0: 3420 3d20 4d75 7461 626c 6548 6173 6854  4 = MutableHashT
│ │ │ │ -000264d0: 6162 6c65 7b2e 2e2e 3130 2e2e 2e7d 2020  able{...10...}  
│ │ │ │ -000264e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000264b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000264c0: 207c 0a7c 6f31 3420 3d20 4d75 7461 626c   |.|o14 = Mutabl
│ │ │ │ +000264d0: 6548 6173 6854 6162 6c65 7b2e 2e2e 3130  eHashTable{...10
│ │ │ │ +000264e0: 2e2e 2e7d 2020 2020 2020 2020 2020 2020  ...}            
│ │ │ │  000264f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026500: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026510: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026550: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00026560: 3420 3a20 4d75 7461 626c 6548 6173 6854  4 : MutableHashT
│ │ │ │ -00026570: 6162 6c65 2020 2020 2020 2020 2020 2020  able            
│ │ │ │ +00026550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026560: 207c 0a7c 6f31 3420 3a20 4d75 7461 626c   |.|o14 : Mutabl
│ │ │ │ +00026570: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │  00026580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000265b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000265a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000265b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000265c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000265d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000265e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00026600: 3520 3a20 7065 656b 2063 736d 584c 6861  5 : peek csmXLha
│ │ │ │ -00026610: 7368 2020 2020 2020 2020 2020 2020 2020  sh              
│ │ │ │ +000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026600: 2d2b 0a7c 6931 3520 3a20 7065 656b 2063  -+.|i15 : peek c
│ │ │ │ +00026610: 736d 584c 6861 7368 2020 2020 2020 2020  smXLhash        
│ │ │ │  00026620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026640: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026690: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000266a0: 3520 3d20 4d75 7461 626c 6548 6173 6854  5 = MutableHashT
│ │ │ │ -000266b0: 6162 6c65 7b47 284a 6163 6f62 6961 6e29  able{G(Jacobian)
│ │ │ │ -000266c0: 7b30 7d20 3d3e 2030 2020 2020 2020 2020  {0} => 0        
│ │ │ │ +00026690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000266a0: 207c 0a7c 6f31 3520 3d20 4d75 7461 626c   |.|o15 = Mutabl
│ │ │ │ +000266b0: 6548 6173 6854 6162 6c65 7b47 284a 6163  eHashTable{G(Jac
│ │ │ │ +000266c0: 6f62 6961 6e29 7b30 7d20 3d3e 2030 2020  obian){0} => 0  
│ │ │ │  000266d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000266f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026700: 2020 2020 2053 6567 7265 284a 6163 6f62       Segre(Jacob
│ │ │ │ -00026710: 6961 6e29 7b30 7d20 3d3e 2030 2020 2020  ian){0} => 0    
│ │ │ │ -00026720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026730: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000266e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000266f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026700: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +00026710: 284a 6163 6f62 6961 6e29 7b30 7d20 3d3e  (Jacobian){0} =>
│ │ │ │ +00026720: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00026730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026740: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026770: 2020 3620 2020 2020 2020 3520 2020 2020    6       5     
│ │ │ │ -00026780: 2020 3420 2020 2020 2020 207c 0a7c 2020    4        |.|  
│ │ │ │ -00026790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000267a0: 2020 2020 2053 6567 7265 284a 6163 6f62       Segre(Jacob
│ │ │ │ -000267b0: 6961 6e29 7b30 2c20 317d 203d 3e20 3339  ian){0, 1} => 39
│ │ │ │ -000267c0: 3068 2020 2d20 3338 3668 2020 2b20 3135  0h  - 386h  + 15
│ │ │ │ -000267d0: 3068 2020 2d20 2020 2020 207c 0a7c 2020  0h  -      |.|  
│ │ │ │ -000267e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026770: 2020 2020 2020 2020 3620 2020 2020 2020          6       
│ │ │ │ +00026780: 3520 2020 2020 2020 3420 2020 2020 2020  5       4       
│ │ │ │ +00026790: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000267a0: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +000267b0: 284a 6163 6f62 6961 6e29 7b30 2c20 317d  (Jacobian){0, 1}
│ │ │ │ +000267c0: 203d 3e20 3339 3068 2020 2d20 3338 3668   => 390h  - 386h
│ │ │ │ +000267d0: 2020 2b20 3135 3068 2020 2d20 2020 2020    + 150h  -     
│ │ │ │ +000267e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000267f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026810: 2020 3120 2020 2020 2020 3120 2020 2020    1       1     
│ │ │ │ -00026820: 2020 3120 2020 2020 2020 207c 0a7c 2020    1        |.|  
│ │ │ │ -00026830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026810: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00026820: 3120 2020 2020 2020 3120 2020 2020 2020  1       1       
│ │ │ │ +00026830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026860: 2036 2020 2020 2020 3520 2020 2020 3320   6      5     3 
│ │ │ │ -00026870: 2020 2020 3220 2020 2020 207c 0a7c 2020      2      |.|  
│ │ │ │ -00026880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026890: 2020 2020 2053 6567 7265 284a 6163 6f62       Segre(Jacob
│ │ │ │ -000268a0: 6961 6e29 7b31 7d20 3d3e 202d 2031 3630  ian){1} => - 160
│ │ │ │ -000268b0: 6820 202b 2033 3268 2020 2d20 3468 2020  h  + 32h  - 4h  
│ │ │ │ -000268c0: 2b20 3268 2020 2020 2020 207c 0a7c 2020  + 2h       |.|  
│ │ │ │ -000268d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026860: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00026870: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ +00026880: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026890: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +000268a0: 284a 6163 6f62 6961 6e29 7b31 7d20 3d3e  (Jacobian){1} =>
│ │ │ │ +000268b0: 202d 2031 3630 6820 202b 2033 3268 2020   - 160h  + 32h  
│ │ │ │ +000268c0: 2d20 3468 2020 2b20 3268 2020 2020 2020  - 4h  + 2h      
│ │ │ │ +000268d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000268e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000268f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026900: 2031 2020 2020 2020 3120 2020 2020 3120   1      1     1 
│ │ │ │ -00026910: 2020 2020 3120 2020 2020 207c 0a7c 2020      1      |.|  
│ │ │ │ -00026920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026900: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00026910: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ +00026920: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026940: 2020 2020 2020 2020 2020 2020 2036 2020               6  
│ │ │ │ -00026950: 2020 2020 3520 2020 2020 2034 2020 2020      5      4    
│ │ │ │ -00026960: 2020 3320 2020 2020 2020 207c 0a7c 2020    3        |.|  
│ │ │ │ -00026970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026980: 2020 2020 2047 284a 6163 6f62 6961 6e29       G(Jacobian)
│ │ │ │ -00026990: 7b30 2c20 317d 203d 3e20 3130 6820 202b  {0, 1} => 10h  +
│ │ │ │ -000269a0: 2031 3068 2020 2b20 3130 6820 202b 2031   10h  + 10h  + 1
│ │ │ │ -000269b0: 3068 2020 2b20 2020 2020 207c 0a7c 2020  0h  +      |.|  
│ │ │ │ -000269c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026950: 2020 2036 2020 2020 2020 3520 2020 2020     6      5     
│ │ │ │ +00026960: 2034 2020 2020 2020 3320 2020 2020 2020   4      3       
│ │ │ │ +00026970: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026980: 2020 2020 2020 2020 2020 2047 284a 6163             G(Jac
│ │ │ │ +00026990: 6f62 6961 6e29 7b30 2c20 317d 203d 3e20  obian){0, 1} => 
│ │ │ │ +000269a0: 3130 6820 202b 2031 3068 2020 2b20 3130  10h  + 10h  + 10
│ │ │ │ +000269b0: 6820 202b 2031 3068 2020 2b20 2020 2020  h  + 10h  +     
│ │ │ │ +000269c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000269d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000269e0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -000269f0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ -00026a00: 2020 3120 2020 2020 2020 207c 0a7c 2020    1        |.|  
│ │ │ │ -00026a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269f0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00026a00: 2031 2020 2020 2020 3120 2020 2020 2020   1      1       
│ │ │ │ +00026a10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a30: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00026a30: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00026a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a70: 2020 2020 2047 284a 6163 6f62 6961 6e29       G(Jacobian)
│ │ │ │ -00026a80: 7b31 7d20 3d3e 2032 6820 202b 2032 6820  {1} => 2h  + 2h 
│ │ │ │ -00026a90: 202b 2031 2020 2020 2020 2020 2020 2020   + 1            
│ │ │ │ -00026aa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026a70: 2020 2020 2020 2020 2020 2047 284a 6163             G(Jac
│ │ │ │ +00026a80: 6f62 6961 6e29 7b31 7d20 3d3e 2032 6820  obian){1} => 2h 
│ │ │ │ +00026a90: 202b 2032 6820 202b 2031 2020 2020 2020   + 2h  + 1      
│ │ │ │ +00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ab0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ad0: 2020 2020 2020 2020 2031 2020 2020 2031           1     1
│ │ │ │ -00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026af0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ad0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00026ae0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ +00026af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b20: 2036 2020 2020 2020 3520 2020 2020 2034   6      5      4
│ │ │ │ -00026b30: 2020 2020 2020 3320 2020 2020 2032 2020        3      2  
│ │ │ │ -00026b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b60: 2020 2020 207b 302c 2031 7d20 3d3e 2032       {0, 1} => 2
│ │ │ │ -00026b70: 6820 202b 2032 3368 2020 2b20 3332 6820  h  + 23h  + 32h 
│ │ │ │ -00026b80: 202b 2033 3368 2020 2b20 3138 6820 202b   + 33h  + 18h  +
│ │ │ │ -00026b90: 2035 6820 2020 2020 2020 207c 0a7c 2020   5h        |.|  
│ │ │ │ -00026ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b20: 2020 2020 2020 2036 2020 2020 2020 3520         6      5 
│ │ │ │ +00026b30: 2020 2020 2034 2020 2020 2020 3320 2020       4      3   
│ │ │ │ +00026b40: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00026b50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026b60: 2020 2020 2020 2020 2020 207b 302c 2031             {0, 1
│ │ │ │ +00026b70: 7d20 3d3e 2032 6820 202b 2032 3368 2020  } => 2h  + 23h  
│ │ │ │ +00026b80: 2b20 3332 6820 202b 2033 3368 2020 2b20  + 32h  + 33h  + 
│ │ │ │ +00026b90: 3138 6820 202b 2035 6820 2020 2020 2020  18h  + 5h       
│ │ │ │ +00026ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026bc0: 2031 2020 2020 2020 3120 2020 2020 2031   1      1      1
│ │ │ │ -00026bd0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00026be0: 2020 2031 2020 2020 2020 207c 0a7c 2020     1       |.|  
│ │ │ │ -00026bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c00: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -00026c10: 2020 2020 2020 3520 2020 2020 2034 2020        5      4  
│ │ │ │ -00026c20: 2020 2020 3320 2020 2020 3220 2020 2020      3     2     
│ │ │ │ -00026c30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c50: 2020 2020 2043 534d 203d 3e20 3130 6820       CSM => 10h 
│ │ │ │ -00026c60: 202b 2031 3268 2020 2b20 3232 6820 202b   + 12h  + 22h  +
│ │ │ │ -00026c70: 2031 3668 2020 2b20 3668 2020 2020 2020   16h  + 6h      
│ │ │ │ -00026c80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ca0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00026cb0: 2020 2020 2020 3120 2020 2020 2031 2020        1      1  
│ │ │ │ -00026cc0: 2020 2020 3120 2020 2020 3120 2020 2020      1     1     
│ │ │ │ -00026cd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026cf0: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -00026d00: 2020 2020 2035 2020 2020 2020 3420 2020       5      4   
│ │ │ │ -00026d10: 2020 2033 2020 2020 2020 3220 2020 2020     3      2     
│ │ │ │ -00026d20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d40: 2020 2020 207b 307d 203d 3e20 3668 2020       {0} => 6h  
│ │ │ │ -00026d50: 2b20 3138 6820 202b 2032 3668 2020 2b20  + 18h  + 26h  + 
│ │ │ │ -00026d60: 3232 6820 202b 2031 3068 2020 2b20 3268  22h  + 10h  + 2h
│ │ │ │ -00026d70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d90: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00026da0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00026db0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00026dc0: 3120 2020 2020 2020 2020 207c 0a7c 2020  1          |.|  
│ │ │ │ -00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026de0: 2020 2020 2020 2020 2020 2020 2020 3620                6 
│ │ │ │ -00026df0: 2020 2020 2035 2020 2020 2020 3420 2020       5      4   
│ │ │ │ -00026e00: 2020 2033 2020 2020 2020 3220 2020 2020     3      2     
│ │ │ │ -00026e10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026e30: 2020 2020 207b 317d 203d 3e20 3668 2020       {1} => 6h  
│ │ │ │ -00026e40: 2b20 3137 6820 202b 2032 3868 2020 2b20  + 17h  + 28h  + 
│ │ │ │ -00026e50: 3237 6820 202b 2031 3468 2020 2b20 3368  27h  + 14h  + 3h
│ │ │ │ -00026e60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026e80: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00026e90: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ -00026ea0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ -00026eb0: 3120 2020 2020 2020 2020 207c 0a7c 2d2d  1          |.|--
│ │ │ │ -00026ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026bc0: 2020 2020 2020 2031 2020 2020 2020 3120         1      1 
│ │ │ │ +00026bd0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00026be0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ +00026bf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c10: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ +00026c20: 2020 2034 2020 2020 2020 3320 2020 2020     4      3     
│ │ │ │ +00026c30: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00026c40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026c50: 2020 2020 2020 2020 2020 2043 534d 203d             CSM =
│ │ │ │ +00026c60: 3e20 3130 6820 202b 2031 3268 2020 2b20  > 10h  + 12h  + 
│ │ │ │ +00026c70: 3232 6820 202b 2031 3668 2020 2b20 3668  22h  + 16h  + 6h
│ │ │ │ +00026c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026cb0: 2020 2020 2031 2020 2020 2020 3120 2020       1      1   
│ │ │ │ +00026cc0: 2020 2031 2020 2020 2020 3120 2020 2020     1      1     
│ │ │ │ +00026cd0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00026ce0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d00: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00026d10: 2020 3420 2020 2020 2033 2020 2020 2020    4      3      
│ │ │ │ +00026d20: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00026d30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026d40: 2020 2020 2020 2020 2020 207b 307d 203d             {0} =
│ │ │ │ +00026d50: 3e20 3668 2020 2b20 3138 6820 202b 2032  > 6h  + 18h  + 2
│ │ │ │ +00026d60: 3668 2020 2b20 3232 6820 202b 2031 3068  6h  + 22h  + 10h
│ │ │ │ +00026d70: 2020 2b20 3268 2020 2020 2020 2020 2020    + 2h          
│ │ │ │ +00026d80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026da0: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00026db0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00026dc0: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ +00026dd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026df0: 2020 2020 3620 2020 2020 2035 2020 2020      6      5    
│ │ │ │ +00026e00: 2020 3420 2020 2020 2033 2020 2020 2020    4      3      
│ │ │ │ +00026e10: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00026e20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026e30: 2020 2020 2020 2020 2020 207b 317d 203d             {1} =
│ │ │ │ +00026e40: 3e20 3668 2020 2b20 3137 6820 202b 2032  > 6h  + 17h  + 2
│ │ │ │ +00026e50: 3868 2020 2b20 3237 6820 202b 2031 3468  8h  + 27h  + 14h
│ │ │ │ +00026e60: 2020 2b20 3368 2020 2020 2020 2020 2020    + 3h          
│ │ │ │ +00026e70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00026e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026e90: 2020 2020 3120 2020 2020 2031 2020 2020      1      1    
│ │ │ │ +00026ea0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ +00026eb0: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ +00026ec0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00026ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00026f10: 2020 2020 2020 2020 2020 207d 2020 2020             }    
│ │ │ │ -00026f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026f10: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00026f20: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │  00026f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026f50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026f60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00026f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026fa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00026fb0: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │ +00026fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026fb0: 207c 0a7c 2020 2033 2020 2020 2032 2020   |.|   3     2  
│ │ │ │  00026fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ff0: 2020 2020 2020 2020 2020 207c 0a7c 3432             |.|42
│ │ │ │ -00027000: 6820 202b 2038 6820 2020 2020 2020 2020  h  + 8h         
│ │ │ │ +00026ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027000: 207c 0a7c 3432 6820 202b 2038 6820 2020   |.|42h  + 8h   
│ │ │ │  00027010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027040: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027050: 2031 2020 2020 2031 2020 2020 2020 2020   1     1        
│ │ │ │ +00027040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027050: 207c 0a7c 2020 2031 2020 2020 2031 2020   |.|   1     1  
│ │ │ │  00027060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027090: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000270a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000270a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000270b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000270c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000270d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000270e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000270f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000270e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000270f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027130: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027140: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027180: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027190: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00027180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027190: 207c 0a7c 2020 3220 2020 2020 2020 2020   |.|  2         
│ │ │ │  000271a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000271b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000271c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000271d0: 2020 2020 2020 2020 2020 207c 0a7c 3868             |.|8h
│ │ │ │ -000271e0: 2020 2b20 3468 2020 2b20 3120 2020 2020    + 4h  + 1     
│ │ │ │ -000271f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000271d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000271e0: 207c 0a7c 3868 2020 2b20 3468 2020 2b20   |.|8h  + 4h  + 
│ │ │ │ +000271f0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00027200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027220: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027230: 3120 2020 2020 3120 2020 2020 2020 2020  1     1         
│ │ │ │ +00027220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027230: 207c 0a7c 2020 3120 2020 2020 3120 2020   |.|  1     1   
│ │ │ │  00027240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027270: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027280: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00027290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000272a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000272b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000272c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000272d0: 3620 3a20 4b3d 6964 6561 6c20 495f 302a  6 : K=ideal I_0*
│ │ │ │ -000272e0: 495f 313b 2020 2020 2020 2020 2020 2020  I_1;            
│ │ │ │ +000272c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000272d0: 2d2b 0a7c 6931 3620 3a20 4b3d 6964 6561  -+.|i16 : K=idea
│ │ │ │ +000272e0: 6c20 495f 302a 495f 313b 2020 2020 2020  l I_0*I_1;      
│ │ │ │  000272f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027310: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027320: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027360: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00027370: 3620 3a20 4964 6561 6c20 6f66 2052 2020  6 : Ideal of R  
│ │ │ │ -00027380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027370: 207c 0a7c 6f31 3620 3a20 4964 6561 6c20   |.|o16 : Ideal 
│ │ │ │ +00027380: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  00027390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000273a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000273b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000273b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000273c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  000273d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00027410: 3720 3a20 4353 4d28 412c 7261 6469 6361  7 : CSM(A,radica
│ │ │ │ -00027420: 6c20 4b29 3d3d 4353 4d28 412c 4b29 2020  l K)==CSM(A,K)  
│ │ │ │ -00027430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027410: 2d2b 0a7c 6931 3720 3a20 4353 4d28 412c  -+.|i17 : CSM(A,
│ │ │ │ +00027420: 7261 6469 6361 6c20 4b29 3d3d 4353 4d28  radical K)==CSM(
│ │ │ │ +00027430: 412c 4b29 2020 2020 2020 2020 2020 2020  A,K)            
│ │ │ │  00027440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027450: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027460: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000274b0: 3720 3d20 7472 7565 2020 2020 2020 2020  7 = true        
│ │ │ │ +000274a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000274b0: 207c 0a7c 6f31 3720 3d20 7472 7565 2020   |.|o17 = true  
│ │ │ │  000274c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000274e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000274f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027500: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00027510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00027550: 3820 3a20 4a3d 6964 6561 6c20 6a61 636f  8 : J=ideal jaco
│ │ │ │ -00027560: 6269 616e 2072 6164 6963 616c 204b 3b20  bian radical K; 
│ │ │ │ -00027570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027550: 2d2b 0a7c 6931 3820 3a20 4a3d 6964 6561  -+.|i18 : J=idea
│ │ │ │ +00027560: 6c20 6a61 636f 6269 616e 2072 6164 6963  l jacobian radic
│ │ │ │ +00027570: 616c 204b 3b20 2020 2020 2020 2020 2020  al K;           
│ │ │ │  00027580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000275a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000275b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000275c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000275d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000275f0: 3820 3a20 4964 6561 6c20 6f66 2052 2020  8 : Ideal of R  
│ │ │ │ -00027600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275f0: 207c 0a7c 6f31 3820 3a20 4964 6561 6c20   |.|o18 : Ideal 
│ │ │ │ +00027600: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │  00027610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027630: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027640: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00027650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00027690: 3920 3a20 7365 674a 3d53 6567 7265 2841  9 : segJ=Segre(A
│ │ │ │ -000276a0: 2c4a 2c4f 7574 7075 743d 3e48 6173 6846  ,J,Output=>HashF
│ │ │ │ -000276b0: 6f72 6d29 2020 2020 2020 2020 2020 2020  orm)            
│ │ │ │ +00027680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027690: 2d2b 0a7c 6931 3920 3a20 7365 674a 3d53  -+.|i19 : segJ=S
│ │ │ │ +000276a0: 6567 7265 2841 2c4a 2c4f 7574 7075 743d  egre(A,J,Output=
│ │ │ │ +000276b0: 3e48 6173 6846 6f72 6d29 2020 2020 2020  >HashForm)      
│ │ │ │  000276c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000276d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000276e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000276d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000276e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000276f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027720: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00027730: 3920 3d20 4d75 7461 626c 6548 6173 6854  9 = MutableHashT
│ │ │ │ -00027740: 6162 6c65 7b2e 2e2e 342e 2e2e 7d20 2020  able{...4...}   
│ │ │ │ -00027750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027730: 207c 0a7c 6f31 3920 3d20 4d75 7461 626c   |.|o19 = Mutabl
│ │ │ │ +00027740: 6548 6173 6854 6162 6c65 7b2e 2e2e 342e  eHashTable{...4.
│ │ │ │ +00027750: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │  00027760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027770: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027780: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000277b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000277d0: 3920 3a20 4d75 7461 626c 6548 6173 6854  9 : MutableHashT
│ │ │ │ -000277e0: 6162 6c65 2020 2020 2020 2020 2020 2020  able            
│ │ │ │ +000277c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000277d0: 207c 0a7c 6f31 3920 3a20 4d75 7461 626c   |.|o19 : Mutabl
│ │ │ │ +000277e0: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │  000277f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027810: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027820: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00027830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00027870: 3020 3a20 6373 6d58 4c68 6173 6823 2822  0 : csmXLhash#("
│ │ │ │ -00027880: 4728 4a61 636f 6269 616e 2922 7c74 6f53  G(Jacobian)"|toS
│ │ │ │ -00027890: 7472 696e 6728 7b30 2c31 7d29 293d 3d73  tring({0,1}))==s
│ │ │ │ -000278a0: 6567 4a23 2247 2220 2020 2020 2020 2020  egJ#"G"         
│ │ │ │ -000278b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000278c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027870: 2d2b 0a7c 6932 3020 3a20 6373 6d58 4c68  -+.|i20 : csmXLh
│ │ │ │ +00027880: 6173 6823 2822 4728 4a61 636f 6269 616e  ash#("G(Jacobian
│ │ │ │ +00027890: 2922 7c74 6f53 7472 696e 6728 7b30 2c31  )"|toString({0,1
│ │ │ │ +000278a0: 7d29 293d 3d73 6567 4a23 2247 2220 2020  }))==segJ#"G"   
│ │ │ │ +000278b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000278d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000278e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000278f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027900: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00027910: 3020 3d20 7472 7565 2020 2020 2020 2020  0 = true        
│ │ │ │ +00027900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027910: 207c 0a7c 6f32 3020 3d20 7472 7565 2020   |.|o20 = true  
│ │ │ │  00027920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027950: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027960: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -000279b0: 3120 3a20 6373 6d58 4c68 6173 6823 2822  1 : csmXLhash#("
│ │ │ │ -000279c0: 5365 6772 6528 4a61 636f 6269 616e 2922  Segre(Jacobian)"
│ │ │ │ -000279d0: 7c74 6f53 7472 696e 6728 7b30 2c31 7d29  |toString({0,1})
│ │ │ │ -000279e0: 293d 3d73 6567 4a23 2253 6567 7265 2220  )==segJ#"Segre" 
│ │ │ │ -000279f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000279b0: 2d2b 0a7c 6932 3120 3a20 6373 6d58 4c68  -+.|i21 : csmXLh
│ │ │ │ +000279c0: 6173 6823 2822 5365 6772 6528 4a61 636f  ash#("Segre(Jaco
│ │ │ │ +000279d0: 6269 616e 2922 7c74 6f53 7472 696e 6728  bian)"|toString(
│ │ │ │ +000279e0: 7b30 2c31 7d29 293d 3d73 6567 4a23 2253  {0,1}))==segJ#"S
│ │ │ │ +000279f0: 6567 7265 2220 2020 2020 2020 2020 2020  egre"           
│ │ │ │ +00027a00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00027a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a40: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00027a50: 3120 3d20 7472 7565 2020 2020 2020 2020  1 = true        
│ │ │ │ +00027a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a50: 207c 0a7c 6f32 3120 3d20 7472 7565 2020   |.|o21 = true  
│ │ │ │  00027a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027aa0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675  -----------+..Fu
│ │ │ │ -00027af0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -00027b00: 696f 6e61 6c20 6172 6775 6d65 6e74 206e  ional argument n
│ │ │ │ -00027b10: 616d 6564 204f 7574 7075 743a 0a3d 3d3d  amed Output:.===
│ │ │ │ -00027b20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00027ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027af0: 2d2b 0a0a 4675 6e63 7469 6f6e 7320 7769  -+..Functions wi
│ │ │ │ +00027b00: 7468 206f 7074 696f 6e61 6c20 6172 6775  th optional argu
│ │ │ │ +00027b10: 6d65 6e74 206e 616d 6564 204f 7574 7075  ment named Outpu
│ │ │ │ +00027b20: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  t:.=============
│ │ │ │  00027b30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00027b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00027b50: 2022 4368 6572 6e28 2e2e 2e2c 4f75 7470   "Chern(...,Outp
│ │ │ │ -00027b60: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565  ut=>...)" -- see
│ │ │ │ -00027b70: 202a 6e6f 7465 2043 6865 726e 3a20 4368   *note Chern: Ch
│ │ │ │ -00027b80: 6572 6e2c 202d 2d20 5468 6520 4368 6572  ern, -- The Cher
│ │ │ │ -00027b90: 6e20 636c 6173 730a 2020 2a20 2243 534d  n class.  * "CSM
│ │ │ │ -00027ba0: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e  (...,Output=>...
│ │ │ │ -00027bb0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -00027bc0: 4353 4d3a 2043 534d 2c20 2d2d 2054 6865  CSM: CSM, -- The
│ │ │ │ -00027bd0: 0a20 2020 2043 6865 726e 2d53 6368 7761  .    Chern-Schwa
│ │ │ │ -00027be0: 7274 7a2d 4d61 6350 6865 7273 6f6e 2063  rtz-MacPherson c
│ │ │ │ -00027bf0: 6c61 7373 0a20 202a 2022 4575 6c65 7228  lass.  * "Euler(
│ │ │ │ -00027c00: 2e2e 2e2c 4f75 7470 7574 3d3e 2e2e 2e29  ...,Output=>...)
│ │ │ │ -00027c10: 2220 2d2d 2073 6565 202a 6e6f 7465 2045  " -- see *note E
│ │ │ │ -00027c20: 756c 6572 3a20 4575 6c65 722c 202d 2d20  uler: Euler, -- 
│ │ │ │ -00027c30: 5468 6520 4575 6c65 720a 2020 2020 4368  The Euler.    Ch
│ │ │ │ -00027c40: 6172 6163 7465 7269 7374 6963 0a20 202a  aracteristic.  *
│ │ │ │ -00027c50: 2022 5365 6772 6528 2e2e 2e2c 4f75 7470   "Segre(...,Outp
│ │ │ │ -00027c60: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565  ut=>...)" -- see
│ │ │ │ -00027c70: 202a 6e6f 7465 2053 6567 7265 3a20 5365   *note Segre: Se
│ │ │ │ -00027c80: 6772 652c 202d 2d20 5468 6520 5365 6772  gre, -- The Segr
│ │ │ │ -00027c90: 6520 636c 6173 7320 6f66 2061 0a20 2020  e class of a.   
│ │ │ │ -00027ca0: 2073 7562 7363 6865 6d65 0a0a 466f 7220   subscheme..For 
│ │ │ │ -00027cb0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -00027cc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00027cd0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -00027ce0: 6f74 6520 4f75 7470 7574 3a20 4f75 7470  ote Output: Outp
│ │ │ │ -00027cf0: 7574 2c20 6973 2061 202a 6e6f 7465 2073  ut, is a *note s
│ │ │ │ -00027d00: 796d 626f 6c3a 2028 4d61 6361 756c 6179  ymbol: (Macaulay
│ │ │ │ -00027d10: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d  2Doc)Symbol,...-
│ │ │ │ -00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00027b50: 3d0a 0a20 202a 2022 4368 6572 6e28 2e2e  =..  * "Chern(..
│ │ │ │ +00027b60: 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 2220  .,Output=>...)" 
│ │ │ │ +00027b70: 2d2d 2073 6565 202a 6e6f 7465 2043 6865  -- see *note Che
│ │ │ │ +00027b80: 726e 3a20 4368 6572 6e2c 202d 2d20 5468  rn: Chern, -- Th
│ │ │ │ +00027b90: 6520 4368 6572 6e20 636c 6173 730a 2020  e Chern class.  
│ │ │ │ +00027ba0: 2a20 2243 534d 282e 2e2e 2c4f 7574 7075  * "CSM(...,Outpu
│ │ │ │ +00027bb0: 743d 3e2e 2e2e 2922 202d 2d20 7365 6520  t=>...)" -- see 
│ │ │ │ +00027bc0: 2a6e 6f74 6520 4353 4d3a 2043 534d 2c20  *note CSM: CSM, 
│ │ │ │ +00027bd0: 2d2d 2054 6865 0a20 2020 2043 6865 726e  -- The.    Chern
│ │ │ │ +00027be0: 2d53 6368 7761 7274 7a2d 4d61 6350 6865  -Schwartz-MacPhe
│ │ │ │ +00027bf0: 7273 6f6e 2063 6c61 7373 0a20 202a 2022  rson class.  * "
│ │ │ │ +00027c00: 4575 6c65 7228 2e2e 2e2c 4f75 7470 7574  Euler(...,Output
│ │ │ │ +00027c10: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a  =>...)" -- see *
│ │ │ │ +00027c20: 6e6f 7465 2045 756c 6572 3a20 4575 6c65  note Euler: Eule
│ │ │ │ +00027c30: 722c 202d 2d20 5468 6520 4575 6c65 720a  r, -- The Euler.
│ │ │ │ +00027c40: 2020 2020 4368 6172 6163 7465 7269 7374      Characterist
│ │ │ │ +00027c50: 6963 0a20 202a 2022 5365 6772 6528 2e2e  ic.  * "Segre(..
│ │ │ │ +00027c60: 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 2220  .,Output=>...)" 
│ │ │ │ +00027c70: 2d2d 2073 6565 202a 6e6f 7465 2053 6567  -- see *note Seg
│ │ │ │ +00027c80: 7265 3a20 5365 6772 652c 202d 2d20 5468  re: Segre, -- Th
│ │ │ │ +00027c90: 6520 5365 6772 6520 636c 6173 7320 6f66  e Segre class of
│ │ │ │ +00027ca0: 2061 0a20 2020 2073 7562 7363 6865 6d65   a.    subscheme
│ │ │ │ +00027cb0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +00027cc0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +00027cd0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +00027ce0: 6563 7420 2a6e 6f74 6520 4f75 7470 7574  ect *note Output
│ │ │ │ +00027cf0: 3a20 4f75 7470 7574 2c20 6973 2061 202a  : Output, is a *
│ │ │ │ +00027d00: 6e6f 7465 2073 796d 626f 6c3a 2028 4d61  note symbol: (Ma
│ │ │ │ +00027d10: 6361 756c 6179 3244 6f63 2953 796d 626f  caulay2Doc)Symbo
│ │ │ │ +00027d20: 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  l,...-----------
│ │ │ │  00027d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -00027d70: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -00027d80: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -00027d90: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -00027da0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -00027db0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ -00027dc0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -00027dd0: 6b61 6765 732f 0a43 6861 7261 6374 6572  kages/.Character
│ │ │ │ -00027de0: 6973 7469 6343 6c61 7373 6573 2e6d 323a  isticClasses.m2:
│ │ │ │ -00027df0: 3234 3639 3a30 2e0a 1f0a 4669 6c65 3a20  2469:0....File: 
│ │ │ │ -00027e00: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ -00027e10: 6173 7365 732e 696e 666f 2c20 4e6f 6465  asses.info, Node
│ │ │ │ -00027e20: 3a20 7072 6f62 6162 696c 6973 7469 6320  : probabilistic 
│ │ │ │ -00027e30: 616c 676f 7269 7468 6d2c 204e 6578 743a  algorithm, Next:
│ │ │ │ -00027e40: 2053 6567 7265 2c20 5072 6576 3a20 4f75   Segre, Prev: Ou
│ │ │ │ -00027e50: 7470 7574 2c20 5570 3a20 546f 700a 0a70  tput, Up: Top..p
│ │ │ │ -00027e60: 726f 6261 6269 6c69 7374 6963 2061 6c67  robabilistic alg
│ │ │ │ -00027e70: 6f72 6974 686d 0a2a 2a2a 2a2a 2a2a 2a2a  orithm.*********
│ │ │ │ -00027e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -00027e90: 5468 6520 616c 676f 7269 7468 6d73 2075  The algorithms u
│ │ │ │ -00027ea0: 7365 6420 666f 7220 7468 6520 636f 6d70  sed for the comp
│ │ │ │ -00027eb0: 7574 6174 696f 6e20 6f66 2063 6861 7261  utation of chara
│ │ │ │ -00027ec0: 6374 6572 6973 7469 6320 636c 6173 7365  cteristic classe
│ │ │ │ -00027ed0: 7320 6172 650a 7072 6f62 6162 696c 6973  s are.probabilis
│ │ │ │ -00027ee0: 7469 632e 2054 6865 6f72 6574 6963 616c  tic. Theoretical
│ │ │ │ -00027ef0: 6c79 2c20 7468 6579 2063 616c 6375 6c61  ly, they calcula
│ │ │ │ -00027f00: 7465 2074 6865 2063 6c61 7373 6573 2063  te the classes c
│ │ │ │ -00027f10: 6f72 7265 6374 6c79 2066 6f72 2061 0a67  orrectly for a.g
│ │ │ │ -00027f20: 656e 6572 616c 2063 686f 6963 6520 6f66  eneral choice of
│ │ │ │ -00027f30: 2063 6572 7461 696e 2070 6f6c 796e 6f6d   certain polynom
│ │ │ │ -00027f40: 6961 6c73 2e20 5468 6174 2069 732c 2074  ials. That is, t
│ │ │ │ -00027f50: 6865 7265 2069 7320 616e 206f 7065 6e20  here is an open 
│ │ │ │ -00027f60: 6465 6e73 6520 5a61 7269 736b 690a 7365  dense Zariski.se
│ │ │ │ -00027f70: 7420 666f 7220 7768 6963 6820 7468 6520  t for which the 
│ │ │ │ -00027f80: 616c 676f 7269 7468 6d20 7969 656c 6473  algorithm yields
│ │ │ │ -00027f90: 2074 6865 2063 6f72 7265 6374 2063 6c61   the correct cla
│ │ │ │ -00027fa0: 7373 2c20 692e 652e 2c20 7468 6520 636f  ss, i.e., the co
│ │ │ │ -00027fb0: 7272 6563 7420 636c 6173 730a 6973 2063  rrect class.is c
│ │ │ │ -00027fc0: 616c 6375 6c61 7465 6420 7769 7468 2070  alculated with p
│ │ │ │ -00027fd0: 726f 6261 6269 6c69 7479 2031 2e20 486f  robability 1. Ho
│ │ │ │ -00027fe0: 7765 7665 722c 2073 696e 6365 2074 6865  wever, since the
│ │ │ │ -00027ff0: 2069 6d70 6c65 6d65 6e74 6174 696f 6e20   implementation 
│ │ │ │ -00028000: 776f 726b 7320 6f76 6572 0a61 2064 6973  works over.a dis
│ │ │ │ -00028010: 6372 6574 6520 7072 6f62 6162 696c 6974  crete probabilit
│ │ │ │ -00028020: 7920 7370 6163 6520 7468 6572 6520 6973  y space there is
│ │ │ │ -00028030: 2061 2076 6572 7920 736d 616c 6c2c 2062   a very small, b
│ │ │ │ -00028040: 7574 206e 6f6e 2d7a 6572 6f2c 2070 726f  ut non-zero, pro
│ │ │ │ -00028050: 6261 6269 6c69 7479 0a6f 6620 6e6f 7420  bability.of not 
│ │ │ │ -00028060: 636f 6d70 7574 696e 6720 7468 6520 636f  computing the co
│ │ │ │ -00028070: 7272 6563 7420 636c 6173 732e 2053 6b65  rrect class. Ske
│ │ │ │ -00028080: 7074 6963 616c 2075 7365 7273 2073 686f  ptical users sho
│ │ │ │ -00028090: 756c 6420 7265 7065 6174 2063 616c 6375  uld repeat calcu
│ │ │ │ -000280a0: 6c61 7469 6f6e 730a 7365 7665 7261 6c20  lations.several 
│ │ │ │ -000280b0: 7469 6d65 7320 746f 2069 6e63 7265 6173  times to increas
│ │ │ │ -000280c0: 6520 7468 6520 7072 6f62 6162 696c 6974  e the probabilit
│ │ │ │ -000280d0: 7920 6f66 2063 6f6d 7075 7469 6e67 2074  y of computing t
│ │ │ │ -000280e0: 6865 2063 6f72 7265 6374 2063 6c61 7373  he correct class
│ │ │ │ -000280f0: 2e0a 0a49 6e20 7468 6520 6361 7365 206f  ...In the case o
│ │ │ │ -00028100: 6620 7468 6520 7379 6d62 6f6c 6963 2069  f the symbolic i
│ │ │ │ -00028110: 6d70 6c65 6d65 6e74 6174 696f 6e20 6f66  mplementation of
│ │ │ │ -00028120: 2074 6865 2050 726f 6a65 6374 6976 6544   the ProjectiveD
│ │ │ │ -00028130: 6567 7265 6520 6d65 7468 6f64 0a70 7261  egree method.pra
│ │ │ │ -00028140: 6374 6963 616c 2065 7870 6572 6965 6e63  ctical experienc
│ │ │ │ -00028150: 6520 616e 6420 616c 676f 7269 7468 6d20  e and algorithm 
│ │ │ │ -00028160: 7465 7374 696e 6720 696e 6469 6361 7465  testing indicate
│ │ │ │ -00028170: 2074 6861 7420 6120 6669 6e69 7465 2066   that a finite f
│ │ │ │ -00028180: 6965 6c64 2077 6974 680a 6f76 6572 2032  ield with.over 2
│ │ │ │ -00028190: 3530 3030 2065 6c65 6d65 6e74 7320 6973  5000 elements is
│ │ │ │ -000281a0: 206d 6f72 6520 7468 616e 2073 7566 6669   more than suffi
│ │ │ │ -000281b0: 6369 656e 7420 746f 2065 7870 6563 7420  cient to expect 
│ │ │ │ -000281c0: 6120 636f 7272 6563 7420 7265 7375 6c74  a correct result
│ │ │ │ -000281d0: 2077 6974 680a 6869 6768 2070 726f 6261   with.high proba
│ │ │ │ -000281e0: 6269 6c69 7479 2c20 692e 652e 2075 7369  bility, i.e. usi
│ │ │ │ -000281f0: 6e67 2074 6865 2066 696e 6974 6520 6669  ng the finite fi
│ │ │ │ -00028200: 656c 6420 6b6b 3d5a 5a2f 3235 3037 3320  eld kk=ZZ/25073 
│ │ │ │ -00028210: 7468 6520 6578 7065 7269 6d65 6e74 616c  the experimental
│ │ │ │ -00028220: 0a63 6861 6e63 6520 6f66 2066 6169 6c75  .chance of failu
│ │ │ │ -00028230: 7265 2077 6974 6820 7468 6520 5072 6f6a  re with the Proj
│ │ │ │ -00028240: 6563 7469 7665 4465 6772 6565 2061 6c67  ectiveDegree alg
│ │ │ │ -00028250: 6f72 6974 686d 206f 6e20 6120 7661 7269  orithm on a vari
│ │ │ │ -00028260: 6574 7920 6f66 2065 7861 6d70 6c65 730a  ety of examples.
│ │ │ │ -00028270: 7761 7320 6c65 7373 2074 6861 6e20 312f  was less than 1/
│ │ │ │ -00028280: 3230 3030 2e20 5573 696e 6720 7468 6520  2000. Using the 
│ │ │ │ -00028290: 6669 6e69 7465 2066 6965 6c64 206b 6b3d  finite field kk=
│ │ │ │ -000282a0: 5a5a 2f33 3237 3439 2072 6573 756c 7465  ZZ/32749 resulte
│ │ │ │ -000282b0: 6420 696e 206e 6f0a 6661 696c 7572 6573  d in no.failures
│ │ │ │ -000282c0: 2069 6e20 6f76 6572 2031 3030 3030 2061   in over 10000 a
│ │ │ │ -000282d0: 7474 656d 7074 7320 6f66 2073 6576 6572  ttempts of sever
│ │ │ │ -000282e0: 616c 2064 6966 6665 7265 6e74 2065 7861  al different exa
│ │ │ │ -000282f0: 6d70 6c65 732e 0a0a 5765 2069 6c6c 7573  mples...We illus
│ │ │ │ -00028300: 7472 6174 6520 7468 6520 7072 6f62 6162  trate the probab
│ │ │ │ -00028310: 696c 6973 7469 6320 6265 6861 7669 6f75  ilistic behaviou
│ │ │ │ -00028320: 7220 7769 7468 2061 6e20 6578 616d 706c  r with an exampl
│ │ │ │ -00028330: 6520 7768 6572 6520 7468 6520 6368 6f73  e where the chos
│ │ │ │ -00028340: 656e 0a72 616e 646f 6d20 7365 6564 206c  en.random seed l
│ │ │ │ -00028350: 6561 6473 2074 6f20 6120 7772 6f6e 6720  eads to a wrong 
│ │ │ │ -00028360: 7265 7375 6c74 2069 6e20 7468 6520 6669  result in the fi
│ │ │ │ -00028370: 7273 7420 6361 6c63 756c 6174 696f 6e2e  rst calculation.
│ │ │ │ -00028380: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027d70: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ +00027d80: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ +00027d90: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ +00027da0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ +00027db0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e  /macaulay2-1.26.
│ │ │ │ +00027dc0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61  06+ds/M2/Macaula
│ │ │ │ +00027dd0: 7932 2f70 6163 6b61 6765 732f 0a43 6861  y2/packages/.Cha
│ │ │ │ +00027de0: 7261 6374 6572 6973 7469 6343 6c61 7373  racteristicClass
│ │ │ │ +00027df0: 6573 2e6d 323a 3234 3639 3a30 2e0a 1f0a  es.m2:2469:0....
│ │ │ │ +00027e00: 4669 6c65 3a20 4368 6172 6163 7465 7269  File: Characteri
│ │ │ │ +00027e10: 7374 6963 436c 6173 7365 732e 696e 666f  sticClasses.info
│ │ │ │ +00027e20: 2c20 4e6f 6465 3a20 7072 6f62 6162 696c  , Node: probabil
│ │ │ │ +00027e30: 6973 7469 6320 616c 676f 7269 7468 6d2c  istic algorithm,
│ │ │ │ +00027e40: 204e 6578 743a 2053 6567 7265 2c20 5072   Next: Segre, Pr
│ │ │ │ +00027e50: 6576 3a20 4f75 7470 7574 2c20 5570 3a20  ev: Output, Up: 
│ │ │ │ +00027e60: 546f 700a 0a70 726f 6261 6269 6c69 7374  Top..probabilist
│ │ │ │ +00027e70: 6963 2061 6c67 6f72 6974 686d 0a2a 2a2a  ic algorithm.***
│ │ │ │ +00027e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00027e90: 2a2a 2a2a 0a0a 5468 6520 616c 676f 7269  ****..The algori
│ │ │ │ +00027ea0: 7468 6d73 2075 7365 6420 666f 7220 7468  thms used for th
│ │ │ │ +00027eb0: 6520 636f 6d70 7574 6174 696f 6e20 6f66  e computation of
│ │ │ │ +00027ec0: 2063 6861 7261 6374 6572 6973 7469 6320   characteristic 
│ │ │ │ +00027ed0: 636c 6173 7365 7320 6172 650a 7072 6f62  classes are.prob
│ │ │ │ +00027ee0: 6162 696c 6973 7469 632e 2054 6865 6f72  abilistic. Theor
│ │ │ │ +00027ef0: 6574 6963 616c 6c79 2c20 7468 6579 2063  etically, they c
│ │ │ │ +00027f00: 616c 6375 6c61 7465 2074 6865 2063 6c61  alculate the cla
│ │ │ │ +00027f10: 7373 6573 2063 6f72 7265 6374 6c79 2066  sses correctly f
│ │ │ │ +00027f20: 6f72 2061 0a67 656e 6572 616c 2063 686f  or a.general cho
│ │ │ │ +00027f30: 6963 6520 6f66 2063 6572 7461 696e 2070  ice of certain p
│ │ │ │ +00027f40: 6f6c 796e 6f6d 6961 6c73 2e20 5468 6174  olynomials. That
│ │ │ │ +00027f50: 2069 732c 2074 6865 7265 2069 7320 616e   is, there is an
│ │ │ │ +00027f60: 206f 7065 6e20 6465 6e73 6520 5a61 7269   open dense Zari
│ │ │ │ +00027f70: 736b 690a 7365 7420 666f 7220 7768 6963  ski.set for whic
│ │ │ │ +00027f80: 6820 7468 6520 616c 676f 7269 7468 6d20  h the algorithm 
│ │ │ │ +00027f90: 7969 656c 6473 2074 6865 2063 6f72 7265  yields the corre
│ │ │ │ +00027fa0: 6374 2063 6c61 7373 2c20 692e 652e 2c20  ct class, i.e., 
│ │ │ │ +00027fb0: 7468 6520 636f 7272 6563 7420 636c 6173  the correct clas
│ │ │ │ +00027fc0: 730a 6973 2063 616c 6375 6c61 7465 6420  s.is calculated 
│ │ │ │ +00027fd0: 7769 7468 2070 726f 6261 6269 6c69 7479  with probability
│ │ │ │ +00027fe0: 2031 2e20 486f 7765 7665 722c 2073 696e   1. However, sin
│ │ │ │ +00027ff0: 6365 2074 6865 2069 6d70 6c65 6d65 6e74  ce the implement
│ │ │ │ +00028000: 6174 696f 6e20 776f 726b 7320 6f76 6572  ation works over
│ │ │ │ +00028010: 0a61 2064 6973 6372 6574 6520 7072 6f62  .a discrete prob
│ │ │ │ +00028020: 6162 696c 6974 7920 7370 6163 6520 7468  ability space th
│ │ │ │ +00028030: 6572 6520 6973 2061 2076 6572 7920 736d  ere is a very sm
│ │ │ │ +00028040: 616c 6c2c 2062 7574 206e 6f6e 2d7a 6572  all, but non-zer
│ │ │ │ +00028050: 6f2c 2070 726f 6261 6269 6c69 7479 0a6f  o, probability.o
│ │ │ │ +00028060: 6620 6e6f 7420 636f 6d70 7574 696e 6720  f not computing 
│ │ │ │ +00028070: 7468 6520 636f 7272 6563 7420 636c 6173  the correct clas
│ │ │ │ +00028080: 732e 2053 6b65 7074 6963 616c 2075 7365  s. Skeptical use
│ │ │ │ +00028090: 7273 2073 686f 756c 6420 7265 7065 6174  rs should repeat
│ │ │ │ +000280a0: 2063 616c 6375 6c61 7469 6f6e 730a 7365   calculations.se
│ │ │ │ +000280b0: 7665 7261 6c20 7469 6d65 7320 746f 2069  veral times to i
│ │ │ │ +000280c0: 6e63 7265 6173 6520 7468 6520 7072 6f62  ncrease the prob
│ │ │ │ +000280d0: 6162 696c 6974 7920 6f66 2063 6f6d 7075  ability of compu
│ │ │ │ +000280e0: 7469 6e67 2074 6865 2063 6f72 7265 6374  ting the correct
│ │ │ │ +000280f0: 2063 6c61 7373 2e0a 0a49 6e20 7468 6520   class...In the 
│ │ │ │ +00028100: 6361 7365 206f 6620 7468 6520 7379 6d62  case of the symb
│ │ │ │ +00028110: 6f6c 6963 2069 6d70 6c65 6d65 6e74 6174  olic implementat
│ │ │ │ +00028120: 696f 6e20 6f66 2074 6865 2050 726f 6a65  ion of the Proje
│ │ │ │ +00028130: 6374 6976 6544 6567 7265 6520 6d65 7468  ctiveDegree meth
│ │ │ │ +00028140: 6f64 0a70 7261 6374 6963 616c 2065 7870  od.practical exp
│ │ │ │ +00028150: 6572 6965 6e63 6520 616e 6420 616c 676f  erience and algo
│ │ │ │ +00028160: 7269 7468 6d20 7465 7374 696e 6720 696e  rithm testing in
│ │ │ │ +00028170: 6469 6361 7465 2074 6861 7420 6120 6669  dicate that a fi
│ │ │ │ +00028180: 6e69 7465 2066 6965 6c64 2077 6974 680a  nite field with.
│ │ │ │ +00028190: 6f76 6572 2032 3530 3030 2065 6c65 6d65  over 25000 eleme
│ │ │ │ +000281a0: 6e74 7320 6973 206d 6f72 6520 7468 616e  nts is more than
│ │ │ │ +000281b0: 2073 7566 6669 6369 656e 7420 746f 2065   sufficient to e
│ │ │ │ +000281c0: 7870 6563 7420 6120 636f 7272 6563 7420  xpect a correct 
│ │ │ │ +000281d0: 7265 7375 6c74 2077 6974 680a 6869 6768  result with.high
│ │ │ │ +000281e0: 2070 726f 6261 6269 6c69 7479 2c20 692e   probability, i.
│ │ │ │ +000281f0: 652e 2075 7369 6e67 2074 6865 2066 696e  e. using the fin
│ │ │ │ +00028200: 6974 6520 6669 656c 6420 6b6b 3d5a 5a2f  ite field kk=ZZ/
│ │ │ │ +00028210: 3235 3037 3320 7468 6520 6578 7065 7269  25073 the experi
│ │ │ │ +00028220: 6d65 6e74 616c 0a63 6861 6e63 6520 6f66  mental.chance of
│ │ │ │ +00028230: 2066 6169 6c75 7265 2077 6974 6820 7468   failure with th
│ │ │ │ +00028240: 6520 5072 6f6a 6563 7469 7665 4465 6772  e ProjectiveDegr
│ │ │ │ +00028250: 6565 2061 6c67 6f72 6974 686d 206f 6e20  ee algorithm on 
│ │ │ │ +00028260: 6120 7661 7269 6574 7920 6f66 2065 7861  a variety of exa
│ │ │ │ +00028270: 6d70 6c65 730a 7761 7320 6c65 7373 2074  mples.was less t
│ │ │ │ +00028280: 6861 6e20 312f 3230 3030 2e20 5573 696e  han 1/2000. Usin
│ │ │ │ +00028290: 6720 7468 6520 6669 6e69 7465 2066 6965  g the finite fie
│ │ │ │ +000282a0: 6c64 206b 6b3d 5a5a 2f33 3237 3439 2072  ld kk=ZZ/32749 r
│ │ │ │ +000282b0: 6573 756c 7465 6420 696e 206e 6f0a 6661  esulted in no.fa
│ │ │ │ +000282c0: 696c 7572 6573 2069 6e20 6f76 6572 2031  ilures in over 1
│ │ │ │ +000282d0: 3030 3030 2061 7474 656d 7074 7320 6f66  0000 attempts of
│ │ │ │ +000282e0: 2073 6576 6572 616c 2064 6966 6665 7265   several differe
│ │ │ │ +000282f0: 6e74 2065 7861 6d70 6c65 732e 0a0a 5765  nt examples...We
│ │ │ │ +00028300: 2069 6c6c 7573 7472 6174 6520 7468 6520   illustrate the 
│ │ │ │ +00028310: 7072 6f62 6162 696c 6973 7469 6320 6265  probabilistic be
│ │ │ │ +00028320: 6861 7669 6f75 7220 7769 7468 2061 6e20  haviour with an 
│ │ │ │ +00028330: 6578 616d 706c 6520 7768 6572 6520 7468  example where th
│ │ │ │ +00028340: 6520 6368 6f73 656e 0a72 616e 646f 6d20  e chosen.random 
│ │ │ │ +00028350: 7365 6564 206c 6561 6473 2074 6f20 6120  seed leads to a 
│ │ │ │ +00028360: 7772 6f6e 6720 7265 7375 6c74 2069 6e20  wrong result in 
│ │ │ │ +00028370: 7468 6520 6669 7273 7420 6361 6c63 756c  the first calcul
│ │ │ │ +00028380: 6174 696f 6e2e 0a0a 2b2d 2d2d 2d2d 2d2d  ation...+-------
│ │ │ │  00028390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000283a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000283b0: 2d2d 2d2b 0a7c 6931 203a 2073 6574 5261  ---+.|i1 : setRa
│ │ │ │ -000283c0: 6e64 6f6d 5365 6564 2031 3231 3b20 2020  ndomSeed 121;   
│ │ │ │ -000283d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283e0: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574        |.| -- set
│ │ │ │ -000283f0: 7469 6e67 2072 616e 646f 6d20 7365 6564  ting random seed
│ │ │ │ -00028400: 2074 6f20 3132 3120 2020 2020 2020 2020   to 121         
│ │ │ │ -00028410: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00028420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000283b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +000283c0: 2073 6574 5261 6e64 6f6d 5365 6564 2031   setRandomSeed 1
│ │ │ │ +000283d0: 3231 3b20 2020 2020 2020 2020 2020 2020  21;             
│ │ │ │ +000283e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000283f0: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ +00028400: 6d20 7365 6564 2074 6f20 3132 3120 2020  m seed to 121   
│ │ │ │ +00028410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028420: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00028430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00028450: 3220 3a20 5220 3d20 5151 5b78 2c79 2c7a  2 : R = QQ[x,y,z
│ │ │ │ -00028460: 2c77 5d20 2020 2020 2020 2020 2020 2020  ,w]             
│ │ │ │ -00028470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028450: 2d2d 2b0a 7c69 3220 3a20 5220 3d20 5151  --+.|i2 : R = QQ
│ │ │ │ +00028460: 5b78 2c79 2c7a 2c77 5d20 2020 2020 2020  [x,y,z,w]       
│ │ │ │ +00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000284a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284b0: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020    |.|o2 = R     
│ │ │ │ -000284c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000284b0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ +000284c0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  000284d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000284e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000284f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028510: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00028520: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ -00028530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00028550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028520: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c  |o2 : Polynomial
│ │ │ │ +00028530: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +00028540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028550: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00028560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00028580: 7c69 3320 3a20 4920 3d20 6d69 6e6f 7273  |i3 : I = minors
│ │ │ │ -00028590: 2832 2c6d 6174 7269 787b 7b78 2c79 2c7a  (2,matrix{{x,y,z
│ │ │ │ -000285a0: 7d2c 7b79 2c7a 2c77 7d7d 2920 2020 2020  },{y,z,w}})     
│ │ │ │ -000285b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00028570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028580: 2d2d 2d2d 2b0a 7c69 3320 3a20 4920 3d20  ----+.|i3 : I = 
│ │ │ │ +00028590: 6d69 6e6f 7273 2832 2c6d 6174 7269 787b  minors(2,matrix{
│ │ │ │ +000285a0: 7b78 2c79 2c7a 7d2c 7b79 2c7a 2c77 7d7d  {x,y,z},{y,z,w}}
│ │ │ │ +000285b0: 2920 2020 2020 207c 0a7c 2020 2020 2020  )      |.|      
│ │ │ │  000285c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000285d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000285e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000285f0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00028600: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00028610: 2020 2020 2020 207c 0a7c 6f33 203d 2069         |.|o3 = i
│ │ │ │ -00028620: 6465 616c 2028 2d20 7920 202b 2078 2a7a  deal (- y  + x*z
│ │ │ │ -00028630: 2c20 2d20 792a 7a20 2b20 782a 772c 202d  , - y*z + x*w, -
│ │ │ │ -00028640: 207a 2020 2b20 792a 7729 7c0a 7c20 2020   z  + y*w)|.|   
│ │ │ │ -00028650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000285e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000285f0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00028600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028610: 2020 2020 2032 2020 2020 2020 207c 0a7c       2       |.|
│ │ │ │ +00028620: 6f33 203d 2069 6465 616c 2028 2d20 7920  o3 = ideal (- y 
│ │ │ │ +00028630: 202b 2078 2a7a 2c20 2d20 792a 7a20 2b20   + x*z, - y*z + 
│ │ │ │ +00028640: 782a 772c 202d 207a 2020 2b20 792a 7729  x*w, - z  + y*w)
│ │ │ │ +00028650: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00028660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028670: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00028680: 6f33 203a 2049 6465 616c 206f 6620 5220  o3 : Ideal of R 
│ │ │ │ -00028690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028680: 2020 207c 0a7c 6f33 203a 2049 6465 616c     |.|o3 : Ideal
│ │ │ │ +00028690: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │  000286a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000286b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000286b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  000286c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000286d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000286e0: 2d2d 2d2b 0a7c 6934 203a 2043 6865 726e  ---+.|i4 : Chern
│ │ │ │ -000286f0: 2028 492c 436f 6d70 4d65 7468 6f64 3d3e   (I,CompMethod=>
│ │ │ │ -00028700: 506e 5265 7369 6475 616c 2920 2020 2020  PnResidual)     
│ │ │ │ -00028710: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000286e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ +000286f0: 2043 6865 726e 2028 492c 436f 6d70 4d65   Chern (I,CompMe
│ │ │ │ +00028700: 7468 6f64 3d3e 506e 5265 7369 6475 616c  thod=>PnResidual
│ │ │ │ +00028710: 2920 2020 2020 2020 2020 2020 7c0a 7c20  )           |.| 
│ │ │ │  00028720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028740: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00028750: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │ +00028740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028750: 0a7c 2020 2020 2020 2033 2020 2020 2032  .|       3     2
│ │ │ │  00028760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028770: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00028780: 3420 3d20 3248 2020 2b20 3348 2020 2020  4 = 2H  + 3H    
│ │ │ │ -00028790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000287a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000287b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028780: 2020 7c0a 7c6f 3420 3d20 3248 2020 2b20    |.|o4 = 2H  + 
│ │ │ │ +00028790: 3348 2020 2020 2020 2020 2020 2020 2020  3H              
│ │ │ │ +000287a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000287b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000287c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000287d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000287e0: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20    |.|     ZZ[H] 
│ │ │ │ -000287f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000287e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000287f0: 5a5a 5b48 5d20 2020 2020 2020 2020 2020  ZZ[H]           
│ │ │ │  00028800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028810: 2020 2020 207c 0a7c 6f34 203a 202d 2d2d       |.|o4 : ---
│ │ │ │ -00028820: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +00028810: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +00028820: 203a 202d 2d2d 2d2d 2020 2020 2020 2020   : -----        
│ │ │ │  00028830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028840: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00028850: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00028840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028850: 7c20 2020 2020 2020 2034 2020 2020 2020  |        4      
│ │ │ │  00028860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028870: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028880: 2020 2020 2048 2020 2020 2020 2020 2020       H          
│ │ │ │ +00028870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028880: 207c 0a7c 2020 2020 2020 2048 2020 2020   |.|       H    
│ │ │ │  00028890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000288a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000288b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000288a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000288b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │  000288c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000288d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000288e0: 2d2b 0a7c 6935 203a 2043 6865 726e 2028  -+.|i5 : Chern (
│ │ │ │ -000288f0: 492c 436f 6d70 4d65 7468 6f64 3d3e 506e  I,CompMethod=>Pn
│ │ │ │ -00028900: 5265 7369 6475 616c 2920 2020 2020 2020  Residual)       
│ │ │ │ -00028910: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000288e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2043  -------+.|i5 : C
│ │ │ │ +000288f0: 6865 726e 2028 492c 436f 6d70 4d65 7468  hern (I,CompMeth
│ │ │ │ +00028900: 6f64 3d3e 506e 5265 7369 6475 616c 2920  od=>PnResidual) 
│ │ │ │ +00028910: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00028920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028940: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028950: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │ +00028940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028950: 2020 2020 2020 2033 2020 2020 2032 2020         3     2  
│ │ │ │  00028960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028970: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00028980: 3d20 3248 2020 2b20 3348 2020 2020 2020  = 2H  + 3H      
│ │ │ │ +00028970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028980: 7c0a 7c6f 3520 3d20 3248 2020 2b20 3348  |.|o5 = 2H  + 3H
│ │ │ │  00028990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000289b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000289a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000289b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000289c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000289d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289e0: 7c0a 7c20 2020 2020 5a5a 5b48 5d20 2020  |.|     ZZ[H]   
│ │ │ │ -000289f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000289e0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a        |.|     ZZ
│ │ │ │ +000289f0: 5b48 5d20 2020 2020 2020 2020 2020 2020  [H]             
│ │ │ │  00028a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a10: 2020 207c 0a7c 6f35 203a 202d 2d2d 2d2d     |.|o5 : -----
│ │ │ │ -00028a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a10: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +00028a20: 202d 2d2d 2d2d 2020 2020 2020 2020 2020   -----          
│ │ │ │  00028a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00028a50: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028a50: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │  00028a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00028a80: 2020 2048 2020 2020 2020 2020 2020 2020     H            
│ │ │ │ +00028a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028a80: 0a7c 2020 2020 2020 2048 2020 2020 2020  .|       H      
│ │ │ │  00028a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028aa0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00028ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ab0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00028ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00028ae0: 0a7c 6936 203a 2043 6865 726e 2028 492c  .|i6 : Chern (I,
│ │ │ │ -00028af0: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265  CompMethod=>PnRe
│ │ │ │ -00028b00: 7369 6475 616c 2920 2020 2020 2020 2020  sidual)         
│ │ │ │ -00028b10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00028ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ae0: 2d2d 2d2d 2d2b 0a7c 6936 203a 2043 6865  -----+.|i6 : Che
│ │ │ │ +00028af0: 726e 2028 492c 436f 6d70 4d65 7468 6f64  rn (I,CompMethod
│ │ │ │ +00028b00: 3d3e 506e 5265 7369 6475 616c 2920 2020  =>PnResidual)   
│ │ │ │ +00028b10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00028b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b40: 2020 2020 207c 0a7c 2020 2020 2020 2033       |.|       3
│ │ │ │ -00028b50: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00028b40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00028b50: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │  00028b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b70: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -00028b80: 3248 2020 2b20 3348 2020 2020 2020 2020  2H  + 3H        
│ │ │ │ +00028b70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028b80: 7c6f 3620 3d20 3248 2020 2b20 3348 2020  |o6 = 2H  + 3H  
│ │ │ │  00028b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028be0: 7c20 2020 2020 5a5a 5b48 5d20 2020 2020  |     ZZ[H]     
│ │ │ │ -00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028be0: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b48      |.|     ZZ[H
│ │ │ │ +00028bf0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  00028c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c10: 207c 0a7c 6f36 203a 202d 2d2d 2d2d 2020   |.|o6 : -----  
│ │ │ │ -00028c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c10: 2020 2020 2020 207c 0a7c 6f36 203a 202d         |.|o6 : -
│ │ │ │ +00028c20: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  00028c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c40: 2020 2020 7c0a 7c20 2020 2020 2020 2034      |.|        4
│ │ │ │ -00028c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028c50: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │  00028c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028c80: 2048 2020 2020 2020 2020 2020 2020 2020   H              
│ │ │ │ +00028c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028c80: 2020 2020 2020 2048 2020 2020 2020 2020         H        
│ │ │ │  00028c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ca0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -00028cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028cb0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │  00028cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00028ce0: 6937 203a 2043 6865 726e 2849 2c43 6f6d  i7 : Chern(I,Com
│ │ │ │ -00028cf0: 704d 6574 686f 643d 3e50 726f 6a65 6374  pMethod=>Project
│ │ │ │ -00028d00: 6976 6544 6567 7265 6529 2020 2020 2020  iveDegree)      
│ │ │ │ -00028d10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00028cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ce0: 2d2d 2d2b 0a7c 6937 203a 2043 6865 726e  ---+.|i7 : Chern
│ │ │ │ +00028cf0: 2849 2c43 6f6d 704d 6574 686f 643d 3e50  (I,CompMethod=>P
│ │ │ │ +00028d00: 726f 6a65 6374 6976 6544 6567 7265 6529  rojectiveDegree)
│ │ │ │ +00028d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00028d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d40: 2020 207c 0a7c 2020 2020 2020 2033 2020     |.|       3  
│ │ │ │ -00028d50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00028d40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00028d50: 2020 2033 2020 2020 2032 2020 2020 2020     3     2      
│ │ │ │  00028d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d70: 2020 2020 2020 7c0a 7c6f 3720 3d20 3268        |.|o7 = 2h
│ │ │ │ -00028d80: 2020 2b20 3368 2020 2020 2020 2020 2020    + 3h          
│ │ │ │ +00028d70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00028d80: 3720 3d20 3268 2020 2b20 3368 2020 2020  7 = 2h  + 3h    
│ │ │ │  00028d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028da0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00028db0: 2020 2031 2020 2020 2031 2020 2020 2020     1     1      
│ │ │ │ +00028da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028db0: 0a7c 2020 2020 2020 2031 2020 2020 2031  .|       1     1
│ │ │ │  00028dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00028de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028de0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00028df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028e10: 0a7c 2020 2020 205a 5a5b 6820 5d20 2020  .|     ZZ[h ]   
│ │ │ │ -00028e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e10: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ +00028e20: 6820 5d20 2020 2020 2020 2020 2020 2020  h ]             
│ │ │ │  00028e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e40: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ -00028e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028e50: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  00028e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e70: 2020 2020 207c 0a7c 6f37 203a 202d 2d2d       |.|o7 : ---
│ │ │ │ -00028e80: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │ +00028e70: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +00028e80: 203a 202d 2d2d 2d2d 2d20 2020 2020 2020   : ------       
│ │ │ │  00028e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ea0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00028eb0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +00028ea0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028eb0: 7c20 2020 2020 2020 2034 2020 2020 2020  |        4      
│ │ │ │  00028ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ed0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028ee0: 2020 2020 2068 2020 2020 2020 2020 2020       h          
│ │ │ │ +00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ee0: 207c 0a7c 2020 2020 2020 2068 2020 2020   |.|       h    
│ │ │ │  00028ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028f10: 7c20 2020 2020 2020 2031 2020 2020 2020  |        1      
│ │ │ │ +00028f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f10: 2020 2020 7c0a 7c20 2020 2020 2020 2031      |.|        1
│ │ │ │  00028f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00028f40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00028f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028f70: 2d2d 2d2d 2b0a 2d2d 2d2d 2d2d 2d2d 2d2d  ----+.----------
│ │ │ │ +00028f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 2d2d 2d2d  ----------+.----
│ │ │ │  00028f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028fc0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -00028fd0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -00028fe0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -00028ff0: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -00029000: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ -00029010: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ -00029020: 6179 322f 7061 636b 6167 6573 2f0a 4368  ay2/packages/.Ch
│ │ │ │ -00029030: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ -00029040: 7365 732e 6d32 3a32 3337 383a 302e 0a1f  ses.m2:2378:0...
│ │ │ │ -00029050: 0a46 696c 653a 2043 6861 7261 6374 6572  .File: Character
│ │ │ │ -00029060: 6973 7469 6343 6c61 7373 6573 2e69 6e66  isticClasses.inf
│ │ │ │ -00029070: 6f2c 204e 6f64 653a 2053 6567 7265 2c20  o, Node: Segre, 
│ │ │ │ -00029080: 4e65 7874 3a20 546f 7269 6343 686f 7752  Next: ToricChowR
│ │ │ │ -00029090: 696e 672c 2050 7265 763a 2070 726f 6261  ing, Prev: proba
│ │ │ │ -000290a0: 6269 6c69 7374 6963 2061 6c67 6f72 6974  bilistic algorit
│ │ │ │ -000290b0: 686d 2c20 5570 3a20 546f 700a 0a53 6567  hm, Up: Top..Seg
│ │ │ │ -000290c0: 7265 202d 2d20 5468 6520 5365 6772 6520  re -- The Segre 
│ │ │ │ -000290d0: 636c 6173 7320 6f66 2061 2073 7562 7363  class of a subsc
│ │ │ │ -000290e0: 6865 6d65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  heme.***********
│ │ │ │ +00028fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00028fd0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00028fe0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00028ff0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00029000: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00029010: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ +00029020: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00029030: 6573 2f0a 4368 6172 6163 7465 7269 7374  es/.Characterist
│ │ │ │ +00029040: 6963 436c 6173 7365 732e 6d32 3a32 3337  icClasses.m2:237
│ │ │ │ +00029050: 383a 302e 0a1f 0a46 696c 653a 2043 6861  8:0....File: Cha
│ │ │ │ +00029060: 7261 6374 6572 6973 7469 6343 6c61 7373  racteristicClass
│ │ │ │ +00029070: 6573 2e69 6e66 6f2c 204e 6f64 653a 2053  es.info, Node: S
│ │ │ │ +00029080: 6567 7265 2c20 4e65 7874 3a20 546f 7269  egre, Next: Tori
│ │ │ │ +00029090: 6343 686f 7752 696e 672c 2050 7265 763a  cChowRing, Prev:
│ │ │ │ +000290a0: 2070 726f 6261 6269 6c69 7374 6963 2061   probabilistic a
│ │ │ │ +000290b0: 6c67 6f72 6974 686d 2c20 5570 3a20 546f  lgorithm, Up: To
│ │ │ │ +000290c0: 700a 0a53 6567 7265 202d 2d20 5468 6520  p..Segre -- The 
│ │ │ │ +000290d0: 5365 6772 6520 636c 6173 7320 6f66 2061  Segre class of a
│ │ │ │ +000290e0: 2073 7562 7363 6865 6d65 0a2a 2a2a 2a2a   subscheme.*****
│ │ │ │  000290f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ -00029110: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -00029120: 2020 5365 6772 6520 490a 2020 2020 2020    Segre I.      
│ │ │ │ -00029130: 2020 5365 6772 6528 412c 4929 0a20 2020    Segre(A,I).   
│ │ │ │ -00029140: 2020 2020 2053 6567 7265 2858 2c4a 290a       Segre(X,J).
│ │ │ │ -00029150: 2020 2020 2020 2020 5365 6772 6528 4368          Segre(Ch
│ │ │ │ -00029160: 2c58 2c4a 290a 2020 2a20 496e 7075 7473  ,X,J).  * Inputs
│ │ │ │ -00029170: 3a0a 2020 2020 2020 2a20 492c 2061 6e20  :.      * I, an 
│ │ │ │ -00029180: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ -00029190: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ -000291a0: 2c2c 2061 206d 756c 7469 2d68 6f6d 6f67  ,, a multi-homog
│ │ │ │ -000291b0: 656e 656f 7573 2069 6465 616c 2069 6e20  eneous ideal in 
│ │ │ │ -000291c0: 610a 2020 2020 2020 2020 6772 6164 6564  a.        graded
│ │ │ │ -000291d0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67   polynomial ring
│ │ │ │ -000291e0: 206f 7665 7220 6120 6669 656c 6420 6465   over a field de
│ │ │ │ -000291f0: 6669 6e69 6e67 2061 2063 6c6f 7365 6420  fining a closed 
│ │ │ │ -00029200: 7375 6273 6368 656d 6520 5620 6f66 0a20  subscheme V of. 
│ │ │ │ -00029210: 2020 2020 2020 205c 5050 5e7b 6e5f 317d         \PP^{n_1}
│ │ │ │ -00029220: 782e 2e2e 785c 5050 5e7b 6e5f 6d7d 0a20  x...x\PP^{n_m}. 
│ │ │ │ -00029230: 2020 2020 202a 2041 2c20 6120 2a6e 6f74       * A, a *not
│ │ │ │ -00029240: 6520 7175 6f74 6965 6e74 2072 696e 673a  e quotient ring:
│ │ │ │ -00029250: 2028 4d61 6361 756c 6179 3244 6f63 2951   (Macaulay2Doc)Q
│ │ │ │ -00029260: 756f 7469 656e 7452 696e 672c 2c0a 2020  uotientRing,,.  
│ │ │ │ -00029270: 2020 2020 2020 413d 5c5a 5a5b 685f 312c        A=\ZZ[h_1,
│ │ │ │ -00029280: 2e2e 2e2c 685f 6d5d 2f28 685f 315e 7b6e  ...,h_m]/(h_1^{n
│ │ │ │ -00029290: 5f31 2b31 7d2c 2e2e 2e2c 685f 6d5e 7b6e  _1+1},...,h_m^{n
│ │ │ │ -000292a0: 5f6d 2b31 7d29 2071 756f 7469 656e 7420  _m+1}) quotient 
│ │ │ │ -000292b0: 7269 6e67 0a20 2020 2020 2020 2072 6570  ring.        rep
│ │ │ │ -000292c0: 7265 7365 6e74 696e 6720 7468 6520 4368  resenting the Ch
│ │ │ │ -000292d0: 6f77 2072 696e 6720 6f66 205c 5050 5e7b  ow ring of \PP^{
│ │ │ │ -000292e0: 6e5f 317d 782e 2e2e 785c 5050 5e7b 6e5f  n_1}x...x\PP^{n_
│ │ │ │ -000292f0: 6d7d 2c20 7468 6973 2072 696e 6720 7368  m}, this ring sh
│ │ │ │ -00029300: 6f75 6c64 0a20 2020 2020 2020 2062 6520  ould.        be 
│ │ │ │ -00029310: 6275 696c 7420 7573 696e 6720 7468 6520  built using the 
│ │ │ │ -00029320: 2a6e 6f74 6520 4368 6f77 5269 6e67 3a20  *note ChowRing: 
│ │ │ │ -00029330: 4368 6f77 5269 6e67 2c20 636f 6d6d 616e  ChowRing, comman
│ │ │ │ -00029340: 640a 2020 2020 2020 2a20 4a2c 2061 6e20  d.      * J, an 
│ │ │ │ -00029350: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61  *note ideal: (Ma
│ │ │ │ -00029360: 6361 756c 6179 3244 6f63 2949 6465 616c  caulay2Doc)Ideal
│ │ │ │ -00029370: 2c2c 2069 6e20 7468 6520 6772 6164 6564  ,, in the graded
│ │ │ │ -00029380: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67   polynomial ring
│ │ │ │ -00029390: 0a20 2020 2020 2020 2077 6869 6368 2069  .        which i
│ │ │ │ -000293a0: 7320 636f 6f72 6469 6e61 7465 2072 696e  s coordinate rin
│ │ │ │ -000293b0: 6720 6f66 2074 6865 204e 6f72 6d61 6c20  g of the Normal 
│ │ │ │ -000293c0: 546f 7269 6320 5661 7269 6574 7920 580a  Toric Variety X.
│ │ │ │ -000293d0: 2020 2020 2020 2a20 582c 2061 202a 6e6f        * X, a *no
│ │ │ │ -000293e0: 7465 206e 6f72 6d61 6c20 746f 7269 6320  te normal toric 
│ │ │ │ -000293f0: 7661 7269 6574 793a 0a20 2020 2020 2020  variety:.       
│ │ │ │ -00029400: 2028 4e6f 726d 616c 546f 7269 6356 6172   (NormalToricVar
│ │ │ │ -00029410: 6965 7469 6573 294e 6f72 6d61 6c54 6f72  ieties)NormalTor
│ │ │ │ -00029420: 6963 5661 7269 6574 792c 2c20 7768 6963  icVariety,, whic
│ │ │ │ -00029430: 6820 6973 2074 6865 2061 6d62 6965 6e74  h is the ambient
│ │ │ │ -00029440: 2073 7061 6365 0a20 2020 2020 2020 2077   space.        w
│ │ │ │ -00029450: 6869 6368 2063 6f6e 7461 696e 7320 5628  hich contains V(
│ │ │ │ -00029460: 4a29 0a20 2020 2020 202a 2043 682c 2061  J).      * Ch, a
│ │ │ │ -00029470: 202a 6e6f 7465 2071 756f 7469 656e 7420   *note quotient 
│ │ │ │ -00029480: 7269 6e67 3a20 284d 6163 6175 6c61 7932  ring: (Macaulay2
│ │ │ │ -00029490: 446f 6329 5175 6f74 6965 6e74 5269 6e67  Doc)QuotientRing
│ │ │ │ -000294a0: 2c2c 2074 6865 2043 686f 7720 7269 6e67  ,, the Chow ring
│ │ │ │ -000294b0: 0a20 2020 2020 2020 206f 6620 7468 6520  .        of the 
│ │ │ │ -000294c0: 746f 7269 6320 7661 7269 6574 7920 582c  toric variety X,
│ │ │ │ -000294d0: 2043 683d 2872 696e 6720 4a29 2f28 5352   Ch=(ring J)/(SR
│ │ │ │ -000294e0: 2b4c 5229 2077 6865 7265 2053 5220 6973  +LR) where SR is
│ │ │ │ -000294f0: 2074 6865 0a20 2020 2020 2020 2053 7461   the.        Sta
│ │ │ │ -00029500: 6e6c 6579 2d52 6569 736e 6572 2069 6465  nley-Reisner ide
│ │ │ │ -00029510: 616c 206f 6620 7468 6520 6661 6e20 6465  al of the fan de
│ │ │ │ -00029520: 6669 6e69 6e67 2058 2061 6e64 204c 5220  fining X and LR 
│ │ │ │ -00029530: 6973 2074 6865 206c 696e 6561 720a 2020  is the linear.  
│ │ │ │ -00029540: 2020 2020 2020 7265 6c61 7469 6f6e 7320        relations 
│ │ │ │ -00029550: 6964 6561 6c2c 2074 6869 7320 7269 6e67  ideal, this ring
│ │ │ │ -00029560: 2073 686f 756c 6420 6265 2062 7569 6c74   should be built
│ │ │ │ -00029570: 2075 7369 6e67 2074 6865 202a 6e6f 7465   using the *note
│ │ │ │ -00029580: 0a20 2020 2020 2020 2054 6f72 6963 4368  .        ToricCh
│ │ │ │ -00029590: 6f77 5269 6e67 3a20 546f 7269 6343 686f  owRing: ToricCho
│ │ │ │ -000295a0: 7752 696e 672c 2063 6f6d 6d61 6e64 0a20  wRing, command. 
│ │ │ │ -000295b0: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -000295c0: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ -000295d0: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ -000295e0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -000295f0: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ -00029600: 2020 2020 202a 2043 6f6d 704d 6574 686f       * CompMetho
│ │ │ │ -00029610: 6420 286d 6973 7369 6e67 2064 6f63 756d  d (missing docum
│ │ │ │ -00029620: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e  entation) => ...
│ │ │ │ -00029630: 2c20 6465 6661 756c 7420 7661 6c75 650a  , default value.
│ │ │ │ -00029640: 2020 2020 2020 2020 5072 6f6a 6563 7469          Projecti
│ │ │ │ -00029650: 7665 4465 6772 6565 2c20 5072 6f6a 6563  veDegree, Projec
│ │ │ │ -00029660: 7469 7665 4465 6772 6565 2c20 7468 6973  tiveDegree, this
│ │ │ │ -00029670: 2061 6c67 6f72 6974 686d 206d 6179 2062   algorithm may b
│ │ │ │ -00029680: 6520 7573 6564 2066 6f72 0a20 2020 2020  e used for.     
│ │ │ │ -00029690: 2020 2073 7562 7363 6865 6d65 7320 6f66     subschemes of
│ │ │ │ -000296a0: 2061 6e79 2061 7070 6c69 6361 626c 6520   any applicable 
│ │ │ │ -000296b0: 746f 7269 6320 7661 7269 6574 7920 2874  toric variety (t
│ │ │ │ -000296c0: 6869 7320 6d61 7920 6265 2063 6865 636b  his may be check
│ │ │ │ -000296d0: 6564 2075 7369 6e67 0a20 2020 2020 2020  ed using.       
│ │ │ │ -000296e0: 2074 6865 202a 6e6f 7465 2043 6865 636b   the *note Check
│ │ │ │ -000296f0: 546f 7269 6356 6172 6965 7479 5661 6c69  ToricVarietyVali
│ │ │ │ -00029700: 643a 2043 6865 636b 546f 7269 6356 6172  d: CheckToricVar
│ │ │ │ -00029710: 6965 7479 5661 6c69 642c 2063 6f6d 6d61  ietyValid, comma
│ │ │ │ -00029720: 6e64 290a 2020 2020 2020 2a20 436f 6d70  nd).      * Comp
│ │ │ │ -00029730: 4d65 7468 6f64 2028 6d69 7373 696e 6720  Method (missing 
│ │ │ │ -00029740: 646f 6375 6d65 6e74 6174 696f 6e29 203d  documentation) =
│ │ │ │ -00029750: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -00029760: 616c 7565 0a20 2020 2020 2020 2050 726f  alue.        Pro
│ │ │ │ -00029770: 6a65 6374 6976 6544 6567 7265 652c 2050  jectiveDegree, P
│ │ │ │ -00029780: 6e52 6573 6964 7561 6c2c 2074 6869 7320  nResidual, this 
│ │ │ │ -00029790: 616c 676f 7269 7468 6d20 6d61 7920 6265  algorithm may be
│ │ │ │ -000297a0: 2075 7365 6420 666f 7220 7375 6273 6368   used for subsch
│ │ │ │ -000297b0: 656d 6573 0a20 2020 2020 2020 206f 6620  emes.        of 
│ │ │ │ -000297c0: 5c50 505e 6e20 6f6e 6c79 0a20 2020 2020  \PP^n only.     
│ │ │ │ -000297d0: 202a 204f 7574 7075 7420 3d3e 202e 2e2e   * Output => ...
│ │ │ │ -000297e0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -000297f0: 4368 6f77 5269 6e67 456c 656d 656e 742c  ChowRingElement,
│ │ │ │ -00029800: 2043 686f 7752 696e 6745 6c65 6d65 6e74   ChowRingElement
│ │ │ │ -00029810: 2c20 7265 7475 726e 730a 2020 2020 2020  , returns.      
│ │ │ │ -00029820: 2020 6120 5269 6e67 456c 656d 656e 7420    a RingElement 
│ │ │ │ -00029830: 696e 2074 6865 2043 686f 7720 7269 6e67  in the Chow ring
│ │ │ │ -00029840: 206f 6620 7468 6520 6170 7072 6f70 7269   of the appropri
│ │ │ │ -00029850: 6174 6520 616d 6269 656e 7420 7370 6163  ate ambient spac
│ │ │ │ -00029860: 650a 2020 2020 2020 2a20 4f75 7470 7574  e.      * Output
│ │ │ │ -00029870: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00029880: 2076 616c 7565 2043 686f 7752 696e 6745   value ChowRingE
│ │ │ │ -00029890: 6c65 6d65 6e74 2c20 4861 7368 466f 726d  lement, HashForm
│ │ │ │ -000298a0: 2c20 4861 7368 466f 726d 0a20 2020 2020  , HashForm.     
│ │ │ │ -000298b0: 2020 2072 6574 7572 6e73 2061 204d 7574     returns a Mut
│ │ │ │ -000298c0: 6162 6c65 4861 7368 5461 626c 6520 636f  ableHashTable co
│ │ │ │ -000298d0: 6e74 6169 6e69 6e67 2074 6865 2066 6f6c  ntaining the fol
│ │ │ │ -000298e0: 6c6f 7769 6e67 206b 6579 733a 2022 4722  lowing keys: "G"
│ │ │ │ -000298f0: 2028 7468 650a 2020 2020 2020 2020 706f   (the.        po
│ │ │ │ -00029900: 6c79 6e6f 6d69 616c 2077 6974 6820 636f  lynomial with co
│ │ │ │ -00029910: 6566 6669 6369 656e 7473 206f 6620 7468  efficients of th
│ │ │ │ -00029920: 6520 6879 7065 7270 6c61 6e65 2063 6c61  e hyperplane cla
│ │ │ │ -00029930: 7373 6573 2072 6570 7265 7365 6e74 696e  sses representin
│ │ │ │ -00029940: 6720 7468 650a 2020 2020 2020 2020 7072  g the.        pr
│ │ │ │ -00029950: 6f6a 6563 7469 7665 2064 6567 7265 6573  ojective degrees
│ │ │ │ -00029960: 292c 2022 476c 6973 7422 2028 7468 6520  ), "Glist" (the 
│ │ │ │ -00029970: 6c69 7374 2066 6f72 6d20 6f66 2022 4722  list form of "G"
│ │ │ │ -00029980: 2920 2c20 2253 6567 7265 2220 2874 6865  ) , "Segre" (the
│ │ │ │ -00029990: 0a20 2020 2020 2020 2074 6f74 616c 2053  .        total S
│ │ │ │ -000299a0: 6567 7265 2063 6c61 7373 206f 6620 7468  egre class of th
│ │ │ │ -000299b0: 6520 696e 7075 7429 2c22 5365 6772 654c  e input),"SegreL
│ │ │ │ -000299c0: 6973 7422 2028 7468 6520 6c69 7374 2066  ist" (the list f
│ │ │ │ -000299d0: 6f72 6d20 6f66 2022 5365 6772 6522 290a  orm of "Segre").
│ │ │ │ -000299e0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -000299f0: 2020 202a 2061 202a 6e6f 7465 2072 696e     * a *note rin
│ │ │ │ -00029a00: 6720 656c 656d 656e 743a 2028 4d61 6361  g element: (Maca
│ │ │ │ -00029a10: 756c 6179 3244 6f63 2952 696e 6745 6c65  ulay2Doc)RingEle
│ │ │ │ -00029a20: 6d65 6e74 2c2c 2074 6865 2070 7573 6866  ment,, the pushf
│ │ │ │ -00029a30: 6f72 7761 7264 206f 660a 2020 2020 2020  orward of.      
│ │ │ │ -00029a40: 2020 7468 6520 746f 7461 6c20 5365 6772    the total Segr
│ │ │ │ -00029a50: 6520 636c 6173 7320 6f66 2074 6865 2073  e class of the s
│ │ │ │ -00029a60: 6368 656d 6520 5620 6465 6669 6e65 6420  cheme V defined 
│ │ │ │ -00029a70: 6279 2074 6865 2069 6e70 7574 2069 6465  by the input ide
│ │ │ │ -00029a80: 616c 2074 6f20 7468 650a 2020 2020 2020  al to the.      
│ │ │ │ -00029a90: 2020 6170 7072 6f70 7269 6174 6520 4368    appropriate Ch
│ │ │ │ -00029aa0: 6f77 2072 696e 670a 0a44 6573 6372 6970  ow ring..Descrip
│ │ │ │ -00029ab0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00029ac0: 0a0a 466f 7220 6120 7375 6273 6368 656d  ..For a subschem
│ │ │ │ -00029ad0: 6520 5620 6f66 2061 6e20 6170 706c 6963  e V of an applic
│ │ │ │ -00029ae0: 6162 6c65 2074 6f72 6963 2076 6172 6965  able toric varie
│ │ │ │ -00029af0: 7479 2058 2074 6869 7320 636f 6d6d 616e  ty X this comman
│ │ │ │ -00029b00: 6420 636f 6d70 7574 6573 2074 6865 0a70  d computes the.p
│ │ │ │ -00029b10: 7573 682d 666f 7277 6172 6420 6f66 2074  ush-forward of t
│ │ │ │ -00029b20: 6865 2074 6f74 616c 2053 6567 7265 2063  he total Segre c
│ │ │ │ -00029b30: 6c61 7373 2073 2856 2c58 2920 6f66 2056  lass s(V,X) of V
│ │ │ │ -00029b40: 2069 6e20 5820 746f 2074 6865 2043 686f   in X to the Cho
│ │ │ │ -00029b50: 7720 7269 6e67 206f 6620 582e 0a0a 2b2d  w ring of X...+-
│ │ │ │ -00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00029110: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +00029120: 2020 2020 2020 2020 5365 6772 6520 490a          Segre I.
│ │ │ │ +00029130: 2020 2020 2020 2020 5365 6772 6528 412c          Segre(A,
│ │ │ │ +00029140: 4929 0a20 2020 2020 2020 2053 6567 7265  I).        Segre
│ │ │ │ +00029150: 2858 2c4a 290a 2020 2020 2020 2020 5365  (X,J).        Se
│ │ │ │ +00029160: 6772 6528 4368 2c58 2c4a 290a 2020 2a20  gre(Ch,X,J).  * 
│ │ │ │ +00029170: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00029180: 492c 2061 6e20 2a6e 6f74 6520 6964 6561  I, an *note idea
│ │ │ │ +00029190: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ +000291a0: 2949 6465 616c 2c2c 2061 206d 756c 7469  )Ideal,, a multi
│ │ │ │ +000291b0: 2d68 6f6d 6f67 656e 656f 7573 2069 6465  -homogeneous ide
│ │ │ │ +000291c0: 616c 2069 6e20 610a 2020 2020 2020 2020  al in a.        
│ │ │ │ +000291d0: 6772 6164 6564 2070 6f6c 796e 6f6d 6961  graded polynomia
│ │ │ │ +000291e0: 6c20 7269 6e67 206f 7665 7220 6120 6669  l ring over a fi
│ │ │ │ +000291f0: 656c 6420 6465 6669 6e69 6e67 2061 2063  eld defining a c
│ │ │ │ +00029200: 6c6f 7365 6420 7375 6273 6368 656d 6520  losed subscheme 
│ │ │ │ +00029210: 5620 6f66 0a20 2020 2020 2020 205c 5050  V of.        \PP
│ │ │ │ +00029220: 5e7b 6e5f 317d 782e 2e2e 785c 5050 5e7b  ^{n_1}x...x\PP^{
│ │ │ │ +00029230: 6e5f 6d7d 0a20 2020 2020 202a 2041 2c20  n_m}.      * A, 
│ │ │ │ +00029240: 6120 2a6e 6f74 6520 7175 6f74 6965 6e74  a *note quotient
│ │ │ │ +00029250: 2072 696e 673a 2028 4d61 6361 756c 6179   ring: (Macaulay
│ │ │ │ +00029260: 3244 6f63 2951 756f 7469 656e 7452 696e  2Doc)QuotientRin
│ │ │ │ +00029270: 672c 2c0a 2020 2020 2020 2020 413d 5c5a  g,,.        A=\Z
│ │ │ │ +00029280: 5a5b 685f 312c 2e2e 2e2c 685f 6d5d 2f28  Z[h_1,...,h_m]/(
│ │ │ │ +00029290: 685f 315e 7b6e 5f31 2b31 7d2c 2e2e 2e2c  h_1^{n_1+1},...,
│ │ │ │ +000292a0: 685f 6d5e 7b6e 5f6d 2b31 7d29 2071 756f  h_m^{n_m+1}) quo
│ │ │ │ +000292b0: 7469 656e 7420 7269 6e67 0a20 2020 2020  tient ring.     
│ │ │ │ +000292c0: 2020 2072 6570 7265 7365 6e74 696e 6720     representing 
│ │ │ │ +000292d0: 7468 6520 4368 6f77 2072 696e 6720 6f66  the Chow ring of
│ │ │ │ +000292e0: 205c 5050 5e7b 6e5f 317d 782e 2e2e 785c   \PP^{n_1}x...x\
│ │ │ │ +000292f0: 5050 5e7b 6e5f 6d7d 2c20 7468 6973 2072  PP^{n_m}, this r
│ │ │ │ +00029300: 696e 6720 7368 6f75 6c64 0a20 2020 2020  ing should.     
│ │ │ │ +00029310: 2020 2062 6520 6275 696c 7420 7573 696e     be built usin
│ │ │ │ +00029320: 6720 7468 6520 2a6e 6f74 6520 4368 6f77  g the *note Chow
│ │ │ │ +00029330: 5269 6e67 3a20 4368 6f77 5269 6e67 2c20  Ring: ChowRing, 
│ │ │ │ +00029340: 636f 6d6d 616e 640a 2020 2020 2020 2a20  command.      * 
│ │ │ │ +00029350: 4a2c 2061 6e20 2a6e 6f74 6520 6964 6561  J, an *note idea
│ │ │ │ +00029360: 6c3a 2028 4d61 6361 756c 6179 3244 6f63  l: (Macaulay2Doc
│ │ │ │ +00029370: 2949 6465 616c 2c2c 2069 6e20 7468 6520  )Ideal,, in the 
│ │ │ │ +00029380: 6772 6164 6564 2070 6f6c 796e 6f6d 6961  graded polynomia
│ │ │ │ +00029390: 6c20 7269 6e67 0a20 2020 2020 2020 2077  l ring.        w
│ │ │ │ +000293a0: 6869 6368 2069 7320 636f 6f72 6469 6e61  hich is coordina
│ │ │ │ +000293b0: 7465 2072 696e 6720 6f66 2074 6865 204e  te ring of the N
│ │ │ │ +000293c0: 6f72 6d61 6c20 546f 7269 6320 5661 7269  ormal Toric Vari
│ │ │ │ +000293d0: 6574 7920 580a 2020 2020 2020 2a20 582c  ety X.      * X,
│ │ │ │ +000293e0: 2061 202a 6e6f 7465 206e 6f72 6d61 6c20   a *note normal 
│ │ │ │ +000293f0: 746f 7269 6320 7661 7269 6574 793a 0a20  toric variety:. 
│ │ │ │ +00029400: 2020 2020 2020 2028 4e6f 726d 616c 546f         (NormalTo
│ │ │ │ +00029410: 7269 6356 6172 6965 7469 6573 294e 6f72  ricVarieties)Nor
│ │ │ │ +00029420: 6d61 6c54 6f72 6963 5661 7269 6574 792c  malToricVariety,
│ │ │ │ +00029430: 2c20 7768 6963 6820 6973 2074 6865 2061  , which is the a
│ │ │ │ +00029440: 6d62 6965 6e74 2073 7061 6365 0a20 2020  mbient space.   
│ │ │ │ +00029450: 2020 2020 2077 6869 6368 2063 6f6e 7461       which conta
│ │ │ │ +00029460: 696e 7320 5628 4a29 0a20 2020 2020 202a  ins V(J).      *
│ │ │ │ +00029470: 2043 682c 2061 202a 6e6f 7465 2071 756f   Ch, a *note quo
│ │ │ │ +00029480: 7469 656e 7420 7269 6e67 3a20 284d 6163  tient ring: (Mac
│ │ │ │ +00029490: 6175 6c61 7932 446f 6329 5175 6f74 6965  aulay2Doc)Quotie
│ │ │ │ +000294a0: 6e74 5269 6e67 2c2c 2074 6865 2043 686f  ntRing,, the Cho
│ │ │ │ +000294b0: 7720 7269 6e67 0a20 2020 2020 2020 206f  w ring.        o
│ │ │ │ +000294c0: 6620 7468 6520 746f 7269 6320 7661 7269  f the toric vari
│ │ │ │ +000294d0: 6574 7920 582c 2043 683d 2872 696e 6720  ety X, Ch=(ring 
│ │ │ │ +000294e0: 4a29 2f28 5352 2b4c 5229 2077 6865 7265  J)/(SR+LR) where
│ │ │ │ +000294f0: 2053 5220 6973 2074 6865 0a20 2020 2020   SR is the.     
│ │ │ │ +00029500: 2020 2053 7461 6e6c 6579 2d52 6569 736e     Stanley-Reisn
│ │ │ │ +00029510: 6572 2069 6465 616c 206f 6620 7468 6520  er ideal of the 
│ │ │ │ +00029520: 6661 6e20 6465 6669 6e69 6e67 2058 2061  fan defining X a
│ │ │ │ +00029530: 6e64 204c 5220 6973 2074 6865 206c 696e  nd LR is the lin
│ │ │ │ +00029540: 6561 720a 2020 2020 2020 2020 7265 6c61  ear.        rela
│ │ │ │ +00029550: 7469 6f6e 7320 6964 6561 6c2c 2074 6869  tions ideal, thi
│ │ │ │ +00029560: 7320 7269 6e67 2073 686f 756c 6420 6265  s ring should be
│ │ │ │ +00029570: 2062 7569 6c74 2075 7369 6e67 2074 6865   built using the
│ │ │ │ +00029580: 202a 6e6f 7465 0a20 2020 2020 2020 2054   *note.        T
│ │ │ │ +00029590: 6f72 6963 4368 6f77 5269 6e67 3a20 546f  oricChowRing: To
│ │ │ │ +000295a0: 7269 6343 686f 7752 696e 672c 2063 6f6d  ricChowRing, com
│ │ │ │ +000295b0: 6d61 6e64 0a20 202a 202a 6e6f 7465 204f  mand.  * *note O
│ │ │ │ +000295c0: 7074 696f 6e61 6c20 696e 7075 7473 3a20  ptional inputs: 
│ │ │ │ +000295d0: 284d 6163 6175 6c61 7932 446f 6329 7573  (Macaulay2Doc)us
│ │ │ │ +000295e0: 696e 6720 6675 6e63 7469 6f6e 7320 7769  ing functions wi
│ │ │ │ +000295f0: 7468 206f 7074 696f 6e61 6c20 696e 7075  th optional inpu
│ │ │ │ +00029600: 7473 2c3a 0a20 2020 2020 202a 2043 6f6d  ts,:.      * Com
│ │ │ │ +00029610: 704d 6574 686f 6420 286d 6973 7369 6e67  pMethod (missing
│ │ │ │ +00029620: 2064 6f63 756d 656e 7461 7469 6f6e 2920   documentation) 
│ │ │ │ +00029630: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00029640: 7661 6c75 650a 2020 2020 2020 2020 5072  value.        Pr
│ │ │ │ +00029650: 6f6a 6563 7469 7665 4465 6772 6565 2c20  ojectiveDegree, 
│ │ │ │ +00029660: 5072 6f6a 6563 7469 7665 4465 6772 6565  ProjectiveDegree
│ │ │ │ +00029670: 2c20 7468 6973 2061 6c67 6f72 6974 686d  , this algorithm
│ │ │ │ +00029680: 206d 6179 2062 6520 7573 6564 2066 6f72   may be used for
│ │ │ │ +00029690: 0a20 2020 2020 2020 2073 7562 7363 6865  .        subsche
│ │ │ │ +000296a0: 6d65 7320 6f66 2061 6e79 2061 7070 6c69  mes of any appli
│ │ │ │ +000296b0: 6361 626c 6520 746f 7269 6320 7661 7269  cable toric vari
│ │ │ │ +000296c0: 6574 7920 2874 6869 7320 6d61 7920 6265  ety (this may be
│ │ │ │ +000296d0: 2063 6865 636b 6564 2075 7369 6e67 0a20   checked using. 
│ │ │ │ +000296e0: 2020 2020 2020 2074 6865 202a 6e6f 7465         the *note
│ │ │ │ +000296f0: 2043 6865 636b 546f 7269 6356 6172 6965   CheckToricVarie
│ │ │ │ +00029700: 7479 5661 6c69 643a 2043 6865 636b 546f  tyValid: CheckTo
│ │ │ │ +00029710: 7269 6356 6172 6965 7479 5661 6c69 642c  ricVarietyValid,
│ │ │ │ +00029720: 2063 6f6d 6d61 6e64 290a 2020 2020 2020   command).      
│ │ │ │ +00029730: 2a20 436f 6d70 4d65 7468 6f64 2028 6d69  * CompMethod (mi
│ │ │ │ +00029740: 7373 696e 6720 646f 6375 6d65 6e74 6174  ssing documentat
│ │ │ │ +00029750: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566  ion) => ..., def
│ │ │ │ +00029760: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ +00029770: 2020 2050 726f 6a65 6374 6976 6544 6567     ProjectiveDeg
│ │ │ │ +00029780: 7265 652c 2050 6e52 6573 6964 7561 6c2c  ree, PnResidual,
│ │ │ │ +00029790: 2074 6869 7320 616c 676f 7269 7468 6d20   this algorithm 
│ │ │ │ +000297a0: 6d61 7920 6265 2075 7365 6420 666f 7220  may be used for 
│ │ │ │ +000297b0: 7375 6273 6368 656d 6573 0a20 2020 2020  subschemes.     
│ │ │ │ +000297c0: 2020 206f 6620 5c50 505e 6e20 6f6e 6c79     of \PP^n only
│ │ │ │ +000297d0: 0a20 2020 2020 202a 204f 7574 7075 7420  .      * Output 
│ │ │ │ +000297e0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +000297f0: 7661 6c75 6520 4368 6f77 5269 6e67 456c  value ChowRingEl
│ │ │ │ +00029800: 656d 656e 742c 2043 686f 7752 696e 6745  ement, ChowRingE
│ │ │ │ +00029810: 6c65 6d65 6e74 2c20 7265 7475 726e 730a  lement, returns.
│ │ │ │ +00029820: 2020 2020 2020 2020 6120 5269 6e67 456c          a RingEl
│ │ │ │ +00029830: 656d 656e 7420 696e 2074 6865 2043 686f  ement in the Cho
│ │ │ │ +00029840: 7720 7269 6e67 206f 6620 7468 6520 6170  w ring of the ap
│ │ │ │ +00029850: 7072 6f70 7269 6174 6520 616d 6269 656e  propriate ambien
│ │ │ │ +00029860: 7420 7370 6163 650a 2020 2020 2020 2a20  t space.      * 
│ │ │ │ +00029870: 4f75 7470 7574 203d 3e20 2e2e 2e2c 2064  Output => ..., d
│ │ │ │ +00029880: 6566 6175 6c74 2076 616c 7565 2043 686f  efault value Cho
│ │ │ │ +00029890: 7752 696e 6745 6c65 6d65 6e74 2c20 4861  wRingElement, Ha
│ │ │ │ +000298a0: 7368 466f 726d 2c20 4861 7368 466f 726d  shForm, HashForm
│ │ │ │ +000298b0: 0a20 2020 2020 2020 2072 6574 7572 6e73  .        returns
│ │ │ │ +000298c0: 2061 204d 7574 6162 6c65 4861 7368 5461   a MutableHashTa
│ │ │ │ +000298d0: 626c 6520 636f 6e74 6169 6e69 6e67 2074  ble containing t
│ │ │ │ +000298e0: 6865 2066 6f6c 6c6f 7769 6e67 206b 6579  he following key
│ │ │ │ +000298f0: 733a 2022 4722 2028 7468 650a 2020 2020  s: "G" (the.    
│ │ │ │ +00029900: 2020 2020 706f 6c79 6e6f 6d69 616c 2077      polynomial w
│ │ │ │ +00029910: 6974 6820 636f 6566 6669 6369 656e 7473  ith coefficients
│ │ │ │ +00029920: 206f 6620 7468 6520 6879 7065 7270 6c61   of the hyperpla
│ │ │ │ +00029930: 6e65 2063 6c61 7373 6573 2072 6570 7265  ne classes repre
│ │ │ │ +00029940: 7365 6e74 696e 6720 7468 650a 2020 2020  senting the.    
│ │ │ │ +00029950: 2020 2020 7072 6f6a 6563 7469 7665 2064      projective d
│ │ │ │ +00029960: 6567 7265 6573 292c 2022 476c 6973 7422  egrees), "Glist"
│ │ │ │ +00029970: 2028 7468 6520 6c69 7374 2066 6f72 6d20   (the list form 
│ │ │ │ +00029980: 6f66 2022 4722 2920 2c20 2253 6567 7265  of "G") , "Segre
│ │ │ │ +00029990: 2220 2874 6865 0a20 2020 2020 2020 2074  " (the.        t
│ │ │ │ +000299a0: 6f74 616c 2053 6567 7265 2063 6c61 7373  otal Segre class
│ │ │ │ +000299b0: 206f 6620 7468 6520 696e 7075 7429 2c22   of the input),"
│ │ │ │ +000299c0: 5365 6772 654c 6973 7422 2028 7468 6520  SegreList" (the 
│ │ │ │ +000299d0: 6c69 7374 2066 6f72 6d20 6f66 2022 5365  list form of "Se
│ │ │ │ +000299e0: 6772 6522 290a 2020 2a20 4f75 7470 7574  gre").  * Output
│ │ │ │ +000299f0: 733a 0a20 2020 2020 202a 2061 202a 6e6f  s:.      * a *no
│ │ │ │ +00029a00: 7465 2072 696e 6720 656c 656d 656e 743a  te ring element:
│ │ │ │ +00029a10: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ +00029a20: 696e 6745 6c65 6d65 6e74 2c2c 2074 6865  ingElement,, the
│ │ │ │ +00029a30: 2070 7573 6866 6f72 7761 7264 206f 660a   pushforward of.
│ │ │ │ +00029a40: 2020 2020 2020 2020 7468 6520 746f 7461          the tota
│ │ │ │ +00029a50: 6c20 5365 6772 6520 636c 6173 7320 6f66  l Segre class of
│ │ │ │ +00029a60: 2074 6865 2073 6368 656d 6520 5620 6465   the scheme V de
│ │ │ │ +00029a70: 6669 6e65 6420 6279 2074 6865 2069 6e70  fined by the inp
│ │ │ │ +00029a80: 7574 2069 6465 616c 2074 6f20 7468 650a  ut ideal to the.
│ │ │ │ +00029a90: 2020 2020 2020 2020 6170 7072 6f70 7269          appropri
│ │ │ │ +00029aa0: 6174 6520 4368 6f77 2072 696e 670a 0a44  ate Chow ring..D
│ │ │ │ +00029ab0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00029ac0: 3d3d 3d3d 3d3d 0a0a 466f 7220 6120 7375  ======..For a su
│ │ │ │ +00029ad0: 6273 6368 656d 6520 5620 6f66 2061 6e20  bscheme V of an 
│ │ │ │ +00029ae0: 6170 706c 6963 6162 6c65 2074 6f72 6963  applicable toric
│ │ │ │ +00029af0: 2076 6172 6965 7479 2058 2074 6869 7320   variety X this 
│ │ │ │ +00029b00: 636f 6d6d 616e 6420 636f 6d70 7574 6573  command computes
│ │ │ │ +00029b10: 2074 6865 0a70 7573 682d 666f 7277 6172   the.push-forwar
│ │ │ │ +00029b20: 6420 6f66 2074 6865 2074 6f74 616c 2053  d of the total S
│ │ │ │ +00029b30: 6567 7265 2063 6c61 7373 2073 2856 2c58  egre class s(V,X
│ │ │ │ +00029b40: 2920 6f66 2056 2069 6e20 5820 746f 2074  ) of V in X to t
│ │ │ │ +00029b50: 6865 2043 686f 7720 7269 6e67 206f 6620  he Chow ring of 
│ │ │ │ +00029b60: 582e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  X...+-----------
│ │ │ │  00029b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00029b90: 3120 3a20 7365 7452 616e 646f 6d53 6565  1 : setRandomSee
│ │ │ │ -00029ba0: 6420 3732 3b20 2020 2020 2020 2020 2020  d 72;           
│ │ │ │ -00029bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029bc0: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ -00029bd0: 6d20 7365 6564 2074 6f20 3732 2020 2020  m seed to 72    
│ │ │ │ -00029be0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00029bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029b90: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ +00029ba0: 646f 6d53 6565 6420 3732 3b20 2020 2020  domSeed 72;     
│ │ │ │ +00029bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029bc0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67    |.| -- setting
│ │ │ │ +00029bd0: 2072 616e 646f 6d20 7365 6564 2074 6f20   random seed to 
│ │ │ │ +00029be0: 3732 2020 2020 2020 2020 2020 2020 2020  72              
│ │ │ │ +00029bf0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00029c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00029c20: 3220 3a20 5220 3d20 5a5a 2f33 3237 3439  2 : R = ZZ/32749
│ │ │ │ -00029c30: 5b77 2c79 2c7a 5d20 2020 2020 2020 2020  [w,y,z]         
│ │ │ │ -00029c40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029c20: 2d2d 2b0a 7c69 3220 3a20 5220 3d20 5a5a  --+.|i2 : R = ZZ
│ │ │ │ +00029c30: 2f33 3237 3439 5b77 2c79 2c7a 5d20 2020  /32749[w,y,z]   
│ │ │ │ +00029c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029c50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00029c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00029c80: 3220 3d20 5220 2020 2020 2020 2020 2020  2 = R           
│ │ │ │ +00029c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029c80: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020    |.|o2 = R     
│ │ │ │  00029c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029cb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00029cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00029ce0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ -00029cf0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00029d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ce0: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f    |.|o2 : Polyno
│ │ │ │ +00029cf0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +00029d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029d10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00029d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00029d40: 3320 3a20 5365 6772 6528 6964 6561 6c28  3 : Segre(ideal(
│ │ │ │ -00029d50: 772a 7929 2c43 6f6d 704d 6574 686f 643d  w*y),CompMethod=
│ │ │ │ -00029d60: 3e50 6e52 6573 6964 7561 6c29 7c0a 7c20  >PnResidual)|.| 
│ │ │ │ -00029d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029d40: 2d2d 2b0a 7c69 3320 3a20 5365 6772 6528  --+.|i3 : Segre(
│ │ │ │ +00029d50: 6964 6561 6c28 772a 7929 2c43 6f6d 704d  ideal(w*y),CompM
│ │ │ │ +00029d60: 6574 686f 643d 3e50 6e52 6573 6964 7561  ethod=>PnResidua
│ │ │ │ +00029d70: 6c29 7c0a 7c20 2020 2020 2020 2020 2020  l)|.|           
│ │ │ │  00029d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029da0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029da0: 2020 7c0a 7c20 2020 2020 2020 2020 3220    |.|         2 
│ │ │ │  00029db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00029dd0: 3320 3d20 2d20 3448 2020 2b20 3248 2020  3 = - 4H  + 2H  
│ │ │ │ -00029de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029dd0: 2020 7c0a 7c6f 3320 3d20 2d20 3448 2020    |.|o3 = - 4H  
│ │ │ │ +00029de0: 2b20 3248 2020 2020 2020 2020 2020 2020  + 2H            
│ │ │ │ +00029df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029e00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00029e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029e30: 2020 2020 5a5a 5b48 5d20 2020 2020 2020      ZZ[H]       
│ │ │ │ +00029e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029e30: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20    |.|     ZZ[H] 
│ │ │ │  00029e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00029e60: 3320 3a20 2d2d 2d2d 2d20 2020 2020 2020  3 : -----       
│ │ │ │ +00029e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029e60: 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d 2d20    |.|o3 : ----- 
│ │ │ │  00029e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029e90: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00029e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029e90: 2020 7c0a 7c20 2020 2020 2020 2033 2020    |.|        3  
│ │ │ │  00029ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029ec0: 2020 2020 2020 4820 2020 2020 2020 2020        H         
│ │ │ │ +00029eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ec0: 2020 7c0a 7c20 2020 2020 2020 4820 2020    |.|       H   
│ │ │ │  00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -00029ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ef0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  00029f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00029f20: 3420 3a20 413d 4368 6f77 5269 6e67 2852  4 : A=ChowRing(R
│ │ │ │ -00029f30: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00029f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029f20: 2d2d 2b0a 7c69 3420 3a20 413d 4368 6f77  --+.|i4 : A=Chow
│ │ │ │ +00029f30: 5269 6e67 2852 2920 2020 2020 2020 2020  Ring(R)         
│ │ │ │ +00029f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00029f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00029f80: 3420 3d20 4120 2020 2020 2020 2020 2020  4 = A           
│ │ │ │ +00029f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f80: 2020 7c0a 7c6f 3420 3d20 4120 2020 2020    |.|o4 = A     
│ │ │ │  00029f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00029fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029fb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00029fe0: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ -00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a000: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029fe0: 2020 7c0a 7c6f 3420 3a20 5175 6f74 6965    |.|o4 : Quotie
│ │ │ │ +00029ff0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +0002a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a010: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002a040: 3520 3a20 5365 6772 6528 412c 6964 6561  5 : Segre(A,idea
│ │ │ │ -0002a050: 6c28 775e 322a 792c 772a 795e 3229 2920  l(w^2*y,w*y^2)) 
│ │ │ │ -0002a060: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0002a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a040: 2d2d 2b0a 7c69 3520 3a20 5365 6772 6528  --+.|i5 : Segre(
│ │ │ │ +0002a050: 412c 6964 6561 6c28 775e 322a 792c 772a  A,ideal(w^2*y,w*
│ │ │ │ +0002a060: 795e 3229 2920 2020 2020 2020 2020 2020  y^2))           
│ │ │ │ +0002a070: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0002a0a0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0002a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0a0: 2020 7c0a 7c20 2020 2020 2020 2020 3220    |.|         2 
│ │ │ │  0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002a0d0: 3520 3d20 2d20 3368 2020 2b20 3268 2020  5 = - 3h  + 2h  
│ │ │ │ -0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0002a100: 2020 2020 2020 2020 3120 2020 2020 3120          1     1 
│ │ │ │ -0002a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a120: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0002a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0d0: 2020 7c0a 7c6f 3520 3d20 2d20 3368 2020    |.|o5 = - 3h  
│ │ │ │ +0002a0e0: 2b20 3268 2020 2020 2020 2020 2020 2020  + 2h            
│ │ │ │ +0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a100: 2020 7c0a 7c20 2020 2020 2020 2020 3120    |.|         1 
│ │ │ │ +0002a110: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +0002a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a130: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a150: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002a160: 3520 3a20 4120 2020 2020 2020 2020 2020  5 : A           
│ │ │ │ +0002a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a160: 2020 7c0a 7c6f 3520 3a20 4120 2020 2020    |.|o5 : A     
│ │ │ │  0002a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a180: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0002a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │  0002a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e  ------------+..N
│ │ │ │ -0002a1c0: 6f77 2063 6f6e 7369 6465 7220 616e 2065  ow consider an e
│ │ │ │ -0002a1d0: 7861 6d70 6c65 2069 6e20 5c50 505e 3220  xample in \PP^2 
│ │ │ │ -0002a1e0: 5c74 696d 6573 205c 5050 5e32 2c20 6966  \times \PP^2, if
│ │ │ │ -0002a1f0: 2077 6520 696e 7075 7420 7468 6520 4368   we input the Ch
│ │ │ │ -0002a200: 6f77 2072 696e 6720 4120 7468 650a 6f75  ow ring A the.ou
│ │ │ │ -0002a210: 7470 7574 2077 696c 6c20 6265 2072 6574  tput will be ret
│ │ │ │ -0002a220: 7572 6e65 6420 696e 2074 6865 2073 616d  urned in the sam
│ │ │ │ -0002a230: 6520 7269 6e67 2e20 546f 2065 6e73 7572  e ring. To ensur
│ │ │ │ -0002a240: 6520 7072 6f70 6572 2066 756e 6374 696f  e proper functio
│ │ │ │ -0002a250: 6e20 6f66 2074 6865 0a6d 6574 686f 6473  n of the.methods
│ │ │ │ -0002a260: 2077 6520 6275 696c 6420 7468 6520 4368   we build the Ch
│ │ │ │ -0002a270: 6f77 2072 696e 6720 7573 696e 6720 7468  ow ring using th
│ │ │ │ -0002a280: 6520 2a6e 6f74 6520 4368 6f77 5269 6e67  e *note ChowRing
│ │ │ │ -0002a290: 3a20 4368 6f77 5269 6e67 2c20 636f 6d6d  : ChowRing, comm
│ │ │ │ -0002a2a0: 616e 642e 2057 650a 6d61 7920 616c 736f  and. We.may also
│ │ │ │ -0002a2b0: 2072 6574 7572 6e20 6120 4d75 7461 626c   return a Mutabl
│ │ │ │ -0002a2c0: 6548 6173 6854 6162 6c65 2e0a 0a2b 2d2d  eHashTable...+--
│ │ │ │ -0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a1c0: 2d2d 2b0a 0a4e 6f77 2063 6f6e 7369 6465  --+..Now conside
│ │ │ │ +0002a1d0: 7220 616e 2065 7861 6d70 6c65 2069 6e20  r an example in 
│ │ │ │ +0002a1e0: 5c50 505e 3220 5c74 696d 6573 205c 5050  \PP^2 \times \PP
│ │ │ │ +0002a1f0: 5e32 2c20 6966 2077 6520 696e 7075 7420  ^2, if we input 
│ │ │ │ +0002a200: 7468 6520 4368 6f77 2072 696e 6720 4120  the Chow ring A 
│ │ │ │ +0002a210: 7468 650a 6f75 7470 7574 2077 696c 6c20  the.output will 
│ │ │ │ +0002a220: 6265 2072 6574 7572 6e65 6420 696e 2074  be returned in t
│ │ │ │ +0002a230: 6865 2073 616d 6520 7269 6e67 2e20 546f  he same ring. To
│ │ │ │ +0002a240: 2065 6e73 7572 6520 7072 6f70 6572 2066   ensure proper f
│ │ │ │ +0002a250: 756e 6374 696f 6e20 6f66 2074 6865 0a6d  unction of the.m
│ │ │ │ +0002a260: 6574 686f 6473 2077 6520 6275 696c 6420  ethods we build 
│ │ │ │ +0002a270: 7468 6520 4368 6f77 2072 696e 6720 7573  the Chow ring us
│ │ │ │ +0002a280: 696e 6720 7468 6520 2a6e 6f74 6520 4368  ing the *note Ch
│ │ │ │ +0002a290: 6f77 5269 6e67 3a20 4368 6f77 5269 6e67  owRing: ChowRing
│ │ │ │ +0002a2a0: 2c20 636f 6d6d 616e 642e 2057 650a 6d61  , command. We.ma
│ │ │ │ +0002a2b0: 7920 616c 736f 2072 6574 7572 6e20 6120  y also return a 
│ │ │ │ +0002a2c0: 4d75 7461 626c 6548 6173 6854 6162 6c65  MutableHashTable
│ │ │ │ +0002a2d0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  0002a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -0002a320: 203a 2052 3d4d 756c 7469 5072 6f6a 436f   : R=MultiProjCo
│ │ │ │ -0002a330: 6f72 6452 696e 6728 7b32 2c32 7d29 2020  ordRing({2,2})  
│ │ │ │ -0002a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a320: 2d2b 0a7c 6936 203a 2052 3d4d 756c 7469  -+.|i6 : R=Multi
│ │ │ │ +0002a330: 5072 6f6a 436f 6f72 6452 696e 6728 7b32  ProjCoordRing({2
│ │ │ │ +0002a340: 2c32 7d29 2020 2020 2020 2020 2020 2020  ,2})            
│ │ │ │  0002a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a360: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a370: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3b0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -0002a3c0: 203d 2052 2020 2020 2020 2020 2020 2020   = R            
│ │ │ │ +0002a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a3c0: 207c 0a7c 6f36 203d 2052 2020 2020 2020   |.|o6 = R      
│ │ │ │  0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a400: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a410: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a450: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -0002a460: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -0002a470: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0002a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a460: 207c 0a7c 6f36 203a 2050 6f6c 796e 6f6d   |.|o6 : Polynom
│ │ │ │ +0002a470: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │  0002a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a4a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a4b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -0002a500: 203a 2072 3d67 656e 7320 5220 2020 2020   : r=gens R     
│ │ │ │ -0002a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a500: 2d2b 0a7c 6937 203a 2072 3d67 656e 7320  -+.|i7 : r=gens 
│ │ │ │ +0002a510: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │  0002a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a540: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a550: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a590: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -0002a5a0: 203d 207b 7820 2c20 7820 2c20 7820 2c20   = {x , x , x , 
│ │ │ │ -0002a5b0: 7820 2c20 7820 2c20 7820 7d20 2020 2020  x , x , x }     
│ │ │ │ -0002a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a5a0: 207c 0a7c 6f37 203d 207b 7820 2c20 7820   |.|o7 = {x , x 
│ │ │ │ +0002a5b0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +0002a5c0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │  0002a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a5e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a5f0: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ -0002a600: 2033 2020 2034 2020 2035 2020 2020 2020   3   4   5      
│ │ │ │ +0002a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a5f0: 207c 0a7c 2020 2020 2020 2030 2020 2031   |.|       0   1
│ │ │ │ +0002a600: 2020 2032 2020 2033 2020 2034 2020 2035     2   3   4   5
│ │ │ │  0002a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a630: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a680: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -0002a690: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +0002a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a690: 207c 0a7c 6f37 203a 204c 6973 7420 2020   |.|o7 : List   
│ │ │ │  0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a6d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002a6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a6e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ -0002a730: 203a 2041 3d43 686f 7752 696e 6728 5229   : A=ChowRing(R)
│ │ │ │ -0002a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a730: 2d2b 0a7c 6938 203a 2041 3d43 686f 7752  -+.|i8 : A=ChowR
│ │ │ │ +0002a740: 696e 6728 5229 2020 2020 2020 2020 2020  ing(R)          
│ │ │ │  0002a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a770: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a780: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7c0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -0002a7d0: 203d 2041 2020 2020 2020 2020 2020 2020   = A            
│ │ │ │ +0002a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a7d0: 207c 0a7c 6f38 203d 2041 2020 2020 2020   |.|o8 = A      
│ │ │ │  0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a820: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a860: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -0002a870: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ -0002a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a870: 207c 0a7c 6f38 203a 2051 756f 7469 656e   |.|o8 : Quotien
│ │ │ │ +0002a880: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │  0002a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a8c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
│ │ │ │ -0002a910: 203a 2049 3d69 6465 616c 2872 5f30 5e32   : I=ideal(r_0^2
│ │ │ │ -0002a920: 2a72 5f33 2d72 5f34 2a72 5f31 2a72 5f32  *r_3-r_4*r_1*r_2
│ │ │ │ -0002a930: 2c72 5f32 5e32 2a72 5f35 2920 2020 2020  ,r_2^2*r_5)     
│ │ │ │ -0002a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a950: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a910: 2d2b 0a7c 6939 203a 2049 3d69 6465 616c  -+.|i9 : I=ideal
│ │ │ │ +0002a920: 2872 5f30 5e32 2a72 5f33 2d72 5f34 2a72  (r_0^2*r_3-r_4*r
│ │ │ │ +0002a930: 5f31 2a72 5f32 2c72 5f32 5e32 2a72 5f35  _1*r_2,r_2^2*r_5
│ │ │ │ +0002a940: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a960: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002a9b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0002a9c0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0002a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a9b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a9c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002a9d0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0002a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a9f0: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -0002aa00: 203d 2069 6465 616c 2028 7820 7820 202d   = ideal (x x  -
│ │ │ │ -0002aa10: 2078 2078 2078 202c 2078 2078 2029 2020   x x x , x x )  
│ │ │ │ -0002aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa00: 207c 0a7c 6f39 203d 2069 6465 616c 2028   |.|o9 = ideal (
│ │ │ │ +0002aa10: 7820 7820 202d 2078 2078 2078 202c 2078  x x  - x x x , x
│ │ │ │ +0002aa20: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │  0002aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002aa50: 2020 2020 2020 2020 2020 2030 2033 2020             0 3  
│ │ │ │ -0002aa60: 2020 3120 3220 3420 2020 3220 3520 2020    1 2 4   2 5   
│ │ │ │ -0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002aa60: 2030 2033 2020 2020 3120 3220 3420 2020   0 3    1 2 4   
│ │ │ │ +0002aa70: 3220 3520 2020 2020 2020 2020 2020 2020  2 5             
│ │ │ │  0002aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aaa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aae0: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -0002aaf0: 203a 2049 6465 616c 206f 6620 5220 2020   : Ideal of R   
│ │ │ │ -0002ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aaf0: 207c 0a7c 6f39 203a 2049 6465 616c 206f   |.|o9 : Ideal o
│ │ │ │ +0002ab00: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │  0002ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ab40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0002ab90: 3020 3a20 5365 6772 6520 4920 2020 2020  0 : Segre I     
│ │ │ │ -0002aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ab90: 2d2b 0a7c 6931 3020 3a20 5365 6772 6520  -+.|i10 : Segre 
│ │ │ │ +0002aba0: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │  0002abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002abd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002abe0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002ac30: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ -0002ac40: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ -0002ac50: 2032 2020 2020 2020 2020 2020 2020 3220   2            2 
│ │ │ │ -0002ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ac70: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002ac80: 3020 3d20 3732 6820 6820 202d 2032 3468  0 = 72h h  - 24h
│ │ │ │ -0002ac90: 2068 2020 2d20 3132 6820 6820 202b 2034   h  - 12h h  + 4
│ │ │ │ -0002aca0: 6820 202b 2034 6820 6820 202b 2068 2020  h  + 4h h  + h  
│ │ │ │ -0002acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002acc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002acd0: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ -0002ace0: 3120 3220 2020 2020 2031 2032 2020 2020  1 2      1 2    
│ │ │ │ -0002acf0: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ -0002ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ac30: 207c 0a7c 2020 2020 2020 2020 2032 2032   |.|         2 2
│ │ │ │ +0002ac40: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0002ac50: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ +0002ac60: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ac80: 207c 0a7c 6f31 3020 3d20 3732 6820 6820   |.|o10 = 72h h 
│ │ │ │ +0002ac90: 202d 2032 3468 2068 2020 2d20 3132 6820   - 24h h  - 12h 
│ │ │ │ +0002aca0: 6820 202b 2034 6820 202b 2034 6820 6820  h  + 4h  + 4h h 
│ │ │ │ +0002acb0: 202b 2068 2020 2020 2020 2020 2020 2020   + h            
│ │ │ │ +0002acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002acd0: 207c 0a7c 2020 2020 2020 2020 2031 2032   |.|         1 2
│ │ │ │ +0002ace0: 2020 2020 2020 3120 3220 2020 2020 2031        1 2      1
│ │ │ │ +0002acf0: 2032 2020 2020 2031 2020 2020 2031 2032   2     1     1 2
│ │ │ │ +0002ad00: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ad20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002ad70: 2020 2020 5a5a 5b68 202e 2e68 205d 2020      ZZ[h ..h ]  
│ │ │ │ -0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ad70: 207c 0a7c 2020 2020 2020 5a5a 5b68 202e   |.|      ZZ[h .
│ │ │ │ +0002ad80: 2e68 205d 2020 2020 2020 2020 2020 2020  .h ]            
│ │ │ │  0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002adb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002adc0: 2020 2020 2020 2020 3120 2020 3220 2020          1   2   
│ │ │ │ -0002add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002adc0: 207c 0a7c 2020 2020 2020 2020 2020 3120   |.|          1 
│ │ │ │ +0002add0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0002ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae00: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002ae10: 3020 3a20 2d2d 2d2d 2d2d 2d2d 2d2d 2020  0 : ----------  
│ │ │ │ -0002ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae10: 207c 0a7c 6f31 3020 3a20 2d2d 2d2d 2d2d   |.|o10 : ------
│ │ │ │ +0002ae20: 2d2d 2d2d 2020 2020 2020 2020 2020 2020  ----            
│ │ │ │  0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002ae60: 2020 2020 2020 2033 2020 2033 2020 2020         3   3    
│ │ │ │ -0002ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae60: 207c 0a7c 2020 2020 2020 2020 2033 2020   |.|         3  
│ │ │ │ +0002ae70: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0002ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aea0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002aeb0: 2020 2020 2028 6820 2c20 6820 2920 2020       (h , h )   
│ │ │ │ -0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aeb0: 207c 0a7c 2020 2020 2020 2028 6820 2c20   |.|       (h , 
│ │ │ │ +0002aec0: 6820 2920 2020 2020 2020 2020 2020 2020  h )             
│ │ │ │  0002aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aef0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002af00: 2020 2020 2020 2031 2020 2032 2020 2020         1   2    
│ │ │ │ -0002af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af00: 207c 0a7c 2020 2020 2020 2020 2031 2020   |.|         1  
│ │ │ │ +0002af10: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0002af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af50: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0002afa0: 3120 3a20 7331 3d53 6567 7265 2841 2c49  1 : s1=Segre(A,I
│ │ │ │ -0002afb0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002afa0: 2d2b 0a7c 6931 3120 3a20 7331 3d53 6567  -+.|i11 : s1=Seg
│ │ │ │ +0002afb0: 7265 2841 2c49 2920 2020 2020 2020 2020  re(A,I)         
│ │ │ │  0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afe0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aff0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b030: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b040: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ -0002b050: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ -0002b060: 2032 2020 2020 2020 2020 2020 2020 3220   2            2 
│ │ │ │ -0002b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b080: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002b090: 3120 3d20 3732 6820 6820 202d 2032 3468  1 = 72h h  - 24h
│ │ │ │ -0002b0a0: 2068 2020 2d20 3132 6820 6820 202b 2034   h  - 12h h  + 4
│ │ │ │ -0002b0b0: 6820 202b 2034 6820 6820 202b 2068 2020  h  + 4h h  + h  
│ │ │ │ -0002b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b0e0: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ -0002b0f0: 3120 3220 2020 2020 2031 2032 2020 2020  1 2      1 2    
│ │ │ │ -0002b100: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ -0002b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b120: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b040: 207c 0a7c 2020 2020 2020 2020 2032 2032   |.|         2 2
│ │ │ │ +0002b050: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0002b060: 2032 2020 2020 2032 2020 2020 2020 2020   2     2        
│ │ │ │ +0002b070: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b090: 207c 0a7c 6f31 3120 3d20 3732 6820 6820   |.|o11 = 72h h 
│ │ │ │ +0002b0a0: 202d 2032 3468 2068 2020 2d20 3132 6820   - 24h h  - 12h 
│ │ │ │ +0002b0b0: 6820 202b 2034 6820 202b 2034 6820 6820  h  + 4h  + 4h h 
│ │ │ │ +0002b0c0: 202b 2068 2020 2020 2020 2020 2020 2020   + h            
│ │ │ │ +0002b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b0e0: 207c 0a7c 2020 2020 2020 2020 2031 2032   |.|         1 2
│ │ │ │ +0002b0f0: 2020 2020 2020 3120 3220 2020 2020 2031        1 2      1
│ │ │ │ +0002b100: 2032 2020 2020 2031 2020 2020 2031 2032   2     1     1 2
│ │ │ │ +0002b110: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b130: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b170: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002b180: 3120 3a20 4120 2020 2020 2020 2020 2020  1 : A           
│ │ │ │ +0002b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b180: 207c 0a7c 6f31 3120 3a20 4120 2020 2020   |.|o11 : A     
│ │ │ │  0002b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b1d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0002b220: 3220 3a20 5365 6748 6173 683d 5365 6772  2 : SegHash=Segr
│ │ │ │ -0002b230: 6528 412c 492c 4f75 7470 7574 3d3e 4861  e(A,I,Output=>Ha
│ │ │ │ -0002b240: 7368 466f 726d 2920 2020 2020 2020 2020  shForm)         
│ │ │ │ +0002b210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b220: 2d2b 0a7c 6931 3220 3a20 5365 6748 6173  -+.|i12 : SegHas
│ │ │ │ +0002b230: 683d 5365 6772 6528 412c 492c 4f75 7470  h=Segre(A,I,Outp
│ │ │ │ +0002b240: 7574 3d3e 4861 7368 466f 726d 2920 2020  ut=>HashForm)   
│ │ │ │  0002b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b260: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b270: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002b2c0: 3220 3d20 4d75 7461 626c 6548 6173 6854  2 = MutableHashT
│ │ │ │ -0002b2d0: 6162 6c65 7b2e 2e2e 342e 2e2e 7d20 2020  able{...4...}   
│ │ │ │ -0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b2c0: 207c 0a7c 6f31 3220 3d20 4d75 7461 626c   |.|o12 = Mutabl
│ │ │ │ +0002b2d0: 6548 6173 6854 6162 6c65 7b2e 2e2e 342e  eHashTable{...4.
│ │ │ │ +0002b2e0: 2e2e 7d20 2020 2020 2020 2020 2020 2020  ..}             
│ │ │ │  0002b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b300: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b310: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b350: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002b360: 3220 3a20 4d75 7461 626c 6548 6173 6854  2 : MutableHashT
│ │ │ │ -0002b370: 6162 6c65 2020 2020 2020 2020 2020 2020  able            
│ │ │ │ +0002b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b360: 207c 0a7c 6f31 3220 3a20 4d75 7461 626c   |.|o12 : Mutabl
│ │ │ │ +0002b370: 6548 6173 6854 6162 6c65 2020 2020 2020  eHashTable      
│ │ │ │  0002b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b3b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002b3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0002b400: 3320 3a20 7065 656b 2053 6567 4861 7368  3 : peek SegHash
│ │ │ │ -0002b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b400: 2d2b 0a7c 6931 3320 3a20 7065 656b 2053  -+.|i13 : peek S
│ │ │ │ +0002b410: 6567 4861 7368 2020 2020 2020 2020 2020  egHash          
│ │ │ │  0002b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b440: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b450: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b490: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002b4a0: 3320 3d20 4d75 7461 626c 6548 6173 6854  3 = MutableHashT
│ │ │ │ -0002b4b0: 6162 6c65 7b47 203d 3e20 3268 2020 2b20  able{G => 2h  + 
│ │ │ │ -0002b4c0: 6820 202b 2031 2020 2020 2020 2020 2020  h  + 1          
│ │ │ │ +0002b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b4a0: 207c 0a7c 6f31 3320 3d20 4d75 7461 626c   |.|o13 = Mutabl
│ │ │ │ +0002b4b0: 6548 6173 6854 6162 6c65 7b47 203d 3e20  eHashTable{G => 
│ │ │ │ +0002b4c0: 3268 2020 2b20 6820 202b 2031 2020 2020  2h  + h  + 1    
│ │ │ │  0002b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b500: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -0002b510: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b4f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b510: 2020 3120 2020 2032 2020 2020 2020 2020    1    2        
│ │ │ │  0002b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b530: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b550: 2020 2020 2047 6c69 7374 203d 3e20 7b31       Glist => {1
│ │ │ │ -0002b560: 2c20 3268 2020 2b20 6820 2c20 302c 2030  , 2h  + h , 0, 0
│ │ │ │ -0002b570: 2c20 307d 2020 2020 2020 2020 2020 2020  , 0}            
│ │ │ │ -0002b580: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b540: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b550: 2020 2020 2020 2020 2020 2047 6c69 7374             Glist
│ │ │ │ +0002b560: 203d 3e20 7b31 2c20 3268 2020 2b20 6820   => {1, 2h  + h 
│ │ │ │ +0002b570: 2c20 302c 2030 2c20 307d 2020 2020 2020  , 0, 0, 0}      
│ │ │ │ +0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b590: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5b0: 2020 2020 3120 2020 2032 2020 2020 2020      1    2      
│ │ │ │ +0002b5b0: 2020 2020 2020 2020 2020 3120 2020 2032            1    2
│ │ │ │  0002b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b5e0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b600: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0002b610: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -0002b620: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b640: 2020 2020 2053 6567 7265 4c69 7374 203d       SegreList =
│ │ │ │ -0002b650: 3e20 7b30 2c20 302c 2034 6820 202b 2034  > {0, 0, 4h  + 4
│ │ │ │ -0002b660: 6820 6820 202b 2068 202c 202d 2020 2020  h h  + h , -    
│ │ │ │ -0002b670: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b610: 2032 2020 2020 2020 2020 2020 2020 3220   2            2 
│ │ │ │ +0002b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b630: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +0002b650: 4c69 7374 203d 3e20 7b30 2c20 302c 2034  List => {0, 0, 4
│ │ │ │ +0002b660: 6820 202b 2034 6820 6820 202b 2068 202c  h  + 4h h  + h ,
│ │ │ │ +0002b670: 202d 2020 2020 2020 2020 2020 2020 2020   -              
│ │ │ │ +0002b680: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6a0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0002b6b0: 2031 2032 2020 2020 3220 2020 2020 2020   1 2    2       
│ │ │ │ -0002b6c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b6b0: 2031 2020 2020 2031 2032 2020 2020 3220   1     1 2    2 
│ │ │ │ +0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b6d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6f0: 2032 2032 2020 2020 2020 3220 2020 2020   2 2      2     
│ │ │ │ -0002b700: 2020 2020 2032 2020 2020 2032 2020 2020       2     2    
│ │ │ │ -0002b710: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b730: 2020 2020 2053 6567 7265 203d 3e20 3732       Segre => 72
│ │ │ │ -0002b740: 6820 6820 202d 2032 3468 2068 2020 2d20  h h  - 24h h  - 
│ │ │ │ -0002b750: 3132 6820 6820 202b 2034 6820 2020 2020  12h h  + 4h     
│ │ │ │ -0002b760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b6f0: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ +0002b700: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ +0002b710: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002b720: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b730: 2020 2020 2020 2020 2020 2053 6567 7265             Segre
│ │ │ │ +0002b740: 203d 3e20 3732 6820 6820 202d 2032 3468   => 72h h  - 24h
│ │ │ │ +0002b750: 2068 2020 2d20 3132 6820 6820 202b 2034   h  - 12h h  + 4
│ │ │ │ +0002b760: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002b770: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b790: 2031 2032 2020 2020 2020 3120 3220 2020   1 2      1 2   
│ │ │ │ -0002b7a0: 2020 2031 2032 2020 2020 2031 2020 2020     1 2     1    
│ │ │ │ -0002b7b0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -0002b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b790: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +0002b7a0: 3120 3220 2020 2020 2031 2032 2020 2020  1 2      1 2    
│ │ │ │ +0002b7b0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +0002b7c0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0002b7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0002b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b820: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0002b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b810: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │  0002b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b860: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b8b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b900: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b940: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b950: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ -0002b960: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +0002b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b950: 207c 0a7c 2020 2032 2020 2020 2020 2020   |.|   2        
│ │ │ │ +0002b960: 2020 3220 2020 2020 3220 3220 2020 2020    2     2 2     
│ │ │ │  0002b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b990: 2020 2020 2020 2020 2020 207c 0a7c 3234             |.|24
│ │ │ │ -0002b9a0: 6820 6820 202d 2031 3268 2068 202c 2037  h h  - 12h h , 7
│ │ │ │ -0002b9b0: 3268 2068 207d 2020 2020 2020 2020 2020  2h h }          
│ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b9a0: 207c 0a7c 3234 6820 6820 202d 2031 3268   |.|24h h  - 12h
│ │ │ │ +0002b9b0: 2068 202c 2037 3268 2068 207d 2020 2020   h , 72h h }    
│ │ │ │  0002b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002b9f0: 2031 2032 2020 2020 2020 3120 3220 2020   1 2      1 2   
│ │ │ │ -0002ba00: 2020 3120 3220 2020 2020 2020 2020 2020    1 2           
│ │ │ │ +0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b9f0: 207c 0a7c 2020 2031 2032 2020 2020 2020   |.|   1 2      
│ │ │ │ +0002ba00: 3120 3220 2020 2020 3120 3220 2020 2020  1 2     1 2     
│ │ │ │  0002ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002ba40: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba40: 207c 0a7c 2020 2020 2020 2020 2020 2032   |.|           2
│ │ │ │  0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba80: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -0002ba90: 3468 2068 2020 2b20 6820 2020 2020 2020  4h h  + h       
│ │ │ │ +0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba90: 207c 0a7c 2b20 3468 2068 2020 2b20 6820   |.|+ 4h h  + h 
│ │ │ │  0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002bae0: 2020 3120 3220 2020 2032 2020 2020 2020    1 2    2      
│ │ │ │ +0002bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bae0: 207c 0a7c 2020 2020 3120 3220 2020 2032   |.|    1 2    2
│ │ │ │  0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002bb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002bb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0002bb80: 3420 3a20 7331 3d3d 5365 6748 6173 6823  4 : s1==SegHash#
│ │ │ │ -0002bb90: 2253 6567 7265 2220 2020 2020 2020 2020  "Segre"         
│ │ │ │ +0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bb80: 2d2b 0a7c 6931 3420 3a20 7331 3d3d 5365  -+.|i14 : s1==Se
│ │ │ │ +0002bb90: 6748 6173 6823 2253 6567 7265 2220 2020  gHash#"Segre"   
│ │ │ │  0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bbc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bbd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc10: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002bc20: 3420 3d20 7472 7565 2020 2020 2020 2020  4 = true        
│ │ │ │ +0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc20: 207c 0a7c 6f31 3420 3d20 7472 7565 2020   |.|o14 = true  
│ │ │ │  0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 496e  -----------+..In
│ │ │ │ -0002bcc0: 2074 6865 2063 6173 6520 7768 6572 6520   the case where 
│ │ │ │ -0002bcd0: 7468 6520 616d 6269 656e 7420 7370 6163  the ambient spac
│ │ │ │ -0002bce0: 6520 6973 2061 2074 6f72 6963 2076 6172  e is a toric var
│ │ │ │ -0002bcf0: 6965 7479 2077 6869 6368 2069 7320 6e6f  iety which is no
│ │ │ │ -0002bd00: 7420 6120 7072 6f64 7563 740a 6f66 2070  t a product.of p
│ │ │ │ -0002bd10: 726f 6a65 6374 6976 6520 7370 6163 6573  rojective spaces
│ │ │ │ -0002bd20: 2077 6520 6d75 7374 206c 6f61 6420 7468   we must load th
│ │ │ │ -0002bd30: 6520 4e6f 726d 616c 546f 7269 6356 6172  e NormalToricVar
│ │ │ │ -0002bd40: 6965 7469 6573 2070 6163 6b61 6765 2061  ieties package a
│ │ │ │ -0002bd50: 6e64 206d 7573 740a 616c 736f 2069 6e70  nd must.also inp
│ │ │ │ -0002bd60: 7574 2074 6865 2074 6f72 6963 2076 6172  ut the toric var
│ │ │ │ -0002bd70: 6965 7479 2e20 4966 2074 6865 2074 6f72  iety. If the tor
│ │ │ │ -0002bd80: 6963 2076 6172 6965 7479 2069 7320 6120  ic variety is a 
│ │ │ │ -0002bd90: 7072 6f64 7563 7420 6f66 2070 726f 6a65  product of proje
│ │ │ │ -0002bda0: 6374 6976 650a 7370 6163 6520 6974 2069  ctive.space it i
│ │ │ │ -0002bdb0: 7320 7265 636f 6d6d 656e 6465 6420 746f  s recommended to
│ │ │ │ -0002bdc0: 2075 7365 2074 6865 2066 6f72 6d20 6162   use the form ab
│ │ │ │ -0002bdd0: 6f76 6520 7261 7468 6572 2074 6861 6e20  ove rather than 
│ │ │ │ -0002bde0: 696e 7075 7474 696e 6720 7468 6520 746f  inputting the to
│ │ │ │ -0002bdf0: 7269 630a 7661 7269 6574 7920 666f 7220  ric.variety for 
│ │ │ │ -0002be00: 6566 6669 6369 656e 6379 2072 6561 736f  efficiency reaso
│ │ │ │ -0002be10: 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ns...+----------
│ │ │ │ +0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bcc0: 2d2b 0a0a 496e 2074 6865 2063 6173 6520  -+..In the case 
│ │ │ │ +0002bcd0: 7768 6572 6520 7468 6520 616d 6269 656e  where the ambien
│ │ │ │ +0002bce0: 7420 7370 6163 6520 6973 2061 2074 6f72  t space is a tor
│ │ │ │ +0002bcf0: 6963 2076 6172 6965 7479 2077 6869 6368  ic variety which
│ │ │ │ +0002bd00: 2069 7320 6e6f 7420 6120 7072 6f64 7563   is not a produc
│ │ │ │ +0002bd10: 740a 6f66 2070 726f 6a65 6374 6976 6520  t.of projective 
│ │ │ │ +0002bd20: 7370 6163 6573 2077 6520 6d75 7374 206c  spaces we must l
│ │ │ │ +0002bd30: 6f61 6420 7468 6520 4e6f 726d 616c 546f  oad the NormalTo
│ │ │ │ +0002bd40: 7269 6356 6172 6965 7469 6573 2070 6163  ricVarieties pac
│ │ │ │ +0002bd50: 6b61 6765 2061 6e64 206d 7573 740a 616c  kage and must.al
│ │ │ │ +0002bd60: 736f 2069 6e70 7574 2074 6865 2074 6f72  so input the tor
│ │ │ │ +0002bd70: 6963 2076 6172 6965 7479 2e20 4966 2074  ic variety. If t
│ │ │ │ +0002bd80: 6865 2074 6f72 6963 2076 6172 6965 7479  he toric variety
│ │ │ │ +0002bd90: 2069 7320 6120 7072 6f64 7563 7420 6f66   is a product of
│ │ │ │ +0002bda0: 2070 726f 6a65 6374 6976 650a 7370 6163   projective.spac
│ │ │ │ +0002bdb0: 6520 6974 2069 7320 7265 636f 6d6d 656e  e it is recommen
│ │ │ │ +0002bdc0: 6465 6420 746f 2075 7365 2074 6865 2066  ded to use the f
│ │ │ │ +0002bdd0: 6f72 6d20 6162 6f76 6520 7261 7468 6572  orm above rather
│ │ │ │ +0002bde0: 2074 6861 6e20 696e 7075 7474 696e 6720   than inputting 
│ │ │ │ +0002bdf0: 7468 6520 746f 7269 630a 7661 7269 6574  the toric.variet
│ │ │ │ +0002be00: 7920 666f 7220 6566 6669 6369 656e 6379  y for efficiency
│ │ │ │ +0002be10: 2072 6561 736f 6e73 2e0a 0a2b 2d2d 2d2d   reasons...+----
│ │ │ │  0002be20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002be30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0002be60: 7c69 3135 203a 206e 6565 6473 5061 636b  |i15 : needsPack
│ │ │ │ -0002be70: 6167 6520 224e 6f72 6d61 6c54 6f72 6963  age "NormalToric
│ │ │ │ -0002be80: 5661 7269 6574 6965 7322 2020 2020 2020  Varieties"      
│ │ │ │ +0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002be60: 2d2d 2d2d 2b0a 7c69 3135 203a 206e 6565  ----+.|i15 : nee
│ │ │ │ +0002be70: 6473 5061 636b 6167 6520 224e 6f72 6d61  dsPackage "Norma
│ │ │ │ +0002be80: 6c54 6f72 6963 5661 7269 6574 6965 7322  lToricVarieties"
│ │ │ │  0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002beb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bef0: 2020 2020 7c0a 7c6f 3135 203d 204e 6f72      |.|o15 = Nor
│ │ │ │ -0002bf00: 6d61 6c54 6f72 6963 5661 7269 6574 6965  malToricVarietie
│ │ │ │ -0002bf10: 7320 2020 2020 2020 2020 2020 2020 2020  s               
│ │ │ │ +0002bef0: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ +0002bf00: 203d 204e 6f72 6d61 6c54 6f72 6963 5661   = NormalToricVa
│ │ │ │ +0002bf10: 7269 6574 6965 7320 2020 2020 2020 2020  rieties         
│ │ │ │  0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002bf40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf80: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -0002bf90: 203a 2050 6163 6b61 6765 2020 2020 2020   : Package      
│ │ │ │ +0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf90: 7c0a 7c6f 3135 203a 2050 6163 6b61 6765  |.|o15 : Package
│ │ │ │  0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfd0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002bfd0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0002bfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c020: 2b0a 7c69 3136 203a 2052 686f 203d 207b  +.|i16 : Rho = {
│ │ │ │ -0002c030: 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 7d2c  {1,0,0},{0,1,0},
│ │ │ │ -0002c040: 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 2c30  {0,0,1},{-1,-1,0
│ │ │ │ -0002c050: 7d2c 7b30 2c30 2c2d 317d 7d20 2020 2020  },{0,0,-1}}     
│ │ │ │ -0002c060: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c020: 2d2d 2d2d 2d2d 2b0a 7c69 3136 203a 2052  ------+.|i16 : R
│ │ │ │ +0002c030: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30  ho = {{1,0,0},{0
│ │ │ │ +0002c040: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d  ,1,0},{0,0,1},{-
│ │ │ │ +0002c050: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d  1,-1,0},{0,0,-1}
│ │ │ │ +0002c060: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002c070: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0b0: 2020 2020 2020 7c0a 7c6f 3136 203d 207b        |.|o16 = {
│ │ │ │ -0002c0c0: 7b31 2c20 302c 2030 7d2c 207b 302c 2031  {1, 0, 0}, {0, 1
│ │ │ │ -0002c0d0: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c  , 0}, {0, 0, 1},
│ │ │ │ -0002c0e0: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30   {-1, -1, 0}, {0
│ │ │ │ -0002c0f0: 2c20 302c 202d 317d 7d20 2020 2020 2020  , 0, -1}}       
│ │ │ │ -0002c100: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002c0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002c0c0: 3136 203d 207b 7b31 2c20 302c 2030 7d2c  16 = {{1, 0, 0},
│ │ │ │ +0002c0d0: 207b 302c 2031 2c20 307d 2c20 7b30 2c20   {0, 1, 0}, {0, 
│ │ │ │ +0002c0e0: 302c 2031 7d2c 207b 2d31 2c20 2d31 2c20  0, 1}, {-1, -1, 
│ │ │ │ +0002c0f0: 307d 2c20 7b30 2c20 302c 202d 317d 7d20  0}, {0, 0, -1}} 
│ │ │ │ +0002c100: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002c150: 3136 203a 204c 6973 7420 2020 2020 2020  16 : List       
│ │ │ │ +0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c150: 2020 7c0a 7c6f 3136 203a 204c 6973 7420    |.|o16 : List 
│ │ │ │  0002c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c190: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002c190: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0002c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c1e0: 2d2d 2b0a 7c69 3137 203a 2053 6967 6d61  --+.|i17 : Sigma
│ │ │ │ -0002c1f0: 203d 207b 7b30 2c31 2c32 7d2c 7b31 2c32   = {{0,1,2},{1,2
│ │ │ │ -0002c200: 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30 2c31  ,3},{0,2,3},{0,1
│ │ │ │ -0002c210: 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30 2c33  ,4},{1,3,4},{0,3
│ │ │ │ -0002c220: 2c34 7d7d 2020 2020 2020 2020 207c 0a7c  ,4}}         |.|
│ │ │ │ -0002c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c1e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a  --------+.|i17 :
│ │ │ │ +0002c1f0: 2053 6967 6d61 203d 207b 7b30 2c31 2c32   Sigma = {{0,1,2
│ │ │ │ +0002c200: 7d2c 7b31 2c32 2c33 7d2c 7b30 2c32 2c33  },{1,2,3},{0,2,3
│ │ │ │ +0002c210: 7d2c 7b30 2c31 2c34 7d2c 7b31 2c33 2c34  },{0,1,4},{1,3,4
│ │ │ │ +0002c220: 7d2c 7b30 2c33 2c34 7d7d 2020 2020 2020  },{0,3,4}}      
│ │ │ │ +0002c230: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0002c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c270: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ -0002c280: 207b 7b30 2c20 312c 2032 7d2c 207b 312c   {{0, 1, 2}, {1,
│ │ │ │ -0002c290: 2032 2c20 337d 2c20 7b30 2c20 322c 2033   2, 3}, {0, 2, 3
│ │ │ │ -0002c2a0: 7d2c 207b 302c 2031 2c20 347d 2c20 7b31  }, {0, 1, 4}, {1
│ │ │ │ -0002c2b0: 2c20 332c 2034 7d2c 207b 302c 2033 2c20  , 3, 4}, {0, 3, 
│ │ │ │ -0002c2c0: 347d 7d7c 0a7c 2020 2020 2020 2020 2020  4}}|.|          
│ │ │ │ +0002c270: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c280: 7c6f 3137 203d 207b 7b30 2c20 312c 2032  |o17 = {{0, 1, 2
│ │ │ │ +0002c290: 7d2c 207b 312c 2032 2c20 337d 2c20 7b30  }, {1, 2, 3}, {0
│ │ │ │ +0002c2a0: 2c20 322c 2033 7d2c 207b 302c 2031 2c20  , 2, 3}, {0, 1, 
│ │ │ │ +0002c2b0: 347d 2c20 7b31 2c20 332c 2034 7d2c 207b  4}, {1, 3, 4}, {
│ │ │ │ +0002c2c0: 302c 2033 2c20 347d 7d7c 0a7c 2020 2020  0, 3, 4}}|.|    
│ │ │ │  0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c310: 7c6f 3137 203a 204c 6973 7420 2020 2020  |o17 : List     
│ │ │ │ -0002c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c310: 2020 2020 7c0a 7c6f 3137 203a 204c 6973      |.|o17 : Lis
│ │ │ │ +0002c320: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │  0002c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c350: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0002c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c3a0: 2d2d 2d2d 2b0a 7c69 3138 203a 2058 203d  ----+.|i18 : X =
│ │ │ │ -0002c3b0: 206e 6f72 6d61 6c54 6f72 6963 5661 7269   normalToricVari
│ │ │ │ -0002c3c0: 6574 7928 5268 6f2c 5369 676d 612c 436f  ety(Rho,Sigma,Co
│ │ │ │ -0002c3d0: 6566 6669 6369 656e 7452 696e 6720 3d3e  efficientRing =>
│ │ │ │ -0002c3e0: 5a5a 2f33 3237 3439 2920 2020 2020 207c  ZZ/32749)      |
│ │ │ │ -0002c3f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ +0002c3b0: 203a 2058 203d 206e 6f72 6d61 6c54 6f72   : X = normalTor
│ │ │ │ +0002c3c0: 6963 5661 7269 6574 7928 5268 6f2c 5369  icVariety(Rho,Si
│ │ │ │ +0002c3d0: 676d 612c 436f 6566 6669 6369 656e 7452  gma,CoefficientR
│ │ │ │ +0002c3e0: 696e 6720 3d3e 5a5a 2f33 3237 3439 2920  ing =>ZZ/32749) 
│ │ │ │ +0002c3f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c430: 2020 2020 2020 2020 2020 7c0a 7c6f 3138            |.|o18
│ │ │ │ -0002c440: 203d 2058 2020 2020 2020 2020 2020 2020   = X            
│ │ │ │ +0002c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c440: 7c0a 7c6f 3138 203d 2058 2020 2020 2020  |.|o18 = X      
│ │ │ │  0002c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002c480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c4d0: 7c0a 7c6f 3138 203a 204e 6f72 6d61 6c54  |.|o18 : NormalT
│ │ │ │ -0002c4e0: 6f72 6963 5661 7269 6574 7920 2020 2020  oricVariety     
│ │ │ │ -0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c4d0: 2020 2020 2020 7c0a 7c6f 3138 203a 204e        |.|o18 : N
│ │ │ │ +0002c4e0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002c4f0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  0002c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c510: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0002c520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c520: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0002c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c560: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2043  ------+.|i19 : C
│ │ │ │ -0002c570: 6865 636b 546f 7269 6356 6172 6965 7479  heckToricVariety
│ │ │ │ -0002c580: 5661 6c69 6428 5829 2020 2020 2020 2020  Valid(X)        
│ │ │ │ +0002c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002c570: 3139 203a 2043 6865 636b 546f 7269 6356  19 : CheckToricV
│ │ │ │ +0002c580: 6172 6965 7479 5661 6c69 6428 5829 2020  arietyValid(X)  
│ │ │ │  0002c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002c5b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002c600: 3139 203d 2074 7275 6520 2020 2020 2020  19 = true       
│ │ │ │ +0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c600: 2020 7c0a 7c6f 3139 203d 2074 7275 6520    |.|o19 = true 
│ │ │ │  0002c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c640: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002c640: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0002c650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c690: 2d2d 2b0a 7c69 3230 203a 2052 3d72 696e  --+.|i20 : R=rin
│ │ │ │ -0002c6a0: 6728 5829 2020 2020 2020 2020 2020 2020  g(X)            
│ │ │ │ +0002c690: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ +0002c6a0: 2052 3d72 696e 6728 5829 2020 2020 2020   R=ring(X)      
│ │ │ │  0002c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c6e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0002c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c720: 2020 2020 2020 2020 7c0a 7c6f 3230 203d          |.|o20 =
│ │ │ │ -0002c730: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0002c720: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c730: 7c6f 3230 203d 2052 2020 2020 2020 2020  |o20 = R        
│ │ │ │  0002c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c770: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002c770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0002c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c7c0: 7c6f 3230 203a 2050 6f6c 796e 6f6d 6961  |o20 : Polynomia
│ │ │ │ -0002c7d0: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +0002c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c7c0: 2020 2020 7c0a 7c6f 3230 203a 2050 6f6c      |.|o20 : Pol
│ │ │ │ +0002c7d0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │  0002c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c800: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0002c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c810: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  0002c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c850: 2d2d 2d2d 2b0a 7c69 3231 203a 2049 3d69  ----+.|i21 : I=i
│ │ │ │ -0002c860: 6465 616c 2852 5f30 5e34 2a52 5f31 2c52  deal(R_0^4*R_1,R
│ │ │ │ -0002c870: 5f30 2a52 5f33 2a52 5f34 2a52 5f32 2d52  _0*R_3*R_4*R_2-R
│ │ │ │ -0002c880: 5f32 5e32 2a52 5f30 5e32 2920 2020 2020  _2^2*R_0^2)     
│ │ │ │ -0002c890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c8a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231  ----------+.|i21
│ │ │ │ +0002c860: 203a 2049 3d69 6465 616c 2852 5f30 5e34   : I=ideal(R_0^4
│ │ │ │ +0002c870: 2a52 5f31 2c52 5f30 2a52 5f33 2a52 5f34  *R_1,R_0*R_3*R_4
│ │ │ │ +0002c880: 2a52 5f32 2d52 5f32 5e32 2a52 5f30 5e32  *R_2-R_2^2*R_0^2
│ │ │ │ +0002c890: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002c8a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c8e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002c8f0: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ -0002c900: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ +0002c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c8f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002c900: 2034 2020 2020 2020 2032 2032 2020 2020   4       2 2    
│ │ │ │  0002c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c930: 2020 2020 207c 0a7c 6f32 3120 3d20 6964       |.|o21 = id
│ │ │ │ -0002c940: 6561 6c20 2878 2078 202c 202d 2078 2078  eal (x x , - x x
│ │ │ │ -0002c950: 2020 2b20 7820 7820 7820 7820 2920 2020    + x x x x )   
│ │ │ │ -0002c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c930: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0002c940: 3120 3d20 6964 6561 6c20 2878 2078 202c  1 = ideal (x x ,
│ │ │ │ +0002c950: 202d 2078 2078 2020 2b20 7820 7820 7820   - x x  + x x x 
│ │ │ │ +0002c960: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │  0002c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c980: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002c990: 2030 2031 2020 2020 2030 2032 2020 2020   0 1     0 2    
│ │ │ │ -0002c9a0: 3020 3220 3320 3420 2020 2020 2020 2020  0 2 3 4         
│ │ │ │ +0002c980: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002c990: 2020 2020 2020 2030 2031 2020 2020 2030         0 1     0
│ │ │ │ +0002c9a0: 2032 2020 2020 3020 3220 3320 3420 2020   2    0 2 3 4   
│ │ │ │  0002c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c9c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c9d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca10: 2020 2020 2020 7c0a 7c6f 3231 203a 2049        |.|o21 : I
│ │ │ │ -0002ca20: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │ +0002ca10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002ca20: 3231 203a 2049 6465 616c 206f 6620 5220  21 : Ideal of R 
│ │ │ │  0002ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002ca60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002cab0: 3232 203a 2053 6567 7265 2858 2c49 2920  22 : Segre(X,I) 
│ │ │ │ -0002cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002cab0: 2d2d 2b0a 7c69 3232 203a 2053 6567 7265  --+.|i22 : Segre
│ │ │ │ +0002cac0: 2858 2c49 2920 2020 2020 2020 2020 2020  (X,I)           
│ │ │ │  0002cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002caf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0002cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb40: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002cb50: 3220 2020 2020 2020 3220 2020 2020 2020  2       2       
│ │ │ │ +0002cb40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002cb50: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │  0002cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002cb90: 6f32 3220 3d20 2d20 3732 7820 7820 202b  o22 = - 72x x  +
│ │ │ │ -0002cba0: 2033 7820 202b 2038 7820 7820 202b 2078   3x  + 8x x  + x
│ │ │ │ -0002cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cb90: 2020 207c 0a7c 6f32 3220 3d20 2d20 3732     |.|o22 = - 72
│ │ │ │ +0002cba0: 7820 7820 202b 2033 7820 202b 2038 7820  x x  + 3x  + 8x 
│ │ │ │ +0002cbb0: 7820 202b 2078 2020 2020 2020 2020 2020  x  + x          
│ │ │ │  0002cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002cbe0: 2020 2020 2020 3320 3420 2020 2020 3320        3 4     3 
│ │ │ │ -0002cbf0: 2020 2020 3320 3420 2020 2033 2020 2020      3 4    3    
│ │ │ │ -0002cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cbd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002cbe0: 7c20 2020 2020 2020 2020 2020 3320 3420  |           3 4 
│ │ │ │ +0002cbf0: 2020 2020 3320 2020 2020 3320 3420 2020      3     3 4   
│ │ │ │ +0002cc00: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  0002cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002cc20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0002cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002cc70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002cc80: 2020 2020 2020 205a 5a5b 7820 2e2e 7820         ZZ[x ..x 
│ │ │ │ -0002cc90: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +0002cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002cc80: 2020 2020 2020 2020 2020 2020 205a 5a5b               ZZ[
│ │ │ │ +0002cc90: 7820 2e2e 7820 5d20 2020 2020 2020 2020  x ..x ]         
│ │ │ │  0002cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccd0: 2020 2020 2020 3020 2020 3420 2020 2020        0   4     
│ │ │ │ -0002cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ccb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002ccc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ccd0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +0002cce0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  0002ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd00: 2020 2020 7c0a 7c6f 3232 203a 202d 2d2d      |.|o22 : ---
│ │ │ │ -0002cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002cd00: 2020 2020 2020 2020 2020 7c0a 7c6f 3232            |.|o22
│ │ │ │ +0002cd10: 203a 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   : -------------
│ │ │ │  0002cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cd30: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020  ------          
│ │ │ │ -0002cd40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cd50: 0a7c 2020 2020 2020 2878 2078 202c 2078  .|      (x x , x
│ │ │ │ -0002cd60: 2078 2078 202c 2078 2020 2d20 7820 2c20   x x , x  - x , 
│ │ │ │ -0002cd70: 7820 202d 2078 202c 2078 2020 2d20 7820  x  - x , x  - x 
│ │ │ │ -0002cd80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0002cd90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002cda0: 2020 2020 2032 2034 2020 2030 2031 2033       2 4   0 1 3
│ │ │ │ -0002cdb0: 2020 2030 2020 2020 3320 2020 3120 2020     0    3   1   
│ │ │ │ -0002cdc0: 2033 2020 2032 2020 2020 3420 2020 2020   3   2    4     
│ │ │ │ -0002cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cde0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020  ------------    
│ │ │ │ +0002cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cd50: 2020 2020 207c 0a7c 2020 2020 2020 2878       |.|      (x
│ │ │ │ +0002cd60: 2078 202c 2078 2078 2078 202c 2078 2020   x , x x x , x  
│ │ │ │ +0002cd70: 2d20 7820 2c20 7820 202d 2078 202c 2078  - x , x  - x , x
│ │ │ │ +0002cd80: 2020 2d20 7820 2920 2020 2020 2020 2020    - x )         
│ │ │ │ +0002cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cda0: 7c0a 7c20 2020 2020 2020 2032 2034 2020  |.|        2 4  
│ │ │ │ +0002cdb0: 2030 2031 2033 2020 2030 2020 2020 3320   0 1 3   0    3 
│ │ │ │ +0002cdc0: 2020 3120 2020 2033 2020 2032 2020 2020    1    3   2    
│ │ │ │ +0002cdd0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0002cde0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ce30: 2b0a 7c69 3233 203a 2043 683d 546f 7269  +.|i23 : Ch=Tori
│ │ │ │ -0002ce40: 6343 686f 7752 696e 6728 5829 2020 2020  cChowRing(X)    
│ │ │ │ -0002ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ce30: 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a 2043  ------+.|i23 : C
│ │ │ │ +0002ce40: 683d 546f 7269 6343 686f 7752 696e 6728  h=ToricChowRing(
│ │ │ │ +0002ce50: 5829 2020 2020 2020 2020 2020 2020 2020  X)              
│ │ │ │  0002ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ce80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cec0: 2020 2020 2020 7c0a 7c6f 3233 203d 2043        |.|o23 = C
│ │ │ │ -0002ced0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │ +0002cec0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002ced0: 3233 203d 2043 6820 2020 2020 2020 2020  23 = Ch         
│ │ │ │  0002cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002cf10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0002cf60: 3233 203a 2051 756f 7469 656e 7452 696e  23 : QuotientRin
│ │ │ │ -0002cf70: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0002cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cf60: 2020 7c0a 7c6f 3233 203a 2051 756f 7469    |.|o23 : Quoti
│ │ │ │ +0002cf70: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │  0002cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cfa0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002cfa0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  0002cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cff0: 2d2d 2b0a 7c69 3234 203a 2073 333d 5365  --+.|i24 : s3=Se
│ │ │ │ -0002d000: 6772 6528 4368 2c58 2c49 2920 2020 2020  gre(Ch,X,I)     
│ │ │ │ -0002d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cff0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a  --------+.|i24 :
│ │ │ │ +0002d000: 2073 333d 5365 6772 6528 4368 2c58 2c49   s3=Segre(Ch,X,I
│ │ │ │ +0002d010: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  0002d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d040: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0002d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d080: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002d090: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │ -0002d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d080: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002d090: 7c20 2020 2020 2020 2020 2020 3220 2020  |           2   
│ │ │ │ +0002d0a0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0002d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0d0: 2020 207c 0a7c 6f32 3420 3d20 2d20 3732     |.|o24 = - 72
│ │ │ │ -0002d0e0: 7820 7820 202b 2033 7820 202b 2038 7820  x x  + 3x  + 8x 
│ │ │ │ -0002d0f0: 7820 202b 2078 2020 2020 2020 2020 2020  x  + x          
│ │ │ │ +0002d0d0: 2020 2020 2020 2020 207c 0a7c 6f32 3420           |.|o24 
│ │ │ │ +0002d0e0: 3d20 2d20 3732 7820 7820 202b 2033 7820  = - 72x x  + 3x 
│ │ │ │ +0002d0f0: 202b 2038 7820 7820 202b 2078 2020 2020   + 8x x  + x    
│ │ │ │  0002d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d110: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002d120: 7c20 2020 2020 2020 2020 2020 3320 3420  |           3 4 
│ │ │ │ -0002d130: 2020 2020 3320 2020 2020 3320 3420 2020      3     3 4   
│ │ │ │ -0002d140: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0002d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d120: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002d130: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +0002d140: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │  0002d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d160: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1b0: 2020 2020 7c0a 7c6f 3234 203a 2043 6820      |.|o24 : Ch 
│ │ │ │ -0002d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d1b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3234            |.|o24
│ │ │ │ +0002d1c0: 203a 2043 6820 2020 2020 2020 2020 2020   : Ch           
│ │ │ │  0002d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d200: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d200: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0002d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6c6c  ----------+..All
│ │ │ │ -0002d250: 2074 6865 2065 7861 6d70 6c65 7320 7765   the examples we
│ │ │ │ -0002d260: 7265 2064 6f6e 6520 7573 696e 6720 7379  re done using sy
│ │ │ │ -0002d270: 6d62 6f6c 6963 2063 6f6d 7075 7461 7469  mbolic computati
│ │ │ │ -0002d280: 6f6e 7320 7769 7468 2047 725c 226f 626e  ons with Gr\"obn
│ │ │ │ -0002d290: 6572 2062 6173 6573 2e0a 4368 616e 6769  er bases..Changi
│ │ │ │ -0002d2a0: 6e67 2074 6865 206f 7074 696f 6e20 2a6e  ng the option *n
│ │ │ │ -0002d2b0: 6f74 6520 436f 6d70 4d65 7468 6f64 3a20  ote CompMethod: 
│ │ │ │ -0002d2c0: 436f 6d70 4d65 7468 6f64 2c20 746f 2062  CompMethod, to b
│ │ │ │ -0002d2d0: 6572 7469 6e69 2077 696c 6c20 646f 2074  ertini will do t
│ │ │ │ -0002d2e0: 6865 206d 6169 6e0a 636f 6d70 7574 6174  he main.computat
│ │ │ │ -0002d2f0: 696f 6e73 206e 756d 6572 6963 616c 6c79  ions numerically
│ │ │ │ -0002d300: 2c20 7072 6f76 6964 6564 2042 6572 7469  , provided Berti
│ │ │ │ -0002d310: 6e69 2069 7320 202a 6e6f 7465 2069 6e73  ni is  *note ins
│ │ │ │ -0002d320: 7461 6c6c 6564 2061 6e64 2063 6f6e 6669  talled and confi
│ │ │ │ -0002d330: 6775 7265 643a 0a63 6f6e 6669 6775 7269  gured:.configuri
│ │ │ │ -0002d340: 6e67 2042 6572 7469 6e69 2c2e 204e 6f74  ng Bertini,. Not
│ │ │ │ -0002d350: 6520 7468 6174 2074 6865 2062 6572 7469  e that the berti
│ │ │ │ -0002d360: 6e69 206f 7074 696f 6e20 6973 206f 6e6c  ni option is onl
│ │ │ │ -0002d370: 7920 6176 6169 6c61 626c 6520 666f 720a  y available for.
│ │ │ │ -0002d380: 7375 6273 6368 656d 6573 206f 6620 5c50  subschemes of \P
│ │ │ │ -0002d390: 505e 6e2e 0a0a 4f62 7365 7276 6520 7468  P^n...Observe th
│ │ │ │ -0002d3a0: 6174 2074 6865 2061 6c67 6f72 6974 686d  at the algorithm
│ │ │ │ -0002d3b0: 2069 7320 6120 7072 6f62 6162 696c 6973   is a probabilis
│ │ │ │ -0002d3c0: 7469 6320 616c 676f 7269 7468 6d20 616e  tic algorithm an
│ │ │ │ -0002d3d0: 6420 6d61 7920 6769 7665 2061 2077 726f  d may give a wro
│ │ │ │ -0002d3e0: 6e67 0a61 6e73 7765 7220 7769 7468 2061  ng.answer with a
│ │ │ │ -0002d3f0: 2073 6d61 6c6c 2062 7574 206e 6f6e 7a65   small but nonze
│ │ │ │ -0002d400: 726f 2070 726f 6261 6269 6c69 7479 2e20  ro probability. 
│ │ │ │ -0002d410: 5265 6164 206d 6f72 6520 756e 6465 7220  Read more under 
│ │ │ │ -0002d420: 2a6e 6f74 650a 7072 6f62 6162 696c 6973  *note.probabilis
│ │ │ │ -0002d430: 7469 6320 616c 676f 7269 7468 6d3a 2070  tic algorithm: p
│ │ │ │ -0002d440: 726f 6261 6269 6c69 7374 6963 2061 6c67  robabilistic alg
│ │ │ │ -0002d450: 6f72 6974 686d 2c2e 0a0a 5761 7973 2074  orithm,...Ways t
│ │ │ │ -0002d460: 6f20 7573 6520 5365 6772 653a 0a3d 3d3d  o use Segre:.===
│ │ │ │ -0002d470: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0002d480: 0a20 202a 2022 5365 6772 6528 4964 6561  .  * "Segre(Idea
│ │ │ │ -0002d490: 6c29 220a 2020 2a20 2253 6567 7265 2849  l)".  * "Segre(I
│ │ │ │ -0002d4a0: 6465 616c 2c53 796d 626f 6c29 220a 2020  deal,Symbol)".  
│ │ │ │ -0002d4b0: 2a20 2253 6567 7265 2851 756f 7469 656e  * "Segre(Quotien
│ │ │ │ -0002d4c0: 7452 696e 672c 4964 6561 6c29 220a 0a46  tRing,Ideal)"..F
│ │ │ │ -0002d4d0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -0002d4e0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -0002d4f0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -0002d500: 202a 6e6f 7465 2053 6567 7265 3a20 5365   *note Segre: Se
│ │ │ │ -0002d510: 6772 652c 2069 7320 6120 2a6e 6f74 6520  gre, is a *note 
│ │ │ │ -0002d520: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ -0002d530: 7769 7468 206f 7074 696f 6e73 3a0a 284d  with options:.(M
│ │ │ │ -0002d540: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -0002d550: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ -0002d560: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d  tions,...-------
│ │ │ │ +0002d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d250: 2b0a 0a41 6c6c 2074 6865 2065 7861 6d70  +..All the examp
│ │ │ │ +0002d260: 6c65 7320 7765 7265 2064 6f6e 6520 7573  les were done us
│ │ │ │ +0002d270: 696e 6720 7379 6d62 6f6c 6963 2063 6f6d  ing symbolic com
│ │ │ │ +0002d280: 7075 7461 7469 6f6e 7320 7769 7468 2047  putations with G
│ │ │ │ +0002d290: 725c 226f 626e 6572 2062 6173 6573 2e0a  r\"obner bases..
│ │ │ │ +0002d2a0: 4368 616e 6769 6e67 2074 6865 206f 7074  Changing the opt
│ │ │ │ +0002d2b0: 696f 6e20 2a6e 6f74 6520 436f 6d70 4d65  ion *note CompMe
│ │ │ │ +0002d2c0: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64  thod: CompMethod
│ │ │ │ +0002d2d0: 2c20 746f 2062 6572 7469 6e69 2077 696c  , to bertini wil
│ │ │ │ +0002d2e0: 6c20 646f 2074 6865 206d 6169 6e0a 636f  l do the main.co
│ │ │ │ +0002d2f0: 6d70 7574 6174 696f 6e73 206e 756d 6572  mputations numer
│ │ │ │ +0002d300: 6963 616c 6c79 2c20 7072 6f76 6964 6564  ically, provided
│ │ │ │ +0002d310: 2042 6572 7469 6e69 2069 7320 202a 6e6f   Bertini is  *no
│ │ │ │ +0002d320: 7465 2069 6e73 7461 6c6c 6564 2061 6e64  te installed and
│ │ │ │ +0002d330: 2063 6f6e 6669 6775 7265 643a 0a63 6f6e   configured:.con
│ │ │ │ +0002d340: 6669 6775 7269 6e67 2042 6572 7469 6e69  figuring Bertini
│ │ │ │ +0002d350: 2c2e 204e 6f74 6520 7468 6174 2074 6865  ,. Note that the
│ │ │ │ +0002d360: 2062 6572 7469 6e69 206f 7074 696f 6e20   bertini option 
│ │ │ │ +0002d370: 6973 206f 6e6c 7920 6176 6169 6c61 626c  is only availabl
│ │ │ │ +0002d380: 6520 666f 720a 7375 6273 6368 656d 6573  e for.subschemes
│ │ │ │ +0002d390: 206f 6620 5c50 505e 6e2e 0a0a 4f62 7365   of \PP^n...Obse
│ │ │ │ +0002d3a0: 7276 6520 7468 6174 2074 6865 2061 6c67  rve that the alg
│ │ │ │ +0002d3b0: 6f72 6974 686d 2069 7320 6120 7072 6f62  orithm is a prob
│ │ │ │ +0002d3c0: 6162 696c 6973 7469 6320 616c 676f 7269  abilistic algori
│ │ │ │ +0002d3d0: 7468 6d20 616e 6420 6d61 7920 6769 7665  thm and may give
│ │ │ │ +0002d3e0: 2061 2077 726f 6e67 0a61 6e73 7765 7220   a wrong.answer 
│ │ │ │ +0002d3f0: 7769 7468 2061 2073 6d61 6c6c 2062 7574  with a small but
│ │ │ │ +0002d400: 206e 6f6e 7a65 726f 2070 726f 6261 6269   nonzero probabi
│ │ │ │ +0002d410: 6c69 7479 2e20 5265 6164 206d 6f72 6520  lity. Read more 
│ │ │ │ +0002d420: 756e 6465 7220 2a6e 6f74 650a 7072 6f62  under *note.prob
│ │ │ │ +0002d430: 6162 696c 6973 7469 6320 616c 676f 7269  abilistic algori
│ │ │ │ +0002d440: 7468 6d3a 2070 726f 6261 6269 6c69 7374  thm: probabilist
│ │ │ │ +0002d450: 6963 2061 6c67 6f72 6974 686d 2c2e 0a0a  ic algorithm,...
│ │ │ │ +0002d460: 5761 7973 2074 6f20 7573 6520 5365 6772  Ways to use Segr
│ │ │ │ +0002d470: 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  e:.=============
│ │ │ │ +0002d480: 3d3d 3d3d 3d0a 0a20 202a 2022 5365 6772  =====..  * "Segr
│ │ │ │ +0002d490: 6528 4964 6561 6c29 220a 2020 2a20 2253  e(Ideal)".  * "S
│ │ │ │ +0002d4a0: 6567 7265 2849 6465 616c 2c53 796d 626f  egre(Ideal,Symbo
│ │ │ │ +0002d4b0: 6c29 220a 2020 2a20 2253 6567 7265 2851  l)".  * "Segre(Q
│ │ │ │ +0002d4c0: 756f 7469 656e 7452 696e 672c 4964 6561  uotientRing,Idea
│ │ │ │ +0002d4d0: 6c29 220a 0a46 6f72 2074 6865 2070 726f  l)"..For the pro
│ │ │ │ +0002d4e0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +0002d4f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +0002d500: 6f62 6a65 6374 202a 6e6f 7465 2053 6567  object *note Seg
│ │ │ │ +0002d510: 7265 3a20 5365 6772 652c 2069 7320 6120  re: Segre, is a 
│ │ │ │ +0002d520: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ +0002d530: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
│ │ │ │ +0002d540: 6e73 3a0a 284d 6163 6175 6c61 7932 446f  ns:.(Macaulay2Do
│ │ │ │ +0002d550: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +0002d560: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d  WithOptions,...-
│ │ │ │  0002d570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d5b0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -0002d5c0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -0002d5d0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -0002d5e0: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -0002d5f0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -0002d600: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ -0002d610: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -0002d620: 0a43 6861 7261 6374 6572 6973 7469 6343  .CharacteristicC
│ │ │ │ -0002d630: 6c61 7373 6573 2e6d 323a 3137 3633 3a30  lasses.m2:1763:0
│ │ │ │ -0002d640: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163  ....File: Charac
│ │ │ │ -0002d650: 7465 7269 7374 6963 436c 6173 7365 732e  teristicClasses.
│ │ │ │ -0002d660: 696e 666f 2c20 4e6f 6465 3a20 546f 7269  info, Node: Tori
│ │ │ │ -0002d670: 6343 686f 7752 696e 672c 2050 7265 763a  cChowRing, Prev:
│ │ │ │ -0002d680: 2053 6567 7265 2c20 5570 3a20 546f 700a   Segre, Up: Top.
│ │ │ │ -0002d690: 0a54 6f72 6963 4368 6f77 5269 6e67 202d  .ToricChowRing -
│ │ │ │ -0002d6a0: 2d20 436f 6d70 7574 6573 2074 6865 2043  - Computes the C
│ │ │ │ -0002d6b0: 686f 7720 7269 6e67 206f 6620 6120 6e6f  how ring of a no
│ │ │ │ -0002d6c0: 726d 616c 2074 6f72 6963 2076 6172 6965  rmal toric varie
│ │ │ │ -0002d6d0: 7479 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ty.*************
│ │ │ │ +0002d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0002d5c0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +0002d5d0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +0002d5e0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0002d5f0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0002d600: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ +0002d610: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +0002d620: 6b61 6765 732f 0a43 6861 7261 6374 6572  kages/.Character
│ │ │ │ +0002d630: 6973 7469 6343 6c61 7373 6573 2e6d 323a  isticClasses.m2:
│ │ │ │ +0002d640: 3137 3633 3a30 2e0a 1f0a 4669 6c65 3a20  1763:0....File: 
│ │ │ │ +0002d650: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ +0002d660: 6173 7365 732e 696e 666f 2c20 4e6f 6465  asses.info, Node
│ │ │ │ +0002d670: 3a20 546f 7269 6343 686f 7752 696e 672c  : ToricChowRing,
│ │ │ │ +0002d680: 2050 7265 763a 2053 6567 7265 2c20 5570   Prev: Segre, Up
│ │ │ │ +0002d690: 3a20 546f 700a 0a54 6f72 6963 4368 6f77  : Top..ToricChow
│ │ │ │ +0002d6a0: 5269 6e67 202d 2d20 436f 6d70 7574 6573  Ring -- Computes
│ │ │ │ +0002d6b0: 2074 6865 2043 686f 7720 7269 6e67 206f   the Chow ring o
│ │ │ │ +0002d6c0: 6620 6120 6e6f 726d 616c 2074 6f72 6963  f a normal toric
│ │ │ │ +0002d6d0: 2076 6172 6965 7479 0a2a 2a2a 2a2a 2a2a   variety.*******
│ │ │ │  0002d6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d710: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -0002d720: 200a 2020 2020 2020 2020 546f 7269 6343   .        ToricC
│ │ │ │ -0002d730: 686f 7752 696e 6720 580a 2020 2a20 496e  howRing X.  * In
│ │ │ │ -0002d740: 7075 7473 3a0a 2020 2020 2020 2a20 522c  puts:.      * R,
│ │ │ │ -0002d750: 2061 202a 6e6f 7465 206e 6f72 6d61 6c20   a *note normal 
│ │ │ │ -0002d760: 746f 7269 6320 7661 7269 6574 793a 0a20  toric variety:. 
│ │ │ │ -0002d770: 2020 2020 2020 2028 4e6f 726d 616c 546f         (NormalTo
│ │ │ │ -0002d780: 7269 6356 6172 6965 7469 6573 294e 6f72  ricVarieties)Nor
│ │ │ │ -0002d790: 6d61 6c54 6f72 6963 5661 7269 6574 792c  malToricVariety,
│ │ │ │ -0002d7a0: 2c20 4120 6e6f 726d 616c 2074 6f72 6963  , A normal toric
│ │ │ │ -0002d7b0: 2076 6172 6965 7479 0a20 202a 204f 7574   variety.  * Out
│ │ │ │ -0002d7c0: 7075 7473 3a0a 2020 2020 2020 2a20 6120  puts:.      * a 
│ │ │ │ -0002d7d0: 2a6e 6f74 6520 7175 6f74 6965 6e74 2072  *note quotient r
│ │ │ │ -0002d7e0: 696e 673a 2028 4d61 6361 756c 6179 3244  ing: (Macaulay2D
│ │ │ │ -0002d7f0: 6f63 2951 756f 7469 656e 7452 696e 672c  oc)QuotientRing,
│ │ │ │ -0002d800: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a  , ..Description.
│ │ │ │ -0002d810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4c 6574  ===========..Let
│ │ │ │ -0002d820: 2058 2062 6520 6120 746f 7269 6320 7661   X be a toric va
│ │ │ │ -0002d830: 7269 6574 7920 7769 7468 2074 6f74 616c  riety with total
│ │ │ │ -0002d840: 2063 6f6f 7264 696e 6174 6520 7269 6e67   coordinate ring
│ │ │ │ -0002d850: 2028 436f 7820 7269 6e67 2920 522e 2054   (Cox ring) R. T
│ │ │ │ -0002d860: 6869 7320 6d65 7468 6f64 0a63 6f6d 7075  his method.compu
│ │ │ │ -0002d870: 7465 7320 7468 6520 4368 6f77 2072 696e  tes the Chow rin
│ │ │ │ -0002d880: 6720 2043 686f 7720 7269 6e67 2043 683d  g  Chow ring Ch=
│ │ │ │ -0002d890: 522f 2853 522b 4c52 2920 6f66 2058 3b20  R/(SR+LR) of X; 
│ │ │ │ -0002d8a0: 6865 7265 2053 5220 6973 2074 6865 0a53  here SR is the.S
│ │ │ │ -0002d8b0: 7461 6e6c 6579 2d52 6569 736e 6572 2069  tanley-Reisner i
│ │ │ │ -0002d8c0: 6465 616c 206f 6620 7468 6520 636f 7272  deal of the corr
│ │ │ │ -0002d8d0: 6573 706f 6e64 696e 6720 6661 6e20 616e  esponding fan an
│ │ │ │ -0002d8e0: 6420 4c52 2069 7320 7468 6520 6964 6561  d LR is the idea
│ │ │ │ -0002d8f0: 6c20 6f66 206c 696e 6561 720a 7265 6c61  l of linear.rela
│ │ │ │ -0002d900: 7469 6f6e 7320 616d 6f75 6e74 2074 6865  tions amount the
│ │ │ │ -0002d910: 2072 6179 732e 2049 7420 6973 206e 6565   rays. It is nee
│ │ │ │ -0002d920: 6465 6420 666f 7220 696e 7075 7420 696e  ded for input in
│ │ │ │ -0002d930: 746f 2074 6865 206d 6574 686f 6473 202a  to the methods *
│ │ │ │ -0002d940: 6e6f 7465 2053 6567 7265 3a0a 5365 6772  note Segre:.Segr
│ │ │ │ -0002d950: 652c 2c20 2a6e 6f74 6520 4368 6572 6e3a  e,, *note Chern:
│ │ │ │ -0002d960: 2043 6865 726e 2c20 616e 6420 2a6e 6f74   Chern, and *not
│ │ │ │ -0002d970: 6520 4353 4d3a 2043 534d 2c20 696e 2074  e CSM: CSM, in t
│ │ │ │ -0002d980: 6865 2063 6173 6573 2077 6865 7265 2061  he cases where a
│ │ │ │ -0002d990: 2074 6f72 6963 0a76 6172 6965 7479 2069   toric.variety i
│ │ │ │ -0002d9a0: 7320 616c 736f 2069 6e70 7574 2074 6f20  s also input to 
│ │ │ │ -0002d9b0: 656e 7375 7265 2074 6861 7420 7468 6573  ensure that thes
│ │ │ │ -0002d9c0: 6520 6d65 7468 6f64 7320 7265 7475 726e  e methods return
│ │ │ │ -0002d9d0: 2072 6573 756c 7473 2069 6e20 7468 6520   results in the 
│ │ │ │ -0002d9e0: 7361 6d65 0a72 696e 672e 2057 6520 6769  same.ring. We gi
│ │ │ │ -0002d9f0: 7665 2061 6e20 6578 616d 706c 6520 6f66  ve an example of
│ │ │ │ -0002da00: 2074 6865 2075 7365 206f 6620 7468 6973   the use of this
│ │ │ │ -0002da10: 206d 6574 686f 6420 746f 2077 6f72 6b20   method to work 
│ │ │ │ -0002da20: 7769 7468 2065 6c65 6d65 6e74 7320 6f66  with elements of
│ │ │ │ -0002da30: 2074 6865 0a43 686f 7720 7269 6e67 206f   the.Chow ring o
│ │ │ │ -0002da40: 6620 6120 746f 7269 6320 7661 7269 6574  f a toric variet
│ │ │ │ -0002da50: 790a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  y..+------------
│ │ │ │ +0002d710: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +0002d720: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0002d730: 546f 7269 6343 686f 7752 696e 6720 580a  ToricChowRing X.
│ │ │ │ +0002d740: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0002d750: 2020 2a20 522c 2061 202a 6e6f 7465 206e    * R, a *note n
│ │ │ │ +0002d760: 6f72 6d61 6c20 746f 7269 6320 7661 7269  ormal toric vari
│ │ │ │ +0002d770: 6574 793a 0a20 2020 2020 2020 2028 4e6f  ety:.        (No
│ │ │ │ +0002d780: 726d 616c 546f 7269 6356 6172 6965 7469  rmalToricVarieti
│ │ │ │ +0002d790: 6573 294e 6f72 6d61 6c54 6f72 6963 5661  es)NormalToricVa
│ │ │ │ +0002d7a0: 7269 6574 792c 2c20 4120 6e6f 726d 616c  riety,, A normal
│ │ │ │ +0002d7b0: 2074 6f72 6963 2076 6172 6965 7479 0a20   toric variety. 
│ │ │ │ +0002d7c0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0002d7d0: 2020 2a20 6120 2a6e 6f74 6520 7175 6f74    * a *note quot
│ │ │ │ +0002d7e0: 6965 6e74 2072 696e 673a 2028 4d61 6361  ient ring: (Maca
│ │ │ │ +0002d7f0: 756c 6179 3244 6f63 2951 756f 7469 656e  ulay2Doc)Quotien
│ │ │ │ +0002d800: 7452 696e 672c 2c20 0a0a 4465 7363 7269  tRing,, ..Descri
│ │ │ │ +0002d810: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +0002d820: 3d0a 0a4c 6574 2058 2062 6520 6120 746f  =..Let X be a to
│ │ │ │ +0002d830: 7269 6320 7661 7269 6574 7920 7769 7468  ric variety with
│ │ │ │ +0002d840: 2074 6f74 616c 2063 6f6f 7264 696e 6174   total coordinat
│ │ │ │ +0002d850: 6520 7269 6e67 2028 436f 7820 7269 6e67  e ring (Cox ring
│ │ │ │ +0002d860: 2920 522e 2054 6869 7320 6d65 7468 6f64  ) R. This method
│ │ │ │ +0002d870: 0a63 6f6d 7075 7465 7320 7468 6520 4368  .computes the Ch
│ │ │ │ +0002d880: 6f77 2072 696e 6720 2043 686f 7720 7269  ow ring  Chow ri
│ │ │ │ +0002d890: 6e67 2043 683d 522f 2853 522b 4c52 2920  ng Ch=R/(SR+LR) 
│ │ │ │ +0002d8a0: 6f66 2058 3b20 6865 7265 2053 5220 6973  of X; here SR is
│ │ │ │ +0002d8b0: 2074 6865 0a53 7461 6e6c 6579 2d52 6569   the.Stanley-Rei
│ │ │ │ +0002d8c0: 736e 6572 2069 6465 616c 206f 6620 7468  sner ideal of th
│ │ │ │ +0002d8d0: 6520 636f 7272 6573 706f 6e64 696e 6720  e corresponding 
│ │ │ │ +0002d8e0: 6661 6e20 616e 6420 4c52 2069 7320 7468  fan and LR is th
│ │ │ │ +0002d8f0: 6520 6964 6561 6c20 6f66 206c 696e 6561  e ideal of linea
│ │ │ │ +0002d900: 720a 7265 6c61 7469 6f6e 7320 616d 6f75  r.relations amou
│ │ │ │ +0002d910: 6e74 2074 6865 2072 6179 732e 2049 7420  nt the rays. It 
│ │ │ │ +0002d920: 6973 206e 6565 6465 6420 666f 7220 696e  is needed for in
│ │ │ │ +0002d930: 7075 7420 696e 746f 2074 6865 206d 6574  put into the met
│ │ │ │ +0002d940: 686f 6473 202a 6e6f 7465 2053 6567 7265  hods *note Segre
│ │ │ │ +0002d950: 3a0a 5365 6772 652c 2c20 2a6e 6f74 6520  :.Segre,, *note 
│ │ │ │ +0002d960: 4368 6572 6e3a 2043 6865 726e 2c20 616e  Chern: Chern, an
│ │ │ │ +0002d970: 6420 2a6e 6f74 6520 4353 4d3a 2043 534d  d *note CSM: CSM
│ │ │ │ +0002d980: 2c20 696e 2074 6865 2063 6173 6573 2077  , in the cases w
│ │ │ │ +0002d990: 6865 7265 2061 2074 6f72 6963 0a76 6172  here a toric.var
│ │ │ │ +0002d9a0: 6965 7479 2069 7320 616c 736f 2069 6e70  iety is also inp
│ │ │ │ +0002d9b0: 7574 2074 6f20 656e 7375 7265 2074 6861  ut to ensure tha
│ │ │ │ +0002d9c0: 7420 7468 6573 6520 6d65 7468 6f64 7320  t these methods 
│ │ │ │ +0002d9d0: 7265 7475 726e 2072 6573 756c 7473 2069  return results i
│ │ │ │ +0002d9e0: 6e20 7468 6520 7361 6d65 0a72 696e 672e  n the same.ring.
│ │ │ │ +0002d9f0: 2057 6520 6769 7665 2061 6e20 6578 616d   We give an exam
│ │ │ │ +0002da00: 706c 6520 6f66 2074 6865 2075 7365 206f  ple of the use o
│ │ │ │ +0002da10: 6620 7468 6973 206d 6574 686f 6420 746f  f this method to
│ │ │ │ +0002da20: 2077 6f72 6b20 7769 7468 2065 6c65 6d65   work with eleme
│ │ │ │ +0002da30: 6e74 7320 6f66 2074 6865 0a43 686f 7720  nts of the.Chow 
│ │ │ │ +0002da40: 7269 6e67 206f 6620 6120 746f 7269 6320  ring of a toric 
│ │ │ │ +0002da50: 7661 7269 6574 790a 0a2b 2d2d 2d2d 2d2d  variety..+------
│ │ │ │  0002da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002daa0: 2d2b 0a7c 6931 203a 206e 6565 6473 5061  -+.|i1 : needsPa
│ │ │ │ -0002dab0: 636b 6167 6520 224e 6f72 6d61 6c54 6f72  ckage "NormalTor
│ │ │ │ -0002dac0: 6963 5661 7269 6574 6965 7322 2020 2020  icVarieties"    
│ │ │ │ -0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002daa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206e  -------+.|i1 : n
│ │ │ │ +0002dab0: 6565 6473 5061 636b 6167 6520 224e 6f72  eedsPackage "Nor
│ │ │ │ +0002dac0: 6d61 6c54 6f72 6963 5661 7269 6574 6965  malToricVarietie
│ │ │ │ +0002dad0: 7322 2020 2020 2020 2020 2020 2020 2020  s"              
│ │ │ │  0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002daf0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002daf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db40: 207c 0a7c 6f31 203d 204e 6f72 6d61 6c54   |.|o1 = NormalT
│ │ │ │ -0002db50: 6f72 6963 5661 7269 6574 6965 7320 2020  oricVarieties   
│ │ │ │ -0002db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db40: 2020 2020 2020 207c 0a7c 6f31 203d 204e         |.|o1 = N
│ │ │ │ +0002db50: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002db60: 6965 7320 2020 2020 2020 2020 2020 2020  ies             
│ │ │ │  0002db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002db90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbe0: 207c 0a7c 6f31 203a 2050 6163 6b61 6765   |.|o1 : Package
│ │ │ │ -0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dbe0: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ +0002dbf0: 6163 6b61 6765 2020 2020 2020 2020 2020  ackage          
│ │ │ │  0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002dc30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dc80: 2d2b 0a7c 6932 203a 2052 686f 203d 207b  -+.|i2 : Rho = {
│ │ │ │ -0002dc90: 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 7d2c  {1,0,0},{0,1,0},
│ │ │ │ -0002dca0: 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 2c30  {0,0,1},{-1,-1,0
│ │ │ │ -0002dcb0: 7d2c 7b30 2c30 2c2d 317d 7d20 2020 2020  },{0,0,-1}}     
│ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dcd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002dc80: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052  -------+.|i2 : R
│ │ │ │ +0002dc90: 686f 203d 207b 7b31 2c30 2c30 7d2c 7b30  ho = {{1,0,0},{0
│ │ │ │ +0002dca0: 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c 7b2d  ,1,0},{0,0,1},{-
│ │ │ │ +0002dcb0: 312c 2d31 2c30 7d2c 7b30 2c30 2c2d 317d  1,-1,0},{0,0,-1}
│ │ │ │ +0002dcc0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002dcd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd20: 207c 0a7c 6f32 203d 207b 7b31 2c20 302c   |.|o2 = {{1, 0,
│ │ │ │ -0002dd30: 2030 7d2c 207b 302c 2031 2c20 307d 2c20   0}, {0, 1, 0}, 
│ │ │ │ -0002dd40: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20  {0, 0, 1}, {-1, 
│ │ │ │ -0002dd50: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d  -1, 0}, {0, 0, -
│ │ │ │ -0002dd60: 317d 7d20 2020 2020 2020 2020 2020 2020  1}}             
│ │ │ │ -0002dd70: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002dd20: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ +0002dd30: 7b31 2c20 302c 2030 7d2c 207b 302c 2031  {1, 0, 0}, {0, 1
│ │ │ │ +0002dd40: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c  , 0}, {0, 0, 1},
│ │ │ │ +0002dd50: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30   {-1, -1, 0}, {0
│ │ │ │ +0002dd60: 2c20 302c 202d 317d 7d20 2020 2020 2020  , 0, -1}}       
│ │ │ │ +0002dd70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddc0: 207c 0a7c 6f32 203a 204c 6973 7420 2020   |.|o2 : List   
│ │ │ │ -0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ddc0: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ +0002ddd0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002de10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de60: 2d2b 0a7c 6933 203a 2053 6967 6d61 203d  -+.|i3 : Sigma =
│ │ │ │ -0002de70: 207b 7b30 2c31 2c32 7d2c 7b31 2c32 2c33   {{0,1,2},{1,2,3
│ │ │ │ -0002de80: 7d2c 7b30 2c32 2c33 7d2c 7b30 2c31 2c34  },{0,2,3},{0,1,4
│ │ │ │ -0002de90: 7d2c 7b31 2c33 2c34 7d2c 7b30 2c33 2c34  },{1,3,4},{0,3,4
│ │ │ │ -0002dea0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ -0002deb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002de60: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2053  -------+.|i3 : S
│ │ │ │ +0002de70: 6967 6d61 203d 207b 7b30 2c31 2c32 7d2c  igma = {{0,1,2},
│ │ │ │ +0002de80: 7b31 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c  {1,2,3},{0,2,3},
│ │ │ │ +0002de90: 7b30 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c  {0,1,4},{1,3,4},
│ │ │ │ +0002dea0: 7b30 2c33 2c34 7d7d 2020 2020 2020 2020  {0,3,4}}        
│ │ │ │ +0002deb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df00: 207c 0a7c 6f33 203d 207b 7b30 2c20 312c   |.|o3 = {{0, 1,
│ │ │ │ -0002df10: 2032 7d2c 207b 312c 2032 2c20 337d 2c20   2}, {1, 2, 3}, 
│ │ │ │ -0002df20: 7b30 2c20 322c 2033 7d2c 207b 302c 2031  {0, 2, 3}, {0, 1
│ │ │ │ -0002df30: 2c20 347d 2c20 7b31 2c20 332c 2034 7d2c  , 4}, {1, 3, 4},
│ │ │ │ -0002df40: 207b 302c 2033 2c20 347d 7d20 2020 2020   {0, 3, 4}}     
│ │ │ │ -0002df50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002df00: 2020 2020 2020 207c 0a7c 6f33 203d 207b         |.|o3 = {
│ │ │ │ +0002df10: 7b30 2c20 312c 2032 7d2c 207b 312c 2032  {0, 1, 2}, {1, 2
│ │ │ │ +0002df20: 2c20 337d 2c20 7b30 2c20 322c 2033 7d2c  , 3}, {0, 2, 3},
│ │ │ │ +0002df30: 207b 302c 2031 2c20 347d 2c20 7b31 2c20   {0, 1, 4}, {1, 
│ │ │ │ +0002df40: 332c 2034 7d2c 207b 302c 2033 2c20 347d  3, 4}, {0, 3, 4}
│ │ │ │ +0002df50: 7d20 2020 2020 207c 0a7c 2020 2020 2020  }      |.|      
│ │ │ │  0002df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfa0: 207c 0a7c 6f33 203a 204c 6973 7420 2020   |.|o3 : List   
│ │ │ │ -0002dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dfa0: 2020 2020 2020 207c 0a7c 6f33 203a 204c         |.|o3 : L
│ │ │ │ +0002dfb0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0002dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dff0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002dff0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e040: 2d2b 0a7c 6934 203a 2058 203d 206e 6f72  -+.|i4 : X = nor
│ │ │ │ -0002e050: 6d61 6c54 6f72 6963 5661 7269 6574 7928  malToricVariety(
│ │ │ │ -0002e060: 5268 6f2c 5369 676d 612c 436f 6566 6669  Rho,Sigma,Coeffi
│ │ │ │ -0002e070: 6369 656e 7452 696e 6720 3d3e 5a5a 2f33  cientRing =>ZZ/3
│ │ │ │ -0002e080: 3237 3439 2920 2020 2020 2020 2020 2020  2749)           
│ │ │ │ -0002e090: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e040: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2058  -------+.|i4 : X
│ │ │ │ +0002e050: 203d 206e 6f72 6d61 6c54 6f72 6963 5661   = normalToricVa
│ │ │ │ +0002e060: 7269 6574 7928 5268 6f2c 5369 676d 612c  riety(Rho,Sigma,
│ │ │ │ +0002e070: 436f 6566 6669 6369 656e 7452 696e 6720  CoefficientRing 
│ │ │ │ +0002e080: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020  =>ZZ/32749)     
│ │ │ │ +0002e090: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0e0: 207c 0a7c 6f34 203d 2058 2020 2020 2020   |.|o4 = X      
│ │ │ │ +0002e0e0: 2020 2020 2020 207c 0a7c 6f34 203d 2058         |.|o4 = X
│ │ │ │  0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e130: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e130: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e180: 207c 0a7c 6f34 203a 204e 6f72 6d61 6c54   |.|o4 : NormalT
│ │ │ │ -0002e190: 6f72 6963 5661 7269 6574 7920 2020 2020  oricVariety     
│ │ │ │ -0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e180: 2020 2020 2020 207c 0a7c 6f34 203a 204e         |.|o4 : N
│ │ │ │ +0002e190: 6f72 6d61 6c54 6f72 6963 5661 7269 6574  ormalToricVariet
│ │ │ │ +0002e1a0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │  0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e1d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002e1d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e220: 2d2b 0a7c 6935 203a 2052 3d72 696e 6720  -+.|i5 : R=ring 
│ │ │ │ -0002e230: 5820 2020 2020 2020 2020 2020 2020 2020  X               
│ │ │ │ +0002e220: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2052  -------+.|i5 : R
│ │ │ │ +0002e230: 3d72 696e 6720 5820 2020 2020 2020 2020  =ring X         
│ │ │ │  0002e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e270: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e270: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2c0: 207c 0a7c 6f35 203d 2052 2020 2020 2020   |.|o5 = R      
│ │ │ │ +0002e2c0: 2020 2020 2020 207c 0a7c 6f35 203d 2052         |.|o5 = R
│ │ │ │  0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e310: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e310: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e360: 207c 0a7c 6f35 203a 2050 6f6c 796e 6f6d   |.|o5 : Polynom
│ │ │ │ -0002e370: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ +0002e360: 2020 2020 2020 207c 0a7c 6f35 203a 2050         |.|o5 : P
│ │ │ │ +0002e370: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │  0002e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002e3b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e400: 2d2b 0a7c 6936 203a 2043 683d 546f 7269  -+.|i6 : Ch=Tori
│ │ │ │ -0002e410: 6343 686f 7752 696e 6728 5829 2020 2020  cChowRing(X)    
│ │ │ │ -0002e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e400: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2043  -------+.|i6 : C
│ │ │ │ +0002e410: 683d 546f 7269 6343 686f 7752 696e 6728  h=ToricChowRing(
│ │ │ │ +0002e420: 5829 2020 2020 2020 2020 2020 2020 2020  X)              
│ │ │ │  0002e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e450: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e450: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4a0: 207c 0a7c 6f36 203d 2043 6820 2020 2020   |.|o6 = Ch     
│ │ │ │ -0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4a0: 2020 2020 2020 207c 0a7c 6f36 203d 2043         |.|o6 = C
│ │ │ │ +0002e4b0: 6820 2020 2020 2020 2020 2020 2020 2020  h               
│ │ │ │  0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e4f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e540: 207c 0a7c 6f36 203a 2051 756f 7469 656e   |.|o6 : Quotien
│ │ │ │ -0002e550: 7452 696e 6720 2020 2020 2020 2020 2020  tRing           
│ │ │ │ +0002e540: 2020 2020 2020 207c 0a7c 6f36 203a 2051         |.|o6 : Q
│ │ │ │ +0002e550: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │  0002e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e590: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002e590: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e5e0: 2d2b 0a7c 6937 203a 2064 6573 6372 6962  -+.|i7 : describ
│ │ │ │ -0002e5f0: 6520 4368 2020 2020 2020 2020 2020 2020  e Ch            
│ │ │ │ +0002e5e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2064  -------+.|i7 : d
│ │ │ │ +0002e5f0: 6573 6372 6962 6520 4368 2020 2020 2020  escribe Ch      
│ │ │ │  0002e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e630: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e630: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e680: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002e690: 2020 2020 2020 2020 205a 5a5b 7820 2e2e           ZZ[x ..
│ │ │ │ -0002e6a0: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │ +0002e680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e690: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +0002e6a0: 5a5b 7820 2e2e 7820 5d20 2020 2020 2020  Z[x ..x ]       
│ │ │ │  0002e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002e6e0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -0002e6f0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +0002e6d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e6f0: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │  0002e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e720: 207c 0a7c 6f37 203d 202d 2d2d 2d2d 2d2d   |.|o7 = -------
│ │ │ │ +0002e720: 2020 2020 2020 207c 0a7c 6f37 203d 202d         |.|o7 = -
│ │ │ │  0002e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e750: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +0002e750: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │  0002e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e770: 207c 0a7c 2020 2020 2028 7820 7820 2c20   |.|     (x x , 
│ │ │ │ -0002e780: 7820 7820 7820 2c20 7820 202d 2078 202c  x x x , x  - x ,
│ │ │ │ -0002e790: 2078 2020 2d20 7820 2c20 7820 202d 2078   x  - x , x  - x
│ │ │ │ -0002e7a0: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │ +0002e770: 2020 2020 2020 207c 0a7c 2020 2020 2028         |.|     (
│ │ │ │ +0002e780: 7820 7820 2c20 7820 7820 7820 2c20 7820  x x , x x x , x 
│ │ │ │ +0002e790: 202d 2078 202c 2078 2020 2d20 7820 2c20   - x , x  - x , 
│ │ │ │ +0002e7a0: 7820 202d 2078 2029 2020 2020 2020 2020  x  - x )        
│ │ │ │  0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7c0: 207c 0a7c 2020 2020 2020 2032 2034 2020   |.|       2 4  
│ │ │ │ -0002e7d0: 2030 2031 2033 2020 2030 2020 2020 3320   0 1 3   0    3 
│ │ │ │ -0002e7e0: 2020 3120 2020 2033 2020 2032 2020 2020    1    3   2    
│ │ │ │ -0002e7f0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0002e7c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e7d0: 2032 2034 2020 2030 2031 2033 2020 2030   2 4   0 1 3   0
│ │ │ │ +0002e7e0: 2020 2020 3320 2020 3120 2020 2033 2020      3   1    3  
│ │ │ │ +0002e7f0: 2032 2020 2020 3420 2020 2020 2020 2020   2    4         
│ │ │ │  0002e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e810: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002e810: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e860: 2d2b 0a7c 6938 203a 2072 3d67 656e 7320  -+.|i8 : r=gens 
│ │ │ │ -0002e870: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +0002e860: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2072  -------+.|i8 : r
│ │ │ │ +0002e870: 3d67 656e 7320 5220 2020 2020 2020 2020  =gens R         
│ │ │ │  0002e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e8b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e8b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e900: 207c 0a7c 6f38 203d 207b 7820 2c20 7820   |.|o8 = {x , x 
│ │ │ │ -0002e910: 2c20 7820 2c20 7820 2c20 7820 7d20 2020  , x , x , x }   
│ │ │ │ -0002e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e900: 2020 2020 2020 207c 0a7c 6f38 203d 207b         |.|o8 = {
│ │ │ │ +0002e910: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +0002e920: 7820 7d20 2020 2020 2020 2020 2020 2020  x }             
│ │ │ │  0002e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e950: 207c 0a7c 2020 2020 2020 2030 2020 2031   |.|       0   1
│ │ │ │ -0002e960: 2020 2032 2020 2033 2020 2034 2020 2020     2   3   4    
│ │ │ │ -0002e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002e960: 2030 2020 2031 2020 2032 2020 2033 2020   0   1   2   3  
│ │ │ │ +0002e970: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │  0002e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002e9a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9f0: 207c 0a7c 6f38 203a 204c 6973 7420 2020   |.|o8 : List   
│ │ │ │ -0002ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e9f0: 2020 2020 2020 207c 0a7c 6f38 203a 204c         |.|o8 : L
│ │ │ │ +0002ea00: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │  0002ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ea40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002ea40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ea90: 2d2b 0a7c 6939 203a 2049 3d69 6465 616c  -+.|i9 : I=ideal
│ │ │ │ -0002eaa0: 2872 616e 646f 6d28 7b31 2c30 7d2c 5229  (random({1,0},R)
│ │ │ │ -0002eab0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002ea90: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2049  -------+.|i9 : I
│ │ │ │ +0002eaa0: 3d69 6465 616c 2872 616e 646f 6d28 7b31  =ideal(random({1
│ │ │ │ +0002eab0: 2c30 7d2c 5229 2920 2020 2020 2020 2020  ,0},R))         
│ │ │ │  0002eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eae0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002eae0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb30: 207c 0a7c 6f39 203d 2069 6465 616c 2831   |.|o9 = ideal(1
│ │ │ │ -0002eb40: 3037 7820 202b 2034 3337 3678 2020 2d20  07x  + 4376x  - 
│ │ │ │ -0002eb50: 3633 3136 7820 2920 2020 2020 2020 2020  6316x )         
│ │ │ │ +0002eb30: 2020 2020 2020 207c 0a7c 6f39 203d 2069         |.|o9 = i
│ │ │ │ +0002eb40: 6465 616c 2831 3037 7820 202b 2034 3337  deal(107x  + 437
│ │ │ │ +0002eb50: 3678 2020 2d20 3633 3136 7820 2920 2020  6x  - 6316x )   
│ │ │ │  0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eb80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002eb90: 2020 2030 2020 2020 2020 2020 3120 2020     0        1   
│ │ │ │ -0002eba0: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +0002eb80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002eb90: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0002eba0: 2020 3120 2020 2020 2020 2033 2020 2020    1        3    
│ │ │ │  0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ebd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002ebd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec20: 207c 0a7c 6f39 203a 2049 6465 616c 206f   |.|o9 : Ideal o
│ │ │ │ -0002ec30: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ +0002ec20: 2020 2020 2020 207c 0a7c 6f39 203a 2049         |.|o9 : I
│ │ │ │ +0002ec30: 6465 616c 206f 6620 5220 2020 2020 2020  deal of R       
│ │ │ │  0002ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ec70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002ec70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ecc0: 2d2b 0a7c 6931 3020 3a20 4b3d 6964 6561  -+.|i10 : K=idea
│ │ │ │ -0002ecd0: 6c28 7261 6e64 6f6d 287b 312c 317d 2c52  l(random({1,1},R
│ │ │ │ -0002ece0: 2929 2020 2020 2020 2020 2020 2020 2020  ))              
│ │ │ │ +0002ecc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ +0002ecd0: 4b3d 6964 6561 6c28 7261 6e64 6f6d 287b  K=ideal(random({
│ │ │ │ +0002ece0: 312c 317d 2c52 2929 2020 2020 2020 2020  1,1},R))        
│ │ │ │  0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002ed10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed60: 207c 0a7c 6f31 3020 3d20 6964 6561 6c28   |.|o10 = ideal(
│ │ │ │ -0002ed70: 3331 3837 7820 7820 202d 2036 3035 3378  3187x x  - 6053x
│ │ │ │ -0002ed80: 2078 2020 2d20 3136 3039 3078 2078 2020   x  - 16090x x  
│ │ │ │ -0002ed90: 2b20 3337 3833 7820 7820 202b 2038 3537  + 3783x x  + 857
│ │ │ │ -0002eda0: 3078 2078 2020 2b20 3834 3434 7820 7820  0x x  + 8444x x 
│ │ │ │ -0002edb0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ -0002edc0: 2020 2020 2030 2032 2020 2020 2020 2020       0 2        
│ │ │ │ -0002edd0: 3120 3220 2020 2020 2020 2020 3220 3320  1 2         2 3 
│ │ │ │ -0002ede0: 2020 2020 2020 2030 2034 2020 2020 2020         0 4      
│ │ │ │ -0002edf0: 2020 3120 3420 2020 2020 2020 2033 2034    1 4        3 4
│ │ │ │ -0002ee00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002ed60: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ +0002ed70: 6964 6561 6c28 3331 3837 7820 7820 202d  ideal(3187x x  -
│ │ │ │ +0002ed80: 2036 3035 3378 2078 2020 2d20 3136 3039   6053x x  - 1609
│ │ │ │ +0002ed90: 3078 2078 2020 2b20 3337 3833 7820 7820  0x x  + 3783x x 
│ │ │ │ +0002eda0: 202b 2038 3537 3078 2078 2020 2b20 3834   + 8570x x  + 84
│ │ │ │ +0002edb0: 3434 7820 7820 297c 0a7c 2020 2020 2020  44x x )|.|      
│ │ │ │ +0002edc0: 2020 2020 2020 2020 2020 2030 2032 2020             0 2  
│ │ │ │ +0002edd0: 2020 2020 2020 3120 3220 2020 2020 2020        1 2       
│ │ │ │ +0002ede0: 2020 3220 3320 2020 2020 2020 2030 2034    2 3        0 4
│ │ │ │ +0002edf0: 2020 2020 2020 2020 3120 3420 2020 2020          1 4     
│ │ │ │ +0002ee00: 2020 2033 2034 207c 0a7c 2020 2020 2020     3 4 |.|      
│ │ │ │  0002ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee50: 207c 0a7c 6f31 3020 3a20 4964 6561 6c20   |.|o10 : Ideal 
│ │ │ │ -0002ee60: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │ +0002ee50: 2020 2020 2020 207c 0a7c 6f31 3020 3a20         |.|o10 : 
│ │ │ │ +0002ee60: 4964 6561 6c20 6f66 2052 2020 2020 2020  Ideal of R      
│ │ │ │  0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eea0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002eea0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eef0: 2d2b 0a7c 6931 3120 3a20 633d 4368 6572  -+.|i11 : c=Cher
│ │ │ │ -0002ef00: 6e28 4368 2c58 2c49 2920 2020 2020 2020  n(Ch,X,I)       
│ │ │ │ +0002eef0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ +0002ef00: 633d 4368 6572 6e28 4368 2c58 2c49 2920  c=Chern(Ch,X,I) 
│ │ │ │  0002ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ef40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002ef40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ef90: 207c 0a7c 2020 2020 2020 2020 3220 2020   |.|        2   
│ │ │ │ -0002efa0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002ef90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002efa0: 2020 3220 2020 2020 2020 3220 2020 2020    2       2     
│ │ │ │  0002efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002efe0: 207c 0a7c 6f31 3120 3d20 3478 2078 2020   |.|o11 = 4x x  
│ │ │ │ -0002eff0: 2b20 3278 2020 2b20 3278 2078 2020 2b20  + 2x  + 2x x  + 
│ │ │ │ -0002f000: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +0002efe0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ +0002eff0: 3478 2078 2020 2b20 3278 2020 2b20 3278  4x x  + 2x  + 2x
│ │ │ │ +0002f000: 2078 2020 2b20 7820 2020 2020 2020 2020   x  + x         
│ │ │ │  0002f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f030: 207c 0a7c 2020 2020 2020 2020 3320 3420   |.|        3 4 
│ │ │ │ -0002f040: 2020 2020 3320 2020 2020 3320 3420 2020      3     3 4   
│ │ │ │ -0002f050: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0002f030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f040: 2020 3320 3420 2020 2020 3320 2020 2020    3 4     3     
│ │ │ │ +0002f050: 3320 3420 2020 2033 2020 2020 2020 2020  3 4    3        
│ │ │ │  0002f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f080: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f080: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f0d0: 207c 0a7c 6f31 3120 3a20 4368 2020 2020   |.|o11 : Ch    
│ │ │ │ -0002f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f0d0: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ +0002f0e0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  0002f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f120: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002f120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002f130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f170: 2d2b 0a7c 6931 3220 3a20 733d 5365 6772  -+.|i12 : s=Segr
│ │ │ │ -0002f180: 6528 4368 2c58 2c4b 2920 2020 2020 2020  e(Ch,X,K)       
│ │ │ │ +0002f170: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ +0002f180: 733d 5365 6772 6528 4368 2c58 2c4b 2920  s=Segre(Ch,X,K) 
│ │ │ │  0002f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f1c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f1c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f210: 207c 0a7c 2020 2020 2020 2020 3220 2020   |.|        2   
│ │ │ │ -0002f220: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0002f210: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f220: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │  0002f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f260: 207c 0a7c 6f31 3220 3d20 3378 2078 2020   |.|o12 = 3x x  
│ │ │ │ -0002f270: 2d20 7820 202d 2032 7820 7820 202b 2078  - x  - 2x x  + x
│ │ │ │ -0002f280: 2020 2b20 7820 2020 2020 2020 2020 2020    + x           
│ │ │ │ +0002f260: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +0002f270: 3378 2078 2020 2d20 7820 202d 2032 7820  3x x  - x  - 2x 
│ │ │ │ +0002f280: 7820 202b 2078 2020 2b20 7820 2020 2020  x  + x  + x     
│ │ │ │  0002f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2b0: 207c 0a7c 2020 2020 2020 2020 3320 3420   |.|        3 4 
│ │ │ │ -0002f2c0: 2020 2033 2020 2020 2033 2034 2020 2020     3     3 4    
│ │ │ │ -0002f2d0: 3320 2020 2034 2020 2020 2020 2020 2020  3    4          
│ │ │ │ +0002f2b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f2c0: 2020 3320 3420 2020 2033 2020 2020 2033    3 4    3     3
│ │ │ │ +0002f2d0: 2034 2020 2020 3320 2020 2034 2020 2020   4    3    4    
│ │ │ │  0002f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f300: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f300: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f350: 207c 0a7c 6f31 3220 3a20 4368 2020 2020   |.|o12 : Ch    
│ │ │ │ -0002f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f350: 2020 2020 2020 207c 0a7c 6f31 3220 3a20         |.|o12 : 
│ │ │ │ +0002f360: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  0002f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002f3a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002f3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f3f0: 2d2b 0a7c 6931 3320 3a20 732d 6320 2020  -+.|i13 : s-c   
│ │ │ │ -0002f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ +0002f400: 732d 6320 2020 2020 2020 2020 2020 2020  s-c             
│ │ │ │  0002f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f440: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f440: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f490: 207c 0a7c 2020 2020 2020 2020 2032 2020   |.|         2  
│ │ │ │ -0002f4a0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0002f490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f4a0: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │  0002f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f4e0: 207c 0a7c 6f31 3320 3d20 2d20 7820 7820   |.|o13 = - x x 
│ │ │ │ -0002f4f0: 202d 2033 7820 202d 2034 7820 7820 202b   - 3x  - 4x x  +
│ │ │ │ -0002f500: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +0002f4e0: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ +0002f4f0: 2d20 7820 7820 202d 2033 7820 202d 2034  - x x  - 3x  - 4
│ │ │ │ +0002f500: 7820 7820 202b 2078 2020 2020 2020 2020  x x  + x        
│ │ │ │  0002f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f530: 207c 0a7c 2020 2020 2020 2020 2033 2034   |.|         3 4
│ │ │ │ -0002f540: 2020 2020 2033 2020 2020 2033 2034 2020       3     3 4  
│ │ │ │ -0002f550: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +0002f530: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f540: 2020 2033 2034 2020 2020 2033 2020 2020     3 4     3    
│ │ │ │ +0002f550: 2033 2034 2020 2020 3420 2020 2020 2020   3 4    4       
│ │ │ │  0002f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f580: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f580: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5d0: 207c 0a7c 6f31 3320 3a20 4368 2020 2020   |.|o13 : Ch    
│ │ │ │ -0002f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f5d0: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ +0002f5e0: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  0002f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f620: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002f620: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f670: 2d2b 0a7c 6931 3420 3a20 732a 6320 2020  -+.|i14 : s*c   
│ │ │ │ -0002f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f670: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ +0002f680: 732a 6320 2020 2020 2020 2020 2020 2020  s*c             
│ │ │ │  0002f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f6c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f6c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f710: 207c 0a7c 2020 2020 2020 2020 3220 2020   |.|        2   
│ │ │ │ -0002f720: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0002f710: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f720: 2020 3220 2020 2020 2032 2020 2020 2020    2      2      
│ │ │ │  0002f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f760: 207c 0a7c 6f31 3420 3d20 3278 2078 2020   |.|o14 = 2x x  
│ │ │ │ -0002f770: 2b20 7820 202b 2078 2078 2020 2020 2020  + x  + x x      
│ │ │ │ +0002f760: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ +0002f770: 3278 2078 2020 2b20 7820 202b 2078 2078  2x x  + x  + x x
│ │ │ │  0002f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f7b0: 207c 0a7c 2020 2020 2020 2020 3320 3420   |.|        3 4 
│ │ │ │ -0002f7c0: 2020 2033 2020 2020 3320 3420 2020 2020     3    3 4     
│ │ │ │ -0002f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f7b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002f7c0: 2020 3320 3420 2020 2033 2020 2020 3320    3 4    3    3 
│ │ │ │ +0002f7d0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │  0002f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f800: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f800: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f850: 207c 0a7c 6f31 3420 3a20 4368 2020 2020   |.|o14 : Ch    
│ │ │ │ -0002f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f850: 2020 2020 2020 207c 0a7c 6f31 3420 3a20         |.|o14 : 
│ │ │ │ +0002f860: 4368 2020 2020 2020 2020 2020 2020 2020  Ch              
│ │ │ │  0002f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f8a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002f8a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  0002f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f8f0: 2d2b 0a0a 466f 7220 7468 6520 7072 6f67  -+..For the prog
│ │ │ │ -0002f900: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -0002f910: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -0002f920: 626a 6563 7420 2a6e 6f74 6520 546f 7269  bject *note Tori
│ │ │ │ -0002f930: 6343 686f 7752 696e 673a 2054 6f72 6963  cChowRing: Toric
│ │ │ │ -0002f940: 4368 6f77 5269 6e67 2c20 6973 2061 202a  ChowRing, is a *
│ │ │ │ -0002f950: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -0002f960: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ -0002f970: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -0002f980: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ +0002f8f0: 2d2d 2d2d 2d2d 2d2b 0a0a 466f 7220 7468  -------+..For th
│ │ │ │ +0002f900: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +0002f910: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0002f920: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +0002f930: 6520 546f 7269 6343 686f 7752 696e 673a  e ToricChowRing:
│ │ │ │ +0002f940: 2054 6f72 6963 4368 6f77 5269 6e67 2c20   ToricChowRing, 
│ │ │ │ +0002f950: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ +0002f960: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ +0002f970: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +0002f980: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │  0002f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f9d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -0002f9e0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -0002f9f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -0002fa00: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -0002fa10: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ -0002fa20: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ -0002fa30: 6179 322f 7061 636b 6167 6573 2f0a 4368  ay2/packages/.Ch
│ │ │ │ -0002fa40: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ -0002fa50: 7365 732e 6d32 3a31 3935 313a 302e 0a1f  ses.m2:1951:0...
│ │ │ │ -0002fa60: 0a54 6167 2054 6162 6c65 3a0a 4e6f 6465  .Tag Table:.Node
│ │ │ │ -0002fa70: 3a20 546f 707f 3239 310a 4e6f 6465 3a20  : Top.291.Node: 
│ │ │ │ -0002fa80: 6265 7274 696e 6943 6865 636b 7f31 3637  bertiniCheck.167
│ │ │ │ -0002fa90: 3035 0a4e 6f64 653a 2043 6865 636b 536d  05.Node: CheckSm
│ │ │ │ -0002faa0: 6f6f 7468 7f31 3739 3232 0a4e 6f64 653a  ooth.17922.Node:
│ │ │ │ -0002fab0: 2043 6865 636b 546f 7269 6356 6172 6965   CheckToricVarie
│ │ │ │ -0002fac0: 7479 5661 6c69 647f 3232 3731 380a 4e6f  tyValid.22718.No
│ │ │ │ -0002fad0: 6465 3a20 4368 6572 6e7f 3333 3730 390a  de: Chern.33709.
│ │ │ │ -0002fae0: 4e6f 6465 3a20 4368 6f77 5269 6e67 7f35  Node: ChowRing.5
│ │ │ │ -0002faf0: 3132 3137 0a4e 6f64 653a 2043 6c61 7373  1217.Node: Class
│ │ │ │ -0002fb00: 496e 4368 6f77 5269 6e67 7f35 3839 3438  InChowRing.58948
│ │ │ │ -0002fb10: 0a4e 6f64 653a 2043 6c61 7373 496e 546f  .Node: ClassInTo
│ │ │ │ -0002fb20: 7269 6343 686f 7752 696e 677f 3631 3330  ricChowRing.6130
│ │ │ │ -0002fb30: 380a 4e6f 6465 3a20 436f 6d70 4d65 7468  8.Node: CompMeth
│ │ │ │ -0002fb40: 6f64 7f36 3635 3531 0a4e 6f64 653a 2063  od.66551.Node: c
│ │ │ │ -0002fb50: 6f6e 6669 6775 7269 6e67 2042 6572 7469  onfiguring Berti
│ │ │ │ -0002fb60: 6e69 7f37 3634 3738 0a4e 6f64 653a 2043  ni.76478.Node: C
│ │ │ │ -0002fb70: 534d 7f37 3831 3133 0a4e 6f64 653a 2045  SM.78113.Node: E
│ │ │ │ -0002fb80: 756c 6572 7f31 3031 3830 320a 4e6f 6465  uler.101802.Node
│ │ │ │ -0002fb90: 3a20 4575 6c65 7241 6666 696e 657f 3131  : EulerAffine.11
│ │ │ │ -0002fba0: 3936 3739 0a4e 6f64 653a 2049 6e64 734f  9679.Node: IndsO
│ │ │ │ -0002fbb0: 6653 6d6f 6f74 687f 3132 3231 3639 0a4e  fSmooth.122169.N
│ │ │ │ -0002fbc0: 6f64 653a 2049 6e70 7574 4973 536d 6f6f  ode: InputIsSmoo
│ │ │ │ -0002fbd0: 7468 7f31 3236 3132 340a 4e6f 6465 3a20  th.126124.Node: 
│ │ │ │ -0002fbe0: 6973 4d75 6c74 6948 6f6d 6f67 656e 656f  isMultiHomogeneo
│ │ │ │ -0002fbf0: 7573 7f31 3330 3130 370a 4e6f 6465 3a20  us.130107.Node: 
│ │ │ │ -0002fc00: 4d65 7468 6f64 7f31 3334 3138 350a 4e6f  Method.134185.No
│ │ │ │ -0002fc10: 6465 3a20 4d75 6c74 6950 726f 6a43 6f6f  de: MultiProjCoo
│ │ │ │ -0002fc20: 7264 5269 6e67 7f31 3338 3133 340a 4e6f  rdRing.138134.No
│ │ │ │ -0002fc30: 6465 3a20 4f75 7470 7574 7f31 3434 3830  de: Output.14480
│ │ │ │ -0002fc40: 320a 4e6f 6465 3a20 7072 6f62 6162 696c  2.Node: probabil
│ │ │ │ -0002fc50: 6973 7469 6320 616c 676f 7269 7468 6d7f  istic algorithm.
│ │ │ │ -0002fc60: 3136 3333 3230 0a4e 6f64 653a 2053 6567  163320.Node: Seg
│ │ │ │ -0002fc70: 7265 7f31 3638 3031 350a 4e6f 6465 3a20  re.168015.Node: 
│ │ │ │ -0002fc80: 546f 7269 6343 686f 7752 696e 677f 3138  ToricChowRing.18
│ │ │ │ -0002fc90: 3539 3232 0a1f 0a45 6e64 2054 6167 2054  5922...End Tag T
│ │ │ │ -0002fca0: 6162 6c65 0a                             able.
│ │ │ │ +0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +0002f9e0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +0002f9f0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +0002fa00: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +0002fa10: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +0002fa20: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ +0002fa30: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +0002fa40: 6573 2f0a 4368 6172 6163 7465 7269 7374  es/.Characterist
│ │ │ │ +0002fa50: 6963 436c 6173 7365 732e 6d32 3a31 3935  icClasses.m2:195
│ │ │ │ +0002fa60: 313a 302e 0a1f 0a54 6167 2054 6162 6c65  1:0....Tag Table
│ │ │ │ +0002fa70: 3a0a 4e6f 6465 3a20 546f 707f 3239 310a  :.Node: Top.291.
│ │ │ │ +0002fa80: 4e6f 6465 3a20 6265 7274 696e 6943 6865  Node: bertiniChe
│ │ │ │ +0002fa90: 636b 7f31 3637 3035 0a4e 6f64 653a 2043  ck.16705.Node: C
│ │ │ │ +0002faa0: 6865 636b 536d 6f6f 7468 7f31 3739 3232  heckSmooth.17922
│ │ │ │ +0002fab0: 0a4e 6f64 653a 2043 6865 636b 546f 7269  .Node: CheckTori
│ │ │ │ +0002fac0: 6356 6172 6965 7479 5661 6c69 647f 3232  cVarietyValid.22
│ │ │ │ +0002fad0: 3731 380a 4e6f 6465 3a20 4368 6572 6e7f  718.Node: Chern.
│ │ │ │ +0002fae0: 3333 3730 390a 4e6f 6465 3a20 4368 6f77  33709.Node: Chow
│ │ │ │ +0002faf0: 5269 6e67 7f35 3132 3137 0a4e 6f64 653a  Ring.51217.Node:
│ │ │ │ +0002fb00: 2043 6c61 7373 496e 4368 6f77 5269 6e67   ClassInChowRing
│ │ │ │ +0002fb10: 7f35 3839 3438 0a4e 6f64 653a 2043 6c61  .58948.Node: Cla
│ │ │ │ +0002fb20: 7373 496e 546f 7269 6343 686f 7752 696e  ssInToricChowRin
│ │ │ │ +0002fb30: 677f 3631 3330 380a 4e6f 6465 3a20 436f  g.61308.Node: Co
│ │ │ │ +0002fb40: 6d70 4d65 7468 6f64 7f36 3635 3531 0a4e  mpMethod.66551.N
│ │ │ │ +0002fb50: 6f64 653a 2063 6f6e 6669 6775 7269 6e67  ode: configuring
│ │ │ │ +0002fb60: 2042 6572 7469 6e69 7f37 3634 3738 0a4e   Bertini.76478.N
│ │ │ │ +0002fb70: 6f64 653a 2043 534d 7f37 3831 3133 0a4e  ode: CSM.78113.N
│ │ │ │ +0002fb80: 6f64 653a 2045 756c 6572 7f31 3031 3735  ode: Euler.10175
│ │ │ │ +0002fb90: 310a 4e6f 6465 3a20 4575 6c65 7241 6666  1.Node: EulerAff
│ │ │ │ +0002fba0: 696e 657f 3131 3936 3238 0a4e 6f64 653a  ine.119628.Node:
│ │ │ │ +0002fbb0: 2049 6e64 734f 6653 6d6f 6f74 687f 3132   IndsOfSmooth.12
│ │ │ │ +0002fbc0: 3231 3430 0a4e 6f64 653a 2049 6e70 7574  2140.Node: Input
│ │ │ │ +0002fbd0: 4973 536d 6f6f 7468 7f31 3236 3039 350a  IsSmooth.126095.
│ │ │ │ +0002fbe0: 4e6f 6465 3a20 6973 4d75 6c74 6948 6f6d  Node: isMultiHom
│ │ │ │ +0002fbf0: 6f67 656e 656f 7573 7f31 3330 3131 330a  ogeneous.130113.
│ │ │ │ +0002fc00: 4e6f 6465 3a20 4d65 7468 6f64 7f31 3334  Node: Method.134
│ │ │ │ +0002fc10: 3139 310a 4e6f 6465 3a20 4d75 6c74 6950  191.Node: MultiP
│ │ │ │ +0002fc20: 726f 6a43 6f6f 7264 5269 6e67 7f31 3338  rojCoordRing.138
│ │ │ │ +0002fc30: 3134 300a 4e6f 6465 3a20 4f75 7470 7574  140.Node: Output
│ │ │ │ +0002fc40: 7f31 3434 3830 380a 4e6f 6465 3a20 7072  .144808.Node: pr
│ │ │ │ +0002fc50: 6f62 6162 696c 6973 7469 6320 616c 676f  obabilistic algo
│ │ │ │ +0002fc60: 7269 7468 6d7f 3136 3333 3236 0a4e 6f64  rithm.163326.Nod
│ │ │ │ +0002fc70: 653a 2053 6567 7265 7f31 3638 3032 310a  e: Segre.168021.
│ │ │ │ +0002fc80: 4e6f 6465 3a20 546f 7269 6343 686f 7752  Node: ToricChowR
│ │ │ │ +0002fc90: 696e 677f 3138 3539 3238 0a1f 0a45 6e64  ing.185928...End
│ │ │ │ +0002fca0: 2054 6167 2054 6162 6c65 0a               Tag Table.
│ │ ├── ./usr/share/info/Chordal.info.gz
│ │ │ ├── Chordal.info
│ │ │ │ @@ -3949,30 +3949,30 @@
│ │ │ │  0000f6c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f710: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │  0000f720: 203d 2045 6c69 6d54 7265 657b 6120 3d3e   = ElimTree{a =>
│ │ │ │ -0000f730: 2062 2020 207d 2020 2020 2020 2020 2020   b   }          
│ │ │ │ +0000f730: 2063 7d20 2020 2020 2020 2020 2020 2020   c}             
│ │ │ │  0000f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f770: 2020 2020 2020 2020 2020 2020 6220 3d3e              b =>
│ │ │ │  0000f780: 2063 2020 2020 2020 2020 2020 2020 2020   c              
│ │ │ │  0000f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f7c0: 2020 2020 2020 2020 2020 2020 6320 3d3e              c =>
│ │ │ │  0000f7d0: 2064 2020 2020 2020 2020 2020 2020 2020   d              
│ │ │ │  0000f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f810: 2020 2020 2020 2020 2020 2020 6420 3d3e              d =>
│ │ │ │ -0000f820: 206e 756c 6c20 2020 2020 2020 2020 2020   null           
│ │ │ │ +0000f820: 2062 2020 2020 2020 2020 2020 2020 2020   b              
│ │ │ │  0000f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4033,31 +4033,31 @@
│ │ │ │  0000fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fc60: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -0000fc70: 203d 2043 686f 7264 616c 4e65 747b 2064   = ChordalNet{ d
│ │ │ │ -0000fc80: 203d 3e20 7b20 2c20 647d 2020 2020 7d20   => { , d}    } 
│ │ │ │ +0000fc70: 203d 2043 686f 7264 616c 4e65 747b 2061   = ChordalNet{ a
│ │ │ │ +0000fc80: 203d 3e20 7b61 2c20 207d 2020 2020 7d20   => {a,  }    } 
│ │ │ │  0000fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ -0000fcd0: 203d 3e20 7b62 2c20 202c 2062 7d20 2020   => {b,  , b}   
│ │ │ │ +0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ +0000fcd0: 203d 3e20 7b20 2c20 637d 2020 2020 2020   => { , c}      
│ │ │ │  0000fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fd10: 2020 2020 2020 2020 2020 2020 2020 2061                 a
│ │ │ │ -0000fd20: 203d 3e20 7b61 2c20 207d 2020 2020 2020   => {a,  }      
│ │ │ │ +0000fd10: 2020 2020 2020 2020 2020 2020 2020 2064                 d
│ │ │ │ +0000fd20: 203d 3e20 7b20 2c20 647d 2020 2020 2020   => { , d}      
│ │ │ │  0000fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0000fd60: 2020 2020 2020 2020 2020 2020 2020 2063                 c
│ │ │ │ -0000fd70: 203d 3e20 7b20 2c20 637d 2020 2020 2020   => { , c}      
│ │ │ │ +0000fd60: 2020 2020 2020 2020 2020 2020 2020 2062                 b
│ │ │ │ +0000fd70: 203d 3e20 7b62 2c20 202c 2062 7d20 2020   => {b,  , b}   
│ │ │ │  0000fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fda0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0000fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5332,18 +5332,18 @@
│ │ │ │  00014d30: 2020 2020 207c 0a7c 6964 6561 6c20 2878       |.|ideal (x
│ │ │ │  00014d40: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │  00014d50: 2029 2c20 6964 6561 6c20 2878 202c 2078   ), ideal (x , x
│ │ │ │  00014d60: 202c 2078 202c 2078 202c 2078 2029 2c20   , x , x , x ), 
│ │ │ │  00014d70: 6964 6561 6c20 2878 202c 2078 202c 2078  ideal (x , x , x
│ │ │ │  00014d80: 202c 2020 207c 0a7c 2020 2020 2020 2020   ,   |.|        
│ │ │ │  00014d90: 3120 2020 3220 2020 3420 2020 3520 2020  1   2   4   5   
│ │ │ │ -00014da0: 3720 2020 2020 2020 2020 2020 3120 2020  7           1   
│ │ │ │ -00014db0: 3320 2020 3520 2020 3620 2020 3820 2020  3   5   6   8   
│ │ │ │ -00014dc0: 2020 2020 2020 2020 3120 2020 3320 2020          1   3   
│ │ │ │ -00014dd0: 3420 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d  4    |.|--------
│ │ │ │ +00014da0: 3720 2020 2020 2020 2020 2020 3220 2020  7           2   
│ │ │ │ +00014db0: 3420 2020 3520 2020 3720 2020 3820 2020  4   5   7   8   
│ │ │ │ +00014dc0: 2020 2020 2020 2020 3220 2020 3320 2020          2   3   
│ │ │ │ +00014dd0: 3520 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d  5    |.|--------
│ │ │ │  00014de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │  00014e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5355,20 +5355,20 @@
│ │ │ │  00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ec0: 2020 2020 207c 0a7c 7820 2c20 7820 292c       |.|x , x ),
│ │ │ │  00014ed0: 2069 6465 616c 2028 7820 2c20 7820 2c20   ideal (x , x , 
│ │ │ │  00014ee0: 7820 2c20 7820 2c20 7820 292c 2069 6465  x , x , x ), ide
│ │ │ │  00014ef0: 616c 2028 7820 2c20 7820 2c20 7820 2c20  al (x , x , x , 
│ │ │ │  00014f00: 7820 2c20 7820 292c 2069 6465 616c 2028  x , x ), ideal (
│ │ │ │ -00014f10: 7820 2c20 207c 0a7c 2036 2020 2038 2020  x ,  |.| 6   8  
│ │ │ │ +00014f10: 7820 2c20 207c 0a7c 2037 2020 2038 2020  x ,  |.| 7   8  
│ │ │ │  00014f20: 2020 2020 2020 2020 2032 2020 2033 2020           2   3  
│ │ │ │  00014f30: 2035 2020 2036 2020 2038 2020 2020 2020   5   6   8      
│ │ │ │ -00014f40: 2020 2020 2032 2020 2034 2020 2035 2020       2   4   5  
│ │ │ │ -00014f50: 2037 2020 2038 2020 2020 2020 2020 2020   7   8          
│ │ │ │ -00014f60: 2032 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d   2   |.|--------
│ │ │ │ +00014f40: 2020 2020 2031 2020 2033 2020 2035 2020       1   3   5  
│ │ │ │ +00014f50: 2036 2020 2038 2020 2020 2020 2020 2020   6   8          
│ │ │ │ +00014f60: 2031 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d   1   |.|--------
│ │ │ │  00014f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014fb0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020  -----|.|        
│ │ │ │  00014fc0: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │  00014fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5380,16 +5380,16 @@
│ │ │ │  00015030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015050: 2020 2020 207c 0a7c 7820 2c20 7820 2c20       |.|x , x , 
│ │ │ │  00015060: 7820 2c20 7820 297d 2020 2020 2020 2020  x , x )}        
│ │ │ │  00015070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150a0: 2020 2020 207c 0a7c 2033 2020 2035 2020       |.| 3   5  
│ │ │ │ -000150b0: 2037 2020 2038 2020 2020 2020 2020 2020   7   8          
│ │ │ │ +000150a0: 2020 2020 207c 0a7c 2033 2020 2034 2020       |.| 3   4  
│ │ │ │ +000150b0: 2036 2020 2038 2020 2020 2020 2020 2020   6   8          
│ │ │ │  000150c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000150f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00015100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/CohomCalg.info.gz
│ │ │ ├── CohomCalg.info
│ │ │ │ @@ -1042,15 +1042,15 @@
│ │ │ │  00004110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00004130: 7c69 3230 203a 2065 6c61 7073 6564 5469  |i20 : elapsedTi
│ │ │ │  00004140: 6d65 2068 7665 6373 203d 2063 6f68 6f6d  me hvecs = cohom
│ │ │ │  00004150: 4361 6c67 2858 2c20 4432 2920 2020 2020  Calg(X, D2)     
│ │ │ │  00004160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004170: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00004180: 7c20 2d2d 2033 2e30 3931 3431 7320 656c  | -- 3.09141s el
│ │ │ │ +00004180: 7c20 2d2d 2033 2e35 3630 3933 7320 656c  | -- 3.56093s el
│ │ │ │  00004190: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  000041a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000041d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000041e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000041f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1677,16 +1677,16 @@
│ │ │ │  000068c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000068d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000068e0: 7c69 3233 203a 2065 6c61 7073 6564 5469  |i23 : elapsedTi
│ │ │ │  000068f0: 6d65 2063 6f68 6f6d 7665 6331 203d 2063  me cohomvec1 = c
│ │ │ │  00006900: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058  ohomCalg(X_3 + X
│ │ │ │  00006910: 5f37 202b 2058 5f38 2920 2020 2020 2020  _7 + X_8)       
│ │ │ │  00006920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00006930: 7c20 2d2d 202e 3338 3039 7320 656c 6170  | -- .3809s elap
│ │ │ │ -00006940: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │ +00006930: 7c20 2d2d 202e 3533 3334 3334 7320 656c  | -- .533434s el
│ │ │ │ +00006940: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00006950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00006980: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00006990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000069a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000069b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1712,15 +1712,15 @@
│ │ │ │  00006af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00006b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00006b10: 7c69 3234 203a 2065 6c61 7073 6564 5469  |i24 : elapsedTi
│ │ │ │  00006b20: 6d65 2063 6f68 6f6d 7665 6332 203d 2066  me cohomvec2 = f
│ │ │ │  00006b30: 6f72 206a 2066 726f 6d20 3020 746f 2064  or j from 0 to d
│ │ │ │  00006b40: 696d 2058 206c 6973 7420 7261 6e6b 2048  im X list rank H
│ │ │ │  00006b50: 485e 6a28 582c 2020 2020 2020 2020 7c0a  H^j(X,        |.
│ │ │ │ -00006b60: 7c20 2d2d 2031 312e 3531 3735 7320 656c  | -- 11.5175s el
│ │ │ │ +00006b60: 7c20 2d2d 2031 312e 3137 3235 7320 656c  | -- 11.1725s el
│ │ │ │  00006b70: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00006b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00006bb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00006bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00006bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1797,15 +1797,15 @@
│ │ │ │  00007040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00007060: 7c69 3237 203a 2065 6c61 7073 6564 5469  |i27 : elapsedTi
│ │ │ │  00007070: 6d65 2063 6f68 6f6d 7665 6331 203d 2063  me cohomvec1 = c
│ │ │ │  00007080: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058  ohomCalg(X_3 + X
│ │ │ │  00007090: 5f37 202d 2058 5f38 2920 2020 2020 2020  _7 - X_8)       
│ │ │ │  000070a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000070b0: 7c20 2d2d 202e 3339 3739 3238 7320 656c  | -- .397928s el
│ │ │ │ +000070b0: 7c20 2d2d 202e 3533 3530 3935 7320 656c  | -- .535095s el
│ │ │ │  000070c0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  000070d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000070e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000070f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00007100: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00007110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1832,20 +1832,20 @@
│ │ │ │  00007270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00007280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00007290: 7c69 3238 203a 2065 6c61 7073 6564 5469  |i28 : elapsedTi
│ │ │ │  000072a0: 6d65 2063 6f68 6f6d 7665 6332 203d 2065  me cohomvec2 = e
│ │ │ │  000072b0: 6c61 7073 6564 5469 6d65 2066 6f72 206a  lapsedTime for j
│ │ │ │  000072c0: 2066 726f 6d20 3020 746f 2064 696d 2058   from 0 to dim X
│ │ │ │  000072d0: 206c 6973 7420 7261 6e6b 2020 2020 7c0a   list rank    |.
│ │ │ │ -000072e0: 7c20 2d2d 202e 3535 3339 3738 7320 656c  | -- .553978s el
│ │ │ │ +000072e0: 7c20 2d2d 202e 3533 3131 3735 7320 656c  | -- .531175s el
│ │ │ │  000072f0: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00007300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00007330: 7c20 2d2d 202e 3535 3430 3039 7320 656c  | -- .554009s el
│ │ │ │ +00007330: 7c20 2d2d 202e 3533 3132 3036 7320 656c  | -- .531206s el
│ │ │ │  00007340: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00007350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00007370: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00007380: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00007390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000073a0: 2020 2020 2020 2020 2020 2020 2020 2020
│ │ ├── ./usr/share/info/CompleteIntersectionResolutions.info.gz
│ │ │ ├── CompleteIntersectionResolutions.info
│ │ │ │ @@ -4343,17 +4343,17 @@
│ │ │ │  00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00010f80: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520  ---+.|i7 : time 
│ │ │ │  00010f90: 4720 3d20 4569 7365 6e62 7564 5368 616d  G = EisenbudSham
│ │ │ │  00010fa0: 6173 6828 6666 2c46 2c6c 656e 2920 2020  ash(ff,F,len)   
│ │ │ │  00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2036     |.| -- used 6
│ │ │ │ -00010fe0: 2e36 3833 3932 7320 2863 7075 293b 2035  .68392s (cpu); 5
│ │ │ │ -00010ff0: 2e30 3138 3533 7320 2874 6872 6561 6429  .01853s (thread)
│ │ │ │ +00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2039     |.| -- used 9
│ │ │ │ +00010fe0: 2e32 3836 3234 7320 2863 7075 293b 2037  .28624s (cpu); 7
│ │ │ │ +00010ff0: 2e32 3639 3639 7320 2874 6872 6561 6429  .26969s (thread)
│ │ │ │  00011000: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  00011010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00011030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00011060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4884,17 +4884,17 @@
│ │ │ │  00013130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013150: 2d2b 0a7c 6932 3020 3a20 4646 203d 2074  -+.|i20 : FF = t
│ │ │ │  00013160: 696d 6520 5368 616d 6173 6828 5231 2c46  ime Shamash(R1,F
│ │ │ │  00013170: 2c34 2920 2020 2020 2020 2020 2020 2020  ,4)             
│ │ │ │  00013180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013190: 2020 207c 0a7c 202d 2d20 7573 6564 2030     |.| -- used 0
│ │ │ │ -000131a0: 2e32 3332 3835 3873 2028 6370 7529 3b20  .232858s (cpu); 
│ │ │ │ -000131b0: 302e 3135 3631 3436 7320 2874 6872 6561  0.156146s (threa
│ │ │ │ -000131c0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +000131a0: 2e30 3937 3133 3833 7320 2863 7075 293b  .0971383s (cpu);
│ │ │ │ +000131b0: 2030 2e30 3936 3934 3437 7320 2874 6872   0.0969447s (thr
│ │ │ │ +000131c0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  000131d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000131e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000131f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013210: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00013220: 2020 3120 2020 2020 2020 3620 2020 2020    1       6     
│ │ │ │  00013230: 2020 3138 2020 2020 2020 2033 3820 2020    18       38   
│ │ │ │ @@ -4925,17 +4925,17 @@
│ │ │ │  000133c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000133d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000133e0: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 4747  -----+.|i21 : GG
│ │ │ │  000133f0: 203d 2074 696d 6520 4569 7365 6e62 7564   = time Eisenbud
│ │ │ │  00013400: 5368 616d 6173 6828 6666 2c46 2c34 2920  Shamash(ff,F,4) 
│ │ │ │  00013410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013420: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00013430: 6564 2030 2e39 3830 3632 3273 2028 6370  ed 0.980622s (cp
│ │ │ │ -00013440: 7529 3b20 302e 3735 3432 3438 7320 2874  u); 0.754248s (t
│ │ │ │ -00013450: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +00013430: 6564 2031 2e35 3131 3333 7320 2863 7075  ed 1.51133s (cpu
│ │ │ │ +00013440: 293b 2031 2e31 3038 3933 7320 2874 6872  ); 1.10893s (thr
│ │ │ │ +00013450: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │  00013460: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00013470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000134a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000134b0: 2020 2020 2f20 525c 3120 2020 2020 2f20      / R\1     / 
│ │ │ │  000134c0: 525c 3620 2020 2020 2f20 525c 3138 2020  R\6     / R\18  
│ │ │ │ @@ -4977,24029 +4977,24027 @@
│ │ │ │  00013700: 5468 6520 6675 6e63 7469 6f6e 2061 6c73  The function als
│ │ │ │  00013710: 6f20 6465 616c 7320 636f 7272 6563 746c  o deals correctl
│ │ │ │  00013720: 7920 7769 7468 2063 6f6d 706c 6578 6573  y with complexes
│ │ │ │  00013730: 2046 2077 6865 7265 206d 696e 2046 2069   F where min F i
│ │ │ │  00013740: 7320 6e6f 7420 303a 0a0a 2b2d 2d2d 2d2d  s not 0:..+-----
│ │ │ │  00013750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013780: 2b0a 7c69 3232 203a 2047 4720 3d20 7469  +.|i22 : GG = ti
│ │ │ │ -00013790: 6d65 2045 6973 656e 6275 6453 6861 6d61  me EisenbudShama
│ │ │ │ -000137a0: 7368 2852 312c 465b 325d 2c34 2920 2020  sh(R1,F[2],4)   
│ │ │ │ -000137b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -000137c0: 7365 6420 302e 3939 3436 3636 7320 2863  sed 0.994666s (c
│ │ │ │ -000137d0: 7075 293b 2030 2e37 3334 3738 3373 2028  pu); 0.734783s (
│ │ │ │ -000137e0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -000137f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00013780: 6932 3220 3a20 4747 203d 2074 696d 6520  i22 : GG = time 
│ │ │ │ +00013790: 4569 7365 6e62 7564 5368 616d 6173 6828  EisenbudShamash(
│ │ │ │ +000137a0: 5231 2c46 5b32 5d2c 3429 2020 2020 2020  R1,F[2],4)      
│ │ │ │ +000137b0: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e    |.| -- used 1.
│ │ │ │ +000137c0: 3339 3733 7320 2863 7075 293b 2031 2e30  3973s (cpu); 1.0
│ │ │ │ +000137d0: 3834 3434 7320 2874 6872 6561 6429 3b20  8444s (thread); 
│ │ │ │ +000137e0: 3073 2028 6763 297c 0a7c 2020 2020 2020  0s (gc)|.|      
│ │ │ │ +000137f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013820: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00013830: 2020 2031 2020 2020 2020 2036 2020 2020     1       6    
│ │ │ │ -00013840: 2020 2031 3820 2020 2020 2020 3338 2020     18       38  
│ │ │ │ -00013850: 2020 2020 2036 3620 2020 2020 2020 2020       66         
│ │ │ │ -00013860: 7c0a 7c6f 3232 203d 2052 3120 203c 2d2d  |.|o22 = R1  <--
│ │ │ │ -00013870: 2052 3120 203c 2d2d 2052 3120 2020 3c2d   R1  <-- R1   <-
│ │ │ │ -00013880: 2d20 5231 2020 203c 2d2d 2052 3120 2020  - R1   <-- R1   
│ │ │ │ -00013890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00013810: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00013820: 2020 2020 2020 2031 2020 2020 2020 2036         1       6
│ │ │ │ +00013830: 2020 2020 2020 2031 3820 2020 2020 2020         18       
│ │ │ │ +00013840: 3338 2020 2020 2020 2036 3620 2020 2020  38       66     
│ │ │ │ +00013850: 207c 0a7c 6f32 3220 3d20 5231 2020 3c2d   |.|o22 = R1  <-
│ │ │ │ +00013860: 2d20 5231 2020 3c2d 2d20 5231 2020 203c  - R1  <-- R1   <
│ │ │ │ +00013870: 2d2d 2052 3120 2020 3c2d 2d20 5231 2020  -- R1   <-- R1  
│ │ │ │ +00013880: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00013890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000138a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000138b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000138c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000138d0: 7c0a 7c20 2020 2020 202d 3220 2020 2020  |.|      -2     
│ │ │ │ -000138e0: 202d 3120 2020 2020 2030 2020 2020 2020   -1      0      
│ │ │ │ -000138f0: 2020 3120 2020 2020 2020 2032 2020 2020    1        2    
│ │ │ │ -00013900: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000138b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000138c0: 2020 2020 2d32 2020 2020 2020 2d31 2020      -2      -1  
│ │ │ │ +000138d0: 2020 2020 3020 2020 2020 2020 2031 2020      0        1  
│ │ │ │ +000138e0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000138f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00013900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00013910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013940: 7c0a 7c6f 3232 203a 2043 6f6d 706c 6578  |.|o22 : Complex
│ │ │ │ -00013950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00013970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00013980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000139a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000139b0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  +..See also.====
│ │ │ │ -000139c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -000139d0: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20  makeHomotopies: 
│ │ │ │ -000139e0: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20  makeHomotopies, 
│ │ │ │ -000139f0: 2d2d 2072 6574 7572 6e73 2061 2073 7973  -- returns a sys
│ │ │ │ -00013a00: 7465 6d20 6f66 2068 6967 6865 720a 2020  tem of higher.  
│ │ │ │ -00013a10: 2020 686f 6d6f 746f 7069 6573 0a20 202a    homotopies.  *
│ │ │ │ -00013a20: 202a 6e6f 7465 2053 6861 6d61 7368 3a20   *note Shamash: 
│ │ │ │ -00013a30: 5368 616d 6173 682c 202d 2d20 436f 6d70  Shamash, -- Comp
│ │ │ │ -00013a40: 7574 6573 2074 6865 2053 6861 6d61 7368  utes the Shamash
│ │ │ │ -00013a50: 2043 6f6d 706c 6578 0a20 202a 202a 6e6f   Complex.  * *no
│ │ │ │ -00013a60: 7465 2065 7870 6f3a 2065 7870 6f2c 202d  te expo: expo, -
│ │ │ │ -00013a70: 2d20 7265 7475 726e 7320 6120 7365 7420  - returns a set 
│ │ │ │ -00013a80: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ -00013a90: 2074 6865 2062 6173 6973 206f 6620 6120   the basis of a 
│ │ │ │ -00013aa0: 6469 7669 6465 640a 2020 2020 706f 7765  divided.    powe
│ │ │ │ -00013ab0: 720a 0a57 6179 7320 746f 2075 7365 2045  r..Ways to use E
│ │ │ │ -00013ac0: 6973 656e 6275 6453 6861 6d61 7368 3a0a  isenbudShamash:.
│ │ │ │ -00013ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00013ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -00013af0: 2a20 2245 6973 656e 6275 6453 6861 6d61  * "EisenbudShama
│ │ │ │ -00013b00: 7368 284d 6174 7269 782c 436f 6d70 6c65  sh(Matrix,Comple
│ │ │ │ -00013b10: 782c 5a5a 2922 0a20 202a 2022 4569 7365  x,ZZ)".  * "Eise
│ │ │ │ -00013b20: 6e62 7564 5368 616d 6173 6828 5269 6e67  nbudShamash(Ring
│ │ │ │ -00013b30: 2c43 6f6d 706c 6578 2c5a 5a29 220a 0a46  ,Complex,ZZ)"..F
│ │ │ │ -00013b40: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00013b50: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00013b60: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00013b70: 202a 6e6f 7465 2045 6973 656e 6275 6453   *note EisenbudS
│ │ │ │ -00013b80: 6861 6d61 7368 3a20 4569 7365 6e62 7564  hamash: Eisenbud
│ │ │ │ -00013b90: 5368 616d 6173 682c 2069 7320 6120 2a6e  Shamash, is a *n
│ │ │ │ -00013ba0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -00013bb0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244  ion:.(Macaulay2D
│ │ │ │ -00013bc0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -00013bd0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  n,...-----------
│ │ │ │ +00013920: 2020 2020 207c 0a7c 6f32 3220 3a20 436f       |.|o22 : Co
│ │ │ │ +00013930: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │ +00013940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00013950: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00013960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00013990: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +000139a0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ +000139b0: 616b 6548 6f6d 6f74 6f70 6965 733a 206d  akeHomotopies: m
│ │ │ │ +000139c0: 616b 6548 6f6d 6f74 6f70 6965 732c 202d  akeHomotopies, -
│ │ │ │ +000139d0: 2d20 7265 7475 726e 7320 6120 7379 7374  - returns a syst
│ │ │ │ +000139e0: 656d 206f 6620 6869 6768 6572 0a20 2020  em of higher.   
│ │ │ │ +000139f0: 2068 6f6d 6f74 6f70 6965 730a 2020 2a20   homotopies.  * 
│ │ │ │ +00013a00: 2a6e 6f74 6520 5368 616d 6173 683a 2053  *note Shamash: S
│ │ │ │ +00013a10: 6861 6d61 7368 2c20 2d2d 2043 6f6d 7075  hamash, -- Compu
│ │ │ │ +00013a20: 7465 7320 7468 6520 5368 616d 6173 6820  tes the Shamash 
│ │ │ │ +00013a30: 436f 6d70 6c65 780a 2020 2a20 2a6e 6f74  Complex.  * *not
│ │ │ │ +00013a40: 6520 6578 706f 3a20 6578 706f 2c20 2d2d  e expo: expo, --
│ │ │ │ +00013a50: 2072 6574 7572 6e73 2061 2073 6574 2063   returns a set c
│ │ │ │ +00013a60: 6f72 7265 7370 6f6e 6469 6e67 2074 6f20  orresponding to 
│ │ │ │ +00013a70: 7468 6520 6261 7369 7320 6f66 2061 2064  the basis of a d
│ │ │ │ +00013a80: 6976 6964 6564 0a20 2020 2070 6f77 6572  ivided.    power
│ │ │ │ +00013a90: 0a0a 5761 7973 2074 6f20 7573 6520 4569  ..Ways to use Ei
│ │ │ │ +00013aa0: 7365 6e62 7564 5368 616d 6173 683a 0a3d  senbudShamash:.=
│ │ │ │ +00013ab0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00013ac0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00013ad0: 2022 4569 7365 6e62 7564 5368 616d 6173   "EisenbudShamas
│ │ │ │ +00013ae0: 6828 4d61 7472 6978 2c43 6f6d 706c 6578  h(Matrix,Complex
│ │ │ │ +00013af0: 2c5a 5a29 220a 2020 2a20 2245 6973 656e  ,ZZ)".  * "Eisen
│ │ │ │ +00013b00: 6275 6453 6861 6d61 7368 2852 696e 672c  budShamash(Ring,
│ │ │ │ +00013b10: 436f 6d70 6c65 782c 5a5a 2922 0a0a 466f  Complex,ZZ)"..Fo
│ │ │ │ +00013b20: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00013b30: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00013b40: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00013b50: 2a6e 6f74 6520 4569 7365 6e62 7564 5368  *note EisenbudSh
│ │ │ │ +00013b60: 616d 6173 683a 2045 6973 656e 6275 6453  amash: EisenbudS
│ │ │ │ +00013b70: 6861 6d61 7368 2c20 6973 2061 202a 6e6f  hamash, is a *no
│ │ │ │ +00013b80: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +00013b90: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ +00013ba0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00013bb0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +00013bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00013bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00013bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00013c20: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -00013c30: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -00013c40: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00013c50: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -00013c60: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e  /macaulay2-1.26.
│ │ │ │ -00013c70: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61  06+ds/M2/Macaula
│ │ │ │ -00013c80: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ -00013c90: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -00013ca0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ -00013cb0: 3438 3432 3a30 2e0a 1f0a 4669 6c65 3a20  4842:0....File: 
│ │ │ │ -00013cc0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00013cd0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00013ce0: 696e 666f 2c20 4e6f 6465 3a20 4569 7365  info, Node: Eise
│ │ │ │ -00013cf0: 6e62 7564 5368 616d 6173 6854 6f74 616c  nbudShamashTotal
│ │ │ │ -00013d00: 2c20 4e65 7874 3a20 6576 656e 4578 744d  , Next: evenExtM
│ │ │ │ -00013d10: 6f64 756c 652c 2050 7265 763a 2045 6973  odule, Prev: Eis
│ │ │ │ -00013d20: 656e 6275 6453 6861 6d61 7368 2c20 5570  enbudShamash, Up
│ │ │ │ -00013d30: 3a20 546f 700a 0a45 6973 656e 6275 6453  : Top..EisenbudS
│ │ │ │ -00013d40: 6861 6d61 7368 546f 7461 6c20 2d2d 2050  hamashTotal -- P
│ │ │ │ -00013d50: 7265 6375 7273 6f72 2063 6f6d 706c 6578  recursor complex
│ │ │ │ -00013d60: 206f 6620 746f 7461 6c20 4578 740a 2a2a   of total Ext.**
│ │ │ │ +00013c00: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +00013c10: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +00013c20: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +00013c30: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +00013c40: 6d61 6361 756c 6179 322d 312e 3236 2e30  macaulay2-1.26.0
│ │ │ │ +00013c50: 362b 6473 2f4d 322f 4d61 6361 756c 6179  6+ds/M2/Macaulay
│ │ │ │ +00013c60: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ +00013c70: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ +00013c80: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34  Resolutions.m2:4
│ │ │ │ +00013c90: 3834 323a 302e 0a1f 0a46 696c 653a 2043  842:0....File: C
│ │ │ │ +00013ca0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +00013cb0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ +00013cc0: 6e66 6f2c 204e 6f64 653a 2045 6973 656e  nfo, Node: Eisen
│ │ │ │ +00013cd0: 6275 6453 6861 6d61 7368 546f 7461 6c2c  budShamashTotal,
│ │ │ │ +00013ce0: 204e 6578 743a 2065 7665 6e45 7874 4d6f   Next: evenExtMo
│ │ │ │ +00013cf0: 6475 6c65 2c20 5072 6576 3a20 4569 7365  dule, Prev: Eise
│ │ │ │ +00013d00: 6e62 7564 5368 616d 6173 682c 2055 703a  nbudShamash, Up:
│ │ │ │ +00013d10: 2054 6f70 0a0a 4569 7365 6e62 7564 5368   Top..EisenbudSh
│ │ │ │ +00013d20: 616d 6173 6854 6f74 616c 202d 2d20 5072  amashTotal -- Pr
│ │ │ │ +00013d30: 6563 7572 736f 7220 636f 6d70 6c65 7820  ecursor complex 
│ │ │ │ +00013d40: 6f66 2074 6f74 616c 2045 7874 0a2a 2a2a  of total Ext.***
│ │ │ │ +00013d50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00013d60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00013d70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013d80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00013da0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -00013db0: 200a 2020 2020 2020 2020 2864 302c 6431   .        (d0,d1
│ │ │ │ -00013dc0: 2920 3d20 2045 6973 656e 6275 6453 6861  ) =  EisenbudSha
│ │ │ │ -00013dd0: 6d61 7368 546f 7461 6c20 4d0a 2020 2a20  mashTotal M.  * 
│ │ │ │ -00013de0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -00013df0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ -00013e00: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -00013e10: 294d 6f64 756c 652c 2c20 6f76 6572 2061  )Module,, over a
│ │ │ │ -00013e20: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -00013e30: 6563 7469 6f6e 0a20 202a 202a 6e6f 7465  ection.  * *note
│ │ │ │ -00013e40: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ -00013e50: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00013e60: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ -00013e70: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ -00013e80: 7075 7473 2c3a 0a20 2020 2020 202a 2043  puts,:.      * C
│ │ │ │ -00013e90: 6865 636b 203d 3e20 2e2e 2e2c 2064 6566  heck => ..., def
│ │ │ │ -00013ea0: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ -00013eb0: 0a20 2020 2020 202a 2047 7261 6469 6e67  .      * Grading
│ │ │ │ -00013ec0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ -00013ed0: 2076 616c 7565 2032 0a20 2020 2020 202a   value 2.      *
│ │ │ │ -00013ee0: 2056 6172 6961 626c 6573 203d 3e20 2e2e   Variables => ..
│ │ │ │ -00013ef0: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00013f00: 2073 0a20 202a 204f 7574 7075 7473 3a0a   s.  * Outputs:.
│ │ │ │ -00013f10: 2020 2020 2020 2a20 6430 2c20 6120 2a6e        * d0, a *n
│ │ │ │ -00013f20: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -00013f30: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -00013f40: 2c2c 206d 6170 206f 6620 6672 6565 206d  ,, map of free m
│ │ │ │ -00013f50: 6f64 756c 6573 206f 7665 7220 616e 0a20  odules over an. 
│ │ │ │ -00013f60: 2020 2020 2020 2065 6e6c 6172 6765 6420         enlarged 
│ │ │ │ -00013f70: 7269 6e67 0a20 2020 2020 202a 2064 312c  ring.      * d1,
│ │ │ │ -00013f80: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ -00013f90: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -00013fa0: 6174 7269 782c 2c20 6d61 7020 6f66 2066  atrix,, map of f
│ │ │ │ -00013fb0: 7265 6520 6d6f 6475 6c65 7320 6f76 6572  ree modules over
│ │ │ │ -00013fc0: 2061 6e0a 2020 2020 2020 2020 656e 6c61   an.        enla
│ │ │ │ -00013fd0: 7267 6564 2072 696e 670a 0a44 6573 6372  rged ring..Descr
│ │ │ │ -00013fe0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00013ff0: 3d3d 0a0a 4173 7375 6d65 2074 6861 7420  ==..Assume that 
│ │ │ │ -00014000: 4d20 6973 2064 6566 696e 6564 206f 7665  M is defined ove
│ │ │ │ -00014010: 7220 6120 7269 6e67 206f 6620 7468 6520  r a ring of the 
│ │ │ │ -00014020: 666f 726d 2052 6261 7220 3d20 522f 2866  form Rbar = R/(f
│ │ │ │ -00014030: 5f30 2e2e 665f 7b63 2d31 7d29 2c20 610a  _0..f_{c-1}), a.
│ │ │ │ -00014040: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ -00014050: 6374 696f 6e2c 2061 6e64 2074 6861 7420  ction, and that 
│ │ │ │ -00014060: 4d20 6861 7320 6120 6669 6e69 7465 2066  M has a finite f
│ │ │ │ -00014070: 7265 6520 7265 736f 6c75 7469 6f6e 2047  ree resolution G
│ │ │ │ -00014080: 206f 7665 7220 522e 2049 6e0a 7468 6973   over R. In.this
│ │ │ │ -00014090: 2063 6173 6520 4d20 6861 7320 6120 6672   case M has a fr
│ │ │ │ -000140a0: 6565 2072 6573 6f6c 7574 696f 6e20 4620  ee resolution F 
│ │ │ │ -000140b0: 6f76 6572 2052 6261 7220 7768 6f73 6520  over Rbar whose 
│ │ │ │ -000140c0: 6475 616c 2c20 465e 2a20 6973 2061 2066  dual, F^* is a f
│ │ │ │ -000140d0: 696e 6974 656c 790a 6765 6e65 7261 7465  initely.generate
│ │ │ │ -000140e0: 642c 205a 2d67 7261 6465 6420 6672 6565  d, Z-graded free
│ │ │ │ -000140f0: 206d 6f64 756c 6520 6f76 6572 2061 2072   module over a r
│ │ │ │ -00014100: 696e 6720 5362 6172 5c63 6f6e 6720 6b6b  ing Sbar\cong kk
│ │ │ │ -00014110: 5b73 5f30 2e2e 735f 7b63 2d31 7d2c 6765  [s_0..s_{c-1},ge
│ │ │ │ -00014120: 6e73 0a52 6261 725d 2c20 7768 6572 6520  ns.Rbar], where 
│ │ │ │ -00014130: 7468 6520 6465 6772 6565 7320 6f66 2074  the degrees of t
│ │ │ │ -00014140: 6865 2073 5f69 2061 7265 207b 2d32 2c20  he s_i are {-2, 
│ │ │ │ -00014150: 2d64 6567 7265 6520 665f 697d 2e20 5468  -degree f_i}. Th
│ │ │ │ -00014160: 6973 2072 6573 6f6c 7574 696f 6e20 6973  is resolution is
│ │ │ │ -00014170: 0a69 7320 636f 6e73 7472 7563 7465 6420  .is constructed 
│ │ │ │ -00014180: 6672 6f6d 2074 6865 2064 7561 6c20 6f66  from the dual of
│ │ │ │ -00014190: 2047 2c20 746f 6765 7468 6572 2077 6974   G, together wit
│ │ │ │ -000141a0: 6820 7468 6520 6475 616c 7320 6f66 2074  h the duals of t
│ │ │ │ -000141b0: 6865 2068 6967 6865 720a 686f 6d6f 746f  he higher.homoto
│ │ │ │ -000141c0: 7069 6573 206f 6e20 4720 6465 6669 6e65  pies on G define
│ │ │ │ -000141d0: 6420 6279 2045 6973 656e 6275 642e 0a0a  d by Eisenbud...
│ │ │ │ -000141e0: 5468 6520 6675 6e63 7469 6f6e 2072 6574  The function ret
│ │ │ │ -000141f0: 7572 6e73 2074 6865 2064 6966 6665 7265  urns the differe
│ │ │ │ -00014200: 6e74 6961 6c73 2064 303a 465e 2a5f 7b65  ntials d0:F^*_{e
│ │ │ │ -00014210: 7665 6e7d 205c 746f 2046 5e2a 5f7b 6f64  ven} \to F^*_{od
│ │ │ │ -00014220: 647d 2061 6e64 0a64 313a 465e 2a5f 7b6f  d} and.d1:F^*_{o
│ │ │ │ -00014230: 6464 7d5c 746f 2046 5e2a 5f7b 6576 656e  dd}\to F^*_{even
│ │ │ │ -00014240: 7d2e 0a0a 5468 6520 6d61 7073 2064 302c  }...The maps d0,
│ │ │ │ -00014250: 6431 2066 6f72 6d20 6120 6d61 7472 6978  d1 form a matrix
│ │ │ │ -00014260: 2066 6163 746f 7269 7a61 7469 6f6e 206f   factorization o
│ │ │ │ -00014270: 6620 7375 6d28 632c 2069 2d3e 735f 692a  f sum(c, i->s_i*
│ │ │ │ -00014280: 665f 6929 2e20 5468 6520 6861 7665 2074  f_i). The have t
│ │ │ │ -00014290: 6865 0a70 726f 7065 7274 7920 7468 6174  he.property that
│ │ │ │ -000142a0: 2066 6f72 2061 6e79 2052 6261 7220 6d6f   for any Rbar mo
│ │ │ │ -000142b0: 6475 6c65 204e 2c0a 0a48 485f 3120 636f  dule N,..HH_1 co
│ │ │ │ -000142c0: 6d70 6c65 7820 5c7b 6430 2a2a 4e2c 2064  mplex \{d0**N, d
│ │ │ │ -000142d0: 312a 2a4e 5c7d 203d 2045 7874 5e7b 6576  1**N\} = Ext^{ev
│ │ │ │ -000142e0: 656e 7d5f 7b52 6261 727d 284d 2c4e 290a  en}_{Rbar}(M,N).
│ │ │ │ -000142f0: 0a53 5e7b 7b31 2c30 7d7d 2a2a 4848 5f31  .S^{{1,0}}**HH_1
│ │ │ │ -00014300: 2063 6f6d 706c 6578 205c 7b53 5e7b 7b2d   complex \{S^{{-
│ │ │ │ -00014310: 322c 307d 7d2a 2a64 312a 2a4e 2c20 6430  2,0}}**d1**N, d0
│ │ │ │ -00014320: 2a2a 4e5c 7d20 3d20 4578 745e 7b6f 6464  **N\} = Ext^{odd
│ │ │ │ -00014330: 7d5f 7b52 6261 727d 284d 2c4e 290a 0a54  }_{Rbar}(M,N)..T
│ │ │ │ -00014340: 6869 7320 6973 2065 6e63 6f64 6564 2069  his is encoded i
│ │ │ │ -00014350: 6e20 7468 6520 7363 7269 7074 206e 6577  n the script new
│ │ │ │ -00014360: 4578 740a 0a4f 7074 696f 6e20 6465 6661  Ext..Option defa
│ │ │ │ -00014370: 756c 7473 3a20 4368 6563 6b3d 3e66 616c  ults: Check=>fal
│ │ │ │ -00014380: 7365 2056 6172 6961 626c 6573 3d3e 6765  se Variables=>ge
│ │ │ │ -00014390: 7453 796d 626f 6c20 2273 222c 2047 7261  tSymbol "s", Gra
│ │ │ │ -000143a0: 6469 6e67 203d 3e32 7d0a 0a49 6620 4772  ding =>2}..If Gr
│ │ │ │ -000143b0: 6164 696e 6720 3d3e 312c 2074 6865 6e20  ading =>1, then 
│ │ │ │ -000143c0: 6120 7369 6e67 6c79 2067 7261 6465 6420  a singly graded 
│ │ │ │ -000143d0: 7265 7375 6c74 2069 7320 7265 7475 726e  result is return
│ │ │ │ -000143e0: 6564 2028 6a75 7374 2066 6f72 6765 7474  ed (just forgett
│ │ │ │ -000143f0: 696e 6720 7468 650a 686f 6d6f 6c6f 6769  ing the.homologi
│ │ │ │ -00014400: 6361 6c20 6772 6164 696e 672e 290a 0a0a  cal grading.)...
│ │ │ │ -00014410: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00013d80: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +00013d90: 0a20 2020 2020 2020 2028 6430 2c64 3129  .        (d0,d1)
│ │ │ │ +00013da0: 203d 2020 4569 7365 6e62 7564 5368 616d   =  EisenbudSham
│ │ │ │ +00013db0: 6173 6854 6f74 616c 204d 0a20 202a 2049  ashTotal M.  * I
│ │ │ │ +00013dc0: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ +00013dd0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +00013de0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00013df0: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120  Module,, over a 
│ │ │ │ +00013e00: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +00013e10: 6374 696f 6e0a 2020 2a20 2a6e 6f74 6520  ction.  * *note 
│ │ │ │ +00013e20: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ +00013e30: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ +00013e40: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
│ │ │ │ +00013e50: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ +00013e60: 7574 732c 3a0a 2020 2020 2020 2a20 4368  uts,:.      * Ch
│ │ │ │ +00013e70: 6563 6b20 3d3e 202e 2e2e 2c20 6465 6661  eck => ..., defa
│ │ │ │ +00013e80: 756c 7420 7661 6c75 6520 6661 6c73 650a  ult value false.
│ │ │ │ +00013e90: 2020 2020 2020 2a20 4772 6164 696e 6720        * Grading 
│ │ │ │ +00013ea0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ +00013eb0: 7661 6c75 6520 320a 2020 2020 2020 2a20  value 2.      * 
│ │ │ │ +00013ec0: 5661 7269 6162 6c65 7320 3d3e 202e 2e2e  Variables => ...
│ │ │ │ +00013ed0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00013ee0: 730a 2020 2a20 4f75 7470 7574 733a 0a20  s.  * Outputs:. 
│ │ │ │ +00013ef0: 2020 2020 202a 2064 302c 2061 202a 6e6f       * d0, a *no
│ │ │ │ +00013f00: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ +00013f10: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ +00013f20: 2c20 6d61 7020 6f66 2066 7265 6520 6d6f  , map of free mo
│ │ │ │ +00013f30: 6475 6c65 7320 6f76 6572 2061 6e0a 2020  dules over an.  
│ │ │ │ +00013f40: 2020 2020 2020 656e 6c61 7267 6564 2072        enlarged r
│ │ │ │ +00013f50: 696e 670a 2020 2020 2020 2a20 6431 2c20  ing.      * d1, 
│ │ │ │ +00013f60: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ +00013f70: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ +00013f80: 7472 6978 2c2c 206d 6170 206f 6620 6672  trix,, map of fr
│ │ │ │ +00013f90: 6565 206d 6f64 756c 6573 206f 7665 7220  ee modules over 
│ │ │ │ +00013fa0: 616e 0a20 2020 2020 2020 2065 6e6c 6172  an.        enlar
│ │ │ │ +00013fb0: 6765 6420 7269 6e67 0a0a 4465 7363 7269  ged ring..Descri
│ │ │ │ +00013fc0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00013fd0: 3d0a 0a41 7373 756d 6520 7468 6174 204d  =..Assume that M
│ │ │ │ +00013fe0: 2069 7320 6465 6669 6e65 6420 6f76 6572   is defined over
│ │ │ │ +00013ff0: 2061 2072 696e 6720 6f66 2074 6865 2066   a ring of the f
│ │ │ │ +00014000: 6f72 6d20 5262 6172 203d 2052 2f28 665f  orm Rbar = R/(f_
│ │ │ │ +00014010: 302e 2e66 5f7b 632d 317d 292c 2061 0a63  0..f_{c-1}), a.c
│ │ │ │ +00014020: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ +00014030: 7469 6f6e 2c20 616e 6420 7468 6174 204d  tion, and that M
│ │ │ │ +00014040: 2068 6173 2061 2066 696e 6974 6520 6672   has a finite fr
│ │ │ │ +00014050: 6565 2072 6573 6f6c 7574 696f 6e20 4720  ee resolution G 
│ │ │ │ +00014060: 6f76 6572 2052 2e20 496e 0a74 6869 7320  over R. In.this 
│ │ │ │ +00014070: 6361 7365 204d 2068 6173 2061 2066 7265  case M has a fre
│ │ │ │ +00014080: 6520 7265 736f 6c75 7469 6f6e 2046 206f  e resolution F o
│ │ │ │ +00014090: 7665 7220 5262 6172 2077 686f 7365 2064  ver Rbar whose d
│ │ │ │ +000140a0: 7561 6c2c 2046 5e2a 2069 7320 6120 6669  ual, F^* is a fi
│ │ │ │ +000140b0: 6e69 7465 6c79 0a67 656e 6572 6174 6564  nitely.generated
│ │ │ │ +000140c0: 2c20 5a2d 6772 6164 6564 2066 7265 6520  , Z-graded free 
│ │ │ │ +000140d0: 6d6f 6475 6c65 206f 7665 7220 6120 7269  module over a ri
│ │ │ │ +000140e0: 6e67 2053 6261 725c 636f 6e67 206b 6b5b  ng Sbar\cong kk[
│ │ │ │ +000140f0: 735f 302e 2e73 5f7b 632d 317d 2c67 656e  s_0..s_{c-1},gen
│ │ │ │ +00014100: 730a 5262 6172 5d2c 2077 6865 7265 2074  s.Rbar], where t
│ │ │ │ +00014110: 6865 2064 6567 7265 6573 206f 6620 7468  he degrees of th
│ │ │ │ +00014120: 6520 735f 6920 6172 6520 7b2d 322c 202d  e s_i are {-2, -
│ │ │ │ +00014130: 6465 6772 6565 2066 5f69 7d2e 2054 6869  degree f_i}. Thi
│ │ │ │ +00014140: 7320 7265 736f 6c75 7469 6f6e 2069 730a  s resolution is.
│ │ │ │ +00014150: 6973 2063 6f6e 7374 7275 6374 6564 2066  is constructed f
│ │ │ │ +00014160: 726f 6d20 7468 6520 6475 616c 206f 6620  rom the dual of 
│ │ │ │ +00014170: 472c 2074 6f67 6574 6865 7220 7769 7468  G, together with
│ │ │ │ +00014180: 2074 6865 2064 7561 6c73 206f 6620 7468   the duals of th
│ │ │ │ +00014190: 6520 6869 6768 6572 0a68 6f6d 6f74 6f70  e higher.homotop
│ │ │ │ +000141a0: 6965 7320 6f6e 2047 2064 6566 696e 6564  ies on G defined
│ │ │ │ +000141b0: 2062 7920 4569 7365 6e62 7564 2e0a 0a54   by Eisenbud...T
│ │ │ │ +000141c0: 6865 2066 756e 6374 696f 6e20 7265 7475  he function retu
│ │ │ │ +000141d0: 726e 7320 7468 6520 6469 6666 6572 656e  rns the differen
│ │ │ │ +000141e0: 7469 616c 7320 6430 3a46 5e2a 5f7b 6576  tials d0:F^*_{ev
│ │ │ │ +000141f0: 656e 7d20 5c74 6f20 465e 2a5f 7b6f 6464  en} \to F^*_{odd
│ │ │ │ +00014200: 7d20 616e 640a 6431 3a46 5e2a 5f7b 6f64  } and.d1:F^*_{od
│ │ │ │ +00014210: 647d 5c74 6f20 465e 2a5f 7b65 7665 6e7d  d}\to F^*_{even}
│ │ │ │ +00014220: 2e0a 0a54 6865 206d 6170 7320 6430 2c64  ...The maps d0,d
│ │ │ │ +00014230: 3120 666f 726d 2061 206d 6174 7269 7820  1 form a matrix 
│ │ │ │ +00014240: 6661 6374 6f72 697a 6174 696f 6e20 6f66  factorization of
│ │ │ │ +00014250: 2073 756d 2863 2c20 692d 3e73 5f69 2a66   sum(c, i->s_i*f
│ │ │ │ +00014260: 5f69 292e 2054 6865 2068 6176 6520 7468  _i). The have th
│ │ │ │ +00014270: 650a 7072 6f70 6572 7479 2074 6861 7420  e.property that 
│ │ │ │ +00014280: 666f 7220 616e 7920 5262 6172 206d 6f64  for any Rbar mod
│ │ │ │ +00014290: 756c 6520 4e2c 0a0a 4848 5f31 2063 6f6d  ule N,..HH_1 com
│ │ │ │ +000142a0: 706c 6578 205c 7b64 302a 2a4e 2c20 6431  plex \{d0**N, d1
│ │ │ │ +000142b0: 2a2a 4e5c 7d20 3d20 4578 745e 7b65 7665  **N\} = Ext^{eve
│ │ │ │ +000142c0: 6e7d 5f7b 5262 6172 7d28 4d2c 4e29 0a0a  n}_{Rbar}(M,N)..
│ │ │ │ +000142d0: 535e 7b7b 312c 307d 7d2a 2a48 485f 3120  S^{{1,0}}**HH_1 
│ │ │ │ +000142e0: 636f 6d70 6c65 7820 5c7b 535e 7b7b 2d32  complex \{S^{{-2
│ │ │ │ +000142f0: 2c30 7d7d 2a2a 6431 2a2a 4e2c 2064 302a  ,0}}**d1**N, d0*
│ │ │ │ +00014300: 2a4e 5c7d 203d 2045 7874 5e7b 6f64 647d  *N\} = Ext^{odd}
│ │ │ │ +00014310: 5f7b 5262 6172 7d28 4d2c 4e29 0a0a 5468  _{Rbar}(M,N)..Th
│ │ │ │ +00014320: 6973 2069 7320 656e 636f 6465 6420 696e  is is encoded in
│ │ │ │ +00014330: 2074 6865 2073 6372 6970 7420 6e65 7745   the script newE
│ │ │ │ +00014340: 7874 0a0a 4f70 7469 6f6e 2064 6566 6175  xt..Option defau
│ │ │ │ +00014350: 6c74 733a 2043 6865 636b 3d3e 6661 6c73  lts: Check=>fals
│ │ │ │ +00014360: 6520 5661 7269 6162 6c65 733d 3e67 6574  e Variables=>get
│ │ │ │ +00014370: 5379 6d62 6f6c 2022 7322 2c20 4772 6164  Symbol "s", Grad
│ │ │ │ +00014380: 696e 6720 3d3e 327d 0a0a 4966 2047 7261  ing =>2}..If Gra
│ │ │ │ +00014390: 6469 6e67 203d 3e31 2c20 7468 656e 2061  ding =>1, then a
│ │ │ │ +000143a0: 2073 696e 676c 7920 6772 6164 6564 2072   singly graded r
│ │ │ │ +000143b0: 6573 756c 7420 6973 2072 6574 7572 6e65  esult is returne
│ │ │ │ +000143c0: 6420 286a 7573 7420 666f 7267 6574 7469  d (just forgetti
│ │ │ │ +000143d0: 6e67 2074 6865 0a68 6f6d 6f6c 6f67 6963  ng the.homologic
│ │ │ │ +000143e0: 616c 2067 7261 6469 6e67 2e29 0a0a 0a0a  al grading.)....
│ │ │ │ +000143f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014460: 0a7c 6931 203a 206e 203d 2033 2020 2020  .|i1 : n = 3    
│ │ │ │ +00014430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014440: 7c69 3120 3a20 6e20 3d20 3320 2020 2020  |i1 : n = 3     
│ │ │ │ +00014450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000144b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014490: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000144a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000144b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000144f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014500: 0a7c 6f31 203d 2033 2020 2020 2020 2020  .|o1 = 3        
│ │ │ │ +000144d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000144e0: 7c6f 3120 3d20 3320 2020 2020 2020 2020  |o1 = 3         
│ │ │ │ +000144f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014550: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014520: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014530: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000145a0: 0a7c 6932 203a 2063 203d 2032 2020 2020  .|i2 : c = 2    
│ │ │ │ +00014570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014580: 7c69 3220 3a20 6320 3d20 3220 2020 2020  |i2 : c = 2     
│ │ │ │ +00014590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000145a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000145b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000145c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000145d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000145e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000145f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000145c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000145d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000145e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000145f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014640: 0a7c 6f32 203d 2032 2020 2020 2020 2020  .|o2 = 2        
│ │ │ │ +00014610: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014620: 7c6f 3220 3d20 3220 2020 2020 2020 2020  |o2 = 2         
│ │ │ │ +00014630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014680: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014690: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014660: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014670: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000146a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000146d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000146e0: 0a7c 6933 203a 206b 6b20 3d20 5a5a 2f31  .|i3 : kk = ZZ/1
│ │ │ │ -000146f0: 3031 2020 2020 2020 2020 2020 2020 2020  01              
│ │ │ │ -00014700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014730: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000146b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000146c0: 7c69 3320 3a20 6b6b 203d 205a 5a2f 3130  |i3 : kk = ZZ/10
│ │ │ │ +000146d0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000146e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000146f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014710: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014780: 0a7c 6f33 203d 206b 6b20 2020 2020 2020  .|o3 = kk       
│ │ │ │ +00014750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014760: 7c6f 3320 3d20 6b6b 2020 2020 2020 2020  |o3 = kk        
│ │ │ │ +00014770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000147a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000147b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000147c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000147d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000147a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000147b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000147c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000147d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000147e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000147f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014820: 0a7c 6f33 203a 2051 756f 7469 656e 7452  .|o3 : QuotientR
│ │ │ │ -00014830: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -00014840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014870: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000147f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014800: 7c6f 3320 3a20 5175 6f74 6965 6e74 5269  |o3 : QuotientRi
│ │ │ │ +00014810: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00014820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014850: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000148a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000148b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000148c0: 0a7c 6934 203a 2052 203d 206b 6b5b 785f  .|i4 : R = kk[x_
│ │ │ │ -000148d0: 302e 2e78 5f28 6e2d 3129 5d20 2020 2020  0..x_(n-1)]     
│ │ │ │ -000148e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000148f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014910: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000148a0: 7c69 3420 3a20 5220 3d20 6b6b 5b78 5f30  |i4 : R = kk[x_0
│ │ │ │ +000148b0: 2e2e 785f 286e 2d31 295d 2020 2020 2020  ..x_(n-1)]      
│ │ │ │ +000148c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000148d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000148e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000148f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014960: 0a7c 6f34 203d 2052 2020 2020 2020 2020  .|o4 = R        
│ │ │ │ +00014930: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014940: 7c6f 3420 3d20 5220 2020 2020 2020 2020  |o4 = R         
│ │ │ │ +00014950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000149b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000149a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000149b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014a00: 0a7c 6f34 203a 2050 6f6c 796e 6f6d 6961  .|o4 : Polynomia
│ │ │ │ -00014a10: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ -00014a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014a50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000149d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000149e0: 7c6f 3420 3a20 506f 6c79 6e6f 6d69 616c  |o4 : Polynomial
│ │ │ │ +000149f0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ +00014a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014a20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014a30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014aa0: 0a7c 6935 203a 2049 203d 2069 6465 616c  .|i5 : I = ideal
│ │ │ │ -00014ab0: 2878 5f30 5e32 2c20 785f 325e 3329 2020  (x_0^2, x_2^3)  
│ │ │ │ -00014ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014af0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014a80: 7c69 3520 3a20 4920 3d20 6964 6561 6c28  |i5 : I = ideal(
│ │ │ │ +00014a90: 785f 305e 322c 2078 5f32 5e33 2920 2020  x_0^2, x_2^3)   
│ │ │ │ +00014aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ac0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014ad0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2032  .|             2
│ │ │ │ -00014b50: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00014b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014b90: 0a7c 6f35 203d 2069 6465 616c 2028 7820  .|o5 = ideal (x 
│ │ │ │ -00014ba0: 2c20 7820 2920 2020 2020 2020 2020 2020  , x )           
│ │ │ │ -00014bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014be0: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -00014bf0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00014c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014c30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014b10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014b20: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ +00014b30: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +00014b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014b60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014b70: 7c6f 3520 3d20 6964 6561 6c20 2878 202c  |o5 = ideal (x ,
│ │ │ │ +00014b80: 2078 2029 2020 2020 2020 2020 2020 2020   x )            
│ │ │ │ +00014b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014bb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014bc0: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +00014bd0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00014be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014c10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014c80: 0a7c 6f35 203a 2049 6465 616c 206f 6620  .|o5 : Ideal of 
│ │ │ │ -00014c90: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ -00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014cd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014c50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014c60: 7c6f 3520 3a20 4964 6561 6c20 6f66 2052  |o5 : Ideal of R
│ │ │ │ +00014c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014cb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014d20: 0a7c 6936 203a 2066 6620 3d20 6765 6e73  .|i6 : ff = gens
│ │ │ │ -00014d30: 2049 2020 2020 2020 2020 2020 2020 2020   I              
│ │ │ │ -00014d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014d70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014d00: 7c69 3620 3a20 6666 203d 2067 656e 7320  |i6 : ff = gens 
│ │ │ │ +00014d10: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │ +00014d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014d40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014d50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014dc0: 0a7c 6f36 203d 207c 2078 5f30 5e32 2078  .|o6 = | x_0^2 x
│ │ │ │ -00014dd0: 5f32 5e33 207c 2020 2020 2020 2020 2020  _2^3 |          
│ │ │ │ -00014de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014e10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014d90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014da0: 7c6f 3620 3d20 7c20 785f 305e 3220 785f  |o6 = | x_0^2 x_
│ │ │ │ +00014db0: 325e 3320 7c20 2020 2020 2020 2020 2020  2^3 |           
│ │ │ │ +00014dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014de0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014df0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014e60: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ -00014e70: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00014e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014eb0: 0a7c 6f36 203a 204d 6174 7269 7820 5220  .|o6 : Matrix R 
│ │ │ │ -00014ec0: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ -00014ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014f00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014e30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014e40: 7c20 2020 2020 2020 2020 2020 2020 3120  |             1 
│ │ │ │ +00014e50: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00014e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014e80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014e90: 7c6f 3620 3a20 4d61 7472 6978 2052 2020  |o6 : Matrix R  
│ │ │ │ +00014ea0: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │ +00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014ee0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00014f50: 0a7c 6937 203a 2052 6261 7220 3d20 522f  .|i7 : Rbar = R/
│ │ │ │ -00014f60: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │ -00014f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014f30: 7c69 3720 3a20 5262 6172 203d 2052 2f49  |i7 : Rbar = R/I
│ │ │ │ +00014f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014f80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00014f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014ff0: 0a7c 6f37 203d 2052 6261 7220 2020 2020  .|o7 = Rbar     
│ │ │ │ +00014fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014fd0: 7c6f 3720 3d20 5262 6172 2020 2020 2020  |o7 = Rbar      
│ │ │ │ +00014fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015020: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015090: 0a7c 6f37 203a 2051 756f 7469 656e 7452  .|o7 : QuotientR
│ │ │ │ -000150a0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -000150b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000150e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015060: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015070: 7c6f 3720 3a20 5175 6f74 6965 6e74 5269  |o7 : QuotientRi
│ │ │ │ +00015080: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00015090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000150b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000150c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000150d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000150e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000150f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00015130: 0a7c 6938 203a 2062 6172 203d 206d 6170  .|i8 : bar = map
│ │ │ │ -00015140: 2852 6261 722c 2052 2920 2020 2020 2020  (Rbar, R)       
│ │ │ │ -00015150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015110: 7c69 3820 3a20 6261 7220 3d20 6d61 7028  |i8 : bar = map(
│ │ │ │ +00015120: 5262 6172 2c20 5229 2020 2020 2020 2020  Rbar, R)        
│ │ │ │ +00015130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015150: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015160: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000151a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000151b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000151c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000151d0: 0a7c 6f38 203d 206d 6170 2028 5262 6172  .|o8 = map (Rbar
│ │ │ │ -000151e0: 2c20 522c 207b 7820 2c20 7820 2c20 7820  , R, {x , x , x 
│ │ │ │ -000151f0: 7d29 2020 2020 2020 2020 2020 2020 2020  })              
│ │ │ │ -00015200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015220: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00015230: 2020 2020 2020 2030 2020 2031 2020 2032         0   1   2
│ │ │ │ -00015240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015270: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000151a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000151b0: 7c6f 3820 3d20 6d61 7020 2852 6261 722c  |o8 = map (Rbar,
│ │ │ │ +000151c0: 2052 2c20 7b78 202c 2078 202c 2078 207d   R, {x , x , x }
│ │ │ │ +000151d0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000151e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000151f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015200: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015210: 2020 2020 2020 3020 2020 3120 2020 3220        0   1   2 
│ │ │ │ +00015220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015250: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000152c0: 0a7c 6f38 203a 2052 696e 674d 6170 2052  .|o8 : RingMap R
│ │ │ │ -000152d0: 6261 7220 3c2d 2d20 5220 2020 2020 2020  bar <-- R       
│ │ │ │ -000152e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000152a0: 7c6f 3820 3a20 5269 6e67 4d61 7020 5262  |o8 : RingMap Rb
│ │ │ │ +000152b0: 6172 203c 2d2d 2052 2020 2020 2020 2020  ar <-- R        
│ │ │ │ +000152c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000152d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000152e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000152f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00015300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00015360: 0a7c 6939 203a 204d 6261 7220 3d20 7072  .|i9 : Mbar = pr
│ │ │ │ -00015370: 756e 6520 636f 6b65 7220 7261 6e64 6f6d  une coker random
│ │ │ │ -00015380: 2852 6261 725e 312c 2052 6261 725e 7b2d  (Rbar^1, Rbar^{-
│ │ │ │ -00015390: 327d 2920 2020 2020 2020 2020 2020 2020  2})             
│ │ │ │ -000153a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000153b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015340: 7c69 3920 3a20 4d62 6172 203d 2070 7275  |i9 : Mbar = pru
│ │ │ │ +00015350: 6e65 2063 6f6b 6572 2072 616e 646f 6d28  ne coker random(
│ │ │ │ +00015360: 5262 6172 5e31 2c20 5262 6172 5e7b 2d32  Rbar^1, Rbar^{-2
│ │ │ │ +00015370: 7d29 2020 2020 2020 2020 2020 2020 2020  })              
│ │ │ │ +00015380: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015390: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000153a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000153b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000153c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015400: 0a7c 6f39 203d 2063 6f6b 6572 6e65 6c20  .|o9 = cokernel 
│ │ │ │ -00015410: 7c20 785f 3078 5f31 2b32 3478 5f31 5e32  | x_0x_1+24x_1^2
│ │ │ │ -00015420: 2b34 3978 5f30 785f 322b 3378 5f31 785f  +49x_0x_2+3x_1x_
│ │ │ │ -00015430: 322b 3578 5f32 5e32 207c 2020 2020 2020  2+5x_2^2 |      
│ │ │ │ -00015440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000153d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000153e0: 7c6f 3920 3d20 636f 6b65 726e 656c 207c  |o9 = cokernel |
│ │ │ │ +000153f0: 2078 5f30 785f 312b 3234 785f 315e 322b   x_0x_1+24x_1^2+
│ │ │ │ +00015400: 3439 785f 3078 5f32 2b33 785f 3178 5f32  49x_0x_2+3x_1x_2
│ │ │ │ +00015410: 2b35 785f 325e 3220 7c20 2020 2020 2020  +5x_2^2 |       
│ │ │ │ +00015420: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015430: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000154a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015480: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000154a0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │  000154b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154c0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -000154d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000154f0: 0a7c 6f39 203a 2052 6261 722d 6d6f 6475  .|o9 : Rbar-modu
│ │ │ │ -00015500: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ -00015510: 5262 6172 2020 2020 2020 2020 2020 2020  Rbar            
│ │ │ │ -00015520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015540: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000154c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000154d0: 7c6f 3920 3a20 5262 6172 2d6d 6f64 756c  |o9 : Rbar-modul
│ │ │ │ +000154e0: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +000154f0: 6261 7220 2020 2020 2020 2020 2020 2020  bar             
│ │ │ │ +00015500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015510: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015520: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00015530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00015590: 0a7c 6931 3020 3a20 2864 302c 6431 2920  .|i10 : (d0,d1) 
│ │ │ │ -000155a0: 3d20 4569 7365 6e62 7564 5368 616d 6173  = EisenbudShamas
│ │ │ │ -000155b0: 6854 6f74 616c 284d 6261 722c 4772 6164  hTotal(Mbar,Grad
│ │ │ │ -000155c0: 696e 6720 3d3e 3129 2020 2020 2020 2020  ing =>1)        
│ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000155e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015570: 7c69 3130 203a 2028 6430 2c64 3129 203d  |i10 : (d0,d1) =
│ │ │ │ +00015580: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ +00015590: 546f 7461 6c28 4d62 6172 2c47 7261 6469  Total(Mbar,Gradi
│ │ │ │ +000155a0: 6e67 203d 3e31 2920 2020 2020 2020 2020  ng =>1)         
│ │ │ │ +000155b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000155c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000155d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000155e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000155f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015630: 0a7c 6f31 3020 3d20 287b 2d32 7d20 7c20  .|o10 = ({-2} | 
│ │ │ │ -00015640: 785f 305e 3220 2020 2020 2020 2020 2020  x_0^2           
│ │ │ │ -00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015660: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00015670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015680: 0a7c 2020 2020 2020 207b 2d32 7d20 7c20  .|       {-2} | 
│ │ │ │ -00015690: 785f 3078 5f31 2b32 3478 5f31 5e32 2b34  x_0x_1+24x_1^2+4
│ │ │ │ -000156a0: 3978 5f30 785f 322b 3378 5f31 785f 322b  9x_0x_2+3x_1x_2+
│ │ │ │ -000156b0: 3578 5f32 5e32 2033 3073 5f30 2020 2020  5x_2^2 30s_0    
│ │ │ │ -000156c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000156d0: 0a7c 2020 2020 2020 207b 2d33 7d20 7c20  .|       {-3} | 
│ │ │ │ -000156e0: 785f 325e 3320 2020 2020 2020 2020 2020  x_2^3           
│ │ │ │ -000156f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015700: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00015710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015720: 0a7c 2020 2020 2020 207b 2d37 7d20 7c20  .|       {-7} | 
│ │ │ │ -00015730: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00015740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015750: 2020 2020 2020 2078 5f32 5e33 2020 2020         x_2^3    
│ │ │ │ -00015760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015770: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00015600: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015610: 7c6f 3130 203d 2028 7b2d 327d 207c 2078  |o10 = ({-2} | x
│ │ │ │ +00015620: 5f30 5e32 2020 2020 2020 2020 2020 2020  _0^2            
│ │ │ │ +00015630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015640: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015660: 7c20 2020 2020 2020 7b2d 327d 207c 2078  |       {-2} | x
│ │ │ │ +00015670: 5f30 785f 312b 3234 785f 315e 322b 3439  _0x_1+24x_1^2+49
│ │ │ │ +00015680: 785f 3078 5f32 2b33 785f 3178 5f32 2b35  x_0x_2+3x_1x_2+5
│ │ │ │ +00015690: 785f 325e 3220 3330 735f 3020 2020 2020  x_2^2 30s_0     
│ │ │ │ +000156a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000156b0: 7c20 2020 2020 2020 7b2d 337d 207c 2078  |       {-3} | x
│ │ │ │ +000156c0: 5f32 5e33 2020 2020 2020 2020 2020 2020  _2^3            
│ │ │ │ +000156d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000156e0: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +000156f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015700: 7c20 2020 2020 2020 7b2d 377d 207c 2030  |       {-7} | 0
│ │ │ │ +00015710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015730: 2020 2020 2020 785f 325e 3320 2020 2020        x_2^3     
│ │ │ │ +00015740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015750: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00015760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000157b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000157c0: 0a7c 2020 2020 2020 2d73 5f31 2020 2020  .|      -s_1    
│ │ │ │ -000157d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157f0: 2020 3020 2020 2020 2020 207c 2c20 7b30    0        |, {0
│ │ │ │ -00015800: 7d20 207c 2020 2020 2020 2020 2020 207c  }  |           |
│ │ │ │ -00015810: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015840: 2020 2d73 5f31 2020 2020 207c 2020 7b2d    -s_1     |  {-
│ │ │ │ -00015850: 347d 207c 2020 2020 2020 2020 2020 207c  4} |           |
│ │ │ │ -00015860: 0a7c 2020 2020 2020 735f 3020 2020 2020  .|      s_0     
│ │ │ │ -00015870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015890: 2020 3020 2020 2020 2020 207c 2020 7b2d    0        |  {-
│ │ │ │ -000158a0: 357d 207c 2020 2020 2020 2020 2020 207c  5} |           |
│ │ │ │ -000158b0: 0a7c 2020 2020 2020 3337 785f 3078 5f31  .|      37x_0x_1
│ │ │ │ -000158c0: 2d32 3178 5f31 5e32 2d35 785f 3078 5f32  -21x_1^2-5x_0x_2
│ │ │ │ -000158d0: 2b31 3078 5f31 785f 322d 3137 785f 325e  +10x_1x_2-17x_2^
│ │ │ │ -000158e0: 3220 2d33 3778 5f30 5e32 207c 2020 7b2d  2 -37x_0^2 |  {-
│ │ │ │ -000158f0: 357d 207c 2020 2020 2020 2020 2020 207c  5} |           |
│ │ │ │ -00015900: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +000157a0: 7c20 2020 2020 202d 735f 3120 2020 2020  |      -s_1     
│ │ │ │ +000157b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000157c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000157d0: 2030 2020 2020 2020 2020 7c2c 207b 307d   0        |, {0}
│ │ │ │ +000157e0: 2020 7c20 2020 2020 2020 2020 2020 7c0a    |           |.
│ │ │ │ +000157f0: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015820: 202d 735f 3120 2020 2020 7c20 207b 2d34   -s_1     |  {-4
│ │ │ │ +00015830: 7d20 7c20 2020 2020 2020 2020 2020 7c0a  } |           |.
│ │ │ │ +00015840: 7c20 2020 2020 2073 5f30 2020 2020 2020  |      s_0      
│ │ │ │ +00015850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015870: 2030 2020 2020 2020 2020 7c20 207b 2d35   0        |  {-5
│ │ │ │ +00015880: 7d20 7c20 2020 2020 2020 2020 2020 7c0a  } |           |.
│ │ │ │ +00015890: 7c20 2020 2020 2033 3778 5f30 785f 312d  |      37x_0x_1-
│ │ │ │ +000158a0: 3231 785f 315e 322d 3578 5f30 785f 322b  21x_1^2-5x_0x_2+
│ │ │ │ +000158b0: 3130 785f 3178 5f32 2d31 3778 5f32 5e32  10x_1x_2-17x_2^2
│ │ │ │ +000158c0: 202d 3337 785f 305e 3220 7c20 207b 2d35   -37x_0^2 |  {-5
│ │ │ │ +000158d0: 7d20 7c20 2020 2020 2020 2020 2020 7c0a  } |           |.
│ │ │ │ +000158e0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +000158f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00015950: 0a7c 2020 2020 2020 735f 3020 2020 2020  .|      s_0     
│ │ │ │ -00015960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015980: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00015990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000159a0: 0a7c 2020 2020 2020 3337 785f 3078 5f31  .|      37x_0x_1
│ │ │ │ -000159b0: 2d32 3178 5f31 5e32 2d35 785f 3078 5f32  -21x_1^2-5x_0x_2
│ │ │ │ -000159c0: 2b31 3078 5f31 785f 322d 3137 785f 325e  +10x_1x_2-17x_2^
│ │ │ │ -000159d0: 3220 2d33 3778 5f30 5e32 2020 2020 2020  2 -37x_0^2      
│ │ │ │ -000159e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000159f0: 0a7c 2020 2020 2020 2d78 5f32 5e33 2020  .|      -x_2^3  
│ │ │ │ -00015a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a20: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00015a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015a40: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a70: 2020 2d78 5f32 5e33 2020 2020 2020 2020    -x_2^3        
│ │ │ │ -00015a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015a90: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00015920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00015930: 7c20 2020 2020 2073 5f30 2020 2020 2020  |      s_0      
│ │ │ │ +00015940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015960: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00015970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015980: 7c20 2020 2020 2033 3778 5f30 785f 312d  |      37x_0x_1-
│ │ │ │ +00015990: 3231 785f 315e 322d 3578 5f30 785f 322b  21x_1^2-5x_0x_2+
│ │ │ │ +000159a0: 3130 785f 3178 5f32 2d31 3778 5f32 5e32  10x_1x_2-17x_2^2
│ │ │ │ +000159b0: 202d 3337 785f 305e 3220 2020 2020 2020   -37x_0^2       
│ │ │ │ +000159c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000159d0: 7c20 2020 2020 202d 785f 325e 3320 2020  |      -x_2^3   
│ │ │ │ +000159e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000159f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a00: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00015a10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015a20: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015a50: 202d 785f 325e 3320 2020 2020 2020 2020   -x_2^3         
│ │ │ │ +00015a60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015a70: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00015a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00015ae0: 0a7c 2020 2020 2020 735f 3120 2020 2020  .|      s_1     
│ │ │ │ -00015af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b00: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00015b10: 2020 2020 207c 2920 2020 2020 2020 2020       |)         
│ │ │ │ -00015b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015b30: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015b50: 2020 2020 2020 2020 2020 2020 2020 2073                 s
│ │ │ │ -00015b60: 5f31 2020 207c 2020 2020 2020 2020 2020  _1   |          
│ │ │ │ -00015b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015b80: 0a7c 2020 2020 2020 785f 305e 3220 2020  .|      x_0^2   
│ │ │ │ -00015b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ba0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00015bb0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ -00015bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015bd0: 0a7c 2020 2020 2020 785f 3078 5f31 2b32  .|      x_0x_1+2
│ │ │ │ -00015be0: 3478 5f31 5e32 2b34 3978 5f30 785f 322b  4x_1^2+49x_0x_2+
│ │ │ │ -00015bf0: 3378 5f31 785f 322b 3578 5f32 5e32 2033  3x_1x_2+5x_2^2 3
│ │ │ │ -00015c00: 3073 5f30 207c 2020 2020 2020 2020 2020  0s_0 |          
│ │ │ │ -00015c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00015ac0: 7c20 2020 2020 2073 5f31 2020 2020 2020  |      s_1      
│ │ │ │ +00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ae0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00015af0: 2020 2020 7c29 2020 2020 2020 2020 2020      |)          
│ │ │ │ +00015b00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015b10: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b30: 2020 2020 2020 2020 2020 2020 2020 735f                s_
│ │ │ │ +00015b40: 3120 2020 7c20 2020 2020 2020 2020 2020  1   |           
│ │ │ │ +00015b50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015b60: 7c20 2020 2020 2078 5f30 5e32 2020 2020  |      x_0^2    
│ │ │ │ +00015b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015b80: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00015b90: 2020 2020 7c20 2020 2020 2020 2020 2020      |           
│ │ │ │ +00015ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015bb0: 7c20 2020 2020 2078 5f30 785f 312b 3234  |      x_0x_1+24
│ │ │ │ +00015bc0: 785f 315e 322b 3439 785f 3078 5f32 2b33  x_1^2+49x_0x_2+3
│ │ │ │ +00015bd0: 785f 3178 5f32 2b35 785f 325e 3220 3330  x_1x_2+5x_2^2 30
│ │ │ │ +00015be0: 735f 3020 7c20 2020 2020 2020 2020 2020  s_0 |           
│ │ │ │ +00015bf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015c00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015c70: 0a7c 6f31 3020 3a20 5365 7175 656e 6365  .|o10 : Sequence
│ │ │ │ +00015c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015c50: 7c6f 3130 203a 2053 6571 7565 6e63 6520  |o10 : Sequence 
│ │ │ │ +00015c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015cc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00015c90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015ca0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00015cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00015d10: 0a7c 6931 3120 3a20 6430 2a64 3120 2020  .|i11 : d0*d1   
│ │ │ │ +00015ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00015cf0: 7c69 3131 203a 2064 302a 6431 2020 2020  |i11 : d0*d1    
│ │ │ │ +00015d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015d30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015d40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00015d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015db0: 0a7c 6f31 3120 3d20 7b2d 327d 207c 2073  .|o11 = {-2} | s
│ │ │ │ -00015dc0: 5f31 785f 325e 332b 735f 3078 5f30 5e32  _1x_2^3+s_0x_0^2
│ │ │ │ -00015dd0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00015de0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ -00015df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015e00: 0a7c 2020 2020 2020 7b2d 327d 207c 2030  .|      {-2} | 0
│ │ │ │ -00015e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e20: 2073 5f31 785f 325e 332b 735f 3078 5f30   s_1x_2^3+s_0x_0
│ │ │ │ -00015e30: 5e32 2030 2020 2020 2020 2020 2020 2020  ^2 0            
│ │ │ │ -00015e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015e50: 0a7c 2020 2020 2020 7b2d 337d 207c 2030  .|      {-3} | 0
│ │ │ │ -00015e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e70: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00015e80: 2020 2073 5f31 785f 325e 332b 735f 3078     s_1x_2^3+s_0x
│ │ │ │ -00015e90: 5f30 5e32 2020 2020 2020 2020 2020 207c  _0^2           |
│ │ │ │ -00015ea0: 0a7c 2020 2020 2020 7b2d 377d 207c 2030  .|      {-7} | 0
│ │ │ │ -00015eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ec0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00015ed0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ -00015ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015ef0: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00015d80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015d90: 7c6f 3131 203d 207b 2d32 7d20 7c20 735f  |o11 = {-2} | s_
│ │ │ │ +00015da0: 3178 5f32 5e33 2b73 5f30 785f 305e 3220  1x_2^3+s_0x_0^2 
│ │ │ │ +00015db0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00015dc0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00015dd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015de0: 7c20 2020 2020 207b 2d32 7d20 7c20 3020  |      {-2} | 0 
│ │ │ │ +00015df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e00: 735f 3178 5f32 5e33 2b73 5f30 785f 305e  s_1x_2^3+s_0x_0^
│ │ │ │ +00015e10: 3220 3020 2020 2020 2020 2020 2020 2020  2 0             
│ │ │ │ +00015e20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015e30: 7c20 2020 2020 207b 2d33 7d20 7c20 3020  |      {-3} | 0 
│ │ │ │ +00015e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015e50: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00015e60: 2020 735f 3178 5f32 5e33 2b73 5f30 785f    s_1x_2^3+s_0x_
│ │ │ │ +00015e70: 305e 3220 2020 2020 2020 2020 2020 7c0a  0^2           |.
│ │ │ │ +00015e80: 7c20 2020 2020 207b 2d37 7d20 7c20 3020  |      {-7} | 0 
│ │ │ │ +00015e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ea0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00015eb0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +00015ec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015ed0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +00015ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00015f40: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015f50: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ -00015f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015f90: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015fa0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ -00015fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00015fe0: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00015ff0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ -00016000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016030: 0a7c 2020 2020 2020 735f 3178 5f32 5e33  .|      s_1x_2^3
│ │ │ │ -00016040: 2b73 5f30 785f 305e 3220 7c20 2020 2020  +s_0x_0^2 |     
│ │ │ │ -00016050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016080: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00015f20: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015f30: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00015f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015f60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015f70: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015f80: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00015f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015fb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00015fc0: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00015fd0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00015fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016000: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016010: 7c20 2020 2020 2073 5f31 785f 325e 332b  |      s_1x_2^3+
│ │ │ │ +00016020: 735f 3078 5f30 5e32 207c 2020 2020 2020  s_0x_0^2 |      
│ │ │ │ +00016030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016060: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000160d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000160e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000160f0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -00016100: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -00016110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016120: 0a7c 6f31 3120 3a20 4d61 7472 6978 2028  .|o11 : Matrix (
│ │ │ │ -00016130: 6b6b 5b73 202e 2e73 202c 2078 202e 2e78  kk[s ..s , x ..x
│ │ │ │ -00016140: 205d 2920 203c 2d2d 2028 6b6b 5b73 202e   ])  <-- (kk[s .
│ │ │ │ -00016150: 2e73 202c 2078 202e 2e78 205d 2920 2020  .s , x ..x ])   
│ │ │ │ -00016160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016180: 2020 2020 3020 2020 3120 2020 3020 2020      0   1   0   
│ │ │ │ -00016190: 3220 2020 2020 2020 2020 2020 2020 3020  2             0 
│ │ │ │ -000161a0: 2020 3120 2020 3020 2020 3220 2020 2020    1   0   2     
│ │ │ │ -000161b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000161c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000160a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000160b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000160c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000160d0: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +000160e0: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +000160f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016100: 7c6f 3131 203a 204d 6174 7269 7820 286b  |o11 : Matrix (k
│ │ │ │ +00016110: 6b5b 7320 2e2e 7320 2c20 7820 2e2e 7820  k[s ..s , x ..x 
│ │ │ │ +00016120: 5d29 2020 3c2d 2d20 286b 6b5b 7320 2e2e  ])  <-- (kk[s ..
│ │ │ │ +00016130: 7320 2c20 7820 2e2e 7820 5d29 2020 2020  s , x ..x ])    
│ │ │ │ +00016140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016150: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016160: 2020 2030 2020 2031 2020 2030 2020 2032     0   1   0   2
│ │ │ │ +00016170: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +00016180: 2031 2020 2030 2020 2032 2020 2020 2020   1   0   2      
│ │ │ │ +00016190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000161a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000161f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016210: 0a7c 6931 3220 3a20 6431 2a64 3020 2020  .|i12 : d1*d0   
│ │ │ │ +000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000161f0: 7c69 3132 203a 2064 312a 6430 2020 2020  |i12 : d1*d0    
│ │ │ │ +00016200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016240: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000162a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000162b0: 0a7c 6f31 3220 3d20 7b30 7d20 207c 2073  .|o12 = {0}  | s
│ │ │ │ -000162c0: 5f31 785f 325e 332b 735f 3078 5f30 5e32  _1x_2^3+s_0x_0^2
│ │ │ │ -000162d0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -000162e0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ -000162f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016300: 0a7c 2020 2020 2020 7b2d 347d 207c 2030  .|      {-4} | 0
│ │ │ │ -00016310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016320: 2073 5f31 785f 325e 332b 735f 3078 5f30   s_1x_2^3+s_0x_0
│ │ │ │ -00016330: 5e32 2030 2020 2020 2020 2020 2020 2020  ^2 0            
│ │ │ │ -00016340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016350: 0a7c 2020 2020 2020 7b2d 357d 207c 2030  .|      {-5} | 0
│ │ │ │ -00016360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016370: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00016380: 2020 2073 5f31 785f 325e 332b 735f 3078     s_1x_2^3+s_0x
│ │ │ │ -00016390: 5f30 5e32 2020 2020 2020 2020 2020 207c  _0^2           |
│ │ │ │ -000163a0: 0a7c 2020 2020 2020 7b2d 357d 207c 2030  .|      {-5} | 0
│ │ │ │ -000163b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000163c0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -000163d0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ -000163e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000163f0: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d  .|      --------
│ │ │ │ +00016280: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016290: 7c6f 3132 203d 207b 307d 2020 7c20 735f  |o12 = {0}  | s_
│ │ │ │ +000162a0: 3178 5f32 5e33 2b73 5f30 785f 305e 3220  1x_2^3+s_0x_0^2 
│ │ │ │ +000162b0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000162c0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000162d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000162e0: 7c20 2020 2020 207b 2d34 7d20 7c20 3020  |      {-4} | 0 
│ │ │ │ +000162f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016300: 735f 3178 5f32 5e33 2b73 5f30 785f 305e  s_1x_2^3+s_0x_0^
│ │ │ │ +00016310: 3220 3020 2020 2020 2020 2020 2020 2020  2 0             
│ │ │ │ +00016320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016330: 7c20 2020 2020 207b 2d35 7d20 7c20 3020  |      {-5} | 0 
│ │ │ │ +00016340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016350: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00016360: 2020 735f 3178 5f32 5e33 2b73 5f30 785f    s_1x_2^3+s_0x_
│ │ │ │ +00016370: 305e 3220 2020 2020 2020 2020 2020 7c0a  0^2           |.
│ │ │ │ +00016380: 7c20 2020 2020 207b 2d35 7d20 7c20 3020  |      {-5} | 0 
│ │ │ │ +00016390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000163a0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000163b0: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ +000163c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000163d0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  |      ---------
│ │ │ │ +000163e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000163f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00016440: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -00016450: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ -00016460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016490: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -000164a0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ -000164b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000164c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000164d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000164e0: 0a7c 2020 2020 2020 3020 2020 2020 2020  .|      0       
│ │ │ │ -000164f0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ -00016500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016530: 0a7c 2020 2020 2020 735f 3178 5f32 5e33  .|      s_1x_2^3
│ │ │ │ -00016540: 2b73 5f30 785f 305e 3220 7c20 2020 2020  +s_0x_0^2 |     
│ │ │ │ -00016550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016580: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a  --------------|.
│ │ │ │ +00016420: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00016430: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00016440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016470: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +00016480: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00016490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000164a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000164b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000164c0: 7c20 2020 2020 2030 2020 2020 2020 2020  |      0        
│ │ │ │ +000164d0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +000164e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000164f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016500: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016510: 7c20 2020 2020 2073 5f31 785f 325e 332b  |      s_1x_2^3+
│ │ │ │ +00016520: 735f 3078 5f30 5e32 207c 2020 2020 2020  s_0x_0^2 |      
│ │ │ │ +00016530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016560: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000165a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000165b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000165c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000165d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000165e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000165f0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -00016600: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -00016610: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016620: 0a7c 6f31 3220 3a20 4d61 7472 6978 2028  .|o12 : Matrix (
│ │ │ │ -00016630: 6b6b 5b73 202e 2e73 202c 2078 202e 2e78  kk[s ..s , x ..x
│ │ │ │ -00016640: 205d 2920 203c 2d2d 2028 6b6b 5b73 202e   ])  <-- (kk[s .
│ │ │ │ -00016650: 2e73 202c 2078 202e 2e78 205d 2920 2020  .s , x ..x ])   
│ │ │ │ -00016660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016680: 2020 2020 3020 2020 3120 2020 3020 2020      0   1   0   
│ │ │ │ -00016690: 3220 2020 2020 2020 2020 2020 2020 3020  2             0 
│ │ │ │ -000166a0: 2020 3120 2020 3020 2020 3220 2020 2020    1   0   2     
│ │ │ │ -000166b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000166c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000165a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000165b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000165c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000165d0: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +000165e0: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +000165f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016600: 7c6f 3132 203a 204d 6174 7269 7820 286b  |o12 : Matrix (k
│ │ │ │ +00016610: 6b5b 7320 2e2e 7320 2c20 7820 2e2e 7820  k[s ..s , x ..x 
│ │ │ │ +00016620: 5d29 2020 3c2d 2d20 286b 6b5b 7320 2e2e  ])  <-- (kk[s ..
│ │ │ │ +00016630: 7320 2c20 7820 2e2e 7820 5d29 2020 2020  s , x ..x ])    
│ │ │ │ +00016640: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016650: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016660: 2020 2030 2020 2031 2020 2030 2020 2032     0   1   0   2
│ │ │ │ +00016670: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +00016680: 2031 2020 2030 2020 2032 2020 2020 2020   1   0   2      
│ │ │ │ +00016690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000166a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000166b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000166c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000166d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000166e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000166f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016710: 0a7c 6931 3320 3a20 5320 3d20 7269 6e67  .|i13 : S = ring
│ │ │ │ -00016720: 2064 3020 2020 2020 2020 2020 2020 2020   d0             
│ │ │ │ -00016730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000166e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000166f0: 7c69 3133 203a 2053 203d 2072 696e 6720  |i13 : S = ring 
│ │ │ │ +00016700: 6430 2020 2020 2020 2020 2020 2020 2020  d0              
│ │ │ │ +00016710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016740: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000167b0: 0a7c 6f31 3320 3d20 5320 2020 2020 2020  .|o13 = S       
│ │ │ │ +00016780: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016790: 7c6f 3133 203d 2053 2020 2020 2020 2020  |o13 = S        
│ │ │ │ +000167a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000167b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000167c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000167d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000167e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000167f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016850: 0a7c 6f31 3320 3a20 506f 6c79 6e6f 6d69  .|o13 : Polynomi
│ │ │ │ -00016860: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ -00016870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000168a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00016820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016830: 7c6f 3133 203a 2050 6f6c 796e 6f6d 6961  |o13 : Polynomia
│ │ │ │ +00016840: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +00016850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016880: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00016890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000168a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000168b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000168c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000168d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000168e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000168f0: 0a7c 6931 3420 3a20 7068 6920 3d20 6d61  .|i14 : phi = ma
│ │ │ │ -00016900: 7028 532c 5229 2020 2020 2020 2020 2020  p(S,R)          
│ │ │ │ -00016910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000168c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000168d0: 7c69 3134 203a 2070 6869 203d 206d 6170  |i14 : phi = map
│ │ │ │ +000168e0: 2853 2c52 2920 2020 2020 2020 2020 2020  (S,R)           
│ │ │ │ +000168f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016910: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016920: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016990: 0a7c 6f31 3420 3d20 6d61 7020 2853 2c20  .|o14 = map (S, 
│ │ │ │ -000169a0: 522c 207b 7820 2c20 7820 2c20 7820 7d29  R, {x , x , x })
│ │ │ │ -000169b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000169c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000169d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000169e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000169f0: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ -00016a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016970: 7c6f 3134 203d 206d 6170 2028 532c 2052  |o14 = map (S, R
│ │ │ │ +00016980: 2c20 7b78 202c 2078 202c 2078 207d 2920  , {x , x , x }) 
│ │ │ │ +00016990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000169a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000169b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000169c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000169d0: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +000169e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000169f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016a10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016a80: 0a7c 6f31 3420 3a20 5269 6e67 4d61 7020  .|o14 : RingMap 
│ │ │ │ -00016a90: 5320 3c2d 2d20 5220 2020 2020 2020 2020  S <-- R         
│ │ │ │ -00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00016a50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016a60: 7c6f 3134 203a 2052 696e 674d 6170 2053  |o14 : RingMap S
│ │ │ │ +00016a70: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ +00016a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016aa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016ab0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00016ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016b20: 0a7c 6931 3520 3a20 4953 203d 2070 6869  .|i15 : IS = phi
│ │ │ │ -00016b30: 2049 2020 2020 2020 2020 2020 2020 2020   I              
│ │ │ │ -00016b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00016b00: 7c69 3135 203a 2049 5320 3d20 7068 6920  |i15 : IS = phi 
│ │ │ │ +00016b10: 4920 2020 2020 2020 2020 2020 2020 2020  I               
│ │ │ │ +00016b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016b40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016b50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016bd0: 3220 2020 3320 2020 2020 2020 2020 2020  2   3           
│ │ │ │ -00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016c10: 0a7c 6f31 3520 3d20 6964 6561 6c20 2878  .|o15 = ideal (x
│ │ │ │ -00016c20: 202c 2078 2029 2020 2020 2020 2020 2020   , x )          
│ │ │ │ -00016c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016c60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00016c70: 3020 2020 3220 2020 2020 2020 2020 2020  0   2           
│ │ │ │ -00016c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016b90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016ba0: 7c20 2020 2020 2020 2020 2020 2020 2032  |              2
│ │ │ │ +00016bb0: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +00016bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016be0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016bf0: 7c6f 3135 203d 2069 6465 616c 2028 7820  |o15 = ideal (x 
│ │ │ │ +00016c00: 2c20 7820 2920 2020 2020 2020 2020 2020  , x )           
│ │ │ │ +00016c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016c40: 7c20 2020 2020 2020 2020 2020 2020 2030  |              0
│ │ │ │ +00016c50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00016c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016c90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016cf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016d00: 0a7c 6f31 3520 3a20 4964 6561 6c20 6f66  .|o15 : Ideal of
│ │ │ │ -00016d10: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │ -00016d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016d50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00016cd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016ce0: 7c6f 3135 203a 2049 6465 616c 206f 6620  |o15 : Ideal of 
│ │ │ │ +00016cf0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00016d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016d30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00016d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016da0: 0a7c 6931 3620 3a20 5362 6172 203d 2053  .|i16 : Sbar = S
│ │ │ │ -00016db0: 2f49 5320 2020 2020 2020 2020 2020 2020  /IS             
│ │ │ │ -00016dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016df0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00016d80: 7c69 3136 203a 2053 6261 7220 3d20 532f  |i16 : Sbar = S/
│ │ │ │ +00016d90: 4953 2020 2020 2020 2020 2020 2020 2020  IS              
│ │ │ │ +00016da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016dc0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016dd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016e40: 0a7c 6f31 3620 3d20 5362 6172 2020 2020  .|o16 = Sbar    
│ │ │ │ +00016e10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016e20: 7c6f 3136 203d 2053 6261 7220 2020 2020  |o16 = Sbar     
│ │ │ │ +00016e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016e60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016e70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016ee0: 0a7c 6f31 3620 3a20 5175 6f74 6965 6e74  .|o16 : Quotient
│ │ │ │ -00016ef0: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ -00016f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016f30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00016eb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016ec0: 7c6f 3136 203a 2051 756f 7469 656e 7452  |o16 : QuotientR
│ │ │ │ +00016ed0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +00016ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016f10: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00016f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00016f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00016f80: 0a7c 6931 3720 3a20 534d 6261 7220 3d20  .|i17 : SMbar = 
│ │ │ │ -00016f90: 5362 6172 2a2a 4d62 6172 2020 2020 2020  Sbar**Mbar      
│ │ │ │ -00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00016fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00016f60: 7c69 3137 203a 2053 4d62 6172 203d 2053  |i17 : SMbar = S
│ │ │ │ +00016f70: 6261 722a 2a4d 6261 7220 2020 2020 2020  bar**Mbar       
│ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00016fb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017020: 0a7c 6f31 3720 3d20 636f 6b65 726e 656c  .|o17 = cokernel
│ │ │ │ -00017030: 207c 2078 5f30 785f 312b 3234 785f 315e   | x_0x_1+24x_1^
│ │ │ │ -00017040: 322b 3439 785f 3078 5f32 2b33 785f 3178  2+49x_0x_2+3x_1x
│ │ │ │ -00017050: 5f32 2b35 785f 325e 3220 7c20 2020 2020  _2+5x_2^2 |     
│ │ │ │ -00017060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00016ff0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017000: 7c6f 3137 203d 2063 6f6b 6572 6e65 6c20  |o17 = cokernel 
│ │ │ │ +00017010: 7c20 785f 3078 5f31 2b32 3478 5f31 5e32  | x_0x_1+24x_1^2
│ │ │ │ +00017020: 2b34 3978 5f30 785f 322b 3378 5f31 785f  +49x_0x_2+3x_1x_
│ │ │ │ +00017030: 322b 3578 5f32 5e32 207c 2020 2020 2020  2+5x_2^2 |      
│ │ │ │ +00017040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017050: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00017060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000170c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00017090: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000170a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000170b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170c0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170e0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -000170f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017110: 0a7c 6f31 3720 3a20 5362 6172 2d6d 6f64  .|o17 : Sbar-mod
│ │ │ │ -00017120: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ -00017130: 2053 6261 7220 2020 2020 2020 2020 2020   Sbar           
│ │ │ │ -00017140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017150: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00017160: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000170e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000170f0: 7c6f 3137 203a 2053 6261 722d 6d6f 6475  |o17 : Sbar-modu
│ │ │ │ +00017100: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ +00017110: 5362 6172 2020 2020 2020 2020 2020 2020  Sbar            
│ │ │ │ +00017120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017130: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00017140: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00017150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000171a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000171b0: 0a0a 486f 6d28 6430 2c53 6261 7229 2061  ..Hom(d0,Sbar) a
│ │ │ │ -000171c0: 6e64 2048 6f6d 2864 312c 5362 6172 2920  nd Hom(d1,Sbar) 
│ │ │ │ -000171d0: 746f 6765 7468 6572 2066 6f72 6d20 7468  together form th
│ │ │ │ -000171e0: 6520 7265 736f 6c75 7469 6f6e 206f 6620  e resolution of 
│ │ │ │ -000171f0: 4d62 6172 3b20 7468 7573 2074 6865 0a68  Mbar; thus the.h
│ │ │ │ -00017200: 6f6d 6f6c 6f67 7920 6f66 206f 6e65 2063  omology of one c
│ │ │ │ -00017210: 6f6d 706f 7369 7469 6f6e 2069 7320 302c  omposition is 0,
│ │ │ │ -00017220: 2077 6869 6c65 2074 6865 206f 7468 6572   while the other
│ │ │ │ -00017230: 2069 7320 4d62 6172 0a0a 2b2d 2d2d 2d2d   is Mbar..+-----
│ │ │ │ +00017180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00017190: 0a48 6f6d 2864 302c 5362 6172 2920 616e  .Hom(d0,Sbar) an
│ │ │ │ +000171a0: 6420 486f 6d28 6431 2c53 6261 7229 2074  d Hom(d1,Sbar) t
│ │ │ │ +000171b0: 6f67 6574 6865 7220 666f 726d 2074 6865  ogether form the
│ │ │ │ +000171c0: 2072 6573 6f6c 7574 696f 6e20 6f66 204d   resolution of M
│ │ │ │ +000171d0: 6261 723b 2074 6875 7320 7468 650a 686f  bar; thus the.ho
│ │ │ │ +000171e0: 6d6f 6c6f 6779 206f 6620 6f6e 6520 636f  mology of one co
│ │ │ │ +000171f0: 6d70 6f73 6974 696f 6e20 6973 2030 2c20  mposition is 0, 
│ │ │ │ +00017200: 7768 696c 6520 7468 6520 6f74 6865 7220  while the other 
│ │ │ │ +00017210: 6973 204d 6261 720a 0a2b 2d2d 2d2d 2d2d  is Mbar..+------
│ │ │ │ +00017220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017280: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a  --------+.|i18 :
│ │ │ │ -00017290: 2070 7275 6e65 2048 485f 3120 636f 6d70   prune HH_1 comp
│ │ │ │ -000172a0: 6c65 787b 6475 616c 2028 5362 6172 2a2a  lex{dual (Sbar**
│ │ │ │ -000172b0: 6430 292c 2064 7561 6c28 5362 6172 2a2a  d0), dual(Sbar**
│ │ │ │ -000172c0: 6431 297d 203d 3d20 3020 2020 2020 2020  d1)} == 0       
│ │ │ │ -000172d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017260: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20  -------+.|i18 : 
│ │ │ │ +00017270: 7072 756e 6520 4848 5f31 2063 6f6d 706c  prune HH_1 compl
│ │ │ │ +00017280: 6578 7b64 7561 6c20 2853 6261 722a 2a64  ex{dual (Sbar**d
│ │ │ │ +00017290: 3029 2c20 6475 616c 2853 6261 722a 2a64  0), dual(Sbar**d
│ │ │ │ +000172a0: 3129 7d20 3d3d 2030 2020 2020 2020 2020  1)} == 0        
│ │ │ │ +000172b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000172c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000172f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017320: 2020 2020 2020 2020 7c0a 7c6f 3138 203d          |.|o18 =
│ │ │ │ -00017330: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +00017300: 2020 2020 2020 207c 0a7c 6f31 3820 3d20         |.|o18 = 
│ │ │ │ +00017310: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00017320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00017350: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00017360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000173a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000173c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a  --------+.|i19 :
│ │ │ │ -000173d0: 204d 6261 7227 203d 2053 6261 725e 312f   Mbar' = Sbar^1/
│ │ │ │ -000173e0: 2853 6261 725f 302c 2053 6261 725f 3129  (Sbar_0, Sbar_1)
│ │ │ │ -000173f0: 2a2a 534d 6261 7220 2020 2020 2020 2020  **SMbar         
│ │ │ │ +000173a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20  -------+.|i19 : 
│ │ │ │ +000173b0: 4d62 6172 2720 3d20 5362 6172 5e31 2f28  Mbar' = Sbar^1/(
│ │ │ │ +000173c0: 5362 6172 5f30 2c20 5362 6172 5f31 292a  Sbar_0, Sbar_1)*
│ │ │ │ +000173d0: 2a53 4d62 6172 2020 2020 2020 2020 2020  *SMbar          
│ │ │ │ +000173e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000173f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00017400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017460: 2020 2020 2020 2020 7c0a 7c6f 3139 203d          |.|o19 =
│ │ │ │ -00017470: 2063 6f6b 6572 6e65 6c20 7c20 785f 3078   cokernel | x_0x
│ │ │ │ -00017480: 5f31 2b32 3478 5f31 5e32 2b34 3978 5f30  _1+24x_1^2+49x_0
│ │ │ │ -00017490: 785f 322b 3378 5f31 785f 322b 3578 5f32  x_2+3x_1x_2+5x_2
│ │ │ │ -000174a0: 5e32 2073 5f30 2073 5f31 207c 2020 2020  ^2 s_0 s_1 |    
│ │ │ │ -000174b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017440: 2020 2020 2020 207c 0a7c 6f31 3920 3d20         |.|o19 = 
│ │ │ │ +00017450: 636f 6b65 726e 656c 207c 2078 5f30 785f  cokernel | x_0x_
│ │ │ │ +00017460: 312b 3234 785f 315e 322b 3439 785f 3078  1+24x_1^2+49x_0x
│ │ │ │ +00017470: 5f32 2b33 785f 3178 5f32 2b35 785f 325e  _2+3x_1x_2+5x_2^
│ │ │ │ +00017480: 3220 735f 3020 735f 3120 7c20 2020 2020  2 s_0 s_1 |     
│ │ │ │ +00017490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000174a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000174c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000174d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000174e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000174f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017500: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017500: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │  00017510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017520: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00017530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017550: 2020 2020 2020 2020 7c0a 7c6f 3139 203a          |.|o19 :
│ │ │ │ -00017560: 2053 6261 722d 6d6f 6475 6c65 2c20 7175   Sbar-module, qu
│ │ │ │ -00017570: 6f74 6965 6e74 206f 6620 5362 6172 2020  otient of Sbar  
│ │ │ │ -00017580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000175a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00017520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017530: 2020 2020 2020 207c 0a7c 6f31 3920 3a20         |.|o19 : 
│ │ │ │ +00017540: 5362 6172 2d6d 6f64 756c 652c 2071 756f  Sbar-module, quo
│ │ │ │ +00017550: 7469 656e 7420 6f66 2053 6261 7220 2020  tient of Sbar   
│ │ │ │ +00017560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017580: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00017590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000175a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000175b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000175c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000175f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ -00017600: 2069 6465 616c 2070 7265 7365 6e74 6174   ideal presentat
│ │ │ │ -00017610: 696f 6e20 7072 756e 6520 4848 5f31 2063  ion prune HH_1 c
│ │ │ │ -00017620: 6f6d 706c 6578 7b64 7561 6c20 2853 6261  omplex{dual (Sba
│ │ │ │ -00017630: 722a 2a64 3129 2c20 6475 616c 2853 6261  r**d1), dual(Sba
│ │ │ │ -00017640: 722a 2a64 3029 7d20 7c0a 7c20 2020 2020  r**d0)} |.|     
│ │ │ │ +000175d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 3a20  -------+.|i20 : 
│ │ │ │ +000175e0: 6964 6561 6c20 7072 6573 656e 7461 7469  ideal presentati
│ │ │ │ +000175f0: 6f6e 2070 7275 6e65 2048 485f 3120 636f  on prune HH_1 co
│ │ │ │ +00017600: 6d70 6c65 787b 6475 616c 2028 5362 6172  mplex{dual (Sbar
│ │ │ │ +00017610: 2a2a 6431 292c 2064 7561 6c28 5362 6172  **d1), dual(Sbar
│ │ │ │ +00017620: 2a2a 6430 297d 207c 0a7c 2020 2020 2020  **d0)} |.|      
│ │ │ │ +00017630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017690: 2020 2020 2020 2020 7c0a 7c6f 3230 203d          |.|o20 =
│ │ │ │ -000176a0: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +00017670: 2020 2020 2020 207c 0a7c 6f32 3020 3d20         |.|o20 = 
│ │ │ │ +00017680: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00017690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000176a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000176b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176e0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +000176c0: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +000176d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000176f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017730: 2d2d 2d2d 2d2d 2d2d 7c0a 7c3d 3d20 6964  --------|.|== id
│ │ │ │ -00017740: 6561 6c20 7072 6573 656e 7461 7469 6f6e  eal presentation
│ │ │ │ -00017750: 204d 6261 7227 2020 2020 2020 2020 2020   Mbar'          
│ │ │ │ -00017760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00017710: 2d2d 2d2d 2d2d 2d7c 0a7c 3d3d 2069 6465  -------|.|== ide
│ │ │ │ +00017720: 616c 2070 7265 7365 6e74 6174 696f 6e20  al presentation 
│ │ │ │ +00017730: 4d62 6172 2720 2020 2020 2020 2020 2020  Mbar'           
│ │ │ │ +00017740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017760: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00017770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000177a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000177b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000177d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -000177e0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -000177f0: 2a20 2a6e 6f74 6520 4578 743a 2028 4d61  * *note Ext: (Ma
│ │ │ │ -00017800: 6361 756c 6179 3244 6f63 2945 7874 2c20  caulay2Doc)Ext, 
│ │ │ │ -00017810: 2d2d 2063 6f6d 7075 7465 2061 6e20 4578  -- compute an Ex
│ │ │ │ -00017820: 7420 6d6f 6475 6c65 0a20 202a 202a 6e6f  t module.  * *no
│ │ │ │ -00017830: 7465 206e 6577 4578 743a 206e 6577 4578  te newExt: newEx
│ │ │ │ -00017840: 742c 202d 2d20 476c 6f62 616c 2045 7874  t, -- Global Ext
│ │ │ │ -00017850: 2066 6f72 206d 6f64 756c 6573 206f 7665   for modules ove
│ │ │ │ -00017860: 7220 6120 636f 6d70 6c65 7465 0a20 2020  r a complete.   
│ │ │ │ -00017870: 2049 6e74 6572 7365 6374 696f 6e0a 2020   Intersection.  
│ │ │ │ -00017880: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f  * *note makeHomo
│ │ │ │ -00017890: 746f 7069 6573 3a20 6d61 6b65 486f 6d6f  topies: makeHomo
│ │ │ │ -000178a0: 746f 7069 6573 2c20 2d2d 2072 6574 7572  topies, -- retur
│ │ │ │ -000178b0: 6e73 2061 2073 7973 7465 6d20 6f66 2068  ns a system of h
│ │ │ │ -000178c0: 6967 6865 720a 2020 2020 686f 6d6f 746f  igher.    homoto
│ │ │ │ -000178d0: 7069 6573 0a0a 5761 7973 2074 6f20 7573  pies..Ways to us
│ │ │ │ -000178e0: 6520 4569 7365 6e62 7564 5368 616d 6173  e EisenbudShamas
│ │ │ │ -000178f0: 6854 6f74 616c 3a0a 3d3d 3d3d 3d3d 3d3d  hTotal:.========
│ │ │ │ -00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00017910: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -00017920: 4569 7365 6e62 7564 5368 616d 6173 6854  EisenbudShamashT
│ │ │ │ -00017930: 6f74 616c 284d 6f64 756c 6529 220a 0a46  otal(Module)"..F
│ │ │ │ -00017940: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00017950: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00017960: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00017970: 202a 6e6f 7465 2045 6973 656e 6275 6453   *note EisenbudS
│ │ │ │ -00017980: 6861 6d61 7368 546f 7461 6c3a 2045 6973  hamashTotal: Eis
│ │ │ │ -00017990: 656e 6275 6453 6861 6d61 7368 546f 7461  enbudShamashTota
│ │ │ │ -000179a0: 6c2c 2069 7320 6120 2a6e 6f74 6520 6d65  l, is a *note me
│ │ │ │ -000179b0: 7468 6f64 0a66 756e 6374 696f 6e20 7769  thod.function wi
│ │ │ │ -000179c0: 7468 206f 7074 696f 6e73 3a20 284d 6163  th options: (Mac
│ │ │ │ -000179d0: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -000179e0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469  FunctionWithOpti
│ │ │ │ -000179f0: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ons,...---------
│ │ │ │ +000177b0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ +000177c0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +000177d0: 202a 6e6f 7465 2045 7874 3a20 284d 6163   *note Ext: (Mac
│ │ │ │ +000177e0: 6175 6c61 7932 446f 6329 4578 742c 202d  aulay2Doc)Ext, -
│ │ │ │ +000177f0: 2d20 636f 6d70 7574 6520 616e 2045 7874  - compute an Ext
│ │ │ │ +00017800: 206d 6f64 756c 650a 2020 2a20 2a6e 6f74   module.  * *not
│ │ │ │ +00017810: 6520 6e65 7745 7874 3a20 6e65 7745 7874  e newExt: newExt
│ │ │ │ +00017820: 2c20 2d2d 2047 6c6f 6261 6c20 4578 7420  , -- Global Ext 
│ │ │ │ +00017830: 666f 7220 6d6f 6475 6c65 7320 6f76 6572  for modules over
│ │ │ │ +00017840: 2061 2063 6f6d 706c 6574 650a 2020 2020   a complete.    
│ │ │ │ +00017850: 496e 7465 7273 6563 7469 6f6e 0a20 202a  Intersection.  *
│ │ │ │ +00017860: 202a 6e6f 7465 206d 616b 6548 6f6d 6f74   *note makeHomot
│ │ │ │ +00017870: 6f70 6965 733a 206d 616b 6548 6f6d 6f74  opies: makeHomot
│ │ │ │ +00017880: 6f70 6965 732c 202d 2d20 7265 7475 726e  opies, -- return
│ │ │ │ +00017890: 7320 6120 7379 7374 656d 206f 6620 6869  s a system of hi
│ │ │ │ +000178a0: 6768 6572 0a20 2020 2068 6f6d 6f74 6f70  gher.    homotop
│ │ │ │ +000178b0: 6965 730a 0a57 6179 7320 746f 2075 7365  ies..Ways to use
│ │ │ │ +000178c0: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ +000178d0: 546f 7461 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d  Total:.=========
│ │ │ │ +000178e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000178f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245  ========..  * "E
│ │ │ │ +00017900: 6973 656e 6275 6453 6861 6d61 7368 546f  isenbudShamashTo
│ │ │ │ +00017910: 7461 6c28 4d6f 6475 6c65 2922 0a0a 466f  tal(Module)"..Fo
│ │ │ │ +00017920: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00017930: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00017940: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00017950: 2a6e 6f74 6520 4569 7365 6e62 7564 5368  *note EisenbudSh
│ │ │ │ +00017960: 616d 6173 6854 6f74 616c 3a20 4569 7365  amashTotal: Eise
│ │ │ │ +00017970: 6e62 7564 5368 616d 6173 6854 6f74 616c  nbudShamashTotal
│ │ │ │ +00017980: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +00017990: 686f 640a 6675 6e63 7469 6f6e 2077 6974  hod.function wit
│ │ │ │ +000179a0: 6820 6f70 7469 6f6e 733a 2028 4d61 6361  h options: (Maca
│ │ │ │ +000179b0: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +000179c0: 756e 6374 696f 6e57 6974 684f 7074 696f  unctionWithOptio
│ │ │ │ +000179d0: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  ns,...----------
│ │ │ │ +000179e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000179f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017a40: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -00017a50: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -00017a60: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -00017a70: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00017a80: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -00017a90: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ -00017aa0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ -00017ab0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -00017ac0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ -00017ad0: 323a 3531 3639 3a30 2e0a 1f0a 4669 6c65  2:5169:0....File
│ │ │ │ -00017ae0: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ -00017af0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -00017b00: 732e 696e 666f 2c20 4e6f 6465 3a20 6576  s.info, Node: ev
│ │ │ │ -00017b10: 656e 4578 744d 6f64 756c 652c 204e 6578  enExtModule, Nex
│ │ │ │ -00017b20: 743a 2065 7870 6f2c 2050 7265 763a 2045  t: expo, Prev: E
│ │ │ │ -00017b30: 6973 656e 6275 6453 6861 6d61 7368 546f  isenbudShamashTo
│ │ │ │ -00017b40: 7461 6c2c 2055 703a 2054 6f70 0a0a 6576  tal, Up: Top..ev
│ │ │ │ -00017b50: 656e 4578 744d 6f64 756c 6520 2d2d 2065  enExtModule -- e
│ │ │ │ -00017b60: 7665 6e20 7061 7274 206f 6620 4578 745e  ven part of Ext^
│ │ │ │ -00017b70: 2a28 4d2c 6b29 206f 7665 7220 6120 636f  *(M,k) over a co
│ │ │ │ -00017b80: 6d70 6c65 7465 2069 6e74 6572 7365 6374  mplete intersect
│ │ │ │ -00017b90: 696f 6e20 6173 206d 6f64 756c 6520 6f76  ion as module ov
│ │ │ │ -00017ba0: 6572 2043 4920 6f70 6572 6174 6f72 2072  er CI operator r
│ │ │ │ -00017bb0: 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ing.************
│ │ │ │ +00017a20: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +00017a30: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +00017a40: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +00017a50: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +00017a60: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ +00017a70: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ +00017a80: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ +00017a90: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +00017aa0: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
│ │ │ │ +00017ab0: 3a35 3136 393a 302e 0a1f 0a46 696c 653a  :5169:0....File:
│ │ │ │ +00017ac0: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +00017ad0: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +00017ae0: 2e69 6e66 6f2c 204e 6f64 653a 2065 7665  .info, Node: eve
│ │ │ │ +00017af0: 6e45 7874 4d6f 6475 6c65 2c20 4e65 7874  nExtModule, Next
│ │ │ │ +00017b00: 3a20 6578 706f 2c20 5072 6576 3a20 4569  : expo, Prev: Ei
│ │ │ │ +00017b10: 7365 6e62 7564 5368 616d 6173 6854 6f74  senbudShamashTot
│ │ │ │ +00017b20: 616c 2c20 5570 3a20 546f 700a 0a65 7665  al, Up: Top..eve
│ │ │ │ +00017b30: 6e45 7874 4d6f 6475 6c65 202d 2d20 6576  nExtModule -- ev
│ │ │ │ +00017b40: 656e 2070 6172 7420 6f66 2045 7874 5e2a  en part of Ext^*
│ │ │ │ +00017b50: 284d 2c6b 2920 6f76 6572 2061 2063 6f6d  (M,k) over a com
│ │ │ │ +00017b60: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ +00017b70: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ +00017b80: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ +00017b90: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ng.*************
│ │ │ │ +00017ba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00017bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00017bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00017bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00017be0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00017bf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00017c10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -00017c20: 7361 6765 3a20 0a20 2020 2020 2020 2045  sage: .        E
│ │ │ │ -00017c30: 203d 2065 7665 6e45 7874 4d6f 6475 6c65   = evenExtModule
│ │ │ │ -00017c40: 204d 0a20 202a 2049 6e70 7574 733a 0a20   M.  * Inputs:. 
│ │ │ │ -00017c50: 2020 2020 202a 204d 2c20 6120 2a6e 6f74       * M, a *not
│ │ │ │ -00017c60: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ -00017c70: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ -00017c80: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ -00017c90: 2069 6e74 6572 7365 6374 696f 6e0a 2020   intersection.  
│ │ │ │ -00017ca0: 2020 2020 2020 7269 6e67 0a20 202a 202a        ring.  * *
│ │ │ │ -00017cb0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e  note Optional in
│ │ │ │ -00017cc0: 7075 7473 3a20 284d 6163 6175 6c61 7932  puts: (Macaulay2
│ │ │ │ -00017cd0: 446f 6329 7573 696e 6720 6675 6e63 7469  Doc)using functi
│ │ │ │ -00017ce0: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ -00017cf0: 6c20 696e 7075 7473 2c3a 0a20 2020 2020  l inputs,:.     
│ │ │ │ -00017d00: 202a 204f 7574 5269 6e67 203d 3e20 2e2e   * OutRing => ..
│ │ │ │ -00017d10: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00017d20: 2030 0a20 202a 204f 7574 7075 7473 3a0a   0.  * Outputs:.
│ │ │ │ -00017d30: 2020 2020 2020 2a20 452c 2061 202a 6e6f        * E, a *no
│ │ │ │ -00017d40: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ -00017d50: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ -00017d60: 2c20 6f76 6572 2061 2070 6f6c 796e 6f6d  , over a polynom
│ │ │ │ -00017d70: 6961 6c20 7269 6e67 2077 6974 680a 2020  ial ring with.  
│ │ │ │ -00017d80: 2020 2020 2020 6765 6e73 2069 6e20 6465        gens in de
│ │ │ │ -00017d90: 6772 6565 2031 0a0a 4465 7363 7269 7074  gree 1..Descript
│ │ │ │ -00017da0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -00017db0: 0a45 7874 7261 6374 7320 7468 6520 6576  .Extracts the ev
│ │ │ │ -00017dc0: 656e 2064 6567 7265 6520 7061 7274 2066  en degree part f
│ │ │ │ -00017dd0: 726f 6d20 4578 744d 6f64 756c 6520 4d20  rom ExtModule M 
│ │ │ │ -00017de0: 4966 2074 6865 206f 7074 696f 6e61 6c20  If the optional 
│ │ │ │ -00017df0: 6172 6775 6d65 6e74 204f 7574 5269 6e67  argument OutRing
│ │ │ │ -00017e00: 0a3d 3e20 5420 6973 2067 6976 656e 2c20  .=> T is given, 
│ │ │ │ -00017e10: 616e 6420 636c 6173 7320 5420 3d3d 3d20  and class T === 
│ │ │ │ -00017e20: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20  PolynomialRing, 
│ │ │ │ -00017e30: 7468 656e 2074 6865 206f 7574 7075 7420  then the output 
│ │ │ │ -00017e40: 7769 6c6c 2062 6520 6120 6d6f 6475 6c65  will be a module
│ │ │ │ -00017e50: 0a6f 7665 7220 542e 0a0a 2b2d 2d2d 2d2d  .over T...+-----
│ │ │ │ +00017bf0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +00017c00: 6167 653a 200a 2020 2020 2020 2020 4520  age: .        E 
│ │ │ │ +00017c10: 3d20 6576 656e 4578 744d 6f64 756c 6520  = evenExtModule 
│ │ │ │ +00017c20: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020  M.  * Inputs:.  
│ │ │ │ +00017c30: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ +00017c40: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +00017c50: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +00017c60: 6f76 6572 2061 2063 6f6d 706c 6574 6520  over a complete 
│ │ │ │ +00017c70: 696e 7465 7273 6563 7469 6f6e 0a20 2020  intersection.   
│ │ │ │ +00017c80: 2020 2020 2072 696e 670a 2020 2a20 2a6e       ring.  * *n
│ │ │ │ +00017c90: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
│ │ │ │ +00017ca0: 7574 733a 2028 4d61 6361 756c 6179 3244  uts: (Macaulay2D
│ │ │ │ +00017cb0: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
│ │ │ │ +00017cc0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ +00017cd0: 2069 6e70 7574 732c 3a0a 2020 2020 2020   inputs,:.      
│ │ │ │ +00017ce0: 2a20 4f75 7452 696e 6720 3d3e 202e 2e2e  * OutRing => ...
│ │ │ │ +00017cf0: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ +00017d00: 300a 2020 2a20 4f75 7470 7574 733a 0a20  0.  * Outputs:. 
│ │ │ │ +00017d10: 2020 2020 202a 2045 2c20 6120 2a6e 6f74       * E, a *not
│ │ │ │ +00017d20: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +00017d30: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +00017d40: 206f 7665 7220 6120 706f 6c79 6e6f 6d69   over a polynomi
│ │ │ │ +00017d50: 616c 2072 696e 6720 7769 7468 0a20 2020  al ring with.   
│ │ │ │ +00017d60: 2020 2020 2067 656e 7320 696e 2064 6567       gens in deg
│ │ │ │ +00017d70: 7265 6520 310a 0a44 6573 6372 6970 7469  ree 1..Descripti
│ │ │ │ +00017d80: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +00017d90: 4578 7472 6163 7473 2074 6865 2065 7665  Extracts the eve
│ │ │ │ +00017da0: 6e20 6465 6772 6565 2070 6172 7420 6672  n degree part fr
│ │ │ │ +00017db0: 6f6d 2045 7874 4d6f 6475 6c65 204d 2049  om ExtModule M I
│ │ │ │ +00017dc0: 6620 7468 6520 6f70 7469 6f6e 616c 2061  f the optional a
│ │ │ │ +00017dd0: 7267 756d 656e 7420 4f75 7452 696e 670a  rgument OutRing.
│ │ │ │ +00017de0: 3d3e 2054 2069 7320 6769 7665 6e2c 2061  => T is given, a
│ │ │ │ +00017df0: 6e64 2063 6c61 7373 2054 203d 3d3d 2050  nd class T === P
│ │ │ │ +00017e00: 6f6c 796e 6f6d 6961 6c52 696e 672c 2074  olynomialRing, t
│ │ │ │ +00017e10: 6865 6e20 7468 6520 6f75 7470 7574 2077  hen the output w
│ │ │ │ +00017e20: 696c 6c20 6265 2061 206d 6f64 756c 650a  ill be a module.
│ │ │ │ +00017e30: 6f76 6572 2054 2e0a 0a2b 2d2d 2d2d 2d2d  over T...+------
│ │ │ │ +00017e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017e90: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20  ----+.|i1 : kk= 
│ │ │ │ -00017ea0: 5a5a 2f31 3031 2020 2020 2020 2020 2020  ZZ/101          
│ │ │ │ -00017eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017e70: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a  ---+.|i1 : kk= Z
│ │ │ │ +00017e80: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │ +00017e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00017eb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00017f10: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ -00017f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017ee0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00017ef0: 203d 206b 6b20 2020 2020 2020 2020 2020   = kk           
│ │ │ │ +00017f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00017f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00017f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f80: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74      |.|o1 : Quot
│ │ │ │ -00017f90: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ -00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00017fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00018000: 3220 3a20 5320 3d20 6b6b 5b78 2c79 2c7a  2 : S = kk[x,y,z
│ │ │ │ -00018010: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ +00017f60: 2020 207c 0a7c 6f31 203a 2051 756f 7469     |.|o1 : Quoti
│ │ │ │ +00017f70: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +00017f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00017fa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00017fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00017fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00017fe0: 203a 2053 203d 206b 6b5b 782c 792c 7a5d   : S = kk[x,y,z]
│ │ │ │ +00017ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018010: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018030: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018050: 2020 207c 0a7c 6f32 203d 2053 2020 2020     |.|o2 = S    
│ │ │ │  00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018070: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020      |.|o2 = S   
│ │ │ │ -00018080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000180a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000180c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000180f0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ -00018100: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -00018110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018120: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000180c0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +000180d0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +000180e0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +000180f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018100: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00018110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018160: 2d2d 2d2d 2b0a 7c69 3320 3a20 4932 203d  ----+.|i3 : I2 =
│ │ │ │ -00018170: 2069 6465 616c 2278 332c 797a 2220 2020   ideal"x3,yz"   
│ │ │ │ -00018180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018140: 2d2d 2d2b 0a7c 6933 203a 2049 3220 3d20  ---+.|i3 : I2 = 
│ │ │ │ +00018150: 6964 6561 6c22 7833 2c79 7a22 2020 2020  ideal"x3,yz"    
│ │ │ │ +00018160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00018190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000181a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000181b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000181c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000181d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000181e0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -000181f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018210: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00018220: 6964 6561 6c20 2878 202c 2079 2a7a 2920  ideal (x , y*z) 
│ │ │ │ -00018230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000181a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000181b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000181c0: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ +000181d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000181e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000181f0: 2020 2020 2020 207c 0a7c 6f33 203d 2069         |.|o3 = i
│ │ │ │ +00018200: 6465 616c 2028 7820 2c20 792a 7a29 2020  deal (x , y*z)  
│ │ │ │ +00018210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018230: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00018240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018250: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018290: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66  |.|o3 : Ideal of
│ │ │ │ -000182a0: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │ -000182b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000182c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00018250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018270: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ +00018280: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00018290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000182b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000182c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000182d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018300: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00018310: 5232 203d 2053 2f49 3220 2020 2020 2020  R2 = S/I2       
│ │ │ │ -00018320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000182e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ +000182f0: 3220 3d20 532f 4932 2020 2020 2020 2020  2 = S/I2        
│ │ │ │ +00018300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018320: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00018330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018340: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018360: 0a7c 6f34 203d 2052 3220 2020 2020 2020  .|o4 = R2       
│ │ │ │  00018370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018380: 7c0a 7c6f 3420 3d20 5232 2020 2020 2020  |.|o4 = R2      
│ │ │ │ -00018390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018390: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000183a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000183b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183f0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -00018400: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ -00018410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018430: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00018440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018470: 2b0a 7c69 3520 3a20 4d32 203d 2052 325e  +.|i5 : M2 = R2^
│ │ │ │ -00018480: 312f 6964 6561 6c22 7832 2c79 2c7a 2220  1/ideal"x2,y,z" 
│ │ │ │ +000183d0: 2020 2020 2020 207c 0a7c 6f34 203a 2051         |.|o4 : Q
│ │ │ │ +000183e0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +000183f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018410: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00018420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00018450: 0a7c 6935 203a 204d 3220 3d20 5232 5e31  .|i5 : M2 = R2^1
│ │ │ │ +00018460: 2f69 6465 616c 2278 322c 792c 7a22 2020  /ideal"x2,y,z"  
│ │ │ │ +00018470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00018490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000184a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000184b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184e0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -000184f0: 636f 6b65 726e 656c 207c 2078 3220 7920  cokernel | x2 y 
│ │ │ │ -00018500: 7a20 7c20 2020 2020 2020 2020 2020 2020  z |             
│ │ │ │ +000184c0: 2020 2020 2020 207c 0a7c 6f35 203d 2063         |.|o5 = c
│ │ │ │ +000184d0: 6f6b 6572 6e65 6c20 7c20 7832 2079 207a  okernel | x2 y z
│ │ │ │ +000184e0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +000184f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018500: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00018510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018520: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018540: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00018550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018560: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018580: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000185a0: 3520 3a20 5232 2d6d 6f64 756c 652c 2071  5 : R2-module, q
│ │ │ │ -000185b0: 756f 7469 656e 7420 6f66 2052 3220 2020  uotient of R2   
│ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00018560: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00018570: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00018580: 203a 2052 322d 6d6f 6475 6c65 2c20 7175   : R2-module, qu
│ │ │ │ +00018590: 6f74 6965 6e74 206f 6620 5232 2020 2020  otient of R2    
│ │ │ │ +000185a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000185b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000185c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000185d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018610: 2d2d 2d2d 2b0a 7c69 3620 3a20 6265 7474  ----+.|i6 : bett
│ │ │ │ -00018620: 6920 6672 6565 5265 736f 6c75 7469 6f6e  i freeResolution
│ │ │ │ -00018630: 2028 4d32 2c20 4c65 6e67 7468 4c69 6d69   (M2, LengthLimi
│ │ │ │ -00018640: 7420 3d3e 3130 2920 2020 2020 2020 2020  t =>10)         
│ │ │ │ -00018650: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018690: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ -000186a0: 2033 2034 2020 3520 2036 2020 3720 2038   3 4  5  6  7  8
│ │ │ │ -000186b0: 2020 3920 3130 2020 2020 2020 2020 2020    9 10          
│ │ │ │ -000186c0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -000186d0: 746f 7461 6c3a 2031 2033 2035 2037 2039  total: 1 3 5 7 9
│ │ │ │ -000186e0: 2031 3120 3133 2031 3520 3137 2031 3920   11 13 15 17 19 
│ │ │ │ -000186f0: 3231 2020 2020 2020 2020 2020 2020 2020  21              
│ │ │ │ -00018700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018710: 303a 2031 2032 2032 2032 2032 2020 3220  0: 1 2 2 2 2  2 
│ │ │ │ -00018720: 2032 2020 3220 2032 2020 3220 2032 2020   2  2  2  2  2  
│ │ │ │ -00018730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018740: 7c0a 7c20 2020 2020 2020 2020 313a 202e  |.|         1: .
│ │ │ │ -00018750: 2031 2033 2034 2034 2020 3420 2034 2020   1 3 4 4  4  4  
│ │ │ │ -00018760: 3420 2034 2020 3420 2034 2020 2020 2020  4  4  4  4      
│ │ │ │ -00018770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00018780: 2020 2020 2020 2020 323a 202e 202e 202e          2: . . .
│ │ │ │ -00018790: 2031 2033 2020 3420 2034 2020 3420 2034   1 3  4  4  4  4
│ │ │ │ -000187a0: 2020 3420 2034 2020 2020 2020 2020 2020    4  4          
│ │ │ │ -000187b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000187c0: 2020 2020 333a 202e 202e 202e 202e 202e      3: . . . . .
│ │ │ │ -000187d0: 2020 3120 2033 2020 3420 2034 2020 3420    1  3  4  4  4 
│ │ │ │ -000187e0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -000187f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018800: 343a 202e 202e 202e 202e 202e 2020 2e20  4: . . . . .  . 
│ │ │ │ -00018810: 202e 2020 3120 2033 2020 3420 2034 2020   .  1  3  4  4  
│ │ │ │ -00018820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018830: 7c0a 7c20 2020 2020 2020 2020 353a 202e  |.|         5: .
│ │ │ │ -00018840: 202e 202e 202e 202e 2020 2e20 202e 2020   . . . .  .  .  
│ │ │ │ -00018850: 2e20 202e 2020 3120 2033 2020 2020 2020  .  .  1  3      
│ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000185f0: 2d2d 2d2b 0a7c 6936 203a 2062 6574 7469  ---+.|i6 : betti
│ │ │ │ +00018600: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ +00018610: 284d 322c 204c 656e 6774 684c 696d 6974  (M2, LengthLimit
│ │ │ │ +00018620: 203d 3e31 3029 2020 2020 2020 2020 207c   =>10)         |
│ │ │ │ +00018630: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00018640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018660: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00018670: 2020 2020 2020 2020 2020 3020 3120 3220            0 1 2 
│ │ │ │ +00018680: 3320 3420 2035 2020 3620 2037 2020 3820  3 4  5  6  7  8 
│ │ │ │ +00018690: 2039 2031 3020 2020 2020 2020 2020 2020   9 10           
│ │ │ │ +000186a0: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ +000186b0: 6f74 616c 3a20 3120 3320 3520 3720 3920  otal: 1 3 5 7 9 
│ │ │ │ +000186c0: 3131 2031 3320 3135 2031 3720 3139 2032  11 13 15 17 19 2
│ │ │ │ +000186d0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000186e0: 2020 207c 0a7c 2020 2020 2020 2020 2030     |.|         0
│ │ │ │ +000186f0: 3a20 3120 3220 3220 3220 3220 2032 2020  : 1 2 2 2 2  2  
│ │ │ │ +00018700: 3220 2032 2020 3220 2032 2020 3220 2020  2  2  2  2  2   
│ │ │ │ +00018710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018720: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ +00018730: 3120 3320 3420 3420 2034 2020 3420 2034  1 3 4 4  4  4  4
│ │ │ │ +00018740: 2020 3420 2034 2020 3420 2020 2020 2020    4  4  4       
│ │ │ │ +00018750: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00018760: 2020 2020 2020 2032 3a20 2e20 2e20 2e20         2: . . . 
│ │ │ │ +00018770: 3120 3320 2034 2020 3420 2034 2020 3420  1 3  4  4  4  4 
│ │ │ │ +00018780: 2034 2020 3420 2020 2020 2020 2020 2020   4  4           
│ │ │ │ +00018790: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000187a0: 2020 2033 3a20 2e20 2e20 2e20 2e20 2e20     3: . . . . . 
│ │ │ │ +000187b0: 2031 2020 3320 2034 2020 3420 2034 2020   1  3  4  4  4  
│ │ │ │ +000187c0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +000187d0: 2020 207c 0a7c 2020 2020 2020 2020 2034     |.|         4
│ │ │ │ +000187e0: 3a20 2e20 2e20 2e20 2e20 2e20 202e 2020  : . . . . .  .  
│ │ │ │ +000187f0: 2e20 2031 2020 3320 2034 2020 3420 2020  .  1  3  4  4   
│ │ │ │ +00018800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018810: 0a7c 2020 2020 2020 2020 2035 3a20 2e20  .|         5: . 
│ │ │ │ +00018820: 2e20 2e20 2e20 2e20 202e 2020 2e20 202e  . . . .  .  .  .
│ │ │ │ +00018830: 2020 2e20 2031 2020 3320 2020 2020 2020    .  1  3       
│ │ │ │ +00018840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00018850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188a0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -000188b0: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ -000188c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000188e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000188f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018920: 2b0a 7c69 3720 3a20 4520 3d20 4578 744d  +.|i7 : E = ExtM
│ │ │ │ -00018930: 6f64 756c 6520 4d32 2020 2020 2020 2020  odule M2        
│ │ │ │ +00018880: 2020 2020 2020 207c 0a7c 6f36 203a 2042         |.|o6 : B
│ │ │ │ +00018890: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +000188a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000188b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000188c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000188d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000188e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000188f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00018900: 0a7c 6937 203a 2045 203d 2045 7874 4d6f  .|i7 : E = ExtMo
│ │ │ │ +00018910: 6475 6c65 204d 3220 2020 2020 2020 2020  dule M2         
│ │ │ │ +00018920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00018940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018950: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018990: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000189a0: 2020 2020 2020 2020 2020 2020 3820 2020              8   
│ │ │ │ -000189b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000189c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000189d0: 2020 2020 7c0a 7c6f 3720 3d20 286b 6b5b      |.|o7 = (kk[
│ │ │ │ -000189e0: 5820 2e2e 5820 5d29 2020 2020 2020 2020  X ..X ])        
│ │ │ │ -000189f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a10: 7c0a 7c20 2020 2020 2020 2020 2030 2020  |.|          0  
│ │ │ │ -00018a20: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00018970: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00018980: 2020 2020 2020 2020 2020 2038 2020 2020             8    
│ │ │ │ +00018990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000189a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000189b0: 2020 207c 0a7c 6f37 203d 2028 6b6b 5b58     |.|o7 = (kk[X
│ │ │ │ +000189c0: 202e 2e58 205d 2920 2020 2020 2020 2020   ..X ])         
│ │ │ │ +000189d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000189e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000189f0: 0a7c 2020 2020 2020 2020 2020 3020 2020  .|          0   
│ │ │ │ +00018a00: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00018a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018a20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00018a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018a80: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -00018a90: 6b6b 5b58 202e 2e58 205d 2d6d 6f64 756c  kk[X ..X ]-modul
│ │ │ │ -00018aa0: 652c 2066 7265 652c 2064 6567 7265 6573  e, free, degrees
│ │ │ │ -00018ab0: 207b 302e 2e31 2c20 323a 312c 2033 3a32   {0..1, 2:1, 3:2
│ │ │ │ -00018ac0: 2c20 337d 7c0a 7c20 2020 2020 2020 2020  , 3}|.|         
│ │ │ │ -00018ad0: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ -00018ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018b00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00018b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00018b40: 3820 3a20 6170 706c 7928 746f 4c69 7374  8 : apply(toList
│ │ │ │ -00018b50: 2830 2e2e 3130 292c 2069 2d3e 6869 6c62  (0..10), i->hilb
│ │ │ │ -00018b60: 6572 7446 756e 6374 696f 6e28 692c 2045  ertFunction(i, E
│ │ │ │ -00018b70: 2929 2020 2020 2020 7c0a 7c20 2020 2020  ))      |.|     
│ │ │ │ +00018a60: 2020 2020 2020 207c 0a7c 6f37 203a 206b         |.|o7 : k
│ │ │ │ +00018a70: 6b5b 5820 2e2e 5820 5d2d 6d6f 6475 6c65  k[X ..X ]-module
│ │ │ │ +00018a80: 2c20 6672 6565 2c20 6465 6772 6565 7320  , free, degrees 
│ │ │ │ +00018a90: 7b30 2e2e 312c 2032 3a31 2c20 333a 322c  {0..1, 2:1, 3:2,
│ │ │ │ +00018aa0: 2033 7d7c 0a7c 2020 2020 2020 2020 2030   3}|.|         0
│ │ │ │ +00018ab0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00018ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018ae0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00018af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ +00018b20: 203a 2061 7070 6c79 2874 6f4c 6973 7428   : apply(toList(
│ │ │ │ +00018b30: 302e 2e31 3029 2c20 692d 3e68 696c 6265  0..10), i->hilbe
│ │ │ │ +00018b40: 7274 4675 6e63 7469 6f6e 2869 2c20 4529  rtFunction(i, E)
│ │ │ │ +00018b50: 2920 2020 2020 207c 0a7c 2020 2020 2020  )      |.|      
│ │ │ │ +00018b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018bb0: 2020 2020 7c0a 7c6f 3820 3d20 7b31 2c20      |.|o8 = {1, 
│ │ │ │ -00018bc0: 332c 2035 2c20 372c 2039 2c20 3131 2c20  3, 5, 7, 9, 11, 
│ │ │ │ -00018bd0: 3133 2c20 3135 2c20 3137 2c20 3139 2c20  13, 15, 17, 19, 
│ │ │ │ -00018be0: 3231 7d20 2020 2020 2020 2020 2020 2020  21}             
│ │ │ │ -00018bf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00018c30: 3820 3a20 4c69 7374 2020 2020 2020 2020  8 : List        
│ │ │ │ -00018c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00018b90: 2020 207c 0a7c 6f38 203d 207b 312c 2033     |.|o8 = {1, 3
│ │ │ │ +00018ba0: 2c20 352c 2037 2c20 392c 2031 312c 2031  , 5, 7, 9, 11, 1
│ │ │ │ +00018bb0: 332c 2031 352c 2031 372c 2031 392c 2032  3, 15, 17, 19, 2
│ │ │ │ +00018bc0: 317d 2020 2020 2020 2020 2020 2020 207c  1}             |
│ │ │ │ +00018bd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00018be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018c00: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +00018c10: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00018c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018c40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00018c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018ca0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4565 7665  ----+.|i9 : Eeve
│ │ │ │ -00018cb0: 6e20 3d20 6576 656e 4578 744d 6f64 756c  n = evenExtModul
│ │ │ │ -00018cc0: 6520 4d32 2020 2020 2020 2020 2020 2020  e M2            
│ │ │ │ +00018c80: 2d2d 2d2b 0a7c 6939 203a 2045 6576 656e  ---+.|i9 : Eeven
│ │ │ │ +00018c90: 203d 2065 7665 6e45 7874 4d6f 6475 6c65   = evenExtModule
│ │ │ │ +00018ca0: 204d 3220 2020 2020 2020 2020 2020 2020   M2             
│ │ │ │ +00018cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00018cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00018ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018cf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00018d00: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ +00018d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d30: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00018d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d50: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -00018d60: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020  (kk[X ..X ])    
│ │ │ │ -00018d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018da0: 2030 2020 2031 2020 2020 2020 2020 2020   0   1          
│ │ │ │ -00018db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018d30: 2020 2020 2020 207c 0a7c 6f39 203d 2028         |.|o9 = (
│ │ │ │ +00018d40: 6b6b 5b58 202e 2e58 205d 2920 2020 2020  kk[X ..X ])     
│ │ │ │ +00018d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018d70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00018d80: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ +00018d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018db0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00018dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00018de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00018e10: 3920 3a20 6b6b 5b58 202e 2e58 205d 2d6d  9 : kk[X ..X ]-m
│ │ │ │ -00018e20: 6f64 756c 652c 2066 7265 652c 2064 6567  odule, free, deg
│ │ │ │ -00018e30: 7265 6573 207b 302e 2e31 2c20 323a 317d  rees {0..1, 2:1}
│ │ │ │ -00018e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00018e50: 2020 2020 3020 2020 3120 2020 2020 2020      0   1       
│ │ │ │ -00018e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018e80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018ec0: 2b0a 7c69 3130 203a 2061 7070 6c79 2874  +.|i10 : apply(t
│ │ │ │ -00018ed0: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e  oList(0..5), i->
│ │ │ │ -00018ee0: 6869 6c62 6572 7446 756e 6374 696f 6e28  hilbertFunction(
│ │ │ │ -00018ef0: 692c 2045 6576 656e 2929 2020 7c0a 7c20  i, Eeven))  |.| 
│ │ │ │ +00018dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018de0: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ +00018df0: 203a 206b 6b5b 5820 2e2e 5820 5d2d 6d6f   : kk[X ..X ]-mo
│ │ │ │ +00018e00: 6475 6c65 2c20 6672 6565 2c20 6465 6772  dule, free, degr
│ │ │ │ +00018e10: 6565 7320 7b30 2e2e 312c 2032 3a31 7d20  ees {0..1, 2:1} 
│ │ │ │ +00018e20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00018e30: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ +00018e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018e60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00018e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00018ea0: 0a7c 6931 3020 3a20 6170 706c 7928 746f  .|i10 : apply(to
│ │ │ │ +00018eb0: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68  List(0..5), i->h
│ │ │ │ +00018ec0: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ +00018ed0: 2c20 4565 7665 6e29 2920 207c 0a7c 2020  , Eeven))  |.|  
│ │ │ │ +00018ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f30: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ -00018f40: 207b 312c 2035 2c20 392c 2031 332c 2031   {1, 5, 9, 13, 1
│ │ │ │ -00018f50: 372c 2032 317d 2020 2020 2020 2020 2020  7, 21}          
│ │ │ │ +00018f10: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ +00018f20: 7b31 2c20 352c 2039 2c20 3133 2c20 3137  {1, 5, 9, 13, 17
│ │ │ │ +00018f30: 2c20 3231 7d20 2020 2020 2020 2020 2020  , 21}           
│ │ │ │ +00018f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018f50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00018f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00018f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00018f90: 0a7c 6f31 3020 3a20 4c69 7374 2020 2020  .|o10 : List    
│ │ │ │  00018fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018fb0: 7c0a 7c6f 3130 203a 204c 6973 7420 2020  |.|o10 : List   
│ │ │ │ -00018fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018fe0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00018fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018fc0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00018fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00018fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019020: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -00019030: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -00019040: 2a20 2a6e 6f74 6520 4578 744d 6f64 756c  * *note ExtModul
│ │ │ │ -00019050: 653a 2045 7874 4d6f 6475 6c65 2c20 2d2d  e: ExtModule, --
│ │ │ │ -00019060: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ -00019070: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ -00019080: 7273 6563 7469 6f6e 2061 730a 2020 2020  rsection as.    
│ │ │ │ -00019090: 6d6f 6475 6c65 206f 7665 7220 4349 206f  module over CI o
│ │ │ │ -000190a0: 7065 7261 746f 7220 7269 6e67 0a20 202a  perator ring.  *
│ │ │ │ -000190b0: 202a 6e6f 7465 206f 6464 4578 744d 6f64   *note oddExtMod
│ │ │ │ -000190c0: 756c 653a 206f 6464 4578 744d 6f64 756c  ule: oddExtModul
│ │ │ │ -000190d0: 652c 202d 2d20 6f64 6420 7061 7274 206f  e, -- odd part o
│ │ │ │ -000190e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ -000190f0: 7220 6120 636f 6d70 6c65 7465 0a20 2020  r a complete.   
│ │ │ │ -00019100: 2069 6e74 6572 7365 6374 696f 6e20 6173   intersection as
│ │ │ │ -00019110: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ -00019120: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ -00019130: 2a20 2a6e 6f74 6520 4f75 7452 696e 673a  * *note OutRing:
│ │ │ │ -00019140: 204f 7574 5269 6e67 2c20 2d2d 204f 7074   OutRing, -- Opt
│ │ │ │ -00019150: 696f 6e20 616c 6c6f 7769 6e67 2073 7065  ion allowing spe
│ │ │ │ -00019160: 6369 6669 6361 7469 6f6e 206f 6620 7468  cification of th
│ │ │ │ -00019170: 6520 7269 6e67 206f 7665 720a 2020 2020  e ring over.    
│ │ │ │ -00019180: 7768 6963 6820 7468 6520 6f75 7470 7574  which the output
│ │ │ │ -00019190: 2069 7320 6465 6669 6e65 640a 0a57 6179   is defined..Way
│ │ │ │ -000191a0: 7320 746f 2075 7365 2065 7665 6e45 7874  s to use evenExt
│ │ │ │ -000191b0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d  Module:.========
│ │ │ │ -000191c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000191d0: 3d3d 0a0a 2020 2a20 2265 7665 6e45 7874  ==..  * "evenExt
│ │ │ │ -000191e0: 4d6f 6475 6c65 284d 6f64 756c 6529 220a  Module(Module)".
│ │ │ │ -000191f0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ -00019200: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ -00019210: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ -00019220: 6374 202a 6e6f 7465 2065 7665 6e45 7874  ct *note evenExt
│ │ │ │ -00019230: 4d6f 6475 6c65 3a20 6576 656e 4578 744d  Module: evenExtM
│ │ │ │ -00019240: 6f64 756c 652c 2069 7320 6120 2a6e 6f74  odule, is a *not
│ │ │ │ -00019250: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ -00019260: 6e20 7769 7468 0a6f 7074 696f 6e73 3a20  n with.options: 
│ │ │ │ -00019270: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -00019280: 7468 6f64 4675 6e63 7469 6f6e 5769 7468  thodFunctionWith
│ │ │ │ -00019290: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d  Options,...-----
│ │ │ │ +00019000: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ +00019010: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +00019020: 202a 6e6f 7465 2045 7874 4d6f 6475 6c65   *note ExtModule
│ │ │ │ +00019030: 3a20 4578 744d 6f64 756c 652c 202d 2d20  : ExtModule, -- 
│ │ │ │ +00019040: 4578 745e 2a28 4d2c 6b29 206f 7665 7220  Ext^*(M,k) over 
│ │ │ │ +00019050: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ +00019060: 7365 6374 696f 6e20 6173 0a20 2020 206d  section as.    m
│ │ │ │ +00019070: 6f64 756c 6520 6f76 6572 2043 4920 6f70  odule over CI op
│ │ │ │ +00019080: 6572 6174 6f72 2072 696e 670a 2020 2a20  erator ring.  * 
│ │ │ │ +00019090: 2a6e 6f74 6520 6f64 6445 7874 4d6f 6475  *note oddExtModu
│ │ │ │ +000190a0: 6c65 3a20 6f64 6445 7874 4d6f 6475 6c65  le: oddExtModule
│ │ │ │ +000190b0: 2c20 2d2d 206f 6464 2070 6172 7420 6f66  , -- odd part of
│ │ │ │ +000190c0: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ +000190d0: 2061 2063 6f6d 706c 6574 650a 2020 2020   a complete.    
│ │ │ │ +000190e0: 696e 7465 7273 6563 7469 6f6e 2061 7320  intersection as 
│ │ │ │ +000190f0: 6d6f 6475 6c65 206f 7665 7220 4349 206f  module over CI o
│ │ │ │ +00019100: 7065 7261 746f 7220 7269 6e67 0a20 202a  perator ring.  *
│ │ │ │ +00019110: 202a 6e6f 7465 204f 7574 5269 6e67 3a20   *note OutRing: 
│ │ │ │ +00019120: 4f75 7452 696e 672c 202d 2d20 4f70 7469  OutRing, -- Opti
│ │ │ │ +00019130: 6f6e 2061 6c6c 6f77 696e 6720 7370 6563  on allowing spec
│ │ │ │ +00019140: 6966 6963 6174 696f 6e20 6f66 2074 6865  ification of the
│ │ │ │ +00019150: 2072 696e 6720 6f76 6572 0a20 2020 2077   ring over.    w
│ │ │ │ +00019160: 6869 6368 2074 6865 206f 7574 7075 7420  hich the output 
│ │ │ │ +00019170: 6973 2064 6566 696e 6564 0a0a 5761 7973  is defined..Ways
│ │ │ │ +00019180: 2074 6f20 7573 6520 6576 656e 4578 744d   to use evenExtM
│ │ │ │ +00019190: 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d 3d3d  odule:.=========
│ │ │ │ +000191a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000191b0: 3d0a 0a20 202a 2022 6576 656e 4578 744d  =..  * "evenExtM
│ │ │ │ +000191c0: 6f64 756c 6528 4d6f 6475 6c65 2922 0a0a  odule(Module)"..
│ │ │ │ +000191d0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ +000191e0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ +000191f0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ +00019200: 7420 2a6e 6f74 6520 6576 656e 4578 744d  t *note evenExtM
│ │ │ │ +00019210: 6f64 756c 653a 2065 7665 6e45 7874 4d6f  odule: evenExtMo
│ │ │ │ +00019220: 6475 6c65 2c20 6973 2061 202a 6e6f 7465  dule, is a *note
│ │ │ │ +00019230: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +00019240: 2077 6974 680a 6f70 7469 6f6e 733a 2028   with.options: (
│ │ │ │ +00019250: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +00019260: 686f 6446 756e 6374 696f 6e57 6974 684f  hodFunctionWithO
│ │ │ │ +00019270: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d  ptions,...------
│ │ │ │ +00019280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000192a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000192b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000192c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000192d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000192e0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ -000192f0: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ -00019300: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ -00019310: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -00019320: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -00019330: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ -00019340: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -00019350: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ -00019360: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00019370: 6e73 2e6d 323a 3336 3433 3a30 2e0a 1f0a  ns.m2:3643:0....
│ │ │ │ -00019380: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ -00019390: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -000193a0: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ -000193b0: 3a20 6578 706f 2c20 4e65 7874 3a20 6578  : expo, Next: ex
│ │ │ │ -000193c0: 7465 7269 6f72 4578 744d 6f64 756c 652c  teriorExtModule,
│ │ │ │ -000193d0: 2050 7265 763a 2065 7665 6e45 7874 4d6f   Prev: evenExtMo
│ │ │ │ -000193e0: 6475 6c65 2c20 5570 3a20 546f 700a 0a65  dule, Up: Top..e
│ │ │ │ -000193f0: 7870 6f20 2d2d 2072 6574 7572 6e73 2061  xpo -- returns a
│ │ │ │ -00019400: 2073 6574 2063 6f72 7265 7370 6f6e 6469   set correspondi
│ │ │ │ -00019410: 6e67 2074 6f20 7468 6520 6261 7369 7320  ng to the basis 
│ │ │ │ -00019420: 6f66 2061 2064 6976 6964 6564 2070 6f77  of a divided pow
│ │ │ │ -00019430: 6572 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  er.*************
│ │ │ │ +000192c0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +000192d0: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +000192e0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +000192f0: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00019300: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00019310: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ +00019320: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +00019330: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +00019340: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00019350: 732e 6d32 3a33 3634 333a 302e 0a1f 0a46  s.m2:3643:0....F
│ │ │ │ +00019360: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +00019370: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00019380: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +00019390: 2065 7870 6f2c 204e 6578 743a 2065 7874   expo, Next: ext
│ │ │ │ +000193a0: 6572 696f 7245 7874 4d6f 6475 6c65 2c20  eriorExtModule, 
│ │ │ │ +000193b0: 5072 6576 3a20 6576 656e 4578 744d 6f64  Prev: evenExtMod
│ │ │ │ +000193c0: 756c 652c 2055 703a 2054 6f70 0a0a 6578  ule, Up: Top..ex
│ │ │ │ +000193d0: 706f 202d 2d20 7265 7475 726e 7320 6120  po -- returns a 
│ │ │ │ +000193e0: 7365 7420 636f 7272 6573 706f 6e64 696e  set correspondin
│ │ │ │ +000193f0: 6720 746f 2074 6865 2062 6173 6973 206f  g to the basis o
│ │ │ │ +00019400: 6620 6120 6469 7669 6465 6420 706f 7765  f a divided powe
│ │ │ │ +00019410: 720a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  r.**************
│ │ │ │ +00019420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00019430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00019440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00019470: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -00019480: 653a 200a 2020 2020 2020 2020 4220 3d20  e: .        B = 
│ │ │ │ -00019490: 6578 706f 2863 2c4e 290a 2020 2020 2020  expo(c,N).      
│ │ │ │ -000194a0: 2020 4220 3d20 6578 706f 2863 2c4c 290a    B = expo(c,L).
│ │ │ │ -000194b0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -000194c0: 2020 2a20 4e2c 2061 6e20 2a6e 6f74 6520    * N, an *note 
│ │ │ │ -000194d0: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -000194e0: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020  ay2Doc)ZZ,, .   
│ │ │ │ -000194f0: 2020 202a 2063 2c20 616e 202a 6e6f 7465     * c, an *note
│ │ │ │ -00019500: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -00019510: 6c61 7932 446f 6329 5a5a 2c2c 200a 2020  lay2Doc)ZZ,, .  
│ │ │ │ -00019520: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465      * L, a *note
│ │ │ │ -00019530: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -00019540: 3244 6f63 294c 6973 742c 2c20 6f66 2063  2Doc)List,, of c
│ │ │ │ -00019550: 206e 6f6e 2d6e 6567 6174 6976 6520 696e   non-negative in
│ │ │ │ -00019560: 7465 6765 7273 0a20 202a 204f 7574 7075  tegers.  * Outpu
│ │ │ │ -00019570: 7473 3a0a 2020 2020 2020 2a20 422c 2061  ts:.      * B, a
│ │ │ │ -00019580: 202a 6e6f 7465 206c 6973 743a 2028 4d61   *note list: (Ma
│ │ │ │ -00019590: 6361 756c 6179 3244 6f63 294c 6973 742c  caulay2Doc)List,
│ │ │ │ -000195a0: 2c20 7061 7274 6974 696f 6e73 2077 6974  , partitions wit
│ │ │ │ -000195b0: 6820 6320 6e6f 6e2d 6e65 6761 7469 7665  h c non-negative
│ │ │ │ -000195c0: 0a20 2020 2020 2020 2070 6172 7473 0a0a  .        parts..
│ │ │ │ -000195d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -000195e0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 2066 6f72  =======..The for
│ │ │ │ -000195f0: 6d20 6578 706f 2863 2c4e 2920 7265 7475  m expo(c,N) retu
│ │ │ │ -00019600: 726e 7320 7061 7274 6974 696f 6e73 206f  rns partitions o
│ │ │ │ -00019610: 6620 4e20 7769 7468 2063 206e 6f6e 2d6e  f N with c non-n
│ │ │ │ -00019620: 6567 6174 6976 6520 7061 7274 732e 2054  egative parts. T
│ │ │ │ -00019630: 6865 2066 6f72 6d0a 6578 706f 2863 2c20  he form.expo(c, 
│ │ │ │ -00019640: 4c29 2072 6574 7572 6e73 2070 6172 7469  L) returns parti
│ │ │ │ -00019650: 7469 6f6e 7320 7769 7468 206e 6f6e 2d6e  tions with non-n
│ │ │ │ -00019660: 6567 6174 6976 6520 7061 7274 7320 7468  egative parts th
│ │ │ │ -00019670: 6174 2061 7265 2063 6f6d 706f 6e65 6e74  at are component
│ │ │ │ -00019680: 7769 7365 203c 3d0a 4c20 2861 6e64 2061  wise <=.L (and a
│ │ │ │ -00019690: 6e79 2073 756d 203c 3d20 7375 6d20 4c29  ny sum <= sum L)
│ │ │ │ -000196a0: 2e0a 0a54 6865 206c 6973 7420 6578 706f  ...The list expo
│ │ │ │ -000196b0: 2863 2c4e 2920 206d 6179 2062 6520 7468  (c,N)  may be th
│ │ │ │ -000196c0: 6f75 6768 7420 6f66 2061 7320 7468 6520  ought of as the 
│ │ │ │ -000196d0: 6c69 7374 206f 6620 6578 706f 6e65 6e74  list of exponent
│ │ │ │ -000196e0: 2076 6563 746f 7273 206f 6620 7468 650a   vectors of the.
│ │ │ │ -000196f0: 6d6f 6e6f 6d69 616c 7320 6f66 2064 6567  monomials of deg
│ │ │ │ -00019700: 7265 6520 4e20 696e 2063 2076 6172 6961  ree N in c varia
│ │ │ │ -00019710: 626c 6573 2e20 5468 6973 2069 7320 7573  bles. This is us
│ │ │ │ -00019720: 6564 2069 6e20 7468 6520 636f 6e73 7472  ed in the constr
│ │ │ │ -00019730: 7563 7469 6f6e 206f 6620 7468 650a 4569  uction of the.Ei
│ │ │ │ -00019740: 7365 6e62 7564 2d53 6861 6d61 7368 2072  senbud-Shamash r
│ │ │ │ -00019750: 6573 6f6c 7574 696f 6e2e 0a0a 5468 6520  esolution...The 
│ │ │ │ -00019760: 6c69 7374 2065 7870 6f28 632c 204c 292c  list expo(c, L),
│ │ │ │ -00019770: 206f 6e20 7468 6520 6f74 6865 7220 6861   on the other ha
│ │ │ │ -00019780: 6e64 2c20 6d61 7920 6265 2074 686f 7567  nd, may be thoug
│ │ │ │ -00019790: 6874 206f 6620 6173 2074 6865 206c 6973  ht of as the lis
│ │ │ │ -000197a0: 7420 6f66 0a64 6976 6973 6f72 7320 6f66  t of.divisors of
│ │ │ │ -000197b0: 2065 5e4c 203d 2065 5f30 5e7b 4c5f 307d   e^L = e_0^{L_0}
│ │ │ │ -000197c0: 202e 2e2e 2065 5f63 5e7b 4c5f 637d 2e20   ... e_c^{L_c}. 
│ │ │ │ -000197d0: 5468 6973 2069 7320 7573 6564 2069 6e20  This is used in 
│ │ │ │ -000197e0: 7468 6520 636f 6e73 7472 7563 7469 6f6e  the construction
│ │ │ │ -000197f0: 206f 660a 7468 6520 6869 6768 6572 2068   of.the higher h
│ │ │ │ -00019800: 6f6d 6f74 6f70 6965 7320 6f6e 2061 2063  omotopies on a c
│ │ │ │ -00019810: 6f6d 706c 6578 2e0a 0a2b 2d2d 2d2d 2d2d  omplex...+------
│ │ │ │ +00019450: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +00019460: 3a20 0a20 2020 2020 2020 2042 203d 2065  : .        B = e
│ │ │ │ +00019470: 7870 6f28 632c 4e29 0a20 2020 2020 2020  xpo(c,N).       
│ │ │ │ +00019480: 2042 203d 2065 7870 6f28 632c 4c29 0a20   B = expo(c,L). 
│ │ │ │ +00019490: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +000194a0: 202a 204e 2c20 616e 202a 6e6f 7465 2069   * N, an *note i
│ │ │ │ +000194b0: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ +000194c0: 7932 446f 6329 5a5a 2c2c 200a 2020 2020  y2Doc)ZZ,, .    
│ │ │ │ +000194d0: 2020 2a20 632c 2061 6e20 2a6e 6f74 6520    * c, an *note 
│ │ │ │ +000194e0: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +000194f0: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020  ay2Doc)ZZ,, .   
│ │ │ │ +00019500: 2020 202a 204c 2c20 6120 2a6e 6f74 6520     * L, a *note 
│ │ │ │ +00019510: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +00019520: 446f 6329 4c69 7374 2c2c 206f 6620 6320  Doc)List,, of c 
│ │ │ │ +00019530: 6e6f 6e2d 6e65 6761 7469 7665 2069 6e74  non-negative int
│ │ │ │ +00019540: 6567 6572 730a 2020 2a20 4f75 7470 7574  egers.  * Output
│ │ │ │ +00019550: 733a 0a20 2020 2020 202a 2042 2c20 6120  s:.      * B, a 
│ │ │ │ +00019560: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163  *note list: (Mac
│ │ │ │ +00019570: 6175 6c61 7932 446f 6329 4c69 7374 2c2c  aulay2Doc)List,,
│ │ │ │ +00019580: 2070 6172 7469 7469 6f6e 7320 7769 7468   partitions with
│ │ │ │ +00019590: 2063 206e 6f6e 2d6e 6567 6174 6976 650a   c non-negative.
│ │ │ │ +000195a0: 2020 2020 2020 2020 7061 7274 730a 0a44          parts..D
│ │ │ │ +000195b0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +000195c0: 3d3d 3d3d 3d3d 0a0a 5468 6520 666f 726d  ======..The form
│ │ │ │ +000195d0: 2065 7870 6f28 632c 4e29 2072 6574 7572   expo(c,N) retur
│ │ │ │ +000195e0: 6e73 2070 6172 7469 7469 6f6e 7320 6f66  ns partitions of
│ │ │ │ +000195f0: 204e 2077 6974 6820 6320 6e6f 6e2d 6e65   N with c non-ne
│ │ │ │ +00019600: 6761 7469 7665 2070 6172 7473 2e20 5468  gative parts. Th
│ │ │ │ +00019610: 6520 666f 726d 0a65 7870 6f28 632c 204c  e form.expo(c, L
│ │ │ │ +00019620: 2920 7265 7475 726e 7320 7061 7274 6974  ) returns partit
│ │ │ │ +00019630: 696f 6e73 2077 6974 6820 6e6f 6e2d 6e65  ions with non-ne
│ │ │ │ +00019640: 6761 7469 7665 2070 6172 7473 2074 6861  gative parts tha
│ │ │ │ +00019650: 7420 6172 6520 636f 6d70 6f6e 656e 7477  t are componentw
│ │ │ │ +00019660: 6973 6520 3c3d 0a4c 2028 616e 6420 616e  ise <=.L (and an
│ │ │ │ +00019670: 7920 7375 6d20 3c3d 2073 756d 204c 292e  y sum <= sum L).
│ │ │ │ +00019680: 0a0a 5468 6520 6c69 7374 2065 7870 6f28  ..The list expo(
│ │ │ │ +00019690: 632c 4e29 2020 6d61 7920 6265 2074 686f  c,N)  may be tho
│ │ │ │ +000196a0: 7567 6874 206f 6620 6173 2074 6865 206c  ught of as the l
│ │ │ │ +000196b0: 6973 7420 6f66 2065 7870 6f6e 656e 7420  ist of exponent 
│ │ │ │ +000196c0: 7665 6374 6f72 7320 6f66 2074 6865 0a6d  vectors of the.m
│ │ │ │ +000196d0: 6f6e 6f6d 6961 6c73 206f 6620 6465 6772  onomials of degr
│ │ │ │ +000196e0: 6565 204e 2069 6e20 6320 7661 7269 6162  ee N in c variab
│ │ │ │ +000196f0: 6c65 732e 2054 6869 7320 6973 2075 7365  les. This is use
│ │ │ │ +00019700: 6420 696e 2074 6865 2063 6f6e 7374 7275  d in the constru
│ │ │ │ +00019710: 6374 696f 6e20 6f66 2074 6865 0a45 6973  ction of the.Eis
│ │ │ │ +00019720: 656e 6275 642d 5368 616d 6173 6820 7265  enbud-Shamash re
│ │ │ │ +00019730: 736f 6c75 7469 6f6e 2e0a 0a54 6865 206c  solution...The l
│ │ │ │ +00019740: 6973 7420 6578 706f 2863 2c20 4c29 2c20  ist expo(c, L), 
│ │ │ │ +00019750: 6f6e 2074 6865 206f 7468 6572 2068 616e  on the other han
│ │ │ │ +00019760: 642c 206d 6179 2062 6520 7468 6f75 6768  d, may be though
│ │ │ │ +00019770: 7420 6f66 2061 7320 7468 6520 6c69 7374  t of as the list
│ │ │ │ +00019780: 206f 660a 6469 7669 736f 7273 206f 6620   of.divisors of 
│ │ │ │ +00019790: 655e 4c20 3d20 655f 305e 7b4c 5f30 7d20  e^L = e_0^{L_0} 
│ │ │ │ +000197a0: 2e2e 2e20 655f 635e 7b4c 5f63 7d2e 2054  ... e_c^{L_c}. T
│ │ │ │ +000197b0: 6869 7320 6973 2075 7365 6420 696e 2074  his is used in t
│ │ │ │ +000197c0: 6865 2063 6f6e 7374 7275 6374 696f 6e20  he construction 
│ │ │ │ +000197d0: 6f66 0a74 6865 2068 6967 6865 7220 686f  of.the higher ho
│ │ │ │ +000197e0: 6d6f 746f 7069 6573 206f 6e20 6120 636f  motopies on a co
│ │ │ │ +000197f0: 6d70 6c65 782e 0a0a 2b2d 2d2d 2d2d 2d2d  mplex...+-------
│ │ │ │ +00019800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019860: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2065  -------+.|i1 : e
│ │ │ │ -00019870: 7870 6f28 332c 3529 2020 2020 2020 2020  xpo(3,5)        
│ │ │ │ +00019840: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6578  ------+.|i1 : ex
│ │ │ │ +00019850: 706f 2833 2c35 2920 2020 2020 2020 2020  po(3,5)         
│ │ │ │ +00019860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019890: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000198a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000198b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000198b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000198c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000198d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000198e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000198f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019900: 2020 2020 2020 207c 0a7c 6f31 203d 207b         |.|o1 = {
│ │ │ │ -00019910: 7b35 2c20 302c 2030 7d2c 207b 342c 2031  {5, 0, 0}, {4, 1
│ │ │ │ -00019920: 2c20 307d 2c20 7b34 2c20 302c 2031 7d2c  , 0}, {4, 0, 1},
│ │ │ │ -00019930: 207b 332c 2032 2c20 307d 2c20 7b33 2c20   {3, 2, 0}, {3, 
│ │ │ │ -00019940: 312c 2031 7d2c 207b 332c 2030 2c20 327d  1, 1}, {3, 0, 2}
│ │ │ │ -00019950: 2c20 7b32 2c20 207c 0a7c 2020 2020 202d  , {2,  |.|     -
│ │ │ │ +000198e0: 2020 2020 2020 7c0a 7c6f 3120 3d20 7b7b        |.|o1 = {{
│ │ │ │ +000198f0: 352c 2030 2c20 307d 2c20 7b34 2c20 312c  5, 0, 0}, {4, 1,
│ │ │ │ +00019900: 2030 7d2c 207b 342c 2030 2c20 317d 2c20   0}, {4, 0, 1}, 
│ │ │ │ +00019910: 7b33 2c20 322c 2030 7d2c 207b 332c 2031  {3, 2, 0}, {3, 1
│ │ │ │ +00019920: 2c20 317d 2c20 7b33 2c20 302c 2032 7d2c  , 1}, {3, 0, 2},
│ │ │ │ +00019930: 207b 322c 2020 7c0a 7c20 2020 2020 2d2d   {2,  |.|     --
│ │ │ │ +00019940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000199a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2033  -------|.|     3
│ │ │ │ -000199b0: 2c20 307d 2c20 7b32 2c20 322c 2031 7d2c  , 0}, {2, 2, 1},
│ │ │ │ -000199c0: 207b 322c 2031 2c20 327d 2c20 7b32 2c20   {2, 1, 2}, {2, 
│ │ │ │ -000199d0: 302c 2033 7d2c 207b 312c 2034 2c20 307d  0, 3}, {1, 4, 0}
│ │ │ │ -000199e0: 2c20 7b31 2c20 332c 2031 7d2c 207b 312c  , {1, 3, 1}, {1,
│ │ │ │ -000199f0: 2032 2c20 327d 2c7c 0a7c 2020 2020 202d   2, 2},|.|     -
│ │ │ │ +00019980: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 332c  ------|.|     3,
│ │ │ │ +00019990: 2030 7d2c 207b 322c 2032 2c20 317d 2c20   0}, {2, 2, 1}, 
│ │ │ │ +000199a0: 7b32 2c20 312c 2032 7d2c 207b 322c 2030  {2, 1, 2}, {2, 0
│ │ │ │ +000199b0: 2c20 337d 2c20 7b31 2c20 342c 2030 7d2c  , 3}, {1, 4, 0},
│ │ │ │ +000199c0: 207b 312c 2033 2c20 317d 2c20 7b31 2c20   {1, 3, 1}, {1, 
│ │ │ │ +000199d0: 322c 2032 7d2c 7c0a 7c20 2020 2020 2d2d  2, 2},|.|     --
│ │ │ │ +000199e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000199f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019a40: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b  -------|.|     {
│ │ │ │ -00019a50: 312c 2031 2c20 337d 2c20 7b31 2c20 302c  1, 1, 3}, {1, 0,
│ │ │ │ -00019a60: 2034 7d2c 207b 302c 2035 2c20 307d 2c20   4}, {0, 5, 0}, 
│ │ │ │ -00019a70: 7b30 2c20 342c 2031 7d2c 207b 302c 2033  {0, 4, 1}, {0, 3
│ │ │ │ -00019a80: 2c20 327d 2c20 7b30 2c20 322c 2033 7d2c  , 2}, {0, 2, 3},
│ │ │ │ -00019a90: 207b 302c 2031 2c7c 0a7c 2020 2020 202d   {0, 1,|.|     -
│ │ │ │ +00019a20: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 7b31  ------|.|     {1
│ │ │ │ +00019a30: 2c20 312c 2033 7d2c 207b 312c 2030 2c20  , 1, 3}, {1, 0, 
│ │ │ │ +00019a40: 347d 2c20 7b30 2c20 352c 2030 7d2c 207b  4}, {0, 5, 0}, {
│ │ │ │ +00019a50: 302c 2034 2c20 317d 2c20 7b30 2c20 332c  0, 4, 1}, {0, 3,
│ │ │ │ +00019a60: 2032 7d2c 207b 302c 2032 2c20 337d 2c20   2}, {0, 2, 3}, 
│ │ │ │ +00019a70: 7b30 2c20 312c 7c0a 7c20 2020 2020 2d2d  {0, 1,|.|     --
│ │ │ │ +00019a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ae0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2034  -------|.|     4
│ │ │ │ -00019af0: 7d2c 207b 302c 2030 2c20 357d 7d20 2020  }, {0, 0, 5}}   
│ │ │ │ +00019ac0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 347d  ------|.|     4}
│ │ │ │ +00019ad0: 2c20 7b30 2c20 302c 2035 7d7d 2020 2020  , {0, 0, 5}}    
│ │ │ │ +00019ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019b10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00019b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00019b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019b80: 2020 2020 2020 207c 0a7c 6f31 203a 204c         |.|o1 : L
│ │ │ │ -00019b90: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00019b60: 2020 2020 2020 7c0a 7c6f 3120 3a20 4c69        |.|o1 : Li
│ │ │ │ +00019b70: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +00019b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00019bb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00019bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2065  -------+.|i2 : e
│ │ │ │ -00019c30: 7870 6f28 332c 207b 332c 322c 317d 2920  xpo(3, {3,2,1}) 
│ │ │ │ +00019c00: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 6578  ------+.|i2 : ex
│ │ │ │ +00019c10: 706f 2833 2c20 7b33 2c32 2c31 7d29 2020  po(3, {3,2,1})  
│ │ │ │ +00019c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00019c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00019c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019cc0: 2020 2020 2020 207c 0a7c 6f32 203d 207b         |.|o2 = {
│ │ │ │ -00019cd0: 7b30 2c20 302c 2030 7d2c 207b 312c 2030  {0, 0, 0}, {1, 0
│ │ │ │ -00019ce0: 2c20 307d 2c20 7b30 2c20 312c 2030 7d2c  , 0}, {0, 1, 0},
│ │ │ │ -00019cf0: 207b 302c 2030 2c20 317d 2c20 7b32 2c20   {0, 0, 1}, {2, 
│ │ │ │ -00019d00: 302c 2030 7d2c 207b 312c 2031 2c20 307d  0, 0}, {1, 1, 0}
│ │ │ │ -00019d10: 2c20 7b31 2c20 207c 0a7c 2020 2020 202d  , {1,  |.|     -
│ │ │ │ +00019ca0: 2020 2020 2020 7c0a 7c6f 3220 3d20 7b7b        |.|o2 = {{
│ │ │ │ +00019cb0: 302c 2030 2c20 307d 2c20 7b31 2c20 302c  0, 0, 0}, {1, 0,
│ │ │ │ +00019cc0: 2030 7d2c 207b 302c 2031 2c20 307d 2c20   0}, {0, 1, 0}, 
│ │ │ │ +00019cd0: 7b30 2c20 302c 2031 7d2c 207b 322c 2030  {0, 0, 1}, {2, 0
│ │ │ │ +00019ce0: 2c20 307d 2c20 7b31 2c20 312c 2030 7d2c  , 0}, {1, 1, 0},
│ │ │ │ +00019cf0: 207b 312c 2020 7c0a 7c20 2020 2020 2d2d   {1,  |.|     --
│ │ │ │ +00019d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d60: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030  -------|.|     0
│ │ │ │ -00019d70: 2c20 317d 2c20 7b30 2c20 322c 2030 7d2c  , 1}, {0, 2, 0},
│ │ │ │ -00019d80: 207b 302c 2031 2c20 317d 2c20 7b33 2c20   {0, 1, 1}, {3, 
│ │ │ │ -00019d90: 302c 2030 7d2c 207b 322c 2031 2c20 307d  0, 0}, {2, 1, 0}
│ │ │ │ -00019da0: 2c20 7b32 2c20 302c 2031 7d2c 207b 312c  , {2, 0, 1}, {1,
│ │ │ │ -00019db0: 2032 2c20 307d 2c7c 0a7c 2020 2020 202d   2, 0},|.|     -
│ │ │ │ +00019d40: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 302c  ------|.|     0,
│ │ │ │ +00019d50: 2031 7d2c 207b 302c 2032 2c20 307d 2c20   1}, {0, 2, 0}, 
│ │ │ │ +00019d60: 7b30 2c20 312c 2031 7d2c 207b 332c 2030  {0, 1, 1}, {3, 0
│ │ │ │ +00019d70: 2c20 307d 2c20 7b32 2c20 312c 2030 7d2c  , 0}, {2, 1, 0},
│ │ │ │ +00019d80: 207b 322c 2030 2c20 317d 2c20 7b31 2c20   {2, 0, 1}, {1, 
│ │ │ │ +00019d90: 322c 2030 7d2c 7c0a 7c20 2020 2020 2d2d  2, 0},|.|     --
│ │ │ │ +00019da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019e00: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b  -------|.|     {
│ │ │ │ -00019e10: 312c 2031 2c20 317d 2c20 7b30 2c20 322c  1, 1, 1}, {0, 2,
│ │ │ │ -00019e20: 2031 7d2c 207b 332c 2031 2c20 307d 2c20   1}, {3, 1, 0}, 
│ │ │ │ -00019e30: 7b33 2c20 302c 2031 7d2c 207b 322c 2032  {3, 0, 1}, {2, 2
│ │ │ │ -00019e40: 2c20 307d 2c20 7b32 2c20 312c 2031 7d2c  , 0}, {2, 1, 1},
│ │ │ │ -00019e50: 207b 312c 2032 2c7c 0a7c 2020 2020 202d   {1, 2,|.|     -
│ │ │ │ +00019de0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 7b31  ------|.|     {1
│ │ │ │ +00019df0: 2c20 312c 2031 7d2c 207b 302c 2032 2c20  , 1, 1}, {0, 2, 
│ │ │ │ +00019e00: 317d 2c20 7b33 2c20 312c 2030 7d2c 207b  1}, {3, 1, 0}, {
│ │ │ │ +00019e10: 332c 2030 2c20 317d 2c20 7b32 2c20 322c  3, 0, 1}, {2, 2,
│ │ │ │ +00019e20: 2030 7d2c 207b 322c 2031 2c20 317d 2c20   0}, {2, 1, 1}, 
│ │ │ │ +00019e30: 7b31 2c20 322c 7c0a 7c20 2020 2020 2d2d  {1, 2,|.|     --
│ │ │ │ +00019e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ea0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2031  -------|.|     1
│ │ │ │ -00019eb0: 7d2c 207b 332c 2032 2c20 307d 2c20 7b33  }, {3, 2, 0}, {3
│ │ │ │ -00019ec0: 2c20 312c 2031 7d2c 207b 322c 2032 2c20  , 1, 1}, {2, 2, 
│ │ │ │ -00019ed0: 317d 7d20 2020 2020 2020 2020 2020 2020  1}}             
│ │ │ │ +00019e80: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 317d  ------|.|     1}
│ │ │ │ +00019e90: 2c20 7b33 2c20 322c 2030 7d2c 207b 332c  , {3, 2, 0}, {3,
│ │ │ │ +00019ea0: 2031 2c20 317d 2c20 7b32 2c20 322c 2031   1, 1}, {2, 2, 1
│ │ │ │ +00019eb0: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ +00019ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019ed0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00019ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ef0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00019ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f40: 2020 2020 2020 207c 0a7c 6f32 203a 204c         |.|o2 : L
│ │ │ │ -00019f50: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00019f20: 2020 2020 2020 7c0a 7c6f 3220 3a20 4c69        |.|o2 : Li
│ │ │ │ +00019f30: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +00019f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00019f70: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00019f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00019f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019fe0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -00019ff0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -0001a000: 202a 6e6f 7465 2045 6973 656e 6275 6453   *note EisenbudS
│ │ │ │ -0001a010: 6861 6d61 7368 3a20 4569 7365 6e62 7564  hamash: Eisenbud
│ │ │ │ -0001a020: 5368 616d 6173 682c 202d 2d20 436f 6d70  Shamash, -- Comp
│ │ │ │ -0001a030: 7574 6573 2074 6865 2045 6973 656e 6275  utes the Eisenbu
│ │ │ │ -0001a040: 642d 5368 616d 6173 680a 2020 2020 436f  d-Shamash.    Co
│ │ │ │ -0001a050: 6d70 6c65 780a 2020 2a20 2a6e 6f74 6520  mplex.  * *note 
│ │ │ │ -0001a060: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20  makeHomotopies: 
│ │ │ │ -0001a070: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20  makeHomotopies, 
│ │ │ │ -0001a080: 2d2d 2072 6574 7572 6e73 2061 2073 7973  -- returns a sys
│ │ │ │ -0001a090: 7465 6d20 6f66 2068 6967 6865 720a 2020  tem of higher.  
│ │ │ │ -0001a0a0: 2020 686f 6d6f 746f 7069 6573 0a0a 5761    homotopies..Wa
│ │ │ │ -0001a0b0: 7973 2074 6f20 7573 6520 6578 706f 3a0a  ys to use expo:.
│ │ │ │ -0001a0c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001a0d0: 3d0a 0a20 202a 2022 6578 706f 285a 5a2c  =..  * "expo(ZZ,
│ │ │ │ -0001a0e0: 4c69 7374 2922 0a20 202a 2022 6578 706f  List)".  * "expo
│ │ │ │ -0001a0f0: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468  (ZZ,ZZ)"..For th
│ │ │ │ -0001a100: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -0001a110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0001a120: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -0001a130: 6520 6578 706f 3a20 6578 706f 2c20 6973  e expo: expo, is
│ │ │ │ -0001a140: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ -0001a150: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ -0001a160: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ -0001a170: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ +00019fc0: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ +00019fd0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00019fe0: 2a6e 6f74 6520 4569 7365 6e62 7564 5368  *note EisenbudSh
│ │ │ │ +00019ff0: 616d 6173 683a 2045 6973 656e 6275 6453  amash: EisenbudS
│ │ │ │ +0001a000: 6861 6d61 7368 2c20 2d2d 2043 6f6d 7075  hamash, -- Compu
│ │ │ │ +0001a010: 7465 7320 7468 6520 4569 7365 6e62 7564  tes the Eisenbud
│ │ │ │ +0001a020: 2d53 6861 6d61 7368 0a20 2020 2043 6f6d  -Shamash.    Com
│ │ │ │ +0001a030: 706c 6578 0a20 202a 202a 6e6f 7465 206d  plex.  * *note m
│ │ │ │ +0001a040: 616b 6548 6f6d 6f74 6f70 6965 733a 206d  akeHomotopies: m
│ │ │ │ +0001a050: 616b 6548 6f6d 6f74 6f70 6965 732c 202d  akeHomotopies, -
│ │ │ │ +0001a060: 2d20 7265 7475 726e 7320 6120 7379 7374  - returns a syst
│ │ │ │ +0001a070: 656d 206f 6620 6869 6768 6572 0a20 2020  em of higher.   
│ │ │ │ +0001a080: 2068 6f6d 6f74 6f70 6965 730a 0a57 6179   homotopies..Way
│ │ │ │ +0001a090: 7320 746f 2075 7365 2065 7870 6f3a 0a3d  s to use expo:.=
│ │ │ │ +0001a0a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001a0b0: 0a0a 2020 2a20 2265 7870 6f28 5a5a 2c4c  ..  * "expo(ZZ,L
│ │ │ │ +0001a0c0: 6973 7429 220a 2020 2a20 2265 7870 6f28  ist)".  * "expo(
│ │ │ │ +0001a0d0: 5a5a 2c5a 5a29 220a 0a46 6f72 2074 6865  ZZ,ZZ)"..For the
│ │ │ │ +0001a0e0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +0001a0f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0001a100: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +0001a110: 2065 7870 6f3a 2065 7870 6f2c 2069 7320   expo: expo, is 
│ │ │ │ +0001a120: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ +0001a130: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ +0001a140: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +0001a150: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +0001a160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a1c0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ -0001a1d0: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ -0001a1e0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -0001a1f0: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -0001a200: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -0001a210: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ -0001a220: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -0001a230: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ -0001a240: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0001a250: 732e 6d32 3a35 3038 363a 302e 0a1f 0a46  s.m2:5086:0....F
│ │ │ │ -0001a260: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -0001a270: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -0001a280: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -0001a290: 2065 7874 6572 696f 7245 7874 4d6f 6475   exteriorExtModu
│ │ │ │ -0001a2a0: 6c65 2c20 4e65 7874 3a20 6578 7465 7269  le, Next: exteri
│ │ │ │ -0001a2b0: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65  orHomologyModule
│ │ │ │ -0001a2c0: 2c20 5072 6576 3a20 6578 706f 2c20 5570  , Prev: expo, Up
│ │ │ │ -0001a2d0: 3a20 546f 700a 0a65 7874 6572 696f 7245  : Top..exteriorE
│ │ │ │ -0001a2e0: 7874 4d6f 6475 6c65 202d 2d20 4578 7428  xtModule -- Ext(
│ │ │ │ -0001a2f0: 4d2c 6b29 206f 7220 4578 7428 4d2c 4e29  M,k) or Ext(M,N)
│ │ │ │ -0001a300: 2061 7320 6120 6d6f 6475 6c65 206f 7665   as a module ove
│ │ │ │ -0001a310: 7220 616e 2065 7874 6572 696f 7220 616c  r an exterior al
│ │ │ │ -0001a320: 6765 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a  gebra.**********
│ │ │ │ +0001a1a0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +0001a1b0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +0001a1c0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +0001a1d0: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +0001a1e0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +0001a1f0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ +0001a200: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +0001a210: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ +0001a220: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0001a230: 2e6d 323a 3530 3836 3a30 2e0a 1f0a 4669  .m2:5086:0....Fi
│ │ │ │ +0001a240: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +0001a250: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0001a260: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +0001a270: 6578 7465 7269 6f72 4578 744d 6f64 756c  exteriorExtModul
│ │ │ │ +0001a280: 652c 204e 6578 743a 2065 7874 6572 696f  e, Next: exterio
│ │ │ │ +0001a290: 7248 6f6d 6f6c 6f67 794d 6f64 756c 652c  rHomologyModule,
│ │ │ │ +0001a2a0: 2050 7265 763a 2065 7870 6f2c 2055 703a   Prev: expo, Up:
│ │ │ │ +0001a2b0: 2054 6f70 0a0a 6578 7465 7269 6f72 4578   Top..exteriorEx
│ │ │ │ +0001a2c0: 744d 6f64 756c 6520 2d2d 2045 7874 284d  tModule -- Ext(M
│ │ │ │ +0001a2d0: 2c6b 2920 6f72 2045 7874 284d 2c4e 2920  ,k) or Ext(M,N) 
│ │ │ │ +0001a2e0: 6173 2061 206d 6f64 756c 6520 6f76 6572  as a module over
│ │ │ │ +0001a2f0: 2061 6e20 6578 7465 7269 6f72 2061 6c67   an exterior alg
│ │ │ │ +0001a300: 6562 7261 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ebra.***********
│ │ │ │ +0001a310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001a320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001a330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001a340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001a350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001a360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001a370: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -0001a380: 200a 2020 2020 2020 2020 4520 3d20 6578   .        E = ex
│ │ │ │ -0001a390: 7465 7269 6f72 4578 744d 6f64 756c 6528  teriorExtModule(
│ │ │ │ -0001a3a0: 662c 4d29 0a20 202a 2049 6e70 7574 733a  f,M).  * Inputs:
│ │ │ │ -0001a3b0: 0a20 2020 2020 202a 2066 2c20 6120 2a6e  .      * f, a *n
│ │ │ │ -0001a3c0: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -0001a3d0: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -0001a3e0: 2c2c 2031 2078 2063 2c20 656e 7472 6965  ,, 1 x c, entrie
│ │ │ │ -0001a3f0: 7320 6d75 7374 2062 650a 2020 2020 2020  s must be.      
│ │ │ │ -0001a400: 2020 686f 6d6f 746f 7069 6320 746f 2030    homotopic to 0
│ │ │ │ -0001a410: 206f 6e20 460a 2020 2020 2020 2a20 4d2c   on F.      * M,
│ │ │ │ -0001a420: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ -0001a430: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -0001a440: 6f64 756c 652c 2c20 616e 6e69 6869 6c61  odule,, annihila
│ │ │ │ -0001a450: 7465 6420 6279 2074 6865 2065 6c65 6d65  ted by the eleme
│ │ │ │ -0001a460: 6e74 730a 2020 2020 2020 2020 6f66 2066  nts.        of f
│ │ │ │ -0001a470: 660a 2020 2020 2020 2a20 4e2c 2061 202a  f.      * N, a *
│ │ │ │ -0001a480: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0001a490: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0001a4a0: 652c 2c20 616e 6e69 6869 6c61 7465 6420  e,, annihilated 
│ │ │ │ -0001a4b0: 6279 2074 6865 2065 6c65 6d65 6e74 730a  by the elements.
│ │ │ │ -0001a4c0: 2020 2020 2020 2020 6f66 2066 660a 2020          of ff.  
│ │ │ │ -0001a4d0: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -0001a4e0: 202a 2045 2c20 6120 2a6e 6f74 6520 6d6f   * E, a *note mo
│ │ │ │ -0001a4f0: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ -0001a500: 446f 6329 4d6f 6475 6c65 2c2c 204d 6f64  Doc)Module,, Mod
│ │ │ │ -0001a510: 756c 6520 6f76 6572 2061 6e20 6578 7465  ule over an exte
│ │ │ │ -0001a520: 7269 6f72 0a20 2020 2020 2020 2061 6c67  rior.        alg
│ │ │ │ -0001a530: 6562 7261 2077 6974 6820 7661 7269 6162  ebra with variab
│ │ │ │ -0001a540: 6c65 7320 636f 7272 6573 706f 6e64 696e  les correspondin
│ │ │ │ -0001a550: 6720 746f 2065 6c65 6d65 6e74 7320 6f66  g to elements of
│ │ │ │ -0001a560: 2066 0a0a 4465 7363 7269 7074 696f 6e0a   f..Description.
│ │ │ │ -0001a570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620  ===========..If 
│ │ │ │ -0001a580: 4d2c 4e20 6172 6520 532d 6d6f 6475 6c65  M,N are S-module
│ │ │ │ -0001a590: 7320 616e 6e69 6869 6c61 7465 6420 6279  s annihilated by
│ │ │ │ -0001a5a0: 2074 6865 2065 6c65 6d65 6e74 7320 6f66   the elements of
│ │ │ │ -0001a5b0: 2074 6865 206d 6174 7269 7820 6666 203d   the matrix ff =
│ │ │ │ -0001a5c0: 2028 665f 312e 2e66 5f63 292c 0a61 6e64   (f_1..f_c),.and
│ │ │ │ -0001a5d0: 206b 2069 7320 7468 6520 7265 7369 6475   k is the residu
│ │ │ │ -0001a5e0: 6520 6669 656c 6420 6f66 2053 2c20 7468  e field of S, th
│ │ │ │ -0001a5f0: 656e 2074 6865 2073 6372 6970 7420 6578  en the script ex
│ │ │ │ -0001a600: 7465 7269 6f72 4578 744d 6f64 756c 6528  teriorExtModule(
│ │ │ │ -0001a610: 662c 4d29 2072 6574 7572 6e73 0a45 7874  f,M) returns.Ext
│ │ │ │ -0001a620: 5f53 284d 2c20 6b29 2061 7320 6120 6d6f  _S(M, k) as a mo
│ │ │ │ -0001a630: 6475 6c65 206f 7665 7220 616e 2065 7874  dule over an ext
│ │ │ │ -0001a640: 6572 696f 7220 616c 6765 6272 6120 4520  erior algebra E 
│ │ │ │ -0001a650: 3d20 6b3c 655f 312c 2e2e 2e2c 655f 633e  = k
│ │ │ │ -0001a660: 2c20 7768 6572 6520 7468 650a 655f 6920  , where the.e_i 
│ │ │ │ -0001a670: 6861 7665 2064 6567 7265 6520 312e 2049  have degree 1. I
│ │ │ │ -0001a680: 7420 6973 2063 6f6d 7075 7465 6420 6173  t is computed as
│ │ │ │ -0001a690: 2074 6865 2045 2d64 7561 6c20 6f66 2065   the E-dual of e
│ │ │ │ -0001a6a0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -0001a6b0: 2e0a 0a54 6865 2073 6372 6970 7420 6578  ...The script ex
│ │ │ │ -0001a6c0: 7465 7269 6f72 546f 724d 6f64 756c 6528  teriorTorModule(
│ │ │ │ -0001a6d0: 662c 4d2c 4e29 2072 6574 7572 6e73 2045  f,M,N) returns E
│ │ │ │ -0001a6e0: 7874 5f53 284d 2c4e 2920 6173 2061 206d  xt_S(M,N) as a m
│ │ │ │ -0001a6f0: 6f64 756c 6520 6f76 6572 2061 0a62 6967  odule over a.big
│ │ │ │ -0001a700: 7261 6465 6420 7269 6e67 2053 4520 3d20  raded ring SE = 
│ │ │ │ -0001a710: 533c 655f 312c 2e2e 2c65 5f63 3e2c 2077  S, w
│ │ │ │ -0001a720: 6865 7265 2074 6865 2065 5f69 2068 6176  here the e_i hav
│ │ │ │ -0001a730: 6520 6465 6772 6565 7320 7b64 5f69 2c31  e degrees {d_i,1
│ │ │ │ -0001a740: 7d2c 2077 6865 7265 2064 5f69 0a69 7320  }, where d_i.is 
│ │ │ │ -0001a750: 7468 6520 6465 6772 6565 206f 6620 665f  the degree of f_
│ │ │ │ -0001a760: 692e 2054 6865 206d 6f64 756c 6520 7374  i. The module st
│ │ │ │ -0001a770: 7275 6374 7572 652c 2069 6e20 6569 7468  ructure, in eith
│ │ │ │ -0001a780: 6572 2063 6173 652c 2069 7320 6465 6669  er case, is defi
│ │ │ │ -0001a790: 6e65 6420 6279 2074 6865 0a68 6f6d 6f74  ned by the.homot
│ │ │ │ -0001a7a0: 6f70 6965 7320 666f 7220 7468 6520 665f  opies for the f_
│ │ │ │ -0001a7b0: 6920 6f6e 2074 6865 2072 6573 6f6c 7574  i on the resolut
│ │ │ │ -0001a7c0: 696f 6e20 6f66 204d 2c20 636f 6d70 7574  ion of M, comput
│ │ │ │ -0001a7d0: 6564 2062 7920 7468 6520 7363 7269 7074  ed by the script
│ │ │ │ -0001a7e0: 0a6d 616b 6548 6f6d 6f74 6f70 6965 7331  .makeHomotopies1
│ │ │ │ -0001a7f0: 2e54 6865 2073 6372 6970 7420 6361 6c6c  .The script call
│ │ │ │ -0001a800: 7320 6d61 6b65 4d6f 6475 6c65 2074 6f20  s makeModule to 
│ │ │ │ -0001a810: 636f 6d70 7574 6520 6120 286e 6f6e 2d6d  compute a (non-m
│ │ │ │ -0001a820: 696e 696d 616c 290a 7072 6573 656e 7461  inimal).presenta
│ │ │ │ -0001a830: 7469 6f6e 206f 6620 7468 6973 206d 6f64  tion of this mod
│ │ │ │ -0001a840: 756c 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ule...+---------
│ │ │ │ +0001a350: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +0001a360: 0a20 2020 2020 2020 2045 203d 2065 7874  .        E = ext
│ │ │ │ +0001a370: 6572 696f 7245 7874 4d6f 6475 6c65 2866  eriorExtModule(f
│ │ │ │ +0001a380: 2c4d 290a 2020 2a20 496e 7075 7473 3a0a  ,M).  * Inputs:.
│ │ │ │ +0001a390: 2020 2020 2020 2a20 662c 2061 202a 6e6f        * f, a *no
│ │ │ │ +0001a3a0: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ +0001a3b0: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ +0001a3c0: 2c20 3120 7820 632c 2065 6e74 7269 6573  , 1 x c, entries
│ │ │ │ +0001a3d0: 206d 7573 7420 6265 0a20 2020 2020 2020   must be.       
│ │ │ │ +0001a3e0: 2068 6f6d 6f74 6f70 6963 2074 6f20 3020   homotopic to 0 
│ │ │ │ +0001a3f0: 6f6e 2046 0a20 2020 2020 202a 204d 2c20  on F.      * M, 
│ │ │ │ +0001a400: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ +0001a410: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
│ │ │ │ +0001a420: 6475 6c65 2c2c 2061 6e6e 6968 696c 6174  dule,, annihilat
│ │ │ │ +0001a430: 6564 2062 7920 7468 6520 656c 656d 656e  ed by the elemen
│ │ │ │ +0001a440: 7473 0a20 2020 2020 2020 206f 6620 6666  ts.        of ff
│ │ │ │ +0001a450: 0a20 2020 2020 202a 204e 2c20 6120 2a6e  .      * N, a *n
│ │ │ │ +0001a460: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +0001a470: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +0001a480: 2c2c 2061 6e6e 6968 696c 6174 6564 2062  ,, annihilated b
│ │ │ │ +0001a490: 7920 7468 6520 656c 656d 656e 7473 0a20  y the elements. 
│ │ │ │ +0001a4a0: 2020 2020 2020 206f 6620 6666 0a20 202a         of ff.  *
│ │ │ │ +0001a4b0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +0001a4c0: 2a20 452c 2061 202a 6e6f 7465 206d 6f64  * E, a *note mod
│ │ │ │ +0001a4d0: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ +0001a4e0: 6f63 294d 6f64 756c 652c 2c20 4d6f 6475  oc)Module,, Modu
│ │ │ │ +0001a4f0: 6c65 206f 7665 7220 616e 2065 7874 6572  le over an exter
│ │ │ │ +0001a500: 696f 720a 2020 2020 2020 2020 616c 6765  ior.        alge
│ │ │ │ +0001a510: 6272 6120 7769 7468 2076 6172 6961 626c  bra with variabl
│ │ │ │ +0001a520: 6573 2063 6f72 7265 7370 6f6e 6469 6e67  es corresponding
│ │ │ │ +0001a530: 2074 6f20 656c 656d 656e 7473 206f 6620   to elements of 
│ │ │ │ +0001a540: 660a 0a44 6573 6372 6970 7469 6f6e 0a3d  f..Description.=
│ │ │ │ +0001a550: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 204d  ==========..If M
│ │ │ │ +0001a560: 2c4e 2061 7265 2053 2d6d 6f64 756c 6573  ,N are S-modules
│ │ │ │ +0001a570: 2061 6e6e 6968 696c 6174 6564 2062 7920   annihilated by 
│ │ │ │ +0001a580: 7468 6520 656c 656d 656e 7473 206f 6620  the elements of 
│ │ │ │ +0001a590: 7468 6520 6d61 7472 6978 2066 6620 3d20  the matrix ff = 
│ │ │ │ +0001a5a0: 2866 5f31 2e2e 665f 6329 2c0a 616e 6420  (f_1..f_c),.and 
│ │ │ │ +0001a5b0: 6b20 6973 2074 6865 2072 6573 6964 7565  k is the residue
│ │ │ │ +0001a5c0: 2066 6965 6c64 206f 6620 532c 2074 6865   field of S, the
│ │ │ │ +0001a5d0: 6e20 7468 6520 7363 7269 7074 2065 7874  n the script ext
│ │ │ │ +0001a5e0: 6572 696f 7245 7874 4d6f 6475 6c65 2866  eriorExtModule(f
│ │ │ │ +0001a5f0: 2c4d 2920 7265 7475 726e 730a 4578 745f  ,M) returns.Ext_
│ │ │ │ +0001a600: 5328 4d2c 206b 2920 6173 2061 206d 6f64  S(M, k) as a mod
│ │ │ │ +0001a610: 756c 6520 6f76 6572 2061 6e20 6578 7465  ule over an exte
│ │ │ │ +0001a620: 7269 6f72 2061 6c67 6562 7261 2045 203d  rior algebra E =
│ │ │ │ +0001a630: 206b 3c65 5f31 2c2e 2e2e 2c65 5f63 3e2c   k,
│ │ │ │ +0001a640: 2077 6865 7265 2074 6865 0a65 5f69 2068   where the.e_i h
│ │ │ │ +0001a650: 6176 6520 6465 6772 6565 2031 2e20 4974  ave degree 1. It
│ │ │ │ +0001a660: 2069 7320 636f 6d70 7574 6564 2061 7320   is computed as 
│ │ │ │ +0001a670: 7468 6520 452d 6475 616c 206f 6620 6578  the E-dual of ex
│ │ │ │ +0001a680: 7465 7269 6f72 546f 724d 6f64 756c 652e  teriorTorModule.
│ │ │ │ +0001a690: 0a0a 5468 6520 7363 7269 7074 2065 7874  ..The script ext
│ │ │ │ +0001a6a0: 6572 696f 7254 6f72 4d6f 6475 6c65 2866  eriorTorModule(f
│ │ │ │ +0001a6b0: 2c4d 2c4e 2920 7265 7475 726e 7320 4578  ,M,N) returns Ex
│ │ │ │ +0001a6c0: 745f 5328 4d2c 4e29 2061 7320 6120 6d6f  t_S(M,N) as a mo
│ │ │ │ +0001a6d0: 6475 6c65 206f 7665 7220 610a 6269 6772  dule over a.bigr
│ │ │ │ +0001a6e0: 6164 6564 2072 696e 6720 5345 203d 2053  aded ring SE = S
│ │ │ │ +0001a6f0: 3c65 5f31 2c2e 2e2c 655f 633e 2c20 7768  , wh
│ │ │ │ +0001a700: 6572 6520 7468 6520 655f 6920 6861 7665  ere the e_i have
│ │ │ │ +0001a710: 2064 6567 7265 6573 207b 645f 692c 317d   degrees {d_i,1}
│ │ │ │ +0001a720: 2c20 7768 6572 6520 645f 690a 6973 2074  , where d_i.is t
│ │ │ │ +0001a730: 6865 2064 6567 7265 6520 6f66 2066 5f69  he degree of f_i
│ │ │ │ +0001a740: 2e20 5468 6520 6d6f 6475 6c65 2073 7472  . The module str
│ │ │ │ +0001a750: 7563 7475 7265 2c20 696e 2065 6974 6865  ucture, in eithe
│ │ │ │ +0001a760: 7220 6361 7365 2c20 6973 2064 6566 696e  r case, is defin
│ │ │ │ +0001a770: 6564 2062 7920 7468 650a 686f 6d6f 746f  ed by the.homoto
│ │ │ │ +0001a780: 7069 6573 2066 6f72 2074 6865 2066 5f69  pies for the f_i
│ │ │ │ +0001a790: 206f 6e20 7468 6520 7265 736f 6c75 7469   on the resoluti
│ │ │ │ +0001a7a0: 6f6e 206f 6620 4d2c 2063 6f6d 7075 7465  on of M, compute
│ │ │ │ +0001a7b0: 6420 6279 2074 6865 2073 6372 6970 740a  d by the script.
│ │ │ │ +0001a7c0: 6d61 6b65 486f 6d6f 746f 7069 6573 312e  makeHomotopies1.
│ │ │ │ +0001a7d0: 5468 6520 7363 7269 7074 2063 616c 6c73  The script calls
│ │ │ │ +0001a7e0: 206d 616b 654d 6f64 756c 6520 746f 2063   makeModule to c
│ │ │ │ +0001a7f0: 6f6d 7075 7465 2061 2028 6e6f 6e2d 6d69  ompute a (non-mi
│ │ │ │ +0001a800: 6e69 6d61 6c29 0a70 7265 7365 6e74 6174  nimal).presentat
│ │ │ │ +0001a810: 696f 6e20 6f66 2074 6869 7320 6d6f 6475  ion of this modu
│ │ │ │ +0001a820: 6c65 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  le...+----------
│ │ │ │ +0001a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a880: 2d2d 2b0a 7c69 3120 3a20 6b6b 203d 205a  --+.|i1 : kk = Z
│ │ │ │ -0001a890: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │ -0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a860: 2d2b 0a7c 6931 203a 206b 6b20 3d20 5a5a  -+.|i1 : kk = ZZ
│ │ │ │ +0001a870: 2f31 3031 2020 2020 2020 2020 2020 2020  /101            
│ │ │ │ +0001a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001a8a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001a900: 7c6f 3120 3d20 6b6b 2020 2020 2020 2020  |o1 = kk        
│ │ │ │ -0001a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001a8e0: 6f31 203d 206b 6b20 2020 2020 2020 2020  o1 = kk         
│ │ │ │ +0001a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a910: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a930: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a970: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0001a980: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ -0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001a950: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ +0001a960: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a990: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001a9f0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320  ------+.|i2 : S 
│ │ │ │ -0001aa00: 3d20 6b6b 5b61 2c62 2c63 5d20 2020 2020  = kk[a,b,c]     
│ │ │ │ -0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a9d0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d  -----+.|i2 : S =
│ │ │ │ +0001a9e0: 206b 6b5b 612c 622c 635d 2020 2020 2020   kk[a,b,c]      
│ │ │ │ +0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa50: 207c 0a7c 6f32 203d 2053 2020 2020 2020   |.|o2 = S      
│ │ │ │  0001aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa70: 2020 7c0a 7c6f 3220 3d20 5320 2020 2020    |.|o2 = S     
│ │ │ │ -0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001aa90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aab0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aae0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001aaf0: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c  |o2 : Polynomial
│ │ │ │ -0001ab00: 5269 6e67 2020 2020 2020 2020 2020 2020  Ring            
│ │ │ │ -0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001aad0: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52  o2 : PolynomialR
│ │ │ │ +0001aae0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ +0001aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ab20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -0001ab70: 3a20 6620 3d20 6d61 7472 6978 2261 342c  : f = matrix"a4,
│ │ │ │ -0001ab80: 6234 2c63 3422 2020 2020 2020 2020 2020  b4,c4"          
│ │ │ │ +0001ab40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +0001ab50: 2066 203d 206d 6174 7269 7822 6134 2c62   f = matrix"a4,b
│ │ │ │ +0001ab60: 342c 6334 2220 2020 2020 2020 2020 2020  4,c4"           
│ │ │ │ +0001ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abe0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7c20        |.|o3 = | 
│ │ │ │ -0001abf0: 6134 2062 3420 6334 207c 2020 2020 2020  a4 b4 c4 |      
│ │ │ │ -0001ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001abc0: 2020 2020 207c 0a7c 6f33 203d 207c 2061       |.|o3 = | a
│ │ │ │ +0001abd0: 3420 6234 2063 3420 7c20 2020 2020 2020  4 b4 c4 |       
│ │ │ │ +0001abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001ac70: 2020 3120 2020 2020 2033 2020 2020 2020    1      3      
│ │ │ │ -0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aca0: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053  |.|o3 : Matrix S
│ │ │ │ -0001acb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -0001acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001acd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ace0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0001acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001ad20: 3420 3a20 5220 3d20 532f 6964 6561 6c20  4 : R = S/ideal 
│ │ │ │ -0001ad30: 6620 2020 2020 2020 2020 2020 2020 2020  f               
│ │ │ │ +0001ac40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001ac50: 2031 2020 2020 2020 3320 2020 2020 2020   1      3       
│ │ │ │ +0001ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001ac80: 0a7c 6f33 203a 204d 6174 7269 7820 5320  .|o3 : Matrix S 
│ │ │ │ +0001ac90: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +0001aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001acb0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001acc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001acd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ace0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ +0001ad00: 203a 2052 203d 2053 2f69 6465 616c 2066   : R = S/ideal f
│ │ │ │ +0001ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad70: 2020 2020 2020 207c 0a7c 6f34 203d 2052         |.|o4 = R
│ │ │ │  0001ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad90: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -0001ada0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ -0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001adb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001add0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae10: 2020 2020 7c0a 7c6f 3420 3a20 5175 6f74      |.|o4 : Quot
│ │ │ │ -0001ae20: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ -0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0001ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ae90: 2b0a 7c69 3520 3a20 7020 3d20 6d61 7028  +.|i5 : p = map(
│ │ │ │ -0001aea0: 522c 5329 2020 2020 2020 2020 2020 2020  R,S)            
│ │ │ │ +0001adf0: 2020 207c 0a7c 6f34 203a 2051 756f 7469     |.|o4 : Quoti
│ │ │ │ +0001ae00: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0001ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001ae40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001ae70: 0a7c 6935 203a 2070 203d 206d 6170 2852  .|i5 : p = map(R
│ │ │ │ +0001ae80: 2c53 2920 2020 2020 2020 2020 2020 2020  ,S)             
│ │ │ │ +0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001aed0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0001aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001af10: 3520 3d20 6d61 7020 2852 2c20 532c 207b  5 = map (R, S, {
│ │ │ │ -0001af20: 612c 2062 2c20 637d 2920 2020 2020 2020  a, b, c})       
│ │ │ │ +0001aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aee0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +0001aef0: 203d 206d 6170 2028 522c 2053 2c20 7b61   = map (R, S, {a
│ │ │ │ +0001af00: 2c20 622c 2063 7d29 2020 2020 2020 2020  , b, c})        
│ │ │ │ +0001af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001af20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af80: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ -0001af90: 5269 6e67 4d61 7020 5220 3c2d 2d20 5320  RingMap R <-- S 
│ │ │ │ -0001afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001afc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001af60: 2020 2020 2020 207c 0a7c 6f35 203a 2052         |.|o5 : R
│ │ │ │ +0001af70: 696e 674d 6170 2052 203c 2d2d 2053 2020  ingMap R <-- S  
│ │ │ │ +0001af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001afa0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b000: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20 3d20  ----+.|i6 : M = 
│ │ │ │ -0001b010: 636f 6b65 7220 6d61 7028 525e 322c 2052  coker map(R^2, R
│ │ │ │ -0001b020: 5e7b 333a 2d31 7d2c 207b 7b61 2c62 2c63  ^{3:-1}, {{a,b,c
│ │ │ │ -0001b030: 7d2c 7b62 2c63 2c61 7d7d 2920 2020 2020  },{b,c,a}})     
│ │ │ │ -0001b040: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b080: 7c0a 7c6f 3620 3d20 636f 6b65 726e 656c  |.|o6 = cokernel
│ │ │ │ -0001b090: 207c 2061 2062 2063 207c 2020 2020 2020   | a b c |      
│ │ │ │ -0001b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b0c0: 7c20 2020 2020 2020 2020 2020 2020 207c  |              |
│ │ │ │ -0001b0d0: 2062 2063 2061 207c 2020 2020 2020 2020   b c a |        
│ │ │ │ +0001afe0: 2d2d 2d2b 0a7c 6936 203a 204d 203d 2063  ---+.|i6 : M = c
│ │ │ │ +0001aff0: 6f6b 6572 206d 6170 2852 5e32 2c20 525e  oker map(R^2, R^
│ │ │ │ +0001b000: 7b33 3a2d 317d 2c20 7b7b 612c 622c 637d  {3:-1}, {{a,b,c}
│ │ │ │ +0001b010: 2c7b 622c 632c 617d 7d29 2020 2020 2020  ,{b,c,a}})      
│ │ │ │ +0001b020: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b060: 0a7c 6f36 203d 2063 6f6b 6572 6e65 6c20  .|o6 = cokernel 
│ │ │ │ +0001b070: 7c20 6120 6220 6320 7c20 2020 2020 2020  | a b c |       
│ │ │ │ +0001b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001b0a0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +0001b0b0: 6220 6320 6120 7c20 2020 2020 2020 2020  b c a |         
│ │ │ │ +0001b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b0d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b110: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b130: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001b130: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b150: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -0001b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b170: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -0001b180: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965  R-module, quotie
│ │ │ │ -0001b190: 6e74 206f 6620 5220 2020 2020 2020 2020  nt of R         
│ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001b150: 2020 2020 2020 207c 0a7c 6f36 203a 2052         |.|o6 : R
│ │ │ │ +0001b160: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +0001b170: 7420 6f66 2052 2020 2020 2020 2020 2020  t of R          
│ │ │ │ +0001b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b190: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b1f0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6265 7474  ----+.|i7 : bett
│ │ │ │ -0001b200: 6920 2846 4620 3d66 7265 6552 6573 6f6c  i (FF =freeResol
│ │ │ │ -0001b210: 7574 696f 6e28 204d 2c20 4c65 6e67 7468  ution( M, Length
│ │ │ │ -0001b220: 4c69 6d69 7420 3d3e 3629 2920 2020 2020  Limit =>6))     
│ │ │ │ -0001b230: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b270: 7c0a 7c20 2020 2020 2020 2020 2020 2030  |.|            0
│ │ │ │ -0001b280: 2031 2032 2033 2034 2020 3520 2036 2020   1 2 3 4  5  6  
│ │ │ │ -0001b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b2b0: 7c6f 3720 3d20 746f 7461 6c3a 2032 2033  |o7 = total: 2 3
│ │ │ │ -0001b2c0: 2034 2036 2039 2031 3320 3138 2020 2020   4 6 9 13 18    
│ │ │ │ -0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001b2f0: 2020 2020 2020 2020 303a 2032 2033 202e          0: 2 3 .
│ │ │ │ -0001b300: 202e 202e 2020 2e20 202e 2020 2020 2020   . .  .  .      
│ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001b330: 2020 2020 2020 313a 202e 202e 2031 202e        1: . . 1 .
│ │ │ │ -0001b340: 202e 2020 2e20 202e 2020 2020 2020 2020   .  .  .        
│ │ │ │ -0001b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b360: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001b370: 2020 2020 323a 202e 202e 2033 2033 202e      2: . . 3 3 .
│ │ │ │ -0001b380: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ -0001b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b3a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b3b0: 2020 333a 202e 202e 202e 2033 2033 2020    3: . . . 3 3  
│ │ │ │ -0001b3c0: 2e20 202e 2020 2020 2020 2020 2020 2020  .  .            
│ │ │ │ -0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b3e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001b3f0: 343a 202e 202e 202e 202e 2033 2020 3320  4: . . . . 3  3 
│ │ │ │ -0001b400: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -0001b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b420: 2020 7c0a 7c20 2020 2020 2020 2020 353a    |.|         5:
│ │ │ │ -0001b430: 202e 202e 202e 202e 2033 2020 3920 2036   . . . . 3  9  6
│ │ │ │ -0001b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b460: 7c0a 7c20 2020 2020 2020 2020 363a 202e  |.|         6: .
│ │ │ │ -0001b470: 202e 202e 202e 202e 2020 2e20 2033 2020   . . . .  .  3  
│ │ │ │ -0001b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b490: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b4a0: 7c20 2020 2020 2020 2020 373a 202e 202e  |         7: . .
│ │ │ │ -0001b4b0: 202e 202e 202e 2020 3120 2039 2020 2020   . . .  1  9    
│ │ │ │ +0001b1d0: 2d2d 2d2b 0a7c 6937 203a 2062 6574 7469  ---+.|i7 : betti
│ │ │ │ +0001b1e0: 2028 4646 203d 6672 6565 5265 736f 6c75   (FF =freeResolu
│ │ │ │ +0001b1f0: 7469 6f6e 2820 4d2c 204c 656e 6774 684c  tion( M, LengthL
│ │ │ │ +0001b200: 696d 6974 203d 3e36 2929 2020 2020 2020  imit =>6))      
│ │ │ │ +0001b210: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b250: 0a7c 2020 2020 2020 2020 2020 2020 3020  .|            0 
│ │ │ │ +0001b260: 3120 3220 3320 3420 2035 2020 3620 2020  1 2 3 4  5  6   
│ │ │ │ +0001b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001b290: 6f37 203d 2074 6f74 616c 3a20 3220 3320  o7 = total: 2 3 
│ │ │ │ +0001b2a0: 3420 3620 3920 3133 2031 3820 2020 2020  4 6 9 13 18     
│ │ │ │ +0001b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b2c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001b2d0: 2020 2020 2020 2030 3a20 3220 3320 2e20         0: 2 3 . 
│ │ │ │ +0001b2e0: 2e20 2e20 202e 2020 2e20 2020 2020 2020  . .  .  .       
│ │ │ │ +0001b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b300: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001b310: 2020 2020 2031 3a20 2e20 2e20 3120 2e20       1: . . 1 . 
│ │ │ │ +0001b320: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ +0001b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b340: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001b350: 2020 2032 3a20 2e20 2e20 3320 3320 2e20     2: . . 3 3 . 
│ │ │ │ +0001b360: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │ +0001b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b380: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001b390: 2033 3a20 2e20 2e20 2e20 3320 3320 202e   3: . . . 3 3  .
│ │ │ │ +0001b3a0: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +0001b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b3c0: 2020 207c 0a7c 2020 2020 2020 2020 2034     |.|         4
│ │ │ │ +0001b3d0: 3a20 2e20 2e20 2e20 2e20 3320 2033 2020  : . . . . 3  3  
│ │ │ │ +0001b3e0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0001b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b400: 207c 0a7c 2020 2020 2020 2020 2035 3a20   |.|         5: 
│ │ │ │ +0001b410: 2e20 2e20 2e20 2e20 3320 2039 2020 3620  . . . . 3  9  6 
│ │ │ │ +0001b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b440: 0a7c 2020 2020 2020 2020 2036 3a20 2e20  .|         6: . 
│ │ │ │ +0001b450: 2e20 2e20 2e20 2e20 202e 2020 3320 2020  . . . .  .  3   
│ │ │ │ +0001b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001b480: 2020 2020 2020 2020 2037 3a20 2e20 2e20           7: . . 
│ │ │ │ +0001b490: 2e20 2e20 2e20 2031 2020 3920 2020 2020  . . .  1  9     
│ │ │ │ +0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b4b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b4d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b510: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0001b520: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ -0001b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b550: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001b4f0: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ +0001b500: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +0001b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b530: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b590: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53  ------+.|i8 : MS
│ │ │ │ -0001b5a0: 203d 2070 7275 6e65 2070 7573 6846 6f72   = prune pushFor
│ │ │ │ -0001b5b0: 7761 7264 2870 2c20 636f 6b65 7220 4646  ward(p, coker FF
│ │ │ │ -0001b5c0: 2e64 645f 3629 3b20 2020 2020 2020 2020  .dd_6);         
│ │ │ │ -0001b5d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001b570: 2d2d 2d2d 2d2b 0a7c 6938 203a 204d 5320  -----+.|i8 : MS 
│ │ │ │ +0001b580: 3d20 7072 756e 6520 7075 7368 466f 7277  = prune pushForw
│ │ │ │ +0001b590: 6172 6428 702c 2063 6f6b 6572 2046 462e  ard(p, coker FF.
│ │ │ │ +0001b5a0: 6464 5f36 293b 2020 2020 2020 2020 2020  dd_6);          
│ │ │ │ +0001b5b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b610: 2d2d 2b0a 7c69 3920 3a20 7265 7346 6c64  --+.|i9 : resFld
│ │ │ │ -0001b620: 203a 3d20 7075 7368 466f 7277 6172 6428   := pushForward(
│ │ │ │ -0001b630: 702c 2063 6f6b 6572 2076 6172 7320 5229  p, coker vars R)
│ │ │ │ -0001b640: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -0001b650: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0001b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001b690: 7c69 3130 203a 2054 203d 2065 7874 6572  |i10 : T = exter
│ │ │ │ -0001b6a0: 696f 7254 6f72 4d6f 6475 6c65 2866 2c4d  iorTorModule(f,M
│ │ │ │ -0001b6b0: 5329 3b20 2020 2020 2020 2020 2020 2020  S);             
│ │ │ │ -0001b6c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001b5f0: 2d2b 0a7c 6939 203a 2072 6573 466c 6420  -+.|i9 : resFld 
│ │ │ │ +0001b600: 3a3d 2070 7573 6846 6f72 7761 7264 2870  := pushForward(p
│ │ │ │ +0001b610: 2c20 636f 6b65 7220 7661 7273 2052 293b  , coker vars R);
│ │ │ │ +0001b620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b630: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001b640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001b670: 6931 3020 3a20 5420 3d20 6578 7465 7269  i10 : T = exteri
│ │ │ │ +0001b680: 6f72 546f 724d 6f64 756c 6528 662c 4d53  orTorModule(f,MS
│ │ │ │ +0001b690: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +0001b6a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b700: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -0001b710: 203a 2045 203d 2065 7874 6572 696f 7245   : E = exteriorE
│ │ │ │ -0001b720: 7874 4d6f 6475 6c65 2866 2c4d 5329 3b20  xtModule(f,MS); 
│ │ │ │ -0001b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b740: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001b6e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +0001b6f0: 3a20 4520 3d20 6578 7465 7269 6f72 4578  : E = exteriorEx
│ │ │ │ +0001b700: 744d 6f64 756c 6528 662c 4d53 293b 2020  tModule(f,MS);  
│ │ │ │ +0001b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b720: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001b730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b780: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2068  ------+.|i12 : h
│ │ │ │ -0001b790: 6628 2d34 2e2e 302c 4529 2020 2020 2020  f(-4..0,E)      
│ │ │ │ -0001b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b760: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 6866  -----+.|i12 : hf
│ │ │ │ +0001b770: 282d 342e 2e30 2c45 2920 2020 2020 2020  (-4..0,E)       
│ │ │ │ +0001b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b7a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b800: 2020 7c0a 7c6f 3132 203d 207b 302c 2039    |.|o12 = {0, 9
│ │ │ │ -0001b810: 2c20 3239 2c20 3333 2c20 3133 7d20 2020  , 29, 33, 13}   
│ │ │ │ -0001b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b7e0: 207c 0a7c 6f31 3220 3d20 7b30 2c20 392c   |.|o12 = {0, 9,
│ │ │ │ +0001b7f0: 2032 392c 2033 332c 2031 337d 2020 2020   29, 33, 13}    
│ │ │ │ +0001b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001b820: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0001b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b840: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001b880: 7c6f 3132 203a 204c 6973 7420 2020 2020  |o12 : List     
│ │ │ │ -0001b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001b860: 6f31 3220 3a20 4c69 7374 2020 2020 2020  o12 : List      
│ │ │ │ +0001b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b890: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001b8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001b8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133  ----------+.|i13
│ │ │ │ -0001b900: 203a 2062 6574 7469 2066 7265 6552 6573   : betti freeRes
│ │ │ │ -0001b910: 6f6c 7574 696f 6e20 4d53 2020 2020 2020  olution MS      
│ │ │ │ +0001b8d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320  ---------+.|i13 
│ │ │ │ +0001b8e0: 3a20 6265 7474 6920 6672 6565 5265 736f  : betti freeReso
│ │ │ │ +0001b8f0: 6c75 7469 6f6e 204d 5320 2020 2020 2020  lution MS       
│ │ │ │ +0001b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b910: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b930: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b970: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b980: 2020 2020 2020 2030 2020 3120 2032 2033         0  1  2 3
│ │ │ │ -0001b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9b0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74      |.|o13 = tot
│ │ │ │ -0001b9c0: 616c 3a20 3133 2033 3320 3239 2039 2020  al: 13 33 29 9  
│ │ │ │ -0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9f0: 2020 7c0a 7c20 2020 2020 2020 2020 2039    |.|          9
│ │ │ │ -0001ba00: 3a20 2033 2020 2e20 202e 202e 2020 2020  :  3  .  . .    
│ │ │ │ -0001ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba30: 7c0a 7c20 2020 2020 2020 2020 3130 3a20  |.|         10: 
│ │ │ │ -0001ba40: 2039 2020 3620 202e 202e 2020 2020 2020   9  6  . .      
│ │ │ │ -0001ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ba70: 7c20 2020 2020 2020 2020 3131 3a20 202e  |         11:  .
│ │ │ │ -0001ba80: 2020 3320 202e 202e 2020 2020 2020 2020    3  . .        
│ │ │ │ -0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001baa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001bab0: 2020 2020 2020 2020 3132 3a20 2031 2031          12:  1 1
│ │ │ │ -0001bac0: 3520 202e 202e 2020 2020 2020 2020 2020  5  . .          
│ │ │ │ -0001bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001baf0: 2020 2020 2020 3133 3a20 202e 2020 3920        13:  .  9 
│ │ │ │ -0001bb00: 2038 202e 2020 2020 2020 2020 2020 2020   8 .            
│ │ │ │ -0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001bb30: 2020 2020 3134 3a20 202e 2020 2e20 2036      14:  .  .  6
│ │ │ │ -0001bb40: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001bb70: 2020 3135 3a20 202e 2020 2e20 3132 202e    15:  .  . 12 .
│ │ │ │ -0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001bbb0: 3136 3a20 202e 2020 2e20 2033 2033 2020  16:  .  .  3 3  
│ │ │ │ -0001bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bbe0: 2020 7c0a 7c20 2020 2020 2020 2020 3137    |.|         17
│ │ │ │ -0001bbf0: 3a20 202e 2020 2e20 202e 2033 2020 2020  :  .  .  . 3    
│ │ │ │ -0001bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc20: 7c0a 7c20 2020 2020 2020 2020 3138 3a20  |.|         18: 
│ │ │ │ -0001bc30: 202e 2020 2e20 202e 2033 2020 2020 2020   .  .  . 3      
│ │ │ │ +0001b950: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001b960: 2020 2020 2020 3020 2031 2020 3220 3320        0  1  2 3 
│ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b990: 2020 207c 0a7c 6f31 3320 3d20 746f 7461     |.|o13 = tota
│ │ │ │ +0001b9a0: 6c3a 2031 3320 3333 2032 3920 3920 2020  l: 13 33 29 9   
│ │ │ │ +0001b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001b9d0: 207c 0a7c 2020 2020 2020 2020 2020 393a   |.|          9:
│ │ │ │ +0001b9e0: 2020 3320 202e 2020 2e20 2e20 2020 2020    3  .  . .     
│ │ │ │ +0001b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ba00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001ba10: 0a7c 2020 2020 2020 2020 2031 303a 2020  .|         10:  
│ │ │ │ +0001ba20: 3920 2036 2020 2e20 2e20 2020 2020 2020  9  6  . .       
│ │ │ │ +0001ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ba40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ba50: 2020 2020 2020 2020 2031 313a 2020 2e20           11:  . 
│ │ │ │ +0001ba60: 2033 2020 2e20 2e20 2020 2020 2020 2020   3  . .         
│ │ │ │ +0001ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ba80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001ba90: 2020 2020 2020 2031 323a 2020 3120 3135         12:  1 15
│ │ │ │ +0001baa0: 2020 2e20 2e20 2020 2020 2020 2020 2020    . .           
│ │ │ │ +0001bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bac0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001bad0: 2020 2020 2031 333a 2020 2e20 2039 2020       13:  .  9  
│ │ │ │ +0001bae0: 3820 2e20 2020 2020 2020 2020 2020 2020  8 .             
│ │ │ │ +0001baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001bb10: 2020 2031 343a 2020 2e20 202e 2020 3620     14:  .  .  6 
│ │ │ │ +0001bb20: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0001bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001bb50: 2031 353a 2020 2e20 202e 2031 3220 2e20   15:  .  . 12 . 
│ │ │ │ +0001bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bb80: 2020 207c 0a7c 2020 2020 2020 2020 2031     |.|         1
│ │ │ │ +0001bb90: 363a 2020 2e20 202e 2020 3320 3320 2020  6:  .  .  3 3   
│ │ │ │ +0001bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bbc0: 207c 0a7c 2020 2020 2020 2020 2031 373a   |.|         17:
│ │ │ │ +0001bbd0: 2020 2e20 202e 2020 2e20 3320 2020 2020    .  .  . 3     
│ │ │ │ +0001bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bbf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001bc00: 0a7c 2020 2020 2020 2020 2031 383a 2020  .|         18:  
│ │ │ │ +0001bc10: 2e20 202e 2020 2e20 3320 2020 2020 2020  .  .  . 3       
│ │ │ │ +0001bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bc30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001bc60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0001bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001bca0: 3133 203a 2042 6574 7469 5461 6c6c 7920  13 : BettiTally 
│ │ │ │ -0001bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bcd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bc70: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001bc80: 3320 3a20 4265 7474 6954 616c 6c79 2020  3 : BettiTally  
│ │ │ │ +0001bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bcb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │ -0001bd20: 2062 6574 7469 2066 7265 6552 6573 6f6c   betti freeResol
│ │ │ │ -0001bd30: 7574 696f 6e20 2850 4520 3d20 7072 756e  ution (PE = prun
│ │ │ │ -0001bd40: 6520 452c 204c 656e 6774 684c 696d 6974  e E, LengthLimit
│ │ │ │ -0001bd50: 203d 3e20 3629 7c0a 7c20 2020 2020 2020   => 6)|.|       
│ │ │ │ +0001bcf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ +0001bd00: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ +0001bd10: 7469 6f6e 2028 5045 203d 2070 7275 6e65  tion (PE = prune
│ │ │ │ +0001bd20: 2045 2c20 4c65 6e67 7468 4c69 6d69 7420   E, LengthLimit 
│ │ │ │ +0001bd30: 3d3e 2036 297c 0a7c 2020 2020 2020 2020  => 6)|.|        
│ │ │ │ +0001bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bd90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001bda0: 2020 2020 2030 2020 3120 2032 2020 3320       0  1  2  3 
│ │ │ │ -0001bdb0: 2034 2020 2035 2020 2036 2020 2020 2020   4   5   6      
│ │ │ │ -0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bdd0: 2020 7c0a 7c6f 3134 203d 2074 6f74 616c    |.|o14 = total
│ │ │ │ -0001bde0: 3a20 3136 2031 3320 3235 2034 3920 3831  : 16 13 25 49 81
│ │ │ │ -0001bdf0: 2031 3231 2031 3639 2020 2020 2020 2020   121 169        
│ │ │ │ -0001be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be10: 7c0a 7c20 2020 2020 2020 2020 2d33 3a20  |.|         -3: 
│ │ │ │ -0001be20: 2039 2020 3420 2033 2020 3320 2033 2020   9  4  3  3  3  
│ │ │ │ -0001be30: 2033 2020 2033 2020 2020 2020 2020 2020   3   3          
│ │ │ │ -0001be40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001be50: 7c20 2020 2020 2020 2020 2d32 3a20 2036  |         -2:  6
│ │ │ │ -0001be60: 2020 3320 202e 2020 2e20 202e 2020 202e    3  .  .  .   .
│ │ │ │ -0001be70: 2020 202e 2020 2020 2020 2020 2020 2020     .            
│ │ │ │ -0001be80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001be90: 2020 2020 2020 2020 2d31 3a20 202e 2020          -1:  .  
│ │ │ │ -0001bea0: 2e20 2037 2031 3820 3333 2020 3532 2020  .  7 18 33  52  
│ │ │ │ -0001beb0: 3735 2020 2020 2020 2020 2020 2020 2020  75              
│ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001bed0: 2020 2020 2020 2030 3a20 2031 2020 3620         0:  1  6 
│ │ │ │ -0001bee0: 3135 2032 3820 3435 2020 3636 2020 3931  15 28 45  66  91
│ │ │ │ +0001bd70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001bd80: 2020 2020 3020 2031 2020 3220 2033 2020      0  1  2  3  
│ │ │ │ +0001bd90: 3420 2020 3520 2020 3620 2020 2020 2020  4   5   6       
│ │ │ │ +0001bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bdb0: 207c 0a7c 6f31 3420 3d20 746f 7461 6c3a   |.|o14 = total:
│ │ │ │ +0001bdc0: 2031 3620 3133 2032 3520 3439 2038 3120   16 13 25 49 81 
│ │ │ │ +0001bdd0: 3132 3120 3136 3920 2020 2020 2020 2020  121 169         
│ │ │ │ +0001bde0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001bdf0: 0a7c 2020 2020 2020 2020 202d 333a 2020  .|         -3:  
│ │ │ │ +0001be00: 3920 2034 2020 3320 2033 2020 3320 2020  9  4  3  3  3   
│ │ │ │ +0001be10: 3320 2020 3320 2020 2020 2020 2020 2020  3   3           
│ │ │ │ +0001be20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001be30: 2020 2020 2020 2020 202d 323a 2020 3620           -2:  6 
│ │ │ │ +0001be40: 2033 2020 2e20 202e 2020 2e20 2020 2e20   3  .  .  .   . 
│ │ │ │ +0001be50: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +0001be60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001be70: 2020 2020 2020 202d 313a 2020 2e20 202e         -1:  .  .
│ │ │ │ +0001be80: 2020 3720 3138 2033 3320 2035 3220 2037    7 18 33  52  7
│ │ │ │ +0001be90: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ +0001bea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001beb0: 2020 2020 2020 303a 2020 3120 2036 2031        0:  1  6 1
│ │ │ │ +0001bec0: 3520 3238 2034 3520 2036 3620 2039 3120  5 28 45  66  91 
│ │ │ │ +0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bee0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf40: 2020 2020 2020 7c0a 7c6f 3134 203a 2042        |.|o14 : B
│ │ │ │ -0001bf50: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ -0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001bf20: 2020 2020 207c 0a7c 6f31 3420 3a20 4265       |.|o14 : Be
│ │ │ │ +0001bf30: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +0001bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bf60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001bf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001bf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bfc0: 2d2d 2b0a 7c69 3135 203a 2062 6574 7469  --+.|i15 : betti
│ │ │ │ -0001bfd0: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ -0001bfe0: 2850 5420 3d20 7072 756e 6520 542c 204c  (PT = prune T, L
│ │ │ │ -0001bff0: 656e 6774 684c 696d 6974 203d 3e20 3629  engthLimit => 6)
│ │ │ │ -0001c000: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001c040: 7c20 2020 2020 2020 2020 2020 2020 2030  |              0
│ │ │ │ -0001c050: 2020 3120 2032 2020 2033 2020 2034 2020    1  2   3   4  
│ │ │ │ -0001c060: 2035 2020 2036 2020 2020 2020 2020 2020   5   6          
│ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001c080: 3135 203d 2074 6f74 616c 3a20 3331 2035  15 = total: 31 5
│ │ │ │ -0001c090: 3520 3837 2031 3237 2031 3735 2032 3331  5 87 127 175 231
│ │ │ │ -0001c0a0: 2032 3935 2020 2020 2020 2020 2020 2020   295            
│ │ │ │ -0001c0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001c0c0: 2020 2020 2020 2030 3a20 3133 2032 3420         0: 13 24 
│ │ │ │ -0001c0d0: 3339 2020 3538 2020 3831 2031 3038 2031  39  58  81 108 1
│ │ │ │ -0001c0e0: 3339 2020 2020 2020 2020 2020 2020 2020  39              
│ │ │ │ -0001c0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001c100: 2020 2020 2031 3a20 3138 2033 3120 3438       1: 18 31 48
│ │ │ │ -0001c110: 2020 3639 2020 3934 2031 3233 2031 3536    69  94 123 156
│ │ │ │ +0001bfa0: 2d2b 0a7c 6931 3520 3a20 6265 7474 6920  -+.|i15 : betti 
│ │ │ │ +0001bfb0: 6672 6565 5265 736f 6c75 7469 6f6e 2028  freeResolution (
│ │ │ │ +0001bfc0: 5054 203d 2070 7275 6e65 2054 2c20 4c65  PT = prune T, Le
│ │ │ │ +0001bfd0: 6e67 7468 4c69 6d69 7420 3d3e 2036 297c  ngthLimit => 6)|
│ │ │ │ +0001bfe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c020: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +0001c030: 2031 2020 3220 2020 3320 2020 3420 2020   1  2   3   4   
│ │ │ │ +0001c040: 3520 2020 3620 2020 2020 2020 2020 2020  5   6           
│ │ │ │ +0001c050: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001c060: 3520 3d20 746f 7461 6c3a 2033 3120 3535  5 = total: 31 55
│ │ │ │ +0001c070: 2038 3720 3132 3720 3137 3520 3233 3120   87 127 175 231 
│ │ │ │ +0001c080: 3239 3520 2020 2020 2020 2020 2020 2020  295             
│ │ │ │ +0001c090: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001c0a0: 2020 2020 2020 303a 2031 3320 3234 2033        0: 13 24 3
│ │ │ │ +0001c0b0: 3920 2035 3820 2038 3120 3130 3820 3133  9  58  81 108 13
│ │ │ │ +0001c0c0: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ +0001c0d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001c0e0: 2020 2020 313a 2031 3820 3331 2034 3820      1: 18 31 48 
│ │ │ │ +0001c0f0: 2036 3920 2039 3420 3132 3320 3135 3620   69  94 123 156 
│ │ │ │ +0001c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c130: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c170: 2020 2020 7c0a 7c6f 3135 203a 2042 6574      |.|o15 : Bet
│ │ │ │ -0001c180: 7469 5461 6c6c 7920 2020 2020 2020 2020  tiTally         
│ │ │ │ -0001c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0001c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c1f0: 2b0a 7c69 3136 203a 2045 3120 3d20 7072  +.|i16 : E1 = pr
│ │ │ │ -0001c200: 756e 6520 6578 7465 7269 6f72 4578 744d  une exteriorExtM
│ │ │ │ -0001c210: 6f64 756c 6528 662c 204d 532c 2072 6573  odule(f, MS, res
│ │ │ │ -0001c220: 466c 6429 3b20 2020 2020 2020 2020 7c0a  Fld);         |.
│ │ │ │ -0001c230: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001c270: 3137 203a 2072 696e 6720 4531 2020 2020  17 : ring E1    
│ │ │ │ -0001c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c150: 2020 207c 0a7c 6f31 3520 3a20 4265 7474     |.|o15 : Bett
│ │ │ │ +0001c160: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ +0001c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c190: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001c1d0: 0a7c 6931 3620 3a20 4531 203d 2070 7275  .|i16 : E1 = pru
│ │ │ │ +0001c1e0: 6e65 2065 7874 6572 696f 7245 7874 4d6f  ne exteriorExtMo
│ │ │ │ +0001c1f0: 6475 6c65 2866 2c20 4d53 2c20 7265 7346  dule(f, MS, resF
│ │ │ │ +0001c200: 6c64 293b 2020 2020 2020 2020 207c 0a2b  ld);         |.+
│ │ │ │ +0001c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001c250: 3720 3a20 7269 6e67 2045 3120 2020 2020  7 : ring E1     
│ │ │ │ +0001c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c280: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2e0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ -0001c2f0: 206b 6b5b 5820 2e2e 5820 2c20 6520 2e2e   kk[X ..X , e ..
│ │ │ │ -0001c300: 6520 5d20 2020 2020 2020 2020 2020 2020  e ]             
│ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001c330: 2020 2030 2020 2032 2020 2030 2020 2032     0   2   0   2
│ │ │ │ -0001c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c2c0: 2020 2020 2020 207c 0a7c 6f31 3720 3d20         |.|o17 = 
│ │ │ │ +0001c2d0: 6b6b 5b58 202e 2e58 202c 2065 202e 2e65  kk[X ..X , e ..e
│ │ │ │ +0001c2e0: 205d 2020 2020 2020 2020 2020 2020 2020   ]              
│ │ │ │ +0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001c310: 2020 3020 2020 3220 2020 3020 2020 3220    0   2   0   2 
│ │ │ │ +0001c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c340: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c360: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3a0: 2020 7c0a 7c6f 3137 203a 2050 6f6c 796e    |.|o17 : Polyn
│ │ │ │ -0001c3b0: 6f6d 6961 6c52 696e 672c 2033 2073 6b65  omialRing, 3 ske
│ │ │ │ -0001c3c0: 7720 636f 6d6d 7574 6174 6976 6520 7661  w commutative va
│ │ │ │ -0001c3d0: 7269 6162 6c65 2873 2920 2020 2020 2020  riable(s)       
│ │ │ │ -0001c3e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0001c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001c420: 7c69 3138 203a 2065 7852 696e 6720 3d20  |i18 : exRing = 
│ │ │ │ -0001c430: 6b6b 5b65 5f30 2c65 5f31 2c65 5f32 2c20  kk[e_0,e_1,e_2, 
│ │ │ │ -0001c440: 536b 6577 436f 6d6d 7574 6174 6976 6520  SkewCommutative 
│ │ │ │ -0001c450: 3d3e 7472 7565 5d20 2020 2020 7c0a 7c20  =>true]     |.| 
│ │ │ │ +0001c380: 207c 0a7c 6f31 3720 3a20 506f 6c79 6e6f   |.|o17 : Polyno
│ │ │ │ +0001c390: 6d69 616c 5269 6e67 2c20 3320 736b 6577  mialRing, 3 skew
│ │ │ │ +0001c3a0: 2063 6f6d 6d75 7461 7469 7665 2076 6172   commutative var
│ │ │ │ +0001c3b0: 6961 626c 6528 7329 2020 2020 2020 207c  iable(s)       |
│ │ │ │ +0001c3c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001c3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001c400: 6931 3820 3a20 6578 5269 6e67 203d 206b  i18 : exRing = k
│ │ │ │ +0001c410: 6b5b 655f 302c 655f 312c 655f 322c 2053  k[e_0,e_1,e_2, S
│ │ │ │ +0001c420: 6b65 7743 6f6d 6d75 7461 7469 7665 203d  kewCommutative =
│ │ │ │ +0001c430: 3e74 7275 655d 2020 2020 207c 0a7c 2020  >true]     |.|  
│ │ │ │ +0001c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c490: 2020 2020 2020 2020 2020 7c0a 7c6f 3138            |.|o18
│ │ │ │ -0001c4a0: 203d 2065 7852 696e 6720 2020 2020 2020   = exRing       
│ │ │ │ -0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c470: 2020 2020 2020 2020 207c 0a7c 6f31 3820           |.|o18 
│ │ │ │ +0001c480: 3d20 6578 5269 6e67 2020 2020 2020 2020  = exRing        
│ │ │ │ +0001c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c4b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c510: 2020 2020 2020 7c0a 7c6f 3138 203a 2050        |.|o18 : P
│ │ │ │ -0001c520: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033  olynomialRing, 3
│ │ │ │ -0001c530: 2073 6b65 7720 636f 6d6d 7574 6174 6976   skew commutativ
│ │ │ │ -0001c540: 6520 7661 7269 6162 6c65 2873 2920 2020  e variable(s)   
│ │ │ │ -0001c550: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001c4f0: 2020 2020 207c 0a7c 6f31 3820 3a20 506f       |.|o18 : Po
│ │ │ │ +0001c500: 6c79 6e6f 6d69 616c 5269 6e67 2c20 3320  lynomialRing, 3 
│ │ │ │ +0001c510: 736b 6577 2063 6f6d 6d75 7461 7469 7665  skew commutative
│ │ │ │ +0001c520: 2076 6172 6961 626c 6528 7329 2020 2020   variable(s)    
│ │ │ │ +0001c530: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c590: 2d2d 2b0a 0a57 6520 6361 6e20 616c 736f  --+..We can also
│ │ │ │ -0001c5a0: 2063 6f6e 7374 7275 6374 2074 6865 2065   construct the e
│ │ │ │ -0001c5b0: 7874 6572 696f 7245 7874 4d6f 6475 6c65  xteriorExtModule
│ │ │ │ -0001c5c0: 2061 7320 6120 6269 6772 6164 6564 206d   as a bigraded m
│ │ │ │ -0001c5d0: 6f64 756c 652c 206f 7665 7220 6120 7269  odule, over a ri
│ │ │ │ -0001c5e0: 6e67 0a53 4520 7468 6174 2068 6173 2062  ng.SE that has b
│ │ │ │ -0001c5f0: 6f74 6820 706f 6c79 6e6f 6d69 616c 2076  oth polynomial v
│ │ │ │ -0001c600: 6172 6961 626c 6573 206c 696b 6520 5320  ariables like S 
│ │ │ │ -0001c610: 616e 6420 6578 7465 7269 6f72 2076 6172  and exterior var
│ │ │ │ -0001c620: 6961 626c 6573 206c 696b 6520 452e 2054  iables like E. T
│ │ │ │ -0001c630: 6865 0a70 6f6c 796e 6f6d 6961 6c20 7661  he.polynomial va
│ │ │ │ -0001c640: 7269 6162 6c65 7320 6861 7665 2064 6567  riables have deg
│ │ │ │ -0001c650: 7265 6573 207b 312c 307d 2e20 5468 6520  rees {1,0}. The 
│ │ │ │ -0001c660: 6578 7465 7269 6f72 2076 6172 6961 626c  exterior variabl
│ │ │ │ -0001c670: 6573 2068 6176 6520 6465 6772 6565 730a  es have degrees.
│ │ │ │ -0001c680: 7b64 6567 2066 665f 692c 2031 7d2e 0a0a  {deg ff_i, 1}...
│ │ │ │ -0001c690: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0001c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c6c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20  -------+.|i19 : 
│ │ │ │ -0001c6d0: 4531 203d 2070 7275 6e65 2065 7874 6572  E1 = prune exter
│ │ │ │ -0001c6e0: 696f 7245 7874 4d6f 6475 6c65 2866 2c20  iorExtModule(f, 
│ │ │ │ -0001c6f0: 4d53 2c20 7265 7346 6c64 293b 2020 2020  MS, resFld);    
│ │ │ │ -0001c700: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0001c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c730: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020  ---------+.|i20 
│ │ │ │ -0001c740: 3a20 7269 6e67 2045 3120 2020 2020 2020  : ring E1       
│ │ │ │ -0001c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c570: 2d2b 0a0a 5765 2063 616e 2061 6c73 6f20  -+..We can also 
│ │ │ │ +0001c580: 636f 6e73 7472 7563 7420 7468 6520 6578  construct the ex
│ │ │ │ +0001c590: 7465 7269 6f72 4578 744d 6f64 756c 6520  teriorExtModule 
│ │ │ │ +0001c5a0: 6173 2061 2062 6967 7261 6465 6420 6d6f  as a bigraded mo
│ │ │ │ +0001c5b0: 6475 6c65 2c20 6f76 6572 2061 2072 696e  dule, over a rin
│ │ │ │ +0001c5c0: 670a 5345 2074 6861 7420 6861 7320 626f  g.SE that has bo
│ │ │ │ +0001c5d0: 7468 2070 6f6c 796e 6f6d 6961 6c20 7661  th polynomial va
│ │ │ │ +0001c5e0: 7269 6162 6c65 7320 6c69 6b65 2053 2061  riables like S a
│ │ │ │ +0001c5f0: 6e64 2065 7874 6572 696f 7220 7661 7269  nd exterior vari
│ │ │ │ +0001c600: 6162 6c65 7320 6c69 6b65 2045 2e20 5468  ables like E. Th
│ │ │ │ +0001c610: 650a 706f 6c79 6e6f 6d69 616c 2076 6172  e.polynomial var
│ │ │ │ +0001c620: 6961 626c 6573 2068 6176 6520 6465 6772  iables have degr
│ │ │ │ +0001c630: 6565 7320 7b31 2c30 7d2e 2054 6865 2065  ees {1,0}. The e
│ │ │ │ +0001c640: 7874 6572 696f 7220 7661 7269 6162 6c65  xterior variable
│ │ │ │ +0001c650: 7320 6861 7665 2064 6567 7265 6573 0a7b  s have degrees.{
│ │ │ │ +0001c660: 6465 6720 6666 5f69 2c20 317d 2e0a 0a2b  deg ff_i, 1}...+
│ │ │ │ +0001c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c6a0: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2045  ------+.|i19 : E
│ │ │ │ +0001c6b0: 3120 3d20 7072 756e 6520 6578 7465 7269  1 = prune exteri
│ │ │ │ +0001c6c0: 6f72 4578 744d 6f64 756c 6528 662c 204d  orExtModule(f, M
│ │ │ │ +0001c6d0: 532c 2072 6573 466c 6429 3b20 2020 207c  S, resFld);    |
│ │ │ │ +0001c6e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c710: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a  --------+.|i20 :
│ │ │ │ +0001c720: 2072 696e 6720 4531 2020 2020 2020 2020   ring E1        
│ │ │ │ +0001c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c750: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c770: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0001c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7a0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0001c7b0: 3020 3d20 6b6b 5b58 202e 2e58 202c 2065  0 = kk[X ..X , e
│ │ │ │ -0001c7c0: 202e 2e65 205d 2020 2020 2020 2020 2020   ..e ]          
│ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c7e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001c7f0: 2030 2020 2032 2020 2030 2020 2032 2020   0   2   0   2  
│ │ │ │ +0001c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c780: 2020 2020 2020 2020 2020 7c0a 7c6f 3230            |.|o20
│ │ │ │ +0001c790: 203d 206b 6b5b 5820 2e2e 5820 2c20 6520   = kk[X ..X , e 
│ │ │ │ +0001c7a0: 2e2e 6520 5d20 2020 2020 2020 2020 2020  ..e ]           
│ │ │ │ +0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c7c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001c7d0: 3020 2020 3220 2020 3020 2020 3220 2020  0   2   0   2   
│ │ │ │ +0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c810: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c850: 2020 2020 2020 7c0a 7c6f 3230 203a 2050        |.|o20 : P
│ │ │ │ -0001c860: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033  olynomialRing, 3
│ │ │ │ -0001c870: 2073 6b65 7720 636f 6d6d 7574 6174 6976   skew commutativ
│ │ │ │ -0001c880: 6520 7661 7269 6162 6c65 2873 2920 207c  e variable(s)  |
│ │ │ │ -0001c890: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0001c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c8c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a  --------+.|i21 :
│ │ │ │ -0001c8d0: 2065 7852 696e 6720 3d20 6b6b 5b65 5f30   exRing = kk[e_0
│ │ │ │ -0001c8e0: 2c65 5f31 2c65 5f32 2c20 536b 6577 436f  ,e_1,e_2, SkewCo
│ │ │ │ -0001c8f0: 6d6d 7574 6174 6976 6520 3d3e 7472 7565  mmutative =>true
│ │ │ │ -0001c900: 5d7c 0a7c 2020 2020 2020 2020 2020 2020  ]|.|            
│ │ │ │ -0001c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c930: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ -0001c940: 203d 2065 7852 696e 6720 2020 2020 2020   = exRing       
│ │ │ │ -0001c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c830: 2020 2020 207c 0a7c 6f32 3020 3a20 506f       |.|o20 : Po
│ │ │ │ +0001c840: 6c79 6e6f 6d69 616c 5269 6e67 2c20 3320  lynomialRing, 3 
│ │ │ │ +0001c850: 736b 6577 2063 6f6d 6d75 7461 7469 7665  skew commutative
│ │ │ │ +0001c860: 2076 6172 6961 626c 6528 7329 2020 7c0a   variable(s)  |.
│ │ │ │ +0001c870: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001c880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c8a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20  -------+.|i21 : 
│ │ │ │ +0001c8b0: 6578 5269 6e67 203d 206b 6b5b 655f 302c  exRing = kk[e_0,
│ │ │ │ +0001c8c0: 655f 312c 655f 322c 2053 6b65 7743 6f6d  e_1,e_2, SkewCom
│ │ │ │ +0001c8d0: 6d75 7461 7469 7665 203d 3e74 7275 655d  mutative =>true]
│ │ │ │ +0001c8e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c910: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +0001c920: 3d20 6578 5269 6e67 2020 2020 2020 2020  = exRing        
│ │ │ │ +0001c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c950: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0001c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c970: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0001c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001c9b0: 3231 203a 2050 6f6c 796e 6f6d 6961 6c52  21 : PolynomialR
│ │ │ │ -0001c9c0: 696e 672c 2033 2073 6b65 7720 636f 6d6d  ing, 3 skew comm
│ │ │ │ -0001c9d0: 7574 6174 6976 6520 7661 7269 6162 6c65  utative variable
│ │ │ │ -0001c9e0: 2873 2920 207c 0a2b 2d2d 2d2d 2d2d 2d2d  (s)  |.+--------
│ │ │ │ -0001c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001ca20: 0a54 6f20 7365 6520 7468 6174 2074 6869  .To see that thi
│ │ │ │ -0001ca30: 7320 6973 2072 6561 6c6c 7920 7468 6520  s is really the 
│ │ │ │ -0001ca40: 7361 6d65 206d 6f64 756c 652c 2077 6974  same module, wit
│ │ │ │ -0001ca50: 6820 6120 6d6f 7265 2063 6f6d 706c 6578  h a more complex
│ │ │ │ -0001ca60: 2067 7261 6469 6e67 2c20 7765 2063 616e   grading, we can
│ │ │ │ -0001ca70: 0a62 7269 6e67 2069 7420 6f76 6572 2074  .bring it over t
│ │ │ │ -0001ca80: 6f20 6120 7075 7265 2065 7874 6572 696f  o a pure exterio
│ │ │ │ -0001ca90: 7220 616c 6765 6272 612e 204e 6f74 6520  r algebra. Note 
│ │ │ │ -0001caa0: 7468 6174 2074 6865 206e 6563 6573 7361  that the necessa
│ │ │ │ -0001cab0: 7279 206d 6170 206f 6620 7269 6e67 730a  ry map of rings.
│ │ │ │ -0001cac0: 6d75 7374 2063 6f6e 7461 696e 2061 2044  must contain a D
│ │ │ │ -0001cad0: 6567 7265 654d 6170 206f 7074 696f 6e2e  egreeMap option.
│ │ │ │ -0001cae0: 2049 6e20 6765 6e65 7261 6c20 7765 2063   In general we c
│ │ │ │ -0001caf0: 6f75 6c64 206f 6e6c 7920 7461 6b65 2074  ould only take t
│ │ │ │ -0001cb00: 6865 2064 6567 7265 6573 206f 660a 7468  he degrees of.th
│ │ │ │ -0001cb10: 6520 6765 6e65 7261 746f 7273 206f 6620  e generators of 
│ │ │ │ -0001cb20: 7468 6520 6578 7465 7269 6f72 2061 6c67  the exterior alg
│ │ │ │ -0001cb30: 6562 7261 2074 6f20 6265 2074 6865 2067  ebra to be the g
│ │ │ │ -0001cb40: 6364 206f 6620 2074 6865 2064 6567 2066  cd of  the deg f
│ │ │ │ -0001cb50: 665f 6920 3b20 696e 2074 6865 0a65 7861  f_i ; in the.exa
│ │ │ │ -0001cb60: 6d70 6c65 2061 626f 7665 2074 6869 7320  mple above this 
│ │ │ │ -0001cb70: 6973 2031 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  is 1...+--------
│ │ │ │ +0001c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001c980: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001c990: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269  1 : PolynomialRi
│ │ │ │ +0001c9a0: 6e67 2c20 3320 736b 6577 2063 6f6d 6d75  ng, 3 skew commu
│ │ │ │ +0001c9b0: 7461 7469 7665 2076 6172 6961 626c 6528  tative variable(
│ │ │ │ +0001c9c0: 7329 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d  s)  |.+---------
│ │ │ │ +0001c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +0001ca00: 546f 2073 6565 2074 6861 7420 7468 6973  To see that this
│ │ │ │ +0001ca10: 2069 7320 7265 616c 6c79 2074 6865 2073   is really the s
│ │ │ │ +0001ca20: 616d 6520 6d6f 6475 6c65 2c20 7769 7468  ame module, with
│ │ │ │ +0001ca30: 2061 206d 6f72 6520 636f 6d70 6c65 7820   a more complex 
│ │ │ │ +0001ca40: 6772 6164 696e 672c 2077 6520 6361 6e0a  grading, we can.
│ │ │ │ +0001ca50: 6272 696e 6720 6974 206f 7665 7220 746f  bring it over to
│ │ │ │ +0001ca60: 2061 2070 7572 6520 6578 7465 7269 6f72   a pure exterior
│ │ │ │ +0001ca70: 2061 6c67 6562 7261 2e20 4e6f 7465 2074   algebra. Note t
│ │ │ │ +0001ca80: 6861 7420 7468 6520 6e65 6365 7373 6172  hat the necessar
│ │ │ │ +0001ca90: 7920 6d61 7020 6f66 2072 696e 6773 0a6d  y map of rings.m
│ │ │ │ +0001caa0: 7573 7420 636f 6e74 6169 6e20 6120 4465  ust contain a De
│ │ │ │ +0001cab0: 6772 6565 4d61 7020 6f70 7469 6f6e 2e20  greeMap option. 
│ │ │ │ +0001cac0: 496e 2067 656e 6572 616c 2077 6520 636f  In general we co
│ │ │ │ +0001cad0: 756c 6420 6f6e 6c79 2074 616b 6520 7468  uld only take th
│ │ │ │ +0001cae0: 6520 6465 6772 6565 7320 6f66 0a74 6865  e degrees of.the
│ │ │ │ +0001caf0: 2067 656e 6572 6174 6f72 7320 6f66 2074   generators of t
│ │ │ │ +0001cb00: 6865 2065 7874 6572 696f 7220 616c 6765  he exterior alge
│ │ │ │ +0001cb10: 6272 6120 746f 2062 6520 7468 6520 6763  bra to be the gc
│ │ │ │ +0001cb20: 6420 6f66 2020 7468 6520 6465 6720 6666  d of  the deg ff
│ │ │ │ +0001cb30: 5f69 203b 2069 6e20 7468 650a 6578 616d  _i ; in the.exam
│ │ │ │ +0001cb40: 706c 6520 6162 6f76 6520 7468 6973 2069  ple above this i
│ │ │ │ +0001cb50: 7320 312e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  s 1...+---------
│ │ │ │ +0001cb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cbc0: 2d2d 2b0a 7c69 3232 203a 2071 203d 206d  --+.|i22 : q = m
│ │ │ │ -0001cbd0: 6170 2865 7852 696e 672c 2072 696e 6720  ap(exRing, ring 
│ │ │ │ -0001cbe0: 4531 2c20 7b33 3a30 2c65 5f30 2c65 5f31  E1, {3:0,e_0,e_1
│ │ │ │ -0001cbf0: 2c65 5f32 7d2c 2044 6567 7265 654d 6170  ,e_2}, DegreeMap
│ │ │ │ -0001cc00: 203d 3e20 6420 2d3e 207b 645f 317d 297c   => d -> {d_1})|
│ │ │ │ -0001cc10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001cba0: 2d2b 0a7c 6932 3220 3a20 7120 3d20 6d61  -+.|i22 : q = ma
│ │ │ │ +0001cbb0: 7028 6578 5269 6e67 2c20 7269 6e67 2045  p(exRing, ring E
│ │ │ │ +0001cbc0: 312c 207b 333a 302c 655f 302c 655f 312c  1, {3:0,e_0,e_1,
│ │ │ │ +0001cbd0: 655f 327d 2c20 4465 6772 6565 4d61 7020  e_2}, DegreeMap 
│ │ │ │ +0001cbe0: 3d3e 2064 202d 3e20 7b64 5f31 7d29 7c0a  => d -> {d_1})|.
│ │ │ │ +0001cbf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001cc60: 3232 203d 206d 6170 2028 6578 5269 6e67  22 = map (exRing
│ │ │ │ -0001cc70: 2c20 6b6b 5b58 202e 2e58 202c 2065 202e  , kk[X ..X , e .
│ │ │ │ -0001cc80: 2e65 205d 2c20 7b30 2c20 302c 2030 2c20  .e ], {0, 0, 0, 
│ │ │ │ -0001cc90: 6520 2c20 6520 2c20 6520 7d29 2020 2020  e , e , e })    
│ │ │ │ -0001cca0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ccc0: 2020 2030 2020 2032 2020 2030 2020 2032     0   2   0   2
│ │ │ │ -0001ccd0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0001cce0: 2020 3120 2020 3220 2020 2020 2020 2020    1   2         
│ │ │ │ -0001ccf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cc30: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001cc40: 3220 3d20 6d61 7020 2865 7852 696e 672c  2 = map (exRing,
│ │ │ │ +0001cc50: 206b 6b5b 5820 2e2e 5820 2c20 6520 2e2e   kk[X ..X , e ..
│ │ │ │ +0001cc60: 6520 5d2c 207b 302c 2030 2c20 302c 2065  e ], {0, 0, 0, e
│ │ │ │ +0001cc70: 202c 2065 202c 2065 207d 2920 2020 2020   , e , e })     
│ │ │ │ +0001cc80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cca0: 2020 3020 2020 3220 2020 3020 2020 3220    0   2   0   2 
│ │ │ │ +0001ccb0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +0001ccc0: 2031 2020 2032 2020 2020 2020 2020 2020   1   2          
│ │ │ │ +0001ccd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd40: 2020 207c 0a7c 6f32 3220 3a20 5269 6e67     |.|o22 : Ring
│ │ │ │ -0001cd50: 4d61 7020 6578 5269 6e67 203c 2d2d 206b  Map exRing <-- k
│ │ │ │ -0001cd60: 6b5b 5820 2e2e 5820 2c20 6520 2e2e 6520  k[X ..X , e ..e 
│ │ │ │ -0001cd70: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd90: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001cd20: 2020 7c0a 7c6f 3232 203a 2052 696e 674d    |.|o22 : RingM
│ │ │ │ +0001cd30: 6170 2065 7852 696e 6720 3c2d 2d20 6b6b  ap exRing <-- kk
│ │ │ │ +0001cd40: 5b58 202e 2e58 202c 2065 202e 2e65 205d  [X ..X , e ..e ]
│ │ │ │ +0001cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cd60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001cd70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001cd80: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +0001cd90: 2020 2032 2020 2030 2020 2032 2020 2020     2   0   2    
│ │ │ │  0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdb0: 3020 2020 3220 2020 3020 2020 3220 2020  0   2   0   2   
│ │ │ │ -0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdd0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001cdb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233  ----------+.|i23
│ │ │ │ -0001ce30: 203a 2045 3220 3d20 636f 6b65 7220 7120   : E2 = coker q 
│ │ │ │ -0001ce40: 7072 6573 656e 7461 7469 6f6e 2045 313b  presentation E1;
│ │ │ │ -0001ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3320  ---------+.|i23 
│ │ │ │ +0001ce10: 3a20 4532 203d 2063 6f6b 6572 2071 2070  : E2 = coker q p
│ │ │ │ +0001ce20: 7265 7365 6e74 6174 696f 6e20 4531 3b20  resentation E1; 
│ │ │ │ +0001ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ce50: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001ce60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cec0: 2d2d 2d2d 2b0a 7c69 3234 203a 2068 6628  ----+.|i24 : hf(
│ │ │ │ -0001ced0: 2d35 2e2e 352c 4532 2920 3d3d 2068 6628  -5..5,E2) == hf(
│ │ │ │ -0001cee0: 2d35 2e2e 352c 4529 2020 2020 2020 2020  -5..5,E)        
│ │ │ │ -0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cea0: 2d2d 2d2b 0a7c 6932 3420 3a20 6866 282d  ---+.|i24 : hf(-
│ │ │ │ +0001ceb0: 352e 2e35 2c45 3229 203d 3d20 6866 282d  5..5,E2) == hf(-
│ │ │ │ +0001cec0: 352e 2e35 2c45 2920 2020 2020 2020 2020  5..5,E)         
│ │ │ │ +0001ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cef0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0001cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf10: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001cf60: 7c6f 3234 203d 2074 7275 6520 2020 2020  |o24 = true     
│ │ │ │ +0001cf30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001cf40: 6f32 3420 3d20 7472 7565 2020 2020 2020  o24 = true      
│ │ │ │ +0001cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cfa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001cf80: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001cf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001cff0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -0001d000: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -0001d010: 2a20 2a6e 6f74 6520 6578 7465 7269 6f72  * *note exterior
│ │ │ │ -0001d020: 546f 724d 6f64 756c 653a 2065 7874 6572  TorModule: exter
│ │ │ │ -0001d030: 696f 7254 6f72 4d6f 6475 6c65 2c20 2d2d  iorTorModule, --
│ │ │ │ -0001d040: 2054 6f72 2061 7320 6120 6d6f 6475 6c65   Tor as a module
│ │ │ │ -0001d050: 206f 7665 7220 616e 0a20 2020 2065 7874   over an.    ext
│ │ │ │ -0001d060: 6572 696f 7220 616c 6765 6272 6120 6f72  erior algebra or
│ │ │ │ -0001d070: 2062 6967 7261 6465 6420 616c 6765 6272   bigraded algebr
│ │ │ │ -0001d080: 610a 2020 2a20 2a6e 6f74 6520 6d61 6b65  a.  * *note make
│ │ │ │ -0001d090: 4d6f 6475 6c65 3a20 6d61 6b65 4d6f 6475  Module: makeModu
│ │ │ │ -0001d0a0: 6c65 2c20 2d2d 206d 616b 6573 2061 204d  le, -- makes a M
│ │ │ │ -0001d0b0: 6f64 756c 6520 6f75 7420 6f66 2061 2063  odule out of a c
│ │ │ │ -0001d0c0: 6f6c 6c65 6374 696f 6e20 6f66 0a20 2020  ollection of.   
│ │ │ │ -0001d0d0: 206d 6f64 756c 6573 2061 6e64 206d 6170   modules and map
│ │ │ │ -0001d0e0: 730a 0a57 6179 7320 746f 2075 7365 2065  s..Ways to use e
│ │ │ │ -0001d0f0: 7874 6572 696f 7245 7874 4d6f 6475 6c65  xteriorExtModule
│ │ │ │ -0001d100: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -0001d110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d120: 0a0a 2020 2a20 2265 7874 6572 696f 7245  ..  * "exteriorE
│ │ │ │ -0001d130: 7874 4d6f 6475 6c65 284d 6174 7269 782c  xtModule(Matrix,
│ │ │ │ -0001d140: 4d6f 6475 6c65 2922 0a20 202a 2022 6578  Module)".  * "ex
│ │ │ │ -0001d150: 7465 7269 6f72 4578 744d 6f64 756c 6528  teriorExtModule(
│ │ │ │ -0001d160: 4d61 7472 6978 2c4d 6f64 756c 652c 4d6f  Matrix,Module,Mo
│ │ │ │ -0001d170: 6475 6c65 2922 0a0a 466f 7220 7468 6520  dule)"..For the 
│ │ │ │ -0001d180: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -0001d190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -0001d1a0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -0001d1b0: 6578 7465 7269 6f72 4578 744d 6f64 756c  exteriorExtModul
│ │ │ │ -0001d1c0: 653a 2065 7874 6572 696f 7245 7874 4d6f  e: exteriorExtMo
│ │ │ │ -0001d1d0: 6475 6c65 2c20 6973 2061 202a 6e6f 7465  dule, is a *note
│ │ │ │ -0001d1e0: 206d 6574 686f 640a 6675 6e63 7469 6f6e   method.function
│ │ │ │ -0001d1f0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -0001d200: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ -0001d210: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │ +0001cfd0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ +0001cfe0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +0001cff0: 202a 6e6f 7465 2065 7874 6572 696f 7254   *note exteriorT
│ │ │ │ +0001d000: 6f72 4d6f 6475 6c65 3a20 6578 7465 7269  orModule: exteri
│ │ │ │ +0001d010: 6f72 546f 724d 6f64 756c 652c 202d 2d20  orTorModule, -- 
│ │ │ │ +0001d020: 546f 7220 6173 2061 206d 6f64 756c 6520  Tor as a module 
│ │ │ │ +0001d030: 6f76 6572 2061 6e0a 2020 2020 6578 7465  over an.    exte
│ │ │ │ +0001d040: 7269 6f72 2061 6c67 6562 7261 206f 7220  rior algebra or 
│ │ │ │ +0001d050: 6269 6772 6164 6564 2061 6c67 6562 7261  bigraded algebra
│ │ │ │ +0001d060: 0a20 202a 202a 6e6f 7465 206d 616b 654d  .  * *note makeM
│ │ │ │ +0001d070: 6f64 756c 653a 206d 616b 654d 6f64 756c  odule: makeModul
│ │ │ │ +0001d080: 652c 202d 2d20 6d61 6b65 7320 6120 4d6f  e, -- makes a Mo
│ │ │ │ +0001d090: 6475 6c65 206f 7574 206f 6620 6120 636f  dule out of a co
│ │ │ │ +0001d0a0: 6c6c 6563 7469 6f6e 206f 660a 2020 2020  llection of.    
│ │ │ │ +0001d0b0: 6d6f 6475 6c65 7320 616e 6420 6d61 7073  modules and maps
│ │ │ │ +0001d0c0: 0a0a 5761 7973 2074 6f20 7573 6520 6578  ..Ways to use ex
│ │ │ │ +0001d0d0: 7465 7269 6f72 4578 744d 6f64 756c 653a  teriorExtModule:
│ │ │ │ +0001d0e0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0001d0f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0001d100: 0a20 202a 2022 6578 7465 7269 6f72 4578  .  * "exteriorEx
│ │ │ │ +0001d110: 744d 6f64 756c 6528 4d61 7472 6978 2c4d  tModule(Matrix,M
│ │ │ │ +0001d120: 6f64 756c 6529 220a 2020 2a20 2265 7874  odule)".  * "ext
│ │ │ │ +0001d130: 6572 696f 7245 7874 4d6f 6475 6c65 284d  eriorExtModule(M
│ │ │ │ +0001d140: 6174 7269 782c 4d6f 6475 6c65 2c4d 6f64  atrix,Module,Mod
│ │ │ │ +0001d150: 756c 6529 220a 0a46 6f72 2074 6865 2070  ule)"..For the p
│ │ │ │ +0001d160: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +0001d170: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +0001d180: 6520 6f62 6a65 6374 202a 6e6f 7465 2065  e object *note e
│ │ │ │ +0001d190: 7874 6572 696f 7245 7874 4d6f 6475 6c65  xteriorExtModule
│ │ │ │ +0001d1a0: 3a20 6578 7465 7269 6f72 4578 744d 6f64  : exteriorExtMod
│ │ │ │ +0001d1b0: 756c 652c 2069 7320 6120 2a6e 6f74 6520  ule, is a *note 
│ │ │ │ +0001d1c0: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a  method.function:
│ │ │ │ +0001d1d0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0001d1e0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ +0001d1f0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +0001d200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d260: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ -0001d270: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ -0001d280: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ -0001d290: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ -0001d2a0: 6361 756c 6179 322d 312e 3236 2e30 362b  caulay2-1.26.06+
│ │ │ │ -0001d2b0: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ -0001d2c0: 7061 636b 6167 6573 2f0a 436f 6d70 6c65  packages/.Comple
│ │ │ │ -0001d2d0: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ -0001d2e0: 736f 6c75 7469 6f6e 732e 6d32 3a34 3236  solutions.m2:426
│ │ │ │ -0001d2f0: 333a 302e 0a1f 0a46 696c 653a 2043 6f6d  3:0....File: Com
│ │ │ │ -0001d300: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -0001d310: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66  nResolutions.inf
│ │ │ │ -0001d320: 6f2c 204e 6f64 653a 2065 7874 6572 696f  o, Node: exterio
│ │ │ │ -0001d330: 7248 6f6d 6f6c 6f67 794d 6f64 756c 652c  rHomologyModule,
│ │ │ │ -0001d340: 204e 6578 743a 2065 7874 6572 696f 7254   Next: exteriorT
│ │ │ │ -0001d350: 6f72 4d6f 6475 6c65 2c20 5072 6576 3a20  orModule, Prev: 
│ │ │ │ -0001d360: 6578 7465 7269 6f72 4578 744d 6f64 756c  exteriorExtModul
│ │ │ │ -0001d370: 652c 2055 703a 2054 6f70 0a0a 6578 7465  e, Up: Top..exte
│ │ │ │ -0001d380: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475  riorHomologyModu
│ │ │ │ -0001d390: 6c65 202d 2d20 4d61 6b65 2074 6865 2068  le -- Make the h
│ │ │ │ -0001d3a0: 6f6d 6f6c 6f67 7920 6f66 2061 2063 6f6d  omology of a com
│ │ │ │ -0001d3b0: 706c 6578 2069 6e74 6f20 6120 6d6f 6475  plex into a modu
│ │ │ │ -0001d3c0: 6c65 206f 7665 7220 616e 2065 7874 6572  le over an exter
│ │ │ │ -0001d3d0: 696f 7220 616c 6765 6272 610a 2a2a 2a2a  ior algebra.****
│ │ │ │ +0001d240: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +0001d250: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ +0001d260: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ +0001d270: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +0001d280: 6175 6c61 7932 2d31 2e32 362e 3036 2b64  aulay2-1.26.06+d
│ │ │ │ +0001d290: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +0001d2a0: 6163 6b61 6765 732f 0a43 6f6d 706c 6574  ackages/.Complet
│ │ │ │ +0001d2b0: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +0001d2c0: 6f6c 7574 696f 6e73 2e6d 323a 3432 3633  olutions.m2:4263
│ │ │ │ +0001d2d0: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70  :0....File: Comp
│ │ │ │ +0001d2e0: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ +0001d2f0: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ +0001d300: 2c20 4e6f 6465 3a20 6578 7465 7269 6f72  , Node: exterior
│ │ │ │ +0001d310: 486f 6d6f 6c6f 6779 4d6f 6475 6c65 2c20  HomologyModule, 
│ │ │ │ +0001d320: 4e65 7874 3a20 6578 7465 7269 6f72 546f  Next: exteriorTo
│ │ │ │ +0001d330: 724d 6f64 756c 652c 2050 7265 763a 2065  rModule, Prev: e
│ │ │ │ +0001d340: 7874 6572 696f 7245 7874 4d6f 6475 6c65  xteriorExtModule
│ │ │ │ +0001d350: 2c20 5570 3a20 546f 700a 0a65 7874 6572  , Up: Top..exter
│ │ │ │ +0001d360: 696f 7248 6f6d 6f6c 6f67 794d 6f64 756c  iorHomologyModul
│ │ │ │ +0001d370: 6520 2d2d 204d 616b 6520 7468 6520 686f  e -- Make the ho
│ │ │ │ +0001d380: 6d6f 6c6f 6779 206f 6620 6120 636f 6d70  mology of a comp
│ │ │ │ +0001d390: 6c65 7820 696e 746f 2061 206d 6f64 756c  lex into a modul
│ │ │ │ +0001d3a0: 6520 6f76 6572 2061 6e20 6578 7465 7269  e over an exteri
│ │ │ │ +0001d3b0: 6f72 2061 6c67 6562 7261 0a2a 2a2a 2a2a  or algebra.*****
│ │ │ │ +0001d3c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001d3d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001d3e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001d3f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001d400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001d430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -0001d440: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0001d450: 204d 203d 2065 7874 6572 696f 7248 6f6d   M = exteriorHom
│ │ │ │ -0001d460: 6f6c 6f67 794d 6f64 756c 6528 6666 2c20  ologyModule(ff, 
│ │ │ │ -0001d470: 4329 0a20 202a 2049 6e70 7574 733a 0a20  C).  * Inputs:. 
│ │ │ │ -0001d480: 2020 2020 202a 2066 662c 2061 202a 6e6f       * ff, a *no
│ │ │ │ -0001d490: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ -0001d4a0: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ -0001d4b0: 2c20 4d61 7472 6978 206f 6620 656c 656d  , Matrix of elem
│ │ │ │ -0001d4c0: 656e 7473 2074 6861 7420 6172 650a 2020  ents that are.  
│ │ │ │ -0001d4d0: 2020 2020 2020 686f 6d6f 746f 7069 6320        homotopic 
│ │ │ │ -0001d4e0: 746f 2030 206f 6e20 430a 2020 2020 2020  to 0 on C.      
│ │ │ │ -0001d4f0: 2a20 432c 2061 202a 6e6f 7465 2063 6f6d  * C, a *note com
│ │ │ │ -0001d500: 706c 6578 3a20 2843 6f6d 706c 6578 6573  plex: (Complexes
│ │ │ │ -0001d510: 2943 6f6d 706c 6578 2c2c 200a 2020 2a20  )Complex,, .  * 
│ │ │ │ -0001d520: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -0001d530: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -0001d540: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -0001d550: 6329 4d6f 6475 6c65 2c2c 200a 0a44 6573  c)Module,, ..Des
│ │ │ │ -0001d560: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0001d570: 3d3d 3d3d 0a0a 4173 7375 6d69 6e67 2074  ====..Assuming t
│ │ │ │ -0001d580: 6861 7420 7468 6520 656c 656d 656e 7473  hat the elements
│ │ │ │ -0001d590: 206f 6620 7468 6520 3178 6320 6d61 7472   of the 1xc matr
│ │ │ │ -0001d5a0: 6978 2066 6620 6172 6520 6e75 6c6c 2d68  ix ff are null-h
│ │ │ │ -0001d5b0: 6f6d 6f74 6f70 6963 206f 6e20 432c 2074  omotopic on C, t
│ │ │ │ -0001d5c0: 6865 0a73 6372 6970 7420 7265 7475 726e  he.script return
│ │ │ │ -0001d5d0: 7320 7468 6520 6469 7265 6374 2073 756d  s the direct sum
│ │ │ │ -0001d5e0: 206f 6620 7468 6520 686f 6d6f 6c6f 6779   of the homology
│ │ │ │ -0001d5f0: 206f 6620 4320 6173 2061 206d 6f64 756c   of C as a modul
│ │ │ │ -0001d600: 6520 6f76 6572 2061 206e 6577 2072 696e  e over a new rin
│ │ │ │ -0001d610: 672c 0a63 6f6e 7369 7374 696e 6720 6f66  g,.consisting of
│ │ │ │ -0001d620: 2072 696e 6720 4320 7769 7468 2063 2065   ring C with c e
│ │ │ │ -0001d630: 7874 6572 696f 7220 7661 7269 6162 6c65  xterior variable
│ │ │ │ -0001d640: 7320 6164 6a6f 696e 6564 2e20 5468 6520  s adjoined. The 
│ │ │ │ -0001d650: 7363 7269 7074 2069 7320 7468 6520 6d61  script is the ma
│ │ │ │ -0001d660: 696e 0a63 6f6d 706f 6e65 6e74 206f 6620  in.component of 
│ │ │ │ -0001d670: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ -0001d680: 650a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  e..See also.====
│ │ │ │ -0001d690: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -0001d6a0: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ -0001d6b0: 653a 2065 7874 6572 696f 7254 6f72 4d6f  e: exteriorTorMo
│ │ │ │ -0001d6c0: 6475 6c65 2c20 2d2d 2054 6f72 2061 7320  dule, -- Tor as 
│ │ │ │ -0001d6d0: 6120 6d6f 6475 6c65 206f 7665 7220 616e  a module over an
│ │ │ │ -0001d6e0: 0a20 2020 2065 7874 6572 696f 7220 616c  .    exterior al
│ │ │ │ -0001d6f0: 6765 6272 6120 6f72 2062 6967 7261 6465  gebra or bigrade
│ │ │ │ -0001d700: 6420 616c 6765 6272 610a 2020 2a20 2a6e  d algebra.  * *n
│ │ │ │ -0001d710: 6f74 6520 6d61 6b65 486f 6d6f 746f 7069  ote makeHomotopi
│ │ │ │ -0001d720: 6573 4f6e 486f 6d6f 6c6f 6779 3a20 6d61  esOnHomology: ma
│ │ │ │ -0001d730: 6b65 486f 6d6f 746f 7069 6573 4f6e 486f  keHomotopiesOnHo
│ │ │ │ -0001d740: 6d6f 6c6f 6779 2c20 2d2d 2048 6f6d 6f6c  mology, -- Homol
│ │ │ │ -0001d750: 6f67 7920 6f66 2061 0a20 2020 2063 6f6d  ogy of a.    com
│ │ │ │ -0001d760: 706c 6578 2061 7320 6578 7465 7269 6f72  plex as exterior
│ │ │ │ -0001d770: 206d 6f64 756c 650a 0a57 6179 7320 746f   module..Ways to
│ │ │ │ -0001d780: 2075 7365 2065 7874 6572 696f 7248 6f6d   use exteriorHom
│ │ │ │ -0001d790: 6f6c 6f67 794d 6f64 756c 653a 0a3d 3d3d  ologyModule:.===
│ │ │ │ -0001d7a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d7b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0001d7c0: 0a0a 2020 2a20 2265 7874 6572 696f 7248  ..  * "exteriorH
│ │ │ │ -0001d7d0: 6f6d 6f6c 6f67 794d 6f64 756c 6528 4d61  omologyModule(Ma
│ │ │ │ -0001d7e0: 7472 6978 2c43 6f6d 706c 6578 2922 0a0a  trix,Complex)"..
│ │ │ │ -0001d7f0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -0001d800: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -0001d810: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -0001d820: 7420 2a6e 6f74 6520 6578 7465 7269 6f72  t *note exterior
│ │ │ │ -0001d830: 486f 6d6f 6c6f 6779 4d6f 6475 6c65 3a20  HomologyModule: 
│ │ │ │ -0001d840: 6578 7465 7269 6f72 486f 6d6f 6c6f 6779  exteriorHomology
│ │ │ │ -0001d850: 4d6f 6475 6c65 2c20 6973 2061 202a 6e6f  Module, is a *no
│ │ │ │ -0001d860: 7465 0a6d 6574 686f 6420 6675 6e63 7469  te.method functi
│ │ │ │ -0001d870: 6f6e 3a20 284d 6163 6175 6c61 7932 446f  on: (Macaulay2Do
│ │ │ │ -0001d880: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -0001d890: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +0001d410: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +0001d420: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0001d430: 4d20 3d20 6578 7465 7269 6f72 486f 6d6f  M = exteriorHomo
│ │ │ │ +0001d440: 6c6f 6779 4d6f 6475 6c65 2866 662c 2043  logyModule(ff, C
│ │ │ │ +0001d450: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +0001d460: 2020 2020 2a20 6666 2c20 6120 2a6e 6f74      * ff, a *not
│ │ │ │ +0001d470: 6520 6d61 7472 6978 3a20 284d 6163 6175  e matrix: (Macau
│ │ │ │ +0001d480: 6c61 7932 446f 6329 4d61 7472 6978 2c2c  lay2Doc)Matrix,,
│ │ │ │ +0001d490: 204d 6174 7269 7820 6f66 2065 6c65 6d65   Matrix of eleme
│ │ │ │ +0001d4a0: 6e74 7320 7468 6174 2061 7265 0a20 2020  nts that are.   
│ │ │ │ +0001d4b0: 2020 2020 2068 6f6d 6f74 6f70 6963 2074       homotopic t
│ │ │ │ +0001d4c0: 6f20 3020 6f6e 2043 0a20 2020 2020 202a  o 0 on C.      *
│ │ │ │ +0001d4d0: 2043 2c20 6120 2a6e 6f74 6520 636f 6d70   C, a *note comp
│ │ │ │ +0001d4e0: 6c65 783a 2028 436f 6d70 6c65 7865 7329  lex: (Complexes)
│ │ │ │ +0001d4f0: 436f 6d70 6c65 782c 2c20 0a20 202a 204f  Complex,, .  * O
│ │ │ │ +0001d500: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +0001d510: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ +0001d520: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +0001d530: 294d 6f64 756c 652c 2c20 0a0a 4465 7363  )Module,, ..Desc
│ │ │ │ +0001d540: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0001d550: 3d3d 3d0a 0a41 7373 756d 696e 6720 7468  ===..Assuming th
│ │ │ │ +0001d560: 6174 2074 6865 2065 6c65 6d65 6e74 7320  at the elements 
│ │ │ │ +0001d570: 6f66 2074 6865 2031 7863 206d 6174 7269  of the 1xc matri
│ │ │ │ +0001d580: 7820 6666 2061 7265 206e 756c 6c2d 686f  x ff are null-ho
│ │ │ │ +0001d590: 6d6f 746f 7069 6320 6f6e 2043 2c20 7468  motopic on C, th
│ │ │ │ +0001d5a0: 650a 7363 7269 7074 2072 6574 7572 6e73  e.script returns
│ │ │ │ +0001d5b0: 2074 6865 2064 6972 6563 7420 7375 6d20   the direct sum 
│ │ │ │ +0001d5c0: 6f66 2074 6865 2068 6f6d 6f6c 6f67 7920  of the homology 
│ │ │ │ +0001d5d0: 6f66 2043 2061 7320 6120 6d6f 6475 6c65  of C as a module
│ │ │ │ +0001d5e0: 206f 7665 7220 6120 6e65 7720 7269 6e67   over a new ring
│ │ │ │ +0001d5f0: 2c0a 636f 6e73 6973 7469 6e67 206f 6620  ,.consisting of 
│ │ │ │ +0001d600: 7269 6e67 2043 2077 6974 6820 6320 6578  ring C with c ex
│ │ │ │ +0001d610: 7465 7269 6f72 2076 6172 6961 626c 6573  terior variables
│ │ │ │ +0001d620: 2061 646a 6f69 6e65 642e 2054 6865 2073   adjoined. The s
│ │ │ │ +0001d630: 6372 6970 7420 6973 2074 6865 206d 6169  cript is the mai
│ │ │ │ +0001d640: 6e0a 636f 6d70 6f6e 656e 7420 6f66 2065  n.component of e
│ │ │ │ +0001d650: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ +0001d660: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +0001d670: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065  ===..  * *note e
│ │ │ │ +0001d680: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ +0001d690: 3a20 6578 7465 7269 6f72 546f 724d 6f64  : exteriorTorMod
│ │ │ │ +0001d6a0: 756c 652c 202d 2d20 546f 7220 6173 2061  ule, -- Tor as a
│ │ │ │ +0001d6b0: 206d 6f64 756c 6520 6f76 6572 2061 6e0a   module over an.
│ │ │ │ +0001d6c0: 2020 2020 6578 7465 7269 6f72 2061 6c67      exterior alg
│ │ │ │ +0001d6d0: 6562 7261 206f 7220 6269 6772 6164 6564  ebra or bigraded
│ │ │ │ +0001d6e0: 2061 6c67 6562 7261 0a20 202a 202a 6e6f   algebra.  * *no
│ │ │ │ +0001d6f0: 7465 206d 616b 6548 6f6d 6f74 6f70 6965  te makeHomotopie
│ │ │ │ +0001d700: 734f 6e48 6f6d 6f6c 6f67 793a 206d 616b  sOnHomology: mak
│ │ │ │ +0001d710: 6548 6f6d 6f74 6f70 6965 734f 6e48 6f6d  eHomotopiesOnHom
│ │ │ │ +0001d720: 6f6c 6f67 792c 202d 2d20 486f 6d6f 6c6f  ology, -- Homolo
│ │ │ │ +0001d730: 6779 206f 6620 610a 2020 2020 636f 6d70  gy of a.    comp
│ │ │ │ +0001d740: 6c65 7820 6173 2065 7874 6572 696f 7220  lex as exterior 
│ │ │ │ +0001d750: 6d6f 6475 6c65 0a0a 5761 7973 2074 6f20  module..Ways to 
│ │ │ │ +0001d760: 7573 6520 6578 7465 7269 6f72 486f 6d6f  use exteriorHomo
│ │ │ │ +0001d770: 6c6f 6779 4d6f 6475 6c65 3a0a 3d3d 3d3d  logyModule:.====
│ │ │ │ +0001d780: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0001d790: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0001d7a0: 0a20 202a 2022 6578 7465 7269 6f72 486f  .  * "exteriorHo
│ │ │ │ +0001d7b0: 6d6f 6c6f 6779 4d6f 6475 6c65 284d 6174  mologyModule(Mat
│ │ │ │ +0001d7c0: 7269 782c 436f 6d70 6c65 7829 220a 0a46  rix,Complex)"..F
│ │ │ │ +0001d7d0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +0001d7e0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +0001d7f0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +0001d800: 202a 6e6f 7465 2065 7874 6572 696f 7248   *note exteriorH
│ │ │ │ +0001d810: 6f6d 6f6c 6f67 794d 6f64 756c 653a 2065  omologyModule: e
│ │ │ │ +0001d820: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d  xteriorHomologyM
│ │ │ │ +0001d830: 6f64 756c 652c 2069 7320 6120 2a6e 6f74  odule, is a *not
│ │ │ │ +0001d840: 650a 6d65 7468 6f64 2066 756e 6374 696f  e.method functio
│ │ │ │ +0001d850: 6e3a 2028 4d61 6361 756c 6179 3244 6f63  n: (Macaulay2Doc
│ │ │ │ +0001d860: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +0001d870: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0001d880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d8e0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -0001d8f0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -0001d900: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -0001d910: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -0001d920: 6d61 6361 756c 6179 322d 312e 3236 2e30  macaulay2-1.26.0
│ │ │ │ -0001d930: 362b 6473 2f4d 322f 4d61 6361 756c 6179  6+ds/M2/Macaulay
│ │ │ │ -0001d940: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ -0001d950: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -0001d960: 5265 736f 6c75 7469 6f6e 732e 6d32 3a32  Resolutions.m2:2
│ │ │ │ -0001d970: 3738 353a 302e 0a1f 0a46 696c 653a 2043  785:0....File: C
│ │ │ │ -0001d980: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0001d990: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -0001d9a0: 6e66 6f2c 204e 6f64 653a 2065 7874 6572  nfo, Node: exter
│ │ │ │ -0001d9b0: 696f 7254 6f72 4d6f 6475 6c65 2c20 4e65  iorTorModule, Ne
│ │ │ │ -0001d9c0: 7874 3a20 6578 7449 734f 6e65 506f 6c79  xt: extIsOnePoly
│ │ │ │ -0001d9d0: 6e6f 6d69 616c 2c20 5072 6576 3a20 6578  nomial, Prev: ex
│ │ │ │ -0001d9e0: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f  teriorHomologyMo
│ │ │ │ -0001d9f0: 6475 6c65 2c20 5570 3a20 546f 700a 0a65  dule, Up: Top..e
│ │ │ │ -0001da00: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -0001da10: 202d 2d20 546f 7220 6173 2061 206d 6f64   -- Tor as a mod
│ │ │ │ -0001da20: 756c 6520 6f76 6572 2061 6e20 6578 7465  ule over an exte
│ │ │ │ -0001da30: 7269 6f72 2061 6c67 6562 7261 206f 7220  rior algebra or 
│ │ │ │ -0001da40: 6269 6772 6164 6564 2061 6c67 6562 7261  bigraded algebra
│ │ │ │ -0001da50: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +0001d8c0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +0001d8d0: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +0001d8e0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +0001d8f0: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +0001d900: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ +0001d910: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +0001d920: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ +0001d930: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +0001d940: 6573 6f6c 7574 696f 6e73 2e6d 323a 3237  esolutions.m2:27
│ │ │ │ +0001d950: 3835 3a30 2e0a 1f0a 4669 6c65 3a20 436f  85:0....File: Co
│ │ │ │ +0001d960: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +0001d970: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ +0001d980: 666f 2c20 4e6f 6465 3a20 6578 7465 7269  fo, Node: exteri
│ │ │ │ +0001d990: 6f72 546f 724d 6f64 756c 652c 204e 6578  orTorModule, Nex
│ │ │ │ +0001d9a0: 743a 2065 7874 4973 4f6e 6550 6f6c 796e  t: extIsOnePolyn
│ │ │ │ +0001d9b0: 6f6d 6961 6c2c 2050 7265 763a 2065 7874  omial, Prev: ext
│ │ │ │ +0001d9c0: 6572 696f 7248 6f6d 6f6c 6f67 794d 6f64  eriorHomologyMod
│ │ │ │ +0001d9d0: 756c 652c 2055 703a 2054 6f70 0a0a 6578  ule, Up: Top..ex
│ │ │ │ +0001d9e0: 7465 7269 6f72 546f 724d 6f64 756c 6520  teriorTorModule 
│ │ │ │ +0001d9f0: 2d2d 2054 6f72 2061 7320 6120 6d6f 6475  -- Tor as a modu
│ │ │ │ +0001da00: 6c65 206f 7665 7220 616e 2065 7874 6572  le over an exter
│ │ │ │ +0001da10: 696f 7220 616c 6765 6272 6120 6f72 2062  ior algebra or b
│ │ │ │ +0001da20: 6967 7261 6465 6420 616c 6765 6272 610a  igraded algebra.
│ │ │ │ +0001da30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001da40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0001da50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001da60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0001da70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001da80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001da90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0001daa0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ -0001dab0: 2020 2020 2020 2020 5420 3d20 6578 7465          T = exte
│ │ │ │ -0001dac0: 7269 6f72 546f 724d 6f64 756c 6528 662c  riorTorModule(f,
│ │ │ │ -0001dad0: 4629 0a20 2020 2020 2020 2054 203d 2065  F).        T = e
│ │ │ │ -0001dae0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -0001daf0: 2866 2c4d 2c4e 290a 2020 2a20 496e 7075  (f,M,N).  * Inpu
│ │ │ │ -0001db00: 7473 3a0a 2020 2020 2020 2a20 662c 2061  ts:.      * f, a
│ │ │ │ -0001db10: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ -0001db20: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ -0001db30: 7269 782c 2c20 3120 7820 632c 2065 6e74  rix,, 1 x c, ent
│ │ │ │ -0001db40: 7269 6573 206d 7573 7420 6265 0a20 2020  ries must be.   
│ │ │ │ -0001db50: 2020 2020 2068 6f6d 6f74 6f70 6963 2074       homotopic t
│ │ │ │ -0001db60: 6f20 3020 6f6e 2046 0a20 2020 2020 202a  o 0 on F.      *
│ │ │ │ -0001db70: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -0001db80: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -0001db90: 6329 4d6f 6475 6c65 2c2c 2053 2d6d 6f64  c)Module,, S-mod
│ │ │ │ -0001dba0: 756c 6520 616e 6e69 6869 6c61 7465 6420  ule annihilated 
│ │ │ │ -0001dbb0: 6279 2069 6465 616c 0a20 2020 2020 2020  by ideal.       
│ │ │ │ -0001dbc0: 2066 0a20 2020 2020 202a 204e 2c20 6120   f.      * N, a 
│ │ │ │ -0001dbd0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ -0001dbe0: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ -0001dbf0: 6c65 2c2c 2053 2d6d 6f64 756c 6520 616e  le,, S-module an
│ │ │ │ -0001dc00: 6e69 6869 6c61 7465 6420 6279 2069 6465  nihilated by ide
│ │ │ │ -0001dc10: 616c 0a20 2020 2020 2020 2066 0a20 202a  al.        f.  *
│ │ │ │ -0001dc20: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -0001dc30: 2a20 542c 2061 202a 6e6f 7465 206d 6f64  * T, a *note mod
│ │ │ │ -0001dc40: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -0001dc50: 6f63 294d 6f64 756c 652c 2c20 546f 725e  oc)Module,, Tor^
│ │ │ │ -0001dc60: 5328 4d2c 4e29 2061 7320 6120 4d6f 6475  S(M,N) as a Modu
│ │ │ │ -0001dc70: 6c65 206f 7665 720a 2020 2020 2020 2020  le over.        
│ │ │ │ -0001dc80: 616e 2065 7874 6572 696f 7220 616c 6765  an exterior alge
│ │ │ │ -0001dc90: 6272 610a 0a44 6573 6372 6970 7469 6f6e  bra..Description
│ │ │ │ -0001dca0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966  .===========..If
│ │ │ │ -0001dcb0: 204d 2c4e 2061 7265 2053 2d6d 6f64 756c   M,N are S-modul
│ │ │ │ -0001dcc0: 6573 2061 6e6e 6968 696c 6174 6564 2062  es annihilated b
│ │ │ │ -0001dcd0: 7920 7468 6520 656c 656d 656e 7473 206f  y the elements o
│ │ │ │ -0001dce0: 6620 7468 6520 6d61 7472 6978 2066 6620  f the matrix ff 
│ │ │ │ -0001dcf0: 3d20 2866 5f31 2e2e 665f 6329 2c0a 616e  = (f_1..f_c),.an
│ │ │ │ -0001dd00: 6420 6b20 6973 2074 6865 2072 6573 6964  d k is the resid
│ │ │ │ -0001dd10: 7565 2066 6965 6c64 206f 6620 532c 2074  ue field of S, t
│ │ │ │ -0001dd20: 6865 6e20 7468 6520 7363 7269 7074 2065  hen the script e
│ │ │ │ -0001dd30: 7874 6572 696f 7254 6f72 4d6f 6475 6c65  xteriorTorModule
│ │ │ │ -0001dd40: 2866 2c4d 2920 7265 7475 726e 730a 546f  (f,M) returns.To
│ │ │ │ -0001dd50: 725e 5328 4d2c 206b 2920 6173 2061 206d  r^S(M, k) as a m
│ │ │ │ -0001dd60: 6f64 756c 6520 6f76 6572 2061 6e20 6578  odule over an ex
│ │ │ │ -0001dd70: 7465 7269 6f72 2061 6c67 6562 7261 206b  terior algebra k
│ │ │ │ -0001dd80: 3c65 5f31 2c2e 2e2e 2c65 5f63 3e2c 2077  , w
│ │ │ │ -0001dd90: 6865 7265 2074 6865 2065 5f69 0a68 6176  here the e_i.hav
│ │ │ │ -0001dda0: 6520 6465 6772 6565 2031 2c20 7768 696c  e degree 1, whil
│ │ │ │ -0001ddb0: 6520 6578 7465 7269 6f72 546f 724d 6f64  e exteriorTorMod
│ │ │ │ -0001ddc0: 756c 6528 662c 4d2c 4e29 2072 6574 7572  ule(f,M,N) retur
│ │ │ │ -0001ddd0: 6e73 2054 6f72 5e53 284d 2c4e 2920 6173  ns Tor^S(M,N) as
│ │ │ │ -0001dde0: 2061 206d 6f64 756c 650a 6f76 6572 2061   a module.over a
│ │ │ │ -0001ddf0: 2062 6967 7261 6465 6420 7269 6e67 2053   bigraded ring S
│ │ │ │ -0001de00: 4520 3d20 533c 655f 312c 2e2e 2c65 5f63  E = S, where the e_i
│ │ │ │ -0001de20: 2068 6176 6520 6465 6772 6565 7320 7b64   have degrees {d
│ │ │ │ -0001de30: 5f69 2c31 7d2c 0a77 6865 7265 2064 5f69  _i,1},.where d_i
│ │ │ │ -0001de40: 2069 7320 7468 6520 6465 6772 6565 206f   is the degree o
│ │ │ │ -0001de50: 6620 665f 692e 2054 6865 206d 6f64 756c  f f_i. The modul
│ │ │ │ -0001de60: 6520 7374 7275 6374 7572 652c 2069 6e20  e structure, in 
│ │ │ │ -0001de70: 6569 7468 6572 2063 6173 652c 2069 730a  either case, is.
│ │ │ │ -0001de80: 6465 6669 6e65 6420 6279 2074 6865 2068  defined by the h
│ │ │ │ -0001de90: 6f6d 6f74 6f70 6965 7320 666f 7220 7468  omotopies for th
│ │ │ │ -0001dea0: 6520 665f 6920 6f6e 2074 6865 2072 6573  e f_i on the res
│ │ │ │ -0001deb0: 6f6c 7574 696f 6e20 6f66 204d 2c20 636f  olution of M, co
│ │ │ │ -0001dec0: 6d70 7574 6564 2062 7920 7468 650a 7363  mputed by the.sc
│ │ │ │ -0001ded0: 7269 7074 206d 616b 6548 6f6d 6f74 6f70  ript makeHomotop
│ │ │ │ -0001dee0: 6965 7331 2e0a 0a54 6865 2073 6372 6970  ies1...The scrip
│ │ │ │ -0001def0: 7473 2063 616c 6c20 6d61 6b65 4d6f 6475  ts call makeModu
│ │ │ │ -0001df00: 6c65 2074 6f20 636f 6d70 7574 6520 6120  le to compute a 
│ │ │ │ -0001df10: 286e 6f6e 2d6d 696e 696d 616c 2920 7072  (non-minimal) pr
│ │ │ │ -0001df20: 6573 656e 7461 7469 6f6e 206f 6620 7468  esentation of th
│ │ │ │ -0001df30: 6973 0a6d 6f64 756c 652e 0a0a 4672 6f6d  is.module...From
│ │ │ │ -0001df40: 2074 6865 2064 6573 6372 6970 7469 6f6e   the description
│ │ │ │ -0001df50: 2062 7920 6d61 7472 6978 2066 6163 746f   by matrix facto
│ │ │ │ -0001df60: 7269 7a61 7469 6f6e 7320 616e 6420 7468  rizations and th
│ │ │ │ -0001df70: 6520 7061 7065 7220 2254 6f72 2061 7320  e paper "Tor as 
│ │ │ │ -0001df80: 6120 6d6f 6475 6c65 0a6f 7665 7220 616e  a module.over an
│ │ │ │ -0001df90: 2065 7874 6572 696f 7220 616c 6765 6272   exterior algebr
│ │ │ │ -0001dfa0: 6122 206f 6620 4569 7365 6e62 7564 2c20  a" of Eisenbud, 
│ │ │ │ -0001dfb0: 5065 6576 6120 616e 6420 5363 6872 6579  Peeva and Schrey
│ │ │ │ -0001dfc0: 6572 2069 7420 666f 6c6c 6f77 7320 7468  er it follows th
│ │ │ │ -0001dfd0: 6174 2077 6865 6e0a 4d20 6973 2061 2068  at when.M is a h
│ │ │ │ -0001dfe0: 6967 6820 7379 7a79 6779 2061 6e64 2046  igh syzygy and F
│ │ │ │ -0001dff0: 2069 7320 6974 7320 7265 736f 6c75 7469   is its resoluti
│ │ │ │ -0001e000: 6f6e 2c20 7468 656e 2074 6865 2070 7265  on, then the pre
│ │ │ │ -0001e010: 7365 6e74 6174 696f 6e20 6f66 0a54 6f72  sentation of.Tor
│ │ │ │ -0001e020: 284d 2c53 5e31 2f6d 6d29 2061 6c77 6179  (M,S^1/mm) alway
│ │ │ │ -0001e030: 7320 6861 7320 6765 6e65 7261 746f 7273  s has generators
│ │ │ │ -0001e040: 2069 6e20 6465 6772 6565 7320 302c 312c   in degrees 0,1,
│ │ │ │ -0001e050: 2063 6f72 7265 7370 6f6e 6469 6e67 2074   corresponding t
│ │ │ │ -0001e060: 6f20 7468 650a 7461 7267 6574 7320 616e  o the.targets an
│ │ │ │ -0001e070: 6420 736f 7572 6365 7320 6f66 2074 6865  d sources of the
│ │ │ │ -0001e080: 2073 7461 636b 206f 6620 6d61 7073 2042   stack of maps B
│ │ │ │ -0001e090: 2869 292c 2061 6e64 2074 6861 7420 7468  (i), and that th
│ │ │ │ -0001e0a0: 6520 7265 736f 6c75 7469 6f6e 2069 730a  e resolution is.
│ │ │ │ -0001e0b0: 636f 6d70 6f6e 656e 7477 6973 6520 6c69  componentwise li
│ │ │ │ -0001e0c0: 6e65 6172 2069 6e20 6120 7375 6974 6162  near in a suitab
│ │ │ │ -0001e0d0: 6c65 2073 656e 7365 2e20 496e 2074 6865  le sense. In the
│ │ │ │ -0001e0e0: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70   following examp
│ │ │ │ -0001e0f0: 6c65 2c20 7468 6573 6520 6661 6374 730a  le, these facts.
│ │ │ │ -0001e100: 6172 6520 7665 7269 6669 6564 2e20 5468  are verified. Th
│ │ │ │ -0001e110: 6520 546f 7220 6d6f 6475 6c65 2064 6f65  e Tor module doe
│ │ │ │ -0001e120: 7320 4e4f 5420 7370 6c69 7420 696e 746f  s NOT split into
│ │ │ │ -0001e130: 2074 6865 2064 6972 6563 7420 7375 6d20   the direct sum 
│ │ │ │ -0001e140: 6f66 2074 6865 0a73 7562 6d6f 6475 6c65  of the.submodule
│ │ │ │ -0001e150: 7320 6765 6e65 7261 7465 6420 696e 2064  s generated in d
│ │ │ │ -0001e160: 6567 7265 6573 2030 2061 6e64 2031 2c20  egrees 0 and 1, 
│ │ │ │ -0001e170: 686f 7765 7665 722e 0a0a 0a0a 2b2d 2d2d  however.....+---
│ │ │ │ +0001da80: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ +0001da90: 2020 2020 2020 2054 203d 2065 7874 6572         T = exter
│ │ │ │ +0001daa0: 696f 7254 6f72 4d6f 6475 6c65 2866 2c46  iorTorModule(f,F
│ │ │ │ +0001dab0: 290a 2020 2020 2020 2020 5420 3d20 6578  ).        T = ex
│ │ │ │ +0001dac0: 7465 7269 6f72 546f 724d 6f64 756c 6528  teriorTorModule(
│ │ │ │ +0001dad0: 662c 4d2c 4e29 0a20 202a 2049 6e70 7574  f,M,N).  * Input
│ │ │ │ +0001dae0: 733a 0a20 2020 2020 202a 2066 2c20 6120  s:.      * f, a 
│ │ │ │ +0001daf0: 2a6e 6f74 6520 6d61 7472 6978 3a20 284d  *note matrix: (M
│ │ │ │ +0001db00: 6163 6175 6c61 7932 446f 6329 4d61 7472  acaulay2Doc)Matr
│ │ │ │ +0001db10: 6978 2c2c 2031 2078 2063 2c20 656e 7472  ix,, 1 x c, entr
│ │ │ │ +0001db20: 6965 7320 6d75 7374 2062 650a 2020 2020  ies must be.    
│ │ │ │ +0001db30: 2020 2020 686f 6d6f 746f 7069 6320 746f      homotopic to
│ │ │ │ +0001db40: 2030 206f 6e20 460a 2020 2020 2020 2a20   0 on F.      * 
│ │ │ │ +0001db50: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ +0001db60: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +0001db70: 294d 6f64 756c 652c 2c20 532d 6d6f 6475  )Module,, S-modu
│ │ │ │ +0001db80: 6c65 2061 6e6e 6968 696c 6174 6564 2062  le annihilated b
│ │ │ │ +0001db90: 7920 6964 6561 6c0a 2020 2020 2020 2020  y ideal.        
│ │ │ │ +0001dba0: 660a 2020 2020 2020 2a20 4e2c 2061 202a  f.      * N, a *
│ │ │ │ +0001dbb0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ +0001dbc0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ +0001dbd0: 652c 2c20 532d 6d6f 6475 6c65 2061 6e6e  e,, S-module ann
│ │ │ │ +0001dbe0: 6968 696c 6174 6564 2062 7920 6964 6561  ihilated by idea
│ │ │ │ +0001dbf0: 6c0a 2020 2020 2020 2020 660a 2020 2a20  l.        f.  * 
│ │ │ │ +0001dc00: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +0001dc10: 2054 2c20 6120 2a6e 6f74 6520 6d6f 6475   T, a *note modu
│ │ │ │ +0001dc20: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +0001dc30: 6329 4d6f 6475 6c65 2c2c 2054 6f72 5e53  c)Module,, Tor^S
│ │ │ │ +0001dc40: 284d 2c4e 2920 6173 2061 204d 6f64 756c  (M,N) as a Modul
│ │ │ │ +0001dc50: 6520 6f76 6572 0a20 2020 2020 2020 2061  e over.        a
│ │ │ │ +0001dc60: 6e20 6578 7465 7269 6f72 2061 6c67 6562  n exterior algeb
│ │ │ │ +0001dc70: 7261 0a0a 4465 7363 7269 7074 696f 6e0a  ra..Description.
│ │ │ │ +0001dc80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620  ===========..If 
│ │ │ │ +0001dc90: 4d2c 4e20 6172 6520 532d 6d6f 6475 6c65  M,N are S-module
│ │ │ │ +0001dca0: 7320 616e 6e69 6869 6c61 7465 6420 6279  s annihilated by
│ │ │ │ +0001dcb0: 2074 6865 2065 6c65 6d65 6e74 7320 6f66   the elements of
│ │ │ │ +0001dcc0: 2074 6865 206d 6174 7269 7820 6666 203d   the matrix ff =
│ │ │ │ +0001dcd0: 2028 665f 312e 2e66 5f63 292c 0a61 6e64   (f_1..f_c),.and
│ │ │ │ +0001dce0: 206b 2069 7320 7468 6520 7265 7369 6475   k is the residu
│ │ │ │ +0001dcf0: 6520 6669 656c 6420 6f66 2053 2c20 7468  e field of S, th
│ │ │ │ +0001dd00: 656e 2074 6865 2073 6372 6970 7420 6578  en the script ex
│ │ │ │ +0001dd10: 7465 7269 6f72 546f 724d 6f64 756c 6528  teriorTorModule(
│ │ │ │ +0001dd20: 662c 4d29 2072 6574 7572 6e73 0a54 6f72  f,M) returns.Tor
│ │ │ │ +0001dd30: 5e53 284d 2c20 6b29 2061 7320 6120 6d6f  ^S(M, k) as a mo
│ │ │ │ +0001dd40: 6475 6c65 206f 7665 7220 616e 2065 7874  dule over an ext
│ │ │ │ +0001dd50: 6572 696f 7220 616c 6765 6272 6120 6b3c  erior algebra k<
│ │ │ │ +0001dd60: 655f 312c 2e2e 2e2c 655f 633e 2c20 7768  e_1,...,e_c>, wh
│ │ │ │ +0001dd70: 6572 6520 7468 6520 655f 690a 6861 7665  ere the e_i.have
│ │ │ │ +0001dd80: 2064 6567 7265 6520 312c 2077 6869 6c65   degree 1, while
│ │ │ │ +0001dd90: 2065 7874 6572 696f 7254 6f72 4d6f 6475   exteriorTorModu
│ │ │ │ +0001dda0: 6c65 2866 2c4d 2c4e 2920 7265 7475 726e  le(f,M,N) return
│ │ │ │ +0001ddb0: 7320 546f 725e 5328 4d2c 4e29 2061 7320  s Tor^S(M,N) as 
│ │ │ │ +0001ddc0: 6120 6d6f 6475 6c65 0a6f 7665 7220 6120  a module.over a 
│ │ │ │ +0001ddd0: 6269 6772 6164 6564 2072 696e 6720 5345  bigraded ring SE
│ │ │ │ +0001dde0: 203d 2053 3c65 5f31 2c2e 2e2c 655f 633e   = S
│ │ │ │ +0001ddf0: 2c20 7768 6572 6520 7468 6520 655f 6920  , where the e_i 
│ │ │ │ +0001de00: 6861 7665 2064 6567 7265 6573 207b 645f  have degrees {d_
│ │ │ │ +0001de10: 692c 317d 2c0a 7768 6572 6520 645f 6920  i,1},.where d_i 
│ │ │ │ +0001de20: 6973 2074 6865 2064 6567 7265 6520 6f66  is the degree of
│ │ │ │ +0001de30: 2066 5f69 2e20 5468 6520 6d6f 6475 6c65   f_i. The module
│ │ │ │ +0001de40: 2073 7472 7563 7475 7265 2c20 696e 2065   structure, in e
│ │ │ │ +0001de50: 6974 6865 7220 6361 7365 2c20 6973 0a64  ither case, is.d
│ │ │ │ +0001de60: 6566 696e 6564 2062 7920 7468 6520 686f  efined by the ho
│ │ │ │ +0001de70: 6d6f 746f 7069 6573 2066 6f72 2074 6865  motopies for the
│ │ │ │ +0001de80: 2066 5f69 206f 6e20 7468 6520 7265 736f   f_i on the reso
│ │ │ │ +0001de90: 6c75 7469 6f6e 206f 6620 4d2c 2063 6f6d  lution of M, com
│ │ │ │ +0001dea0: 7075 7465 6420 6279 2074 6865 0a73 6372  puted by the.scr
│ │ │ │ +0001deb0: 6970 7420 6d61 6b65 486f 6d6f 746f 7069  ipt makeHomotopi
│ │ │ │ +0001dec0: 6573 312e 0a0a 5468 6520 7363 7269 7074  es1...The script
│ │ │ │ +0001ded0: 7320 6361 6c6c 206d 616b 654d 6f64 756c  s call makeModul
│ │ │ │ +0001dee0: 6520 746f 2063 6f6d 7075 7465 2061 2028  e to compute a (
│ │ │ │ +0001def0: 6e6f 6e2d 6d69 6e69 6d61 6c29 2070 7265  non-minimal) pre
│ │ │ │ +0001df00: 7365 6e74 6174 696f 6e20 6f66 2074 6869  sentation of thi
│ │ │ │ +0001df10: 730a 6d6f 6475 6c65 2e0a 0a46 726f 6d20  s.module...From 
│ │ │ │ +0001df20: 7468 6520 6465 7363 7269 7074 696f 6e20  the description 
│ │ │ │ +0001df30: 6279 206d 6174 7269 7820 6661 6374 6f72  by matrix factor
│ │ │ │ +0001df40: 697a 6174 696f 6e73 2061 6e64 2074 6865  izations and the
│ │ │ │ +0001df50: 2070 6170 6572 2022 546f 7220 6173 2061   paper "Tor as a
│ │ │ │ +0001df60: 206d 6f64 756c 650a 6f76 6572 2061 6e20   module.over an 
│ │ │ │ +0001df70: 6578 7465 7269 6f72 2061 6c67 6562 7261  exterior algebra
│ │ │ │ +0001df80: 2220 6f66 2045 6973 656e 6275 642c 2050  " of Eisenbud, P
│ │ │ │ +0001df90: 6565 7661 2061 6e64 2053 6368 7265 7965  eeva and Schreye
│ │ │ │ +0001dfa0: 7220 6974 2066 6f6c 6c6f 7773 2074 6861  r it follows tha
│ │ │ │ +0001dfb0: 7420 7768 656e 0a4d 2069 7320 6120 6869  t when.M is a hi
│ │ │ │ +0001dfc0: 6768 2073 797a 7967 7920 616e 6420 4620  gh syzygy and F 
│ │ │ │ +0001dfd0: 6973 2069 7473 2072 6573 6f6c 7574 696f  is its resolutio
│ │ │ │ +0001dfe0: 6e2c 2074 6865 6e20 7468 6520 7072 6573  n, then the pres
│ │ │ │ +0001dff0: 656e 7461 7469 6f6e 206f 660a 546f 7228  entation of.Tor(
│ │ │ │ +0001e000: 4d2c 535e 312f 6d6d 2920 616c 7761 7973  M,S^1/mm) always
│ │ │ │ +0001e010: 2068 6173 2067 656e 6572 6174 6f72 7320   has generators 
│ │ │ │ +0001e020: 696e 2064 6567 7265 6573 2030 2c31 2c20  in degrees 0,1, 
│ │ │ │ +0001e030: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ +0001e040: 2074 6865 0a74 6172 6765 7473 2061 6e64   the.targets and
│ │ │ │ +0001e050: 2073 6f75 7263 6573 206f 6620 7468 6520   sources of the 
│ │ │ │ +0001e060: 7374 6163 6b20 6f66 206d 6170 7320 4228  stack of maps B(
│ │ │ │ +0001e070: 6929 2c20 616e 6420 7468 6174 2074 6865  i), and that the
│ │ │ │ +0001e080: 2072 6573 6f6c 7574 696f 6e20 6973 0a63   resolution is.c
│ │ │ │ +0001e090: 6f6d 706f 6e65 6e74 7769 7365 206c 696e  omponentwise lin
│ │ │ │ +0001e0a0: 6561 7220 696e 2061 2073 7569 7461 626c  ear in a suitabl
│ │ │ │ +0001e0b0: 6520 7365 6e73 652e 2049 6e20 7468 6520  e sense. In the 
│ │ │ │ +0001e0c0: 666f 6c6c 6f77 696e 6720 6578 616d 706c  following exampl
│ │ │ │ +0001e0d0: 652c 2074 6865 7365 2066 6163 7473 0a61  e, these facts.a
│ │ │ │ +0001e0e0: 7265 2076 6572 6966 6965 642e 2054 6865  re verified. The
│ │ │ │ +0001e0f0: 2054 6f72 206d 6f64 756c 6520 646f 6573   Tor module does
│ │ │ │ +0001e100: 204e 4f54 2073 706c 6974 2069 6e74 6f20   NOT split into 
│ │ │ │ +0001e110: 7468 6520 6469 7265 6374 2073 756d 206f  the direct sum o
│ │ │ │ +0001e120: 6620 7468 650a 7375 626d 6f64 756c 6573  f the.submodules
│ │ │ │ +0001e130: 2067 656e 6572 6174 6564 2069 6e20 6465   generated in de
│ │ │ │ +0001e140: 6772 6565 7320 3020 616e 6420 312c 2068  grees 0 and 1, h
│ │ │ │ +0001e150: 6f77 6576 6572 2e0a 0a0a 0a2b 2d2d 2d2d  owever.....+----
│ │ │ │ +0001e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e1c0: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f  +.|i1 : kk = ZZ/
│ │ │ │ -0001e1d0: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │ -0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001e1a0: 0a7c 6931 203a 206b 6b20 3d20 5a5a 2f31  .|i1 : kk = ZZ/1
│ │ │ │ +0001e1b0: 3031 2020 2020 2020 2020 2020 2020 2020  01              
│ │ │ │ +0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e1e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e200: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001e250: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +0001e220: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0001e230: 203d 206b 6b20 2020 2020 2020 2020 2020   = kk           
│ │ │ │ +0001e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e270: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e290: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2d0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -0001e2e0: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ -0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e310: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e320: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001e2b0: 2020 2020 2020 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ +0001e2c0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +0001e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e2f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001e300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e360: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20  ----+.|i2 : S = 
│ │ │ │ -0001e370: 6b6b 5b61 2c62 2c63 5d20 2020 2020 2020  kk[a,b,c]       
│ │ │ │ -0001e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e340: 2d2d 2d2b 0a7c 6932 203a 2053 203d 206b  ---+.|i2 : S = k
│ │ │ │ +0001e350: 6b5b 612c 622c 635d 2020 2020 2020 2020  k[a,b,c]        
│ │ │ │ +0001e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e380: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e3d0: 0a7c 6f32 203d 2053 2020 2020 2020 2020  .|o2 = S        
│ │ │ │  0001e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e3f0: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020  |.|o2 = S       
│ │ │ │ +0001e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e410: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e430: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e470: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001e480: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ -0001e490: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -0001e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e4c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0001e450: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0001e460: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ +0001e470: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0001e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e4a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0001e4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -0001e510: 6620 3d20 6d61 7472 6978 2261 342c 6234  f = matrix"a4,b4
│ │ │ │ -0001e520: 2c63 3422 2020 2020 2020 2020 2020 2020  ,c4"            
│ │ │ │ +0001e4e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066  -------+.|i3 : f
│ │ │ │ +0001e4f0: 203d 206d 6174 7269 7822 6134 2c62 342c   = matrix"a4,b4,
│ │ │ │ +0001e500: 6334 2220 2020 2020 2020 2020 2020 2020  c4"             
│ │ │ │ +0001e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0001e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e540: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e550: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0001e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e590: 2020 2020 7c0a 7c6f 3320 3d20 7c20 6134      |.|o3 = | a4
│ │ │ │ -0001e5a0: 2062 3420 6334 207c 2020 2020 2020 2020   b4 c4 |        
│ │ │ │ -0001e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e570: 2020 207c 0a7c 6f33 203d 207c 2061 3420     |.|o3 = | a4 
│ │ │ │ +0001e580: 6234 2063 3420 7c20 2020 2020 2020 2020  b4 c4 |         
│ │ │ │ +0001e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e5b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e620: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0001e630: 3120 2020 2020 2033 2020 2020 2020 2020  1      3        
│ │ │ │ -0001e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e660: 2020 2020 2020 7c0a 7c6f 3320 3a20 4d61        |.|o3 : Ma
│ │ │ │ -0001e670: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
│ │ │ │ -0001e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e6a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0001e5f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e600: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ +0001e610: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0001e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e640: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ +0001e650: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +0001e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e680: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0001e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e6f0: 2d2d 2b0a 7c69 3420 3a20 5220 3d20 532f  --+.|i4 : R = S/
│ │ │ │ -0001e700: 6964 6561 6c20 6620 2020 2020 2020 2020  ideal f         
│ │ │ │ -0001e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e6d0: 2d2b 0a7c 6934 203a 2052 203d 2053 2f69  -+.|i4 : R = S/i
│ │ │ │ +0001e6e0: 6465 616c 2066 2020 2020 2020 2020 2020  deal f          
│ │ │ │ +0001e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e710: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e730: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e770: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e780: 7c6f 3420 3d20 5220 2020 2020 2020 2020  |o4 = R         
│ │ │ │ +0001e750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e760: 6f34 203d 2052 2020 2020 2020 2020 2020  o4 = R          
│ │ │ │ +0001e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e7a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e800: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -0001e810: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ -0001e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e850: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0001e7e0: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ +0001e7f0: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +0001e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001e830: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e890: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7020  ------+.|i5 : p 
│ │ │ │ -0001e8a0: 3d20 6d61 7028 522c 5329 2020 2020 2020  = map(R,S)      
│ │ │ │ -0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e870: 2d2d 2d2d 2d2b 0a7c 6935 203a 2070 203d  -----+.|i5 : p =
│ │ │ │ +0001e880: 206d 6170 2852 2c53 2920 2020 2020 2020   map(R,S)       
│ │ │ │ +0001e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e8b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e920: 2020 7c0a 7c6f 3520 3d20 6d61 7020 2852    |.|o5 = map (R
│ │ │ │ -0001e930: 2c20 532c 207b 612c 2062 2c20 637d 2920  , S, {a, b, c}) 
│ │ │ │ -0001e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e900: 207c 0a7c 6f35 203d 206d 6170 2028 522c   |.|o5 = map (R,
│ │ │ │ +0001e910: 2053 2c20 7b61 2c20 622c 2063 7d29 2020   S, {a, b, c})  
│ │ │ │ +0001e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e940: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e960: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001e9b0: 7c6f 3520 3a20 5269 6e67 4d61 7020 5220  |o5 : RingMap R 
│ │ │ │ -0001e9c0: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ -0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e9f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001e980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001e990: 6f35 203a 2052 696e 674d 6170 2052 203c  o5 : RingMap R <
│ │ │ │ +0001e9a0: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ +0001e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e9d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -0001ea40: 3a20 4d20 3d20 636f 6b65 7220 6d61 7028  : M = coker map(
│ │ │ │ -0001ea50: 525e 322c 2052 5e7b 333a 2d31 7d2c 207b  R^2, R^{3:-1}, {
│ │ │ │ -0001ea60: 7b61 2c62 2c63 7d2c 7b62 2c63 2c61 7d7d  {a,b,c},{b,c,a}}
│ │ │ │ -0001ea70: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001ea80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0001ea20: 204d 203d 2063 6f6b 6572 206d 6170 2852   M = coker map(R
│ │ │ │ +0001ea30: 5e32 2c20 525e 7b33 3a2d 317d 2c20 7b7b  ^2, R^{3:-1}, {{
│ │ │ │ +0001ea40: 612c 622c 637d 2c7b 622c 632c 617d 7d29  a,b,c},{b,c,a}})
│ │ │ │ +0001ea50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001ea60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eac0: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f        |.|o6 = co
│ │ │ │ -0001ead0: 6b65 726e 656c 207c 2061 2062 2063 207c  kernel | a b c |
│ │ │ │ -0001eae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001eb10: 2020 2020 2020 2020 2020 2020 207c 2062               | b
│ │ │ │ -0001eb20: 2063 2061 207c 2020 2020 2020 2020 2020   c a |          
│ │ │ │ -0001eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eaa0: 2020 2020 207c 0a7c 6f36 203d 2063 6f6b       |.|o6 = cok
│ │ │ │ +0001eab0: 6572 6e65 6c20 7c20 6120 6220 6320 7c20  ernel | a b c | 
│ │ │ │ +0001eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001eaf0: 2020 2020 2020 2020 2020 2020 7c20 6220              | b 
│ │ │ │ +0001eb00: 6320 6120 7c20 2020 2020 2020 2020 2020  c a |           
│ │ │ │ +0001eb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eb30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eb70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0001eb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0001eb90: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0001eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ebb0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0001ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ebd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ebe0: 7c6f 3620 3a20 522d 6d6f 6475 6c65 2c20  |o6 : R-module, 
│ │ │ │ -0001ebf0: 7175 6f74 6965 6e74 206f 6620 5220 2020  quotient of R   
│ │ │ │ -0001ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ec20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0001ebb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ebc0: 6f36 203a 2052 2d6d 6f64 756c 652c 2071  o6 : R-module, q
│ │ │ │ +0001ebd0: 756f 7469 656e 7420 6f66 2052 2020 2020  uotient of R    
│ │ │ │ +0001ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ec00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0001ec10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ -0001ec70: 3a20 6265 7474 6920 2846 4620 3d66 7265  : betti (FF =fre
│ │ │ │ -0001ec80: 6552 6573 6f6c 7574 696f 6e28 204d 2c20  eResolution( M, 
│ │ │ │ -0001ec90: 4c65 6e67 7468 4c69 6d69 7420 3d3e 3629  LengthLimit =>6)
│ │ │ │ -0001eca0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -0001ecb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0001ec40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ +0001ec50: 2062 6574 7469 2028 4646 203d 6672 6565   betti (FF =free
│ │ │ │ +0001ec60: 5265 736f 6c75 7469 6f6e 2820 4d2c 204c  Resolution( M, L
│ │ │ │ +0001ec70: 656e 6774 684c 696d 6974 203d 3e36 2929  engthLimit =>6))
│ │ │ │ +0001ec80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001ec90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ecf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ed00: 2020 2020 2030 2031 2032 2033 2034 2020       0 1 2 3 4  
│ │ │ │ -0001ed10: 3520 2036 2020 2020 2020 2020 2020 2020  5  6            
│ │ │ │ -0001ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0001ed40: 3720 3d20 746f 7461 6c3a 2032 2033 2034  7 = total: 2 3 4
│ │ │ │ -0001ed50: 2036 2039 2031 3320 3138 2020 2020 2020   6 9 13 18      
│ │ │ │ -0001ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed80: 2020 7c0a 7c20 2020 2020 2020 2020 303a    |.|         0:
│ │ │ │ -0001ed90: 2032 2033 202e 202e 202e 2020 2e20 202e   2 3 . . .  .  .
│ │ │ │ -0001eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001edb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001edc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001edd0: 2020 2020 313a 202e 202e 2031 202e 202e      1: . . 1 . .
│ │ │ │ -0001ede0: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ -0001edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ee10: 7c20 2020 2020 2020 2020 323a 202e 202e  |         2: . .
│ │ │ │ -0001ee20: 2033 2033 202e 2020 2e20 202e 2020 2020   3 3 .  .  .    
│ │ │ │ -0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001ee60: 333a 202e 202e 202e 2033 2033 2020 2e20  3: . . . 3 3  . 
│ │ │ │ -0001ee70: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001eea0: 2020 2020 2020 343a 202e 202e 202e 202e        4: . . . .
│ │ │ │ -0001eeb0: 2033 2020 3320 202e 2020 2020 2020 2020   3  3  .        
│ │ │ │ -0001eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eee0: 7c0a 7c20 2020 2020 2020 2020 353a 202e  |.|         5: .
│ │ │ │ -0001eef0: 202e 202e 202e 2033 2020 3920 2036 2020   . . . 3  9  6  
│ │ │ │ -0001ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ef30: 2020 363a 202e 202e 202e 202e 202e 2020    6: . . . . .  
│ │ │ │ -0001ef40: 2e20 2033 2020 2020 2020 2020 2020 2020  .  3            
│ │ │ │ -0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001ef70: 2020 2020 2020 2020 373a 202e 202e 202e          7: . . .
│ │ │ │ -0001ef80: 202e 202e 2020 3120 2039 2020 2020 2020   . .  1  9      
│ │ │ │ -0001ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ecd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ece0: 2020 2020 3020 3120 3220 3320 3420 2035      0 1 2 3 4  5
│ │ │ │ +0001ecf0: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ +0001ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ed10: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ +0001ed20: 203d 2074 6f74 616c 3a20 3220 3320 3420   = total: 2 3 4 
│ │ │ │ +0001ed30: 3620 3920 3133 2031 3820 2020 2020 2020  6 9 13 18       
│ │ │ │ +0001ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ed60: 207c 0a7c 2020 2020 2020 2020 2030 3a20   |.|         0: 
│ │ │ │ +0001ed70: 3220 3320 2e20 2e20 2e20 202e 2020 2e20  2 3 . . .  .  . 
│ │ │ │ +0001ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eda0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001edb0: 2020 2031 3a20 2e20 2e20 3120 2e20 2e20     1: . . 1 . . 
│ │ │ │ +0001edc0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │ +0001edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ede0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001edf0: 2020 2020 2020 2020 2032 3a20 2e20 2e20           2: . . 
│ │ │ │ +0001ee00: 3320 3320 2e20 202e 2020 2e20 2020 2020  3 3 .  .  .     
│ │ │ │ +0001ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee30: 2020 207c 0a7c 2020 2020 2020 2020 2033     |.|         3
│ │ │ │ +0001ee40: 3a20 2e20 2e20 2e20 3320 3320 202e 2020  : . . . 3 3  .  
│ │ │ │ +0001ee50: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0001ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001ee80: 2020 2020 2034 3a20 2e20 2e20 2e20 2e20       4: . . . . 
│ │ │ │ +0001ee90: 3320 2033 2020 2e20 2020 2020 2020 2020  3  3  .         
│ │ │ │ +0001eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eeb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001eec0: 0a7c 2020 2020 2020 2020 2035 3a20 2e20  .|         5: . 
│ │ │ │ +0001eed0: 2e20 2e20 2e20 3320 2039 2020 3620 2020  . . . 3  9  6   
│ │ │ │ +0001eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001eef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ef10: 2036 3a20 2e20 2e20 2e20 2e20 2e20 202e   6: . . . . .  .
│ │ │ │ +0001ef20: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001ef50: 2020 2020 2020 2037 3a20 2e20 2e20 2e20         7: . . . 
│ │ │ │ +0001ef60: 2e20 2e20 2031 2020 3920 2020 2020 2020  . .  1  9       
│ │ │ │ +0001ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ef90: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001eff0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -0001f000: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ -0001f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f030: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f040: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001efd0: 2020 2020 2020 207c 0a7c 6f37 203a 2042         |.|o7 : B
│ │ │ │ +0001efe0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +0001eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f010: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001f020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f080: 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53 203d  ----+.|i8 : MS =
│ │ │ │ -0001f090: 2070 7275 6e65 2070 7573 6846 6f72 7761   prune pushForwa
│ │ │ │ -0001f0a0: 7264 2870 2c20 636f 6b65 7220 4646 2e64  rd(p, coker FF.d
│ │ │ │ -0001f0b0: 645f 3629 3b20 2020 2020 2020 2020 2020  d_6);           
│ │ │ │ -0001f0c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0001f060: 2d2d 2d2b 0a7c 6938 203a 204d 5320 3d20  ---+.|i8 : MS = 
│ │ │ │ +0001f070: 7072 756e 6520 7075 7368 466f 7277 6172  prune pushForwar
│ │ │ │ +0001f080: 6428 702c 2063 6f6b 6572 2046 462e 6464  d(p, coker FF.dd
│ │ │ │ +0001f090: 5f36 293b 2020 2020 2020 2020 2020 2020  _6);            
│ │ │ │ +0001f0a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0001f0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f110: 2b0a 7c69 3920 3a20 5420 3d20 6578 7465  +.|i9 : T = exte
│ │ │ │ -0001f120: 7269 6f72 546f 724d 6f64 756c 6528 662c  riorTorModule(f,
│ │ │ │ -0001f130: 4d53 293b 2020 2020 2020 2020 2020 2020  MS);            
│ │ │ │ -0001f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f150: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001f0f0: 0a7c 6939 203a 2054 203d 2065 7874 6572  .|i9 : T = exter
│ │ │ │ +0001f100: 696f 7254 6f72 4d6f 6475 6c65 2866 2c4d  iorTorModule(f,M
│ │ │ │ +0001f110: 5329 3b20 2020 2020 2020 2020 2020 2020  S);             
│ │ │ │ +0001f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f130: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001f1a0: 3130 203a 2062 6574 7469 2054 2020 2020  10 : betti T    
│ │ │ │ +0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001f180: 3020 3a20 6265 7474 6920 5420 2020 2020  0 : betti T     
│ │ │ │ +0001f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f1c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f1e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f220: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f230: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ -0001f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f270: 7c6f 3130 203d 2074 6f74 616c 3a20 3834  |o10 = total: 84
│ │ │ │ -0001f280: 2032 3532 2020 2020 2020 2020 2020 2020   252            
│ │ │ │ -0001f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001f2c0: 2030 3a20 3133 2020 3339 2020 2020 2020   0: 13  39      
│ │ │ │ -0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f2f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f300: 2020 2020 2020 2031 3a20 3333 2020 3939         1: 33  99
│ │ │ │ -0001f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f340: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ -0001f350: 3239 2020 3837 2020 2020 2020 2020 2020  29  87          
│ │ │ │ -0001f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f380: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f390: 2020 2033 3a20 2039 2020 3237 2020 2020     3:  9  27    
│ │ │ │ -0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f200: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f210: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +0001f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001f250: 6f31 3020 3d20 746f 7461 6c3a 2038 3420  o10 = total: 84 
│ │ │ │ +0001f260: 3235 3220 2020 2020 2020 2020 2020 2020  252             
│ │ │ │ +0001f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001f2a0: 303a 2031 3320 2033 3920 2020 2020 2020  0: 13  39       
│ │ │ │ +0001f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f2d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f2e0: 2020 2020 2020 313a 2033 3320 2039 3920        1: 33  99 
│ │ │ │ +0001f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f320: 0a7c 2020 2020 2020 2020 2020 323a 2032  .|          2: 2
│ │ │ │ +0001f330: 3920 2038 3720 2020 2020 2020 2020 2020  9  87           
│ │ │ │ +0001f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f370: 2020 333a 2020 3920 2032 3720 2020 2020    3:  9  27     
│ │ │ │ +0001f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f3a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f410: 2020 7c0a 7c6f 3130 203a 2042 6574 7469    |.|o10 : Betti
│ │ │ │ -0001f420: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ -0001f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f450: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0001f3f0: 207c 0a7c 6f31 3020 3a20 4265 7474 6954   |.|o10 : BettiT
│ │ │ │ +0001f400: 616c 6c79 2020 2020 2020 2020 2020 2020  ally            
│ │ │ │ +0001f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f430: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0001f440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0001f4a0: 7c69 3131 203a 2062 6574 7469 2066 7265  |i11 : betti fre
│ │ │ │ -0001f4b0: 6552 6573 6f6c 7574 696f 6e20 2850 5420  eResolution (PT 
│ │ │ │ -0001f4c0: 3d20 7072 756e 6520 542c 204c 656e 6774  = prune T, Lengt
│ │ │ │ -0001f4d0: 684c 696d 6974 203d 3e20 3429 2020 2020  hLimit => 4)    
│ │ │ │ -0001f4e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0001f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0001f480: 6931 3120 3a20 6265 7474 6920 6672 6565  i11 : betti free
│ │ │ │ +0001f490: 5265 736f 6c75 7469 6f6e 2028 5054 203d  Resolution (PT =
│ │ │ │ +0001f4a0: 2070 7275 6e65 2054 2c20 4c65 6e67 7468   prune T, Length
│ │ │ │ +0001f4b0: 4c69 6d69 7420 3d3e 2034 2920 2020 2020  Limit => 4)     
│ │ │ │ +0001f4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f520: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0001f530: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ -0001f540: 2032 2020 2033 2020 2034 2020 2020 2020   2   3   4      
│ │ │ │ -0001f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f570: 7c0a 7c6f 3131 203d 2074 6f74 616c 3a20  |.|o11 = total: 
│ │ │ │ -0001f580: 3331 2035 3520 3837 2031 3237 2031 3735  31 55 87 127 175
│ │ │ │ -0001f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f5c0: 2020 2030 3a20 3133 2032 3420 3339 2020     0: 13 24 39  
│ │ │ │ -0001f5d0: 3538 2020 3831 2020 2020 2020 2020 2020  58  81          
│ │ │ │ -0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0001f600: 2020 2020 2020 2020 2031 3a20 3138 2033           1: 18 3
│ │ │ │ -0001f610: 3120 3438 2020 3639 2020 3934 2020 2020  1 48  69  94    
│ │ │ │ -0001f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f500: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0001f510: 2020 2020 2020 2020 2020 3020 2031 2020            0  1  
│ │ │ │ +0001f520: 3220 2020 3320 2020 3420 2020 2020 2020  2   3   4       
│ │ │ │ +0001f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f550: 0a7c 6f31 3120 3d20 746f 7461 6c3a 2033  .|o11 = total: 3
│ │ │ │ +0001f560: 3120 3535 2038 3720 3132 3720 3137 3520  1 55 87 127 175 
│ │ │ │ +0001f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f5a0: 2020 303a 2031 3320 3234 2033 3920 2035    0: 13 24 39  5
│ │ │ │ +0001f5b0: 3820 2038 3120 2020 2020 2020 2020 2020  8  81           
│ │ │ │ +0001f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f5d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0001f5e0: 2020 2020 2020 2020 313a 2031 3820 3331          1: 18 31
│ │ │ │ +0001f5f0: 2034 3820 2036 3920 2039 3420 2020 2020   48  69  94     
│ │ │ │ +0001f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f620: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0001f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f640: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f680: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ -0001f690: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ -0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f6d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001f660: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ +0001f670: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0001f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6a0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f710: 2d2d 2d2d 2b0a 7c69 3132 203a 2061 6e6e  ----+.|i12 : ann
│ │ │ │ -0001f720: 2050 5420 2020 2020 2020 2020 2020 2020   PT             
│ │ │ │ -0001f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6f0: 2d2d 2d2b 0a7c 6931 3220 3a20 616e 6e20  ---+.|i12 : ann 
│ │ │ │ +0001f700: 5054 2020 2020 2020 2020 2020 2020 2020  PT              
│ │ │ │ +0001f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f730: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f750: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7a0: 7c0a 7c6f 3132 203d 2069 6465 616c 2865  |.|o12 = ideal(e
│ │ │ │ -0001f7b0: 2065 2065 2029 2020 2020 2020 2020 2020   e e )          
│ │ │ │ -0001f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001f7f0: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ -0001f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001f780: 0a7c 6f31 3220 3d20 6964 6561 6c28 6520  .|o12 = ideal(e 
│ │ │ │ +0001f790: 6520 6520 2920 2020 2020 2020 2020 2020  e e )           
│ │ │ │ +0001f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f7c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001f7d0: 2020 2020 2030 2031 2032 2020 2020 2020       0 1 2      
│ │ │ │ +0001f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0001f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f870: 2020 7c0a 7c6f 3132 203a 2049 6465 616c    |.|o12 : Ideal
│ │ │ │ -0001f880: 206f 6620 6b6b 5b65 202e 2e65 205d 2020   of kk[e ..e ]  
│ │ │ │ -0001f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001f8c0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0001f8d0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0001f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001f900: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0001f850: 207c 0a7c 6f31 3220 3a20 4964 6561 6c20   |.|o12 : Ideal 
│ │ │ │ +0001f860: 6f66 206b 6b5b 6520 2e2e 6520 5d20 2020  of kk[e ..e ]   
│ │ │ │ +0001f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f890: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001f8a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +0001f8b0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0001f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f8d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0001f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f940: 2d2d 2d2d 2b0a 7c69 3133 203a 2050 5430  ----+.|i13 : PT0
│ │ │ │ -0001f950: 203d 2069 6d61 6765 2028 696e 6475 6365   = image (induce
│ │ │ │ -0001f960: 644d 6170 2850 542c 636f 7665 7220 5054  dMap(PT,cover PT
│ │ │ │ -0001f970: 292a 2028 2863 6f76 6572 2050 5429 5f7b  )* ((cover PT)_{
│ │ │ │ -0001f980: 302e 2e31 327d 2929 3b20 7c0a 2b2d 2d2d  0..12})); |.+---
│ │ │ │ +0001f920: 2d2d 2d2b 0a7c 6931 3320 3a20 5054 3020  ---+.|i13 : PT0 
│ │ │ │ +0001f930: 3d20 696d 6167 6520 2869 6e64 7563 6564  = image (induced
│ │ │ │ +0001f940: 4d61 7028 5054 2c63 6f76 6572 2050 5429  Map(PT,cover PT)
│ │ │ │ +0001f950: 2a20 2828 636f 7665 7220 5054 295f 7b30  * ((cover PT)_{0
│ │ │ │ +0001f960: 2e2e 3132 7d29 293b 207c 0a2b 2d2d 2d2d  ..12})); |.+----
│ │ │ │ +0001f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f9d0: 2b0a 7c69 3134 203a 2050 5431 203d 2069  +.|i14 : PT1 = i
│ │ │ │ -0001f9e0: 6d61 6765 2028 696e 6475 6365 644d 6170  mage (inducedMap
│ │ │ │ -0001f9f0: 2850 542c 636f 7665 7220 5054 292a 2028  (PT,cover PT)* (
│ │ │ │ -0001fa00: 2863 6f76 6572 2050 5429 5f7b 3133 2e2e  (cover PT)_{13..
│ │ │ │ -0001fa10: 3330 7d29 293b 7c0a 2b2d 2d2d 2d2d 2d2d  30}));|.+-------
│ │ │ │ +0001f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0001f9b0: 0a7c 6931 3420 3a20 5054 3120 3d20 696d  .|i14 : PT1 = im
│ │ │ │ +0001f9c0: 6167 6520 2869 6e64 7563 6564 4d61 7028  age (inducedMap(
│ │ │ │ +0001f9d0: 5054 2c63 6f76 6572 2050 5429 2a20 2828  PT,cover PT)* ((
│ │ │ │ +0001f9e0: 636f 7665 7220 5054 295f 7b31 332e 2e33  cover PT)_{13..3
│ │ │ │ +0001f9f0: 307d 2929 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d  0}));|.+--------
│ │ │ │ +0001fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001fa60: 3135 203a 2062 6574 7469 2066 7265 6552  15 : betti freeR
│ │ │ │ -0001fa70: 6573 6f6c 7574 696f 6e28 7072 756e 6520  esolution(prune 
│ │ │ │ -0001fa80: 5054 302c 204c 656e 6774 684c 696d 6974  PT0, LengthLimit
│ │ │ │ -0001fa90: 203d 3e20 3429 2020 2020 2020 2020 2020   => 4)          
│ │ │ │ -0001faa0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001fa40: 3520 3a20 6265 7474 6920 6672 6565 5265  5 : betti freeRe
│ │ │ │ +0001fa50: 736f 6c75 7469 6f6e 2870 7275 6e65 2050  solution(prune P
│ │ │ │ +0001fa60: 5430 2c20 4c65 6e67 7468 4c69 6d69 7420  T0, LengthLimit 
│ │ │ │ +0001fa70: 3d3e 2034 2920 2020 2020 2020 2020 2020  => 4)           
│ │ │ │ +0001fa80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001faf0: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ -0001fb00: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ -0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001fb30: 7c6f 3135 203d 2074 6f74 616c 3a20 3133  |o15 = total: 13
│ │ │ │ -0001fb40: 2032 3420 3339 2035 3820 3831 2020 2020   24 39 58 81    
│ │ │ │ -0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb70: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fb80: 2030 3a20 3133 2032 3420 3339 2035 3820   0: 13 24 39 58 
│ │ │ │ -0001fb90: 3831 2020 2020 2020 2020 2020 2020 2020  81              
│ │ │ │ +0001fac0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001fad0: 2020 2020 2020 2020 3020 2031 2020 3220          0  1  2 
│ │ │ │ +0001fae0: 2033 2020 3420 2020 2020 2020 2020 2020   3  4           
│ │ │ │ +0001faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001fb10: 6f31 3520 3d20 746f 7461 6c3a 2031 3320  o15 = total: 13 
│ │ │ │ +0001fb20: 3234 2033 3920 3538 2038 3120 2020 2020  24 39 58 81     
│ │ │ │ +0001fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001fb60: 303a 2031 3320 3234 2033 3920 3538 2038  0: 13 24 39 58 8
│ │ │ │ +0001fb70: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0001fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc00: 7c0a 7c6f 3135 203a 2042 6574 7469 5461  |.|o15 : BettiTa
│ │ │ │ -0001fc10: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ -0001fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001fbd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fbe0: 0a7c 6f31 3520 3a20 4265 7474 6954 616c  .|o15 : BettiTal
│ │ │ │ +0001fbf0: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ +0001fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001fc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001fc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001fc90: 3136 203a 2062 6574 7469 2066 7265 6552  16 : betti freeR
│ │ │ │ -0001fca0: 6573 6f6c 7574 696f 6e28 7072 756e 6520  esolution(prune 
│ │ │ │ -0001fcb0: 5054 312c 204c 656e 6774 684c 696d 6974  PT1, LengthLimit
│ │ │ │ -0001fcc0: 203d 3e20 3429 2020 2020 2020 2020 2020   => 4)          
│ │ │ │ -0001fcd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001fc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001fc70: 3620 3a20 6265 7474 6920 6672 6565 5265  6 : betti freeRe
│ │ │ │ +0001fc80: 736f 6c75 7469 6f6e 2870 7275 6e65 2050  solution(prune P
│ │ │ │ +0001fc90: 5431 2c20 4c65 6e67 7468 4c69 6d69 7420  T1, LengthLimit 
│ │ │ │ +0001fca0: 3d3e 2034 2920 2020 2020 2020 2020 2020  => 4)           
│ │ │ │ +0001fcb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001fd20: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ -0001fd30: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ -0001fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001fd60: 7c6f 3136 203d 2074 6f74 616c 3a20 3138  |o16 = total: 18
│ │ │ │ -0001fd70: 2032 3820 3339 2035 3120 3634 2020 2020   28 39 51 64    
│ │ │ │ -0001fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fda0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001fdb0: 2031 3a20 3138 2032 3820 3339 2035 3120   1: 18 28 39 51 
│ │ │ │ -0001fdc0: 3634 2020 2020 2020 2020 2020 2020 2020  64              
│ │ │ │ +0001fcf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001fd00: 2020 2020 2020 2020 3020 2031 2020 3220          0  1  2 
│ │ │ │ +0001fd10: 2033 2020 3420 2020 2020 2020 2020 2020   3  4           
│ │ │ │ +0001fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fd30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001fd40: 6f31 3620 3d20 746f 7461 6c3a 2031 3820  o16 = total: 18 
│ │ │ │ +0001fd50: 3238 2033 3920 3531 2036 3420 2020 2020  28 39 51 64     
│ │ │ │ +0001fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fd80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001fd90: 313a 2031 3820 3238 2033 3920 3531 2036  1: 18 28 39 51 6
│ │ │ │ +0001fda0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0001fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fdc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0001fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fde0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0001fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe30: 7c0a 7c6f 3136 203a 2042 6574 7469 5461  |.|o16 : BettiTa
│ │ │ │ -0001fe40: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ -0001fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe70: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001fe00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0001fe10: 0a7c 6f31 3620 3a20 4265 7474 6954 616c  .|o16 : BettiTal
│ │ │ │ +0001fe20: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ +0001fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fe50: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0001fe60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001feb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0001fec0: 3137 203a 2062 6574 7469 2066 7265 6552  17 : betti freeR
│ │ │ │ -0001fed0: 6573 6f6c 7574 696f 6e28 7072 756e 6520  esolution(prune 
│ │ │ │ -0001fee0: 5054 2c20 4c65 6e67 7468 4c69 6d69 7420  PT, LengthLimit 
│ │ │ │ -0001fef0: 3d3e 2034 2920 2020 2020 2020 2020 2020  => 4)           
│ │ │ │ -0001ff00: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0001fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0001fea0: 3720 3a20 6265 7474 6920 6672 6565 5265  7 : betti freeRe
│ │ │ │ +0001feb0: 736f 6c75 7469 6f6e 2870 7275 6e65 2050  solution(prune P
│ │ │ │ +0001fec0: 542c 204c 656e 6774 684c 696d 6974 203d  T, LengthLimit =
│ │ │ │ +0001fed0: 3e20 3429 2020 2020 2020 2020 2020 2020  > 4)            
│ │ │ │ +0001fee0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0001fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0001ff50: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ -0001ff60: 2020 2033 2020 2034 2020 2020 2020 2020     3   4        
│ │ │ │ -0001ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0001ff90: 7c6f 3137 203d 2074 6f74 616c 3a20 3331  |o17 = total: 31
│ │ │ │ -0001ffa0: 2035 3520 3837 2031 3237 2031 3735 2020   55 87 127 175  
│ │ │ │ -0001ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ffd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0001ffe0: 2030 3a20 3133 2032 3420 3339 2020 3538   0: 13 24 39  58
│ │ │ │ -0001fff0: 2020 3831 2020 2020 2020 2020 2020 2020    81            
│ │ │ │ -00020000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020010: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00020020: 2020 2020 2020 2031 3a20 3138 2033 3120         1: 18 31 
│ │ │ │ -00020030: 3438 2020 3639 2020 3934 2020 2020 2020  48  69  94      
│ │ │ │ -00020040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ff20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0001ff30: 2020 2020 2020 2020 3020 2031 2020 3220          0  1  2 
│ │ │ │ +0001ff40: 2020 3320 2020 3420 2020 2020 2020 2020    3   4         
│ │ │ │ +0001ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ff60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0001ff70: 6f31 3720 3d20 746f 7461 6c3a 2033 3120  o17 = total: 31 
│ │ │ │ +0001ff80: 3535 2038 3720 3132 3720 3137 3520 2020  55 87 127 175   
│ │ │ │ +0001ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ffb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0001ffc0: 303a 2031 3320 3234 2033 3920 2035 3820  0: 13 24 39  58 
│ │ │ │ +0001ffd0: 2038 3120 2020 2020 2020 2020 2020 2020   81             
│ │ │ │ +0001ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fff0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00020000: 2020 2020 2020 313a 2031 3820 3331 2034        1: 18 31 4
│ │ │ │ +00020010: 3820 2036 3920 2039 3420 2020 2020 2020  8  69  94       
│ │ │ │ +00020020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00020050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020060: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00020060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200a0: 2020 2020 2020 7c0a 7c6f 3137 203a 2042        |.|o17 : B
│ │ │ │ -000200b0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ -000200c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00020080: 2020 2020 207c 0a7c 6f31 3720 3a20 4265       |.|o17 : Be
│ │ │ │ +00020090: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +000200a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000200b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000200c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000200d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000200e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000200f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020130: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ -00020140: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -00020150: 6520 6d61 6b65 4d6f 6475 6c65 3a20 6d61  e makeModule: ma
│ │ │ │ -00020160: 6b65 4d6f 6475 6c65 2c20 2d2d 206d 616b  keModule, -- mak
│ │ │ │ -00020170: 6573 2061 204d 6f64 756c 6520 6f75 7420  es a Module out 
│ │ │ │ -00020180: 6f66 2061 2063 6f6c 6c65 6374 696f 6e20  of a collection 
│ │ │ │ -00020190: 6f66 0a20 2020 206d 6f64 756c 6573 2061  of.    modules a
│ │ │ │ -000201a0: 6e64 206d 6170 730a 0a57 6179 7320 746f  nd maps..Ways to
│ │ │ │ -000201b0: 2075 7365 2065 7874 6572 696f 7254 6f72   use exteriorTor
│ │ │ │ -000201c0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d  Module:.========
│ │ │ │ -000201d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000201e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 7874  ======..  * "ext
│ │ │ │ -000201f0: 6572 696f 7254 6f72 4d6f 6475 6c65 284d  eriorTorModule(M
│ │ │ │ -00020200: 6174 7269 782c 4d6f 6475 6c65 2922 0a20  atrix,Module)". 
│ │ │ │ -00020210: 202a 2022 6578 7465 7269 6f72 546f 724d   * "exteriorTorM
│ │ │ │ -00020220: 6f64 756c 6528 4d61 7472 6978 2c4d 6f64  odule(Matrix,Mod
│ │ │ │ -00020230: 756c 652c 4d6f 6475 6c65 2922 0a0a 466f  ule,Module)"..Fo
│ │ │ │ -00020240: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00020250: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00020260: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00020270: 2a6e 6f74 6520 6578 7465 7269 6f72 546f  *note exteriorTo
│ │ │ │ -00020280: 724d 6f64 756c 653a 2065 7874 6572 696f  rModule: exterio
│ │ │ │ -00020290: 7254 6f72 4d6f 6475 6c65 2c20 6973 2061  rTorModule, is a
│ │ │ │ -000202a0: 202a 6e6f 7465 206d 6574 686f 640a 6675   *note method.fu
│ │ │ │ -000202b0: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61  nction: (Macaula
│ │ │ │ -000202c0: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -000202d0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  tion,...--------
│ │ │ │ +00020110: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ +00020120: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +00020130: 206d 616b 654d 6f64 756c 653a 206d 616b   makeModule: mak
│ │ │ │ +00020140: 654d 6f64 756c 652c 202d 2d20 6d61 6b65  eModule, -- make
│ │ │ │ +00020150: 7320 6120 4d6f 6475 6c65 206f 7574 206f  s a Module out o
│ │ │ │ +00020160: 6620 6120 636f 6c6c 6563 7469 6f6e 206f  f a collection o
│ │ │ │ +00020170: 660a 2020 2020 6d6f 6475 6c65 7320 616e  f.    modules an
│ │ │ │ +00020180: 6420 6d61 7073 0a0a 5761 7973 2074 6f20  d maps..Ways to 
│ │ │ │ +00020190: 7573 6520 6578 7465 7269 6f72 546f 724d  use exteriorTorM
│ │ │ │ +000201a0: 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d 3d3d  odule:.=========
│ │ │ │ +000201b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000201c0: 3d3d 3d3d 3d0a 0a20 202a 2022 6578 7465  =====..  * "exte
│ │ │ │ +000201d0: 7269 6f72 546f 724d 6f64 756c 6528 4d61  riorTorModule(Ma
│ │ │ │ +000201e0: 7472 6978 2c4d 6f64 756c 6529 220a 2020  trix,Module)".  
│ │ │ │ +000201f0: 2a20 2265 7874 6572 696f 7254 6f72 4d6f  * "exteriorTorMo
│ │ │ │ +00020200: 6475 6c65 284d 6174 7269 782c 4d6f 6475  dule(Matrix,Modu
│ │ │ │ +00020210: 6c65 2c4d 6f64 756c 6529 220a 0a46 6f72  le,Module)"..For
│ │ │ │ +00020220: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +00020230: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00020240: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +00020250: 6e6f 7465 2065 7874 6572 696f 7254 6f72  note exteriorTor
│ │ │ │ +00020260: 4d6f 6475 6c65 3a20 6578 7465 7269 6f72  Module: exterior
│ │ │ │ +00020270: 546f 724d 6f64 756c 652c 2069 7320 6120  TorModule, is a 
│ │ │ │ +00020280: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ +00020290: 6374 696f 6e3a 2028 4d61 6361 756c 6179  ction: (Macaulay
│ │ │ │ +000202a0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +000202b0: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020320: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00020330: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -00020340: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -00020350: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -00020360: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00020370: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ -00020380: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -00020390: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -000203a0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -000203b0: 6d32 3a34 3138 323a 302e 0a1f 0a46 696c  m2:4182:0....Fil
│ │ │ │ -000203c0: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -000203d0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -000203e0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2065  ns.info, Node: e
│ │ │ │ -000203f0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961  xtIsOnePolynomia
│ │ │ │ -00020400: 6c2c 204e 6578 743a 2045 7874 4d6f 6475  l, Next: ExtModu
│ │ │ │ -00020410: 6c65 2c20 5072 6576 3a20 6578 7465 7269  le, Prev: exteri
│ │ │ │ -00020420: 6f72 546f 724d 6f64 756c 652c 2055 703a  orTorModule, Up:
│ │ │ │ -00020430: 2054 6f70 0a0a 6578 7449 734f 6e65 506f   Top..extIsOnePo
│ │ │ │ -00020440: 6c79 6e6f 6d69 616c 202d 2d20 6368 6563  lynomial -- chec
│ │ │ │ -00020450: 6b20 7768 6574 6865 7220 7468 6520 4869  k whether the Hi
│ │ │ │ -00020460: 6c62 6572 7420 6675 6e63 7469 6f6e 206f  lbert function o
│ │ │ │ -00020470: 6620 4578 7428 4d2c 6b29 2069 7320 6f6e  f Ext(M,k) is on
│ │ │ │ -00020480: 6520 706f 6c79 6e6f 6d69 616c 0a2a 2a2a  e polynomial.***
│ │ │ │ +00020300: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +00020310: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +00020320: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00020330: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00020340: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00020350: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ +00020360: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ +00020370: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +00020380: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ +00020390: 323a 3431 3832 3a30 2e0a 1f0a 4669 6c65  2:4182:0....File
│ │ │ │ +000203a0: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ +000203b0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +000203c0: 732e 696e 666f 2c20 4e6f 6465 3a20 6578  s.info, Node: ex
│ │ │ │ +000203d0: 7449 734f 6e65 506f 6c79 6e6f 6d69 616c  tIsOnePolynomial
│ │ │ │ +000203e0: 2c20 4e65 7874 3a20 4578 744d 6f64 756c  , Next: ExtModul
│ │ │ │ +000203f0: 652c 2050 7265 763a 2065 7874 6572 696f  e, Prev: exterio
│ │ │ │ +00020400: 7254 6f72 4d6f 6475 6c65 2c20 5570 3a20  rTorModule, Up: 
│ │ │ │ +00020410: 546f 700a 0a65 7874 4973 4f6e 6550 6f6c  Top..extIsOnePol
│ │ │ │ +00020420: 796e 6f6d 6961 6c20 2d2d 2063 6865 636b  ynomial -- check
│ │ │ │ +00020430: 2077 6865 7468 6572 2074 6865 2048 696c   whether the Hil
│ │ │ │ +00020440: 6265 7274 2066 756e 6374 696f 6e20 6f66  bert function of
│ │ │ │ +00020450: 2045 7874 284d 2c6b 2920 6973 206f 6e65   Ext(M,k) is one
│ │ │ │ +00020460: 2070 6f6c 796e 6f6d 6961 6c0a 2a2a 2a2a   polynomial.****
│ │ │ │ +00020470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00020480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00020490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000204a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000204b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000204c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000204d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000204e0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -000204f0: 0a20 2020 2020 2020 2028 702c 7429 203d  .        (p,t) =
│ │ │ │ -00020500: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d   extIsOnePolynom
│ │ │ │ -00020510: 6961 6c20 4d0a 2020 2a20 496e 7075 7473  ial M.  * Inputs
│ │ │ │ -00020520: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a  :.      * M, a *
│ │ │ │ -00020530: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00020540: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00020550: 652c 2c20 6d6f 6475 6c65 206f 7665 7220  e,, module over 
│ │ │ │ -00020560: 6120 636f 6d70 6c65 7465 0a20 2020 2020  a complete.     
│ │ │ │ -00020570: 2020 2069 6e74 6572 7365 6374 696f 6e0a     intersection.
│ │ │ │ -00020580: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00020590: 2020 202a 2070 2c20 6120 2a6e 6f74 6520     * p, a *note 
│ │ │ │ -000205a0: 7269 6e67 2065 6c65 6d65 6e74 3a20 284d  ring element: (M
│ │ │ │ -000205b0: 6163 6175 6c61 7932 446f 6329 5269 6e67  acaulay2Doc)Ring
│ │ │ │ -000205c0: 456c 656d 656e 742c 2c20 7028 7a29 3d70  Element,, p(z)=p
│ │ │ │ -000205d0: 6528 7a2f 3229 2c0a 2020 2020 2020 2020  e(z/2),.        
│ │ │ │ -000205e0: 7768 6572 6520 7065 2069 7320 7468 6520  where pe is the 
│ │ │ │ -000205f0: 4869 6c62 6572 7420 706f 6c79 206f 6620  Hilbert poly of 
│ │ │ │ -00020600: 4578 745e 7b65 7665 6e7d 284d 2c6b 290a  Ext^{even}(M,k).
│ │ │ │ -00020610: 2020 2020 2020 2a20 742c 2061 202a 6e6f        * t, a *no
│ │ │ │ -00020620: 7465 2042 6f6f 6c65 616e 2076 616c 7565  te Boolean value
│ │ │ │ -00020630: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00020640: 426f 6f6c 6561 6e2c 2c20 7472 7565 2069  Boolean,, true i
│ │ │ │ -00020650: 6620 7468 6520 6576 656e 2061 6e64 0a20  f the even and. 
│ │ │ │ -00020660: 2020 2020 2020 206f 6464 2070 6f6c 796e         odd polyn
│ │ │ │ -00020670: 6f6d 6961 6c73 206d 6174 6368 2074 6f20  omials match to 
│ │ │ │ -00020680: 666f 726d 206f 6e65 2070 6f6c 796e 6f6d  form one polynom
│ │ │ │ -00020690: 6961 6c0a 0a44 6573 6372 6970 7469 6f6e  ial..Description
│ │ │ │ -000206a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 436f  .===========..Co
│ │ │ │ -000206b0: 6d70 7574 6573 2074 6865 2048 696c 6265  mputes the Hilbe
│ │ │ │ -000206c0: 7274 2070 6f6c 796e 6f6d 6961 6c73 2070  rt polynomials p
│ │ │ │ -000206d0: 6528 7a29 2c20 706f 287a 2920 6f66 2065  e(z), po(z) of e
│ │ │ │ -000206e0: 7665 6e45 7874 4d6f 6475 6c65 2061 6e64  venExtModule and
│ │ │ │ -000206f0: 0a6f 6464 4578 744d 6f64 756c 652e 2049  .oddExtModule. I
│ │ │ │ -00020700: 7420 7265 7475 726e 7320 7065 287a 2f32  t returns pe(z/2
│ │ │ │ -00020710: 292c 2061 6e64 2063 6f6d 7061 7265 7320  ), and compares 
│ │ │ │ -00020720: 746f 2073 6565 2077 6865 7468 6572 2074  to see whether t
│ │ │ │ -00020730: 6869 7320 6973 2065 7175 616c 2074 6f0a  his is equal to.
│ │ │ │ -00020740: 706f 287a 2f32 2d31 2f32 292e 2041 7672  po(z/2-1/2). Avr
│ │ │ │ -00020750: 616d 6f76 2c20 5365 6365 6c65 616e 7520  amov, Seceleanu 
│ │ │ │ -00020760: 616e 6420 5a68 656e 6720 6861 7665 2070  and Zheng have p
│ │ │ │ -00020770: 726f 7665 6e20 7468 6174 2069 6620 7468  roven that if th
│ │ │ │ -00020780: 6520 6964 6561 6c20 6f66 0a71 7561 6472  e ideal of.quadr
│ │ │ │ -00020790: 6174 6963 206c 6561 6469 6e67 2066 6f72  atic leading for
│ │ │ │ -000207a0: 6d73 206f 6620 6120 636f 6d70 6c65 7465  ms of a complete
│ │ │ │ -000207b0: 2069 6e74 6572 7365 6374 696f 6e20 6f66   intersection of
│ │ │ │ -000207c0: 2063 6f64 696d 656e 7369 6f6e 2063 2067   codimension c g
│ │ │ │ -000207d0: 656e 6572 6174 6520 616e 0a69 6465 616c  enerate an.ideal
│ │ │ │ -000207e0: 206f 6620 636f 6469 6d65 6e73 696f 6e20   of codimension 
│ │ │ │ -000207f0: 6174 206c 6561 7374 2063 2d31 2c20 7468  at least c-1, th
│ │ │ │ -00020800: 656e 2074 6865 2042 6574 7469 206e 756d  en the Betti num
│ │ │ │ -00020810: 6265 7273 206f 6620 616e 7920 6d6f 6475  bers of any modu
│ │ │ │ -00020820: 6c65 2067 726f 772c 0a65 7665 6e74 7561  le grow,.eventua
│ │ │ │ -00020830: 6c6c 792c 2061 7320 6120 7369 6e67 6c65  lly, as a single
│ │ │ │ -00020840: 2070 6f6c 796e 6f6d 6961 6c20 2869 6e73   polynomial (ins
│ │ │ │ -00020850: 7465 6164 206f 6620 7265 7175 6972 696e  tead of requirin
│ │ │ │ -00020860: 6720 7365 7061 7261 7465 2070 6f6c 796e  g separate polyn
│ │ │ │ -00020870: 6f6d 6961 6c73 0a66 6f72 2065 7665 6e20  omials.for even 
│ │ │ │ -00020880: 616e 6420 6f64 6420 7465 726d 732e 2920  and odd terms.) 
│ │ │ │ -00020890: 5468 6973 2073 6372 6970 7420 6368 6563  This script chec
│ │ │ │ -000208a0: 6b73 2074 6865 2072 6573 756c 7420 696e  ks the result in
│ │ │ │ -000208b0: 2074 6865 2068 6f6d 6f67 656e 656f 7573   the homogeneous
│ │ │ │ -000208c0: 2063 6173 650a 2869 6e20 7768 6963 6820   case.(in which 
│ │ │ │ -000208d0: 6361 7365 2074 6865 2063 6f6e 6469 7469  case the conditi
│ │ │ │ -000208e0: 6f6e 2069 7320 6e65 6365 7373 6172 7920  on is necessary 
│ │ │ │ -000208f0: 616e 6420 7375 6666 6963 6965 6e74 2e29  and sufficient.)
│ │ │ │ -00020900: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ -00020910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00020940: 203a 2052 313d 5a5a 2f31 3031 5b61 2c62   : R1=ZZ/101[a,b
│ │ │ │ -00020950: 2c63 5d2f 6964 6561 6c28 615e 322c 625e  ,c]/ideal(a^2,b^
│ │ │ │ -00020960: 322c 635e 3529 2020 2020 2020 2020 2020  2,c^5)          
│ │ │ │ -00020970: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000204c0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +000204d0: 2020 2020 2020 2020 2870 2c74 2920 3d20          (p,t) = 
│ │ │ │ +000204e0: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69  extIsOnePolynomi
│ │ │ │ +000204f0: 616c 204d 0a20 202a 2049 6e70 7574 733a  al M.  * Inputs:
│ │ │ │ +00020500: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
│ │ │ │ +00020510: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +00020520: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +00020530: 2c2c 206d 6f64 756c 6520 6f76 6572 2061  ,, module over a
│ │ │ │ +00020540: 2063 6f6d 706c 6574 650a 2020 2020 2020   complete.      
│ │ │ │ +00020550: 2020 696e 7465 7273 6563 7469 6f6e 0a20    intersection. 
│ │ │ │ +00020560: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00020570: 2020 2a20 702c 2061 202a 6e6f 7465 2072    * p, a *note r
│ │ │ │ +00020580: 696e 6720 656c 656d 656e 743a 2028 4d61  ing element: (Ma
│ │ │ │ +00020590: 6361 756c 6179 3244 6f63 2952 696e 6745  caulay2Doc)RingE
│ │ │ │ +000205a0: 6c65 6d65 6e74 2c2c 2070 287a 293d 7065  lement,, p(z)=pe
│ │ │ │ +000205b0: 287a 2f32 292c 0a20 2020 2020 2020 2077  (z/2),.        w
│ │ │ │ +000205c0: 6865 7265 2070 6520 6973 2074 6865 2048  here pe is the H
│ │ │ │ +000205d0: 696c 6265 7274 2070 6f6c 7920 6f66 2045  ilbert poly of E
│ │ │ │ +000205e0: 7874 5e7b 6576 656e 7d28 4d2c 6b29 0a20  xt^{even}(M,k). 
│ │ │ │ +000205f0: 2020 2020 202a 2074 2c20 6120 2a6e 6f74       * t, a *not
│ │ │ │ +00020600: 6520 426f 6f6c 6561 6e20 7661 6c75 653a  e Boolean value:
│ │ │ │ +00020610: 2028 4d61 6361 756c 6179 3244 6f63 2942   (Macaulay2Doc)B
│ │ │ │ +00020620: 6f6f 6c65 616e 2c2c 2074 7275 6520 6966  oolean,, true if
│ │ │ │ +00020630: 2074 6865 2065 7665 6e20 616e 640a 2020   the even and.  
│ │ │ │ +00020640: 2020 2020 2020 6f64 6420 706f 6c79 6e6f        odd polyno
│ │ │ │ +00020650: 6d69 616c 7320 6d61 7463 6820 746f 2066  mials match to f
│ │ │ │ +00020660: 6f72 6d20 6f6e 6520 706f 6c79 6e6f 6d69  orm one polynomi
│ │ │ │ +00020670: 616c 0a0a 4465 7363 7269 7074 696f 6e0a  al..Description.
│ │ │ │ +00020680: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a43 6f6d  ===========..Com
│ │ │ │ +00020690: 7075 7465 7320 7468 6520 4869 6c62 6572  putes the Hilber
│ │ │ │ +000206a0: 7420 706f 6c79 6e6f 6d69 616c 7320 7065  t polynomials pe
│ │ │ │ +000206b0: 287a 292c 2070 6f28 7a29 206f 6620 6576  (z), po(z) of ev
│ │ │ │ +000206c0: 656e 4578 744d 6f64 756c 6520 616e 640a  enExtModule and.
│ │ │ │ +000206d0: 6f64 6445 7874 4d6f 6475 6c65 2e20 4974  oddExtModule. It
│ │ │ │ +000206e0: 2072 6574 7572 6e73 2070 6528 7a2f 3229   returns pe(z/2)
│ │ │ │ +000206f0: 2c20 616e 6420 636f 6d70 6172 6573 2074  , and compares t
│ │ │ │ +00020700: 6f20 7365 6520 7768 6574 6865 7220 7468  o see whether th
│ │ │ │ +00020710: 6973 2069 7320 6571 7561 6c20 746f 0a70  is is equal to.p
│ │ │ │ +00020720: 6f28 7a2f 322d 312f 3229 2e20 4176 7261  o(z/2-1/2). Avra
│ │ │ │ +00020730: 6d6f 762c 2053 6563 656c 6561 6e75 2061  mov, Seceleanu a
│ │ │ │ +00020740: 6e64 205a 6865 6e67 2068 6176 6520 7072  nd Zheng have pr
│ │ │ │ +00020750: 6f76 656e 2074 6861 7420 6966 2074 6865  oven that if the
│ │ │ │ +00020760: 2069 6465 616c 206f 660a 7175 6164 7261   ideal of.quadra
│ │ │ │ +00020770: 7469 6320 6c65 6164 696e 6720 666f 726d  tic leading form
│ │ │ │ +00020780: 7320 6f66 2061 2063 6f6d 706c 6574 6520  s of a complete 
│ │ │ │ +00020790: 696e 7465 7273 6563 7469 6f6e 206f 6620  intersection of 
│ │ │ │ +000207a0: 636f 6469 6d65 6e73 696f 6e20 6320 6765  codimension c ge
│ │ │ │ +000207b0: 6e65 7261 7465 2061 6e0a 6964 6561 6c20  nerate an.ideal 
│ │ │ │ +000207c0: 6f66 2063 6f64 696d 656e 7369 6f6e 2061  of codimension a
│ │ │ │ +000207d0: 7420 6c65 6173 7420 632d 312c 2074 6865  t least c-1, the
│ │ │ │ +000207e0: 6e20 7468 6520 4265 7474 6920 6e75 6d62  n the Betti numb
│ │ │ │ +000207f0: 6572 7320 6f66 2061 6e79 206d 6f64 756c  ers of any modul
│ │ │ │ +00020800: 6520 6772 6f77 2c0a 6576 656e 7475 616c  e grow,.eventual
│ │ │ │ +00020810: 6c79 2c20 6173 2061 2073 696e 676c 6520  ly, as a single 
│ │ │ │ +00020820: 706f 6c79 6e6f 6d69 616c 2028 696e 7374  polynomial (inst
│ │ │ │ +00020830: 6561 6420 6f66 2072 6571 7569 7269 6e67  ead of requiring
│ │ │ │ +00020840: 2073 6570 6172 6174 6520 706f 6c79 6e6f   separate polyno
│ │ │ │ +00020850: 6d69 616c 730a 666f 7220 6576 656e 2061  mials.for even a
│ │ │ │ +00020860: 6e64 206f 6464 2074 6572 6d73 2e29 2054  nd odd terms.) T
│ │ │ │ +00020870: 6869 7320 7363 7269 7074 2063 6865 636b  his script check
│ │ │ │ +00020880: 7320 7468 6520 7265 7375 6c74 2069 6e20  s the result in 
│ │ │ │ +00020890: 7468 6520 686f 6d6f 6765 6e65 6f75 7320  the homogeneous 
│ │ │ │ +000208a0: 6361 7365 0a28 696e 2077 6869 6368 2063  case.(in which c
│ │ │ │ +000208b0: 6173 6520 7468 6520 636f 6e64 6974 696f  ase the conditio
│ │ │ │ +000208c0: 6e20 6973 206e 6563 6573 7361 7279 2061  n is necessary a
│ │ │ │ +000208d0: 6e64 2073 7566 6669 6369 656e 742e 290a  nd sufficient.).
│ │ │ │ +000208e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000208f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020910: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00020920: 3a20 5231 3d5a 5a2f 3130 315b 612c 622c  : R1=ZZ/101[a,b,
│ │ │ │ +00020930: 635d 2f69 6465 616c 2861 5e32 2c62 5e32  c]/ideal(a^2,b^2
│ │ │ │ +00020940: 2c63 5e35 2920 2020 2020 2020 2020 2020  ,c^5)           
│ │ │ │ +00020950: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020990: 7c0a 7c6f 3120 3d20 5231 2020 2020 2020  |.|o1 = R1      
│ │ │ │  000209a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209b0: 207c 0a7c 6f31 203d 2052 3120 2020 2020   |.|o1 = R1     
│ │ │ │ -000209c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000209b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000209c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000209d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000209e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000209f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a20: 2020 2020 2020 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ -00020a30: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ -00020a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00020a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00020aa0: 6932 203a 2052 323d 5a5a 2f31 3031 5b61  i2 : R2=ZZ/101[a
│ │ │ │ -00020ab0: 2c62 2c63 5d2f 6964 6561 6c28 615e 332c  ,b,c]/ideal(a^3,
│ │ │ │ -00020ac0: 625e 3329 2020 2020 2020 2020 2020 2020  b^3)            
│ │ │ │ -00020ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00020a00: 2020 2020 2020 7c0a 7c6f 3120 3a20 5175        |.|o1 : Qu
│ │ │ │ +00020a10: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +00020a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00020a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00020a80: 3220 3a20 5232 3d5a 5a2f 3130 315b 612c  2 : R2=ZZ/101[a,
│ │ │ │ +00020a90: 622c 635d 2f69 6465 616c 2861 5e33 2c62  b,c]/ideal(a^3,b
│ │ │ │ +00020aa0: 5e33 2920 2020 2020 2020 2020 2020 2020  ^3)             
│ │ │ │ +00020ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020af0: 2020 7c0a 7c6f 3220 3d20 5232 2020 2020    |.|o2 = R2    
│ │ │ │  00020b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b10: 2020 207c 0a7c 6f32 203d 2052 3220 2020     |.|o2 = R2   
│ │ │ │ -00020b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020b20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00020b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020b50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00020b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b80: 2020 2020 2020 2020 207c 0a7c 6f32 203a           |.|o2 :
│ │ │ │ -00020b90: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ -00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00020bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00020c00: 0a7c 6933 203a 2065 7874 4973 4f6e 6550  .|i3 : extIsOneP
│ │ │ │ -00020c10: 6f6c 796e 6f6d 6961 6c20 636f 6b65 7220  olynomial coker 
│ │ │ │ -00020c20: 7261 6e64 6f6d 2852 315e 7b30 2c31 7d2c  random(R1^{0,1},
│ │ │ │ -00020c30: 5231 5e7b 333a 2d31 7d29 7c0a 7c20 2020  R1^{3:-1})|.|   
│ │ │ │ +00020b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020b60: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ +00020b70: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00020b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020ba0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00020bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00020be0: 7c69 3320 3a20 6578 7449 734f 6e65 506f  |i3 : extIsOnePo
│ │ │ │ +00020bf0: 6c79 6e6f 6d69 616c 2063 6f6b 6572 2072  lynomial coker r
│ │ │ │ +00020c00: 616e 646f 6d28 5231 5e7b 302c 317d 2c52  andom(R1^{0,1},R
│ │ │ │ +00020c10: 315e 7b33 3a2d 317d 297c 0a7c 2020 2020  1^{3:-1})|.|    
│ │ │ │ +00020c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c70: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ -00020c80: 3220 2020 3120 2020 2020 2020 2020 2020  2   1           
│ │ │ │ -00020c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cb0: 7c0a 7c6f 3320 3d20 282d 7a20 202d 202d  |.|o3 = (-z  - -
│ │ │ │ -00020cc0: 7a20 2b20 332c 2074 7275 6529 2020 2020  z + 3, true)    
│ │ │ │ -00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ce0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00020cf0: 2020 2020 3220 2020 2020 3220 2020 2020      2     2     
│ │ │ │ -00020d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020c50: 2020 2020 7c0a 7c20 2020 2020 2031 2032      |.|      1 2
│ │ │ │ +00020c60: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00020c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00020c90: 0a7c 6f33 203d 2028 2d7a 2020 2d20 2d7a  .|o3 = (-z  - -z
│ │ │ │ +00020ca0: 202b 2033 2c20 7472 7565 2920 2020 2020   + 3, true)     
│ │ │ │ +00020cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020cc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00020cd0: 2020 2032 2020 2020 2032 2020 2020 2020     2     2      
│ │ │ │ +00020ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00020d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d40: 7c0a 7c6f 3320 3a20 5365 7175 656e 6365  |.|o3 : Sequence
│ │ │ │  00020d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d60: 207c 0a7c 6f33 203a 2053 6571 7565 6e63   |.|o3 : Sequenc
│ │ │ │ -00020d70: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ -00020d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00020d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00020d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020dd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2065  -------+.|i4 : e
│ │ │ │ -00020de0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961  xtIsOnePolynomia
│ │ │ │ -00020df0: 6c20 636f 6b65 7220 7261 6e64 6f6d 2852  l coker random(R
│ │ │ │ -00020e00: 325e 7b30 2c31 7d2c 5232 5e7b 333a 2d31  2^{0,1},R2^{3:-1
│ │ │ │ -00020e10: 7d29 7c0a 7c20 2020 2020 2020 2020 2020  })|.|           
│ │ │ │ -00020e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00020e50: 6f34 203d 2028 337a 202d 2032 2c20 6661  o4 = (3z - 2, fa
│ │ │ │ -00020e60: 6c73 6529 2020 2020 2020 2020 2020 2020  lse)            
│ │ │ │ +00020db0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6578  ------+.|i4 : ex
│ │ │ │ +00020dc0: 7449 734f 6e65 506f 6c79 6e6f 6d69 616c  tIsOnePolynomial
│ │ │ │ +00020dd0: 2063 6f6b 6572 2072 616e 646f 6d28 5232   coker random(R2
│ │ │ │ +00020de0: 5e7b 302c 317d 2c52 325e 7b33 3a2d 317d  ^{0,1},R2^{3:-1}
│ │ │ │ +00020df0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +00020e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020e20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00020e30: 3420 3d20 2833 7a20 2d20 322c 2066 616c  4 = (3z - 2, fal
│ │ │ │ +00020e40: 7365 2920 2020 2020 2020 2020 2020 2020  se)             
│ │ │ │ +00020e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020e60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00020e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00020e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ec0: 2020 207c 0a7c 6f34 203a 2053 6571 7565     |.|o4 : Seque
│ │ │ │ -00020ed0: 6e63 6520 2020 2020 2020 2020 2020 2020  nce             
│ │ │ │ -00020ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00020f00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00020f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00020f30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ -00020f40: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ -00020f50: 202a 202a 6e6f 7465 2065 7665 6e45 7874   * *note evenExt
│ │ │ │ -00020f60: 4d6f 6475 6c65 3a20 6576 656e 4578 744d  Module: evenExtM
│ │ │ │ -00020f70: 6f64 756c 652c 202d 2d20 6576 656e 2070  odule, -- even p
│ │ │ │ -00020f80: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ -00020f90: 2920 6f76 6572 2061 0a20 2020 2063 6f6d  ) over a.    com
│ │ │ │ -00020fa0: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ -00020fb0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ -00020fc0: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ -00020fd0: 6e67 0a20 202a 202a 6e6f 7465 206f 6464  ng.  * *note odd
│ │ │ │ -00020fe0: 4578 744d 6f64 756c 653a 206f 6464 4578  ExtModule: oddEx
│ │ │ │ -00020ff0: 744d 6f64 756c 652c 202d 2d20 6f64 6420  tModule, -- odd 
│ │ │ │ -00021000: 7061 7274 206f 6620 4578 745e 2a28 4d2c  part of Ext^*(M,
│ │ │ │ -00021010: 6b29 206f 7665 7220 6120 636f 6d70 6c65  k) over a comple
│ │ │ │ -00021020: 7465 0a20 2020 2069 6e74 6572 7365 6374  te.    intersect
│ │ │ │ -00021030: 696f 6e20 6173 206d 6f64 756c 6520 6f76  ion as module ov
│ │ │ │ -00021040: 6572 2043 4920 6f70 6572 6174 6f72 2072  er CI operator r
│ │ │ │ -00021050: 696e 670a 0a57 6179 7320 746f 2075 7365  ing..Ways to use
│ │ │ │ -00021060: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d   extIsOnePolynom
│ │ │ │ -00021070: 6961 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ial:.===========
│ │ │ │ -00021080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00021090: 3d3d 3d3d 0a0a 2020 2a20 2265 7874 4973  ====..  * "extIs
│ │ │ │ -000210a0: 4f6e 6550 6f6c 796e 6f6d 6961 6c28 4d6f  OnePolynomial(Mo
│ │ │ │ -000210b0: 6475 6c65 2922 0a0a 466f 7220 7468 6520  dule)"..For the 
│ │ │ │ -000210c0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -000210d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -000210e0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -000210f0: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69  extIsOnePolynomi
│ │ │ │ -00021100: 616c 3a20 6578 7449 734f 6e65 506f 6c79  al: extIsOnePoly
│ │ │ │ -00021110: 6e6f 6d69 616c 2c20 6973 2061 202a 6e6f  nomial, is a *no
│ │ │ │ -00021120: 7465 206d 6574 686f 640a 6675 6e63 7469  te method.functi
│ │ │ │ -00021130: 6f6e 3a20 284d 6163 6175 6c61 7932 446f  on: (Macaulay2Do
│ │ │ │ -00021140: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -00021150: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +00020ea0: 2020 7c0a 7c6f 3420 3a20 5365 7175 656e    |.|o4 : Sequen
│ │ │ │ +00020eb0: 6365 2020 2020 2020 2020 2020 2020 2020  ce              
│ │ │ │ +00020ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020ed0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00020ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00020f10: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ +00020f20: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ +00020f30: 2a20 2a6e 6f74 6520 6576 656e 4578 744d  * *note evenExtM
│ │ │ │ +00020f40: 6f64 756c 653a 2065 7665 6e45 7874 4d6f  odule: evenExtMo
│ │ │ │ +00020f50: 6475 6c65 2c20 2d2d 2065 7665 6e20 7061  dule, -- even pa
│ │ │ │ +00020f60: 7274 206f 6620 4578 745e 2a28 4d2c 6b29  rt of Ext^*(M,k)
│ │ │ │ +00020f70: 206f 7665 7220 610a 2020 2020 636f 6d70   over a.    comp
│ │ │ │ +00020f80: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +00020f90: 6e20 6173 206d 6f64 756c 6520 6f76 6572  n as module over
│ │ │ │ +00020fa0: 2043 4920 6f70 6572 6174 6f72 2072 696e   CI operator rin
│ │ │ │ +00020fb0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445  g.  * *note oddE
│ │ │ │ +00020fc0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874  xtModule: oddExt
│ │ │ │ +00020fd0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070  Module, -- odd p
│ │ │ │ +00020fe0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ +00020ff0: 2920 6f76 6572 2061 2063 6f6d 706c 6574  ) over a complet
│ │ │ │ +00021000: 650a 2020 2020 696e 7465 7273 6563 7469  e.    intersecti
│ │ │ │ +00021010: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ +00021020: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ +00021030: 6e67 0a0a 5761 7973 2074 6f20 7573 6520  ng..Ways to use 
│ │ │ │ +00021040: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69  extIsOnePolynomi
│ │ │ │ +00021050: 616c 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  al:.============
│ │ │ │ +00021060: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00021070: 3d3d 3d0a 0a20 202a 2022 6578 7449 734f  ===..  * "extIsO
│ │ │ │ +00021080: 6e65 506f 6c79 6e6f 6d69 616c 284d 6f64  nePolynomial(Mod
│ │ │ │ +00021090: 756c 6529 220a 0a46 6f72 2074 6865 2070  ule)"..For the p
│ │ │ │ +000210a0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +000210b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +000210c0: 6520 6f62 6a65 6374 202a 6e6f 7465 2065  e object *note e
│ │ │ │ +000210d0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961  xtIsOnePolynomia
│ │ │ │ +000210e0: 6c3a 2065 7874 4973 4f6e 6550 6f6c 796e  l: extIsOnePolyn
│ │ │ │ +000210f0: 6f6d 6961 6c2c 2069 7320 6120 2a6e 6f74  omial, is a *not
│ │ │ │ +00021100: 6520 6d65 7468 6f64 0a66 756e 6374 696f  e method.functio
│ │ │ │ +00021110: 6e3a 2028 4d61 6361 756c 6179 3244 6f63  n: (Macaulay2Doc
│ │ │ │ +00021120: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +00021130: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +00021140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000211a0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -000211b0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -000211c0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -000211d0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -000211e0: 6d61 6361 756c 6179 322d 312e 3236 2e30  macaulay2-1.26.0
│ │ │ │ -000211f0: 362b 6473 2f4d 322f 4d61 6361 756c 6179  6+ds/M2/Macaulay
│ │ │ │ -00021200: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ -00021210: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -00021220: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34  Resolutions.m2:4
│ │ │ │ -00021230: 3933 313a 302e 0a1f 0a46 696c 653a 2043  931:0....File: C
│ │ │ │ -00021240: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -00021250: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -00021260: 6e66 6f2c 204e 6f64 653a 2045 7874 4d6f  nfo, Node: ExtMo
│ │ │ │ -00021270: 6475 6c65 2c20 4e65 7874 3a20 4578 744d  dule, Next: ExtM
│ │ │ │ -00021280: 6f64 756c 6544 6174 612c 2050 7265 763a  oduleData, Prev:
│ │ │ │ -00021290: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d   extIsOnePolynom
│ │ │ │ -000212a0: 6961 6c2c 2055 703a 2054 6f70 0a0a 4578  ial, Up: Top..Ex
│ │ │ │ -000212b0: 744d 6f64 756c 6520 2d2d 2045 7874 5e2a  tModule -- Ext^*
│ │ │ │ -000212c0: 284d 2c6b 2920 6f76 6572 2061 2063 6f6d  (M,k) over a com
│ │ │ │ -000212d0: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ -000212e0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ -000212f0: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ -00021300: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ng.*************
│ │ │ │ +00021180: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +00021190: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +000211a0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +000211b0: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +000211c0: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ +000211d0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +000211e0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ +000211f0: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +00021200: 6573 6f6c 7574 696f 6e73 2e6d 323a 3439  esolutions.m2:49
│ │ │ │ +00021210: 3331 3a30 2e0a 1f0a 4669 6c65 3a20 436f  31:0....File: Co
│ │ │ │ +00021220: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +00021230: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ +00021240: 666f 2c20 4e6f 6465 3a20 4578 744d 6f64  fo, Node: ExtMod
│ │ │ │ +00021250: 756c 652c 204e 6578 743a 2045 7874 4d6f  ule, Next: ExtMo
│ │ │ │ +00021260: 6475 6c65 4461 7461 2c20 5072 6576 3a20  duleData, Prev: 
│ │ │ │ +00021270: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69  extIsOnePolynomi
│ │ │ │ +00021280: 616c 2c20 5570 3a20 546f 700a 0a45 7874  al, Up: Top..Ext
│ │ │ │ +00021290: 4d6f 6475 6c65 202d 2d20 4578 745e 2a28  Module -- Ext^*(
│ │ │ │ +000212a0: 4d2c 6b29 206f 7665 7220 6120 636f 6d70  M,k) over a comp
│ │ │ │ +000212b0: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +000212c0: 6e20 6173 206d 6f64 756c 6520 6f76 6572  n as module over
│ │ │ │ +000212d0: 2043 4920 6f70 6572 6174 6f72 2072 696e   CI operator rin
│ │ │ │ +000212e0: 670a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  g.**************
│ │ │ │ +000212f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00021300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00021320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00021330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00021340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00021350: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -00021360: 6765 3a20 0a20 2020 2020 2020 2045 203d  ge: .        E =
│ │ │ │ -00021370: 2045 7874 4d6f 6475 6c65 204d 0a20 202a   ExtModule M.  *
│ │ │ │ -00021380: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00021390: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -000213a0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -000213b0: 6329 4d6f 6475 6c65 2c2c 206f 7665 7220  c)Module,, over 
│ │ │ │ -000213c0: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ -000213d0: 7365 6374 696f 6e0a 2020 2020 2020 2020  section.        
│ │ │ │ -000213e0: 7269 6e67 0a20 202a 204f 7574 7075 7473  ring.  * Outputs
│ │ │ │ -000213f0: 3a0a 2020 2020 2020 2a20 452c 2061 202a  :.      * E, a *
│ │ │ │ -00021400: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00021410: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00021420: 652c 2c20 6f76 6572 2061 2070 6f6c 796e  e,, over a polyn
│ │ │ │ -00021430: 6f6d 6961 6c20 7269 6e67 2077 6974 680a  omial ring with.
│ │ │ │ -00021440: 2020 2020 2020 2020 6765 6e73 2069 6e20          gens in 
│ │ │ │ -00021450: 6576 656e 2064 6567 7265 650a 0a44 6573  even degree..Des
│ │ │ │ -00021460: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00021470: 3d3d 3d3d 0a0a 5573 6573 2063 6f64 6520  ====..Uses code 
│ │ │ │ -00021480: 6f66 2041 7672 616d 6f76 2d47 7261 7973  of Avramov-Grays
│ │ │ │ -00021490: 6f6e 2064 6573 6372 6962 6564 2069 6e20  on described in 
│ │ │ │ -000214a0: 4d61 6361 756c 6179 3220 626f 6f6b 0a0a  Macaulay2 book..
│ │ │ │ -000214b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -000214c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000214d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000214e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000214f0: 203a 206b 6b3d 205a 5a2f 3130 3120 2020   : kk= ZZ/101   
│ │ │ │ -00021500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021330: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ +00021340: 653a 200a 2020 2020 2020 2020 4520 3d20  e: .        E = 
│ │ │ │ +00021350: 4578 744d 6f64 756c 6520 4d0a 2020 2a20  ExtModule M.  * 
│ │ │ │ +00021360: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00021370: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ +00021380: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +00021390: 294d 6f64 756c 652c 2c20 6f76 6572 2061  )Module,, over a
│ │ │ │ +000213a0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ +000213b0: 6563 7469 6f6e 0a20 2020 2020 2020 2072  ection.        r
│ │ │ │ +000213c0: 696e 670a 2020 2a20 4f75 7470 7574 733a  ing.  * Outputs:
│ │ │ │ +000213d0: 0a20 2020 2020 202a 2045 2c20 6120 2a6e  .      * E, a *n
│ │ │ │ +000213e0: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +000213f0: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +00021400: 2c2c 206f 7665 7220 6120 706f 6c79 6e6f  ,, over a polyno
│ │ │ │ +00021410: 6d69 616c 2072 696e 6720 7769 7468 0a20  mial ring with. 
│ │ │ │ +00021420: 2020 2020 2020 2067 656e 7320 696e 2065         gens in e
│ │ │ │ +00021430: 7665 6e20 6465 6772 6565 0a0a 4465 7363  ven degree..Desc
│ │ │ │ +00021440: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00021450: 3d3d 3d0a 0a55 7365 7320 636f 6465 206f  ===..Uses code o
│ │ │ │ +00021460: 6620 4176 7261 6d6f 762d 4772 6179 736f  f Avramov-Grayso
│ │ │ │ +00021470: 6e20 6465 7363 7269 6265 6420 696e 204d  n described in M
│ │ │ │ +00021480: 6163 6175 6c61 7932 2062 6f6f 6b0a 0a2b  acaulay2 book..+
│ │ │ │ +00021490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000214a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000214b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000214c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +000214d0: 3a20 6b6b 3d20 5a5a 2f31 3031 2020 2020  : kk= ZZ/101    
│ │ │ │ +000214e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000214f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021500: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00021510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021520: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021540: 2020 2020 7c0a 7c6f 3120 3d20 6b6b 2020      |.|o1 = kk  
│ │ │ │  00021550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021560: 2020 2020 207c 0a7c 6f31 203d 206b 6b20       |.|o1 = kk 
│ │ │ │ +00021560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021580: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00021590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000215b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000215e0: 0a7c 6f31 203a 2051 756f 7469 656e 7452  .|o1 : QuotientR
│ │ │ │ -000215f0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021610: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000215a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000215b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000215c0: 7c6f 3120 3a20 5175 6f74 6965 6e74 5269  |o1 : QuotientRi
│ │ │ │ +000215d0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +000215e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000215f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00021600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021650: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -00021660: 2053 203d 206b 6b5b 782c 792c 7a5d 2020   S = kk[x,y,z]  
│ │ │ │ -00021670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021630: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +00021640: 5320 3d20 6b6b 5b78 2c79 2c7a 5d20 2020  S = kk[x,y,z]   
│ │ │ │ +00021650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021690: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00021690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216b0: 2020 7c0a 7c6f 3220 3d20 5320 2020 2020    |.|o2 = S     
│ │ │ │  000216c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216d0: 2020 207c 0a7c 6f32 203d 2053 2020 2020     |.|o2 = S    
│ │ │ │ -000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000216e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000216f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00021700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021710: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00021720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00021750: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52  o2 : PolynomialR
│ │ │ │ -00021760: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -00021770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021780: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00021710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021720: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00021730: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ +00021740: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00021750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021760: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00021770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000217a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000217c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049  -------+.|i3 : I
│ │ │ │ -000217d0: 3120 3d20 6964 6561 6c20 2278 3379 2220  1 = ideal "x3y" 
│ │ │ │ -000217e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217a0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 4931  ------+.|i3 : I1
│ │ │ │ +000217b0: 203d 2069 6465 616c 2022 7833 7922 2020   = ideal "x3y"  
│ │ │ │ +000217c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000217f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021800: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021820: 7c0a 7c20 2020 2020 2020 2020 2020 2033  |.|            3
│ │ │ │  00021830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021840: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00021850: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00021860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021880: 7c6f 3320 3d20 6964 6561 6c28 7820 7929  |o3 = ideal(x y)
│ │ │ │ -00021890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021860: 6f33 203d 2069 6465 616c 2878 2079 2920  o3 = ideal(x y) 
│ │ │ │ +00021870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021890: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000218a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000218b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000218c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00021900: 4964 6561 6c20 6f66 2053 2020 2020 2020  Ideal of S      
│ │ │ │ -00021910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021930: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000218d0: 2020 2020 2020 207c 0a7c 6f33 203a 2049         |.|o3 : I
│ │ │ │ +000218e0: 6465 616c 206f 6620 5320 2020 2020 2020  deal of S       
│ │ │ │ +000218f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021910: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00021920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021970: 2d2d 2b0a 7c69 3420 3a20 5231 203d 2053  --+.|i4 : R1 = S
│ │ │ │ -00021980: 2f49 3120 2020 2020 2020 2020 2020 2020  /I1             
│ │ │ │ -00021990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000219b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000219c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -000219f0: 3420 3d20 5231 2020 2020 2020 2020 2020  4 = R1          
│ │ │ │ -00021a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021950: 2d2b 0a7c 6934 203a 2052 3120 3d20 532f  -+.|i4 : R1 = S/
│ │ │ │ +00021960: 4931 2020 2020 2020 2020 2020 2020 2020  I1              
│ │ │ │ +00021970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000219a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000219b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000219c0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +000219d0: 203d 2052 3120 2020 2020 2020 2020 2020   = R1           
│ │ │ │ +000219e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000219f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00021a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a20: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00021a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a60: 2020 2020 2020 7c0a 7c6f 3420 3a20 5175        |.|o4 : Qu
│ │ │ │ -00021a70: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ -00021a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021ae0: 2b0a 7c69 3520 3a20 4d31 203d 2052 315e  +.|i5 : M1 = R1^
│ │ │ │ -00021af0: 312f 6964 6561 6c28 785e 3229 2020 2020  1/ideal(x^2)    
│ │ │ │ +00021a40: 2020 2020 207c 0a7c 6f34 203a 2051 756f       |.|o4 : Quo
│ │ │ │ +00021a50: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +00021a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00021a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00021ac0: 0a7c 6935 203a 204d 3120 3d20 5231 5e31  .|i5 : M1 = R1^1
│ │ │ │ +00021ad0: 2f69 6465 616c 2878 5e32 2920 2020 2020  /ideal(x^2)     
│ │ │ │ +00021ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00021b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b50: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00021b60: 3d20 636f 6b65 726e 656c 207c 2078 3220  = cokernel | x2 
│ │ │ │ -00021b70: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00021b30: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ +00021b40: 2063 6f6b 6572 6e65 6c20 7c20 7832 207c   cokernel | x2 |
│ │ │ │ +00021b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021b70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00021b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00021b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021bb0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00021bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021bd0: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │  00021be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021bf0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -00021c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c10: 207c 0a7c 6f35 203a 2052 312d 6d6f 6475   |.|o5 : R1-modu
│ │ │ │ -00021c20: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ -00021c30: 5231 2020 2020 2020 2020 2020 2020 2020  R1              
│ │ │ │ -00021c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00021c50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00021c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -00021c90: 203a 2062 6574 7469 2066 7265 6552 6573   : betti freeRes
│ │ │ │ -00021ca0: 6f6c 7574 696f 6e20 284d 312c 204c 656e  olution (M1, Len
│ │ │ │ -00021cb0: 6774 684c 696d 6974 203d 3e35 2920 2020  gthLimit =>5)   
│ │ │ │ -00021cc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021bf0: 7c0a 7c6f 3520 3a20 5231 2d6d 6f64 756c  |.|o5 : R1-modul
│ │ │ │ +00021c00: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +00021c10: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00021c20: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00021c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ +00021c70: 3a20 6265 7474 6920 6672 6565 5265 736f  : betti freeReso
│ │ │ │ +00021c80: 6c75 7469 6f6e 2028 4d31 2c20 4c65 6e67  lution (M1, Leng
│ │ │ │ +00021c90: 7468 4c69 6d69 7420 3d3e 3529 2020 2020  thLimit =>5)    
│ │ │ │ +00021ca0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00021cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00021d10: 2020 2020 3020 3120 3220 3320 3420 3520      0 1 2 3 4 5 
│ │ │ │ -00021d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d40: 2020 7c0a 7c6f 3620 3d20 746f 7461 6c3a    |.|o6 = total:
│ │ │ │ -00021d50: 2031 2031 2031 2031 2031 2031 2020 2020   1 1 1 1 1 1    
│ │ │ │ -00021d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021d80: 0a7c 2020 2020 2020 2020 2030 3a20 3120  .|         0: 1 
│ │ │ │ -00021d90: 2e20 2e20 2e20 2e20 2e20 2020 2020 2020  . . . . .       
│ │ │ │ -00021da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00021dc0: 2020 2020 2020 2020 313a 202e 2031 202e          1: . 1 .
│ │ │ │ -00021dd0: 202e 202e 202e 2020 2020 2020 2020 2020   . . .          
│ │ │ │ -00021de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021df0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00021e00: 2020 2020 2032 3a20 2e20 2e20 3120 2e20       2: . . 1 . 
│ │ │ │ -00021e10: 2e20 2e20 2020 2020 2020 2020 2020 2020  . .             
│ │ │ │ -00021e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00021e40: 2020 333a 202e 202e 202e 2031 202e 202e    3: . . . 1 . .
│ │ │ │ -00021e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e70: 2020 207c 0a7c 2020 2020 2020 2020 2034     |.|         4
│ │ │ │ -00021e80: 3a20 2e20 2e20 2e20 2e20 3120 2e20 2020  : . . . . 1 .   
│ │ │ │ -00021e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021eb0: 7c0a 7c20 2020 2020 2020 2020 353a 202e  |.|         5: .
│ │ │ │ -00021ec0: 202e 202e 202e 202e 2031 2020 2020 2020   . . . . 1      
│ │ │ │ +00021ce0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00021cf0: 2020 2030 2031 2032 2033 2034 2035 2020     0 1 2 3 4 5  
│ │ │ │ +00021d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021d20: 207c 0a7c 6f36 203d 2074 6f74 616c 3a20   |.|o6 = total: 
│ │ │ │ +00021d30: 3120 3120 3120 3120 3120 3120 2020 2020  1 1 1 1 1 1     
│ │ │ │ +00021d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021d50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00021d60: 7c20 2020 2020 2020 2020 303a 2031 202e  |         0: 1 .
│ │ │ │ +00021d70: 202e 202e 202e 202e 2020 2020 2020 2020   . . . .        
│ │ │ │ +00021d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021d90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00021da0: 2020 2020 2020 2031 3a20 2e20 3120 2e20         1: . 1 . 
│ │ │ │ +00021db0: 2e20 2e20 2e20 2020 2020 2020 2020 2020  . . .           
│ │ │ │ +00021dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021dd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00021de0: 2020 2020 323a 202e 202e 2031 202e 202e      2: . . 1 . .
│ │ │ │ +00021df0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ +00021e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021e20: 2033 3a20 2e20 2e20 2e20 3120 2e20 2e20   3: . . . 1 . . 
│ │ │ │ +00021e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021e50: 2020 7c0a 7c20 2020 2020 2020 2020 343a    |.|         4:
│ │ │ │ +00021e60: 202e 202e 202e 202e 2031 202e 2020 2020   . . . . 1 .    
│ │ │ │ +00021e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021e80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00021e90: 0a7c 2020 2020 2020 2020 2035 3a20 2e20  .|         5: . 
│ │ │ │ +00021ea0: 2e20 2e20 2e20 2e20 3120 2020 2020 2020  . . . . 1       
│ │ │ │ +00021eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021ec0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00021ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00021ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f20: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00021f30: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ -00021f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00021f00: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ +00021f10: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +00021f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021f40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00021f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021fa0: 2d2d 2d2d 2b0a 7c69 3720 3a20 4520 3d20  ----+.|i7 : E = 
│ │ │ │ -00021fb0: 4578 744d 6f64 756c 6520 4d31 2020 2020  ExtModule M1    
│ │ │ │ -00021fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021f80: 2d2d 2d2b 0a7c 6937 203a 2045 203d 2045  ---+.|i7 : E = E
│ │ │ │ +00021f90: 7874 4d6f 6475 6c65 204d 3120 2020 2020  xtModule M1     
│ │ │ │ +00021fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021fc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00021fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021fe0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00021ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022010: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00022020: 7c20 2020 2020 2020 2020 2020 2020 3220  |             2 
│ │ │ │ -00022030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022050: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -00022060: 203d 2028 6b6b 5b58 205d 2920 2020 2020   = (kk[X ])     
│ │ │ │ -00022070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022090: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000220a0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -000220b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022000: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00022010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022030: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +00022040: 3d20 286b 6b5b 5820 5d29 2020 2020 2020  = (kk[X ])      
│ │ │ │ +00022050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022070: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022080: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ +00022090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000220a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000220b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000220c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000220d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000220e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000220f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022110: 2020 7c0a 7c6f 3720 3a20 6b6b 5b58 205d    |.|o7 : kk[X ]
│ │ │ │ -00022120: 2d6d 6f64 756c 652c 2066 7265 652c 2064  -module, free, d
│ │ │ │ -00022130: 6567 7265 6573 207b 302e 2e31 7d20 2020  egrees {0..1}   
│ │ │ │ -00022140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022150: 0a7c 2020 2020 2020 2020 2030 2020 2020  .|         0    
│ │ │ │ -00022160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022180: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000220f0: 207c 0a7c 6f37 203a 206b 6b5b 5820 5d2d   |.|o7 : kk[X ]-
│ │ │ │ +00022100: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465  module, free, de
│ │ │ │ +00022110: 6772 6565 7320 7b30 2e2e 317d 2020 2020  grees {0..1}    
│ │ │ │ +00022120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00022130: 7c20 2020 2020 2020 2020 3020 2020 2020  |         0     
│ │ │ │ +00022140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022160: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00022170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000221a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000221b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000221c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -000221d0: 2061 7070 6c79 2874 6f4c 6973 7428 302e   apply(toList(0.
│ │ │ │ -000221e0: 2e31 3029 2c20 692d 3e68 696c 6265 7274  .10), i->hilbert
│ │ │ │ -000221f0: 4675 6e63 7469 6f6e 2869 2c20 4529 2920  Function(i, E)) 
│ │ │ │ -00022200: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000221a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ +000221b0: 6170 706c 7928 746f 4c69 7374 2830 2e2e  apply(toList(0..
│ │ │ │ +000221c0: 3130 292c 2069 2d3e 6869 6c62 6572 7446  10), i->hilbertF
│ │ │ │ +000221d0: 756e 6374 696f 6e28 692c 2045 2929 2020  unction(i, E))  
│ │ │ │ +000221e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000221f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022240: 2020 207c 0a7c 6f38 203d 207b 312c 2031     |.|o8 = {1, 1
│ │ │ │ -00022250: 2c20 312c 2031 2c20 312c 2031 2c20 312c  , 1, 1, 1, 1, 1,
│ │ │ │ -00022260: 2031 2c20 312c 2031 2c20 317d 2020 2020   1, 1, 1, 1}    
│ │ │ │ +00022220: 2020 7c0a 7c6f 3820 3d20 7b31 2c20 312c    |.|o8 = {1, 1,
│ │ │ │ +00022230: 2031 2c20 312c 2031 2c20 312c 2031 2c20   1, 1, 1, 1, 1, 
│ │ │ │ +00022240: 312c 2031 2c20 312c 2031 7d20 2020 2020  1, 1, 1, 1}     
│ │ │ │ +00022250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00022270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022280: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00022290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000222c0: 6f38 203a 204c 6973 7420 2020 2020 2020  o8 : List       
│ │ │ │ -000222d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000222f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00022280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022290: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000222a0: 3820 3a20 4c69 7374 2020 2020 2020 2020  8 : List        
│ │ │ │ +000222b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000222c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000222d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000222e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000222f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022330: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2045  -------+.|i9 : E
│ │ │ │ -00022340: 6576 656e 203d 2065 7665 6e45 7874 4d6f  even = evenExtMo
│ │ │ │ -00022350: 6475 6c65 284d 3129 2020 2020 2020 2020  dule(M1)        
│ │ │ │ +00022310: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 4565  ------+.|i9 : Ee
│ │ │ │ +00022320: 7665 6e20 3d20 6576 656e 4578 744d 6f64  ven = evenExtMod
│ │ │ │ +00022330: 756c 6528 4d31 2920 2020 2020 2020 2020  ule(M1)         
│ │ │ │ +00022340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022350: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00022360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022370: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000223c0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -000223d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000223e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000223f0: 7c6f 3920 3d20 286b 6b5b 5820 5d29 2020  |o9 = (kk[X ])  
│ │ │ │ -00022400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00022430: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -00022440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022390: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000223a0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000223b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000223c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000223d0: 6f39 203d 2028 6b6b 5b58 205d 2920 2020  o9 = (kk[X ])   
│ │ │ │ +000223e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000223f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022400: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022410: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +00022420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022440: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00022450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00022460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224a0: 2020 2020 207c 0a7c 6f39 203a 206b 6b5b       |.|o9 : kk[
│ │ │ │ -000224b0: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565  X ]-module, free
│ │ │ │ -000224c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022480: 2020 2020 7c0a 7c6f 3920 3a20 6b6b 5b58      |.|o9 : kk[X
│ │ │ │ +00022490: 205d 2d6d 6f64 756c 652c 2066 7265 6520   ]-module, free 
│ │ │ │ +000224a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224c0: 207c 0a7c 2020 2020 2020 2020 2030 2020   |.|         0  
│ │ │ │  000224d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224e0: 2020 7c0a 7c20 2020 2020 2020 2020 3020    |.|         0 
│ │ │ │ -000224f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022520: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00022530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00022560: 3130 203a 2061 7070 6c79 2874 6f4c 6973  10 : apply(toLis
│ │ │ │ -00022570: 7428 302e 2e35 292c 2069 2d3e 6869 6c62  t(0..5), i->hilb
│ │ │ │ -00022580: 6572 7446 756e 6374 696f 6e28 692c 2045  ertFunction(i, E
│ │ │ │ -00022590: 6576 656e 2929 2020 207c 0a7c 2020 2020  even))   |.|    
│ │ │ │ +000224e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00022500: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00022510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00022540: 3020 3a20 6170 706c 7928 746f 4c69 7374  0 : apply(toList
│ │ │ │ +00022550: 2830 2e2e 3529 2c20 692d 3e68 696c 6265  (0..5), i->hilbe
│ │ │ │ +00022560: 7274 4675 6e63 7469 6f6e 2869 2c20 4565  rtFunction(i, Ee
│ │ │ │ +00022570: 7665 6e29 2920 2020 7c0a 7c20 2020 2020  ven))   |.|     
│ │ │ │ +00022580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225d0: 2020 2020 2020 7c0a 7c6f 3130 203d 207b        |.|o10 = {
│ │ │ │ -000225e0: 312c 2031 2c20 312c 2031 2c20 312c 2031  1, 1, 1, 1, 1, 1
│ │ │ │ -000225f0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +000225b0: 2020 2020 207c 0a7c 6f31 3020 3d20 7b31       |.|o10 = {1
│ │ │ │ +000225c0: 2c20 312c 2031 2c20 312c 2031 2c20 317d  , 1, 1, 1, 1, 1}
│ │ │ │ +000225d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000225e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000225f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00022600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022610: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00022620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022630: 0a7c 6f31 3020 3a20 4c69 7374 2020 2020  .|o10 : List    
│ │ │ │  00022640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022650: 7c0a 7c6f 3130 203a 204c 6973 7420 2020  |.|o10 : List   
│ │ │ │ -00022660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022680: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022660: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00022670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000226a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000226b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000226c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131  ----------+.|i11
│ │ │ │ -000226d0: 203a 2045 6f64 6420 3d20 6f64 6445 7874   : Eodd = oddExt
│ │ │ │ -000226e0: 4d6f 6475 6c65 284d 3129 2020 2020 2020  Module(M1)      
│ │ │ │ +000226a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ +000226b0: 3a20 456f 6464 203d 206f 6464 4578 744d  : Eodd = oddExtM
│ │ │ │ +000226c0: 6f64 756c 6528 4d31 2920 2020 2020 2020  odule(M1)       
│ │ │ │ +000226d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000226e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000226f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022700: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00022700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022740: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00022750: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │ -00022760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022780: 207c 0a7c 6f31 3120 3d20 286b 6b5b 5820   |.|o11 = (kk[X 
│ │ │ │ -00022790: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ -000227a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000227b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000227c0: 7c20 2020 2020 2020 2020 2020 3020 2020  |           0   
│ │ │ │ -000227d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022720: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00022730: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ +00022740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022760: 7c0a 7c6f 3131 203d 2028 6b6b 5b58 205d  |.|o11 = (kk[X ]
│ │ │ │ +00022770: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00022780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022790: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000227a0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ +000227b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000227c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000227d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000227e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000227f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000227f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022830: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ -00022840: 206b 6b5b 5820 5d2d 6d6f 6475 6c65 2c20   kk[X ]-module, 
│ │ │ │ -00022850: 6672 6565 2020 2020 2020 2020 2020 2020  free            
│ │ │ │ -00022860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022870: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00022880: 2020 3020 2020 2020 2020 2020 2020 2020    0             
│ │ │ │ -00022890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -000228c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000228d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000228e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000228f0: 0a7c 6931 3220 3a20 6170 706c 7928 746f  .|i12 : apply(to
│ │ │ │ -00022900: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68  List(0..5), i->h
│ │ │ │ -00022910: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ -00022920: 2c20 456f 6464 2929 2020 2020 7c0a 7c20  , Eodd))    |.| 
│ │ │ │ +00022810: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ +00022820: 6b6b 5b58 205d 2d6d 6f64 756c 652c 2066  kk[X ]-module, f
│ │ │ │ +00022830: 7265 6520 2020 2020 2020 2020 2020 2020  ree             
│ │ │ │ +00022840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022860: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00022870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022890: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000228a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000228b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000228c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000228d0: 7c69 3132 203a 2061 7070 6c79 2874 6f4c  |i12 : apply(toL
│ │ │ │ +000228e0: 6973 7428 302e 2e35 292c 2069 2d3e 6869  ist(0..5), i->hi
│ │ │ │ +000228f0: 6c62 6572 7446 756e 6374 696f 6e28 692c  lbertFunction(i,
│ │ │ │ +00022900: 2045 6f64 6429 2920 2020 207c 0a7c 2020   Eodd))    |.|  
│ │ │ │ +00022910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022960: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ -00022970: 3d20 7b31 2c20 312c 2031 2c20 312c 2031  = {1, 1, 1, 1, 1
│ │ │ │ -00022980: 2c20 317d 2020 2020 2020 2020 2020 2020  , 1}            
│ │ │ │ +00022940: 2020 2020 2020 2020 7c0a 7c6f 3132 203d          |.|o12 =
│ │ │ │ +00022950: 207b 312c 2031 2c20 312c 2031 2c20 312c   {1, 1, 1, 1, 1,
│ │ │ │ +00022960: 2031 7d20 2020 2020 2020 2020 2020 2020   1}             
│ │ │ │ +00022970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000229a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000229c0: 2020 7c0a 7c6f 3132 203a 204c 6973 7420    |.|o12 : List 
│ │ │ │  000229d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229e0: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374     |.|o12 : List
│ │ │ │ -000229f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00022a60: 6931 3320 3a20 7573 6520 5320 2020 2020  i13 : use S     
│ │ │ │ -00022a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000229e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000229f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022a00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00022a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00022a40: 3133 203a 2075 7365 2053 2020 2020 2020  13 : use S      
│ │ │ │ +00022a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00022a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00022a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ab0: 2020 2020 2020 7c0a 7c6f 3133 203d 2053        |.|o13 = S
│ │ │ │  00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ad0: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ -00022ae0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -00022af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022af0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00022b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b50: 207c 0a7c 6f31 3320 3a20 506f 6c79 6e6f   |.|o13 : Polyno
│ │ │ │ -00022b60: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ -00022b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00022b90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00022ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00022bd0: 3420 3a20 4932 203d 2069 6465 616c 2278  4 : I2 = ideal"x
│ │ │ │ -00022be0: 332c 797a 2220 2020 2020 2020 2020 2020  3,yz"           
│ │ │ │ +00022b30: 7c0a 7c6f 3133 203a 2050 6f6c 796e 6f6d  |.|o13 : Polynom
│ │ │ │ +00022b40: 6961 6c52 696e 6720 2020 2020 2020 2020  ialRing         
│ │ │ │ +00022b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022b60: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00022b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134  ----------+.|i14
│ │ │ │ +00022bb0: 203a 2049 3220 3d20 6964 6561 6c22 7833   : I2 = ideal"x3
│ │ │ │ +00022bc0: 2c79 7a22 2020 2020 2020 2020 2020 2020  ,yz"            
│ │ │ │ +00022bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022be0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00022c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00022c50: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ -00022c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c80: 2020 7c0a 7c6f 3134 203d 2069 6465 616c    |.|o14 = ideal
│ │ │ │ -00022c90: 2028 7820 2c20 792a 7a29 2020 2020 2020   (x , y*z)      
│ │ │ │ -00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00022d00: 3134 203a 2049 6465 616c 206f 6620 5320  14 : Ideal of S 
│ │ │ │ -00022d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00022c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00022c30: 2020 2020 2033 2020 2020 2020 2020 2020       3          
│ │ │ │ +00022c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022c60: 207c 0a7c 6f31 3420 3d20 6964 6561 6c20   |.|o14 = ideal 
│ │ │ │ +00022c70: 2878 202c 2079 2a7a 2920 2020 2020 2020  (x , y*z)       
│ │ │ │ +00022c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022c90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00022ca0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022cd0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00022ce0: 3420 3a20 4964 6561 6c20 6f66 2053 2020  4 : Ideal of S  
│ │ │ │ +00022cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d10: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00022d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022d70: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2052  ------+.|i15 : R
│ │ │ │ -00022d80: 3220 3d20 532f 4932 2020 2020 2020 2020  2 = S/I2        
│ │ │ │ -00022d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d50: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 5232  -----+.|i15 : R2
│ │ │ │ +00022d60: 203d 2053 2f49 3220 2020 2020 2020 2020   = S/I2         
│ │ │ │ +00022d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d90: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00022da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022db0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00022dd0: 0a7c 6f31 3520 3d20 5232 2020 2020 2020  .|o15 = R2      
│ │ │ │  00022de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022df0: 7c0a 7c6f 3135 203d 2052 3220 2020 2020  |.|o15 = R2     
│ │ │ │ -00022e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00022e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e60: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ -00022e70: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ -00022e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ea0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00022e40: 2020 2020 2020 2020 207c 0a7c 6f31 3520           |.|o15 
│ │ │ │ +00022e50: 3a20 5175 6f74 6965 6e74 5269 6e67 2020  : QuotientRing  
│ │ │ │ +00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00022e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00022ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022ee0: 2d2d 2d2d 2b0a 7c69 3136 203a 204d 3220  ----+.|i16 : M2 
│ │ │ │ -00022ef0: 3d20 5232 5e31 2f69 6465 616c 2278 322c  = R2^1/ideal"x2,
│ │ │ │ -00022f00: 792c 7a22 2020 2020 2020 2020 2020 2020  y,z"            
│ │ │ │ +00022ec0: 2d2d 2d2b 0a7c 6931 3620 3a20 4d32 203d  ---+.|i16 : M2 =
│ │ │ │ +00022ed0: 2052 325e 312f 6964 6561 6c22 7832 2c79   R2^1/ideal"x2,y
│ │ │ │ +00022ee0: 2c7a 2220 2020 2020 2020 2020 2020 2020  ,z"             
│ │ │ │ +00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00022f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00022f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00022f60: 7c6f 3136 203d 2063 6f6b 6572 6e65 6c20  |o16 = cokernel 
│ │ │ │ -00022f70: 7c20 7832 2079 207a 207c 2020 2020 2020  | x2 y z |      
│ │ │ │ +00022f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00022f40: 6f31 3620 3d20 636f 6b65 726e 656c 207c  o16 = cokernel |
│ │ │ │ +00022f50: 2078 3220 7920 7a20 7c20 2020 2020 2020   x2 y z |       
│ │ │ │ +00022f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00022f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00022f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022fb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00022fd0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │  00022fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ff0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00023000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023010: 2020 2020 207c 0a7c 6f31 3620 3a20 5232       |.|o16 : R2
│ │ │ │ -00023020: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00023030: 7420 6f66 2052 3220 2020 2020 2020 2020  t of R2         
│ │ │ │ -00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023050: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00023060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00023090: 0a7c 6931 3720 3a20 6265 7474 6920 6672  .|i17 : betti fr
│ │ │ │ -000230a0: 6565 5265 736f 6c75 7469 6f6e 2028 4d32  eeResolution (M2
│ │ │ │ -000230b0: 2c20 4c65 6e67 7468 4c69 6d69 7420 3d3e  , LengthLimit =>
│ │ │ │ -000230c0: 3130 2920 2020 2020 2020 2020 7c0a 7c20  10)         |.| 
│ │ │ │ +00022ff0: 2020 2020 7c0a 7c6f 3136 203a 2052 322d      |.|o16 : R2-
│ │ │ │ +00023000: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +00023010: 206f 6620 5232 2020 2020 2020 2020 2020   of R2          
│ │ │ │ +00023020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023030: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00023040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00023070: 7c69 3137 203a 2062 6574 7469 2066 7265  |i17 : betti fre
│ │ │ │ +00023080: 6552 6573 6f6c 7574 696f 6e20 284d 322c  eResolution (M2,
│ │ │ │ +00023090: 204c 656e 6774 684c 696d 6974 203d 3e31   LengthLimit =>1
│ │ │ │ +000230a0: 3029 2020 2020 2020 2020 207c 0a7c 2020  0)         |.|  
│ │ │ │ +000230b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000230c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000230d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00023110: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ -00023120: 2034 2020 3520 2036 2020 3720 2038 2020   4  5  6  7  8  
│ │ │ │ -00023130: 3920 3130 2020 2020 2020 2020 2020 2020  9 10            
│ │ │ │ -00023140: 2020 2020 2020 7c0a 7c6f 3137 203d 2074        |.|o17 = t
│ │ │ │ -00023150: 6f74 616c 3a20 3120 3320 3520 3720 3920  otal: 1 3 5 7 9 
│ │ │ │ -00023160: 3131 2031 3320 3135 2031 3720 3139 2032  11 13 15 17 19 2
│ │ │ │ -00023170: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00023180: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00023190: 303a 2031 2032 2032 2032 2032 2020 3220  0: 1 2 2 2 2  2 
│ │ │ │ -000231a0: 2032 2020 3220 2032 2020 3220 2032 2020   2  2  2  2  2  
│ │ │ │ -000231b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000231c0: 7c0a 7c20 2020 2020 2020 2020 2031 3a20  |.|          1: 
│ │ │ │ -000231d0: 2e20 3120 3320 3420 3420 2034 2020 3420  . 1 3 4 4  4  4 
│ │ │ │ -000231e0: 2034 2020 3420 2034 2020 3420 2020 2020   4  4  4  4     
│ │ │ │ -000231f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023200: 2020 2020 2020 2020 2020 323a 202e 202e            2: . .
│ │ │ │ -00023210: 202e 2031 2033 2020 3420 2034 2020 3420   . 1 3  4  4  4 
│ │ │ │ -00023220: 2034 2020 3420 2034 2020 2020 2020 2020   4  4  4        
│ │ │ │ -00023230: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00023240: 2020 2020 2020 2033 3a20 2e20 2e20 2e20         3: . . . 
│ │ │ │ -00023250: 2e20 2e20 2031 2020 3320 2034 2020 3420  . .  1  3  4  4 
│ │ │ │ -00023260: 2034 2020 3420 2020 2020 2020 2020 2020   4  4           
│ │ │ │ -00023270: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00023280: 2020 2020 343a 202e 202e 202e 202e 202e      4: . . . . .
│ │ │ │ -00023290: 2020 2e20 202e 2020 3120 2033 2020 3420    .  .  1  3  4 
│ │ │ │ -000232a0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -000232b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000232c0: 2035 3a20 2e20 2e20 2e20 2e20 2e20 202e   5: . . . . .  .
│ │ │ │ -000232d0: 2020 2e20 202e 2020 2e20 2031 2020 3320    .  .  .  1  3 
│ │ │ │ +000230e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000230f0: 2020 2020 2020 2020 3020 3120 3220 3320          0 1 2 3 
│ │ │ │ +00023100: 3420 2035 2020 3620 2037 2020 3820 2039  4  5  6  7  8  9
│ │ │ │ +00023110: 2031 3020 2020 2020 2020 2020 2020 2020   10             
│ │ │ │ +00023120: 2020 2020 207c 0a7c 6f31 3720 3d20 746f       |.|o17 = to
│ │ │ │ +00023130: 7461 6c3a 2031 2033 2035 2037 2039 2031  tal: 1 3 5 7 9 1
│ │ │ │ +00023140: 3120 3133 2031 3520 3137 2031 3920 3231  1 13 15 17 19 21
│ │ │ │ +00023150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023160: 2020 7c0a 7c20 2020 2020 2020 2020 2030    |.|          0
│ │ │ │ +00023170: 3a20 3120 3220 3220 3220 3220 2032 2020  : 1 2 2 2 2  2  
│ │ │ │ +00023180: 3220 2032 2020 3220 2032 2020 3220 2020  2  2  2  2  2   
│ │ │ │ +00023190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000231a0: 0a7c 2020 2020 2020 2020 2020 313a 202e  .|          1: .
│ │ │ │ +000231b0: 2031 2033 2034 2034 2020 3420 2034 2020   1 3 4 4  4  4  
│ │ │ │ +000231c0: 3420 2034 2020 3420 2034 2020 2020 2020  4  4  4  4      
│ │ │ │ +000231d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000231e0: 2020 2020 2020 2020 2032 3a20 2e20 2e20           2: . . 
│ │ │ │ +000231f0: 2e20 3120 3320 2034 2020 3420 2034 2020  . 1 3  4  4  4  
│ │ │ │ +00023200: 3420 2034 2020 3420 2020 2020 2020 2020  4  4  4         
│ │ │ │ +00023210: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00023220: 2020 2020 2020 333a 202e 202e 202e 202e        3: . . . .
│ │ │ │ +00023230: 202e 2020 3120 2033 2020 3420 2034 2020   .  1  3  4  4  
│ │ │ │ +00023240: 3420 2034 2020 2020 2020 2020 2020 2020  4  4            
│ │ │ │ +00023250: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00023260: 2020 2034 3a20 2e20 2e20 2e20 2e20 2e20     4: . . . . . 
│ │ │ │ +00023270: 202e 2020 2e20 2031 2020 3320 2034 2020   .  .  1  3  4  
│ │ │ │ +00023280: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +00023290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000232a0: 353a 202e 202e 202e 202e 202e 2020 2e20  5: . . . . .  . 
│ │ │ │ +000232b0: 202e 2020 2e20 202e 2020 3120 2033 2020   .  .  .  1  3  
│ │ │ │ +000232c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000232d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000232e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00023300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00023330: 7c6f 3137 203a 2042 6574 7469 5461 6c6c  |o17 : BettiTall
│ │ │ │ -00023340: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ -00023350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023360: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000232f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00023310: 6f31 3720 3a20 4265 7474 6954 616c 6c79  o17 : BettiTally
│ │ │ │ +00023320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023340: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00023350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000233a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a  --------+.|i18 :
│ │ │ │ -000233b0: 2045 203d 2045 7874 4d6f 6475 6c65 204d   E = ExtModule M
│ │ │ │ -000233c0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00023380: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20  -------+.|i18 : 
│ │ │ │ +00023390: 4520 3d20 4578 744d 6f64 756c 6520 4d32  E = ExtModule M2
│ │ │ │ +000233a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000233b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000233c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000233d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000233e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000233e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000233f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023420: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00023430: 2020 2020 2020 2038 2020 2020 2020 2020         8        
│ │ │ │ -00023440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023460: 0a7c 6f31 3820 3d20 286b 6b5b 5820 2e2e  .|o18 = (kk[X ..
│ │ │ │ -00023470: 5820 5d29 2020 2020 2020 2020 2020 2020  X ])            
│ │ │ │ -00023480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023490: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -000234a0: 2020 2020 2020 2020 2020 3020 2020 3120            0   1 
│ │ │ │ -000234b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023400: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00023410: 2020 2020 2020 3820 2020 2020 2020 2020        8         
│ │ │ │ +00023420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00023440: 7c6f 3138 203d 2028 6b6b 5b58 202e 2e58  |o18 = (kk[X ..X
│ │ │ │ +00023450: 205d 2920 2020 2020 2020 2020 2020 2020   ])             
│ │ │ │ +00023460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023470: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00023480: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ +00023490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000234a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000234b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000234c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000234d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000234e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023510: 2020 2020 2020 7c0a 7c6f 3138 203a 206b        |.|o18 : k
│ │ │ │ -00023520: 6b5b 5820 2e2e 5820 5d2d 6d6f 6475 6c65  k[X ..X ]-module
│ │ │ │ -00023530: 2c20 6672 6565 2c20 6465 6772 6565 7320  , free, degrees 
│ │ │ │ -00023540: 7b30 2e2e 312c 2032 3a31 2c20 333a 322c  {0..1, 2:1, 3:2,
│ │ │ │ -00023550: 2033 7d7c 0a7c 2020 2020 2020 2020 2020   3}|.|          
│ │ │ │ -00023560: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ -00023570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023590: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -000235a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000235b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000235c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000235d0: 6931 3920 3a20 6170 706c 7928 746f 4c69  i19 : apply(toLi
│ │ │ │ -000235e0: 7374 2830 2e2e 3130 292c 2069 2d3e 6869  st(0..10), i->hi
│ │ │ │ -000235f0: 6c62 6572 7446 756e 6374 696f 6e28 692c  lbertFunction(i,
│ │ │ │ -00023600: 2045 2929 2020 2020 2020 7c0a 7c20 2020   E))      |.|   
│ │ │ │ +000234f0: 2020 2020 207c 0a7c 6f31 3820 3a20 6b6b       |.|o18 : kk
│ │ │ │ +00023500: 5b58 202e 2e58 205d 2d6d 6f64 756c 652c  [X ..X ]-module,
│ │ │ │ +00023510: 2066 7265 652c 2064 6567 7265 6573 207b   free, degrees {
│ │ │ │ +00023520: 302e 2e31 2c20 323a 312c 2033 3a32 2c20  0..1, 2:1, 3:2, 
│ │ │ │ +00023530: 337d 7c0a 7c20 2020 2020 2020 2020 2030  3}|.|          0
│ │ │ │ +00023540: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00023550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00023570: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00023580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000235a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000235b0: 3139 203a 2061 7070 6c79 2874 6f4c 6973  19 : apply(toLis
│ │ │ │ +000235c0: 7428 302e 2e31 3029 2c20 692d 3e68 696c  t(0..10), i->hil
│ │ │ │ +000235d0: 6265 7274 4675 6e63 7469 6f6e 2869 2c20  bertFunction(i, 
│ │ │ │ +000235e0: 4529 2920 2020 2020 207c 0a7c 2020 2020  E))      |.|    
│ │ │ │ +000235f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023640: 2020 2020 2020 207c 0a7c 6f31 3920 3d20         |.|o19 = 
│ │ │ │ -00023650: 7b31 2c20 332c 2035 2c20 372c 2039 2c20  {1, 3, 5, 7, 9, 
│ │ │ │ -00023660: 3131 2c20 3133 2c20 3135 2c20 3137 2c20  11, 13, 15, 17, 
│ │ │ │ -00023670: 3139 2c20 3231 7d20 2020 2020 2020 2020  19, 21}         
│ │ │ │ -00023680: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00023620: 2020 2020 2020 7c0a 7c6f 3139 203d 207b        |.|o19 = {
│ │ │ │ +00023630: 312c 2033 2c20 352c 2037 2c20 392c 2031  1, 3, 5, 7, 9, 1
│ │ │ │ +00023640: 312c 2031 332c 2031 352c 2031 372c 2031  1, 13, 15, 17, 1
│ │ │ │ +00023650: 392c 2032 317d 2020 2020 2020 2020 2020  9, 21}          
│ │ │ │ +00023660: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00023670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000236a0: 7c0a 7c6f 3139 203a 204c 6973 7420 2020  |.|o19 : List   
│ │ │ │  000236b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236c0: 207c 0a7c 6f31 3920 3a20 4c69 7374 2020   |.|o19 : List  
│ │ │ │ -000236d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00023700: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00023710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00023740: 3020 3a20 4565 7665 6e20 3d20 6576 656e  0 : Eeven = even
│ │ │ │ -00023750: 4578 744d 6f64 756c 6520 4d32 2020 2020  ExtModule M2    
│ │ │ │ +000236c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000236d0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000236e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000236f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023710: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230  ----------+.|i20
│ │ │ │ +00023720: 203a 2045 6576 656e 203d 2065 7665 6e45   : Eeven = evenE
│ │ │ │ +00023730: 7874 4d6f 6475 6c65 204d 3220 2020 2020  xtModule M2     
│ │ │ │ +00023740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023750: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00023760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023770: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00023770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000237c0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ -000237d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237f0: 2020 7c0a 7c6f 3230 203d 2028 6b6b 5b58    |.|o20 = (kk[X
│ │ │ │ -00023800: 202e 2e58 205d 2920 2020 2020 2020 2020   ..X ])         
│ │ │ │ -00023810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023830: 0a7c 2020 2020 2020 2020 2020 2030 2020  .|           0  
│ │ │ │ -00023840: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ +00023790: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000237a0: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ +000237b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000237c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000237d0: 207c 0a7c 6f32 3020 3d20 286b 6b5b 5820   |.|o20 = (kk[X 
│ │ │ │ +000237e0: 2e2e 5820 5d29 2020 2020 2020 2020 2020  ..X ])          
│ │ │ │ +000237f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023800: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00023810: 7c20 2020 2020 2020 2020 2020 3020 2020  |           0   
│ │ │ │ +00023820: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00023830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00023850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00023860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238a0: 2020 2020 2020 2020 207c 0a7c 6f32 3020           |.|o20 
│ │ │ │ -000238b0: 3a20 6b6b 5b58 202e 2e58 205d 2d6d 6f64  : kk[X ..X ]-mod
│ │ │ │ -000238c0: 756c 652c 2066 7265 652c 2064 6567 7265  ule, free, degre
│ │ │ │ -000238d0: 6573 207b 302e 2e31 2c20 323a 317d 2020  es {0..1, 2:1}  
│ │ │ │ -000238e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000238f0: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ -00023900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023920: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00023930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023960: 2b0a 7c69 3231 203a 2061 7070 6c79 2874  +.|i21 : apply(t
│ │ │ │ -00023970: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e  oList(0..5), i->
│ │ │ │ -00023980: 6869 6c62 6572 7446 756e 6374 696f 6e28  hilbertFunction(
│ │ │ │ -00023990: 692c 2045 6576 656e 2929 2020 207c 0a7c  i, Eeven))   |.|
│ │ │ │ +00023880: 2020 2020 2020 2020 7c0a 7c6f 3230 203a          |.|o20 :
│ │ │ │ +00023890: 206b 6b5b 5820 2e2e 5820 5d2d 6d6f 6475   kk[X ..X ]-modu
│ │ │ │ +000238a0: 6c65 2c20 6672 6565 2c20 6465 6772 6565  le, free, degree
│ │ │ │ +000238b0: 7320 7b30 2e2e 312c 2032 3a31 7d20 2020  s {0..1, 2:1}   
│ │ │ │ +000238c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000238d0: 2020 3020 2020 3120 2020 2020 2020 2020    0   1         
│ │ │ │ +000238e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000238f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023900: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00023910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00023940: 0a7c 6932 3120 3a20 6170 706c 7928 746f  .|i21 : apply(to
│ │ │ │ +00023950: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68  List(0..5), i->h
│ │ │ │ +00023960: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ +00023970: 2c20 4565 7665 6e29 2920 2020 7c0a 7c20  , Eeven))   |.| 
│ │ │ │ +00023980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ -000239e0: 203d 207b 312c 2035 2c20 392c 2031 332c   = {1, 5, 9, 13,
│ │ │ │ -000239f0: 2031 372c 2032 317d 2020 2020 2020 2020   17, 21}        
│ │ │ │ +000239b0: 2020 2020 2020 2020 207c 0a7c 6f32 3120           |.|o21 
│ │ │ │ +000239c0: 3d20 7b31 2c20 352c 2039 2c20 3133 2c20  = {1, 5, 9, 13, 
│ │ │ │ +000239d0: 3137 2c20 3231 7d20 2020 2020 2020 2020  17, 21}         
│ │ │ │ +000239e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000239f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00023a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00023a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a30: 2020 207c 0a7c 6f32 3120 3a20 4c69 7374     |.|o21 : List
│ │ │ │  00023a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a50: 2020 2020 7c0a 7c6f 3231 203a 204c 6973      |.|o21 : Lis
│ │ │ │ -00023a60: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -00023a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00023ad0: 7c69 3232 203a 2045 6f64 6420 3d20 6f64  |i22 : Eodd = od
│ │ │ │ -00023ae0: 6445 7874 4d6f 6475 6c65 204d 3220 2020  dExtModule M2   
│ │ │ │ +00023a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00023a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00023ab0: 6932 3220 3a20 456f 6464 203d 206f 6464  i22 : Eodd = odd
│ │ │ │ +00023ac0: 4578 744d 6f64 756c 6520 4d32 2020 2020  ExtModule M2    
│ │ │ │ +00023ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00023af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00023b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00023b50: 2020 2020 2020 2020 2020 2020 2034 2020               4  
│ │ │ │ -00023b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b80: 2020 2020 207c 0a7c 6f32 3220 3d20 286b       |.|o22 = (k
│ │ │ │ -00023b90: 6b5b 5820 2e2e 5820 5d29 2020 2020 2020  k[X ..X ])      
│ │ │ │ -00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00023bd0: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ -00023be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023c00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00023c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023c30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00023c40: 3232 203a 206b 6b5b 5820 2e2e 5820 5d2d  22 : kk[X ..X ]-
│ │ │ │ -00023c50: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465  module, free, de
│ │ │ │ -00023c60: 6772 6565 7320 7b33 3a30 2c20 317d 2020  grees {3:0, 1}  
│ │ │ │ -00023c70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00023c80: 2020 2020 2020 3020 2020 3120 2020 2020        0   1     
│ │ │ │ -00023c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023cb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00023b20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00023b30: 2020 2020 2020 2020 2020 2020 3420 2020              4   
│ │ │ │ +00023b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b60: 2020 2020 7c0a 7c6f 3232 203d 2028 6b6b      |.|o22 = (kk
│ │ │ │ +00023b70: 5b58 202e 2e58 205d 2920 2020 2020 2020  [X ..X ])       
│ │ │ │ +00023b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023ba0: 207c 0a7c 2020 2020 2020 2020 2020 2030   |.|           0
│ │ │ │ +00023bb0: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023bd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00023be0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00023bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c10: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00023c20: 3220 3a20 6b6b 5b58 202e 2e58 205d 2d6d  2 : kk[X ..X ]-m
│ │ │ │ +00023c30: 6f64 756c 652c 2066 7265 652c 2064 6567  odule, free, deg
│ │ │ │ +00023c40: 7265 6573 207b 333a 302c 2031 7d20 2020  rees {3:0, 1}   
│ │ │ │ +00023c50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00023c60: 2020 2020 2030 2020 2031 2020 2020 2020       0   1      
│ │ │ │ +00023c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023c90: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00023ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023cf0: 2d2d 2d2b 0a7c 6932 3320 3a20 6170 706c  ---+.|i23 : appl
│ │ │ │ -00023d00: 7928 746f 4c69 7374 2830 2e2e 3529 2c20  y(toList(0..5), 
│ │ │ │ -00023d10: 692d 3e68 696c 6265 7274 4675 6e63 7469  i->hilbertFuncti
│ │ │ │ -00023d20: 6f6e 2869 2c20 456f 6464 2929 2020 2020  on(i, Eodd))    
│ │ │ │ -00023d30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00023d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00023d70: 6f32 3320 3d20 7b33 2c20 372c 2031 312c  o23 = {3, 7, 11,
│ │ │ │ -00023d80: 2031 352c 2031 392c 2032 337d 2020 2020   15, 19, 23}    
│ │ │ │ +00023cd0: 2d2d 2b0a 7c69 3233 203a 2061 7070 6c79  --+.|i23 : apply
│ │ │ │ +00023ce0: 2874 6f4c 6973 7428 302e 2e35 292c 2069  (toList(0..5), i
│ │ │ │ +00023cf0: 2d3e 6869 6c62 6572 7446 756e 6374 696f  ->hilbertFunctio
│ │ │ │ +00023d00: 6e28 692c 2045 6f64 6429 2920 2020 207c  n(i, Eodd))    |
│ │ │ │ +00023d10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00023d50: 3233 203d 207b 332c 2037 2c20 3131 2c20  23 = {3, 7, 11, 
│ │ │ │ +00023d60: 3135 2c20 3139 2c20 3233 7d20 2020 2020  15, 19, 23}     
│ │ │ │ +00023d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00023d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023da0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00023da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023de0: 2020 2020 2020 207c 0a7c 6f32 3320 3a20         |.|o23 : 
│ │ │ │ -00023df0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ -00023e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00023dc0: 2020 2020 2020 7c0a 7c6f 3233 203a 204c        |.|o23 : L
│ │ │ │ +00023dd0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00023de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00023e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023e60: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -00023e70: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00023e80: 2065 7665 6e45 7874 4d6f 6475 6c65 3a20   evenExtModule: 
│ │ │ │ -00023e90: 6576 656e 4578 744d 6f64 756c 652c 202d  evenExtModule, -
│ │ │ │ -00023ea0: 2d20 6576 656e 2070 6172 7420 6f66 2045  - even part of E
│ │ │ │ -00023eb0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -00023ec0: 0a20 2020 2063 6f6d 706c 6574 6520 696e  .    complete in
│ │ │ │ -00023ed0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f  tersection as mo
│ │ │ │ -00023ee0: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ -00023ef0: 7261 746f 7220 7269 6e67 0a20 202a 202a  rator ring.  * *
│ │ │ │ -00023f00: 6e6f 7465 206f 6464 4578 744d 6f64 756c  note oddExtModul
│ │ │ │ -00023f10: 653a 206f 6464 4578 744d 6f64 756c 652c  e: oddExtModule,
│ │ │ │ -00023f20: 202d 2d20 6f64 6420 7061 7274 206f 6620   -- odd part of 
│ │ │ │ -00023f30: 4578 745e 2a28 4d2c 6b29 206f 7665 7220  Ext^*(M,k) over 
│ │ │ │ -00023f40: 6120 636f 6d70 6c65 7465 0a20 2020 2069  a complete.    i
│ │ │ │ -00023f50: 6e74 6572 7365 6374 696f 6e20 6173 206d  ntersection as m
│ │ │ │ -00023f60: 6f64 756c 6520 6f76 6572 2043 4920 6f70  odule over CI op
│ │ │ │ -00023f70: 6572 6174 6f72 2072 696e 670a 0a57 6179  erator ring..Way
│ │ │ │ -00023f80: 7320 746f 2075 7365 2045 7874 4d6f 6475  s to use ExtModu
│ │ │ │ -00023f90: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  le:.============
│ │ │ │ -00023fa0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00023fb0: 2245 7874 4d6f 6475 6c65 284d 6f64 756c  "ExtModule(Modul
│ │ │ │ -00023fc0: 6529 220a 0a46 6f72 2074 6865 2070 726f  e)"..For the pro
│ │ │ │ -00023fd0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -00023fe0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -00023ff0: 6f62 6a65 6374 202a 6e6f 7465 2045 7874  object *note Ext
│ │ │ │ -00024000: 4d6f 6475 6c65 3a20 4578 744d 6f64 756c  Module: ExtModul
│ │ │ │ -00024010: 652c 2069 7320 6120 2a6e 6f74 6520 6d65  e, is a *note me
│ │ │ │ -00024020: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ -00024030: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ -00024040: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │ +00023e40: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  +..See also.====
│ │ │ │ +00023e50: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +00023e60: 6576 656e 4578 744d 6f64 756c 653a 2065  evenExtModule: e
│ │ │ │ +00023e70: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d  venExtModule, --
│ │ │ │ +00023e80: 2065 7665 6e20 7061 7274 206f 6620 4578   even part of Ex
│ │ │ │ +00023e90: 745e 2a28 4d2c 6b29 206f 7665 7220 610a  t^*(M,k) over a.
│ │ │ │ +00023ea0: 2020 2020 636f 6d70 6c65 7465 2069 6e74      complete int
│ │ │ │ +00023eb0: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ +00023ec0: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ +00023ed0: 6174 6f72 2072 696e 670a 2020 2a20 2a6e  ator ring.  * *n
│ │ │ │ +00023ee0: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65  ote oddExtModule
│ │ │ │ +00023ef0: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20  : oddExtModule, 
│ │ │ │ +00023f00: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ +00023f10: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ +00023f20: 2063 6f6d 706c 6574 650a 2020 2020 696e   complete.    in
│ │ │ │ +00023f30: 7465 7273 6563 7469 6f6e 2061 7320 6d6f  tersection as mo
│ │ │ │ +00023f40: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ +00023f50: 7261 746f 7220 7269 6e67 0a0a 5761 7973  rator ring..Ways
│ │ │ │ +00023f60: 2074 6f20 7573 6520 4578 744d 6f64 756c   to use ExtModul
│ │ │ │ +00023f70: 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  e:.=============
│ │ │ │ +00023f80: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +00023f90: 4578 744d 6f64 756c 6528 4d6f 6475 6c65  ExtModule(Module
│ │ │ │ +00023fa0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00023fb0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00023fc0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00023fd0: 626a 6563 7420 2a6e 6f74 6520 4578 744d  bject *note ExtM
│ │ │ │ +00023fe0: 6f64 756c 653a 2045 7874 4d6f 6475 6c65  odule: ExtModule
│ │ │ │ +00023ff0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +00024000: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ +00024010: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +00024020: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ +00024030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00024060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -000240a0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -000240b0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -000240c0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -000240d0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -000240e0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ -000240f0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -00024100: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ -00024110: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -00024120: 7574 696f 6e73 2e6d 323a 3335 3936 3a30  utions.m2:3596:0
│ │ │ │ -00024130: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ -00024140: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ -00024150: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ -00024160: 4e6f 6465 3a20 4578 744d 6f64 756c 6544  Node: ExtModuleD
│ │ │ │ -00024170: 6174 612c 204e 6578 743a 2065 7874 5673  ata, Next: extVs
│ │ │ │ -00024180: 436f 686f 6d6f 6c6f 6779 2c20 5072 6576  Cohomology, Prev
│ │ │ │ -00024190: 3a20 4578 744d 6f64 756c 652c 2055 703a  : ExtModule, Up:
│ │ │ │ -000241a0: 2054 6f70 0a0a 4578 744d 6f64 756c 6544   Top..ExtModuleD
│ │ │ │ -000241b0: 6174 6120 2d2d 2045 7665 6e20 616e 6420  ata -- Even and 
│ │ │ │ -000241c0: 6f64 6420 4578 7420 6d6f 6475 6c65 7320  odd Ext modules 
│ │ │ │ -000241d0: 616e 6420 7468 6569 7220 7265 6775 6c61  and their regula
│ │ │ │ -000241e0: 7269 7479 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  rity.***********
│ │ │ │ +00024070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +00024080: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +00024090: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +000240a0: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +000240b0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +000240c0: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ +000240d0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +000240e0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ +000240f0: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +00024100: 7469 6f6e 732e 6d32 3a33 3539 363a 302e  tions.m2:3596:0.
│ │ │ │ +00024110: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ +00024120: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +00024130: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ +00024140: 6f64 653a 2045 7874 4d6f 6475 6c65 4461  ode: ExtModuleDa
│ │ │ │ +00024150: 7461 2c20 4e65 7874 3a20 6578 7456 7343  ta, Next: extVsC
│ │ │ │ +00024160: 6f68 6f6d 6f6c 6f67 792c 2050 7265 763a  ohomology, Prev:
│ │ │ │ +00024170: 2045 7874 4d6f 6475 6c65 2c20 5570 3a20   ExtModule, Up: 
│ │ │ │ +00024180: 546f 700a 0a45 7874 4d6f 6475 6c65 4461  Top..ExtModuleDa
│ │ │ │ +00024190: 7461 202d 2d20 4576 656e 2061 6e64 206f  ta -- Even and o
│ │ │ │ +000241a0: 6464 2045 7874 206d 6f64 756c 6573 2061  dd Ext modules a
│ │ │ │ +000241b0: 6e64 2074 6865 6972 2072 6567 756c 6172  nd their regular
│ │ │ │ +000241c0: 6974 790a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ity.************
│ │ │ │ +000241d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000241e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000241f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00024200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00024210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00024220: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -00024230: 0a20 2020 2020 2020 204c 203d 2045 7874  .        L = Ext
│ │ │ │ -00024240: 4d6f 6475 6c65 4461 7461 204d 0a20 202a  ModuleData M.  *
│ │ │ │ -00024250: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00024260: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -00024270: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -00024280: 6329 4d6f 6475 6c65 2c2c 204d 6f64 756c  c)Module,, Modul
│ │ │ │ -00024290: 6520 6f76 6572 2061 2063 6f6d 706c 6574  e over a complet
│ │ │ │ -000242a0: 650a 2020 2020 2020 2020 696e 7465 7273  e.        inters
│ │ │ │ -000242b0: 6563 7469 6f6e 2053 0a20 202a 204f 7574  ection S.  * Out
│ │ │ │ -000242c0: 7075 7473 3a0a 2020 2020 2020 2a20 4c2c  puts:.      * L,
│ │ │ │ -000242d0: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ -000242e0: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ -000242f0: 742c 2c20 4c20 3d20 5c7b 6576 656e 4578  t,, L = \{evenEx
│ │ │ │ -00024300: 744d 6f64 756c 652c 0a20 2020 2020 2020  tModule,.       
│ │ │ │ -00024310: 206f 6464 4578 744d 6f64 756c 652c 2072   oddExtModule, r
│ │ │ │ -00024320: 6567 302c 2072 6567 315c 7d0a 0a44 6573  eg0, reg1\}..Des
│ │ │ │ -00024330: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00024340: 3d3d 3d3d 0a0a 5375 7070 6f73 6520 7468  ====..Suppose th
│ │ │ │ -00024350: 6174 204d 2069 7320 6120 6d6f 6475 6c65  at M is a module
│ │ │ │ -00024360: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ -00024370: 2069 6e74 6572 7365 6374 696f 6e20 5220   intersection R 
│ │ │ │ -00024380: 736f 2074 6861 740a 0a45 203a 3d20 4578  so that..E := Ex
│ │ │ │ -00024390: 744d 6f64 756c 6520 4d0a 0a69 7320 6120  tModule M..is a 
│ │ │ │ -000243a0: 6d6f 6475 6c65 2067 656e 6572 6174 6564  module generated
│ │ │ │ -000243b0: 2069 6e20 6465 6772 6565 7320 3e3d 3020   in degrees >=0 
│ │ │ │ -000243c0: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961  over a polynomia
│ │ │ │ -000243d0: 6c20 7269 6e67 2054 2720 6765 6e65 7261  l ring T' genera
│ │ │ │ -000243e0: 7465 6420 696e 0a64 6567 7265 6520 322c  ted in.degree 2,
│ │ │ │ -000243f0: 2061 6e64 0a0a 4530 203a 3d20 6576 656e   and..E0 := even
│ │ │ │ -00024400: 4578 744d 6f64 756c 6520 4d20 616e 6420  ExtModule M and 
│ │ │ │ -00024410: 4531 203a 3d20 6f64 6445 7874 4d6f 6475  E1 := oddExtModu
│ │ │ │ -00024420: 6c65 204d 0a0a 6172 6520 6d6f 6475 6c65  le M..are module
│ │ │ │ -00024430: 7320 6765 6e65 7261 7465 6420 696e 2064  s generated in d
│ │ │ │ -00024440: 6567 7265 6520 3e3d 2030 206f 7665 7220  egree >= 0 over 
│ │ │ │ -00024450: 6120 706f 6c79 6e6f 6d69 616c 2072 696e  a polynomial rin
│ │ │ │ -00024460: 6720 5420 7769 7468 2067 656e 6572 6174  g T with generat
│ │ │ │ -00024470: 6f72 730a 696e 2064 6567 7265 6520 312e  ors.in degree 1.
│ │ │ │ -00024480: 0a0a 5468 6520 7363 7269 7074 2072 6574  ..The script ret
│ │ │ │ -00024490: 7572 6e73 0a0a 4c20 3d20 5c7b 4530 2c45  urns..L = \{E0,E
│ │ │ │ -000244a0: 312c 2072 6567 756c 6172 6974 7920 4530  1, regularity E0
│ │ │ │ -000244b0: 2c20 7265 6775 6c61 7269 7479 2045 315c  , regularity E1\
│ │ │ │ -000244c0: 7d0a 0a61 6e64 2070 7269 6e74 7320 6120  }..and prints a 
│ │ │ │ -000244d0: 6d65 7373 6167 6520 6966 207c 7265 6730  message if |reg0
│ │ │ │ -000244e0: 2d72 6567 317c 3e31 2e0a 0a49 6620 7765  -reg1|>1...If we
│ │ │ │ -000244f0: 2073 6574 2072 203d 206d 6178 2832 2a72   set r = max(2*r
│ │ │ │ -00024500: 6567 302c 2031 2b32 2a72 6567 3129 2c20  eg0, 1+2*reg1), 
│ │ │ │ -00024510: 616e 6420 4620 6973 2061 2072 6573 6f6c  and F is a resol
│ │ │ │ -00024520: 7574 696f 6e20 6f66 204d 2c20 7468 656e  ution of M, then
│ │ │ │ -00024530: 2063 6f6b 6572 0a46 2e64 645f 7b28 722b   coker.F.dd_{(r+
│ │ │ │ -00024540: 3129 7d20 6973 2074 6865 2066 6972 7374  1)} is the first
│ │ │ │ -00024550: 2073 7a79 6779 206d 6f64 756c 6520 6f66   szygy module of
│ │ │ │ -00024560: 204d 2073 7563 6820 7468 6174 2072 6567   M such that reg
│ │ │ │ -00024570: 756c 6172 6974 7920 6576 656e 4578 744d  ularity evenExtM
│ │ │ │ -00024580: 6f64 756c 650a 4d20 3d30 2041 4e44 2072  odule.M =0 AND r
│ │ │ │ -00024590: 6567 756c 6172 6974 7920 6f64 6445 7874  egularity oddExt
│ │ │ │ -000245a0: 4d6f 6475 6c65 204d 203d 300a 0a57 6520  Module M =0..We 
│ │ │ │ -000245b0: 6861 7665 2062 6565 6e20 7573 696e 6720  have been using 
│ │ │ │ -000245c0: 7265 6775 6c61 7269 7479 2045 7874 4d6f  regularity ExtMo
│ │ │ │ -000245d0: 6475 6c65 204d 2061 7320 6120 7375 6273  dule M as a subs
│ │ │ │ -000245e0: 7469 7475 7465 2066 6f72 2072 2c20 6275  titute for r, bu
│ │ │ │ -000245f0: 7420 7468 6174 2773 206e 6f74 0a61 6c77  t that's not.alw
│ │ │ │ -00024600: 6179 7320 7468 6520 7361 6d65 2e0a 0a54  ays the same...T
│ │ │ │ -00024610: 6865 2072 6567 756c 6172 6974 6965 7320  he regularities 
│ │ │ │ -00024620: 6f66 2074 6865 2065 7665 6e20 616e 6420  of the even and 
│ │ │ │ -00024630: 6f64 6420 4578 7420 6d6f 6475 6c65 7320  odd Ext modules 
│ │ │ │ -00024640: 2a63 616e 2a20 6469 6666 6572 2062 7920  *can* differ by 
│ │ │ │ -00024650: 6d6f 7265 2074 6861 6e20 312e 0a41 6e20  more than 1..An 
│ │ │ │ -00024660: 6578 616d 706c 6520 6361 6e20 6265 2070  example can be p
│ │ │ │ -00024670: 726f 6475 6365 6420 7769 7468 2073 6574  roduced with set
│ │ │ │ -00024680: 5261 6e64 6f6d 5365 6564 2030 2053 203d  RandomSeed 0 S =
│ │ │ │ -00024690: 205a 5a2f 3130 315b 612c 622c 632c 645d   ZZ/101[a,b,c,d]
│ │ │ │ -000246a0: 2066 660a 3d6d 6174 7269 7822 6134 2c62   ff.=matrix"a4,b
│ │ │ │ -000246b0: 342c 6334 2c64 3422 2052 203d 2053 2f69  4,c4,d4" R = S/i
│ │ │ │ -000246c0: 6465 616c 2066 6620 4e20 3d20 636f 6b65  deal ff N = coke
│ │ │ │ -000246d0: 7220 7261 6e64 6f6d 2852 5e7b 302c 317d  r random(R^{0,1}
│ │ │ │ -000246e0: 2c20 525e 7b20 2d31 2c2d 322c 2d33 2c2d  , R^{ -1,-2,-3,-
│ │ │ │ -000246f0: 347d 290a 2d2d 6769 7665 7320 7265 6720  4}).--gives reg 
│ │ │ │ -00024700: 4578 745e 6576 656e 203d 2034 2c20 7265  Ext^even = 4, re
│ │ │ │ -00024710: 6720 4578 745e 6f64 6420 3d20 3320 4c20  g Ext^odd = 3 L 
│ │ │ │ -00024720: 3d20 4578 744d 6f64 756c 6544 6174 6120  = ExtModuleData 
│ │ │ │ -00024730: 4e3b 2062 7574 2074 616b 6573 2073 6f6d  N; but takes som
│ │ │ │ -00024740: 650a 7469 6d65 2074 6f20 636f 6d70 7574  e.time to comput
│ │ │ │ -00024750: 652e 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  e.....+---------
│ │ │ │ -00024760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00024790: 203a 2073 6574 5261 6e64 6f6d 5365 6564   : setRandomSeed
│ │ │ │ -000247a0: 2031 3030 2020 2020 2020 2020 2020 2020   100            
│ │ │ │ -000247b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247c0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67    |.| -- setting
│ │ │ │ -000247d0: 2072 616e 646f 6d20 7365 6564 2074 6f20   random seed to 
│ │ │ │ -000247e0: 3130 3020 2020 2020 2020 2020 2020 2020  100             
│ │ │ │ -000247f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024200: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +00024210: 2020 2020 2020 2020 4c20 3d20 4578 744d          L = ExtM
│ │ │ │ +00024220: 6f64 756c 6544 6174 6120 4d0a 2020 2a20  oduleData M.  * 
│ │ │ │ +00024230: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00024240: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ +00024250: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +00024260: 294d 6f64 756c 652c 2c20 4d6f 6475 6c65  )Module,, Module
│ │ │ │ +00024270: 206f 7665 7220 6120 636f 6d70 6c65 7465   over a complete
│ │ │ │ +00024280: 0a20 2020 2020 2020 2069 6e74 6572 7365  .        interse
│ │ │ │ +00024290: 6374 696f 6e20 530a 2020 2a20 4f75 7470  ction S.  * Outp
│ │ │ │ +000242a0: 7574 733a 0a20 2020 2020 202a 204c 2c20  uts:.      * L, 
│ │ │ │ +000242b0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ +000242c0: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ +000242d0: 2c2c 204c 203d 205c 7b65 7665 6e45 7874  ,, L = \{evenExt
│ │ │ │ +000242e0: 4d6f 6475 6c65 2c0a 2020 2020 2020 2020  Module,.        
│ │ │ │ +000242f0: 6f64 6445 7874 4d6f 6475 6c65 2c20 7265  oddExtModule, re
│ │ │ │ +00024300: 6730 2c20 7265 6731 5c7d 0a0a 4465 7363  g0, reg1\}..Desc
│ │ │ │ +00024310: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00024320: 3d3d 3d0a 0a53 7570 706f 7365 2074 6861  ===..Suppose tha
│ │ │ │ +00024330: 7420 4d20 6973 2061 206d 6f64 756c 6520  t M is a module 
│ │ │ │ +00024340: 6f76 6572 2061 2063 6f6d 706c 6574 6520  over a complete 
│ │ │ │ +00024350: 696e 7465 7273 6563 7469 6f6e 2052 2073  intersection R s
│ │ │ │ +00024360: 6f20 7468 6174 0a0a 4520 3a3d 2045 7874  o that..E := Ext
│ │ │ │ +00024370: 4d6f 6475 6c65 204d 0a0a 6973 2061 206d  Module M..is a m
│ │ │ │ +00024380: 6f64 756c 6520 6765 6e65 7261 7465 6420  odule generated 
│ │ │ │ +00024390: 696e 2064 6567 7265 6573 203e 3d30 206f  in degrees >=0 o
│ │ │ │ +000243a0: 7665 7220 6120 706f 6c79 6e6f 6d69 616c  ver a polynomial
│ │ │ │ +000243b0: 2072 696e 6720 5427 2067 656e 6572 6174   ring T' generat
│ │ │ │ +000243c0: 6564 2069 6e0a 6465 6772 6565 2032 2c20  ed in.degree 2, 
│ │ │ │ +000243d0: 616e 640a 0a45 3020 3a3d 2065 7665 6e45  and..E0 := evenE
│ │ │ │ +000243e0: 7874 4d6f 6475 6c65 204d 2061 6e64 2045  xtModule M and E
│ │ │ │ +000243f0: 3120 3a3d 206f 6464 4578 744d 6f64 756c  1 := oddExtModul
│ │ │ │ +00024400: 6520 4d0a 0a61 7265 206d 6f64 756c 6573  e M..are modules
│ │ │ │ +00024410: 2067 656e 6572 6174 6564 2069 6e20 6465   generated in de
│ │ │ │ +00024420: 6772 6565 203e 3d20 3020 6f76 6572 2061  gree >= 0 over a
│ │ │ │ +00024430: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67   polynomial ring
│ │ │ │ +00024440: 2054 2077 6974 6820 6765 6e65 7261 746f   T with generato
│ │ │ │ +00024450: 7273 0a69 6e20 6465 6772 6565 2031 2e0a  rs.in degree 1..
│ │ │ │ +00024460: 0a54 6865 2073 6372 6970 7420 7265 7475  .The script retu
│ │ │ │ +00024470: 726e 730a 0a4c 203d 205c 7b45 302c 4531  rns..L = \{E0,E1
│ │ │ │ +00024480: 2c20 7265 6775 6c61 7269 7479 2045 302c  , regularity E0,
│ │ │ │ +00024490: 2072 6567 756c 6172 6974 7920 4531 5c7d   regularity E1\}
│ │ │ │ +000244a0: 0a0a 616e 6420 7072 696e 7473 2061 206d  ..and prints a m
│ │ │ │ +000244b0: 6573 7361 6765 2069 6620 7c72 6567 302d  essage if |reg0-
│ │ │ │ +000244c0: 7265 6731 7c3e 312e 0a0a 4966 2077 6520  reg1|>1...If we 
│ │ │ │ +000244d0: 7365 7420 7220 3d20 6d61 7828 322a 7265  set r = max(2*re
│ │ │ │ +000244e0: 6730 2c20 312b 322a 7265 6731 292c 2061  g0, 1+2*reg1), a
│ │ │ │ +000244f0: 6e64 2046 2069 7320 6120 7265 736f 6c75  nd F is a resolu
│ │ │ │ +00024500: 7469 6f6e 206f 6620 4d2c 2074 6865 6e20  tion of M, then 
│ │ │ │ +00024510: 636f 6b65 720a 462e 6464 5f7b 2872 2b31  coker.F.dd_{(r+1
│ │ │ │ +00024520: 297d 2069 7320 7468 6520 6669 7273 7420  )} is the first 
│ │ │ │ +00024530: 737a 7967 7920 6d6f 6475 6c65 206f 6620  szygy module of 
│ │ │ │ +00024540: 4d20 7375 6368 2074 6861 7420 7265 6775  M such that regu
│ │ │ │ +00024550: 6c61 7269 7479 2065 7665 6e45 7874 4d6f  larity evenExtMo
│ │ │ │ +00024560: 6475 6c65 0a4d 203d 3020 414e 4420 7265  dule.M =0 AND re
│ │ │ │ +00024570: 6775 6c61 7269 7479 206f 6464 4578 744d  gularity oddExtM
│ │ │ │ +00024580: 6f64 756c 6520 4d20 3d30 0a0a 5765 2068  odule M =0..We h
│ │ │ │ +00024590: 6176 6520 6265 656e 2075 7369 6e67 2072  ave been using r
│ │ │ │ +000245a0: 6567 756c 6172 6974 7920 4578 744d 6f64  egularity ExtMod
│ │ │ │ +000245b0: 756c 6520 4d20 6173 2061 2073 7562 7374  ule M as a subst
│ │ │ │ +000245c0: 6974 7574 6520 666f 7220 722c 2062 7574  itute for r, but
│ │ │ │ +000245d0: 2074 6861 7427 7320 6e6f 740a 616c 7761   that's not.alwa
│ │ │ │ +000245e0: 7973 2074 6865 2073 616d 652e 0a0a 5468  ys the same...Th
│ │ │ │ +000245f0: 6520 7265 6775 6c61 7269 7469 6573 206f  e regularities o
│ │ │ │ +00024600: 6620 7468 6520 6576 656e 2061 6e64 206f  f the even and o
│ │ │ │ +00024610: 6464 2045 7874 206d 6f64 756c 6573 202a  dd Ext modules *
│ │ │ │ +00024620: 6361 6e2a 2064 6966 6665 7220 6279 206d  can* differ by m
│ │ │ │ +00024630: 6f72 6520 7468 616e 2031 2e0a 416e 2065  ore than 1..An e
│ │ │ │ +00024640: 7861 6d70 6c65 2063 616e 2062 6520 7072  xample can be pr
│ │ │ │ +00024650: 6f64 7563 6564 2077 6974 6820 7365 7452  oduced with setR
│ │ │ │ +00024660: 616e 646f 6d53 6565 6420 3020 5320 3d20  andomSeed 0 S = 
│ │ │ │ +00024670: 5a5a 2f31 3031 5b61 2c62 2c63 2c64 5d20  ZZ/101[a,b,c,d] 
│ │ │ │ +00024680: 6666 0a3d 6d61 7472 6978 2261 342c 6234  ff.=matrix"a4,b4
│ │ │ │ +00024690: 2c63 342c 6434 2220 5220 3d20 532f 6964  ,c4,d4" R = S/id
│ │ │ │ +000246a0: 6561 6c20 6666 204e 203d 2063 6f6b 6572  eal ff N = coker
│ │ │ │ +000246b0: 2072 616e 646f 6d28 525e 7b30 2c31 7d2c   random(R^{0,1},
│ │ │ │ +000246c0: 2052 5e7b 202d 312c 2d32 2c2d 332c 2d34   R^{ -1,-2,-3,-4
│ │ │ │ +000246d0: 7d29 0a2d 2d67 6976 6573 2072 6567 2045  }).--gives reg E
│ │ │ │ +000246e0: 7874 5e65 7665 6e20 3d20 342c 2072 6567  xt^even = 4, reg
│ │ │ │ +000246f0: 2045 7874 5e6f 6464 203d 2033 204c 203d   Ext^odd = 3 L =
│ │ │ │ +00024700: 2045 7874 4d6f 6475 6c65 4461 7461 204e   ExtModuleData N
│ │ │ │ +00024710: 3b20 6275 7420 7461 6b65 7320 736f 6d65  ; but takes some
│ │ │ │ +00024720: 0a74 696d 6520 746f 2063 6f6d 7075 7465  .time to compute
│ │ │ │ +00024730: 2e0a 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  .....+----------
│ │ │ │ +00024740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024760: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ +00024770: 3a20 7365 7452 616e 646f 6d53 6565 6420  : setRandomSeed 
│ │ │ │ +00024780: 3130 3020 2020 2020 2020 2020 2020 2020  100             
│ │ │ │ +00024790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000247a0: 207c 0a7c 202d 2d20 7365 7474 696e 6720   |.| -- setting 
│ │ │ │ +000247b0: 7261 6e64 6f6d 2073 6565 6420 746f 2031  random seed to 1
│ │ │ │ +000247c0: 3030 2020 2020 2020 2020 2020 2020 2020  00              
│ │ │ │ +000247d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000247e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000247f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024810: 0a7c 6f31 203d 2031 3030 2020 2020 2020  .|o1 = 100      
│ │ │ │  00024820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024830: 7c0a 7c6f 3120 3d20 3130 3020 2020 2020  |.|o1 = 100     
│ │ │ │ -00024840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024860: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00024870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000248a0: 7c69 3220 3a20 5320 3d20 5a5a 2f31 3031  |i2 : S = ZZ/101
│ │ │ │ -000248b0: 5b61 2c62 2c63 2c64 5d3b 2020 2020 2020  [a,b,c,d];      
│ │ │ │ -000248c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -000248e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000248f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00024910: 3320 3a20 6620 3d20 6d61 7028 535e 312c  3 : f = map(S^1,
│ │ │ │ -00024920: 2053 5e34 2c20 2869 2c6a 2920 2d3e 2053   S^4, (i,j) -> S
│ │ │ │ -00024930: 5f6a 5e33 2920 2020 2020 2020 2020 2020  _j^3)           
│ │ │ │ -00024940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00024950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024970: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -00024980: 3d20 7c20 6133 2062 3320 6333 2064 3320  = | a3 b3 c3 d3 
│ │ │ │ -00024990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00024830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024840: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00024850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00024880: 6932 203a 2053 203d 205a 5a2f 3130 315b  i2 : S = ZZ/101[
│ │ │ │ +00024890: 612c 622c 632c 645d 3b20 2020 2020 2020  a,b,c,d];       
│ │ │ │ +000248a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000248b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000248c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000248d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000248e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +000248f0: 203a 2066 203d 206d 6170 2853 5e31 2c20   : f = map(S^1, 
│ │ │ │ +00024900: 535e 342c 2028 692c 6a29 202d 3e20 535f  S^4, (i,j) -> S_
│ │ │ │ +00024910: 6a5e 3329 2020 2020 2020 2020 2020 2020  j^3)            
│ │ │ │ +00024920: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00024930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024950: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ +00024960: 207c 2061 3320 6233 2063 3320 6433 207c   | a3 b3 c3 d3 |
│ │ │ │ +00024970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024990: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000249a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000249c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000249f0: 2020 2020 2020 2020 3120 2020 2020 2034          1      4
│ │ │ │ -00024a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024a20: 0a7c 6f33 203a 204d 6174 7269 7820 5320  .|o3 : Matrix S 
│ │ │ │ -00024a30: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ -00024a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a50: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00024a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00024a90: 6934 203a 2052 203d 2053 2f69 6465 616c  i4 : R = S/ideal
│ │ │ │ -00024aa0: 2066 3b20 2020 2020 2020 2020 2020 2020   f;             
│ │ │ │ -00024ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00024ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -00024b00: 203a 204d 203d 2052 5e31 2f69 6465 616c   : M = R^1/ideal
│ │ │ │ -00024b10: 2261 6232 2b63 6432 223b 2020 2020 2020  "ab2+cd2";      
│ │ │ │ -00024b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00024b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024b60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ -00024b70: 2062 6574 7469 2028 4620 3d20 6672 6565   betti (F = free
│ │ │ │ -00024b80: 5265 736f 6c75 7469 6f6e 284d 2c20 4c65  Resolution(M, Le
│ │ │ │ -00024b90: 6e67 7468 4c69 6d69 7420 3d3e 2035 2929  ngthLimit => 5))
│ │ │ │ -00024ba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00024be0: 2020 2020 2020 3020 3120 3220 2033 2020        0 1 2  3  
│ │ │ │ -00024bf0: 3420 2035 2020 2020 2020 2020 2020 2020  4  5            
│ │ │ │ -00024c00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00024c10: 7c6f 3620 3d20 746f 7461 6c3a 2031 2031  |o6 = total: 1 1
│ │ │ │ -00024c20: 2035 2031 3620 3335 2036 3420 2020 2020   5 16 35 64     
│ │ │ │ -00024c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00024c50: 2030 3a20 3120 2e20 2e20 202e 2020 2e20   0: 1 . .  .  . 
│ │ │ │ -00024c60: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -00024c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00024c80: 2020 2020 2020 2020 313a 202e 202e 202e          1: . . .
│ │ │ │ -00024c90: 2020 2e20 202e 2020 2e20 2020 2020 2020    .  .  .       
│ │ │ │ -00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024cb0: 2020 207c 0a7c 2020 2020 2020 2020 2032     |.|         2
│ │ │ │ -00024cc0: 3a20 2e20 3120 2e20 202e 2020 2e20 202e  : . 1 .  .  .  .
│ │ │ │ -00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00024cf0: 2020 2020 2020 333a 202e 202e 2031 2020        3: . . 1  
│ │ │ │ -00024d00: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ -00024d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d20: 207c 0a7c 2020 2020 2020 2020 2034 3a20   |.|         4: 
│ │ │ │ -00024d30: 2e20 2e20 3320 2038 2020 3520 202e 2020  . . 3  8  5  .  
│ │ │ │ -00024d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00024d60: 2020 2020 353a 202e 202e 2031 2020 3820      5: . . 1  8 
│ │ │ │ -00024d70: 3235 2033 3220 2020 2020 2020 2020 2020  25 32           
│ │ │ │ -00024d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024d90: 0a7c 2020 2020 2020 2020 2036 3a20 2e20  .|         6: . 
│ │ │ │ -00024da0: 2e20 2e20 202e 2020 3520 3332 2020 2020  . .  .  5 32    
│ │ │ │ +000249b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000249c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000249d0: 2020 2020 2020 2031 2020 2020 2020 3420         1      4 
│ │ │ │ +000249e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000249f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00024a00: 7c6f 3320 3a20 4d61 7472 6978 2053 2020  |o3 : Matrix S  
│ │ │ │ +00024a10: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ +00024a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00024a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00024a70: 3420 3a20 5220 3d20 532f 6964 6561 6c20  4 : R = S/ideal 
│ │ │ │ +00024a80: 663b 2020 2020 2020 2020 2020 2020 2020  f;              
│ │ │ │ +00024a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +00024ae0: 3a20 4d20 3d20 525e 312f 6964 6561 6c22  : M = R^1/ideal"
│ │ │ │ +00024af0: 6162 322b 6364 3222 3b20 2020 2020 2020  ab2+cd2";       
│ │ │ │ +00024b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00024b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024b40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +00024b50: 6265 7474 6920 2846 203d 2066 7265 6552  betti (F = freeR
│ │ │ │ +00024b60: 6573 6f6c 7574 696f 6e28 4d2c 204c 656e  esolution(M, Len
│ │ │ │ +00024b70: 6774 684c 696d 6974 203d 3e20 3529 297c  gthLimit => 5))|
│ │ │ │ +00024b80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024bb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00024bc0: 2020 2020 2030 2031 2032 2020 3320 2034       0 1 2  3  4
│ │ │ │ +00024bd0: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │ +00024be0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00024bf0: 6f36 203d 2074 6f74 616c 3a20 3120 3120  o6 = total: 1 1 
│ │ │ │ +00024c00: 3520 3136 2033 3520 3634 2020 2020 2020  5 16 35 64      
│ │ │ │ +00024c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00024c30: 303a 2031 202e 202e 2020 2e20 202e 2020  0: 1 . .  .  .  
│ │ │ │ +00024c40: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +00024c50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00024c60: 2020 2020 2020 2031 3a20 2e20 2e20 2e20         1: . . . 
│ │ │ │ +00024c70: 202e 2020 2e20 202e 2020 2020 2020 2020   .  .  .        
│ │ │ │ +00024c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c90: 2020 7c0a 7c20 2020 2020 2020 2020 323a    |.|         2:
│ │ │ │ +00024ca0: 202e 2031 202e 2020 2e20 202e 2020 2e20   . 1 .  .  .  . 
│ │ │ │ +00024cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024cc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00024cd0: 2020 2020 2033 3a20 2e20 2e20 3120 202e       3: . . 1  .
│ │ │ │ +00024ce0: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +00024cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d00: 7c0a 7c20 2020 2020 2020 2020 343a 202e  |.|         4: .
│ │ │ │ +00024d10: 202e 2033 2020 3820 2035 2020 2e20 2020   . 3  8  5  .   
│ │ │ │ +00024d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00024d40: 2020 2035 3a20 2e20 2e20 3120 2038 2032     5: . . 1  8 2
│ │ │ │ +00024d50: 3520 3332 2020 2020 2020 2020 2020 2020  5 32            
│ │ │ │ +00024d60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00024d70: 7c20 2020 2020 2020 2020 363a 202e 202e  |         6: . .
│ │ │ │ +00024d80: 202e 2020 2e20 2035 2033 3220 2020 2020   .  .  5 32     
│ │ │ │ +00024d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024da0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024df0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00024e00: 6f36 203a 2042 6574 7469 5461 6c6c 7920  o6 : BettiTally 
│ │ │ │ -00024e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00024e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -00024e70: 203a 2045 203d 2045 7874 4d6f 6475 6c65   : E = ExtModule
│ │ │ │ -00024e80: 4461 7461 204d 3b20 2020 2020 2020 2020  Data M;         
│ │ │ │ -00024e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00024eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024ed0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -00024ee0: 2045 5f32 2020 2020 2020 2020 2020 2020   E_2            
│ │ │ │ -00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00024de0: 3620 3a20 4265 7474 6954 616c 6c79 2020  6 : BettiTally  
│ │ │ │ +00024df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00024e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +00024e50: 3a20 4520 3d20 4578 744d 6f64 756c 6544  : E = ExtModuleD
│ │ │ │ +00024e60: 6174 6120 4d3b 2020 2020 2020 2020 2020  ata M;          
│ │ │ │ +00024e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e80: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00024e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024eb0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ +00024ec0: 455f 3220 2020 2020 2020 2020 2020 2020  E_2             
│ │ │ │ +00024ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00024ef0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00024f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00024f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024f20: 2020 2020 2020 7c0a 7c6f 3820 3d20 3220        |.|o8 = 2 
│ │ │ │  00024f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f40: 2020 2020 2020 207c 0a7c 6f38 203d 2032         |.|o8 = 2
│ │ │ │ -00024f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00024f80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00024f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00024fb0: 2d2d 2d2d 2d2b 0a7c 6939 203a 2045 5f33  -----+.|i9 : E_3
│ │ │ │ -00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024f50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00024f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00024f90: 2d2d 2d2d 2b0a 7c69 3920 3a20 455f 3320  ----+.|i9 : E_3 
│ │ │ │ +00024fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00024fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025000: 2020 7c0a 7c6f 3920 3d20 3120 2020 2020    |.|o9 = 1     
│ │ │ │  00025010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025020: 2020 207c 0a7c 6f39 203d 2031 2020 2020     |.|o9 = 1    
│ │ │ │ -00025030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025050: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00025020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025030: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00025040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025090: 2d2b 0a7c 6931 3020 3a20 7220 3d20 6d61  -+.|i10 : r = ma
│ │ │ │ -000250a0: 7828 322a 455f 322c 322a 455f 332b 3129  x(2*E_2,2*E_3+1)
│ │ │ │ +00025070: 2b0a 7c69 3130 203a 2072 203d 206d 6178  +.|i10 : r = max
│ │ │ │ +00025080: 2832 2a45 5f32 2c32 2a45 5f33 2b31 2920  (2*E_2,2*E_3+1) 
│ │ │ │ +00025090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000250a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000250b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000250d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025100: 0a7c 6f31 3020 3d20 3420 2020 2020 2020  .|o10 = 4       
│ │ │ │ -00025110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025130: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00025140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00025170: 6931 3120 3a20 4572 203d 2045 7874 4d6f  i11 : Er = ExtMo
│ │ │ │ -00025180: 6475 6c65 4461 7461 2063 6f6b 6572 2046  duleData coker F
│ │ │ │ -00025190: 2e64 645f 723b 2020 2020 2020 2020 2020  .dd_r;          
│ │ │ │ -000251a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000251c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000251d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000251e0: 3220 3a20 7265 6775 6c61 7269 7479 2045  2 : regularity E
│ │ │ │ -000251f0: 725f 3020 2020 2020 2020 2020 2020 2020  r_0             
│ │ │ │ +000250c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000250d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000250e0: 7c6f 3130 203d 2034 2020 2020 2020 2020  |o10 = 4        
│ │ │ │ +000250f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025110: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00025120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00025150: 3131 203a 2045 7220 3d20 4578 744d 6f64  11 : Er = ExtMod
│ │ │ │ +00025160: 756c 6544 6174 6120 636f 6b65 7220 462e  uleData coker F.
│ │ │ │ +00025170: 6464 5f72 3b20 2020 2020 2020 2020 2020  dd_r;           
│ │ │ │ +00025180: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00025190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000251a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ +000251c0: 203a 2072 6567 756c 6172 6974 7920 4572   : regularity Er
│ │ │ │ +000251d0: 5f30 2020 2020 2020 2020 2020 2020 2020  _0              
│ │ │ │ +000251e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000251f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00025200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025210: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00025220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025240: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ -00025250: 3d20 3020 2020 2020 2020 2020 2020 2020  = 0             
│ │ │ │ -00025260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025280: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00025290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000252a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000252b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ -000252c0: 7265 6775 6c61 7269 7479 2045 725f 3120  regularity Er_1 
│ │ │ │ +00025210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025220: 2020 2020 2020 2020 7c0a 7c6f 3132 203d          |.|o12 =
│ │ │ │ +00025230: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00025240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00025260: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00025270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025290: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2072  ------+.|i13 : r
│ │ │ │ +000252a0: 6567 756c 6172 6974 7920 4572 5f31 2020  egularity Er_1  
│ │ │ │ +000252b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000252c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000252d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000252f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00025300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000252e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025300: 2020 2020 7c0a 7c6f 3133 203d 2030 2020      |.|o13 = 0  
│ │ │ │  00025310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025320: 2020 2020 207c 0a7c 6f31 3320 3d20 3020       |.|o13 = 0 
│ │ │ │ -00025330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025350: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00025320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025330: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025390: 2d2d 2d2b 0a7c 6931 3420 3a20 7265 6775  ---+.|i14 : regu
│ │ │ │ -000253a0: 6c61 7269 7479 2065 7665 6e45 7874 4d6f  larity evenExtMo
│ │ │ │ -000253b0: 6475 6c65 2863 6f6b 6572 2046 2e64 645f  dule(coker F.dd_
│ │ │ │ -000253c0: 2872 2d31 2929 2020 2020 7c0a 7c20 2020  (r-1))    |.|   
│ │ │ │ +00025370: 2d2d 2b0a 7c69 3134 203a 2072 6567 756c  --+.|i14 : regul
│ │ │ │ +00025380: 6172 6974 7920 6576 656e 4578 744d 6f64  arity evenExtMod
│ │ │ │ +00025390: 756c 6528 636f 6b65 7220 462e 6464 5f28  ule(coker F.dd_(
│ │ │ │ +000253a0: 722d 3129 2920 2020 207c 0a7c 2020 2020  r-1))    |.|    
│ │ │ │ +000253b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000253c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000253e0: 7c0a 7c6f 3134 203d 2031 2020 2020 2020  |.|o14 = 1      
│ │ │ │  000253f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025400: 207c 0a7c 6f31 3420 3d20 3120 2020 2020   |.|o14 = 1     
│ │ │ │ -00025410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025430: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -00025440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00025470: 0a7c 6931 3520 3a20 6666 203d 2066 2a72  .|i15 : ff = f*r
│ │ │ │ -00025480: 616e 646f 6d28 736f 7572 6365 2066 2c20  andom(source f, 
│ │ │ │ -00025490: 736f 7572 6365 2066 293b 2020 2020 2020  source f);      
│ │ │ │ -000254a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000254b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000254e0: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -000254f0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │ -00025500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025510: 2020 2020 7c0a 7c6f 3135 203a 204d 6174      |.|o15 : Mat
│ │ │ │ -00025520: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ -00025530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00025400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025410: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00025420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00025450: 7c69 3135 203a 2066 6620 3d20 662a 7261  |i15 : ff = f*ra
│ │ │ │ +00025460: 6e64 6f6d 2873 6f75 7263 6520 662c 2073  ndom(source f, s
│ │ │ │ +00025470: 6f75 7263 6520 6629 3b20 2020 2020 2020  ource f);       
│ │ │ │ +00025480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00025490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000254a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000254b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000254c0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +000254d0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ +000254e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000254f0: 2020 207c 0a7c 6f31 3520 3a20 4d61 7472     |.|o15 : Matr
│ │ │ │ +00025500: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ +00025510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025520: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00025530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025580: 2d2d 2b0a 7c69 3136 203a 206d 6174 7269  --+.|i16 : matri
│ │ │ │ -00025590: 7846 6163 746f 7269 7a61 7469 6f6e 2866  xFactorization(f
│ │ │ │ -000255a0: 662c 2063 6f6b 6572 2046 2e64 645f 2872  f, coker F.dd_(r
│ │ │ │ -000255b0: 2b31 2929 3b20 2020 207c 0a2b 2d2d 2d2d  +1));    |.+----
│ │ │ │ -000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000255d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000255e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000255f0: 2b0a 0a54 6869 7320 7375 6363 6565 6473  +..This succeeds
│ │ │ │ -00025600: 2c20 6275 7420 7765 2063 6f75 6c64 2067  , but we could g
│ │ │ │ -00025610: 6574 2061 6e20 6572 726f 7220 6672 6f6d  et an error from
│ │ │ │ -00025620: 0a0a 6d61 7472 6978 4661 6374 6f72 697a  ..matrixFactoriz
│ │ │ │ -00025630: 6174 696f 6e28 6666 2c20 636f 6b65 7220  ation(ff, coker 
│ │ │ │ -00025640: 462e 6464 5f72 290a 0a69 6620 6f6e 6520  F.dd_r)..if one 
│ │ │ │ -00025650: 6f66 2074 6865 2043 4920 6f70 6572 6174  of the CI operat
│ │ │ │ -00025660: 6f72 7320 7765 7265 206e 6f74 2073 7572  ors were not sur
│ │ │ │ -00025670: 6a65 6374 6976 652e 0a0a 4361 7665 6174  jective...Caveat
│ │ │ │ -00025680: 0a3d 3d3d 3d3d 3d0a 0a45 7874 4d6f 6475  .======..ExtModu
│ │ │ │ -00025690: 6c65 2063 7265 6174 6573 2061 2072 696e  le creates a rin
│ │ │ │ -000256a0: 6720 696e 7369 6465 2074 6865 2073 6372  g inside the scr
│ │ │ │ -000256b0: 6970 742c 2073 6f20 6966 2069 7427 7320  ipt, so if it's 
│ │ │ │ -000256c0: 7275 6e20 7477 6963 6520 796f 7520 6765  run twice you ge
│ │ │ │ -000256d0: 740a 6d6f 6475 6c65 7320 6f76 6572 2064  t.modules over d
│ │ │ │ -000256e0: 6966 6665 7265 6e74 2072 696e 6773 2e20  ifferent rings. 
│ │ │ │ -000256f0: 5468 6973 2073 686f 756c 6420 6265 2063  This should be c
│ │ │ │ -00025700: 6861 6e67 6564 2e0a 0a53 6565 2061 6c73  hanged...See als
│ │ │ │ -00025710: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -00025720: 2a6e 6f74 6520 4578 744d 6f64 756c 653a  *note ExtModule:
│ │ │ │ -00025730: 2045 7874 4d6f 6475 6c65 2c20 2d2d 2045   ExtModule, -- E
│ │ │ │ -00025740: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -00025750: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -00025760: 6563 7469 6f6e 2061 730a 2020 2020 6d6f  ection as.    mo
│ │ │ │ -00025770: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ -00025780: 7261 746f 7220 7269 6e67 0a20 202a 202a  rator ring.  * *
│ │ │ │ -00025790: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475  note evenExtModu
│ │ │ │ -000257a0: 6c65 3a20 6576 656e 4578 744d 6f64 756c  le: evenExtModul
│ │ │ │ -000257b0: 652c 202d 2d20 6576 656e 2070 6172 7420  e, -- even part 
│ │ │ │ -000257c0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76  of Ext^*(M,k) ov
│ │ │ │ -000257d0: 6572 2061 0a20 2020 2063 6f6d 706c 6574  er a.    complet
│ │ │ │ -000257e0: 6520 696e 7465 7273 6563 7469 6f6e 2061  e intersection a
│ │ │ │ -000257f0: 7320 6d6f 6475 6c65 206f 7665 7220 4349  s module over CI
│ │ │ │ -00025800: 206f 7065 7261 746f 7220 7269 6e67 0a20   operator ring. 
│ │ │ │ -00025810: 202a 202a 6e6f 7465 206f 6464 4578 744d   * *note oddExtM
│ │ │ │ -00025820: 6f64 756c 653a 206f 6464 4578 744d 6f64  odule: oddExtMod
│ │ │ │ -00025830: 756c 652c 202d 2d20 6f64 6420 7061 7274  ule, -- odd part
│ │ │ │ -00025840: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ -00025850: 7665 7220 6120 636f 6d70 6c65 7465 0a20  ver a complete. 
│ │ │ │ -00025860: 2020 2069 6e74 6572 7365 6374 696f 6e20     intersection 
│ │ │ │ -00025870: 6173 206d 6f64 756c 6520 6f76 6572 2043  as module over C
│ │ │ │ -00025880: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ -00025890: 0a57 6179 7320 746f 2075 7365 2045 7874  .Ways to use Ext
│ │ │ │ -000258a0: 4d6f 6475 6c65 4461 7461 3a0a 3d3d 3d3d  ModuleData:.====
│ │ │ │ -000258b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000258c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 7874  ======..  * "Ext
│ │ │ │ -000258d0: 4d6f 6475 6c65 4461 7461 284d 6f64 756c  ModuleData(Modul
│ │ │ │ -000258e0: 6529 220a 0a46 6f72 2074 6865 2070 726f  e)"..For the pro
│ │ │ │ -000258f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -00025900: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -00025910: 6f62 6a65 6374 202a 6e6f 7465 2045 7874  object *note Ext
│ │ │ │ -00025920: 4d6f 6475 6c65 4461 7461 3a20 4578 744d  ModuleData: ExtM
│ │ │ │ -00025930: 6f64 756c 6544 6174 612c 2069 7320 6120  oduleData, is a 
│ │ │ │ -00025940: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e  *note method fun
│ │ │ │ -00025950: 6374 696f 6e3a 0a28 4d61 6361 756c 6179  ction:.(Macaulay
│ │ │ │ -00025960: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -00025970: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +00025560: 2d2b 0a7c 6931 3620 3a20 6d61 7472 6978  -+.|i16 : matrix
│ │ │ │ +00025570: 4661 6374 6f72 697a 6174 696f 6e28 6666  Factorization(ff
│ │ │ │ +00025580: 2c20 636f 6b65 7220 462e 6464 5f28 722b  , coker F.dd_(r+
│ │ │ │ +00025590: 3129 293b 2020 2020 7c0a 2b2d 2d2d 2d2d  1));    |.+-----
│ │ │ │ +000255a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000255b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000255d0: 0a0a 5468 6973 2073 7563 6365 6564 732c  ..This succeeds,
│ │ │ │ +000255e0: 2062 7574 2077 6520 636f 756c 6420 6765   but we could ge
│ │ │ │ +000255f0: 7420 616e 2065 7272 6f72 2066 726f 6d0a  t an error from.
│ │ │ │ +00025600: 0a6d 6174 7269 7846 6163 746f 7269 7a61  .matrixFactoriza
│ │ │ │ +00025610: 7469 6f6e 2866 662c 2063 6f6b 6572 2046  tion(ff, coker F
│ │ │ │ +00025620: 2e64 645f 7229 0a0a 6966 206f 6e65 206f  .dd_r)..if one o
│ │ │ │ +00025630: 6620 7468 6520 4349 206f 7065 7261 746f  f the CI operato
│ │ │ │ +00025640: 7273 2077 6572 6520 6e6f 7420 7375 726a  rs were not surj
│ │ │ │ +00025650: 6563 7469 7665 2e0a 0a43 6176 6561 740a  ective...Caveat.
│ │ │ │ +00025660: 3d3d 3d3d 3d3d 0a0a 4578 744d 6f64 756c  ======..ExtModul
│ │ │ │ +00025670: 6520 6372 6561 7465 7320 6120 7269 6e67  e creates a ring
│ │ │ │ +00025680: 2069 6e73 6964 6520 7468 6520 7363 7269   inside the scri
│ │ │ │ +00025690: 7074 2c20 736f 2069 6620 6974 2773 2072  pt, so if it's r
│ │ │ │ +000256a0: 756e 2074 7769 6365 2079 6f75 2067 6574  un twice you get
│ │ │ │ +000256b0: 0a6d 6f64 756c 6573 206f 7665 7220 6469  .modules over di
│ │ │ │ +000256c0: 6666 6572 656e 7420 7269 6e67 732e 2054  fferent rings. T
│ │ │ │ +000256d0: 6869 7320 7368 6f75 6c64 2062 6520 6368  his should be ch
│ │ │ │ +000256e0: 616e 6765 642e 0a0a 5365 6520 616c 736f  anged...See also
│ │ │ │ +000256f0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +00025700: 6e6f 7465 2045 7874 4d6f 6475 6c65 3a20  note ExtModule: 
│ │ │ │ +00025710: 4578 744d 6f64 756c 652c 202d 2d20 4578  ExtModule, -- Ex
│ │ │ │ +00025720: 745e 2a28 4d2c 6b29 206f 7665 7220 6120  t^*(M,k) over a 
│ │ │ │ +00025730: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +00025740: 6374 696f 6e20 6173 0a20 2020 206d 6f64  ction as.    mod
│ │ │ │ +00025750: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ +00025760: 6174 6f72 2072 696e 670a 2020 2a20 2a6e  ator ring.  * *n
│ │ │ │ +00025770: 6f74 6520 6576 656e 4578 744d 6f64 756c  ote evenExtModul
│ │ │ │ +00025780: 653a 2065 7665 6e45 7874 4d6f 6475 6c65  e: evenExtModule
│ │ │ │ +00025790: 2c20 2d2d 2065 7665 6e20 7061 7274 206f  , -- even part o
│ │ │ │ +000257a0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ +000257b0: 7220 610a 2020 2020 636f 6d70 6c65 7465  r a.    complete
│ │ │ │ +000257c0: 2069 6e74 6572 7365 6374 696f 6e20 6173   intersection as
│ │ │ │ +000257d0: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ +000257e0: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ +000257f0: 2a20 2a6e 6f74 6520 6f64 6445 7874 4d6f  * *note oddExtMo
│ │ │ │ +00025800: 6475 6c65 3a20 6f64 6445 7874 4d6f 6475  dule: oddExtModu
│ │ │ │ +00025810: 6c65 2c20 2d2d 206f 6464 2070 6172 7420  le, -- odd part 
│ │ │ │ +00025820: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76  of Ext^*(M,k) ov
│ │ │ │ +00025830: 6572 2061 2063 6f6d 706c 6574 650a 2020  er a complete.  
│ │ │ │ +00025840: 2020 696e 7465 7273 6563 7469 6f6e 2061    intersection a
│ │ │ │ +00025850: 7320 6d6f 6475 6c65 206f 7665 7220 4349  s module over CI
│ │ │ │ +00025860: 206f 7065 7261 746f 7220 7269 6e67 0a0a   operator ring..
│ │ │ │ +00025870: 5761 7973 2074 6f20 7573 6520 4578 744d  Ways to use ExtM
│ │ │ │ +00025880: 6f64 756c 6544 6174 613a 0a3d 3d3d 3d3d  oduleData:.=====
│ │ │ │ +00025890: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000258a0: 3d3d 3d3d 3d0a 0a20 202a 2022 4578 744d  =====..  * "ExtM
│ │ │ │ +000258b0: 6f64 756c 6544 6174 6128 4d6f 6475 6c65  oduleData(Module
│ │ │ │ +000258c0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +000258d0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +000258e0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +000258f0: 626a 6563 7420 2a6e 6f74 6520 4578 744d  bject *note ExtM
│ │ │ │ +00025900: 6f64 756c 6544 6174 613a 2045 7874 4d6f  oduleData: ExtMo
│ │ │ │ +00025910: 6475 6c65 4461 7461 2c20 6973 2061 202a  duleData, is a *
│ │ │ │ +00025920: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ +00025930: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ +00025940: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +00025950: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ +00025960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000259a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000259b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000259c0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -000259d0: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -000259e0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -000259f0: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00025a00: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -00025a10: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ -00025a20: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ -00025a30: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -00025a40: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ -00025a50: 323a 3334 3433 3a30 2e0a 1f0a 4669 6c65  2:3443:0....File
│ │ │ │ -00025a60: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ -00025a70: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -00025a80: 732e 696e 666f 2c20 4e6f 6465 3a20 6578  s.info, Node: ex
│ │ │ │ -00025a90: 7456 7343 6f68 6f6d 6f6c 6f67 792c 204e  tVsCohomology, N
│ │ │ │ -00025aa0: 6578 743a 2066 696e 6974 6542 6574 7469  ext: finiteBetti
│ │ │ │ -00025ab0: 4e75 6d62 6572 732c 2050 7265 763a 2045  Numbers, Prev: E
│ │ │ │ -00025ac0: 7874 4d6f 6475 6c65 4461 7461 2c20 5570  xtModuleData, Up
│ │ │ │ -00025ad0: 3a20 546f 700a 0a65 7874 5673 436f 686f  : Top..extVsCoho
│ │ │ │ -00025ae0: 6d6f 6c6f 6779 202d 2d20 636f 6d70 6172  mology -- compar
│ │ │ │ -00025af0: 6573 2045 7874 5f53 284d 2c6b 2920 6173  es Ext_S(M,k) as
│ │ │ │ -00025b00: 2065 7874 6572 696f 7220 6d6f 6475 6c65   exterior module
│ │ │ │ -00025b10: 2077 6974 6820 636f 6820 7461 626c 6520   with coh table 
│ │ │ │ -00025b20: 6f66 2073 6865 6166 2045 7874 5f52 284d  of sheaf Ext_R(M
│ │ │ │ -00025b30: 2c6b 290a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ,k).************
│ │ │ │ +000259a0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +000259b0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +000259c0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +000259d0: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +000259e0: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ +000259f0: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ +00025a00: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ +00025a10: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +00025a20: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
│ │ │ │ +00025a30: 3a33 3434 333a 302e 0a1f 0a46 696c 653a  :3443:0....File:
│ │ │ │ +00025a40: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +00025a50: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +00025a60: 2e69 6e66 6f2c 204e 6f64 653a 2065 7874  .info, Node: ext
│ │ │ │ +00025a70: 5673 436f 686f 6d6f 6c6f 6779 2c20 4e65  VsCohomology, Ne
│ │ │ │ +00025a80: 7874 3a20 6669 6e69 7465 4265 7474 694e  xt: finiteBettiN
│ │ │ │ +00025a90: 756d 6265 7273 2c20 5072 6576 3a20 4578  umbers, Prev: Ex
│ │ │ │ +00025aa0: 744d 6f64 756c 6544 6174 612c 2055 703a  tModuleData, Up:
│ │ │ │ +00025ab0: 2054 6f70 0a0a 6578 7456 7343 6f68 6f6d   Top..extVsCohom
│ │ │ │ +00025ac0: 6f6c 6f67 7920 2d2d 2063 6f6d 7061 7265  ology -- compare
│ │ │ │ +00025ad0: 7320 4578 745f 5328 4d2c 6b29 2061 7320  s Ext_S(M,k) as 
│ │ │ │ +00025ae0: 6578 7465 7269 6f72 206d 6f64 756c 6520  exterior module 
│ │ │ │ +00025af0: 7769 7468 2063 6f68 2074 6162 6c65 206f  with coh table o
│ │ │ │ +00025b00: 6620 7368 6561 6620 4578 745f 5228 4d2c  f sheaf Ext_R(M,
│ │ │ │ +00025b10: 6b29 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  k).*************
│ │ │ │ +00025b20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00025b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00025b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00025b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00025b70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00025b80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00025b90: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -00025ba0: 2020 2020 2020 2845 2c54 2920 3d20 6578        (E,T) = ex
│ │ │ │ -00025bb0: 7456 7343 6f68 6f6d 6f6c 6f67 7928 6666  tVsCohomology(ff
│ │ │ │ -00025bc0: 2c4e 290a 2020 2a20 496e 7075 7473 3a0a  ,N).  * Inputs:.
│ │ │ │ -00025bd0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e        * ff, a *n
│ │ │ │ -00025be0: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -00025bf0: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -00025c00: 2c2c 2072 6567 756c 6172 2073 6571 7565  ,, regular seque
│ │ │ │ -00025c10: 6e63 6520 696e 2061 0a20 2020 2020 2020  nce in a.       
│ │ │ │ -00025c20: 2072 6567 756c 6172 2072 696e 6720 530a   regular ring S.
│ │ │ │ -00025c30: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f        * N, a *no
│ │ │ │ -00025c40: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ -00025c50: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ -00025c60: 2c20 6772 6164 6564 206d 6f64 756c 6520  , graded module 
│ │ │ │ -00025c70: 6f76 6572 2052 203d 0a20 2020 2020 2020  over R =.       
│ │ │ │ -00025c80: 2053 2f69 6465 616c 2866 6629 2028 7573   S/ideal(ff) (us
│ │ │ │ -00025c90: 7561 6c6c 7920 6120 6869 6768 2073 797a  ually a high syz
│ │ │ │ -00025ca0: 7967 7929 0a20 202a 204f 7574 7075 7473  ygy).  * Outputs
│ │ │ │ -00025cb0: 3a0a 2020 2020 2020 2a20 452c 2061 202a  :.      * E, a *
│ │ │ │ -00025cc0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00025cd0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00025ce0: 652c 2c20 0a20 2020 2020 202a 2054 2c20  e,, .      * T, 
│ │ │ │ -00025cf0: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ -00025d00: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
│ │ │ │ -00025d10: 6475 6c65 2c2c 2045 7874 2061 6e64 2054  dule,, Ext and T
│ │ │ │ -00025d20: 6f72 2061 7320 6578 7465 7269 6f72 0a20  or as exterior. 
│ │ │ │ -00025d30: 2020 2020 2020 206d 6f64 756c 6573 0a0a         modules..
│ │ │ │ -00025d40: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00025d50: 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e 2061  =======..Given a
│ │ │ │ -00025d60: 206d 6174 7269 7820 6666 2063 6f6e 7461   matrix ff conta
│ │ │ │ -00025d70: 696e 696e 6720 6120 7265 6775 6c61 7220  ining a regular 
│ │ │ │ -00025d80: 7365 7175 656e 6365 2069 6e20 6120 706f  sequence in a po
│ │ │ │ -00025d90: 6c79 6e6f 6d69 616c 2072 696e 6720 5320  lynomial ring S 
│ │ │ │ -00025da0: 6f76 6572 206b 2c0a 7365 7420 5220 3d20  over k,.set R = 
│ │ │ │ -00025db0: 532f 2869 6465 616c 2066 6629 2e20 4966  S/(ideal ff). If
│ │ │ │ -00025dc0: 204e 2069 7320 6120 6772 6164 6564 2052   N is a graded R
│ │ │ │ -00025dd0: 2d6d 6f64 756c 652c 2061 6e64 204d 2069  -module, and M i
│ │ │ │ -00025de0: 7320 7468 6520 6d6f 6475 6c65 204e 2072  s the module N r
│ │ │ │ -00025df0: 6567 6172 6465 640a 6173 2061 6e20 532d  egarded.as an S-
│ │ │ │ -00025e00: 6d6f 6475 6c65 2c20 7468 6520 7363 7269  module, the scri
│ │ │ │ -00025e10: 7074 2072 6574 7572 6e73 2045 203d 2045  pt returns E = E
│ │ │ │ -00025e20: 7874 5f53 284d 2c6b 2920 616e 6420 5420  xt_S(M,k) and T 
│ │ │ │ -00025e30: 3d20 546f 725e 5328 4d2c 6b29 2061 7320  = Tor^S(M,k) as 
│ │ │ │ -00025e40: 6d6f 6475 6c65 730a 6f76 6572 2061 6e20  modules.over an 
│ │ │ │ -00025e50: 6578 7465 7269 6f72 2061 6c67 6562 7261  exterior algebra
│ │ │ │ -00025e60: 2e0a 0a54 6865 2073 6372 6970 7420 7072  ...The script pr
│ │ │ │ -00025e70: 696e 7473 2074 6865 2054 6174 6520 7265  ints the Tate re
│ │ │ │ -00025e80: 736f 6c75 7469 6f6e 206f 6620 453b 2061  solution of E; a
│ │ │ │ -00025e90: 6e64 2074 6865 2063 6f68 6f6d 6f6c 6f67  nd the cohomolog
│ │ │ │ -00025ea0: 7920 7461 626c 6520 6f66 2074 6865 0a73  y table of the.s
│ │ │ │ -00025eb0: 6865 6166 2061 7373 6f63 6961 7465 6420  heaf associated 
│ │ │ │ -00025ec0: 746f 2045 7874 5f52 284e 2c6b 2920 6f76  to Ext_R(N,k) ov
│ │ │ │ -00025ed0: 6572 2074 6865 2072 696e 6720 6f66 2043  er the ring of C
│ │ │ │ -00025ee0: 4920 6f70 6572 6174 6f72 732c 2077 6869  I operators, whi
│ │ │ │ -00025ef0: 6368 2069 7320 610a 706f 6c79 6e6f 6d69  ch is a.polynomi
│ │ │ │ -00025f00: 616c 2072 696e 6720 6f76 6572 206b 206f  al ring over k o
│ │ │ │ -00025f10: 6e20 6320 7661 7269 6162 6c65 732e 0a0a  n c variables...
│ │ │ │ -00025f20: 5468 6520 6f75 7470 7574 2063 616e 2062  The output can b
│ │ │ │ -00025f30: 6520 7573 6564 2074 6f20 2873 6f6d 6574  e used to (somet
│ │ │ │ -00025f40: 696d 6573 2920 6368 6563 6b20 7768 6574  imes) check whet
│ │ │ │ -00025f50: 6865 7220 7468 6520 7375 626d 6f64 756c  her the submodul
│ │ │ │ -00025f60: 6520 6f66 2045 7874 5f53 284d 2c6b 290a  e of Ext_S(M,k).
│ │ │ │ -00025f70: 6765 6e65 7261 7465 6420 696e 2064 6567  generated in deg
│ │ │ │ -00025f80: 7265 6520 3020 7370 6c69 7473 2028 6173  ree 0 splits (as
│ │ │ │ -00025f90: 2061 6e20 6578 7465 7269 6f72 206d 6f64   an exterior mod
│ │ │ │ -00025fa0: 756c 650a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ule..+----------
│ │ │ │ +00025b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +00025b70: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00025b80: 2020 2020 2028 452c 5429 203d 2065 7874       (E,T) = ext
│ │ │ │ +00025b90: 5673 436f 686f 6d6f 6c6f 6779 2866 662c  VsCohomology(ff,
│ │ │ │ +00025ba0: 4e29 0a20 202a 2049 6e70 7574 733a 0a20  N).  * Inputs:. 
│ │ │ │ +00025bb0: 2020 2020 202a 2066 662c 2061 202a 6e6f       * ff, a *no
│ │ │ │ +00025bc0: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ +00025bd0: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ +00025be0: 2c20 7265 6775 6c61 7220 7365 7175 656e  , regular sequen
│ │ │ │ +00025bf0: 6365 2069 6e20 610a 2020 2020 2020 2020  ce in a.        
│ │ │ │ +00025c00: 7265 6775 6c61 7220 7269 6e67 2053 0a20  regular ring S. 
│ │ │ │ +00025c10: 2020 2020 202a 204e 2c20 6120 2a6e 6f74       * N, a *not
│ │ │ │ +00025c20: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +00025c30: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +00025c40: 2067 7261 6465 6420 6d6f 6475 6c65 206f   graded module o
│ │ │ │ +00025c50: 7665 7220 5220 3d0a 2020 2020 2020 2020  ver R =.        
│ │ │ │ +00025c60: 532f 6964 6561 6c28 6666 2920 2875 7375  S/ideal(ff) (usu
│ │ │ │ +00025c70: 616c 6c79 2061 2068 6967 6820 7379 7a79  ally a high syzy
│ │ │ │ +00025c80: 6779 290a 2020 2a20 4f75 7470 7574 733a  gy).  * Outputs:
│ │ │ │ +00025c90: 0a20 2020 2020 202a 2045 2c20 6120 2a6e  .      * E, a *n
│ │ │ │ +00025ca0: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +00025cb0: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +00025cc0: 2c2c 200a 2020 2020 2020 2a20 542c 2061  ,, .      * T, a
│ │ │ │ +00025cd0: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ +00025ce0: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ +00025cf0: 756c 652c 2c20 4578 7420 616e 6420 546f  ule,, Ext and To
│ │ │ │ +00025d00: 7220 6173 2065 7874 6572 696f 720a 2020  r as exterior.  
│ │ │ │ +00025d10: 2020 2020 2020 6d6f 6475 6c65 730a 0a44        modules..D
│ │ │ │ +00025d20: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00025d30: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120  ======..Given a 
│ │ │ │ +00025d40: 6d61 7472 6978 2066 6620 636f 6e74 6169  matrix ff contai
│ │ │ │ +00025d50: 6e69 6e67 2061 2072 6567 756c 6172 2073  ning a regular s
│ │ │ │ +00025d60: 6571 7565 6e63 6520 696e 2061 2070 6f6c  equence in a pol
│ │ │ │ +00025d70: 796e 6f6d 6961 6c20 7269 6e67 2053 206f  ynomial ring S o
│ │ │ │ +00025d80: 7665 7220 6b2c 0a73 6574 2052 203d 2053  ver k,.set R = S
│ │ │ │ +00025d90: 2f28 6964 6561 6c20 6666 292e 2049 6620  /(ideal ff). If 
│ │ │ │ +00025da0: 4e20 6973 2061 2067 7261 6465 6420 522d  N is a graded R-
│ │ │ │ +00025db0: 6d6f 6475 6c65 2c20 616e 6420 4d20 6973  module, and M is
│ │ │ │ +00025dc0: 2074 6865 206d 6f64 756c 6520 4e20 7265   the module N re
│ │ │ │ +00025dd0: 6761 7264 6564 0a61 7320 616e 2053 2d6d  garded.as an S-m
│ │ │ │ +00025de0: 6f64 756c 652c 2074 6865 2073 6372 6970  odule, the scrip
│ │ │ │ +00025df0: 7420 7265 7475 726e 7320 4520 3d20 4578  t returns E = Ex
│ │ │ │ +00025e00: 745f 5328 4d2c 6b29 2061 6e64 2054 203d  t_S(M,k) and T =
│ │ │ │ +00025e10: 2054 6f72 5e53 284d 2c6b 2920 6173 206d   Tor^S(M,k) as m
│ │ │ │ +00025e20: 6f64 756c 6573 0a6f 7665 7220 616e 2065  odules.over an e
│ │ │ │ +00025e30: 7874 6572 696f 7220 616c 6765 6272 612e  xterior algebra.
│ │ │ │ +00025e40: 0a0a 5468 6520 7363 7269 7074 2070 7269  ..The script pri
│ │ │ │ +00025e50: 6e74 7320 7468 6520 5461 7465 2072 6573  nts the Tate res
│ │ │ │ +00025e60: 6f6c 7574 696f 6e20 6f66 2045 3b20 616e  olution of E; an
│ │ │ │ +00025e70: 6420 7468 6520 636f 686f 6d6f 6c6f 6779  d the cohomology
│ │ │ │ +00025e80: 2074 6162 6c65 206f 6620 7468 650a 7368   table of the.sh
│ │ │ │ +00025e90: 6561 6620 6173 736f 6369 6174 6564 2074  eaf associated t
│ │ │ │ +00025ea0: 6f20 4578 745f 5228 4e2c 6b29 206f 7665  o Ext_R(N,k) ove
│ │ │ │ +00025eb0: 7220 7468 6520 7269 6e67 206f 6620 4349  r the ring of CI
│ │ │ │ +00025ec0: 206f 7065 7261 746f 7273 2c20 7768 6963   operators, whic
│ │ │ │ +00025ed0: 6820 6973 2061 0a70 6f6c 796e 6f6d 6961  h is a.polynomia
│ │ │ │ +00025ee0: 6c20 7269 6e67 206f 7665 7220 6b20 6f6e  l ring over k on
│ │ │ │ +00025ef0: 2063 2076 6172 6961 626c 6573 2e0a 0a54   c variables...T
│ │ │ │ +00025f00: 6865 206f 7574 7075 7420 6361 6e20 6265  he output can be
│ │ │ │ +00025f10: 2075 7365 6420 746f 2028 736f 6d65 7469   used to (someti
│ │ │ │ +00025f20: 6d65 7329 2063 6865 636b 2077 6865 7468  mes) check wheth
│ │ │ │ +00025f30: 6572 2074 6865 2073 7562 6d6f 6475 6c65  er the submodule
│ │ │ │ +00025f40: 206f 6620 4578 745f 5328 4d2c 6b29 0a67   of Ext_S(M,k).g
│ │ │ │ +00025f50: 656e 6572 6174 6564 2069 6e20 6465 6772  enerated in degr
│ │ │ │ +00025f60: 6565 2030 2073 706c 6974 7320 2861 7320  ee 0 splits (as 
│ │ │ │ +00025f70: 616e 2065 7874 6572 696f 7220 6d6f 6475  an exterior modu
│ │ │ │ +00025f80: 6c65 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  le..+-----------
│ │ │ │ +00025f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00025fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00025fe0: 2d2d 2d2b 0a7c 6931 203a 2053 203d 205a  ---+.|i1 : S = Z
│ │ │ │ -00025ff0: 5a2f 3130 315b 612c 622c 635d 2020 2020  Z/101[a,b,c]    
│ │ │ │ -00026000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025fc0: 2d2d 2b0a 7c69 3120 3a20 5320 3d20 5a5a  --+.|i1 : S = ZZ
│ │ │ │ +00025fd0: 2f31 3031 5b61 2c62 2c63 5d20 2020 2020  /101[a,b,c]     
│ │ │ │ +00025fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026000: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00026010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026040: 2020 7c0a 7c6f 3120 3d20 5320 2020 2020    |.|o1 = S     
│ │ │ │  00026050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026060: 2020 207c 0a7c 6f31 203d 2053 2020 2020     |.|o1 = S    
│ │ │ │ +00026060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026080: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00026090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000260a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000260b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260e0: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e     |.|o1 : Polyn
│ │ │ │ -000260f0: 6f6d 6961 6c52 696e 6720 2020 2020 2020  omialRing       
│ │ │ │ -00026100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026120: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000260c0: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f    |.|o1 : Polyno
│ │ │ │ +000260d0: 6d69 616c 5269 6e67 2020 2020 2020 2020  mialRing        
│ │ │ │ +000260e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000260f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026100: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00026110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026160: 2d2d 2d2b 0a7c 6932 203a 2066 6620 3d20  ---+.|i2 : ff = 
│ │ │ │ -00026170: 6d61 7472 6978 2022 6132 2c62 322c 6332  matrix "a2,b2,c2
│ │ │ │ -00026180: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +00026140: 2d2d 2b0a 7c69 3220 3a20 6666 203d 206d  --+.|i2 : ff = m
│ │ │ │ +00026150: 6174 7269 7820 2261 322c 6232 2c63 3222  atrix "a2,b2,c2"
│ │ │ │ +00026160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026180: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00026190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000261a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000261b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261e0: 2020 207c 0a7c 6f32 203d 207c 2061 3220     |.|o2 = | a2 
│ │ │ │ -000261f0: 6232 2063 3220 7c20 2020 2020 2020 2020  b2 c2 |         
│ │ │ │ -00026200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000261c0: 2020 7c0a 7c6f 3220 3d20 7c20 6132 2062    |.|o2 = | a2 b
│ │ │ │ +000261d0: 3220 6332 207c 2020 2020 2020 2020 2020  2 c2 |          
│ │ │ │ +000261e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000261f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026200: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00026210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026220: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026260: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00026270: 2020 2031 2020 2020 2020 3320 2020 2020     1      3     
│ │ │ │ -00026280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262a0: 2020 207c 0a7c 6f32 203a 204d 6174 7269     |.|o2 : Matri
│ │ │ │ -000262b0: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ -000262c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026240: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00026250: 2020 3120 2020 2020 2033 2020 2020 2020    1      3      
│ │ │ │ +00026260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026280: 2020 7c0a 7c6f 3220 3a20 4d61 7472 6978    |.|o2 : Matrix
│ │ │ │ +00026290: 2053 2020 3c2d 2d20 5320 2020 2020 2020   S  <-- S       
│ │ │ │ +000262a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000262b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000262c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000262d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000262e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000262f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026320: 2d2d 2d2b 0a7c 6933 203a 2052 203d 2053  ---+.|i3 : R = S
│ │ │ │ -00026330: 2f28 6964 6561 6c20 6666 2920 2020 2020  /(ideal ff)     
│ │ │ │ -00026340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026300: 2d2d 2b0a 7c69 3320 3a20 5220 3d20 532f  --+.|i3 : R = S/
│ │ │ │ +00026310: 2869 6465 616c 2066 6629 2020 2020 2020  (ideal ff)      
│ │ │ │ +00026320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026340: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00026350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026360: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026380: 2020 7c0a 7c6f 3320 3d20 5220 2020 2020    |.|o3 = R     
│ │ │ │  00026390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263a0: 2020 207c 0a7c 6f33 203d 2052 2020 2020     |.|o3 = R    
│ │ │ │ +000263a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000263b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000263c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000263d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000263e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000263f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026420: 2020 207c 0a7c 6f33 203a 2051 756f 7469     |.|o3 : Quoti
│ │ │ │ -00026430: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ -00026440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026460: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026400: 2020 7c0a 7c6f 3320 3a20 5175 6f74 6965    |.|o3 : Quotie
│ │ │ │ +00026410: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +00026420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026440: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00026450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000264a0: 2d2d 2d2b 0a7c 6934 203a 204e 203d 2068  ---+.|i4 : N = h
│ │ │ │ -000264b0: 6967 6853 797a 7967 7928 525e 312f 6964  ighSyzygy(R^1/id
│ │ │ │ -000264c0: 6561 6c28 612a 622c 6329 2920 2020 2020  eal(a*b,c))     
│ │ │ │ +00026480: 2d2d 2b0a 7c69 3420 3a20 4e20 3d20 6869  --+.|i4 : N = hi
│ │ │ │ +00026490: 6768 5379 7a79 6779 2852 5e31 2f69 6465  ghSyzygy(R^1/ide
│ │ │ │ +000264a0: 616c 2861 2a62 2c63 2929 2020 2020 2020  al(a*b,c))      
│ │ │ │ +000264b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000264c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000264d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000264e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026520: 2020 207c 0a7c 6f34 203d 2063 6f6b 6572     |.|o4 = coker
│ │ │ │ -00026530: 6e65 6c20 7b34 7d20 7c20 6320 2d61 6220  nel {4} | c -ab 
│ │ │ │ -00026540: 3020 3020 3020 2030 2020 3020 2030 2030  0 0 0  0  0  0 0
│ │ │ │ -00026550: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -00026560: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026570: 2020 2020 7b35 7d20 7c20 3020 6320 2020      {5} | 0 c   
│ │ │ │ -00026580: 6220 6120 3020 2030 2020 3020 2030 2030  b a 0  0  0  0 0
│ │ │ │ -00026590: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -000265a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000265b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -000265c0: 6320 3020 2d62 2061 2020 3020 2030 2030  c 0 -b a  0  0 0
│ │ │ │ -000265d0: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -000265e0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000265f0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026600: 3020 6320 3020 202d 6220 2d61 2030 2030  0 c 0  -b -a 0 0
│ │ │ │ -00026610: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -00026620: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026630: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026640: 3020 3020 6320 2030 2020 3020 2062 2061  0 0 c  0  0  b a
│ │ │ │ -00026650: 2030 2030 2020 3020 3020 2030 2020 3020   0 0  0 0  0  0 
│ │ │ │ -00026660: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026670: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026680: 3020 3020 3020 2063 2020 3020 2030 2062  0 0 0  c  0  0 b
│ │ │ │ -00026690: 2030 2030 2020 3020 2d61 2030 2020 3020   0 0  0 -a 0  0 
│ │ │ │ -000266a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000266b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -000266c0: 3020 3020 3020 2030 2020 6320 2030 2030  0 0 0  0  c  0 0
│ │ │ │ -000266d0: 2030 2030 2020 3020 6220 2030 2020 6120   0 0  0 b  0  a 
│ │ │ │ -000266e0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000266f0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026700: 3020 3020 3020 2030 2020 3020 2063 2030  0 0 0  0  0  c 0
│ │ │ │ -00026710: 2062 202d 6120 3020 3020 2030 2020 3020   b -a 0 0  0  0 
│ │ │ │ -00026720: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026730: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026740: 3020 3020 3020 2030 2020 3020 2030 2063  0 0 0  0  0  0 c
│ │ │ │ -00026750: 2030 2062 2020 6120 3020 2030 2020 3020   0 b  a 0  0  0 
│ │ │ │ -00026760: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -00026770: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -00026780: 3020 3020 3020 2030 2020 3020 2030 2030  0 0 0  0  0  0 0
│ │ │ │ -00026790: 2030 2030 2020 6220 6320 202d 6120 3020   0 0  b c  -a 0 
│ │ │ │ -000267a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020  0 ||.|          
│ │ │ │ -000267b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020      {5} | 0 0   
│ │ │ │ -000267c0: 3020 3020 3020 2030 2020 3020 2030 2030  0 0 0  0  0  0 0
│ │ │ │ -000267d0: 2030 2030 2020 3020 3020 2062 2020 6320   0 0  0 0  b  c 
│ │ │ │ -000267e0: 6120 7c7c 0a7c 2020 2020 2020 2020 2020  a ||.|          
│ │ │ │ +00026500: 2020 7c0a 7c6f 3420 3d20 636f 6b65 726e    |.|o4 = cokern
│ │ │ │ +00026510: 656c 207b 347d 207c 2063 202d 6162 2030  el {4} | c -ab 0
│ │ │ │ +00026520: 2030 2030 2020 3020 2030 2020 3020 3020   0 0  0  0  0 0 
│ │ │ │ +00026530: 3020 3020 2030 2030 2020 3020 2030 2030  0 0  0 0  0  0 0
│ │ │ │ +00026540: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00026550: 2020 207b 357d 207c 2030 2063 2020 2062     {5} | 0 c   b
│ │ │ │ +00026560: 2061 2030 2020 3020 2030 2020 3020 3020   a 0  0  0  0 0 
│ │ │ │ +00026570: 3020 3020 2030 2030 2020 3020 2030 2030  0 0  0 0  0  0 0
│ │ │ │ +00026580: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00026590: 2020 207b 357d 207c 2030 2030 2020 2063     {5} | 0 0   c
│ │ │ │ +000265a0: 2030 202d 6220 6120 2030 2020 3020 3020   0 -b a  0  0 0 
│ │ │ │ +000265b0: 3020 3020 2030 2030 2020 3020 2030 2030  0 0  0 0  0  0 0
│ │ │ │ +000265c0: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +000265d0: 2020 207b 357d 207c 2030 2030 2020 2030     {5} | 0 0   0
│ │ │ │ +000265e0: 2063 2030 2020 2d62 202d 6120 3020 3020   c 0  -b -a 0 0 
│ │ │ │ +000265f0: 3020 3020 2030 2030 2020 3020 2030 2030  0 0  0 0  0  0 0
│ │ │ │ +00026600: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00026610: 2020 207b 357d 207c 2030 2030 2020 2030     {5} | 0 0   0
│ │ │ │ +00026620: 2030 2063 2020 3020 2030 2020 6220 6120   0 c  0  0  b a 
│ │ │ │ +00026630: 3020 3020 2030 2030 2020 3020 2030 2030  0 0  0 0  0  0 0
│ │ │ │ +00026640: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00026650: 2020 207b 357d 207c 2030 2030 2020 2030     {5} | 0 0   0
│ │ │ │ +00026660: 2030 2030 2020 6320 2030 2020 3020 6220   0 0  c  0  0 b 
│ │ │ │ +00026670: 3020 3020 2030 202d 6120 3020 2030 2030  0 0  0 -a 0  0 0
│ │ │ │ +00026680: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00026690: 2020 207b 357d 207c 2030 2030 2020 2030     {5} | 0 0   0
│ │ │ │ +000266a0: 2030 2030 2020 3020 2063 2020 3020 3020   0 0  0  c  0 0 
│ │ │ │ +000266b0: 3020 3020 2030 2062 2020 3020 2061 2030  0 0  0 b  0  a 0
│ │ │ │ +000266c0: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +000266d0: 2020 207b 357d 207c 2030 2030 2020 2030     {5} | 0 0   0
│ │ │ │ +000266e0: 2030 2030 2020 3020 2030 2020 6320 3020   0 0  0  0  c 0 
│ │ │ │ +000266f0: 6220 2d61 2030 2030 2020 3020 2030 2030  b -a 0 0  0  0 0
│ │ │ │ +00026700: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00026710: 2020 207b 357d 207c 2030 2030 2020 2030     {5} | 0 0   0
│ │ │ │ +00026720: 2030 2030 2020 3020 2030 2020 3020 6320   0 0  0  0  0 c 
│ │ │ │ +00026730: 3020 6220 2061 2030 2020 3020 2030 2030  0 b  a 0  0  0 0
│ │ │ │ +00026740: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00026750: 2020 207b 357d 207c 2030 2030 2020 2030     {5} | 0 0   0
│ │ │ │ +00026760: 2030 2030 2020 3020 2030 2020 3020 3020   0 0  0  0  0 0 
│ │ │ │ +00026770: 3020 3020 2062 2063 2020 2d61 2030 2030  0 0  b c  -a 0 0
│ │ │ │ +00026780: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +00026790: 2020 207b 357d 207c 2030 2030 2020 2030     {5} | 0 0   0
│ │ │ │ +000267a0: 2030 2030 2020 3020 2030 2020 3020 3020   0 0  0  0  0 0 
│ │ │ │ +000267b0: 3020 3020 2030 2030 2020 6220 2063 2061  0 0  0 0  b  c a
│ │ │ │ +000267c0: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │ +000267d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000267f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026800: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00026810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026820: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00026820: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │  00026830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026840: 2020 3131 2020 2020 2020 2020 2020 2020    11            
│ │ │ │ -00026850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026860: 2020 207c 0a7c 6f34 203a 2052 2d6d 6f64     |.|o4 : R-mod
│ │ │ │ -00026870: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ -00026880: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ -00026890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000268a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026840: 2020 7c0a 7c6f 3420 3a20 522d 6d6f 6475    |.|o4 : R-modu
│ │ │ │ +00026850: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ +00026860: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00026870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026880: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00026890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000268a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000268b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000268c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000268d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000268e0: 2d2d 2d2b 0a7c 6935 203a 2045 203d 2065  ---+.|i5 : E = e
│ │ │ │ -000268f0: 7874 5673 436f 686f 6d6f 6c6f 6779 2866  xtVsCohomology(f
│ │ │ │ -00026900: 662c 6869 6768 5379 7a79 6779 204e 293b  f,highSyzygy N);
│ │ │ │ -00026910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026920: 2020 207c 0a7c 5461 7465 2052 6573 6f6c     |.|Tate Resol
│ │ │ │ -00026930: 7574 696f 6e20 6f66 2045 7874 5f53 284d  ution of Ext_S(M
│ │ │ │ -00026940: 2c6b 2920 6173 2065 7874 6572 696f 7220  ,k) as exterior 
│ │ │ │ -00026950: 6d6f 6475 6c65 3a20 2020 2020 2020 2020  module:         
│ │ │ │ -00026960: 2020 207c 0a7c 4e6f 7465 2074 6861 7420     |.|Note that 
│ │ │ │ -00026970: 6d61 7073 2067 6f20 6c65 6674 2074 6f20  maps go left to 
│ │ │ │ -00026980: 7269 6768 7420 2020 2020 2020 2020 2020  right           
│ │ │ │ -00026990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000269a0: 2020 207c 0a7c 2020 2020 2020 202d 3131     |.|       -11
│ │ │ │ -000269b0: 202d 3130 2020 2d39 202d 3820 2d37 202d   -10  -9 -8 -7 -
│ │ │ │ -000269c0: 3620 2d35 202d 3420 2d33 202d 3220 202d  6 -5 -4 -3 -2  -
│ │ │ │ -000269d0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -000269e0: 2020 207c 0a7c 746f 7461 6c3a 2031 3938     |.|total: 198
│ │ │ │ -000269f0: 2031 3436 2031 3032 2036 3620 3338 2031   146 102 66 38 1
│ │ │ │ -00026a00: 3820 2039 2031 3620 3336 2036 3420 3130  8  9 16 36 64 10
│ │ │ │ -00026a10: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00026a20: 2020 207c 0a7c 2020 2020 383a 2031 3036     |.|    8: 106
│ │ │ │ -00026a30: 2020 3739 2020 3536 2033 3720 3232 2031    79  56 37 22 1
│ │ │ │ -00026a40: 3120 2034 2020 3120 2031 2020 3120 2020  1  4  1  1  1   
│ │ │ │ -00026a50: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00026a60: 2020 207c 0a7c 2020 2020 393a 2020 3932     |.|    9:  92
│ │ │ │ -00026a70: 2020 3637 2020 3436 2032 3920 3136 2020    67  46 29 16  
│ │ │ │ -00026a80: 3720 2032 2020 2e20 202e 2020 2e20 2020  7  2  .  .  .   
│ │ │ │ -00026a90: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -00026aa0: 2020 207c 0a7c 2020 2031 303a 2020 202e     |.|   10:   .
│ │ │ │ -00026ab0: 2020 202e 2020 202e 2020 2e20 202e 2020     .   .  .  .  
│ │ │ │ -00026ac0: 2e20 202e 2020 3520 3134 2032 3720 2034  .  .  5 14 27  4
│ │ │ │ -00026ad0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00026ae0: 2020 207c 0a7c 2020 2031 313a 2020 202e     |.|   11:   .
│ │ │ │ -00026af0: 2020 202e 2020 202e 2020 2e20 202e 2020     .   .  .  .  
│ │ │ │ -00026b00: 2e20 2033 2031 3020 3231 2033 3620 2035  .  3 10 21 36  5
│ │ │ │ -00026b10: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ -00026b20: 2020 207c 0a7c 2d2d 2d20 2020 2020 2020     |.|---       
│ │ │ │ +000268c0: 2d2d 2b0a 7c69 3520 3a20 4520 3d20 6578  --+.|i5 : E = ex
│ │ │ │ +000268d0: 7456 7343 6f68 6f6d 6f6c 6f67 7928 6666  tVsCohomology(ff
│ │ │ │ +000268e0: 2c68 6967 6853 797a 7967 7920 4e29 3b20  ,highSyzygy N); 
│ │ │ │ +000268f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026900: 2020 7c0a 7c54 6174 6520 5265 736f 6c75    |.|Tate Resolu
│ │ │ │ +00026910: 7469 6f6e 206f 6620 4578 745f 5328 4d2c  tion of Ext_S(M,
│ │ │ │ +00026920: 6b29 2061 7320 6578 7465 7269 6f72 206d  k) as exterior m
│ │ │ │ +00026930: 6f64 756c 653a 2020 2020 2020 2020 2020  odule:          
│ │ │ │ +00026940: 2020 7c0a 7c4e 6f74 6520 7468 6174 206d    |.|Note that m
│ │ │ │ +00026950: 6170 7320 676f 206c 6566 7420 746f 2072  aps go left to r
│ │ │ │ +00026960: 6967 6874 2020 2020 2020 2020 2020 2020  ight            
│ │ │ │ +00026970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026980: 2020 7c0a 7c20 2020 2020 2020 2d31 3120    |.|       -11 
│ │ │ │ +00026990: 2d31 3020 202d 3920 2d38 202d 3720 2d36  -10  -9 -8 -7 -6
│ │ │ │ +000269a0: 202d 3520 2d34 202d 3320 2d32 2020 2d31   -5 -4 -3 -2  -1
│ │ │ │ +000269b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269c0: 2020 7c0a 7c74 6f74 616c 3a20 3139 3820    |.|total: 198 
│ │ │ │ +000269d0: 3134 3620 3130 3220 3636 2033 3820 3138  146 102 66 38 18
│ │ │ │ +000269e0: 2020 3920 3136 2033 3620 3634 2031 3030    9 16 36 64 100
│ │ │ │ +000269f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a00: 2020 7c0a 7c20 2020 2038 3a20 3130 3620    |.|    8: 106 
│ │ │ │ +00026a10: 2037 3920 2035 3620 3337 2032 3220 3131   79  56 37 22 11
│ │ │ │ +00026a20: 2020 3420 2031 2020 3120 2031 2020 2031    4  1  1  1   1
│ │ │ │ +00026a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a40: 2020 7c0a 7c20 2020 2039 3a20 2039 3220    |.|    9:  92 
│ │ │ │ +00026a50: 2036 3720 2034 3620 3239 2031 3620 2037   67  46 29 16  7
│ │ │ │ +00026a60: 2020 3220 202e 2020 2e20 202e 2020 202e    2  .  .  .   .
│ │ │ │ +00026a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a80: 2020 7c0a 7c20 2020 3130 3a20 2020 2e20    |.|   10:   . 
│ │ │ │ +00026a90: 2020 2e20 2020 2e20 202e 2020 2e20 202e    .   .  .  .  .
│ │ │ │ +00026aa0: 2020 2e20 2035 2031 3420 3237 2020 3434    .  5 14 27  44
│ │ │ │ +00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ac0: 2020 7c0a 7c20 2020 3131 3a20 2020 2e20    |.|   11:   . 
│ │ │ │ +00026ad0: 2020 2e20 2020 2e20 202e 2020 2e20 202e    .   .  .  .  .
│ │ │ │ +00026ae0: 2020 3320 3130 2032 3120 3336 2020 3535    3 10 21 36  55
│ │ │ │ +00026af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b00: 2020 7c0a 7c2d 2d2d 2020 2020 2020 2020    |.|---        
│ │ │ │ +00026b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b60: 2020 207c 0a7c 436f 686f 6d6f 6c6f 6779     |.|Cohomology
│ │ │ │ -00026b70: 2074 6162 6c65 206f 6620 6576 656e 4578   table of evenEx
│ │ │ │ -00026b80: 744d 6f64 756c 6520 4d3a 2020 2020 2020  tModule M:      
│ │ │ │ -00026b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ba0: 2020 207c 0a7c 2020 202d 3520 2d34 202d     |.|   -5 -4 -
│ │ │ │ -00026bb0: 3320 2d32 202d 3120 2030 2020 3120 2032  3 -2 -1  0  1  2
│ │ │ │ -00026bc0: 2020 3320 2034 2020 2035 2020 2020 2020    3  4   5      
│ │ │ │ -00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026be0: 2020 207c 0a7c 323a 2033 3620 3231 2031     |.|2: 36 21 1
│ │ │ │ -00026bf0: 3020 2033 2020 2e20 202e 2020 2e20 202e  0  3  .  .  .  .
│ │ │ │ -00026c00: 2020 2e20 202e 2020 202e 2020 2020 2020    .  .   .      
│ │ │ │ -00026c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c20: 2020 207c 0a7c 313a 2020 2e20 202e 2020     |.|1:  .  .  
│ │ │ │ -00026c30: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00026c40: 2020 2e20 202e 2020 202e 2020 2020 2020    .  .   .      
│ │ │ │ -00026c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c60: 2020 207c 0a7c 303a 2020 3120 2031 2020     |.|0:  1  1  
│ │ │ │ -00026c70: 3120 2032 2020 3720 3136 2032 3920 3436  1  2  7 16 29 46
│ │ │ │ -00026c80: 2036 3720 3932 2031 3231 2020 2020 2020   67 92 121      
│ │ │ │ +00026b40: 2020 7c0a 7c43 6f68 6f6d 6f6c 6f67 7920    |.|Cohomology 
│ │ │ │ +00026b50: 7461 626c 6520 6f66 2065 7665 6e45 7874  table of evenExt
│ │ │ │ +00026b60: 4d6f 6475 6c65 204d 3a20 2020 2020 2020  Module M:       
│ │ │ │ +00026b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b80: 2020 7c0a 7c20 2020 2d35 202d 3420 2d33    |.|   -5 -4 -3
│ │ │ │ +00026b90: 202d 3220 2d31 2020 3020 2031 2020 3220   -2 -1  0  1  2 
│ │ │ │ +00026ba0: 2033 2020 3420 2020 3520 2020 2020 2020   3  4   5       
│ │ │ │ +00026bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026bc0: 2020 7c0a 7c32 3a20 3336 2032 3120 3130    |.|2: 36 21 10
│ │ │ │ +00026bd0: 2020 3320 202e 2020 2e20 202e 2020 2e20    3  .  .  .  . 
│ │ │ │ +00026be0: 202e 2020 2e20 2020 2e20 2020 2020 2020   .  .   .       
│ │ │ │ +00026bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c00: 2020 7c0a 7c31 3a20 202e 2020 2e20 202e    |.|1:  .  .  .
│ │ │ │ +00026c10: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00026c20: 202e 2020 2e20 2020 2e20 2020 2020 2020   .  .   .       
│ │ │ │ +00026c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c40: 2020 7c0a 7c30 3a20 2031 2020 3120 2031    |.|0:  1  1  1
│ │ │ │ +00026c50: 2020 3220 2037 2031 3620 3239 2034 3620    2  7 16 29 46 
│ │ │ │ +00026c60: 3637 2039 3220 3132 3120 2020 2020 2020  67 92 121       
│ │ │ │ +00026c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c80: 2020 7c0a 7c2d 2d2d 2020 2020 2020 2020    |.|---        
│ │ │ │  00026c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ca0: 2020 207c 0a7c 2d2d 2d20 2020 2020 2020     |.|---       
│ │ │ │ +00026ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ce0: 2020 207c 0a7c 436f 686f 6d6f 6c6f 6779     |.|Cohomology
│ │ │ │ -00026cf0: 2074 6162 6c65 206f 6620 6f64 6445 7874   table of oddExt
│ │ │ │ -00026d00: 4d6f 6475 6c65 204d 3a20 2020 2020 2020  Module M:       
│ │ │ │ -00026d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d20: 2020 207c 0a7c 2020 202d 3520 2d34 202d     |.|   -5 -4 -
│ │ │ │ -00026d30: 3320 2d32 202d 3120 2030 2020 3120 2032  3 -2 -1  0  1  2
│ │ │ │ -00026d40: 2020 3320 2020 3420 2020 3520 2020 2020    3   4   5     
│ │ │ │ -00026d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d60: 2020 207c 0a7c 323a 2032 3820 3135 2020     |.|2: 28 15  
│ │ │ │ -00026d70: 3620 2031 2020 2e20 202e 2020 2e20 202e  6  1  .  .  .  .
│ │ │ │ -00026d80: 2020 2e20 2020 2e20 2020 2e20 2020 2020    .   .   .     
│ │ │ │ -00026d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026da0: 2020 207c 0a7c 313a 2020 2e20 202e 2020     |.|1:  .  .  
│ │ │ │ -00026db0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -00026dc0: 2020 2e20 2020 2e20 2020 2e20 2020 2020    .   .   .     
│ │ │ │ -00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026de0: 2020 207c 0a7c 303a 2020 3120 2031 2020     |.|0:  1  1  
│ │ │ │ -00026df0: 3120 2034 2031 3120 3232 2033 3720 3536  1  4 11 22 37 56
│ │ │ │ -00026e00: 2037 3920 3130 3620 3133 3720 2020 2020   79 106 137     
│ │ │ │ -00026e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026e20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00026cc0: 2020 7c0a 7c43 6f68 6f6d 6f6c 6f67 7920    |.|Cohomology 
│ │ │ │ +00026cd0: 7461 626c 6520 6f66 206f 6464 4578 744d  table of oddExtM
│ │ │ │ +00026ce0: 6f64 756c 6520 4d3a 2020 2020 2020 2020  odule M:        
│ │ │ │ +00026cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d00: 2020 7c0a 7c20 2020 2d35 202d 3420 2d33    |.|   -5 -4 -3
│ │ │ │ +00026d10: 202d 3220 2d31 2020 3020 2031 2020 3220   -2 -1  0  1  2 
│ │ │ │ +00026d20: 2033 2020 2034 2020 2035 2020 2020 2020   3   4   5      
│ │ │ │ +00026d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d40: 2020 7c0a 7c32 3a20 3238 2031 3520 2036    |.|2: 28 15  6
│ │ │ │ +00026d50: 2020 3120 202e 2020 2e20 202e 2020 2e20    1  .  .  .  . 
│ │ │ │ +00026d60: 202e 2020 202e 2020 202e 2020 2020 2020   .   .   .      
│ │ │ │ +00026d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d80: 2020 7c0a 7c31 3a20 202e 2020 2e20 202e    |.|1:  .  .  .
│ │ │ │ +00026d90: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +00026da0: 202e 2020 202e 2020 202e 2020 2020 2020   .   .   .      
│ │ │ │ +00026db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026dc0: 2020 7c0a 7c30 3a20 2031 2020 3120 2031    |.|0:  1  1  1
│ │ │ │ +00026dd0: 2020 3420 3131 2032 3220 3337 2035 3620    4 11 22 37 56 
│ │ │ │ +00026de0: 3739 2031 3036 2031 3337 2020 2020 2020  79 106 137      
│ │ │ │ +00026df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026e00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00026e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026e60: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d  ---+..See also.=
│ │ │ │ -00026e70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f  =======..  * *no
│ │ │ │ -00026e80: 7465 2068 6967 6853 797a 7967 793a 2068  te highSyzygy: h
│ │ │ │ -00026e90: 6967 6853 797a 7967 792c 202d 2d20 5265  ighSyzygy, -- Re
│ │ │ │ -00026ea0: 7475 726e 7320 6120 7379 7a79 6779 206d  turns a syzygy m
│ │ │ │ -00026eb0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64  odule one beyond
│ │ │ │ -00026ec0: 2074 6865 0a20 2020 2072 6567 756c 6172   the.    regular
│ │ │ │ -00026ed0: 6974 7920 6f66 2045 7874 284d 2c6b 290a  ity of Ext(M,k).
│ │ │ │ -00026ee0: 2020 2a20 2a6e 6f74 6520 6578 7465 7269    * *note exteri
│ │ │ │ -00026ef0: 6f72 4578 744d 6f64 756c 653a 2065 7874  orExtModule: ext
│ │ │ │ -00026f00: 6572 696f 7245 7874 4d6f 6475 6c65 2c20  eriorExtModule, 
│ │ │ │ -00026f10: 2d2d 2045 7874 284d 2c6b 2920 6f72 2045  -- Ext(M,k) or E
│ │ │ │ -00026f20: 7874 284d 2c4e 2920 6173 2061 0a20 2020  xt(M,N) as a.   
│ │ │ │ -00026f30: 206d 6f64 756c 6520 6f76 6572 2061 6e20   module over an 
│ │ │ │ -00026f40: 6578 7465 7269 6f72 2061 6c67 6562 7261  exterior algebra
│ │ │ │ -00026f50: 0a0a 5761 7973 2074 6f20 7573 6520 6578  ..Ways to use ex
│ │ │ │ -00026f60: 7456 7343 6f68 6f6d 6f6c 6f67 793a 0a3d  tVsCohomology:.=
│ │ │ │ -00026f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00026f80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00026f90: 2022 6578 7456 7343 6f68 6f6d 6f6c 6f67   "extVsCohomolog
│ │ │ │ -00026fa0: 7928 4d61 7472 6978 2c4d 6f64 756c 6529  y(Matrix,Module)
│ │ │ │ -00026fb0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00026fc0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00026fd0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -00026fe0: 6a65 6374 202a 6e6f 7465 2065 7874 5673  ject *note extVs
│ │ │ │ -00026ff0: 436f 686f 6d6f 6c6f 6779 3a20 6578 7456  Cohomology: extV
│ │ │ │ -00027000: 7343 6f68 6f6d 6f6c 6f67 792c 2069 7320  sCohomology, is 
│ │ │ │ -00027010: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ -00027020: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ -00027030: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -00027040: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +00026e40: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ +00026e50: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ +00026e60: 6520 6869 6768 5379 7a79 6779 3a20 6869  e highSyzygy: hi
│ │ │ │ +00026e70: 6768 5379 7a79 6779 2c20 2d2d 2052 6574  ghSyzygy, -- Ret
│ │ │ │ +00026e80: 7572 6e73 2061 2073 797a 7967 7920 6d6f  urns a syzygy mo
│ │ │ │ +00026e90: 6475 6c65 206f 6e65 2062 6579 6f6e 6420  dule one beyond 
│ │ │ │ +00026ea0: 7468 650a 2020 2020 7265 6775 6c61 7269  the.    regulari
│ │ │ │ +00026eb0: 7479 206f 6620 4578 7428 4d2c 6b29 0a20  ty of Ext(M,k). 
│ │ │ │ +00026ec0: 202a 202a 6e6f 7465 2065 7874 6572 696f   * *note exterio
│ │ │ │ +00026ed0: 7245 7874 4d6f 6475 6c65 3a20 6578 7465  rExtModule: exte
│ │ │ │ +00026ee0: 7269 6f72 4578 744d 6f64 756c 652c 202d  riorExtModule, -
│ │ │ │ +00026ef0: 2d20 4578 7428 4d2c 6b29 206f 7220 4578  - Ext(M,k) or Ex
│ │ │ │ +00026f00: 7428 4d2c 4e29 2061 7320 610a 2020 2020  t(M,N) as a.    
│ │ │ │ +00026f10: 6d6f 6475 6c65 206f 7665 7220 616e 2065  module over an e
│ │ │ │ +00026f20: 7874 6572 696f 7220 616c 6765 6272 610a  xterior algebra.
│ │ │ │ +00026f30: 0a57 6179 7320 746f 2075 7365 2065 7874  .Ways to use ext
│ │ │ │ +00026f40: 5673 436f 686f 6d6f 6c6f 6779 3a0a 3d3d  VsCohomology:.==
│ │ │ │ +00026f50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00026f60: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00026f70: 2265 7874 5673 436f 686f 6d6f 6c6f 6779  "extVsCohomology
│ │ │ │ +00026f80: 284d 6174 7269 782c 4d6f 6475 6c65 2922  (Matrix,Module)"
│ │ │ │ +00026f90: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +00026fa0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +00026fb0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +00026fc0: 6563 7420 2a6e 6f74 6520 6578 7456 7343  ect *note extVsC
│ │ │ │ +00026fd0: 6f68 6f6d 6f6c 6f67 793a 2065 7874 5673  ohomology: extVs
│ │ │ │ +00026fe0: 436f 686f 6d6f 6c6f 6779 2c20 6973 2061  Cohomology, is a
│ │ │ │ +00026ff0: 202a 6e6f 7465 206d 6574 686f 6420 6675   *note method fu
│ │ │ │ +00027000: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61  nction:.(Macaula
│ │ │ │ +00027010: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ +00027020: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  tion,...--------
│ │ │ │ +00027030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027090: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -000270a0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -000270b0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -000270c0: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -000270d0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -000270e0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ -000270f0: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -00027100: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ -00027110: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ -00027120: 2e6d 323a 3238 3236 3a30 2e0a 1f0a 4669  .m2:2826:0....Fi
│ │ │ │ -00027130: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ -00027140: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -00027150: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ -00027160: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ -00027170: 7273 2c20 4e65 7874 3a20 6672 6565 4578  rs, Next: freeEx
│ │ │ │ -00027180: 7465 7269 6f72 5375 6d6d 616e 642c 2050  teriorSummand, P
│ │ │ │ -00027190: 7265 763a 2065 7874 5673 436f 686f 6d6f  rev: extVsCohomo
│ │ │ │ -000271a0: 6c6f 6779 2c20 5570 3a20 546f 700a 0a66  logy, Up: Top..f
│ │ │ │ -000271b0: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ -000271c0: 7320 2d2d 2062 6574 7469 206e 756d 6265  s -- betti numbe
│ │ │ │ -000271d0: 7273 206f 6620 6669 6e69 7465 2072 6573  rs of finite res
│ │ │ │ -000271e0: 6f6c 7574 696f 6e20 636f 6d70 7574 6564  olution computed
│ │ │ │ -000271f0: 2066 726f 6d20 6120 6d61 7472 6978 2066   from a matrix f
│ │ │ │ -00027200: 6163 746f 7269 7a61 7469 6f6e 0a2a 2a2a  actorization.***
│ │ │ │ +00027070: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ +00027080: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ +00027090: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ +000270a0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ +000270b0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ +000270c0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ +000270d0: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ +000270e0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +000270f0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +00027100: 6d32 3a32 3832 363a 302e 0a1f 0a46 696c  m2:2826:0....Fil
│ │ │ │ +00027110: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ +00027120: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +00027130: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2066  ns.info, Node: f
│ │ │ │ +00027140: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +00027150: 732c 204e 6578 743a 2066 7265 6545 7874  s, Next: freeExt
│ │ │ │ +00027160: 6572 696f 7253 756d 6d61 6e64 2c20 5072  eriorSummand, Pr
│ │ │ │ +00027170: 6576 3a20 6578 7456 7343 6f68 6f6d 6f6c  ev: extVsCohomol
│ │ │ │ +00027180: 6f67 792c 2055 703a 2054 6f70 0a0a 6669  ogy, Up: Top..fi
│ │ │ │ +00027190: 6e69 7465 4265 7474 694e 756d 6265 7273  niteBettiNumbers
│ │ │ │ +000271a0: 202d 2d20 6265 7474 6920 6e75 6d62 6572   -- betti number
│ │ │ │ +000271b0: 7320 6f66 2066 696e 6974 6520 7265 736f  s of finite reso
│ │ │ │ +000271c0: 6c75 7469 6f6e 2063 6f6d 7075 7465 6420  lution computed 
│ │ │ │ +000271d0: 6672 6f6d 2061 206d 6174 7269 7820 6661  from a matrix fa
│ │ │ │ +000271e0: 6374 6f72 697a 6174 696f 6e0a 2a2a 2a2a  ctorization.****
│ │ │ │ +000271f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00027200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00027210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00027220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00027230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00027260: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -00027270: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -00027280: 4c20 3d20 6669 6e69 7465 4265 7474 694e  L = finiteBettiN
│ │ │ │ -00027290: 756d 6265 7273 204d 460a 2020 2a20 496e  umbers MF.  * In
│ │ │ │ -000272a0: 7075 7473 3a0a 2020 2020 2020 2a20 4d46  puts:.      * MF
│ │ │ │ -000272b0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -000272c0: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -000272d0: 7374 2c2c 204c 6973 7420 6f66 2048 6173  st,, List of Has
│ │ │ │ -000272e0: 6854 6162 6c65 7320 6173 2063 6f6d 7075  hTables as compu
│ │ │ │ -000272f0: 7465 640a 2020 2020 2020 2020 6279 2022  ted.        by "
│ │ │ │ -00027300: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ -00027310: 696f 6e22 0a20 202a 204f 7574 7075 7473  ion".  * Outputs
│ │ │ │ -00027320: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a  :.      * L, a *
│ │ │ │ -00027330: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ -00027340: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ -00027350: 4c69 7374 206f 6620 6265 7474 6920 6e75  List of betti nu
│ │ │ │ -00027360: 6d62 6572 730a 0a44 6573 6372 6970 7469  mbers..Descripti
│ │ │ │ -00027370: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00027380: 5573 6573 2074 6865 2072 616e 6b73 206f  Uses the ranks o
│ │ │ │ -00027390: 6620 7468 6520 4220 6d61 7472 6963 6573  f the B matrices
│ │ │ │ -000273a0: 2069 6e20 6120 6d61 7472 6978 2066 6163   in a matrix fac
│ │ │ │ -000273b0: 746f 7269 7a61 7469 6f6e 2066 6f72 2061  torization for a
│ │ │ │ -000273c0: 206d 6f64 756c 6520 4d20 6f76 6572 0a53   module M over.S
│ │ │ │ -000273d0: 2f28 665f 312c 2e2e 2c66 5f63 2920 746f  /(f_1,..,f_c) to
│ │ │ │ -000273e0: 2063 6f6d 7075 7465 2074 6865 2062 6574   compute the bet
│ │ │ │ -000273f0: 7469 206e 756d 6265 7273 206f 6620 7468  ti numbers of th
│ │ │ │ -00027400: 6520 6d69 6e69 6d61 6c20 7265 736f 6c75  e minimal resolu
│ │ │ │ -00027410: 7469 6f6e 206f 6620 4d20 6f76 6572 0a53  tion of M over.S
│ │ │ │ -00027420: 2c20 7768 6963 6820 6973 2074 6865 2073  , which is the s
│ │ │ │ -00027430: 756d 206f 6620 7468 6520 4b6f 737a 756c  um of the Koszul
│ │ │ │ -00027440: 2063 6f6d 706c 6578 6573 204b 2866 5f31   complexes K(f_1
│ │ │ │ -00027450: 2e2e 665f 7b6a 2d31 7d29 2074 656e 736f  ..f_{j-1}) tenso
│ │ │ │ -00027460: 7265 6420 7769 7468 2042 286a 290a 0a2b  red with B(j)..+
│ │ │ │ +00027240: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ +00027250: 7361 6765 3a20 0a20 2020 2020 2020 204c  sage: .        L
│ │ │ │ +00027260: 203d 2066 696e 6974 6542 6574 7469 4e75   = finiteBettiNu
│ │ │ │ +00027270: 6d62 6572 7320 4d46 0a20 202a 2049 6e70  mbers MF.  * Inp
│ │ │ │ +00027280: 7574 733a 0a20 2020 2020 202a 204d 462c  uts:.      * MF,
│ │ │ │ +00027290: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ +000272a0: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ +000272b0: 742c 2c20 4c69 7374 206f 6620 4861 7368  t,, List of Hash
│ │ │ │ +000272c0: 5461 626c 6573 2061 7320 636f 6d70 7574  Tables as comput
│ │ │ │ +000272d0: 6564 0a20 2020 2020 2020 2062 7920 226d  ed.        by "m
│ │ │ │ +000272e0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +000272f0: 6f6e 220a 2020 2a20 4f75 7470 7574 733a  on".  * Outputs:
│ │ │ │ +00027300: 0a20 2020 2020 202a 204c 2c20 6120 2a6e  .      * L, a *n
│ │ │ │ +00027310: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +00027320: 6c61 7932 446f 6329 4c69 7374 2c2c 204c  lay2Doc)List,, L
│ │ │ │ +00027330: 6973 7420 6f66 2062 6574 7469 206e 756d  ist of betti num
│ │ │ │ +00027340: 6265 7273 0a0a 4465 7363 7269 7074 696f  bers..Descriptio
│ │ │ │ +00027350: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a55  n.===========..U
│ │ │ │ +00027360: 7365 7320 7468 6520 7261 6e6b 7320 6f66  ses the ranks of
│ │ │ │ +00027370: 2074 6865 2042 206d 6174 7269 6365 7320   the B matrices 
│ │ │ │ +00027380: 696e 2061 206d 6174 7269 7820 6661 6374  in a matrix fact
│ │ │ │ +00027390: 6f72 697a 6174 696f 6e20 666f 7220 6120  orization for a 
│ │ │ │ +000273a0: 6d6f 6475 6c65 204d 206f 7665 720a 532f  module M over.S/
│ │ │ │ +000273b0: 2866 5f31 2c2e 2e2c 665f 6329 2074 6f20  (f_1,..,f_c) to 
│ │ │ │ +000273c0: 636f 6d70 7574 6520 7468 6520 6265 7474  compute the bett
│ │ │ │ +000273d0: 6920 6e75 6d62 6572 7320 6f66 2074 6865  i numbers of the
│ │ │ │ +000273e0: 206d 696e 696d 616c 2072 6573 6f6c 7574   minimal resolut
│ │ │ │ +000273f0: 696f 6e20 6f66 204d 206f 7665 720a 532c  ion of M over.S,
│ │ │ │ +00027400: 2077 6869 6368 2069 7320 7468 6520 7375   which is the su
│ │ │ │ +00027410: 6d20 6f66 2074 6865 204b 6f73 7a75 6c20  m of the Koszul 
│ │ │ │ +00027420: 636f 6d70 6c65 7865 7320 4b28 665f 312e  complexes K(f_1.
│ │ │ │ +00027430: 2e66 5f7b 6a2d 317d 2920 7465 6e73 6f72  .f_{j-1}) tensor
│ │ │ │ +00027440: 6564 2077 6974 6820 4228 6a29 0a0a 2b2d  ed with B(j)..+-
│ │ │ │ +00027450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000274a0: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ -000274b0: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ -000274c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000274d0: 2020 2020 2020 207c 0a7c 202d 2d20 7365         |.| -- se
│ │ │ │ -000274e0: 7474 696e 6720 7261 6e64 6f6d 2073 6565  tting random see
│ │ │ │ -000274f0: 6420 746f 2030 2020 2020 2020 2020 2020  d to 0          
│ │ │ │ -00027500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00027480: 2d2b 0a7c 6931 203a 2073 6574 5261 6e64  -+.|i1 : setRand
│ │ │ │ +00027490: 6f6d 5365 6564 2030 2020 2020 2020 2020  omSeed 0        
│ │ │ │ +000274a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000274b0: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574        |.| -- set
│ │ │ │ +000274c0: 7469 6e67 2072 616e 646f 6d20 7365 6564  ting random seed
│ │ │ │ +000274d0: 2074 6f20 3020 2020 2020 2020 2020 2020   to 0           
│ │ │ │ +000274e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000274f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027520: 7c0a 7c6f 3120 3d20 3020 2020 2020 2020  |.|o1 = 0       
│ │ │ │  00027530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027540: 207c 0a7c 6f31 203d 2030 2020 2020 2020   |.|o1 = 0      
│ │ │ │ -00027550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027570: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00027580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -000275b0: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020   : kk = ZZ/101  
│ │ │ │ -000275c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027550: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00027560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027580: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00027590: 3a20 6b6b 203d 205a 5a2f 3130 3120 2020  : kk = ZZ/101   
│ │ │ │ +000275a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000275c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000275d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000275e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000275f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000275f0: 2020 2020 7c0a 7c6f 3220 3d20 6b6b 2020      |.|o2 = kk  
│ │ │ │  00027600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027610: 2020 2020 207c 0a7c 6f32 203d 206b 6b20       |.|o2 = kk 
│ │ │ │ -00027620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027620: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00027630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027640: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00027650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00027680: 0a7c 6f32 203a 2051 756f 7469 656e 7452  .|o2 : QuotientR
│ │ │ │ -00027690: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -000276a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000276b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000276c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000276e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -000276f0: 2053 203d 206b 6b5b 612c 622c 752c 765d   S = kk[a,b,u,v]
│ │ │ │ +00027640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027650: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027660: 7c6f 3220 3a20 5175 6f74 6965 6e74 5269  |o2 : QuotientRi
│ │ │ │ +00027670: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +00027680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027690: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000276a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000276b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000276c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +000276d0: 5320 3d20 6b6b 5b61 2c62 2c75 2c76 5d20  S = kk[a,b,u,v] 
│ │ │ │ +000276e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000276f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00027700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00027720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00027730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027730: 2020 7c0a 7c6f 3320 3d20 5320 2020 2020    |.|o3 = S     
│ │ │ │  00027740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027750: 2020 207c 0a7c 6f33 203d 2053 2020 2020     |.|o3 = S    
│ │ │ │ -00027760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027760: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00027770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027780: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00027790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000277c0: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -000277d0: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -000277e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000277f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -00027800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027820: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066  -------+.|i4 : f
│ │ │ │ -00027830: 6620 3d20 6d61 7472 6978 2261 752c 6276  f = matrix"au,bv
│ │ │ │ -00027840: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -00027850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00027780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027790: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000277a0: 3320 3a20 506f 6c79 6e6f 6d69 616c 5269  3 : PolynomialRi
│ │ │ │ +000277b0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +000277c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000277d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000277e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000277f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027800: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6666  ------+.|i4 : ff
│ │ │ │ +00027810: 203d 206d 6174 7269 7822 6175 2c62 7622   = matrix"au,bv"
│ │ │ │ +00027820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027830: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00027840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027890: 207c 0a7c 6f34 203d 207c 2061 7520 6276   |.|o4 = | au bv
│ │ │ │ -000278a0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00027870: 7c0a 7c6f 3420 3d20 7c20 6175 2062 7620  |.|o4 = | au bv 
│ │ │ │ +00027880: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00027890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000278b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000278c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000278d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000278e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000278f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00027900: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00027910: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00027920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027930: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053  |.|o4 : Matrix S
│ │ │ │ -00027940: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -00027950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027960: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -000279a0: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ -000279b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000279c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000279d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000279e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000278d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000278e0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +000278f0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00027900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00027910: 0a7c 6f34 203a 204d 6174 7269 7820 5320  .|o4 : Matrix S 
│ │ │ │ +00027920: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +00027930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00027950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027970: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +00027980: 2052 203d 2053 2f69 6465 616c 2066 6620   R = S/ideal ff 
│ │ │ │ +00027990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000279a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000279b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000279c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000279d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000279e0: 2020 207c 0a7c 6f35 203d 2052 2020 2020     |.|o5 = R    
│ │ │ │  000279f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a00: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020      |.|o5 = R   
│ │ │ │ -00027a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00027a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00027a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027a60: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00027a70: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269  |o5 : QuotientRi
│ │ │ │ -00027a80: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -00027a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ad0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00027ae0: 4d30 203d 2052 5e31 2f69 6465 616c 2261  M0 = R^1/ideal"a
│ │ │ │ -00027af0: 2c62 2220 2020 2020 2020 2020 2020 2020  ,b"             
│ │ │ │ -00027b00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00027a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00027a50: 6f35 203a 2051 756f 7469 656e 7452 696e  o5 : QuotientRin
│ │ │ │ +00027a60: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +00027a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027a80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00027a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027ab0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d  -------+.|i6 : M
│ │ │ │ +00027ac0: 3020 3d20 525e 312f 6964 6561 6c22 612c  0 = R^1/ideal"a,
│ │ │ │ +00027ad0: 6222 2020 2020 2020 2020 2020 2020 2020  b"              
│ │ │ │ +00027ae0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00027af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00027b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b40: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e    |.|o6 = cokern
│ │ │ │ -00027b50: 656c 207c 2061 2062 207c 2020 2020 2020  el | a b |      
│ │ │ │ +00027b20: 207c 0a7c 6f36 203d 2063 6f6b 6572 6e65   |.|o6 = cokerne
│ │ │ │ +00027b30: 6c20 7c20 6120 6220 7c20 2020 2020 2020  l | a b |       
│ │ │ │ +00027b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027b50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00027b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027b70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00027b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027b80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00027b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00027ba0: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │  00027bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027bc0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -00027bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027be0: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c   |.|o6 : R-modul
│ │ │ │ -00027bf0: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -00027c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027c10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00027c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -00027c50: 203a 2046 203d 2066 7265 6552 6573 6f6c   : F = freeResol
│ │ │ │ -00027c60: 7574 696f 6e28 4d30 2c20 4c65 6e67 7468  ution(M0, Length
│ │ │ │ -00027c70: 4c69 6d69 7420 3d3e 3329 2020 2020 2020  Limit =>3)      
│ │ │ │ -00027c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00027c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027cb0: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ -00027cc0: 2020 2020 2032 2020 2020 2020 3320 2020       2      3   
│ │ │ │ -00027cd0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -00027ce0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -00027cf0: 3d20 5220 203c 2d2d 2052 2020 3c2d 2d20  = R  <-- R  <-- 
│ │ │ │ -00027d00: 5220 203c 2d2d 2052 2020 2020 2020 2020  R  <-- R        
│ │ │ │ -00027d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00027d20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00027d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027d50: 2020 2020 7c0a 7c20 2020 2020 3020 2020      |.|     0   
│ │ │ │ -00027d60: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -00027d70: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00027d80: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00027d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027db0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00027dc0: 7c6f 3720 3a20 436f 6d70 6c65 7820 2020  |o7 : Complex   
│ │ │ │ -00027dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027df0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00027e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -00027e30: 4d20 3d20 636f 6b65 7220 462e 6464 5f33  M = coker F.dd_3
│ │ │ │ -00027e40: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -00027e50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00027bc0: 7c0a 7c6f 3620 3a20 522d 6d6f 6475 6c65  |.|o6 : R-module
│ │ │ │ +00027bd0: 2c20 7175 6f74 6965 6e74 206f 6620 5220  , quotient of R 
│ │ │ │ +00027be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027bf0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00027c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +00027c30: 3a20 4620 3d20 6672 6565 5265 736f 6c75  : F = freeResolu
│ │ │ │ +00027c40: 7469 6f6e 284d 302c 204c 656e 6774 684c  tion(M0, LengthL
│ │ │ │ +00027c50: 696d 6974 203d 3e33 2920 2020 2020 207c  imit =>3)      |
│ │ │ │ +00027c60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00027c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027c90: 2020 2020 7c0a 7c20 2020 2020 2031 2020      |.|      1  
│ │ │ │ +00027ca0: 2020 2020 3220 2020 2020 2033 2020 2020      2      3    
│ │ │ │ +00027cb0: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +00027cc0: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ +00027cd0: 2052 2020 3c2d 2d20 5220 203c 2d2d 2052   R  <-- R  <-- R
│ │ │ │ +00027ce0: 2020 3c2d 2d20 5220 2020 2020 2020 2020    <-- R         
│ │ │ │ +00027cf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00027d00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00027d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027d30: 2020 207c 0a7c 2020 2020 2030 2020 2020     |.|     0    
│ │ │ │ +00027d40: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ +00027d50: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00027d60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00027d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00027da0: 6f37 203a 2043 6f6d 706c 6578 2020 2020  o7 : Complex    
│ │ │ │ +00027db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027dd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00027de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027e00: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 204d  -------+.|i8 : M
│ │ │ │ +00027e10: 203d 2063 6f6b 6572 2046 2e64 645f 333b   = coker F.dd_3;
│ │ │ │ +00027e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027e30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00027e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e90: 2d2d 2b0a 7c69 3920 3a20 4d46 203d 206d  --+.|i9 : MF = m
│ │ │ │ -00027ea0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -00027eb0: 6f6e 2866 662c 4d29 3b20 2020 2020 2020  on(ff,M);       
│ │ │ │ -00027ec0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00027ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00027f00: 3130 203a 2062 6574 7469 2066 7265 6552  10 : betti freeR
│ │ │ │ -00027f10: 6573 6f6c 7574 696f 6e20 7075 7368 466f  esolution pushFo
│ │ │ │ -00027f20: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ -00027f30: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ -00027f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00027f70: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ -00027f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027f90: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00027fa0: 3020 3d20 746f 7461 6c3a 2033 2035 2032  0 = total: 3 5 2
│ │ │ │ -00027fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00027fd0: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ -00027fe0: 3320 3420 2e20 2020 2020 2020 2020 2020  3 4 .           
│ │ │ │ -00027ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028010: 2020 333a 202e 2031 2032 2020 2020 2020    3: . 1 2      
│ │ │ │ +00027e70: 2d2b 0a7c 6939 203a 204d 4620 3d20 6d61  -+.|i9 : MF = ma
│ │ │ │ +00027e80: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +00027e90: 6e28 6666 2c4d 293b 2020 2020 2020 2020  n(ff,M);        
│ │ │ │ +00027ea0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00027eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00027ee0: 3020 3a20 6265 7474 6920 6672 6565 5265  0 : betti freeRe
│ │ │ │ +00027ef0: 736f 6c75 7469 6f6e 2070 7573 6846 6f72  solution pushFor
│ │ │ │ +00027f00: 7761 7264 286d 6170 2852 2c53 292c 4d29  ward(map(R,S),M)
│ │ │ │ +00027f10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00027f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027f40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00027f50: 2020 2020 2030 2031 2032 2020 2020 2020       0 1 2      
│ │ │ │ +00027f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027f70: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ +00027f80: 203d 2074 6f74 616c 3a20 3320 3520 3220   = total: 3 5 2 
│ │ │ │ +00027f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00027fb0: 0a7c 2020 2020 2020 2020 2020 323a 2033  .|          2: 3
│ │ │ │ +00027fc0: 2034 202e 2020 2020 2020 2020 2020 2020   4 .            
│ │ │ │ +00027fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00027fe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00027ff0: 2033 3a20 2e20 3120 3220 2020 2020 2020   3: . 1 2       
│ │ │ │ +00028000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028010: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00028020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028030: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00028040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028070: 0a7c 6f31 3020 3a20 4265 7474 6954 616c  .|o10 : BettiTal
│ │ │ │ -00028080: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ -00028090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000280a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -000280b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ -000280e0: 3a20 6669 6e69 7465 4265 7474 694e 756d  : finiteBettiNum
│ │ │ │ -000280f0: 6265 7273 204d 4620 2020 2020 2020 2020  bers MF         
│ │ │ │ -00028100: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00028120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028140: 2020 207c 0a7c 6f31 3120 3d20 7b33 2c20     |.|o11 = {3, 
│ │ │ │ -00028150: 352c 2032 7d20 2020 2020 2020 2020 2020  5, 2}           
│ │ │ │ +00028030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028040: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028050: 7c6f 3130 203a 2042 6574 7469 5461 6c6c  |o10 : BettiTall
│ │ │ │ +00028060: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +00028070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028080: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00028090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000280a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000280b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ +000280c0: 2066 696e 6974 6542 6574 7469 4e75 6d62   finiteBettiNumb
│ │ │ │ +000280d0: 6572 7320 4d46 2020 2020 2020 2020 2020  ers MF          
│ │ │ │ +000280e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000280f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028120: 2020 7c0a 7c6f 3131 203d 207b 332c 2035    |.|o11 = {3, 5
│ │ │ │ +00028130: 2c20 327d 2020 2020 2020 2020 2020 2020  , 2}            
│ │ │ │ +00028140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028150: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00028160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00028180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000281a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000281b0: 6f31 3120 3a20 4c69 7374 2020 2020 2020  o11 : List      
│ │ │ │ -000281c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000281d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000281e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -000281f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028210: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -00028220: 696e 6669 6e69 7465 4265 7474 694e 756d  infiniteBettiNum
│ │ │ │ -00028230: 6265 7273 284d 462c 3529 2020 2020 2020  bers(MF,5)      
│ │ │ │ -00028240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028180: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00028190: 3131 203a 204c 6973 7420 2020 2020 2020  11 : List       
│ │ │ │ +000281a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000281b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000281c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000281d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000281e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000281f0: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2069  ------+.|i12 : i
│ │ │ │ +00028200: 6e66 696e 6974 6542 6574 7469 4e75 6d62  nfiniteBettiNumb
│ │ │ │ +00028210: 6572 7328 4d46 2c35 2920 2020 2020 2020  ers(MF,5)       
│ │ │ │ +00028220: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00028230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028280: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c   |.|o12 = {3, 4,
│ │ │ │ -00028290: 2035 2c20 362c 2037 2c20 387d 2020 2020   5, 6, 7, 8}    
│ │ │ │ +00028260: 7c0a 7c6f 3132 203d 207b 332c 2034 2c20  |.|o12 = {3, 4, 
│ │ │ │ +00028270: 352c 2036 2c20 372c 2038 7d20 2020 2020  5, 6, 7, 8}     
│ │ │ │ +00028280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000282a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000282b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000282c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000282d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000282e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000282f0: 3220 3a20 4c69 7374 2020 2020 2020 2020  2 : List        
│ │ │ │ -00028300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028320: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -00028330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028350: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6265  -----+.|i13 : be
│ │ │ │ -00028360: 7474 6920 6672 6565 5265 736f 6c75 7469  tti freeResoluti
│ │ │ │ -00028370: 6f6e 2028 4d2c 204c 656e 6774 684c 696d  on (M, LengthLim
│ │ │ │ -00028380: 6974 203d 3e20 3529 2020 7c0a 7c20 2020  it => 5)  |.|   
│ │ │ │ -00028390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000283c0: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -000283d0: 2031 2032 2033 2034 2035 2020 2020 2020   1 2 3 4 5      
│ │ │ │ -000283e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000283f0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74      |.|o13 = tot
│ │ │ │ -00028400: 616c 3a20 3320 3420 3520 3620 3720 3820  al: 3 4 5 6 7 8 
│ │ │ │ -00028410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028420: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00028430: 2020 2020 2020 323a 2033 2034 2035 2036        2: 3 4 5 6
│ │ │ │ -00028440: 2037 2038 2020 2020 2020 2020 2020 2020   7 8            
│ │ │ │ -00028450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00028460: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00028470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028490: 2020 207c 0a7c 6f31 3320 3a20 4265 7474     |.|o13 : Bett
│ │ │ │ -000284a0: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ -000284b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000284c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -000284d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000284e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000284f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00028500: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -00028510: 3d0a 0a20 202a 202a 6e6f 7465 206d 6174  =..  * *note mat
│ │ │ │ -00028520: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00028530: 3a20 6d61 7472 6978 4661 6374 6f72 697a  : matrixFactoriz
│ │ │ │ -00028540: 6174 696f 6e2c 202d 2d20 4d61 7073 2069  ation, -- Maps i
│ │ │ │ -00028550: 6e20 6120 6869 6768 6572 0a20 2020 2063  n a higher.    c
│ │ │ │ -00028560: 6f64 696d 656e 7369 6f6e 206d 6174 7269  odimension matri
│ │ │ │ -00028570: 7820 6661 6374 6f72 697a 6174 696f 6e0a  x factorization.
│ │ │ │ -00028580: 2020 2a20 2a6e 6f74 6520 696e 6669 6e69    * *note infini
│ │ │ │ -00028590: 7465 4265 7474 694e 756d 6265 7273 3a20  teBettiNumbers: 
│ │ │ │ -000285a0: 696e 6669 6e69 7465 4265 7474 694e 756d  infiniteBettiNum
│ │ │ │ -000285b0: 6265 7273 2c20 2d2d 2062 6574 7469 206e  bers, -- betti n
│ │ │ │ -000285c0: 756d 6265 7273 206f 660a 2020 2020 6669  umbers of.    fi
│ │ │ │ -000285d0: 6e69 7465 2072 6573 6f6c 7574 696f 6e20  nite resolution 
│ │ │ │ -000285e0: 636f 6d70 7574 6564 2066 726f 6d20 6120  computed from a 
│ │ │ │ -000285f0: 6d61 7472 6978 2066 6163 746f 7269 7a61  matrix factoriza
│ │ │ │ -00028600: 7469 6f6e 0a0a 5761 7973 2074 6f20 7573  tion..Ways to us
│ │ │ │ -00028610: 6520 6669 6e69 7465 4265 7474 694e 756d  e finiteBettiNum
│ │ │ │ -00028620: 6265 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  bers:.==========
│ │ │ │ -00028630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00028640: 3d3d 3d3d 3d0a 0a20 202a 2022 6669 6e69  =====..  * "fini
│ │ │ │ -00028650: 7465 4265 7474 694e 756d 6265 7273 284c  teBettiNumbers(L
│ │ │ │ -00028660: 6973 7429 220a 0a46 6f72 2074 6865 2070  ist)"..For the p
│ │ │ │ -00028670: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -00028680: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -00028690: 6520 6f62 6a65 6374 202a 6e6f 7465 2066  e object *note f
│ │ │ │ -000286a0: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ -000286b0: 733a 2066 696e 6974 6542 6574 7469 4e75  s: finiteBettiNu
│ │ │ │ -000286c0: 6d62 6572 732c 2069 7320 6120 2a6e 6f74  mbers, is a *not
│ │ │ │ -000286d0: 6520 6d65 7468 6f64 0a66 756e 6374 696f  e method.functio
│ │ │ │ -000286e0: 6e3a 2028 4d61 6361 756c 6179 3244 6f63  n: (Macaulay2Doc
│ │ │ │ -000286f0: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ -00028700: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +000282b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000282c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ +000282d0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +000282e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000282f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00028300: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00028310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028330: 2d2d 2d2d 2b0a 7c69 3133 203a 2062 6574  ----+.|i13 : bet
│ │ │ │ +00028340: 7469 2066 7265 6552 6573 6f6c 7574 696f  ti freeResolutio
│ │ │ │ +00028350: 6e20 284d 2c20 4c65 6e67 7468 4c69 6d69  n (M, LengthLimi
│ │ │ │ +00028360: 7420 3d3e 2035 2920 207c 0a7c 2020 2020  t => 5)  |.|    
│ │ │ │ +00028370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000283a0: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +000283b0: 3120 3220 3320 3420 3520 2020 2020 2020  1 2 3 4 5       
│ │ │ │ +000283c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000283d0: 2020 207c 0a7c 6f31 3320 3d20 746f 7461     |.|o13 = tota
│ │ │ │ +000283e0: 6c3a 2033 2034 2035 2036 2037 2038 2020  l: 3 4 5 6 7 8  
│ │ │ │ +000283f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028400: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00028410: 2020 2020 2032 3a20 3320 3420 3520 3620       2: 3 4 5 6 
│ │ │ │ +00028420: 3720 3820 2020 2020 2020 2020 2020 2020  7 8             
│ │ │ │ +00028430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028470: 2020 7c0a 7c6f 3133 203a 2042 6574 7469    |.|o13 : Betti
│ │ │ │ +00028480: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +00028490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000284a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000284b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000284c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000284d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +000284e0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +000284f0: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 7472  ..  * *note matr
│ │ │ │ +00028500: 6978 4661 6374 6f72 697a 6174 696f 6e3a  ixFactorization:
│ │ │ │ +00028510: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +00028520: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e  tion, -- Maps in
│ │ │ │ +00028530: 2061 2068 6967 6865 720a 2020 2020 636f   a higher.    co
│ │ │ │ +00028540: 6469 6d65 6e73 696f 6e20 6d61 7472 6978  dimension matrix
│ │ │ │ +00028550: 2066 6163 746f 7269 7a61 7469 6f6e 0a20   factorization. 
│ │ │ │ +00028560: 202a 202a 6e6f 7465 2069 6e66 696e 6974   * *note infinit
│ │ │ │ +00028570: 6542 6574 7469 4e75 6d62 6572 733a 2069  eBettiNumbers: i
│ │ │ │ +00028580: 6e66 696e 6974 6542 6574 7469 4e75 6d62  nfiniteBettiNumb
│ │ │ │ +00028590: 6572 732c 202d 2d20 6265 7474 6920 6e75  ers, -- betti nu
│ │ │ │ +000285a0: 6d62 6572 7320 6f66 0a20 2020 2066 696e  mbers of.    fin
│ │ │ │ +000285b0: 6974 6520 7265 736f 6c75 7469 6f6e 2063  ite resolution c
│ │ │ │ +000285c0: 6f6d 7075 7465 6420 6672 6f6d 2061 206d  omputed from a m
│ │ │ │ +000285d0: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ +000285e0: 696f 6e0a 0a57 6179 7320 746f 2075 7365  ion..Ways to use
│ │ │ │ +000285f0: 2066 696e 6974 6542 6574 7469 4e75 6d62   finiteBettiNumb
│ │ │ │ +00028600: 6572 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ers:.===========
│ │ │ │ +00028610: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00028620: 3d3d 3d3d 0a0a 2020 2a20 2266 696e 6974  ====..  * "finit
│ │ │ │ +00028630: 6542 6574 7469 4e75 6d62 6572 7328 4c69  eBettiNumbers(Li
│ │ │ │ +00028640: 7374 2922 0a0a 466f 7220 7468 6520 7072  st)"..For the pr
│ │ │ │ +00028650: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +00028660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +00028670: 206f 626a 6563 7420 2a6e 6f74 6520 6669   object *note fi
│ │ │ │ +00028680: 6e69 7465 4265 7474 694e 756d 6265 7273  niteBettiNumbers
│ │ │ │ +00028690: 3a20 6669 6e69 7465 4265 7474 694e 756d  : finiteBettiNum
│ │ │ │ +000286a0: 6265 7273 2c20 6973 2061 202a 6e6f 7465  bers, is a *note
│ │ │ │ +000286b0: 206d 6574 686f 640a 6675 6e63 7469 6f6e   method.function
│ │ │ │ +000286c0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000286d0: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ +000286e0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │ +000286f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028750: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -00028760: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -00028770: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -00028780: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -00028790: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ -000287a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -000287b0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ -000287c0: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -000287d0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3430  esolutions.m2:40
│ │ │ │ -000287e0: 3732 3a30 2e0a 1f0a 4669 6c65 3a20 436f  72:0....File: Co
│ │ │ │ -000287f0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -00028800: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ -00028810: 666f 2c20 4e6f 6465 3a20 6672 6565 4578  fo, Node: freeEx
│ │ │ │ -00028820: 7465 7269 6f72 5375 6d6d 616e 642c 204e  teriorSummand, N
│ │ │ │ -00028830: 6578 743a 2047 7261 6469 6e67 2c20 5072  ext: Grading, Pr
│ │ │ │ -00028840: 6576 3a20 6669 6e69 7465 4265 7474 694e  ev: finiteBettiN
│ │ │ │ -00028850: 756d 6265 7273 2c20 5570 3a20 546f 700a  umbers, Up: Top.
│ │ │ │ -00028860: 0a66 7265 6545 7874 6572 696f 7253 756d  .freeExteriorSum
│ │ │ │ -00028870: 6d61 6e64 202d 2d20 6669 6e64 2074 6865  mand -- find the
│ │ │ │ -00028880: 2066 7265 6520 7375 6d6d 616e 6473 206f   free summands o
│ │ │ │ -00028890: 6620 6120 6d6f 6475 6c65 206f 7665 7220  f a module over 
│ │ │ │ -000288a0: 616e 2065 7874 6572 696f 7220 616c 6765  an exterior alge
│ │ │ │ -000288b0: 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  bra.************
│ │ │ │ +00028730: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ +00028740: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ +00028750: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ +00028760: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ +00028770: 6361 756c 6179 322d 312e 3236 2e30 362b  caulay2-1.26.06+
│ │ │ │ +00028780: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ +00028790: 7061 636b 6167 6573 2f0a 436f 6d70 6c65  packages/.Comple
│ │ │ │ +000287a0: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +000287b0: 736f 6c75 7469 6f6e 732e 6d32 3a34 3037  solutions.m2:407
│ │ │ │ +000287c0: 323a 302e 0a1f 0a46 696c 653a 2043 6f6d  2:0....File: Com
│ │ │ │ +000287d0: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ +000287e0: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66  nResolutions.inf
│ │ │ │ +000287f0: 6f2c 204e 6f64 653a 2066 7265 6545 7874  o, Node: freeExt
│ │ │ │ +00028800: 6572 696f 7253 756d 6d61 6e64 2c20 4e65  eriorSummand, Ne
│ │ │ │ +00028810: 7874 3a20 4772 6164 696e 672c 2050 7265  xt: Grading, Pre
│ │ │ │ +00028820: 763a 2066 696e 6974 6542 6574 7469 4e75  v: finiteBettiNu
│ │ │ │ +00028830: 6d62 6572 732c 2055 703a 2054 6f70 0a0a  mbers, Up: Top..
│ │ │ │ +00028840: 6672 6565 4578 7465 7269 6f72 5375 6d6d  freeExteriorSumm
│ │ │ │ +00028850: 616e 6420 2d2d 2066 696e 6420 7468 6520  and -- find the 
│ │ │ │ +00028860: 6672 6565 2073 756d 6d61 6e64 7320 6f66  free summands of
│ │ │ │ +00028870: 2061 206d 6f64 756c 6520 6f76 6572 2061   a module over a
│ │ │ │ +00028880: 6e20 6578 7465 7269 6f72 2061 6c67 6562  n exterior algeb
│ │ │ │ +00028890: 7261 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ra.*************
│ │ │ │ +000288a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000288b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000288c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000288d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000288e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000288f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00028900: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -00028910: 653a 200a 2020 2020 2020 2020 4620 3d20  e: .        F = 
│ │ │ │ -00028920: 6672 6565 4578 7465 7269 6f72 5375 6d6d  freeExteriorSumm
│ │ │ │ -00028930: 616e 6420 4d0a 2020 2a20 496e 7075 7473  and M.  * Inputs
│ │ │ │ -00028940: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a  :.      * M, a *
│ │ │ │ -00028950: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -00028960: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -00028970: 652c 2c20 6f76 6572 2061 6e20 6578 7465  e,, over an exte
│ │ │ │ -00028980: 7269 6f72 2061 6c67 6562 7261 0a20 202a  rior algebra.  *
│ │ │ │ -00028990: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -000289a0: 2a20 462c 2061 202a 6e6f 7465 206d 6174  * F, a *note mat
│ │ │ │ -000289b0: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ -000289c0: 6f63 294d 6174 7269 782c 2c20 4d61 7020  oc)Matrix,, Map 
│ │ │ │ -000289d0: 6672 6f6d 2061 2066 7265 6520 6d6f 6475  from a free modu
│ │ │ │ -000289e0: 6c65 2074 6f20 4d2e 0a20 2020 2020 2020  le to M..       
│ │ │ │ -000289f0: 2049 6d61 6765 2069 7320 7468 6520 6c61   Image is the la
│ │ │ │ -00028a00: 7267 6573 7420 6672 6565 2073 756d 6d61  rgest free summa
│ │ │ │ -00028a10: 6e64 0a0a 4465 7363 7269 7074 696f 6e0a  nd..Description.
│ │ │ │ -00028a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d  ===========..+--
│ │ │ │ +000288e0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +000288f0: 3a20 0a20 2020 2020 2020 2046 203d 2066  : .        F = f
│ │ │ │ +00028900: 7265 6545 7874 6572 696f 7253 756d 6d61  reeExteriorSumma
│ │ │ │ +00028910: 6e64 204d 0a20 202a 2049 6e70 7574 733a  nd M.  * Inputs:
│ │ │ │ +00028920: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
│ │ │ │ +00028930: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +00028940: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +00028950: 2c2c 206f 7665 7220 616e 2065 7874 6572  ,, over an exter
│ │ │ │ +00028960: 696f 7220 616c 6765 6272 610a 2020 2a20  ior algebra.  * 
│ │ │ │ +00028970: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00028980: 2046 2c20 6120 2a6e 6f74 6520 6d61 7472   F, a *note matr
│ │ │ │ +00028990: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ +000289a0: 6329 4d61 7472 6978 2c2c 204d 6170 2066  c)Matrix,, Map f
│ │ │ │ +000289b0: 726f 6d20 6120 6672 6565 206d 6f64 756c  rom a free modul
│ │ │ │ +000289c0: 6520 746f 204d 2e0a 2020 2020 2020 2020  e to M..        
│ │ │ │ +000289d0: 496d 6167 6520 6973 2074 6865 206c 6172  Image is the lar
│ │ │ │ +000289e0: 6765 7374 2066 7265 6520 7375 6d6d 616e  gest free summan
│ │ │ │ +000289f0: 640a 0a44 6573 6372 6970 7469 6f6e 0a3d  d..Description.=
│ │ │ │ +00028a00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d  ==========..+---
│ │ │ │ +00028a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028a60: 2d2b 0a7c 6931 203a 206b 6b3d 205a 5a2f  -+.|i1 : kk= ZZ/
│ │ │ │ -00028a70: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │ +00028a40: 2b0a 7c69 3120 3a20 6b6b 3d20 5a5a 2f31  +.|i1 : kk= ZZ/1
│ │ │ │ +00028a50: 3031 2020 2020 2020 2020 2020 2020 2020  01              
│ │ │ │ +00028a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00028a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00028ad0: 6f31 203d 206b 6b20 2020 2020 2020 2020  o1 = kk         
│ │ │ │ -00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00028ab0: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +00028ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ae0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00028af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00028b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b30: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ -00028b40: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ -00028b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028b70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00028b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028ba0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2045 203d  -----+.|i2 : E =
│ │ │ │ -00028bb0: 206b 6b5b 652c 662c 672c 2053 6b65 7743   kk[e,f,g, SkewC
│ │ │ │ -00028bc0: 6f6d 6d75 7461 7469 7665 203d 3e20 7472  ommutative => tr
│ │ │ │ -00028bd0: 7565 5d20 2020 2020 2020 207c 0a7c 2020  ue]        |.|  
│ │ │ │ +00028b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b10: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +00028b20: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00028b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00028b50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00028b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028b80: 2d2d 2d2d 2b0a 7c69 3220 3a20 4520 3d20  ----+.|i2 : E = 
│ │ │ │ +00028b90: 6b6b 5b65 2c66 2c67 2c20 536b 6577 436f  kk[e,f,g, SkewCo
│ │ │ │ +00028ba0: 6d6d 7574 6174 6976 6520 3d3e 2074 7275  mmutative => tru
│ │ │ │ +00028bb0: 655d 2020 2020 2020 2020 7c0a 7c20 2020  e]        |.|   
│ │ │ │ +00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028bf0: 7c0a 7c6f 3220 3d20 4520 2020 2020 2020  |.|o2 = E       
│ │ │ │  00028c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c10: 207c 0a7c 6f32 203d 2045 2020 2020 2020   |.|o2 = E      
│ │ │ │ -00028c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00028c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00028c80: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52  o2 : PolynomialR
│ │ │ │ -00028c90: 696e 672c 2033 2073 6b65 7720 636f 6d6d  ing, 3 skew comm
│ │ │ │ -00028ca0: 7574 6174 6976 6520 7661 7269 6162 6c65  utative variable
│ │ │ │ -00028cb0: 2873 297c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  (s)|.+----------
│ │ │ │ -00028cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028ce0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -00028cf0: 204d 203d 2045 5e31 2b2b 6d6f 6475 6c65   M = E^1++module
│ │ │ │ -00028d00: 2069 6465 616c 2076 6172 7320 452b 2b45   ideal vars E++E
│ │ │ │ -00028d10: 5e7b 2d31 7d20 2020 2020 2020 2020 207c  ^{-1}          |
│ │ │ │ -00028d20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00028d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d50: 2020 2020 207c 0a7c 6f33 203d 2069 6d61       |.|o3 = ima
│ │ │ │ -00028d60: 6765 207b 307d 207c 2031 2030 2030 2030  ge {0} | 1 0 0 0
│ │ │ │ -00028d70: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ -00028d80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00028d90: 2020 2020 2020 2020 207b 307d 207c 2030           {0} | 0
│ │ │ │ -00028da0: 2065 2066 2067 2030 207c 2020 2020 2020   e f g 0 |      
│ │ │ │ -00028db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028dc0: 207c 0a7c 2020 2020 2020 2020 2020 207b   |.|           {
│ │ │ │ -00028dd0: 317d 207c 2030 2030 2030 2030 2031 207c  1} | 0 0 0 0 1 |
│ │ │ │ +00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00028c60: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269  2 : PolynomialRi
│ │ │ │ +00028c70: 6e67 2c20 3320 736b 6577 2063 6f6d 6d75  ng, 3 skew commu
│ │ │ │ +00028c80: 7461 7469 7665 2076 6172 6961 626c 6528  tative variable(
│ │ │ │ +00028c90: 7329 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d  s)|.+-----------
│ │ │ │ +00028ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028cc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +00028cd0: 4d20 3d20 455e 312b 2b6d 6f64 756c 6520  M = E^1++module 
│ │ │ │ +00028ce0: 6964 6561 6c20 7661 7273 2045 2b2b 455e  ideal vars E++E^
│ │ │ │ +00028cf0: 7b2d 317d 2020 2020 2020 2020 2020 7c0a  {-1}          |.
│ │ │ │ +00028d00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00028d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028d30: 2020 2020 7c0a 7c6f 3320 3d20 696d 6167      |.|o3 = imag
│ │ │ │ +00028d40: 6520 7b30 7d20 7c20 3120 3020 3020 3020  e {0} | 1 0 0 0 
│ │ │ │ +00028d50: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00028d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00028d70: 2020 2020 2020 2020 7b30 7d20 7c20 3020          {0} | 0 
│ │ │ │ +00028d80: 6520 6620 6720 3020 7c20 2020 2020 2020  e f g 0 |       
│ │ │ │ +00028d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028da0: 7c0a 7c20 2020 2020 2020 2020 2020 7b31  |.|           {1
│ │ │ │ +00028db0: 7d20 7c20 3020 3020 3020 3020 3120 7c20  } | 0 0 0 0 1 | 
│ │ │ │ +00028dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028dd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00028de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028df0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00028e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00028e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00028e20: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │  00028e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e40: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ -00028e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e60: 2020 207c 0a7c 6f33 203a 2045 2d6d 6f64     |.|o3 : E-mod
│ │ │ │ -00028e70: 756c 652c 2073 7562 6d6f 6475 6c65 206f  ule, submodule o
│ │ │ │ -00028e80: 6620 4520 2020 2020 2020 2020 2020 2020  f E             
│ │ │ │ -00028e90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00028ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00028ed0: 0a7c 6934 203a 2066 7265 6545 7874 6572  .|i4 : freeExter
│ │ │ │ -00028ee0: 696f 7253 756d 6d61 6e64 204d 2020 2020  iorSummand M    
│ │ │ │ +00028e40: 2020 7c0a 7c6f 3320 3a20 452d 6d6f 6475    |.|o3 : E-modu
│ │ │ │ +00028e50: 6c65 2c20 7375 626d 6f64 756c 6520 6f66  le, submodule of
│ │ │ │ +00028e60: 2045 2020 2020 2020 2020 2020 2020 2020   E              
│ │ │ │ +00028e70: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00028e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00028eb0: 7c69 3420 3a20 6672 6565 4578 7465 7269  |i4 : freeExteri
│ │ │ │ +00028ec0: 6f72 5375 6d6d 616e 6420 4d20 2020 2020  orSummand M     
│ │ │ │ +00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ee0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00028ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00028f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f30: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -00028f40: 203d 207b 307d 207c 2031 2030 207c 2020   = {0} | 1 0 |  
│ │ │ │ -00028f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f70: 207c 0a7c 2020 2020 207b 317d 207c 2030   |.|     {1} | 0
│ │ │ │ -00028f80: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ -00028f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028fa0: 2020 2020 2020 207c 0a7c 2020 2020 207b         |.|     {
│ │ │ │ -00028fb0: 317d 207c 2030 2030 207c 2020 2020 2020  1} | 0 0 |      
│ │ │ │ -00028fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00028fe0: 2020 2020 207b 317d 207c 2030 2030 207c       {1} | 0 0 |
│ │ │ │ -00028ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029010: 2020 207c 0a7c 2020 2020 207b 317d 207c     |.|     {1} |
│ │ │ │ -00029020: 2030 2031 207c 2020 2020 2020 2020 2020   0 1 |          
│ │ │ │ +00028f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f10: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +00028f20: 3d20 7b30 7d20 7c20 3120 3020 7c20 2020  = {0} | 1 0 |   
│ │ │ │ +00028f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f50: 7c0a 7c20 2020 2020 7b31 7d20 7c20 3020  |.|     {1} | 0 
│ │ │ │ +00028f60: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ +00028f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f80: 2020 2020 2020 7c0a 7c20 2020 2020 7b31        |.|     {1
│ │ │ │ +00028f90: 7d20 7c20 3020 3020 7c20 2020 2020 2020  } | 0 0 |       
│ │ │ │ +00028fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028fb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00028fc0: 2020 2020 7b31 7d20 7c20 3020 3020 7c20      {1} | 0 0 | 
│ │ │ │ +00028fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028ff0: 2020 7c0a 7c20 2020 2020 7b31 7d20 7c20    |.|     {1} | 
│ │ │ │ +00029000: 3020 3120 7c20 2020 2020 2020 2020 2020  0 1 |           
│ │ │ │ +00029010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029020: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  00029030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029040: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00029050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029080: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029090: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -000290a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000290b0: 2020 2020 207c 0a7c 6f34 203a 204d 6174       |.|o4 : Mat
│ │ │ │ -000290c0: 7269 7820 4d20 3c2d 2d20 4520 2020 2020  rix M <-- E     
│ │ │ │ -000290d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000290e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00029040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00029060: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00029070: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00029080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029090: 2020 2020 7c0a 7c6f 3420 3a20 4d61 7472      |.|o4 : Matr
│ │ │ │ +000290a0: 6978 204d 203c 2d2d 2045 2020 2020 2020  ix M <-- E      
│ │ │ │ +000290b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000290c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000290d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000290e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000290f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029120: 2d2b 0a0a 5761 7973 2074 6f20 7573 6520  -+..Ways to use 
│ │ │ │ -00029130: 6672 6565 4578 7465 7269 6f72 5375 6d6d  freeExteriorSumm
│ │ │ │ -00029140: 616e 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  and:.===========
│ │ │ │ -00029150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00029160: 3d3d 3d3d 3d0a 0a20 202a 2022 6672 6565  =====..  * "free
│ │ │ │ -00029170: 4578 7465 7269 6f72 5375 6d6d 616e 6428  ExteriorSummand(
│ │ │ │ -00029180: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ -00029190: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -000291a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -000291b0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -000291c0: 6520 6672 6565 4578 7465 7269 6f72 5375  e freeExteriorSu
│ │ │ │ -000291d0: 6d6d 616e 643a 2066 7265 6545 7874 6572  mmand: freeExter
│ │ │ │ -000291e0: 696f 7253 756d 6d61 6e64 2c20 6973 2061  iorSummand, is a
│ │ │ │ -000291f0: 202a 6e6f 7465 206d 6574 686f 640a 6675   *note method.fu
│ │ │ │ -00029200: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61  nction: (Macaula
│ │ │ │ -00029210: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -00029220: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  tion,...--------
│ │ │ │ +00029100: 2b0a 0a57 6179 7320 746f 2075 7365 2066  +..Ways to use f
│ │ │ │ +00029110: 7265 6545 7874 6572 696f 7253 756d 6d61  reeExteriorSumma
│ │ │ │ +00029120: 6e64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  nd:.============
│ │ │ │ +00029130: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00029140: 3d3d 3d3d 0a0a 2020 2a20 2266 7265 6545  ====..  * "freeE
│ │ │ │ +00029150: 7874 6572 696f 7253 756d 6d61 6e64 284d  xteriorSummand(M
│ │ │ │ +00029160: 6f64 756c 6529 220a 0a46 6f72 2074 6865  odule)"..For the
│ │ │ │ +00029170: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +00029180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00029190: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +000291a0: 2066 7265 6545 7874 6572 696f 7253 756d   freeExteriorSum
│ │ │ │ +000291b0: 6d61 6e64 3a20 6672 6565 4578 7465 7269  mand: freeExteri
│ │ │ │ +000291c0: 6f72 5375 6d6d 616e 642c 2069 7320 6120  orSummand, is a 
│ │ │ │ +000291d0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ +000291e0: 6374 696f 6e3a 2028 4d61 6361 756c 6179  ction: (Macaulay
│ │ │ │ +000291f0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +00029200: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +00029210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029270: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00029280: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -00029290: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -000292a0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -000292b0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -000292c0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ -000292d0: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -000292e0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -000292f0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00029300: 6d32 3a34 3238 353a 302e 0a1f 0a46 696c  m2:4285:0....Fil
│ │ │ │ -00029310: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -00029320: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00029330: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2047  ns.info, Node: G
│ │ │ │ -00029340: 7261 6469 6e67 2c20 4e65 7874 3a20 6866  rading, Next: hf
│ │ │ │ -00029350: 2c20 5072 6576 3a20 6672 6565 4578 7465  , Prev: freeExte
│ │ │ │ -00029360: 7269 6f72 5375 6d6d 616e 642c 2055 703a  riorSummand, Up:
│ │ │ │ -00029370: 2054 6f70 0a0a 4772 6164 696e 6720 2d2d   Top..Grading --
│ │ │ │ -00029380: 204f 7074 696f 6e20 666f 7220 4569 7365   Option for Eise
│ │ │ │ -00029390: 6e62 7564 5368 616d 6173 6854 6f74 616c  nbudShamashTotal
│ │ │ │ -000293a0: 2c20 6e65 7745 7874 0a2a 2a2a 2a2a 2a2a  , newExt.*******
│ │ │ │ -000293b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000293c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000293d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -000293e0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -000293f0: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ -00029400: 546f 7461 6c28 4d62 6172 2c47 7261 6469  Total(Mbar,Gradi
│ │ │ │ -00029410: 6e67 203d 3e20 3229 0a20 202a 2049 6e70  ng => 2).  * Inp
│ │ │ │ -00029420: 7574 733a 0a20 2020 2020 202a 2043 6865  uts:.      * Che
│ │ │ │ -00029430: 636b 2c20 616e 202a 6e6f 7465 2069 6e74  ck, an *note int
│ │ │ │ -00029440: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ -00029450: 446f 6329 5a5a 2c2c 200a 0a44 6573 6372  Doc)ZZ,, ..Descr
│ │ │ │ -00029460: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00029470: 3d3d 0a0a 6966 2047 7261 6469 6e67 203d  ==..if Grading =
│ │ │ │ -00029480: 3e31 2c20 7468 656e 2074 6865 206f 7574  >1, then the out
│ │ │ │ -00029490: 7075 7420 6973 2063 6f6e 7665 7274 6564  put is converted
│ │ │ │ -000294a0: 2074 6f20 7369 6e67 6c65 2d67 7261 6469   to single-gradi
│ │ │ │ -000294b0: 6e67 2c20 7573 6566 756c 2069 6e20 7468  ng, useful in th
│ │ │ │ -000294c0: 650a 7061 636b 6167 6520 436c 6966 666f  e.package Cliffo
│ │ │ │ -000294d0: 7264 0a0a 5365 6520 616c 736f 0a3d 3d3d  rd..See also.===
│ │ │ │ -000294e0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -000294f0: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ -00029500: 546f 7461 6c3a 2045 6973 656e 6275 6453  Total: EisenbudS
│ │ │ │ -00029510: 6861 6d61 7368 546f 7461 6c2c 202d 2d20  hamashTotal, -- 
│ │ │ │ -00029520: 5072 6563 7572 736f 7220 636f 6d70 6c65  Precursor comple
│ │ │ │ -00029530: 7820 6f66 0a20 2020 2074 6f74 616c 2045  x of.    total E
│ │ │ │ -00029540: 7874 0a20 202a 202a 6e6f 7465 206e 6577  xt.  * *note new
│ │ │ │ -00029550: 4578 743a 206e 6577 4578 742c 202d 2d20  Ext: newExt, -- 
│ │ │ │ -00029560: 476c 6f62 616c 2045 7874 2066 6f72 206d  Global Ext for m
│ │ │ │ -00029570: 6f64 756c 6573 206f 7665 7220 6120 636f  odules over a co
│ │ │ │ -00029580: 6d70 6c65 7465 0a20 2020 2049 6e74 6572  mplete.    Inter
│ │ │ │ -00029590: 7365 6374 696f 6e0a 0a46 756e 6374 696f  section..Functio
│ │ │ │ -000295a0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ -000295b0: 2061 7267 756d 656e 7420 6e61 6d65 6420   argument named 
│ │ │ │ -000295c0: 4772 6164 696e 673a 0a3d 3d3d 3d3d 3d3d  Grading:.=======
│ │ │ │ -000295d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000295e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000295f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245  ========..  * "E
│ │ │ │ -00029600: 6973 656e 6275 6453 6861 6d61 7368 546f  isenbudShamashTo
│ │ │ │ -00029610: 7461 6c28 2e2e 2e2c 4772 6164 696e 673d  tal(...,Grading=
│ │ │ │ -00029620: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e  >...)" -- see *n
│ │ │ │ -00029630: 6f74 6520 4569 7365 6e62 7564 5368 616d  ote EisenbudSham
│ │ │ │ -00029640: 6173 6854 6f74 616c 3a0a 2020 2020 4569  ashTotal:.    Ei
│ │ │ │ -00029650: 7365 6e62 7564 5368 616d 6173 6854 6f74  senbudShamashTot
│ │ │ │ -00029660: 616c 2c20 2d2d 2050 7265 6375 7273 6f72  al, -- Precursor
│ │ │ │ -00029670: 2063 6f6d 706c 6578 206f 6620 746f 7461   complex of tota
│ │ │ │ -00029680: 6c20 4578 740a 2020 2a20 226e 6577 4578  l Ext.  * "newEx
│ │ │ │ -00029690: 7428 2e2e 2e2c 4772 6164 696e 673d 3e2e  t(...,Grading=>.
│ │ │ │ -000296a0: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74  ..)" -- see *not
│ │ │ │ -000296b0: 6520 6e65 7745 7874 3a20 6e65 7745 7874  e newExt: newExt
│ │ │ │ -000296c0: 2c20 2d2d 2047 6c6f 6261 6c20 4578 7420  , -- Global Ext 
│ │ │ │ -000296d0: 666f 720a 2020 2020 6d6f 6475 6c65 7320  for.    modules 
│ │ │ │ -000296e0: 6f76 6572 2061 2063 6f6d 706c 6574 6520  over a complete 
│ │ │ │ -000296f0: 496e 7465 7273 6563 7469 6f6e 0a0a 466f  Intersection..Fo
│ │ │ │ -00029700: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00029710: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00029720: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00029730: 2a6e 6f74 6520 4772 6164 696e 673a 2047  *note Grading: G
│ │ │ │ -00029740: 7261 6469 6e67 2c20 6973 2061 202a 6e6f  rading, is a *no
│ │ │ │ -00029750: 7465 2073 796d 626f 6c3a 2028 4d61 6361  te symbol: (Maca
│ │ │ │ -00029760: 756c 6179 3244 6f63 2953 796d 626f 6c2c  ulay2Doc)Symbol,
│ │ │ │ -00029770: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +00029250: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +00029260: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +00029270: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00029280: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00029290: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +000292a0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ +000292b0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ +000292c0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +000292d0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ +000292e0: 323a 3432 3835 3a30 2e0a 1f0a 4669 6c65  2:4285:0....File
│ │ │ │ +000292f0: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ +00029300: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00029310: 732e 696e 666f 2c20 4e6f 6465 3a20 4772  s.info, Node: Gr
│ │ │ │ +00029320: 6164 696e 672c 204e 6578 743a 2068 662c  ading, Next: hf,
│ │ │ │ +00029330: 2050 7265 763a 2066 7265 6545 7874 6572   Prev: freeExter
│ │ │ │ +00029340: 696f 7253 756d 6d61 6e64 2c20 5570 3a20  iorSummand, Up: 
│ │ │ │ +00029350: 546f 700a 0a47 7261 6469 6e67 202d 2d20  Top..Grading -- 
│ │ │ │ +00029360: 4f70 7469 6f6e 2066 6f72 2045 6973 656e  Option for Eisen
│ │ │ │ +00029370: 6275 6453 6861 6d61 7368 546f 7461 6c2c  budShamashTotal,
│ │ │ │ +00029380: 206e 6577 4578 740a 2a2a 2a2a 2a2a 2a2a   newExt.********
│ │ │ │ +00029390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000293a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000293b0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +000293c0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +000293d0: 4569 7365 6e62 7564 5368 616d 6173 6854  EisenbudShamashT
│ │ │ │ +000293e0: 6f74 616c 284d 6261 722c 4772 6164 696e  otal(Mbar,Gradin
│ │ │ │ +000293f0: 6720 3d3e 2032 290a 2020 2a20 496e 7075  g => 2).  * Inpu
│ │ │ │ +00029400: 7473 3a0a 2020 2020 2020 2a20 4368 6563  ts:.      * Chec
│ │ │ │ +00029410: 6b2c 2061 6e20 2a6e 6f74 6520 696e 7465  k, an *note inte
│ │ │ │ +00029420: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ +00029430: 6f63 295a 5a2c 2c20 0a0a 4465 7363 7269  oc)ZZ,, ..Descri
│ │ │ │ +00029440: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +00029450: 3d0a 0a69 6620 4772 6164 696e 6720 3d3e  =..if Grading =>
│ │ │ │ +00029460: 312c 2074 6865 6e20 7468 6520 6f75 7470  1, then the outp
│ │ │ │ +00029470: 7574 2069 7320 636f 6e76 6572 7465 6420  ut is converted 
│ │ │ │ +00029480: 746f 2073 696e 676c 652d 6772 6164 696e  to single-gradin
│ │ │ │ +00029490: 672c 2075 7365 6675 6c20 696e 2074 6865  g, useful in the
│ │ │ │ +000294a0: 0a70 6163 6b61 6765 2043 6c69 6666 6f72  .package Cliffor
│ │ │ │ +000294b0: 640a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  d..See also.====
│ │ │ │ +000294c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +000294d0: 4569 7365 6e62 7564 5368 616d 6173 6854  EisenbudShamashT
│ │ │ │ +000294e0: 6f74 616c 3a20 4569 7365 6e62 7564 5368  otal: EisenbudSh
│ │ │ │ +000294f0: 616d 6173 6854 6f74 616c 2c20 2d2d 2050  amashTotal, -- P
│ │ │ │ +00029500: 7265 6375 7273 6f72 2063 6f6d 706c 6578  recursor complex
│ │ │ │ +00029510: 206f 660a 2020 2020 746f 7461 6c20 4578   of.    total Ex
│ │ │ │ +00029520: 740a 2020 2a20 2a6e 6f74 6520 6e65 7745  t.  * *note newE
│ │ │ │ +00029530: 7874 3a20 6e65 7745 7874 2c20 2d2d 2047  xt: newExt, -- G
│ │ │ │ +00029540: 6c6f 6261 6c20 4578 7420 666f 7220 6d6f  lobal Ext for mo
│ │ │ │ +00029550: 6475 6c65 7320 6f76 6572 2061 2063 6f6d  dules over a com
│ │ │ │ +00029560: 706c 6574 650a 2020 2020 496e 7465 7273  plete.    Inters
│ │ │ │ +00029570: 6563 7469 6f6e 0a0a 4675 6e63 7469 6f6e  ection..Function
│ │ │ │ +00029580: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ +00029590: 6172 6775 6d65 6e74 206e 616d 6564 2047  argument named G
│ │ │ │ +000295a0: 7261 6469 6e67 3a0a 3d3d 3d3d 3d3d 3d3d  rading:.========
│ │ │ │ +000295b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000295c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000295d0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4569  =======..  * "Ei
│ │ │ │ +000295e0: 7365 6e62 7564 5368 616d 6173 6854 6f74  senbudShamashTot
│ │ │ │ +000295f0: 616c 282e 2e2e 2c47 7261 6469 6e67 3d3e  al(...,Grading=>
│ │ │ │ +00029600: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f  ...)" -- see *no
│ │ │ │ +00029610: 7465 2045 6973 656e 6275 6453 6861 6d61  te EisenbudShama
│ │ │ │ +00029620: 7368 546f 7461 6c3a 0a20 2020 2045 6973  shTotal:.    Eis
│ │ │ │ +00029630: 656e 6275 6453 6861 6d61 7368 546f 7461  enbudShamashTota
│ │ │ │ +00029640: 6c2c 202d 2d20 5072 6563 7572 736f 7220  l, -- Precursor 
│ │ │ │ +00029650: 636f 6d70 6c65 7820 6f66 2074 6f74 616c  complex of total
│ │ │ │ +00029660: 2045 7874 0a20 202a 2022 6e65 7745 7874   Ext.  * "newExt
│ │ │ │ +00029670: 282e 2e2e 2c47 7261 6469 6e67 3d3e 2e2e  (...,Grading=>..
│ │ │ │ +00029680: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465  .)" -- see *note
│ │ │ │ +00029690: 206e 6577 4578 743a 206e 6577 4578 742c   newExt: newExt,
│ │ │ │ +000296a0: 202d 2d20 476c 6f62 616c 2045 7874 2066   -- Global Ext f
│ │ │ │ +000296b0: 6f72 0a20 2020 206d 6f64 756c 6573 206f  or.    modules o
│ │ │ │ +000296c0: 7665 7220 6120 636f 6d70 6c65 7465 2049  ver a complete I
│ │ │ │ +000296d0: 6e74 6572 7365 6374 696f 6e0a 0a46 6f72  ntersection..For
│ │ │ │ +000296e0: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +000296f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00029700: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +00029710: 6e6f 7465 2047 7261 6469 6e67 3a20 4772  note Grading: Gr
│ │ │ │ +00029720: 6164 696e 672c 2069 7320 6120 2a6e 6f74  ading, is a *not
│ │ │ │ +00029730: 6520 7379 6d62 6f6c 3a20 284d 6163 6175  e symbol: (Macau
│ │ │ │ +00029740: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e  lay2Doc)Symbol,.
│ │ │ │ +00029750: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │ +00029760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000297a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000297b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000297c0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -000297d0: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -000297e0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -000297f0: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -00029800: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ -00029810: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -00029820: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ -00029830: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -00029840: 6573 6f6c 7574 696f 6e73 2e6d 323a 3332  esolutions.m2:32
│ │ │ │ -00029850: 3136 3a30 2e0a 1f0a 4669 6c65 3a20 436f  16:0....File: Co
│ │ │ │ -00029860: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -00029870: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ -00029880: 666f 2c20 4e6f 6465 3a20 6866 2c20 4e65  fo, Node: hf, Ne
│ │ │ │ -00029890: 7874 3a20 6866 4d6f 6475 6c65 4173 4578  xt: hfModuleAsEx
│ │ │ │ -000298a0: 742c 2050 7265 763a 2047 7261 6469 6e67  t, Prev: Grading
│ │ │ │ -000298b0: 2c20 5570 3a20 546f 700a 0a68 6620 2d2d  , Up: Top..hf --
│ │ │ │ -000298c0: 2043 6f6d 7075 7465 7320 7468 6520 6869   Computes the hi
│ │ │ │ -000298d0: 6c62 6572 7420 6675 6e63 7469 6f6e 2069  lbert function i
│ │ │ │ -000298e0: 6e20 6120 7261 6e67 6520 6f66 2064 6567  n a range of deg
│ │ │ │ -000298f0: 7265 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  rees.***********
│ │ │ │ -00029900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -00029930: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00029940: 2020 2020 4820 3d20 6866 2873 2c50 290a      H = hf(s,P).
│ │ │ │ -00029950: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ -00029960: 2020 2a20 732c 2061 202a 6e6f 7465 2073    * s, a *note s
│ │ │ │ -00029970: 6571 7565 6e63 653a 2028 4d61 6361 756c  equence: (Macaul
│ │ │ │ -00029980: 6179 3244 6f63 2953 6571 7565 6e63 652c  ay2Doc)Sequence,
│ │ │ │ -00029990: 2c20 6f72 204c 6973 740a 2020 2020 2020  , or List.      
│ │ │ │ -000299a0: 2a20 502c 2061 202a 6e6f 7465 206d 6f64  * P, a *note mod
│ │ │ │ -000299b0: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -000299c0: 6f63 294d 6f64 756c 652c 2c20 6772 6164  oc)Module,, grad
│ │ │ │ -000299d0: 6564 206d 6f64 756c 650a 2020 2a20 4f75  ed module.  * Ou
│ │ │ │ -000299e0: 7470 7574 733a 0a20 2020 2020 202a 2048  tputs:.      * H
│ │ │ │ -000299f0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -00029a00: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -00029a10: 7374 2c2c 200a 0a57 6179 7320 746f 2075  st,, ..Ways to u
│ │ │ │ -00029a20: 7365 2068 663a 0a3d 3d3d 3d3d 3d3d 3d3d  se hf:.=========
│ │ │ │ -00029a30: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2268 6628  ======..  * "hf(
│ │ │ │ -00029a40: 4c69 7374 2c4d 6f64 756c 6529 220a 2020  List,Module)".  
│ │ │ │ -00029a50: 2a20 2268 6628 5365 7175 656e 6365 2c4d  * "hf(Sequence,M
│ │ │ │ -00029a60: 6f64 756c 6529 220a 0a46 6f72 2074 6865  odule)"..For the
│ │ │ │ -00029a70: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00029a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00029a90: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00029aa0: 2068 663a 2068 662c 2069 7320 6120 2a6e   hf: hf, is a *n
│ │ │ │ -00029ab0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -00029ac0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244  ion:.(Macaulay2D
│ │ │ │ -00029ad0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -00029ae0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  n,...-----------
│ │ │ │ +000297a0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ +000297b0: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ +000297c0: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ +000297d0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ +000297e0: 6361 756c 6179 322d 312e 3236 2e30 362b  caulay2-1.26.06+
│ │ │ │ +000297f0: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ +00029800: 7061 636b 6167 6573 2f0a 436f 6d70 6c65  packages/.Comple
│ │ │ │ +00029810: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +00029820: 736f 6c75 7469 6f6e 732e 6d32 3a33 3231  solutions.m2:321
│ │ │ │ +00029830: 363a 302e 0a1f 0a46 696c 653a 2043 6f6d  6:0....File: Com
│ │ │ │ +00029840: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ +00029850: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66  nResolutions.inf
│ │ │ │ +00029860: 6f2c 204e 6f64 653a 2068 662c 204e 6578  o, Node: hf, Nex
│ │ │ │ +00029870: 743a 2068 664d 6f64 756c 6541 7345 7874  t: hfModuleAsExt
│ │ │ │ +00029880: 2c20 5072 6576 3a20 4772 6164 696e 672c  , Prev: Grading,
│ │ │ │ +00029890: 2055 703a 2054 6f70 0a0a 6866 202d 2d20   Up: Top..hf -- 
│ │ │ │ +000298a0: 436f 6d70 7574 6573 2074 6865 2068 696c  Computes the hil
│ │ │ │ +000298b0: 6265 7274 2066 756e 6374 696f 6e20 696e  bert function in
│ │ │ │ +000298c0: 2061 2072 616e 6765 206f 6620 6465 6772   a range of degr
│ │ │ │ +000298d0: 6565 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ees.************
│ │ │ │ +000298e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000298f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00029900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ +00029910: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +00029920: 2020 2048 203d 2068 6628 732c 5029 0a20     H = hf(s,P). 
│ │ │ │ +00029930: 202a 2049 6e70 7574 733a 0a20 2020 2020   * Inputs:.     
│ │ │ │ +00029940: 202a 2073 2c20 6120 2a6e 6f74 6520 7365   * s, a *note se
│ │ │ │ +00029950: 7175 656e 6365 3a20 284d 6163 6175 6c61  quence: (Macaula
│ │ │ │ +00029960: 7932 446f 6329 5365 7175 656e 6365 2c2c  y2Doc)Sequence,,
│ │ │ │ +00029970: 206f 7220 4c69 7374 0a20 2020 2020 202a   or List.      *
│ │ │ │ +00029980: 2050 2c20 6120 2a6e 6f74 6520 6d6f 6475   P, a *note modu
│ │ │ │ +00029990: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +000299a0: 6329 4d6f 6475 6c65 2c2c 2067 7261 6465  c)Module,, grade
│ │ │ │ +000299b0: 6420 6d6f 6475 6c65 0a20 202a 204f 7574  d module.  * Out
│ │ │ │ +000299c0: 7075 7473 3a0a 2020 2020 2020 2a20 482c  puts:.      * H,
│ │ │ │ +000299d0: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ +000299e0: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ +000299f0: 742c 2c20 0a0a 5761 7973 2074 6f20 7573  t,, ..Ways to us
│ │ │ │ +00029a00: 6520 6866 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  e hf:.==========
│ │ │ │ +00029a10: 3d3d 3d3d 3d0a 0a20 202a 2022 6866 284c  =====..  * "hf(L
│ │ │ │ +00029a20: 6973 742c 4d6f 6475 6c65 2922 0a20 202a  ist,Module)".  *
│ │ │ │ +00029a30: 2022 6866 2853 6571 7565 6e63 652c 4d6f   "hf(Sequence,Mo
│ │ │ │ +00029a40: 6475 6c65 2922 0a0a 466f 7220 7468 6520  dule)"..For the 
│ │ │ │ +00029a50: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00029a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00029a70: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +00029a80: 6866 3a20 6866 2c20 6973 2061 202a 6e6f  hf: hf, is a *no
│ │ │ │ +00029a90: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +00029aa0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ +00029ab0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00029ac0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +00029ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029b30: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -00029b40: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -00029b50: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00029b60: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -00029b70: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e  /macaulay2-1.26.
│ │ │ │ -00029b80: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61  06+ds/M2/Macaula
│ │ │ │ -00029b90: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ -00029ba0: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -00029bb0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ -00029bc0: 3435 3932 3a30 2e0a 1f0a 4669 6c65 3a20  4592:0....File: 
│ │ │ │ -00029bd0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00029be0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00029bf0: 696e 666f 2c20 4e6f 6465 3a20 6866 4d6f  info, Node: hfMo
│ │ │ │ -00029c00: 6475 6c65 4173 4578 742c 204e 6578 743a  duleAsExt, Next:
│ │ │ │ -00029c10: 2068 6967 6853 797a 7967 792c 2050 7265   highSyzygy, Pre
│ │ │ │ -00029c20: 763a 2068 662c 2055 703a 2054 6f70 0a0a  v: hf, Up: Top..
│ │ │ │ -00029c30: 6866 4d6f 6475 6c65 4173 4578 7420 2d2d  hfModuleAsExt --
│ │ │ │ -00029c40: 2070 7265 6469 6374 2062 6574 7469 206e   predict betti n
│ │ │ │ -00029c50: 756d 6265 7273 206f 6620 6d6f 6475 6c65  umbers of module
│ │ │ │ -00029c60: 4173 4578 7428 4d2c 5229 0a2a 2a2a 2a2a  AsExt(M,R).*****
│ │ │ │ +00029b10: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +00029b20: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +00029b30: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +00029b40: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +00029b50: 6d61 6361 756c 6179 322d 312e 3236 2e30  macaulay2-1.26.0
│ │ │ │ +00029b60: 362b 6473 2f4d 322f 4d61 6361 756c 6179  6+ds/M2/Macaulay
│ │ │ │ +00029b70: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ +00029b80: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ +00029b90: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34  Resolutions.m2:4
│ │ │ │ +00029ba0: 3539 323a 302e 0a1f 0a46 696c 653a 2043  592:0....File: C
│ │ │ │ +00029bb0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +00029bc0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ +00029bd0: 6e66 6f2c 204e 6f64 653a 2068 664d 6f64  nfo, Node: hfMod
│ │ │ │ +00029be0: 756c 6541 7345 7874 2c20 4e65 7874 3a20  uleAsExt, Next: 
│ │ │ │ +00029bf0: 6869 6768 5379 7a79 6779 2c20 5072 6576  highSyzygy, Prev
│ │ │ │ +00029c00: 3a20 6866 2c20 5570 3a20 546f 700a 0a68  : hf, Up: Top..h
│ │ │ │ +00029c10: 664d 6f64 756c 6541 7345 7874 202d 2d20  fModuleAsExt -- 
│ │ │ │ +00029c20: 7072 6564 6963 7420 6265 7474 6920 6e75  predict betti nu
│ │ │ │ +00029c30: 6d62 6572 7320 6f66 206d 6f64 756c 6541  mbers of moduleA
│ │ │ │ +00029c40: 7345 7874 284d 2c52 290a 2a2a 2a2a 2a2a  sExt(M,R).******
│ │ │ │ +00029c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00029c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00029c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00029ca0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -00029cb0: 3a20 0a20 2020 2020 2020 2073 6571 203d  : .        seq =
│ │ │ │ -00029cc0: 2068 664d 6f64 756c 6541 7345 7874 286e   hfModuleAsExt(n
│ │ │ │ -00029cd0: 756d 5661 6c75 6573 2c4d 2c6e 756d 6765  umValues,M,numge
│ │ │ │ -00029ce0: 6e73 5229 0a20 202a 2049 6e70 7574 733a  nsR).  * Inputs:
│ │ │ │ -00029cf0: 0a20 2020 2020 202a 206e 756d 5661 6c75  .      * numValu
│ │ │ │ -00029d00: 6573 2c20 616e 202a 6e6f 7465 2069 6e74  es, an *note int
│ │ │ │ -00029d10: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ -00029d20: 446f 6329 5a5a 2c2c 206e 756d 6265 7220  Doc)ZZ,, number 
│ │ │ │ -00029d30: 6f66 2076 616c 7565 7320 746f 0a20 2020  of values to.   
│ │ │ │ -00029d40: 2020 2020 2063 6f6d 7075 7465 0a20 2020       compute.   
│ │ │ │ -00029d50: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ -00029d60: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ -00029d70: 7932 446f 6329 4d6f 6475 6c65 2c2c 206d  y2Doc)Module,, m
│ │ │ │ -00029d80: 6f64 756c 6520 6f76 6572 2074 6865 2072  odule over the r
│ │ │ │ -00029d90: 696e 6720 6f66 0a20 2020 2020 2020 206f  ing of.        o
│ │ │ │ -00029da0: 7065 7261 746f 7273 0a20 2020 2020 202a  perators.      *
│ │ │ │ -00029db0: 206e 756d 6765 6e73 522c 2061 6e20 2a6e   numgensR, an *n
│ │ │ │ -00029dc0: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -00029dd0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -00029de0: 6e75 6d62 6572 206f 6620 6765 6e65 7261  number of genera
│ │ │ │ -00029df0: 746f 7273 206f 660a 2020 2020 2020 2020  tors of.        
│ │ │ │ -00029e00: 7468 6520 7461 7267 6574 2072 696e 670a  the target ring.
│ │ │ │ -00029e10: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00029e20: 2020 202a 2073 6571 2c20 6120 2a6e 6f74     * seq, a *not
│ │ │ │ -00029e30: 6520 7365 7175 656e 6365 3a20 284d 6163  e sequence: (Mac
│ │ │ │ -00029e40: 6175 6c61 7932 446f 6329 5365 7175 656e  aulay2Doc)Sequen
│ │ │ │ -00029e50: 6365 2c2c 2073 6571 7565 6e63 6520 6f66  ce,, sequence of
│ │ │ │ -00029e60: 206e 756d 5661 6c75 6573 0a20 2020 2020   numValues.     
│ │ │ │ -00029e70: 2020 2069 6e74 6567 6572 732c 2074 6865     integers, the
│ │ │ │ -00029e80: 2065 7870 6563 7465 6420 746f 7461 6c20   expected total 
│ │ │ │ -00029e90: 4265 7474 6920 6e75 6d62 6572 730a 0a44  Betti numbers..D
│ │ │ │ -00029ea0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00029eb0: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120  ======..Given a 
│ │ │ │ -00029ec0: 6d6f 6475 6c65 204d 206f 7665 7220 7468  module M over th
│ │ │ │ -00029ed0: 6520 7269 6e67 206f 6620 6f70 6572 6174  e ring of operat
│ │ │ │ -00029ee0: 6f72 7320 246b 5b78 5f31 2e2e 785f 635d  ors $k[x_1..x_c]
│ │ │ │ -00029ef0: 242c 2074 6865 2063 616c 6c20 244e 203d  $, the call $N =
│ │ │ │ -00029f00: 0a6d 6f64 756c 6541 7345 7874 284d 2c52  .moduleAsExt(M,R
│ │ │ │ -00029f10: 2924 2070 726f 6475 6365 7320 6120 6d6f  )$ produces a mo
│ │ │ │ -00029f20: 6475 6c65 204e 206f 7665 7220 7468 6520  dule N over the 
│ │ │ │ -00029f30: 7269 6e67 2052 2077 686f 7365 2065 7874  ring R whose ext
│ │ │ │ -00029f40: 206d 6f64 756c 6520 6973 2074 6865 0a65   module is the.e
│ │ │ │ -00029f50: 7874 6572 696f 7220 616c 6765 6272 6120  xterior algebra 
│ │ │ │ -00029f60: 6f6e 206e 3d6e 756d 6765 6e73 5220 6765  on n=numgensR ge
│ │ │ │ -00029f70: 6e65 7261 746f 7273 2074 656e 736f 7265  nerators tensore
│ │ │ │ -00029f80: 6420 7769 7468 204d 2e20 5468 6973 2073  d with M. This s
│ │ │ │ -00029f90: 6372 6970 7420 636f 6d70 7574 6573 0a6e  cript computes.n
│ │ │ │ -00029fa0: 756d 5661 6c75 6573 2076 616c 7565 7320  umValues values 
│ │ │ │ -00029fb0: 6f66 2074 6865 2048 696c 6265 7274 2066  of the Hilbert f
│ │ │ │ -00029fc0: 756e 6374 696f 6e20 6f66 2024 2420 4d20  unction of $$ M 
│ │ │ │ -00029fd0: 5c6f 7469 6d65 7320 5c77 6564 6765 206b  \otimes \wedge k
│ │ │ │ -00029fe0: 5e6e 2c20 2424 2077 6869 6368 0a73 686f  ^n, $$ which.sho
│ │ │ │ -00029ff0: 756c 6420 6265 2065 7175 616c 2074 6f20  uld be equal to 
│ │ │ │ -0002a000: 7468 6520 746f 7461 6c20 6265 7474 6920  the total betti 
│ │ │ │ -0002a010: 6e75 6d62 6572 7320 6f66 204e 2e0a 0a2b  numbers of N...+
│ │ │ │ -0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a050: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f  +.|i1 : kk = ZZ/
│ │ │ │ -0002a060: 3130 313b 2020 2020 2020 2020 2020 2020  101;            
│ │ │ │ -0002a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a080: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0002a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a0b0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320  ------+.|i2 : S 
│ │ │ │ -0002a0c0: 3d20 6b6b 5b61 2c62 2c63 5d3b 2020 2020  = kk[a,b,c];    
│ │ │ │ -0002a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -0002a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002a120: 3320 3a20 6666 203d 206d 6174 7269 787b  3 : ff = matrix{
│ │ │ │ -0002a130: 7b61 5e34 2c20 625e 342c 635e 347d 7d3b  {a^4, b^4,c^4}};
│ │ │ │ -0002a140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a150: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a180: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002a190: 2020 3120 2020 2020 2033 2020 2020 2020    1      3      
│ │ │ │ -0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a1b0: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ -0002a1c0: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ -0002a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a1e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0002a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -0002a220: 203a 2052 203d 2053 2f69 6465 616c 2066   : R = S/ideal f
│ │ │ │ -0002a230: 663b 2020 2020 2020 2020 2020 2020 2020  f;              
│ │ │ │ -0002a240: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002a250: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a280: 2d2b 0a7c 6935 203a 204f 7073 203d 206b  -+.|i5 : Ops = k
│ │ │ │ -0002a290: 6b5b 785f 312c 785f 322c 785f 335d 3b20  k[x_1,x_2,x_3]; 
│ │ │ │ -0002a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a2b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d  -------+.|i6 : M
│ │ │ │ -0002a2f0: 4d20 3d20 4f70 735e 312f 2878 5f31 2a69  M = Ops^1/(x_1*i
│ │ │ │ -0002a300: 6465 616c 2878 5f32 5e32 2c78 5f33 2929  deal(x_2^2,x_3))
│ │ │ │ -0002a310: 3b20 2020 2020 2020 2020 7c0a 2b2d 2d2d  ;         |.+---
│ │ │ │ -0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002a350: 6937 203a 204e 203d 206d 6f64 756c 6541  i7 : N = moduleA
│ │ │ │ -0002a360: 7345 7874 284d 4d2c 5229 3b20 2020 2020  sExt(MM,R);     
│ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a380: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0002a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a3b0: 2d2d 2d2b 0a7c 6938 203a 2062 6574 7469  ---+.|i8 : betti
│ │ │ │ -0002a3c0: 2066 7265 6552 6573 6f6c 7574 696f 6e28   freeResolution(
│ │ │ │ -0002a3d0: 204e 2c20 4c65 6e67 7468 4c69 6d69 7420   N, LengthLimit 
│ │ │ │ -0002a3e0: 3d3e 2031 3029 7c0a 7c20 2020 2020 2020  => 10)|.|       
│ │ │ │ -0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a410: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002a420: 2020 2020 2020 2020 2030 2020 3120 2032           0  1  2
│ │ │ │ -0002a430: 2020 3320 2034 2020 3520 2036 2020 3720    3  4  5  6  7 
│ │ │ │ -0002a440: 2038 2020 3920 3130 2020 2020 7c0a 7c6f   8  9 10    |.|o
│ │ │ │ -0002a450: 3820 3d20 746f 7461 6c3a 2033 3620 3237  8 = total: 36 27
│ │ │ │ -0002a460: 2032 3920 3331 2033 3320 3335 2033 3720   29 31 33 35 37 
│ │ │ │ -0002a470: 3339 2034 3120 3433 2034 3520 2020 207c  39 41 43 45    |
│ │ │ │ -0002a480: 0a7c 2020 2020 2020 2020 2d36 3a20 3138  .|        -6: 18
│ │ │ │ -0002a490: 2020 3620 202e 2020 2e20 202e 2020 2e20    6  .  .  .  . 
│ │ │ │ +00029c80: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +00029c90: 200a 2020 2020 2020 2020 7365 7120 3d20   .        seq = 
│ │ │ │ +00029ca0: 6866 4d6f 6475 6c65 4173 4578 7428 6e75  hfModuleAsExt(nu
│ │ │ │ +00029cb0: 6d56 616c 7565 732c 4d2c 6e75 6d67 656e  mValues,M,numgen
│ │ │ │ +00029cc0: 7352 290a 2020 2a20 496e 7075 7473 3a0a  sR).  * Inputs:.
│ │ │ │ +00029cd0: 2020 2020 2020 2a20 6e75 6d56 616c 7565        * numValue
│ │ │ │ +00029ce0: 732c 2061 6e20 2a6e 6f74 6520 696e 7465  s, an *note inte
│ │ │ │ +00029cf0: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ +00029d00: 6f63 295a 5a2c 2c20 6e75 6d62 6572 206f  oc)ZZ,, number o
│ │ │ │ +00029d10: 6620 7661 6c75 6573 2074 6f0a 2020 2020  f values to.    
│ │ │ │ +00029d20: 2020 2020 636f 6d70 7574 650a 2020 2020      compute.    
│ │ │ │ +00029d30: 2020 2a20 4d2c 2061 202a 6e6f 7465 206d    * M, a *note m
│ │ │ │ +00029d40: 6f64 756c 653a 2028 4d61 6361 756c 6179  odule: (Macaulay
│ │ │ │ +00029d50: 3244 6f63 294d 6f64 756c 652c 2c20 6d6f  2Doc)Module,, mo
│ │ │ │ +00029d60: 6475 6c65 206f 7665 7220 7468 6520 7269  dule over the ri
│ │ │ │ +00029d70: 6e67 206f 660a 2020 2020 2020 2020 6f70  ng of.        op
│ │ │ │ +00029d80: 6572 6174 6f72 730a 2020 2020 2020 2a20  erators.      * 
│ │ │ │ +00029d90: 6e75 6d67 656e 7352 2c20 616e 202a 6e6f  numgensR, an *no
│ │ │ │ +00029da0: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +00029db0: 6175 6c61 7932 446f 6329 5a5a 2c2c 206e  aulay2Doc)ZZ,, n
│ │ │ │ +00029dc0: 756d 6265 7220 6f66 2067 656e 6572 6174  umber of generat
│ │ │ │ +00029dd0: 6f72 7320 6f66 0a20 2020 2020 2020 2074  ors of.        t
│ │ │ │ +00029de0: 6865 2074 6172 6765 7420 7269 6e67 0a20  he target ring. 
│ │ │ │ +00029df0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +00029e00: 2020 2a20 7365 712c 2061 202a 6e6f 7465    * seq, a *note
│ │ │ │ +00029e10: 2073 6571 7565 6e63 653a 2028 4d61 6361   sequence: (Maca
│ │ │ │ +00029e20: 756c 6179 3244 6f63 2953 6571 7565 6e63  ulay2Doc)Sequenc
│ │ │ │ +00029e30: 652c 2c20 7365 7175 656e 6365 206f 6620  e,, sequence of 
│ │ │ │ +00029e40: 6e75 6d56 616c 7565 730a 2020 2020 2020  numValues.      
│ │ │ │ +00029e50: 2020 696e 7465 6765 7273 2c20 7468 6520    integers, the 
│ │ │ │ +00029e60: 6578 7065 6374 6564 2074 6f74 616c 2042  expected total B
│ │ │ │ +00029e70: 6574 7469 206e 756d 6265 7273 0a0a 4465  etti numbers..De
│ │ │ │ +00029e80: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +00029e90: 3d3d 3d3d 3d0a 0a47 6976 656e 2061 206d  =====..Given a m
│ │ │ │ +00029ea0: 6f64 756c 6520 4d20 6f76 6572 2074 6865  odule M over the
│ │ │ │ +00029eb0: 2072 696e 6720 6f66 206f 7065 7261 746f   ring of operato
│ │ │ │ +00029ec0: 7273 2024 6b5b 785f 312e 2e78 5f63 5d24  rs $k[x_1..x_c]$
│ │ │ │ +00029ed0: 2c20 7468 6520 6361 6c6c 2024 4e20 3d0a  , the call $N =.
│ │ │ │ +00029ee0: 6d6f 6475 6c65 4173 4578 7428 4d2c 5229  moduleAsExt(M,R)
│ │ │ │ +00029ef0: 2420 7072 6f64 7563 6573 2061 206d 6f64  $ produces a mod
│ │ │ │ +00029f00: 756c 6520 4e20 6f76 6572 2074 6865 2072  ule N over the r
│ │ │ │ +00029f10: 696e 6720 5220 7768 6f73 6520 6578 7420  ing R whose ext 
│ │ │ │ +00029f20: 6d6f 6475 6c65 2069 7320 7468 650a 6578  module is the.ex
│ │ │ │ +00029f30: 7465 7269 6f72 2061 6c67 6562 7261 206f  terior algebra o
│ │ │ │ +00029f40: 6e20 6e3d 6e75 6d67 656e 7352 2067 656e  n n=numgensR gen
│ │ │ │ +00029f50: 6572 6174 6f72 7320 7465 6e73 6f72 6564  erators tensored
│ │ │ │ +00029f60: 2077 6974 6820 4d2e 2054 6869 7320 7363   with M. This sc
│ │ │ │ +00029f70: 7269 7074 2063 6f6d 7075 7465 730a 6e75  ript computes.nu
│ │ │ │ +00029f80: 6d56 616c 7565 7320 7661 6c75 6573 206f  mValues values o
│ │ │ │ +00029f90: 6620 7468 6520 4869 6c62 6572 7420 6675  f the Hilbert fu
│ │ │ │ +00029fa0: 6e63 7469 6f6e 206f 6620 2424 204d 205c  nction of $$ M \
│ │ │ │ +00029fb0: 6f74 696d 6573 205c 7765 6467 6520 6b5e  otimes \wedge k^
│ │ │ │ +00029fc0: 6e2c 2024 2420 7768 6963 680a 7368 6f75  n, $$ which.shou
│ │ │ │ +00029fd0: 6c64 2062 6520 6571 7561 6c20 746f 2074  ld be equal to t
│ │ │ │ +00029fe0: 6865 2074 6f74 616c 2062 6574 7469 206e  he total betti n
│ │ │ │ +00029ff0: 756d 6265 7273 206f 6620 4e2e 0a0a 2b2d  umbers of N...+-
│ │ │ │ +0002a000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002a030: 0a7c 6931 203a 206b 6b20 3d20 5a5a 2f31  .|i1 : kk = ZZ/1
│ │ │ │ +0002a040: 3031 3b20 2020 2020 2020 2020 2020 2020  01;             
│ │ │ │ +0002a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a060: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a090: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d  -----+.|i2 : S =
│ │ │ │ +0002a0a0: 206b 6b5b 612c 622c 635d 3b20 2020 2020   kk[a,b,c];     
│ │ │ │ +0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002a0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ +0002a100: 203a 2066 6620 3d20 6d61 7472 6978 7b7b   : ff = matrix{{
│ │ │ │ +0002a110: 615e 342c 2062 5e34 2c63 5e34 7d7d 3b20  a^4, b^4,c^4}}; 
│ │ │ │ +0002a120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002a130: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a160: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002a170: 2031 2020 2020 2020 3320 2020 2020 2020   1      3       
│ │ │ │ +0002a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a190: 2020 2020 7c0a 7c6f 3320 3a20 4d61 7472      |.|o3 : Matr
│ │ │ │ +0002a1a0: 6978 2053 2020 3c2d 2d20 5320 2020 2020  ix S  <-- S     
│ │ │ │ +0002a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a1c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +0002a200: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ +0002a210: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +0002a220: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a260: 2b0a 7c69 3520 3a20 4f70 7320 3d20 6b6b  +.|i5 : Ops = kk
│ │ │ │ +0002a270: 5b78 5f31 2c78 5f32 2c78 5f33 5d3b 2020  [x_1,x_2,x_3];  
│ │ │ │ +0002a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a290: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002a2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a2c0: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4d4d  ------+.|i6 : MM
│ │ │ │ +0002a2d0: 203d 204f 7073 5e31 2f28 785f 312a 6964   = Ops^1/(x_1*id
│ │ │ │ +0002a2e0: 6561 6c28 785f 325e 322c 785f 3329 293b  eal(x_2^2,x_3));
│ │ │ │ +0002a2f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002a330: 3720 3a20 4e20 3d20 6d6f 6475 6c65 4173  7 : N = moduleAs
│ │ │ │ +0002a340: 4578 7428 4d4d 2c52 293b 2020 2020 2020  Ext(MM,R);      
│ │ │ │ +0002a350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002a360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a390: 2d2d 2b0a 7c69 3820 3a20 6265 7474 6920  --+.|i8 : betti 
│ │ │ │ +0002a3a0: 6672 6565 5265 736f 6c75 7469 6f6e 2820  freeResolution( 
│ │ │ │ +0002a3b0: 4e2c 204c 656e 6774 684c 696d 6974 203d  N, LengthLimit =
│ │ │ │ +0002a3c0: 3e20 3130 297c 0a7c 2020 2020 2020 2020  > 10)|.|        
│ │ │ │ +0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a3f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002a400: 2020 2020 2020 2020 3020 2031 2020 3220          0  1  2 
│ │ │ │ +0002a410: 2033 2020 3420 2035 2020 3620 2037 2020   3  4  5  6  7  
│ │ │ │ +0002a420: 3820 2039 2031 3020 2020 207c 0a7c 6f38  8  9 10    |.|o8
│ │ │ │ +0002a430: 203d 2074 6f74 616c 3a20 3336 2032 3720   = total: 36 27 
│ │ │ │ +0002a440: 3239 2033 3120 3333 2033 3520 3337 2033  29 31 33 35 37 3
│ │ │ │ +0002a450: 3920 3431 2034 3320 3435 2020 2020 7c0a  9 41 43 45    |.
│ │ │ │ +0002a460: 7c20 2020 2020 2020 202d 363a 2031 3820  |        -6: 18 
│ │ │ │ +0002a470: 2036 2020 2e20 202e 2020 2e20 202e 2020   6  .  .  .  .  
│ │ │ │ +0002a480: 2e20 202e 2020 2e20 202e 2020 2e20 2020  .  .  .  .  .   
│ │ │ │ +0002a490: 207c 0a7c 2020 2020 2020 2020 2d35 3a20   |.|        -5: 
│ │ │ │  0002a4a0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a4b0: 2020 7c0a 7c20 2020 2020 2020 202d 353a    |.|        -5:
│ │ │ │ -0002a4c0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0002a4d0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a4e0: 2e20 2020 207c 0a7c 2020 2020 2020 2020  .    |.|        
│ │ │ │ -0002a4f0: 2d34 3a20 3138 2032 3120 3231 2020 3720  -4: 18 21 21  7 
│ │ │ │ -0002a500: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a510: 2e20 202e 2020 2020 7c0a 7c20 2020 2020  .  .    |.|     
│ │ │ │ -0002a520: 2020 202d 333a 2020 2e20 202e 2020 2e20     -3:  .  .  . 
│ │ │ │ -0002a530: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a540: 2e20 202e 2020 2e20 2020 207c 0a7c 2020  .  .  .    |.|  
│ │ │ │ -0002a550: 2020 2020 2020 2d32 3a20 202e 2020 2e20        -2:  .  . 
│ │ │ │ -0002a560: 2038 2032 3420 3234 2020 3820 202e 2020   8 24 24  8  .  
│ │ │ │ -0002a570: 2e20 202e 2020 2e20 202e 2020 2020 7c0a  .  .  .  .    |.
│ │ │ │ -0002a580: 7c20 2020 2020 2020 202d 313a 2020 2e20  |        -1:  . 
│ │ │ │ -0002a590: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0002a5a0: 2e20 202e 2020 2e20 202e 2020 2e20 2020  .  .  .  .  .   
│ │ │ │ -0002a5b0: 207c 0a7c 2020 2020 2020 2020 2030 3a20   |.|         0: 
│ │ │ │ -0002a5c0: 202e 2020 2e20 202e 2020 2e20 2039 2032   .  .  .  .  9 2
│ │ │ │ -0002a5d0: 3720 3237 2020 3920 202e 2020 2e20 202e  7 27  9  .  .  .
│ │ │ │ -0002a5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002a5f0: 313a 2020 2e20 202e 2020 2e20 202e 2020  1:  .  .  .  .  
│ │ │ │ -0002a600: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0002a610: 2020 2e20 2020 207c 0a7c 2020 2020 2020    .    |.|      
│ │ │ │ -0002a620: 2020 2032 3a20 202e 2020 2e20 202e 2020     2:  .  .  .  
│ │ │ │ -0002a630: 2e20 202e 2020 2e20 3130 2033 3020 3330  .  .  . 10 30 30
│ │ │ │ -0002a640: 2031 3020 202e 2020 2020 7c0a 7c20 2020   10  .    |.|   
│ │ │ │ -0002a650: 2020 2020 2020 333a 2020 2e20 202e 2020        3:  .  .  
│ │ │ │ -0002a660: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0002a670: 2020 2e20 202e 2020 2e20 2020 207c 0a7c    .  .  .    |.|
│ │ │ │ -0002a680: 2020 2020 2020 2020 2034 3a20 202e 2020           4:  .  
│ │ │ │ -0002a690: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0002a6a0: 2020 2e20 3131 2033 3320 3333 2020 2020    . 11 33 33    
│ │ │ │ -0002a6b0: 7c0a 7c20 2020 2020 2020 2020 353a 2020  |.|         5:  
│ │ │ │ -0002a6c0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0002a4b0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0002a4c0: 2020 2020 7c0a 7c20 2020 2020 2020 202d      |.|        -
│ │ │ │ +0002a4d0: 343a 2031 3820 3231 2032 3120 2037 2020  4: 18 21 21  7  
│ │ │ │ +0002a4e0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0002a4f0: 2020 2e20 2020 207c 0a7c 2020 2020 2020    .    |.|      
│ │ │ │ +0002a500: 2020 2d33 3a20 202e 2020 2e20 202e 2020    -3:  .  .  .  
│ │ │ │ +0002a510: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0002a520: 2020 2e20 202e 2020 2020 7c0a 7c20 2020    .  .    |.|   
│ │ │ │ +0002a530: 2020 2020 202d 323a 2020 2e20 202e 2020       -2:  .  .  
│ │ │ │ +0002a540: 3820 3234 2032 3420 2038 2020 2e20 202e  8 24 24  8  .  .
│ │ │ │ +0002a550: 2020 2e20 202e 2020 2e20 2020 207c 0a7c    .  .  .    |.|
│ │ │ │ +0002a560: 2020 2020 2020 2020 2d31 3a20 202e 2020          -1:  .  
│ │ │ │ +0002a570: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0002a580: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +0002a590: 7c0a 7c20 2020 2020 2020 2020 303a 2020  |.|         0:  
│ │ │ │ +0002a5a0: 2e20 202e 2020 2e20 202e 2020 3920 3237  .  .  .  .  9 27
│ │ │ │ +0002a5b0: 2032 3720 2039 2020 2e20 202e 2020 2e20   27  9  .  .  . 
│ │ │ │ +0002a5c0: 2020 207c 0a7c 2020 2020 2020 2020 2031     |.|         1
│ │ │ │ +0002a5d0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ +0002a5e0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a5f0: 202e 2020 2020 7c0a 7c20 2020 2020 2020   .    |.|       
│ │ │ │ +0002a600: 2020 323a 2020 2e20 202e 2020 2e20 202e    2:  .  .  .  .
│ │ │ │ +0002a610: 2020 2e20 202e 2031 3020 3330 2033 3020    .  . 10 30 30 
│ │ │ │ +0002a620: 3130 2020 2e20 2020 207c 0a7c 2020 2020  10  .    |.|    
│ │ │ │ +0002a630: 2020 2020 2033 3a20 202e 2020 2e20 202e       3:  .  .  .
│ │ │ │ +0002a640: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a650: 202e 2020 2e20 202e 2020 2020 7c0a 7c20   .  .  .    |.| 
│ │ │ │ +0002a660: 2020 2020 2020 2020 343a 2020 2e20 202e          4:  .  .
│ │ │ │ +0002a670: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a680: 202e 2031 3120 3333 2033 3320 2020 207c   . 11 33 33    |
│ │ │ │ +0002a690: 0a7c 2020 2020 2020 2020 2035 3a20 202e  .|         5:  .
│ │ │ │ +0002a6a0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0002a6b0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0002a6c0: 2020 7c0a 7c20 2020 2020 2020 2020 363a    |.|         6:
│ │ │ │  0002a6d0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0002a6e0: 2020 207c 0a7c 2020 2020 2020 2020 2036     |.|         6
│ │ │ │ -0002a6f0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0002a700: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0002a710: 3132 2020 2020 7c0a 7c20 2020 2020 2020  12    |.|       
│ │ │ │ -0002a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a740: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │ -0002a750: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ -0002a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a770: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0002a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002a7b0: 0a7c 6939 203a 2068 664d 6f64 756c 6541  .|i9 : hfModuleA
│ │ │ │ -0002a7c0: 7345 7874 2831 322c 4d4d 2c33 2920 2020  sExt(12,MM,3)   
│ │ │ │ +0002a6e0: 202e 2020 2e20 202e 2020 2e20 202e 2031   .  .  .  .  . 1
│ │ │ │ +0002a6f0: 3220 2020 207c 0a7c 2020 2020 2020 2020  2    |.|        
│ │ │ │ +0002a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a720: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ +0002a730: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0002a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a750: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002a760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002a790: 7c69 3920 3a20 6866 4d6f 6475 6c65 4173  |i9 : hfModuleAs
│ │ │ │ +0002a7a0: 4578 7428 3132 2c4d 4d2c 3329 2020 2020  Ext(12,MM,3)    
│ │ │ │ +0002a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a7c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a7e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a810: 2020 2020 207c 0a7c 6f39 203d 2028 3233       |.|o9 = (23
│ │ │ │ -0002a820: 2c20 3235 2c20 3237 2c20 3239 2c20 3331  , 25, 27, 29, 31
│ │ │ │ -0002a830: 2c20 3333 2c20 3335 2c20 3337 2c20 3339  , 33, 35, 37, 39
│ │ │ │ -0002a840: 2c20 3431 2920 2020 7c0a 7c20 2020 2020  , 41)   |.|     
│ │ │ │ -0002a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a870: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -0002a880: 203a 2053 6571 7565 6e63 6520 2020 2020   : Sequence     
│ │ │ │ -0002a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002a8b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002a8e0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ -0002a8f0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -0002a900: 206d 6f64 756c 6541 7345 7874 3a20 6d6f   moduleAsExt: mo
│ │ │ │ -0002a910: 6475 6c65 4173 4578 742c 202d 2d20 4669  duleAsExt, -- Fi
│ │ │ │ -0002a920: 6e64 2061 206d 6f64 756c 6520 7769 7468  nd a module with
│ │ │ │ -0002a930: 2067 6976 656e 2061 7379 6d70 746f 7469   given asymptoti
│ │ │ │ -0002a940: 630a 2020 2020 7265 736f 6c75 7469 6f6e  c.    resolution
│ │ │ │ -0002a950: 0a0a 5761 7973 2074 6f20 7573 6520 6866  ..Ways to use hf
│ │ │ │ -0002a960: 4d6f 6475 6c65 4173 4578 743a 0a3d 3d3d  ModuleAsExt:.===
│ │ │ │ -0002a970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002a980: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6866  =======..  * "hf
│ │ │ │ -0002a990: 4d6f 6475 6c65 4173 4578 7428 5a5a 2c4d  ModuleAsExt(ZZ,M
│ │ │ │ -0002a9a0: 6f64 756c 652c 5a5a 2922 0a0a 466f 7220  odule,ZZ)"..For 
│ │ │ │ -0002a9b0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -0002a9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002a9d0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0002a9e0: 6f74 6520 6866 4d6f 6475 6c65 4173 4578  ote hfModuleAsEx
│ │ │ │ -0002a9f0: 743a 2068 664d 6f64 756c 6541 7345 7874  t: hfModuleAsExt
│ │ │ │ -0002aa00: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -0002aa10: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ -0002aa20: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -0002aa30: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ +0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a7f0: 2020 2020 7c0a 7c6f 3920 3d20 2832 332c      |.|o9 = (23,
│ │ │ │ +0002a800: 2032 352c 2032 372c 2032 392c 2033 312c   25, 27, 29, 31,
│ │ │ │ +0002a810: 2033 332c 2033 352c 2033 372c 2033 392c   33, 35, 37, 39,
│ │ │ │ +0002a820: 2034 3129 2020 207c 0a7c 2020 2020 2020   41)   |.|      
│ │ │ │ +0002a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a850: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ +0002a860: 3a20 5365 7175 656e 6365 2020 2020 2020  : Sequence      
│ │ │ │ +0002a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a880: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002a890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002a8c0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  +..See also.====
│ │ │ │ +0002a8d0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +0002a8e0: 6d6f 6475 6c65 4173 4578 743a 206d 6f64  moduleAsExt: mod
│ │ │ │ +0002a8f0: 756c 6541 7345 7874 2c20 2d2d 2046 696e  uleAsExt, -- Fin
│ │ │ │ +0002a900: 6420 6120 6d6f 6475 6c65 2077 6974 6820  d a module with 
│ │ │ │ +0002a910: 6769 7665 6e20 6173 796d 7074 6f74 6963  given asymptotic
│ │ │ │ +0002a920: 0a20 2020 2072 6573 6f6c 7574 696f 6e0a  .    resolution.
│ │ │ │ +0002a930: 0a57 6179 7320 746f 2075 7365 2068 664d  .Ways to use hfM
│ │ │ │ +0002a940: 6f64 756c 6541 7345 7874 3a0a 3d3d 3d3d  oduleAsExt:.====
│ │ │ │ +0002a950: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002a960: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2268 664d  ======..  * "hfM
│ │ │ │ +0002a970: 6f64 756c 6541 7345 7874 285a 5a2c 4d6f  oduleAsExt(ZZ,Mo
│ │ │ │ +0002a980: 6475 6c65 2c5a 5a29 220a 0a46 6f72 2074  dule,ZZ)"..For t
│ │ │ │ +0002a990: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +0002a9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002a9b0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +0002a9c0: 7465 2068 664d 6f64 756c 6541 7345 7874  te hfModuleAsExt
│ │ │ │ +0002a9d0: 3a20 6866 4d6f 6475 6c65 4173 4578 742c  : hfModuleAsExt,
│ │ │ │ +0002a9e0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +0002a9f0: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ +0002aa00: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +0002aa10: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d  dFunction,...---
│ │ │ │ +0002aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002aa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002aa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002aa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002aa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -0002aa90: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -0002aaa0: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -0002aab0: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -0002aac0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -0002aad0: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ -0002aae0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -0002aaf0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ -0002ab00: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -0002ab10: 7469 6f6e 732e 6d32 3a33 3134 313a 302e  tions.m2:3141:0.
│ │ │ │ -0002ab20: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ -0002ab30: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -0002ab40: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ -0002ab50: 6f64 653a 2068 6967 6853 797a 7967 792c  ode: highSyzygy,
│ │ │ │ -0002ab60: 204e 6578 743a 2068 4d61 7073 2c20 5072   Next: hMaps, Pr
│ │ │ │ -0002ab70: 6576 3a20 6866 4d6f 6475 6c65 4173 4578  ev: hfModuleAsEx
│ │ │ │ -0002ab80: 742c 2055 703a 2054 6f70 0a0a 6869 6768  t, Up: Top..high
│ │ │ │ -0002ab90: 5379 7a79 6779 202d 2d20 5265 7475 726e  Syzygy -- Return
│ │ │ │ -0002aba0: 7320 6120 7379 7a79 6779 206d 6f64 756c  s a syzygy modul
│ │ │ │ -0002abb0: 6520 6f6e 6520 6265 796f 6e64 2074 6865  e one beyond the
│ │ │ │ -0002abc0: 2072 6567 756c 6172 6974 7920 6f66 2045   regularity of E
│ │ │ │ -0002abd0: 7874 284d 2c6b 290a 2a2a 2a2a 2a2a 2a2a  xt(M,k).********
│ │ │ │ +0002aa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +0002aa70: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +0002aa80: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +0002aa90: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +0002aaa0: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +0002aab0: 7932 2d31 2e32 362e 3036 2b64 732f 4d32  y2-1.26.06+ds/M2
│ │ │ │ +0002aac0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +0002aad0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74  ges/.CompleteInt
│ │ │ │ +0002aae0: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +0002aaf0: 696f 6e73 2e6d 323a 3331 3431 3a30 2e0a  ions.m2:3141:0..
│ │ │ │ +0002ab00: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +0002ab10: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +0002ab20: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +0002ab30: 6465 3a20 6869 6768 5379 7a79 6779 2c20  de: highSyzygy, 
│ │ │ │ +0002ab40: 4e65 7874 3a20 684d 6170 732c 2050 7265  Next: hMaps, Pre
│ │ │ │ +0002ab50: 763a 2068 664d 6f64 756c 6541 7345 7874  v: hfModuleAsExt
│ │ │ │ +0002ab60: 2c20 5570 3a20 546f 700a 0a68 6967 6853  , Up: Top..highS
│ │ │ │ +0002ab70: 797a 7967 7920 2d2d 2052 6574 7572 6e73  yzygy -- Returns
│ │ │ │ +0002ab80: 2061 2073 797a 7967 7920 6d6f 6475 6c65   a syzygy module
│ │ │ │ +0002ab90: 206f 6e65 2062 6579 6f6e 6420 7468 6520   one beyond the 
│ │ │ │ +0002aba0: 7265 6775 6c61 7269 7479 206f 6620 4578  regularity of Ex
│ │ │ │ +0002abb0: 7428 4d2c 6b29 0a2a 2a2a 2a2a 2a2a 2a2a  t(M,k).*********
│ │ │ │ +0002abc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002abd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002abe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002abf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ac00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ac10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ac20: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -0002ac30: 0a20 2020 2020 2020 204d 203d 2068 6967  .        M = hig
│ │ │ │ -0002ac40: 6853 797a 7967 7920 4d30 0a20 202a 2049  hSyzygy M0.  * I
│ │ │ │ -0002ac50: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ -0002ac60: 302c 2061 202a 6e6f 7465 206d 6f64 756c  0, a *note modul
│ │ │ │ -0002ac70: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -0002ac80: 294d 6f64 756c 652c 2c20 6f76 6572 2061  )Module,, over a
│ │ │ │ -0002ac90: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -0002aca0: 6563 7469 6f6e 0a20 2020 2020 2020 2072  ection.        r
│ │ │ │ -0002acb0: 696e 670a 2020 2a20 2a6e 6f74 6520 4f70  ing.  * *note Op
│ │ │ │ -0002acc0: 7469 6f6e 616c 2069 6e70 7574 733a 2028  tional inputs: (
│ │ │ │ -0002acd0: 4d61 6361 756c 6179 3244 6f63 2975 7369  Macaulay2Doc)usi
│ │ │ │ -0002ace0: 6e67 2066 756e 6374 696f 6e73 2077 6974  ng functions wit
│ │ │ │ -0002acf0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574  h optional input
│ │ │ │ -0002ad00: 732c 3a0a 2020 2020 2020 2a20 4f70 7469  s,:.      * Opti
│ │ │ │ -0002ad10: 6d69 736d 203d 3e20 2e2e 2e2c 2064 6566  mism => ..., def
│ │ │ │ -0002ad20: 6175 6c74 2076 616c 7565 2030 0a20 202a  ault value 0.  *
│ │ │ │ -0002ad30: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -0002ad40: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64  * M, a *note mod
│ │ │ │ -0002ad50: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -0002ad60: 6f63 294d 6f64 756c 652c 2c20 6120 7379  oc)Module,, a sy
│ │ │ │ -0002ad70: 7a79 6779 206d 6f64 756c 6520 6f66 204d  zygy module of M
│ │ │ │ -0002ad80: 300a 0a44 6573 6372 6970 7469 6f6e 0a3d  0..Description.=
│ │ │ │ -0002ad90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120 2268  ==========..A "h
│ │ │ │ -0002ada0: 6967 6820 7379 7a79 6779 2220 6f76 6572  igh syzygy" over
│ │ │ │ -0002adb0: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ -0002adc0: 7273 6563 7469 6f6e 2069 7320 6f6e 6520  rsection is one 
│ │ │ │ -0002add0: 7375 6368 2074 6861 7420 6765 6e65 7261  such that genera
│ │ │ │ -0002ade0: 6c0a 6369 2d6f 7065 7261 746f 7273 2068  l.ci-operators h
│ │ │ │ -0002adf0: 6176 6520 7370 6c69 7420 6b65 726e 656c  ave split kernel
│ │ │ │ -0002ae00: 7320 7768 656e 2061 7070 6c69 6564 2072  s when applied r
│ │ │ │ -0002ae10: 6563 7572 7369 7665 6c79 206f 6e20 636f  ecursively on co
│ │ │ │ -0002ae20: 7379 7a79 6779 2063 6861 696e 7320 6f66  syzygy chains of
│ │ │ │ -0002ae30: 0a70 7265 7669 6f75 7320 6b65 726e 656c  .previous kernel
│ │ │ │ -0002ae40: 732e 0a0a 4966 2070 203d 206d 6642 6f75  s...If p = mfBou
│ │ │ │ -0002ae50: 6e64 204d 302c 2074 6865 6e20 6869 6768  nd M0, then high
│ │ │ │ -0002ae60: 5379 7a79 6779 204d 3020 7265 7475 726e  Syzygy M0 return
│ │ │ │ -0002ae70: 7320 7468 6520 702d 7468 2073 797a 7967  s the p-th syzyg
│ │ │ │ -0002ae80: 7920 6f66 204d 302e 2028 6966 2046 2069  y of M0. (if F i
│ │ │ │ -0002ae90: 7320 610a 7265 736f 6c75 7469 6f6e 206f  s a.resolution o
│ │ │ │ -0002aea0: 6620 4d20 7468 6973 2069 7320 7468 6520  f M this is the 
│ │ │ │ -0002aeb0: 636f 6b65 726e 656c 206f 6620 462e 6464  cokernel of F.dd
│ │ │ │ -0002aec0: 5f7b 702b 317d 292e 204f 7074 696d 6973  _{p+1}). Optimis
│ │ │ │ -0002aed0: 6d20 3d3e 2072 2061 7320 6f70 7469 6f6e  m => r as option
│ │ │ │ -0002aee0: 616c 0a61 7267 756d 656e 742c 2068 6967  al.argument, hig
│ │ │ │ -0002aef0: 6853 797a 7967 7928 4d30 2c4f 7074 696d  hSyzygy(M0,Optim
│ │ │ │ -0002af00: 6973 6d3d 3e72 2920 7265 7475 726e 7320  ism=>r) returns 
│ │ │ │ -0002af10: 7468 6520 2870 2d72 292d 7468 2073 797a  the (p-r)-th syz
│ │ │ │ -0002af20: 7967 792e 2054 6865 2073 6372 6970 7420  ygy. The script 
│ │ │ │ -0002af30: 6973 0a75 7365 6675 6c20 7769 7468 206d  is.useful with m
│ │ │ │ -0002af40: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -0002af50: 6f6e 2866 662c 2068 6967 6853 797a 7967  on(ff, highSyzyg
│ │ │ │ -0002af60: 7920 4d30 292e 0a0a 2b2d 2d2d 2d2d 2d2d  y M0)...+-------
│ │ │ │ +0002ac00: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +0002ac10: 2020 2020 2020 2020 4d20 3d20 6869 6768          M = high
│ │ │ │ +0002ac20: 5379 7a79 6779 204d 300a 2020 2a20 496e  Syzygy M0.  * In
│ │ │ │ +0002ac30: 7075 7473 3a0a 2020 2020 2020 2a20 4d30  puts:.      * M0
│ │ │ │ +0002ac40: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0002ac50: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0002ac60: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120  Module,, over a 
│ │ │ │ +0002ac70: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +0002ac80: 6374 696f 6e0a 2020 2020 2020 2020 7269  ction.        ri
│ │ │ │ +0002ac90: 6e67 0a20 202a 202a 6e6f 7465 204f 7074  ng.  * *note Opt
│ │ │ │ +0002aca0: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +0002acb0: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ +0002acc0: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ +0002acd0: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +0002ace0: 2c3a 0a20 2020 2020 202a 204f 7074 696d  ,:.      * Optim
│ │ │ │ +0002acf0: 6973 6d20 3d3e 202e 2e2e 2c20 6465 6661  ism => ..., defa
│ │ │ │ +0002ad00: 756c 7420 7661 6c75 6520 300a 2020 2a20  ult value 0.  * 
│ │ │ │ +0002ad10: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +0002ad20: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ +0002ad30: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +0002ad40: 6329 4d6f 6475 6c65 2c2c 2061 2073 797a  c)Module,, a syz
│ │ │ │ +0002ad50: 7967 7920 6d6f 6475 6c65 206f 6620 4d30  ygy module of M0
│ │ │ │ +0002ad60: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +0002ad70: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 2022 6869  =========..A "hi
│ │ │ │ +0002ad80: 6768 2073 797a 7967 7922 206f 7665 7220  gh syzygy" over 
│ │ │ │ +0002ad90: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ +0002ada0: 7365 6374 696f 6e20 6973 206f 6e65 2073  section is one s
│ │ │ │ +0002adb0: 7563 6820 7468 6174 2067 656e 6572 616c  uch that general
│ │ │ │ +0002adc0: 0a63 692d 6f70 6572 6174 6f72 7320 6861  .ci-operators ha
│ │ │ │ +0002add0: 7665 2073 706c 6974 206b 6572 6e65 6c73  ve split kernels
│ │ │ │ +0002ade0: 2077 6865 6e20 6170 706c 6965 6420 7265   when applied re
│ │ │ │ +0002adf0: 6375 7273 6976 656c 7920 6f6e 2063 6f73  cursively on cos
│ │ │ │ +0002ae00: 797a 7967 7920 6368 6169 6e73 206f 660a  yzygy chains of.
│ │ │ │ +0002ae10: 7072 6576 696f 7573 206b 6572 6e65 6c73  previous kernels
│ │ │ │ +0002ae20: 2e0a 0a49 6620 7020 3d20 6d66 426f 756e  ...If p = mfBoun
│ │ │ │ +0002ae30: 6420 4d30 2c20 7468 656e 2068 6967 6853  d M0, then highS
│ │ │ │ +0002ae40: 797a 7967 7920 4d30 2072 6574 7572 6e73  yzygy M0 returns
│ │ │ │ +0002ae50: 2074 6865 2070 2d74 6820 7379 7a79 6779   the p-th syzygy
│ │ │ │ +0002ae60: 206f 6620 4d30 2e20 2869 6620 4620 6973   of M0. (if F is
│ │ │ │ +0002ae70: 2061 0a72 6573 6f6c 7574 696f 6e20 6f66   a.resolution of
│ │ │ │ +0002ae80: 204d 2074 6869 7320 6973 2074 6865 2063   M this is the c
│ │ │ │ +0002ae90: 6f6b 6572 6e65 6c20 6f66 2046 2e64 645f  okernel of F.dd_
│ │ │ │ +0002aea0: 7b70 2b31 7d29 2e20 4f70 7469 6d69 736d  {p+1}). Optimism
│ │ │ │ +0002aeb0: 203d 3e20 7220 6173 206f 7074 696f 6e61   => r as optiona
│ │ │ │ +0002aec0: 6c0a 6172 6775 6d65 6e74 2c20 6869 6768  l.argument, high
│ │ │ │ +0002aed0: 5379 7a79 6779 284d 302c 4f70 7469 6d69  Syzygy(M0,Optimi
│ │ │ │ +0002aee0: 736d 3d3e 7229 2072 6574 7572 6e73 2074  sm=>r) returns t
│ │ │ │ +0002aef0: 6865 2028 702d 7229 2d74 6820 7379 7a79  he (p-r)-th syzy
│ │ │ │ +0002af00: 6779 2e20 5468 6520 7363 7269 7074 2069  gy. The script i
│ │ │ │ +0002af10: 730a 7573 6566 756c 2077 6974 6820 6d61  s.useful with ma
│ │ │ │ +0002af20: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0002af30: 6e28 6666 2c20 6869 6768 5379 7a79 6779  n(ff, highSyzygy
│ │ │ │ +0002af40: 204d 3029 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d   M0)...+--------
│ │ │ │ +0002af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002afa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073  -------+.|i1 : s
│ │ │ │ -0002afb0: 6574 5261 6e64 6f6d 5365 6564 2031 3030  etRandomSeed 100
│ │ │ │ -0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afe0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2073          |.| -- s
│ │ │ │ -0002aff0: 6574 7469 6e67 2072 616e 646f 6d20 7365  etting random se
│ │ │ │ -0002b000: 6564 2074 6f20 3130 3020 2020 2020 2020  ed to 100       
│ │ │ │ +0002af80: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365  ------+.|i1 : se
│ │ │ │ +0002af90: 7452 616e 646f 6d53 6565 6420 3130 3020  tRandomSeed 100 
│ │ │ │ +0002afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002afc0: 2020 2020 2020 207c 0a7c 202d 2d20 7365         |.| -- se
│ │ │ │ +0002afd0: 7474 696e 6720 7261 6e64 6f6d 2073 6565  tting random see
│ │ │ │ +0002afe0: 6420 746f 2031 3030 2020 2020 2020 2020  d to 100        
│ │ │ │ +0002aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0002b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b060: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ -0002b070: 3d20 3130 3020 2020 2020 2020 2020 2020  = 100           
│ │ │ │ -0002b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002b040: 2020 2020 2020 2020 207c 0a7c 6f31 203d           |.|o1 =
│ │ │ │ +0002b050: 2031 3030 2020 2020 2020 2020 2020 2020   100            
│ │ │ │ +0002b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b080: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002b0f0: 3220 3a20 5320 3d20 5a5a 2f31 3031 5b78  2 : S = ZZ/101[x
│ │ │ │ -0002b100: 2c79 2c7a 5d20 2020 2020 2020 2020 2020  ,y,z]           
│ │ │ │ +0002b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0002b0d0: 203a 2053 203d 205a 5a2f 3130 315b 782c   : S = ZZ/101[x,
│ │ │ │ +0002b0e0: 792c 7a5d 2020 2020 2020 2020 2020 2020  y,z]            
│ │ │ │ +0002b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b100: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0002b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002b170: 7c6f 3220 3d20 5320 2020 2020 2020 2020  |o2 = S         
│ │ │ │ -0002b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b1b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1f0: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69  |.|o2 : Polynomi
│ │ │ │ -0002b200: 616c 5269 6e67 2020 2020 2020 2020 2020  alRing          
│ │ │ │ -0002b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b230: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002b140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002b150: 6f32 203d 2053 2020 2020 2020 2020 2020  o2 = S          
│ │ │ │ +0002b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b190: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b1c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b1d0: 0a7c 6f32 203a 2050 6f6c 796e 6f6d 6961  .|o2 : Polynomia
│ │ │ │ +0002b1e0: 6c52 696e 6720 2020 2020 2020 2020 2020  lRing           
│ │ │ │ +0002b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b210: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0002b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b270: 2d2d 2b0a 7c69 3320 3a20 6620 3d20 6d61  --+.|i3 : f = ma
│ │ │ │ -0002b280: 7472 6978 2278 332c 7933 2b78 332c 7a33  trix"x3,y3+x3,z3
│ │ │ │ -0002b290: 2b78 332b 7933 2220 2020 2020 2020 2020  +x3+y3"         
│ │ │ │ +0002b250: 2d2b 0a7c 6933 203a 2066 203d 206d 6174  -+.|i3 : f = mat
│ │ │ │ +0002b260: 7269 7822 7833 2c79 332b 7833 2c7a 332b  rix"x3,y3+x3,z3+
│ │ │ │ +0002b270: 7833 2b79 3322 2020 2020 2020 2020 2020  x3+y3"          
│ │ │ │ +0002b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b290: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2f0: 2020 2020 7c0a 7c6f 3320 3d20 7c20 7833      |.|o3 = | x3
│ │ │ │ -0002b300: 2078 332b 7933 2078 332b 7933 2b7a 3320   x3+y3 x3+y3+z3 
│ │ │ │ -0002b310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002b2d0: 2020 207c 0a7c 6f33 203d 207c 2078 3320     |.|o3 = | x3 
│ │ │ │ +0002b2e0: 7833 2b79 3320 7833 2b79 332b 7a33 207c  x3+y3 x3+y3+z3 |
│ │ │ │ +0002b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b310: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0002b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002b380: 2020 2020 2020 3120 2020 2020 2033 2020        1      3  
│ │ │ │ -0002b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3b0: 2020 2020 2020 207c 0a7c 6f33 203a 204d         |.|o3 : M
│ │ │ │ -0002b3c0: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ -0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0002b350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002b360: 2020 2020 2031 2020 2020 2020 3320 2020       1      3   
│ │ │ │ +0002b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b390: 2020 2020 2020 7c0a 7c6f 3320 3a20 4d61        |.|o3 : Ma
│ │ │ │ +0002b3a0: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
│ │ │ │ +0002b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b3d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b430: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -0002b440: 2066 6620 3d20 662a 7261 6e64 6f6d 2873   ff = f*random(s
│ │ │ │ -0002b450: 6f75 7263 6520 662c 2073 6f75 7263 6520  ource f, source 
│ │ │ │ -0002b460: 6629 2020 2020 2020 2020 2020 2020 2020  f)              
│ │ │ │ -0002b470: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002b410: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ +0002b420: 6666 203d 2066 2a72 616e 646f 6d28 736f  ff = f*random(so
│ │ │ │ +0002b430: 7572 6365 2066 2c20 736f 7572 6365 2066  urce f, source f
│ │ │ │ +0002b440: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0002b450: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -0002b4c0: 203d 207c 2031 3078 332d 3232 7933 2d34   = | 10x3-22y3-4
│ │ │ │ -0002b4d0: 7a33 202d 3230 7833 2d32 3079 332d 367a  z3 -20x3-20y3-6z
│ │ │ │ -0002b4e0: 3320 2d32 3778 332d 3431 7933 2b7a 3320  3 -27x3-41y3+z3 
│ │ │ │ -0002b4f0: 7c20 2020 2020 2020 2020 2020 7c0a 7c20  |           |.| 
│ │ │ │ +0002b490: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ +0002b4a0: 3d20 7c20 3130 7833 2d32 3279 332d 347a  = | 10x3-22y3-4z
│ │ │ │ +0002b4b0: 3320 2d32 3078 332d 3230 7933 2d36 7a33  3 -20x3-20y3-6z3
│ │ │ │ +0002b4c0: 202d 3237 7833 2d34 3179 332b 7a33 207c   -27x3-41y3+z3 |
│ │ │ │ +0002b4d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002b540: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0002b550: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ -0002b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b570: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002b580: 7c6f 3420 3a20 4d61 7472 6978 2053 2020  |o4 : Matrix S  
│ │ │ │ -0002b590: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ -0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b5c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0002b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b600: 2b0a 7c69 3520 3a20 5220 3d20 532f 6964  +.|i5 : R = S/id
│ │ │ │ -0002b610: 6561 6c20 6620 2020 2020 2020 2020 2020  eal f           
│ │ │ │ -0002b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b510: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002b520: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +0002b530: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ +0002b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002b560: 6f34 203a 204d 6174 7269 7820 5320 203c  o4 : Matrix S  <
│ │ │ │ +0002b570: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ +0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b590: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b5a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0002b5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0002b5e0: 0a7c 6935 203a 2052 203d 2053 2f69 6465  .|i5 : R = S/ide
│ │ │ │ +0002b5f0: 616c 2066 2020 2020 2020 2020 2020 2020  al f            
│ │ │ │ +0002b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b620: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0002b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b640: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b660: 207c 0a7c 6f35 203d 2052 2020 2020 2020   |.|o5 = R      
│ │ │ │  0002b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b680: 2020 7c0a 7c6f 3520 3d20 5220 2020 2020    |.|o5 = R     
│ │ │ │ +0002b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b6a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0002b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b700: 2020 2020 7c0a 7c6f 3520 3a20 5175 6f74      |.|o5 : Quot
│ │ │ │ -0002b710: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ -0002b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b740: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002b6e0: 2020 207c 0a7c 6f35 203a 2051 756f 7469     |.|o5 : Quoti
│ │ │ │ +0002b6f0: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ +0002b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b720: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002b730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b780: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4d30  ------+.|i6 : M0
│ │ │ │ -0002b790: 203d 2052 5e31 2f69 6465 616c 2278 327a   = R^1/ideal"x2z
│ │ │ │ -0002b7a0: 322c 7879 7a22 2020 2020 2020 2020 2020  2,xyz"          
│ │ │ │ +0002b760: 2d2d 2d2d 2d2b 0a7c 6936 203a 204d 3020  -----+.|i6 : M0 
│ │ │ │ +0002b770: 3d20 525e 312f 6964 6561 6c22 7832 7a32  = R^1/ideal"x2z2
│ │ │ │ +0002b780: 2c78 797a 2220 2020 2020 2020 2020 2020  ,xyz"           
│ │ │ │ +0002b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b7a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b7c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0002b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b800: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -0002b810: 636f 6b65 726e 656c 207c 2078 327a 3220  cokernel | x2z2 
│ │ │ │ -0002b820: 7879 7a20 7c20 2020 2020 2020 2020 2020  xyz |           
│ │ │ │ +0002b7e0: 2020 2020 2020 207c 0a7c 6f36 203d 2063         |.|o6 = c
│ │ │ │ +0002b7f0: 6f6b 6572 6e65 6c20 7c20 7832 7a32 2078  okernel | x2z2 x
│ │ │ │ +0002b800: 797a 207c 2020 2020 2020 2020 2020 2020  yz |            
│ │ │ │ +0002b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b820: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0002b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b840: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b860: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0002b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b880: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002b880: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │  0002b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8a0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b8c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -0002b8d0: 203a 2052 2d6d 6f64 756c 652c 2071 756f   : R-module, quo
│ │ │ │ -0002b8e0: 7469 656e 7420 6f66 2052 2020 2020 2020  tient of R      
│ │ │ │ -0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002b8a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ +0002b8b0: 3a20 522d 6d6f 6475 6c65 2c20 7175 6f74  : R-module, quot
│ │ │ │ +0002b8c0: 6965 6e74 206f 6620 5220 2020 2020 2020  ient of R       
│ │ │ │ +0002b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b8e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0002b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002b900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002b910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0002b950: 6937 203a 2062 6574 7469 2066 7265 6552  i7 : betti freeR
│ │ │ │ -0002b960: 6573 6f6c 7574 696f 6e20 284d 302c 204c  esolution (M0, L
│ │ │ │ -0002b970: 656e 6774 684c 696d 6974 203d 3e20 3729  engthLimit => 7)
│ │ │ │ -0002b980: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002b990: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b9c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b9d0: 0a7c 2020 2020 2020 2020 2020 2020 3020  .|            0 
│ │ │ │ -0002b9e0: 3120 3220 2033 2020 3420 2035 2020 3620  1 2  3  4  5  6 
│ │ │ │ -0002b9f0: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │ -0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba10: 7c0a 7c6f 3720 3d20 746f 7461 6c3a 2031  |.|o7 = total: 1
│ │ │ │ -0002ba20: 2032 2036 2031 3120 3138 2032 3620 3336   2 6 11 18 26 36
│ │ │ │ -0002ba30: 2034 3720 2020 2020 2020 2020 2020 2020   47             
│ │ │ │ -0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba50: 207c 0a7c 2020 2020 2020 2020 2030 3a20   |.|         0: 
│ │ │ │ -0002ba60: 3120 2e20 2e20 202e 2020 2e20 202e 2020  1 . .  .  .  .  
│ │ │ │ -0002ba70: 2e20 202e 2020 2020 2020 2020 2020 2020  .  .            
│ │ │ │ -0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba90: 2020 7c0a 7c20 2020 2020 2020 2020 313a    |.|         1:
│ │ │ │ -0002baa0: 202e 202e 202e 2020 2e20 202e 2020 2e20   . . .  .  .  . 
│ │ │ │ -0002bab0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │ -0002bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bad0: 2020 207c 0a7c 2020 2020 2020 2020 2032     |.|         2
│ │ │ │ -0002bae0: 3a20 2e20 3120 2e20 202e 2020 2e20 202e  : . 1 .  .  .  .
│ │ │ │ -0002baf0: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ -0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb10: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002bb20: 333a 202e 2031 2036 2020 3620 202e 2020  3: . 1 6  6  .  
│ │ │ │ -0002bb30: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ -0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002bb60: 2034 3a20 2e20 2e20 2e20 2035 2031 3820   4: . . .  5 18 
│ │ │ │ -0002bb70: 3134 2020 2e20 202e 2020 2020 2020 2020  14  .  .        
│ │ │ │ -0002bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002bba0: 2020 353a 202e 202e 202e 2020 2e20 202e    5: . . .  .  .
│ │ │ │ -0002bbb0: 2031 3220 3336 2032 3520 2020 2020 2020   12 36 25       
│ │ │ │ -0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bbd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002bbe0: 2020 2036 3a20 2e20 2e20 2e20 202e 2020     6: . . .  .  
│ │ │ │ -0002bbf0: 2e20 202e 2020 2e20 3232 2020 2020 2020  .  .  . 22      
│ │ │ │ +0002b920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0002b930: 3720 3a20 6265 7474 6920 6672 6565 5265  7 : betti freeRe
│ │ │ │ +0002b940: 736f 6c75 7469 6f6e 2028 4d30 2c20 4c65  solution (M0, Le
│ │ │ │ +0002b950: 6e67 7468 4c69 6d69 7420 3d3e 2037 2920  ngthLimit => 7) 
│ │ │ │ +0002b960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002b9b0: 7c20 2020 2020 2020 2020 2020 2030 2031  |            0 1
│ │ │ │ +0002b9c0: 2032 2020 3320 2034 2020 3520 2036 2020   2  3  4  5  6  
│ │ │ │ +0002b9d0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +0002b9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002b9f0: 0a7c 6f37 203d 2074 6f74 616c 3a20 3120  .|o7 = total: 1 
│ │ │ │ +0002ba00: 3220 3620 3131 2031 3820 3236 2033 3620  2 6 11 18 26 36 
│ │ │ │ +0002ba10: 3437 2020 2020 2020 2020 2020 2020 2020  47              
│ │ │ │ +0002ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba30: 7c0a 7c20 2020 2020 2020 2020 303a 2031  |.|         0: 1
│ │ │ │ +0002ba40: 202e 202e 2020 2e20 202e 2020 2e20 202e   . .  .  .  .  .
│ │ │ │ +0002ba50: 2020 2e20 2020 2020 2020 2020 2020 2020    .             
│ │ │ │ +0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba70: 207c 0a7c 2020 2020 2020 2020 2031 3a20   |.|         1: 
│ │ │ │ +0002ba80: 2e20 2e20 2e20 202e 2020 2e20 202e 2020  . . .  .  .  .  
│ │ │ │ +0002ba90: 2e20 202e 2020 2020 2020 2020 2020 2020  .  .            
│ │ │ │ +0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bab0: 2020 7c0a 7c20 2020 2020 2020 2020 323a    |.|         2:
│ │ │ │ +0002bac0: 202e 2031 202e 2020 2e20 202e 2020 2e20   . 1 .  .  .  . 
│ │ │ │ +0002bad0: 202e 2020 2e20 2020 2020 2020 2020 2020   .  .           
│ │ │ │ +0002bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002baf0: 2020 207c 0a7c 2020 2020 2020 2020 2033     |.|         3
│ │ │ │ +0002bb00: 3a20 2e20 3120 3620 2036 2020 2e20 202e  : . 1 6  6  .  .
│ │ │ │ +0002bb10: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bb40: 343a 202e 202e 202e 2020 3520 3138 2031  4: . . .  5 18 1
│ │ │ │ +0002bb50: 3420 202e 2020 2e20 2020 2020 2020 2020  4  .  .         
│ │ │ │ +0002bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002bb80: 2035 3a20 2e20 2e20 2e20 202e 2020 2e20   5: . . .  .  . 
│ │ │ │ +0002bb90: 3132 2033 3620 3235 2020 2020 2020 2020  12 36 25        
│ │ │ │ +0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bbb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002bbc0: 2020 363a 202e 202e 202e 2020 2e20 202e    6: . . .  .  .
│ │ │ │ +0002bbd0: 2020 2e20 202e 2032 3220 2020 2020 2020    .  . 22       
│ │ │ │ +0002bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bbf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc50: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ -0002bc60: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ -0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bc90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002bc30: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ +0002bc40: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0002bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bc70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ -0002bce0: 203a 206d 6642 6f75 6e64 204d 3020 2020   : mfBound M0   
│ │ │ │ -0002bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +0002bcc0: 3a20 6d66 426f 756e 6420 4d30 2020 2020  : mfBound M0    
│ │ │ │ +0002bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bcf0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002bd60: 6f38 203d 2033 2020 2020 2020 2020 2020  o8 = 3          
│ │ │ │ -0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002bda0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002bde0: 0a7c 6939 203a 204d 203d 2062 6574 7469  .|i9 : M = betti
│ │ │ │ -0002bdf0: 2066 7265 6552 6573 6f6c 7574 696f 6e28   freeResolution(
│ │ │ │ -0002be00: 6869 6768 5379 7a79 6779 204d 302c 204c  highSyzygy M0, L
│ │ │ │ -0002be10: 656e 6774 684c 696d 6974 203d 3e20 3729  engthLimit => 7)
│ │ │ │ -0002be20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002bd30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002bd40: 3820 3d20 3320 2020 2020 2020 2020 2020  8 = 3           
│ │ │ │ +0002bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bd70: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002bd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0002bdc0: 7c69 3920 3a20 4d20 3d20 6265 7474 6920  |i9 : M = betti 
│ │ │ │ +0002bdd0: 6672 6565 5265 736f 6c75 7469 6f6e 2868  freeResolution(h
│ │ │ │ +0002bde0: 6967 6853 797a 7967 7920 4d30 2c20 4c65  ighSyzygy M0, Le
│ │ │ │ +0002bdf0: 6e67 7468 4c69 6d69 7420 3d3e 2037 297c  ngthLimit => 7)|
│ │ │ │ +0002be00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0002be70: 2030 2020 3120 2032 2020 3320 2034 2020   0  1  2  3  4  
│ │ │ │ -0002be80: 3520 2036 2020 3720 2020 2020 2020 2020  5  6  7         
│ │ │ │ -0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bea0: 2020 7c0a 7c6f 3920 3d20 746f 7461 6c3a    |.|o9 = total:
│ │ │ │ -0002beb0: 2031 3120 3138 2032 3620 3336 2034 3720   11 18 26 36 47 
│ │ │ │ -0002bec0: 3630 2037 3420 3930 2020 2020 2020 2020  60 74 90        
│ │ │ │ -0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bee0: 2020 207c 0a7c 2020 2020 2020 2020 2036     |.|         6
│ │ │ │ -0002bef0: 3a20 2036 2020 2e20 202e 2020 2e20 202e  :  6  .  .  .  .
│ │ │ │ -0002bf00: 2020 2e20 202e 2020 2e20 2020 2020 2020    .  .  .       
│ │ │ │ -0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0002bf30: 373a 2020 3520 3138 2031 3420 202e 2020  7:  5 18 14  .  
│ │ │ │ -0002bf40: 2e20 202e 2020 2e20 202e 2020 2020 2020  .  .  .  .      
│ │ │ │ -0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bf60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002bf70: 2038 3a20 202e 2020 2e20 3132 2033 3620   8:  .  . 12 36 
│ │ │ │ -0002bf80: 3235 2020 2e20 202e 2020 2e20 2020 2020  25  .  .  .     
│ │ │ │ -0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002bfb0: 2020 393a 2020 2e20 202e 2020 2e20 202e    9:  .  .  .  .
│ │ │ │ -0002bfc0: 2032 3220 3630 2033 3920 202e 2020 2020   22 60 39  .    
│ │ │ │ -0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfe0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002bff0: 2020 3130 3a20 202e 2020 2e20 202e 2020    10:  .  .  .  
│ │ │ │ -0002c000: 2e20 202e 2020 2e20 3335 2039 3020 2020  .  .  . 35 90   
│ │ │ │ +0002be40: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002be50: 3020 2031 2020 3220 2033 2020 3420 2035  0  1  2  3  4  5
│ │ │ │ +0002be60: 2020 3620 2037 2020 2020 2020 2020 2020    6  7          
│ │ │ │ +0002be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002be80: 207c 0a7c 6f39 203d 2074 6f74 616c 3a20   |.|o9 = total: 
│ │ │ │ +0002be90: 3131 2031 3820 3236 2033 3620 3437 2036  11 18 26 36 47 6
│ │ │ │ +0002bea0: 3020 3734 2039 3020 2020 2020 2020 2020  0 74 90         
│ │ │ │ +0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bec0: 2020 7c0a 7c20 2020 2020 2020 2020 363a    |.|         6:
│ │ │ │ +0002bed0: 2020 3620 202e 2020 2e20 202e 2020 2e20    6  .  .  .  . 
│ │ │ │ +0002bee0: 202e 2020 2e20 202e 2020 2020 2020 2020   .  .  .        
│ │ │ │ +0002bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf00: 2020 207c 0a7c 2020 2020 2020 2020 2037     |.|         7
│ │ │ │ +0002bf10: 3a20 2035 2031 3820 3134 2020 2e20 202e  :  5 18 14  .  .
│ │ │ │ +0002bf20: 2020 2e20 202e 2020 2e20 2020 2020 2020    .  .  .       
│ │ │ │ +0002bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002bf50: 383a 2020 2e20 202e 2031 3220 3336 2032  8:  .  . 12 36 2
│ │ │ │ +0002bf60: 3520 202e 2020 2e20 202e 2020 2020 2020  5  .  .  .      
│ │ │ │ +0002bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bf80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002bf90: 2039 3a20 202e 2020 2e20 202e 2020 2e20   9:  .  .  .  . 
│ │ │ │ +0002bfa0: 3232 2036 3020 3339 2020 2e20 2020 2020  22 60 39  .     
│ │ │ │ +0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bfc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0002bfd0: 2031 303a 2020 2e20 202e 2020 2e20 202e   10:  .  .  .  .
│ │ │ │ +0002bfe0: 2020 2e20 202e 2033 3520 3930 2020 2020    .  . 35 90    
│ │ │ │ +0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c000: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c020: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c060: 2020 2020 2020 2020 207c 0a7c 6f39 203a           |.|o9 :
│ │ │ │ -0002c070: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ -0002c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0002c040: 2020 2020 2020 2020 7c0a 7c6f 3920 3a20          |.|o9 : 
│ │ │ │ +0002c050: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +0002c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c080: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002c090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -0002c0f0: 3020 3a20 6e65 744c 6973 7420 4252 616e  0 : netList BRan
│ │ │ │ -0002c100: 6b73 206d 6174 7269 7846 6163 746f 7269  ks matrixFactori
│ │ │ │ -0002c110: 7a61 7469 6f6e 2866 662c 2068 6967 6853  zation(ff, highS
│ │ │ │ -0002c120: 797a 7967 7920 4d30 2920 2020 7c0a 7c20  yzygy M0)   |.| 
│ │ │ │ +0002c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ +0002c0d0: 203a 206e 6574 4c69 7374 2042 5261 6e6b   : netList BRank
│ │ │ │ +0002c0e0: 7320 6d61 7472 6978 4661 6374 6f72 697a  s matrixFactoriz
│ │ │ │ +0002c0f0: 6174 696f 6e28 6666 2c20 6869 6768 5379  ation(ff, highSy
│ │ │ │ +0002c100: 7a79 6779 204d 3029 2020 207c 0a7c 2020  zygy M0)   |.|  
│ │ │ │ +0002c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002c170: 2020 2020 2020 2b2d 2b2d 2b20 2020 2020        +-+-+     
│ │ │ │ -0002c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002c1b0: 7c6f 3130 203d 207c 367c 367c 2020 2020  |o10 = |6|6|    
│ │ │ │ -0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c1f0: 0a7c 2020 2020 2020 2b2d 2b2d 2b20 2020  .|      +-+-+   
│ │ │ │ -0002c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c140: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002c150: 2020 2020 202b 2d2b 2d2b 2020 2020 2020       +-+-+      
│ │ │ │ +0002c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c180: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002c190: 6f31 3020 3d20 7c36 7c36 7c20 2020 2020  o10 = |6|6|     
│ │ │ │ +0002c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c1c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002c1d0: 7c20 2020 2020 202b 2d2b 2d2b 2020 2020  |      +-+-+    
│ │ │ │ +0002c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002c210: 0a7c 2020 2020 2020 7c33 7c36 7c20 2020  .|      |3|6|   
│ │ │ │  0002c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c230: 7c0a 7c20 2020 2020 207c 337c 367c 2020  |.|      |3|6|  
│ │ │ │ +0002c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c250: 7c0a 7c20 2020 2020 202b 2d2b 2d2b 2020  |.|      +-+-+  
│ │ │ │  0002c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c270: 207c 0a7c 2020 2020 2020 2b2d 2b2d 2b20   |.|      +-+-+ 
│ │ │ │ +0002c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c290: 207c 0a7c 2020 2020 2020 7c32 7c36 7c20   |.|      |2|6| 
│ │ │ │  0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2b0: 2020 7c0a 7c20 2020 2020 207c 327c 367c    |.|      |2|6|
│ │ │ │ +0002c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c2d0: 2020 7c0a 7c20 2020 2020 202b 2d2b 2d2b    |.|      +-+-+
│ │ │ │  0002c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2f0: 2020 207c 0a7c 2020 2020 2020 2b2d 2b2d     |.|      +-+-
│ │ │ │ -0002c300: 2b20 2020 2020 2020 2020 2020 2020 2020  +               
│ │ │ │ -0002c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c330: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c310: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002c320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c370: 2d2d 2d2d 2d2b 0a0a 496e 2074 6869 7320  -----+..In this 
│ │ │ │ -0002c380: 6361 7365 2061 7320 696e 2061 6c6c 206f  case as in all o
│ │ │ │ -0002c390: 7468 6572 7320 7765 2068 6176 6520 6578  thers we have ex
│ │ │ │ -0002c3a0: 616d 696e 6564 2c20 6772 6561 7465 7220  amined, greater 
│ │ │ │ -0002c3b0: 224f 7074 696d 6973 6d22 2069 7320 6e6f  "Optimism" is no
│ │ │ │ -0002c3c0: 740a 6a75 7374 6966 6965 642c 2061 6e64  t.justified, and
│ │ │ │ -0002c3d0: 2074 6875 7320 6d61 7472 6978 4661 6374   thus matrixFact
│ │ │ │ -0002c3e0: 6f72 697a 6174 696f 6e28 6666 2c20 6869  orization(ff, hi
│ │ │ │ -0002c3f0: 6768 5379 7a79 6779 284d 302c 204f 7074  ghSyzygy(M0, Opt
│ │ │ │ -0002c400: 696d 6973 6d3d 3e31 2929 3b20 776f 756c  imism=>1)); woul
│ │ │ │ -0002c410: 640a 7072 6f64 7563 6520 616e 2065 7272  d.produce an err
│ │ │ │ -0002c420: 6f72 2e0a 0a43 6176 6561 740a 3d3d 3d3d  or...Caveat.====
│ │ │ │ -0002c430: 3d3d 0a0a 4120 6275 6720 696e 2074 6865  ==..A bug in the
│ │ │ │ -0002c440: 2074 6f74 616c 2045 7874 2073 6372 6970   total Ext scrip
│ │ │ │ -0002c450: 7420 6d65 616e 7320 7468 6174 2074 6865  t means that the
│ │ │ │ -0002c460: 206f 6464 4578 744d 6f64 756c 6520 6973   oddExtModule is
│ │ │ │ -0002c470: 2073 6f6d 6574 696d 6573 207a 6572 6f2c   sometimes zero,
│ │ │ │ -0002c480: 0a61 6e64 2074 6869 7320 6361 6e20 6361  .and this can ca
│ │ │ │ -0002c490: 7573 6520 6120 7772 6f6e 6720 7661 6c75  use a wrong valu
│ │ │ │ -0002c4a0: 6520 746f 2062 6520 7265 7475 726e 6564  e to be returned
│ │ │ │ -0002c4b0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ...See also.====
│ │ │ │ -0002c4c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -0002c4d0: 6576 656e 4578 744d 6f64 756c 653a 2065  evenExtModule: e
│ │ │ │ -0002c4e0: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d  venExtModule, --
│ │ │ │ -0002c4f0: 2065 7665 6e20 7061 7274 206f 6620 4578   even part of Ex
│ │ │ │ -0002c500: 745e 2a28 4d2c 6b29 206f 7665 7220 610a  t^*(M,k) over a.
│ │ │ │ -0002c510: 2020 2020 636f 6d70 6c65 7465 2069 6e74      complete int
│ │ │ │ -0002c520: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ -0002c530: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ -0002c540: 6174 6f72 2072 696e 670a 2020 2a20 2a6e  ator ring.  * *n
│ │ │ │ -0002c550: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65  ote oddExtModule
│ │ │ │ -0002c560: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20  : oddExtModule, 
│ │ │ │ -0002c570: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ -0002c580: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -0002c590: 2063 6f6d 706c 6574 650a 2020 2020 696e   complete.    in
│ │ │ │ -0002c5a0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f  tersection as mo
│ │ │ │ -0002c5b0: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ -0002c5c0: 7261 746f 7220 7269 6e67 0a20 202a 202a  rator ring.  * *
│ │ │ │ -0002c5d0: 6e6f 7465 206d 6642 6f75 6e64 3a20 6d66  note mfBound: mf
│ │ │ │ -0002c5e0: 426f 756e 642c 202d 2d20 6465 7465 726d  Bound, -- determ
│ │ │ │ -0002c5f0: 696e 6573 2068 6f77 2068 6967 6820 6120  ines how high a 
│ │ │ │ -0002c600: 7379 7a79 6779 2074 6f20 7461 6b65 2066  syzygy to take f
│ │ │ │ -0002c610: 6f72 0a20 2020 2022 6d61 7472 6978 4661  or.    "matrixFa
│ │ │ │ -0002c620: 6374 6f72 697a 6174 696f 6e22 0a20 202a  ctorization".  *
│ │ │ │ -0002c630: 202a 6e6f 7465 206d 6174 7269 7846 6163   *note matrixFac
│ │ │ │ -0002c640: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472  torization: matr
│ │ │ │ -0002c650: 6978 4661 6374 6f72 697a 6174 696f 6e2c  ixFactorization,
│ │ │ │ -0002c660: 202d 2d20 4d61 7073 2069 6e20 6120 6869   -- Maps in a hi
│ │ │ │ -0002c670: 6768 6572 0a20 2020 2063 6f64 696d 656e  gher.    codimen
│ │ │ │ -0002c680: 7369 6f6e 206d 6174 7269 7820 6661 6374  sion matrix fact
│ │ │ │ -0002c690: 6f72 697a 6174 696f 6e0a 0a57 6179 7320  orization..Ways 
│ │ │ │ -0002c6a0: 746f 2075 7365 2068 6967 6853 797a 7967  to use highSyzyg
│ │ │ │ -0002c6b0: 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  y:.=============
│ │ │ │ -0002c6c0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0002c6d0: 2268 6967 6853 797a 7967 7928 4d6f 6475  "highSyzygy(Modu
│ │ │ │ -0002c6e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072  le)"..For the pr
│ │ │ │ -0002c6f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -0002c700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -0002c710: 206f 626a 6563 7420 2a6e 6f74 6520 6869   object *note hi
│ │ │ │ -0002c720: 6768 5379 7a79 6779 3a20 6869 6768 5379  ghSyzygy: highSy
│ │ │ │ -0002c730: 7a79 6779 2c20 6973 2061 202a 6e6f 7465  zygy, is a *note
│ │ │ │ -0002c740: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ -0002c750: 2077 6974 680a 6f70 7469 6f6e 733a 2028   with.options: (
│ │ │ │ -0002c760: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ -0002c770: 686f 6446 756e 6374 696f 6e57 6974 684f  hodFunctionWithO
│ │ │ │ -0002c780: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d  ptions,...------
│ │ │ │ +0002c350: 2d2d 2d2d 2b0a 0a49 6e20 7468 6973 2063  ----+..In this c
│ │ │ │ +0002c360: 6173 6520 6173 2069 6e20 616c 6c20 6f74  ase as in all ot
│ │ │ │ +0002c370: 6865 7273 2077 6520 6861 7665 2065 7861  hers we have exa
│ │ │ │ +0002c380: 6d69 6e65 642c 2067 7265 6174 6572 2022  mined, greater "
│ │ │ │ +0002c390: 4f70 7469 6d69 736d 2220 6973 206e 6f74  Optimism" is not
│ │ │ │ +0002c3a0: 0a6a 7573 7469 6669 6564 2c20 616e 6420  .justified, and 
│ │ │ │ +0002c3b0: 7468 7573 206d 6174 7269 7846 6163 746f  thus matrixFacto
│ │ │ │ +0002c3c0: 7269 7a61 7469 6f6e 2866 662c 2068 6967  rization(ff, hig
│ │ │ │ +0002c3d0: 6853 797a 7967 7928 4d30 2c20 4f70 7469  hSyzygy(M0, Opti
│ │ │ │ +0002c3e0: 6d69 736d 3d3e 3129 293b 2077 6f75 6c64  mism=>1)); would
│ │ │ │ +0002c3f0: 0a70 726f 6475 6365 2061 6e20 6572 726f  .produce an erro
│ │ │ │ +0002c400: 722e 0a0a 4361 7665 6174 0a3d 3d3d 3d3d  r...Caveat.=====
│ │ │ │ +0002c410: 3d0a 0a41 2062 7567 2069 6e20 7468 6520  =..A bug in the 
│ │ │ │ +0002c420: 746f 7461 6c20 4578 7420 7363 7269 7074  total Ext script
│ │ │ │ +0002c430: 206d 6561 6e73 2074 6861 7420 7468 6520   means that the 
│ │ │ │ +0002c440: 6f64 6445 7874 4d6f 6475 6c65 2069 7320  oddExtModule is 
│ │ │ │ +0002c450: 736f 6d65 7469 6d65 7320 7a65 726f 2c0a  sometimes zero,.
│ │ │ │ +0002c460: 616e 6420 7468 6973 2063 616e 2063 6175  and this can cau
│ │ │ │ +0002c470: 7365 2061 2077 726f 6e67 2076 616c 7565  se a wrong value
│ │ │ │ +0002c480: 2074 6f20 6265 2072 6574 7572 6e65 642e   to be returned.
│ │ │ │ +0002c490: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +0002c4a0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065  ===..  * *note e
│ │ │ │ +0002c4b0: 7665 6e45 7874 4d6f 6475 6c65 3a20 6576  venExtModule: ev
│ │ │ │ +0002c4c0: 656e 4578 744d 6f64 756c 652c 202d 2d20  enExtModule, -- 
│ │ │ │ +0002c4d0: 6576 656e 2070 6172 7420 6f66 2045 7874  even part of Ext
│ │ │ │ +0002c4e0: 5e2a 284d 2c6b 2920 6f76 6572 2061 0a20  ^*(M,k) over a. 
│ │ │ │ +0002c4f0: 2020 2063 6f6d 706c 6574 6520 696e 7465     complete inte
│ │ │ │ +0002c500: 7273 6563 7469 6f6e 2061 7320 6d6f 6475  rsection as modu
│ │ │ │ +0002c510: 6c65 206f 7665 7220 4349 206f 7065 7261  le over CI opera
│ │ │ │ +0002c520: 746f 7220 7269 6e67 0a20 202a 202a 6e6f  tor ring.  * *no
│ │ │ │ +0002c530: 7465 206f 6464 4578 744d 6f64 756c 653a  te oddExtModule:
│ │ │ │ +0002c540: 206f 6464 4578 744d 6f64 756c 652c 202d   oddExtModule, -
│ │ │ │ +0002c550: 2d20 6f64 6420 7061 7274 206f 6620 4578  - odd part of Ex
│ │ │ │ +0002c560: 745e 2a28 4d2c 6b29 206f 7665 7220 6120  t^*(M,k) over a 
│ │ │ │ +0002c570: 636f 6d70 6c65 7465 0a20 2020 2069 6e74  complete.    int
│ │ │ │ +0002c580: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ +0002c590: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ +0002c5a0: 6174 6f72 2072 696e 670a 2020 2a20 2a6e  ator ring.  * *n
│ │ │ │ +0002c5b0: 6f74 6520 6d66 426f 756e 643a 206d 6642  ote mfBound: mfB
│ │ │ │ +0002c5c0: 6f75 6e64 2c20 2d2d 2064 6574 6572 6d69  ound, -- determi
│ │ │ │ +0002c5d0: 6e65 7320 686f 7720 6869 6768 2061 2073  nes how high a s
│ │ │ │ +0002c5e0: 797a 7967 7920 746f 2074 616b 6520 666f  yzygy to take fo
│ │ │ │ +0002c5f0: 720a 2020 2020 226d 6174 7269 7846 6163  r.    "matrixFac
│ │ │ │ +0002c600: 746f 7269 7a61 7469 6f6e 220a 2020 2a20  torization".  * 
│ │ │ │ +0002c610: 2a6e 6f74 6520 6d61 7472 6978 4661 6374  *note matrixFact
│ │ │ │ +0002c620: 6f72 697a 6174 696f 6e3a 206d 6174 7269  orization: matri
│ │ │ │ +0002c630: 7846 6163 746f 7269 7a61 7469 6f6e 2c20  xFactorization, 
│ │ │ │ +0002c640: 2d2d 204d 6170 7320 696e 2061 2068 6967  -- Maps in a hig
│ │ │ │ +0002c650: 6865 720a 2020 2020 636f 6469 6d65 6e73  her.    codimens
│ │ │ │ +0002c660: 696f 6e20 6d61 7472 6978 2066 6163 746f  ion matrix facto
│ │ │ │ +0002c670: 7269 7a61 7469 6f6e 0a0a 5761 7973 2074  rization..Ways t
│ │ │ │ +0002c680: 6f20 7573 6520 6869 6768 5379 7a79 6779  o use highSyzygy
│ │ │ │ +0002c690: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +0002c6a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +0002c6b0: 6869 6768 5379 7a79 6779 284d 6f64 756c  highSyzygy(Modul
│ │ │ │ +0002c6c0: 6529 220a 0a46 6f72 2074 6865 2070 726f  e)"..For the pro
│ │ │ │ +0002c6d0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +0002c6e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +0002c6f0: 6f62 6a65 6374 202a 6e6f 7465 2068 6967  object *note hig
│ │ │ │ +0002c700: 6853 797a 7967 793a 2068 6967 6853 797a  hSyzygy: highSyz
│ │ │ │ +0002c710: 7967 792c 2069 7320 6120 2a6e 6f74 6520  ygy, is a *note 
│ │ │ │ +0002c720: 6d65 7468 6f64 2066 756e 6374 696f 6e20  method function 
│ │ │ │ +0002c730: 7769 7468 0a6f 7074 696f 6e73 3a20 284d  with.options: (M
│ │ │ │ +0002c740: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +0002c750: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70  odFunctionWithOp
│ │ │ │ +0002c760: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d  tions,...-------
│ │ │ │ +0002c770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002c7d0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ -0002c7e0: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ -0002c7f0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -0002c800: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -0002c810: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -0002c820: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ -0002c830: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -0002c840: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ -0002c850: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0002c860: 732e 6d32 3a33 3330 393a 302e 0a1f 0a46  s.m2:3309:0....F
│ │ │ │ -0002c870: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -0002c880: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -0002c890: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -0002c8a0: 2068 4d61 7073 2c20 4e65 7874 3a20 486f   hMaps, Next: Ho
│ │ │ │ -0002c8b0: 6d57 6974 6843 6f6d 706f 6e65 6e74 732c  mWithComponents,
│ │ │ │ -0002c8c0: 2050 7265 763a 2068 6967 6853 797a 7967   Prev: highSyzyg
│ │ │ │ -0002c8d0: 792c 2055 703a 2054 6f70 0a0a 684d 6170  y, Up: Top..hMap
│ │ │ │ -0002c8e0: 7320 2d2d 206c 6973 7420 7468 6520 6d61  s -- list the ma
│ │ │ │ -0002c8f0: 7073 2020 6828 7029 3a20 415f 3028 7029  ps  h(p): A_0(p)
│ │ │ │ -0002c900: 2d2d 3e20 415f 3128 7029 2069 6e20 6120  --> A_1(p) in a 
│ │ │ │ -0002c910: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ -0002c920: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ion.************
│ │ │ │ +0002c7b0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +0002c7c0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +0002c7d0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +0002c7e0: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +0002c7f0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +0002c800: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ +0002c810: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +0002c820: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ +0002c830: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0002c840: 2e6d 323a 3333 3039 3a30 2e0a 1f0a 4669  .m2:3309:0....Fi
│ │ │ │ +0002c850: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +0002c860: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0002c870: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +0002c880: 684d 6170 732c 204e 6578 743a 2048 6f6d  hMaps, Next: Hom
│ │ │ │ +0002c890: 5769 7468 436f 6d70 6f6e 656e 7473 2c20  WithComponents, 
│ │ │ │ +0002c8a0: 5072 6576 3a20 6869 6768 5379 7a79 6779  Prev: highSyzygy
│ │ │ │ +0002c8b0: 2c20 5570 3a20 546f 700a 0a68 4d61 7073  , Up: Top..hMaps
│ │ │ │ +0002c8c0: 202d 2d20 6c69 7374 2074 6865 206d 6170   -- list the map
│ │ │ │ +0002c8d0: 7320 2068 2870 293a 2041 5f30 2870 292d  s  h(p): A_0(p)-
│ │ │ │ +0002c8e0: 2d3e 2041 5f31 2870 2920 696e 2061 206d  -> A_1(p) in a m
│ │ │ │ +0002c8f0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +0002c900: 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  on.*************
│ │ │ │ +0002c910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002c920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002c930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002c940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002c950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002c960: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -0002c970: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0002c980: 2068 4d61 7073 203d 2068 4d61 7073 206d   hMaps = hMaps m
│ │ │ │ -0002c990: 660a 2020 2a20 496e 7075 7473 3a0a 2020  f.  * Inputs:.  
│ │ │ │ -0002c9a0: 2020 2020 2a20 6d66 2c20 6120 2a6e 6f74      * mf, a *not
│ │ │ │ -0002c9b0: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ -0002c9c0: 7932 446f 6329 4c69 7374 2c2c 206f 7574  y2Doc)List,, out
│ │ │ │ -0002c9d0: 7075 7420 6f66 2061 206d 6174 7269 7846  put of a matrixF
│ │ │ │ -0002c9e0: 6163 746f 7269 7a61 7469 6f6e 0a20 2020  actorization.   
│ │ │ │ -0002c9f0: 2020 2020 2063 6f6d 7075 7461 7469 6f6e       computation
│ │ │ │ -0002ca00: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -0002ca10: 2020 2020 2a20 684d 6170 732c 2061 202a      * hMaps, a *
│ │ │ │ -0002ca20: 6e6f 7465 206c 6973 743a 2028 4d61 6361  note list: (Maca
│ │ │ │ -0002ca30: 756c 6179 3244 6f63 294c 6973 742c 2c20  ulay2Doc)List,, 
│ │ │ │ -0002ca40: 6c69 7374 206d 6174 7269 6365 7320 2468  list matrices $h
│ │ │ │ -0002ca50: 5f70 3a20 415f 3028 7029 5c74 6f0a 2020  _p: A_0(p)\to.  
│ │ │ │ -0002ca60: 2020 2020 2020 415f 3128 7029 242e 2054        A_1(p)$. T
│ │ │ │ -0002ca70: 6865 2073 6f75 7263 6573 2061 6e64 2074  he sources and t
│ │ │ │ -0002ca80: 6172 6765 7473 206f 6620 7468 6573 6520  argets of these 
│ │ │ │ -0002ca90: 6d61 7073 2068 6176 6520 7468 6520 636f  maps have the co
│ │ │ │ -0002caa0: 6d70 6f6e 656e 7473 0a20 2020 2020 2020  mponents.       
│ │ │ │ -0002cab0: 2042 5f73 2870 292e 0a0a 4465 7363 7269   B_s(p)...Descri
│ │ │ │ -0002cac0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -0002cad0: 3d0a 0a53 6565 2074 6865 2064 6f63 756d  =..See the docum
│ │ │ │ -0002cae0: 656e 7461 7469 6f6e 2066 6f72 206d 6174  entation for mat
│ │ │ │ -0002caf0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0002cb00: 2066 6f72 2061 6e20 6578 616d 706c 652e   for an example.
│ │ │ │ -0002cb10: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0002cb20: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ -0002cb30: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -0002cb40: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72  on: matrixFactor
│ │ │ │ -0002cb50: 697a 6174 696f 6e2c 202d 2d20 4d61 7073  ization, -- Maps
│ │ │ │ -0002cb60: 2069 6e20 6120 6869 6768 6572 0a20 2020   in a higher.   
│ │ │ │ -0002cb70: 2063 6f64 696d 656e 7369 6f6e 206d 6174   codimension mat
│ │ │ │ -0002cb80: 7269 7820 6661 6374 6f72 697a 6174 696f  rix factorizatio
│ │ │ │ -0002cb90: 6e0a 2020 2a20 2a6e 6f74 6520 644d 6170  n.  * *note dMap
│ │ │ │ -0002cba0: 733a 2064 4d61 7073 2c20 2d2d 206c 6973  s: dMaps, -- lis
│ │ │ │ -0002cbb0: 7420 7468 6520 6d61 7073 2020 6428 7029  t the maps  d(p)
│ │ │ │ -0002cbc0: 3a41 5f31 2870 292d 2d3e 2041 5f30 2870  :A_1(p)--> A_0(p
│ │ │ │ -0002cbd0: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ -0002cbe0: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ -0002cbf0: 202a 202a 6e6f 7465 2042 5261 6e6b 733a   * *note BRanks:
│ │ │ │ -0002cc00: 2042 5261 6e6b 732c 202d 2d20 7261 6e6b   BRanks, -- rank
│ │ │ │ -0002cc10: 7320 6f66 2074 6865 206d 6f64 756c 6573  s of the modules
│ │ │ │ -0002cc20: 2042 5f69 2864 2920 696e 2061 0a20 2020   B_i(d) in a.   
│ │ │ │ -0002cc30: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -0002cc40: 7469 6f6e 0a20 202a 202a 6e6f 7465 2062  tion.  * *note b
│ │ │ │ -0002cc50: 4d61 7073 3a20 624d 6170 732c 202d 2d20  Maps: bMaps, -- 
│ │ │ │ -0002cc60: 6c69 7374 2074 6865 206d 6170 7320 2064  list the maps  d
│ │ │ │ -0002cc70: 5f70 3a42 5f31 2870 292d 2d3e 425f 3028  _p:B_1(p)-->B_0(
│ │ │ │ -0002cc80: 7029 2069 6e20 610a 2020 2020 6d61 7472  p) in a.    matr
│ │ │ │ -0002cc90: 6978 4661 6374 6f72 697a 6174 696f 6e0a  ixFactorization.
│ │ │ │ -0002cca0: 2020 2a20 2a6e 6f74 6520 7073 694d 6170    * *note psiMap
│ │ │ │ -0002ccb0: 733a 2070 7369 4d61 7073 2c20 2d2d 206c  s: psiMaps, -- l
│ │ │ │ -0002ccc0: 6973 7420 7468 6520 6d61 7073 2020 7073  ist the maps  ps
│ │ │ │ -0002ccd0: 6928 7029 3a20 425f 3128 7029 202d 2d3e  i(p): B_1(p) -->
│ │ │ │ -0002cce0: 2041 5f30 2870 2d31 2920 696e 2061 0a20   A_0(p-1) in a. 
│ │ │ │ -0002ccf0: 2020 206d 6174 7269 7846 6163 746f 7269     matrixFactori
│ │ │ │ -0002cd00: 7a61 7469 6f6e 0a0a 5761 7973 2074 6f20  zation..Ways to 
│ │ │ │ -0002cd10: 7573 6520 684d 6170 733a 0a3d 3d3d 3d3d  use hMaps:.=====
│ │ │ │ -0002cd20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -0002cd30: 202a 2022 684d 6170 7328 4c69 7374 2922   * "hMaps(List)"
│ │ │ │ -0002cd40: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0002cd50: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0002cd60: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0002cd70: 6563 7420 2a6e 6f74 6520 684d 6170 733a  ect *note hMaps:
│ │ │ │ -0002cd80: 2068 4d61 7073 2c20 6973 2061 202a 6e6f   hMaps, is a *no
│ │ │ │ -0002cd90: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ -0002cda0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ -0002cdb0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -0002cdc0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +0002c940: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +0002c950: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0002c960: 684d 6170 7320 3d20 684d 6170 7320 6d66  hMaps = hMaps mf
│ │ │ │ +0002c970: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +0002c980: 2020 202a 206d 662c 2061 202a 6e6f 7465     * mf, a *note
│ │ │ │ +0002c990: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ +0002c9a0: 3244 6f63 294c 6973 742c 2c20 6f75 7470  2Doc)List,, outp
│ │ │ │ +0002c9b0: 7574 206f 6620 6120 6d61 7472 6978 4661  ut of a matrixFa
│ │ │ │ +0002c9c0: 6374 6f72 697a 6174 696f 6e0a 2020 2020  ctorization.    
│ │ │ │ +0002c9d0: 2020 2020 636f 6d70 7574 6174 696f 6e0a      computation.
│ │ │ │ +0002c9e0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +0002c9f0: 2020 202a 2068 4d61 7073 2c20 6120 2a6e     * hMaps, a *n
│ │ │ │ +0002ca00: 6f74 6520 6c69 7374 3a20 284d 6163 6175  ote list: (Macau
│ │ │ │ +0002ca10: 6c61 7932 446f 6329 4c69 7374 2c2c 206c  lay2Doc)List,, l
│ │ │ │ +0002ca20: 6973 7420 6d61 7472 6963 6573 2024 685f  ist matrices $h_
│ │ │ │ +0002ca30: 703a 2041 5f30 2870 295c 746f 0a20 2020  p: A_0(p)\to.   
│ │ │ │ +0002ca40: 2020 2020 2041 5f31 2870 2924 2e20 5468       A_1(p)$. Th
│ │ │ │ +0002ca50: 6520 736f 7572 6365 7320 616e 6420 7461  e sources and ta
│ │ │ │ +0002ca60: 7267 6574 7320 6f66 2074 6865 7365 206d  rgets of these m
│ │ │ │ +0002ca70: 6170 7320 6861 7665 2074 6865 2063 6f6d  aps have the com
│ │ │ │ +0002ca80: 706f 6e65 6e74 730a 2020 2020 2020 2020  ponents.        
│ │ │ │ +0002ca90: 425f 7328 7029 2e0a 0a44 6573 6372 6970  B_s(p)...Descrip
│ │ │ │ +0002caa0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +0002cab0: 0a0a 5365 6520 7468 6520 646f 6375 6d65  ..See the docume
│ │ │ │ +0002cac0: 6e74 6174 696f 6e20 666f 7220 6d61 7472  ntation for matr
│ │ │ │ +0002cad0: 6978 4661 6374 6f72 697a 6174 696f 6e20  ixFactorization 
│ │ │ │ +0002cae0: 666f 7220 616e 2065 7861 6d70 6c65 2e0a  for an example..
│ │ │ │ +0002caf0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0002cb00: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6d61  ==..  * *note ma
│ │ │ │ +0002cb10: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0002cb20: 6e3a 206d 6174 7269 7846 6163 746f 7269  n: matrixFactori
│ │ │ │ +0002cb30: 7a61 7469 6f6e 2c20 2d2d 204d 6170 7320  zation, -- Maps 
│ │ │ │ +0002cb40: 696e 2061 2068 6967 6865 720a 2020 2020  in a higher.    
│ │ │ │ +0002cb50: 636f 6469 6d65 6e73 696f 6e20 6d61 7472  codimension matr
│ │ │ │ +0002cb60: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ +0002cb70: 0a20 202a 202a 6e6f 7465 2064 4d61 7073  .  * *note dMaps
│ │ │ │ +0002cb80: 3a20 644d 6170 732c 202d 2d20 6c69 7374  : dMaps, -- list
│ │ │ │ +0002cb90: 2074 6865 206d 6170 7320 2064 2870 293a   the maps  d(p):
│ │ │ │ +0002cba0: 415f 3128 7029 2d2d 3e20 415f 3028 7029  A_1(p)--> A_0(p)
│ │ │ │ +0002cbb0: 2069 6e20 610a 2020 2020 6d61 7472 6978   in a.    matrix
│ │ │ │ +0002cbc0: 4661 6374 6f72 697a 6174 696f 6e0a 2020  Factorization.  
│ │ │ │ +0002cbd0: 2a20 2a6e 6f74 6520 4252 616e 6b73 3a20  * *note BRanks: 
│ │ │ │ +0002cbe0: 4252 616e 6b73 2c20 2d2d 2072 616e 6b73  BRanks, -- ranks
│ │ │ │ +0002cbf0: 206f 6620 7468 6520 6d6f 6475 6c65 7320   of the modules 
│ │ │ │ +0002cc00: 425f 6928 6429 2069 6e20 610a 2020 2020  B_i(d) in a.    
│ │ │ │ +0002cc10: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +0002cc20: 696f 6e0a 2020 2a20 2a6e 6f74 6520 624d  ion.  * *note bM
│ │ │ │ +0002cc30: 6170 733a 2062 4d61 7073 2c20 2d2d 206c  aps: bMaps, -- l
│ │ │ │ +0002cc40: 6973 7420 7468 6520 6d61 7073 2020 645f  ist the maps  d_
│ │ │ │ +0002cc50: 703a 425f 3128 7029 2d2d 3e42 5f30 2870  p:B_1(p)-->B_0(p
│ │ │ │ +0002cc60: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ +0002cc70: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ +0002cc80: 202a 202a 6e6f 7465 2070 7369 4d61 7073   * *note psiMaps
│ │ │ │ +0002cc90: 3a20 7073 694d 6170 732c 202d 2d20 6c69  : psiMaps, -- li
│ │ │ │ +0002cca0: 7374 2074 6865 206d 6170 7320 2070 7369  st the maps  psi
│ │ │ │ +0002ccb0: 2870 293a 2042 5f31 2870 2920 2d2d 3e20  (p): B_1(p) --> 
│ │ │ │ +0002ccc0: 415f 3028 702d 3129 2069 6e20 610a 2020  A_0(p-1) in a.  
│ │ │ │ +0002ccd0: 2020 6d61 7472 6978 4661 6374 6f72 697a    matrixFactoriz
│ │ │ │ +0002cce0: 6174 696f 6e0a 0a57 6179 7320 746f 2075  ation..Ways to u
│ │ │ │ +0002ccf0: 7365 2068 4d61 7073 3a0a 3d3d 3d3d 3d3d  se hMaps:.======
│ │ │ │ +0002cd00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +0002cd10: 2a20 2268 4d61 7073 284c 6973 7429 220a  * "hMaps(List)".
│ │ │ │ +0002cd20: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +0002cd30: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +0002cd40: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +0002cd50: 6374 202a 6e6f 7465 2068 4d61 7073 3a20  ct *note hMaps: 
│ │ │ │ +0002cd60: 684d 6170 732c 2069 7320 6120 2a6e 6f74  hMaps, is a *not
│ │ │ │ +0002cd70: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ +0002cd80: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ +0002cd90: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +0002cda0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0002cdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ce10: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -0002ce20: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -0002ce30: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -0002ce40: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -0002ce50: 6d61 6361 756c 6179 322d 312e 3236 2e30  macaulay2-1.26.0
│ │ │ │ -0002ce60: 362b 6473 2f4d 322f 4d61 6361 756c 6179  6+ds/M2/Macaulay
│ │ │ │ -0002ce70: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ -0002ce80: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -0002ce90: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34  Resolutions.m2:4
│ │ │ │ -0002cea0: 3435 373a 302e 0a1f 0a46 696c 653a 2043  457:0....File: C
│ │ │ │ -0002ceb0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0002cec0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -0002ced0: 6e66 6f2c 204e 6f64 653a 2048 6f6d 5769  nfo, Node: HomWi
│ │ │ │ -0002cee0: 7468 436f 6d70 6f6e 656e 7473 2c20 4e65  thComponents, Ne
│ │ │ │ -0002cef0: 7874 3a20 696e 6669 6e69 7465 4265 7474  xt: infiniteBett
│ │ │ │ -0002cf00: 694e 756d 6265 7273 2c20 5072 6576 3a20  iNumbers, Prev: 
│ │ │ │ -0002cf10: 684d 6170 732c 2055 703a 2054 6f70 0a0a  hMaps, Up: Top..
│ │ │ │ -0002cf20: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74  HomWithComponent
│ │ │ │ -0002cf30: 7320 2d2d 2063 6f6d 7075 7465 7320 486f  s -- computes Ho
│ │ │ │ -0002cf40: 6d2c 2070 7265 7365 7276 696e 6720 6469  m, preserving di
│ │ │ │ -0002cf50: 7265 6374 2073 756d 2069 6e66 6f72 6d61  rect sum informa
│ │ │ │ -0002cf60: 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  tion.***********
│ │ │ │ +0002cdf0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +0002ce00: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +0002ce10: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +0002ce20: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +0002ce30: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ +0002ce40: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +0002ce50: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ +0002ce60: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +0002ce70: 6573 6f6c 7574 696f 6e73 2e6d 323a 3434  esolutions.m2:44
│ │ │ │ +0002ce80: 3537 3a30 2e0a 1f0a 4669 6c65 3a20 436f  57:0....File: Co
│ │ │ │ +0002ce90: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +0002cea0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ +0002ceb0: 666f 2c20 4e6f 6465 3a20 486f 6d57 6974  fo, Node: HomWit
│ │ │ │ +0002cec0: 6843 6f6d 706f 6e65 6e74 732c 204e 6578  hComponents, Nex
│ │ │ │ +0002ced0: 743a 2069 6e66 696e 6974 6542 6574 7469  t: infiniteBetti
│ │ │ │ +0002cee0: 4e75 6d62 6572 732c 2050 7265 763a 2068  Numbers, Prev: h
│ │ │ │ +0002cef0: 4d61 7073 2c20 5570 3a20 546f 700a 0a48  Maps, Up: Top..H
│ │ │ │ +0002cf00: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473  omWithComponents
│ │ │ │ +0002cf10: 202d 2d20 636f 6d70 7574 6573 2048 6f6d   -- computes Hom
│ │ │ │ +0002cf20: 2c20 7072 6573 6572 7669 6e67 2064 6972  , preserving dir
│ │ │ │ +0002cf30: 6563 7420 7375 6d20 696e 666f 726d 6174  ect sum informat
│ │ │ │ +0002cf40: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ion.************
│ │ │ │ +0002cf50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002cf60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002cf70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002cf80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002cf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002cfa0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -0002cfb0: 7361 6765 3a20 0a20 2020 2020 2020 2048  sage: .        H
│ │ │ │ -0002cfc0: 203d 2048 6f6d 284d 2c4e 290a 2020 2a20   = Hom(M,N).  * 
│ │ │ │ -0002cfd0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -0002cfe0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ -0002cff0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -0002d000: 294d 6f64 756c 652c 2c20 0a20 2020 2020  )Module,, .     
│ │ │ │ -0002d010: 202a 204e 2c20 6120 2a6e 6f74 6520 6d6f   * N, a *note mo
│ │ │ │ -0002d020: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ -0002d030: 446f 6329 4d6f 6475 6c65 2c2c 200a 2020  Doc)Module,, .  
│ │ │ │ -0002d040: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -0002d050: 202a 2048 2c20 6120 2a6e 6f74 6520 6d6f   * H, a *note mo
│ │ │ │ -0002d060: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ -0002d070: 446f 6329 4d6f 6475 6c65 2c2c 200a 0a44  Doc)Module,, ..D
│ │ │ │ -0002d080: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -0002d090: 3d3d 3d3d 3d3d 0a0a 4966 204d 2061 6e64  ======..If M and
│ │ │ │ -0002d0a0: 2f6f 7220 4e20 6172 6520 6469 7265 6374  /or N are direct
│ │ │ │ -0002d0b0: 2073 756d 206d 6f64 756c 6573 2028 6973   sum modules (is
│ │ │ │ -0002d0c0: 4469 7265 6374 5375 6d20 4d20 3d3d 2074  DirectSum M == t
│ │ │ │ -0002d0d0: 7275 6529 2074 6865 6e20 4820 6973 2074  rue) then H is t
│ │ │ │ -0002d0e0: 6865 0a64 6972 6563 7420 7375 6d20 6f66  he.direct sum of
│ │ │ │ -0002d0f0: 2074 6865 2048 6f6d 7320 6265 7477 6565   the Homs betwee
│ │ │ │ -0002d100: 6e20 7468 6520 636f 6d70 6f6e 656e 7473  n the components
│ │ │ │ -0002d110: 2e20 5468 6973 2053 484f 554c 4420 6265  . This SHOULD be
│ │ │ │ -0002d120: 2062 7569 6c74 2069 6e74 6f0a 486f 6d28   built into.Hom(
│ │ │ │ -0002d130: 4d2c 4e29 2c20 6275 7420 6973 6e27 7420  M,N), but isn't 
│ │ │ │ -0002d140: 6173 206f 6620 4d32 2c20 762e 2031 2e37  as of M2, v. 1.7
│ │ │ │ -0002d150: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -0002d160: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2074  ===..  * *note t
│ │ │ │ -0002d170: 656e 736f 7257 6974 6843 6f6d 706f 6e65  ensorWithCompone
│ │ │ │ -0002d180: 6e74 733a 2074 656e 736f 7257 6974 6843  nts: tensorWithC
│ │ │ │ -0002d190: 6f6d 706f 6e65 6e74 732c 202d 2d20 666f  omponents, -- fo
│ │ │ │ -0002d1a0: 726d 7320 7468 6520 7465 6e73 6f72 0a20  rms the tensor. 
│ │ │ │ -0002d1b0: 2020 2070 726f 6475 6374 2c20 7072 6573     product, pres
│ │ │ │ -0002d1c0: 6572 7669 6e67 2064 6972 6563 7420 7375  erving direct su
│ │ │ │ -0002d1d0: 6d20 696e 666f 726d 6174 696f 6e0a 2020  m information.  
│ │ │ │ -0002d1e0: 2a20 2a6e 6f74 6520 6475 616c 5769 7468  * *note dualWith
│ │ │ │ -0002d1f0: 436f 6d70 6f6e 656e 7473 3a20 6475 616c  Components: dual
│ │ │ │ -0002d200: 5769 7468 436f 6d70 6f6e 656e 7473 2c20  WithComponents, 
│ │ │ │ -0002d210: 2d2d 2064 7561 6c20 6d6f 6475 6c65 2070  -- dual module p
│ │ │ │ -0002d220: 7265 7365 7276 696e 670a 2020 2020 6469  reserving.    di
│ │ │ │ -0002d230: 7265 6374 2073 756d 2069 6e66 6f72 6d61  rect sum informa
│ │ │ │ -0002d240: 7469 6f6e 0a0a 5761 7973 2074 6f20 7573  tion..Ways to us
│ │ │ │ -0002d250: 6520 486f 6d57 6974 6843 6f6d 706f 6e65  e HomWithCompone
│ │ │ │ -0002d260: 6e74 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  nts:.===========
│ │ │ │ -0002d270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002d280: 3d3d 3d0a 0a20 202a 2022 486f 6d57 6974  ===..  * "HomWit
│ │ │ │ -0002d290: 6843 6f6d 706f 6e65 6e74 7328 4d6f 6475  hComponents(Modu
│ │ │ │ -0002d2a0: 6c65 2c4d 6f64 756c 6529 220a 0a46 6f72  le,Module)"..For
│ │ │ │ -0002d2b0: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ -0002d2c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002d2d0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ -0002d2e0: 6e6f 7465 2048 6f6d 5769 7468 436f 6d70  note HomWithComp
│ │ │ │ -0002d2f0: 6f6e 656e 7473 3a20 486f 6d57 6974 6843  onents: HomWithC
│ │ │ │ -0002d300: 6f6d 706f 6e65 6e74 732c 2069 7320 6120  omponents, is a 
│ │ │ │ -0002d310: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ -0002d320: 6374 696f 6e3a 2028 4d61 6361 756c 6179  ction: (Macaulay
│ │ │ │ -0002d330: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ -0002d340: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +0002cf80: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +0002cf90: 6167 653a 200a 2020 2020 2020 2020 4820  age: .        H 
│ │ │ │ +0002cfa0: 3d20 486f 6d28 4d2c 4e29 0a20 202a 2049  = Hom(M,N).  * I
│ │ │ │ +0002cfb0: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ +0002cfc0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0002cfd0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0002cfe0: 4d6f 6475 6c65 2c2c 200a 2020 2020 2020  Module,, .      
│ │ │ │ +0002cff0: 2a20 4e2c 2061 202a 6e6f 7465 206d 6f64  * N, a *note mod
│ │ │ │ +0002d000: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ +0002d010: 6f63 294d 6f64 756c 652c 2c20 0a20 202a  oc)Module,, .  *
│ │ │ │ +0002d020: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ +0002d030: 2a20 482c 2061 202a 6e6f 7465 206d 6f64  * H, a *note mod
│ │ │ │ +0002d040: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ +0002d050: 6f63 294d 6f64 756c 652c 2c20 0a0a 4465  oc)Module,, ..De
│ │ │ │ +0002d060: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0002d070: 3d3d 3d3d 3d0a 0a49 6620 4d20 616e 642f  =====..If M and/
│ │ │ │ +0002d080: 6f72 204e 2061 7265 2064 6972 6563 7420  or N are direct 
│ │ │ │ +0002d090: 7375 6d20 6d6f 6475 6c65 7320 2869 7344  sum modules (isD
│ │ │ │ +0002d0a0: 6972 6563 7453 756d 204d 203d 3d20 7472  irectSum M == tr
│ │ │ │ +0002d0b0: 7565 2920 7468 656e 2048 2069 7320 7468  ue) then H is th
│ │ │ │ +0002d0c0: 650a 6469 7265 6374 2073 756d 206f 6620  e.direct sum of 
│ │ │ │ +0002d0d0: 7468 6520 486f 6d73 2062 6574 7765 656e  the Homs between
│ │ │ │ +0002d0e0: 2074 6865 2063 6f6d 706f 6e65 6e74 732e   the components.
│ │ │ │ +0002d0f0: 2054 6869 7320 5348 4f55 4c44 2062 6520   This SHOULD be 
│ │ │ │ +0002d100: 6275 696c 7420 696e 746f 0a48 6f6d 284d  built into.Hom(M
│ │ │ │ +0002d110: 2c4e 292c 2062 7574 2069 736e 2774 2061  ,N), but isn't a
│ │ │ │ +0002d120: 7320 6f66 204d 322c 2076 2e20 312e 370a  s of M2, v. 1.7.
│ │ │ │ +0002d130: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +0002d140: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7465  ==..  * *note te
│ │ │ │ +0002d150: 6e73 6f72 5769 7468 436f 6d70 6f6e 656e  nsorWithComponen
│ │ │ │ +0002d160: 7473 3a20 7465 6e73 6f72 5769 7468 436f  ts: tensorWithCo
│ │ │ │ +0002d170: 6d70 6f6e 656e 7473 2c20 2d2d 2066 6f72  mponents, -- for
│ │ │ │ +0002d180: 6d73 2074 6865 2074 656e 736f 720a 2020  ms the tensor.  
│ │ │ │ +0002d190: 2020 7072 6f64 7563 742c 2070 7265 7365    product, prese
│ │ │ │ +0002d1a0: 7276 696e 6720 6469 7265 6374 2073 756d  rving direct sum
│ │ │ │ +0002d1b0: 2069 6e66 6f72 6d61 7469 6f6e 0a20 202a   information.  *
│ │ │ │ +0002d1c0: 202a 6e6f 7465 2064 7561 6c57 6974 6843   *note dualWithC
│ │ │ │ +0002d1d0: 6f6d 706f 6e65 6e74 733a 2064 7561 6c57  omponents: dualW
│ │ │ │ +0002d1e0: 6974 6843 6f6d 706f 6e65 6e74 732c 202d  ithComponents, -
│ │ │ │ +0002d1f0: 2d20 6475 616c 206d 6f64 756c 6520 7072  - dual module pr
│ │ │ │ +0002d200: 6573 6572 7669 6e67 0a20 2020 2064 6972  eserving.    dir
│ │ │ │ +0002d210: 6563 7420 7375 6d20 696e 666f 726d 6174  ect sum informat
│ │ │ │ +0002d220: 696f 6e0a 0a57 6179 7320 746f 2075 7365  ion..Ways to use
│ │ │ │ +0002d230: 2048 6f6d 5769 7468 436f 6d70 6f6e 656e   HomWithComponen
│ │ │ │ +0002d240: 7473 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ts:.============
│ │ │ │ +0002d250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002d260: 3d3d 0a0a 2020 2a20 2248 6f6d 5769 7468  ==..  * "HomWith
│ │ │ │ +0002d270: 436f 6d70 6f6e 656e 7473 284d 6f64 756c  Components(Modul
│ │ │ │ +0002d280: 652c 4d6f 6475 6c65 2922 0a0a 466f 7220  e,Module)"..For 
│ │ │ │ +0002d290: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +0002d2a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002d2b0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +0002d2c0: 6f74 6520 486f 6d57 6974 6843 6f6d 706f  ote HomWithCompo
│ │ │ │ +0002d2d0: 6e65 6e74 733a 2048 6f6d 5769 7468 436f  nents: HomWithCo
│ │ │ │ +0002d2e0: 6d70 6f6e 656e 7473 2c20 6973 2061 202a  mponents, is a *
│ │ │ │ +0002d2f0: 6e6f 7465 206d 6574 686f 640a 6675 6e63  note method.func
│ │ │ │ +0002d300: 7469 6f6e 3a20 284d 6163 6175 6c61 7932  tion: (Macaulay2
│ │ │ │ +0002d310: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ +0002d320: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ +0002d330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d390: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -0002d3a0: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -0002d3b0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -0002d3c0: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -0002d3d0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -0002d3e0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ -0002d3f0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ -0002d400: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0002d410: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ -0002d420: 323a 3236 3435 3a30 2e0a 1f0a 4669 6c65  2:2645:0....File
│ │ │ │ -0002d430: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ -0002d440: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0002d450: 732e 696e 666f 2c20 4e6f 6465 3a20 696e  s.info, Node: in
│ │ │ │ -0002d460: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ -0002d470: 7273 2c20 4e65 7874 3a20 6973 4c69 6e65  rs, Next: isLine
│ │ │ │ -0002d480: 6172 2c20 5072 6576 3a20 486f 6d57 6974  ar, Prev: HomWit
│ │ │ │ -0002d490: 6843 6f6d 706f 6e65 6e74 732c 2055 703a  hComponents, Up:
│ │ │ │ -0002d4a0: 2054 6f70 0a0a 696e 6669 6e69 7465 4265   Top..infiniteBe
│ │ │ │ -0002d4b0: 7474 694e 756d 6265 7273 202d 2d20 6265  ttiNumbers -- be
│ │ │ │ -0002d4c0: 7474 6920 6e75 6d62 6572 7320 6f66 2066  tti numbers of f
│ │ │ │ -0002d4d0: 696e 6974 6520 7265 736f 6c75 7469 6f6e  inite resolution
│ │ │ │ -0002d4e0: 2063 6f6d 7075 7465 6420 6672 6f6d 2061   computed from a
│ │ │ │ -0002d4f0: 206d 6174 7269 7820 6661 6374 6f72 697a   matrix factoriz
│ │ │ │ -0002d500: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a  ation.**********
│ │ │ │ +0002d370: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +0002d380: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +0002d390: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +0002d3a0: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +0002d3b0: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ +0002d3c0: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ +0002d3d0: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ +0002d3e0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +0002d3f0: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
│ │ │ │ +0002d400: 3a32 3634 353a 302e 0a1f 0a46 696c 653a  :2645:0....File:
│ │ │ │ +0002d410: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +0002d420: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0002d430: 2e69 6e66 6f2c 204e 6f64 653a 2069 6e66  .info, Node: inf
│ │ │ │ +0002d440: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +0002d450: 732c 204e 6578 743a 2069 734c 696e 6561  s, Next: isLinea
│ │ │ │ +0002d460: 722c 2050 7265 763a 2048 6f6d 5769 7468  r, Prev: HomWith
│ │ │ │ +0002d470: 436f 6d70 6f6e 656e 7473 2c20 5570 3a20  Components, Up: 
│ │ │ │ +0002d480: 546f 700a 0a69 6e66 696e 6974 6542 6574  Top..infiniteBet
│ │ │ │ +0002d490: 7469 4e75 6d62 6572 7320 2d2d 2062 6574  tiNumbers -- bet
│ │ │ │ +0002d4a0: 7469 206e 756d 6265 7273 206f 6620 6669  ti numbers of fi
│ │ │ │ +0002d4b0: 6e69 7465 2072 6573 6f6c 7574 696f 6e20  nite resolution 
│ │ │ │ +0002d4c0: 636f 6d70 7574 6564 2066 726f 6d20 6120  computed from a 
│ │ │ │ +0002d4d0: 6d61 7472 6978 2066 6163 746f 7269 7a61  matrix factoriza
│ │ │ │ +0002d4e0: 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  tion.***********
│ │ │ │ +0002d4f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002d500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002d530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002d560: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -0002d570: 3a20 0a20 2020 2020 2020 204c 203d 2066  : .        L = f
│ │ │ │ -0002d580: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ -0002d590: 7320 284d 462c 6c65 6e29 0a20 202a 2049  s (MF,len).  * I
│ │ │ │ -0002d5a0: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ -0002d5b0: 462c 2061 202a 6e6f 7465 206c 6973 743a  F, a *note list:
│ │ │ │ -0002d5c0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -0002d5d0: 6973 742c 2c20 4c69 7374 206f 6620 4861  ist,, List of Ha
│ │ │ │ -0002d5e0: 7368 5461 626c 6573 2061 7320 636f 6d70  shTables as comp
│ │ │ │ -0002d5f0: 7574 6564 0a20 2020 2020 2020 2062 7920  uted.        by 
│ │ │ │ -0002d600: 226d 6174 7269 7846 6163 746f 7269 7a61  "matrixFactoriza
│ │ │ │ -0002d610: 7469 6f6e 220a 2020 2020 2020 2a20 6c65  tion".      * le
│ │ │ │ -0002d620: 6e2c 2061 6e20 2a6e 6f74 6520 696e 7465  n, an *note inte
│ │ │ │ -0002d630: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ -0002d640: 6f63 295a 5a2c 2c20 6c65 6e67 7468 206f  oc)ZZ,, length o
│ │ │ │ -0002d650: 6620 6265 7474 6920 6e75 6d62 6572 0a20  f betti number. 
│ │ │ │ -0002d660: 2020 2020 2020 2073 6571 7565 6e63 6520         sequence 
│ │ │ │ -0002d670: 746f 2070 726f 6475 6365 0a20 202a 204f  to produce.  * O
│ │ │ │ -0002d680: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -0002d690: 4c2c 2061 202a 6e6f 7465 206c 6973 743a  L, a *note list:
│ │ │ │ -0002d6a0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -0002d6b0: 6973 742c 2c20 4c69 7374 206f 6620 6265  ist,, List of be
│ │ │ │ -0002d6c0: 7474 6920 6e75 6d62 6572 730a 0a44 6573  tti numbers..Des
│ │ │ │ -0002d6d0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0002d6e0: 3d3d 3d3d 0a0a 5573 6573 2074 6865 2072  ====..Uses the r
│ │ │ │ -0002d6f0: 616e 6b73 206f 6620 7468 6520 4220 6d61  anks of the B ma
│ │ │ │ -0002d700: 7472 6963 6573 2069 6e20 6120 6d61 7472  trices in a matr
│ │ │ │ -0002d710: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ -0002d720: 2066 6f72 2061 206d 6f64 756c 6520 4d20   for a module M 
│ │ │ │ -0002d730: 6f76 6572 0a53 2f28 665f 312c 2e2e 2c66  over.S/(f_1,..,f
│ │ │ │ -0002d740: 5f63 2920 746f 2063 6f6d 7075 7465 2074  _c) to compute t
│ │ │ │ -0002d750: 6865 2062 6574 7469 206e 756d 6265 7273  he betti numbers
│ │ │ │ -0002d760: 206f 6620 7468 6520 6d69 6e69 6d61 6c20   of the minimal 
│ │ │ │ -0002d770: 7265 736f 6c75 7469 6f6e 206f 6620 4d20  resolution of M 
│ │ │ │ -0002d780: 6f76 6572 0a52 2c20 7768 6963 6820 6973  over.R, which is
│ │ │ │ -0002d790: 2074 6865 2073 756d 206f 6620 7468 6520   the sum of the 
│ │ │ │ -0002d7a0: 6469 7669 6465 6420 706f 7765 7220 616c  divided power al
│ │ │ │ -0002d7b0: 6765 6272 6173 206f 6e20 632d 6a2b 3120  gebras on c-j+1 
│ │ │ │ -0002d7c0: 7661 7269 6162 6c65 7320 7465 6e73 6f72  variables tensor
│ │ │ │ -0002d7d0: 6564 0a77 6974 6820 4228 6a29 2e0a 0a2b  ed.with B(j)...+
│ │ │ │ +0002d540: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +0002d550: 200a 2020 2020 2020 2020 4c20 3d20 6669   .        L = fi
│ │ │ │ +0002d560: 6e69 7465 4265 7474 694e 756d 6265 7273  niteBettiNumbers
│ │ │ │ +0002d570: 2028 4d46 2c6c 656e 290a 2020 2a20 496e   (MF,len).  * In
│ │ │ │ +0002d580: 7075 7473 3a0a 2020 2020 2020 2a20 4d46  puts:.      * MF
│ │ │ │ +0002d590: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ +0002d5a0: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ +0002d5b0: 7374 2c2c 204c 6973 7420 6f66 2048 6173  st,, List of Has
│ │ │ │ +0002d5c0: 6854 6162 6c65 7320 6173 2063 6f6d 7075  hTables as compu
│ │ │ │ +0002d5d0: 7465 640a 2020 2020 2020 2020 6279 2022  ted.        by "
│ │ │ │ +0002d5e0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +0002d5f0: 696f 6e22 0a20 2020 2020 202a 206c 656e  ion".      * len
│ │ │ │ +0002d600: 2c20 616e 202a 6e6f 7465 2069 6e74 6567  , an *note integ
│ │ │ │ +0002d610: 6572 3a20 284d 6163 6175 6c61 7932 446f  er: (Macaulay2Do
│ │ │ │ +0002d620: 6329 5a5a 2c2c 206c 656e 6774 6820 6f66  c)ZZ,, length of
│ │ │ │ +0002d630: 2062 6574 7469 206e 756d 6265 720a 2020   betti number.  
│ │ │ │ +0002d640: 2020 2020 2020 7365 7175 656e 6365 2074        sequence t
│ │ │ │ +0002d650: 6f20 7072 6f64 7563 650a 2020 2a20 4f75  o produce.  * Ou
│ │ │ │ +0002d660: 7470 7574 733a 0a20 2020 2020 202a 204c  tputs:.      * L
│ │ │ │ +0002d670: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ +0002d680: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ +0002d690: 7374 2c2c 204c 6973 7420 6f66 2062 6574  st,, List of bet
│ │ │ │ +0002d6a0: 7469 206e 756d 6265 7273 0a0a 4465 7363  ti numbers..Desc
│ │ │ │ +0002d6b0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0002d6c0: 3d3d 3d0a 0a55 7365 7320 7468 6520 7261  ===..Uses the ra
│ │ │ │ +0002d6d0: 6e6b 7320 6f66 2074 6865 2042 206d 6174  nks of the B mat
│ │ │ │ +0002d6e0: 7269 6365 7320 696e 2061 206d 6174 7269  rices in a matri
│ │ │ │ +0002d6f0: 7820 6661 6374 6f72 697a 6174 696f 6e20  x factorization 
│ │ │ │ +0002d700: 666f 7220 6120 6d6f 6475 6c65 204d 206f  for a module M o
│ │ │ │ +0002d710: 7665 720a 532f 2866 5f31 2c2e 2e2c 665f  ver.S/(f_1,..,f_
│ │ │ │ +0002d720: 6329 2074 6f20 636f 6d70 7574 6520 7468  c) to compute th
│ │ │ │ +0002d730: 6520 6265 7474 6920 6e75 6d62 6572 7320  e betti numbers 
│ │ │ │ +0002d740: 6f66 2074 6865 206d 696e 696d 616c 2072  of the minimal r
│ │ │ │ +0002d750: 6573 6f6c 7574 696f 6e20 6f66 204d 206f  esolution of M o
│ │ │ │ +0002d760: 7665 720a 522c 2077 6869 6368 2069 7320  ver.R, which is 
│ │ │ │ +0002d770: 7468 6520 7375 6d20 6f66 2074 6865 2064  the sum of the d
│ │ │ │ +0002d780: 6976 6964 6564 2070 6f77 6572 2061 6c67  ivided power alg
│ │ │ │ +0002d790: 6562 7261 7320 6f6e 2063 2d6a 2b31 2076  ebras on c-j+1 v
│ │ │ │ +0002d7a0: 6172 6961 626c 6573 2074 656e 736f 7265  ariables tensore
│ │ │ │ +0002d7b0: 640a 7769 7468 2042 286a 292e 0a0a 2b2d  d.with B(j)...+-
│ │ │ │ +0002d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d810: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ -0002d820: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ -0002d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d840: 2020 2020 2020 207c 0a7c 202d 2d20 7365         |.| -- se
│ │ │ │ -0002d850: 7474 696e 6720 7261 6e64 6f6d 2073 6565  tting random see
│ │ │ │ -0002d860: 6420 746f 2030 2020 2020 2020 2020 2020  d to 0          
│ │ │ │ -0002d870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002d7f0: 2d2b 0a7c 6931 203a 2073 6574 5261 6e64  -+.|i1 : setRand
│ │ │ │ +0002d800: 6f6d 5365 6564 2030 2020 2020 2020 2020  omSeed 0        
│ │ │ │ +0002d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d820: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574        |.| -- set
│ │ │ │ +0002d830: 7469 6e67 2072 616e 646f 6d20 7365 6564  ting random seed
│ │ │ │ +0002d840: 2074 6f20 3020 2020 2020 2020 2020 2020   to 0           
│ │ │ │ +0002d850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d890: 7c0a 7c6f 3120 3d20 3020 2020 2020 2020  |.|o1 = 0       
│ │ │ │  0002d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8b0: 207c 0a7c 6f31 203d 2030 2020 2020 2020   |.|o1 = 0      
│ │ │ │ -0002d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d8e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0002d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -0002d920: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020   : kk = ZZ/101  
│ │ │ │ -0002d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d8c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +0002d900: 3a20 6b6b 203d 205a 5a2f 3130 3120 2020  : kk = ZZ/101   
│ │ │ │ +0002d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002d930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0002d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d950: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d960: 2020 2020 7c0a 7c6f 3220 3d20 6b6b 2020      |.|o2 = kk  
│ │ │ │  0002d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d980: 2020 2020 207c 0a7c 6f32 203d 206b 6b20       |.|o2 = kk 
│ │ │ │ -0002d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d990: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d9f0: 0a7c 6f32 203a 2051 756f 7469 656e 7452  .|o2 : QuotientR
│ │ │ │ -0002da00: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0002da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002da50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0002da60: 2053 203d 206b 6b5b 612c 622c 752c 765d   S = kk[a,b,u,v]
│ │ │ │ +0002d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d9c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002d9d0: 7c6f 3220 3a20 5175 6f74 6965 6e74 5269  |o2 : QuotientRi
│ │ │ │ +0002d9e0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0002d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002da00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002da30: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0002da40: 5320 3d20 6b6b 5b61 2c62 2c75 2c76 5d20  S = kk[a,b,u,v] 
│ │ │ │ +0002da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002da60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0002da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002da80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002da90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002daa0: 2020 7c0a 7c6f 3320 3d20 5320 2020 2020    |.|o3 = S     
│ │ │ │  0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dac0: 2020 207c 0a7c 6f33 203d 2053 2020 2020     |.|o3 = S    
│ │ │ │ -0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dad0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002daf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002db30: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -0002db40: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0002db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0002db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002db90: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066  -------+.|i4 : f
│ │ │ │ -0002dba0: 6620 3d20 6d61 7472 6978 2261 752c 6276  f = matrix"au,bv
│ │ │ │ -0002dbb0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002db10: 3320 3a20 506f 6c79 6e6f 6d69 616c 5269  3 : PolynomialRi
│ │ │ │ +0002db20: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0002db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002db50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002db60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002db70: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6666  ------+.|i4 : ff
│ │ │ │ +0002db80: 203d 206d 6174 7269 7822 6175 2c62 7622   = matrix"au,bv"
│ │ │ │ +0002db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc00: 207c 0a7c 6f34 203d 207c 2061 7520 6276   |.|o4 = | au bv
│ │ │ │ -0002dc10: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0002dbe0: 7c0a 7c6f 3420 3d20 7c20 6175 2062 7620  |.|o4 = | au bv 
│ │ │ │ +0002dbf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0002dc70: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0002dc80: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dca0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053  |.|o4 : Matrix S
│ │ │ │ -0002dcb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dcd0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -0002dce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -0002dd10: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ -0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dd40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0002dc50: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +0002dc60: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0002dc70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002dc80: 0a7c 6f34 203a 204d 6174 7269 7820 5320  .|o4 : Matrix S 
│ │ │ │ +0002dc90: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dcb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0002dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002dce0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +0002dcf0: 2052 203d 2053 2f69 6465 616c 2066 6620   R = S/ideal ff 
│ │ │ │ +0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002dd20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd50: 2020 207c 0a7c 6f35 203d 2052 2020 2020     |.|o5 = R    
│ │ │ │  0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd70: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020      |.|o5 = R   
│ │ │ │ -0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dda0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ddd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002dde0: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269  |o5 : QuotientRi
│ │ │ │ -0002ddf0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002de40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -0002de50: 4d30 203d 2052 5e31 2f69 6465 616c 2261  M0 = R^1/ideal"a
│ │ │ │ -0002de60: 2c62 2220 2020 2020 2020 2020 2020 2020  ,b"             
│ │ │ │ -0002de70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ddb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002ddc0: 6f35 203a 2051 756f 7469 656e 7452 696e  o5 : QuotientRin
│ │ │ │ +0002ddd0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ddf0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002de20: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d  -------+.|i6 : M
│ │ │ │ +0002de30: 3020 3d20 525e 312f 6964 6561 6c22 612c  0 = R^1/ideal"a,
│ │ │ │ +0002de40: 6222 2020 2020 2020 2020 2020 2020 2020  b"              
│ │ │ │ +0002de50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002deb0: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e    |.|o6 = cokern
│ │ │ │ -0002dec0: 656c 207c 2061 2062 207c 2020 2020 2020  el | a b |      
│ │ │ │ +0002de90: 207c 0a7c 6f36 203d 2063 6f6b 6572 6e65   |.|o6 = cokerne
│ │ │ │ +0002dea0: 6c20 7c20 6120 6220 7c20 2020 2020 2020  l | a b |       
│ │ │ │ +0002deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dec0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dee0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0002def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002def0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0002df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002df10: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │  0002df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df30: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0002df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df50: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c   |.|o6 : R-modul
│ │ │ │ -0002df60: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -0002df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0002df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -0002dfc0: 203a 2046 203d 2066 7265 6552 6573 6f6c   : F = freeResol
│ │ │ │ -0002dfd0: 7574 696f 6e28 4d30 2c20 4c65 6e67 7468  ution(M0, Length
│ │ │ │ -0002dfe0: 4c69 6d69 7420 3d3e 3329 2020 2020 2020  Limit =>3)      
│ │ │ │ -0002dff0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0002e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e020: 2020 2020 207c 0a7c 2020 2020 2020 3120       |.|      1 
│ │ │ │ -0002e030: 2020 2020 2032 2020 2020 2020 3320 2020       2      3   
│ │ │ │ -0002e040: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ -0002e050: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0002e060: 3d20 5220 203c 2d2d 2052 2020 3c2d 2d20  = R  <-- R  <-- 
│ │ │ │ -0002e070: 5220 203c 2d2d 2052 2020 2020 2020 2020  R  <-- R        
│ │ │ │ -0002e080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0c0: 2020 2020 7c0a 7c20 2020 2020 3020 2020      |.|     0   
│ │ │ │ -0002e0d0: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -0002e0e0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -0002e0f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002e130: 7c6f 3720 3a20 436f 6d70 6c65 7820 2020  |o7 : Complex   
│ │ │ │ -0002e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e160: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0002e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -0002e1a0: 4d20 3d20 636f 6b65 7220 462e 6464 5f33  M = coker F.dd_3
│ │ │ │ -0002e1b0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -0002e1c0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002df30: 7c0a 7c6f 3620 3a20 522d 6d6f 6475 6c65  |.|o6 : R-module
│ │ │ │ +0002df40: 2c20 7175 6f74 6965 6e74 206f 6620 5220  , quotient of R 
│ │ │ │ +0002df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002df60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +0002dfa0: 3a20 4620 3d20 6672 6565 5265 736f 6c75  : F = freeResolu
│ │ │ │ +0002dfb0: 7469 6f6e 284d 302c 204c 656e 6774 684c  tion(M0, LengthL
│ │ │ │ +0002dfc0: 696d 6974 203d 3e33 2920 2020 2020 207c  imit =>3)      |
│ │ │ │ +0002dfd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e000: 2020 2020 7c0a 7c20 2020 2020 2031 2020      |.|      1  
│ │ │ │ +0002e010: 2020 2020 3220 2020 2020 2033 2020 2020      2      3    
│ │ │ │ +0002e020: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +0002e030: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ +0002e040: 2052 2020 3c2d 2d20 5220 203c 2d2d 2052   R  <-- R  <-- R
│ │ │ │ +0002e050: 2020 3c2d 2d20 5220 2020 2020 2020 2020    <-- R         
│ │ │ │ +0002e060: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002e070: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0002e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e0a0: 2020 207c 0a7c 2020 2020 2030 2020 2020     |.|     0    
│ │ │ │ +0002e0b0: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ +0002e0c0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0002e0d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002e110: 6f37 203a 2043 6f6d 706c 6578 2020 2020  o7 : Complex    
│ │ │ │ +0002e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e140: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0002e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e170: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 204d  -------+.|i8 : M
│ │ │ │ +0002e180: 203d 2063 6f6b 6572 2046 2e64 645f 333b   = coker F.dd_3;
│ │ │ │ +0002e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e1a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0002e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e200: 2d2d 2b0a 7c69 3920 3a20 4d46 203d 206d  --+.|i9 : MF = m
│ │ │ │ -0002e210: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -0002e220: 6f6e 2866 662c 4d29 3b20 2020 2020 2020  on(ff,M);       
│ │ │ │ -0002e230: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0002e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0002e270: 3130 203a 2062 6574 7469 2066 7265 6552  10 : betti freeR
│ │ │ │ -0002e280: 6573 6f6c 7574 696f 6e20 7075 7368 466f  esolution pushFo
│ │ │ │ -0002e290: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ -0002e2a0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ -0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002e2e0: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ -0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e300: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002e310: 3020 3d20 746f 7461 6c3a 2033 2035 2032  0 = total: 3 5 2
│ │ │ │ -0002e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e340: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ -0002e350: 3320 3420 2e20 2020 2020 2020 2020 2020  3 4 .           
│ │ │ │ -0002e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e370: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0002e380: 2020 333a 202e 2031 2032 2020 2020 2020    3: . 1 2      
│ │ │ │ +0002e1e0: 2d2b 0a7c 6939 203a 204d 4620 3d20 6d61  -+.|i9 : MF = ma
│ │ │ │ +0002e1f0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0002e200: 6e28 6666 2c4d 293b 2020 2020 2020 2020  n(ff,M);        
│ │ │ │ +0002e210: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0002e220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0002e250: 3020 3a20 6265 7474 6920 6672 6565 5265  0 : betti freeRe
│ │ │ │ +0002e260: 736f 6c75 7469 6f6e 2070 7573 6846 6f72  solution pushFor
│ │ │ │ +0002e270: 7761 7264 286d 6170 2852 2c53 292c 4d29  ward(map(R,S),M)
│ │ │ │ +0002e280: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0002e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e2b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e2c0: 2020 2020 2030 2031 2032 2020 2020 2020       0 1 2      
│ │ │ │ +0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e2e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ +0002e2f0: 203d 2074 6f74 616c 3a20 3320 3520 3220   = total: 3 5 2 
│ │ │ │ +0002e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e320: 0a7c 2020 2020 2020 2020 2020 323a 2033  .|          2: 3
│ │ │ │ +0002e330: 2034 202e 2020 2020 2020 2020 2020 2020   4 .            
│ │ │ │ +0002e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e350: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0002e360: 2033 3a20 2e20 3120 3220 2020 2020 2020   3: . 1 2       
│ │ │ │ +0002e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e380: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0002e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e3e0: 0a7c 6f31 3020 3a20 4265 7474 6954 616c  .|o10 : BettiTal
│ │ │ │ -0002e3f0: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ -0002e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e410: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0002e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e440: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120  ---------+.|i11 
│ │ │ │ -0002e450: 3a20 6669 6e69 7465 4265 7474 694e 756d  : finiteBettiNum
│ │ │ │ -0002e460: 6265 7273 204d 4620 2020 2020 2020 2020  bers MF         
│ │ │ │ -0002e470: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002e480: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4b0: 2020 207c 0a7c 6f31 3120 3d20 7b33 2c20     |.|o11 = {3, 
│ │ │ │ -0002e4c0: 352c 2032 7d20 2020 2020 2020 2020 2020  5, 2}           
│ │ │ │ +0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e3b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002e3c0: 7c6f 3130 203a 2042 6574 7469 5461 6c6c  |o10 : BettiTall
│ │ │ │ +0002e3d0: 7920 2020 2020 2020 2020 2020 2020 2020  y               
│ │ │ │ +0002e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e3f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0002e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e420: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a  --------+.|i11 :
│ │ │ │ +0002e430: 2066 696e 6974 6542 6574 7469 4e75 6d62   finiteBettiNumb
│ │ │ │ +0002e440: 6572 7320 4d46 2020 2020 2020 2020 2020  ers MF          
│ │ │ │ +0002e450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e490: 2020 7c0a 7c6f 3131 203d 207b 332c 2035    |.|o11 = {3, 5
│ │ │ │ +0002e4a0: 2c20 327d 2020 2020 2020 2020 2020 2020  , 2}            
│ │ │ │ +0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e4e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0002e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0002e520: 6f31 3120 3a20 4c69 7374 2020 2020 2020  o11 : List      
│ │ │ │ -0002e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e550: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0002e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e580: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -0002e590: 696e 6669 6e69 7465 4265 7474 694e 756d  infiniteBettiNum
│ │ │ │ -0002e5a0: 6265 7273 284d 462c 3529 2020 2020 2020  bers(MF,5)      
│ │ │ │ -0002e5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0002e500: 3131 203a 204c 6973 7420 2020 2020 2020  11 : List       
│ │ │ │ +0002e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e530: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002e540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e560: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2069  ------+.|i12 : i
│ │ │ │ +0002e570: 6e66 696e 6974 6542 6574 7469 4e75 6d62  nfiniteBettiNumb
│ │ │ │ +0002e580: 6572 7328 4d46 2c35 2920 2020 2020 2020  ers(MF,5)       
│ │ │ │ +0002e590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0002e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e5f0: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c   |.|o12 = {3, 4,
│ │ │ │ -0002e600: 2035 2c20 362c 2037 2c20 387d 2020 2020   5, 6, 7, 8}    
│ │ │ │ +0002e5d0: 7c0a 7c6f 3132 203d 207b 332c 2034 2c20  |.|o12 = {3, 4, 
│ │ │ │ +0002e5e0: 352c 2036 2c20 372c 2038 7d20 2020 2020  5, 6, 7, 8}     
│ │ │ │ +0002e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e600: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e620: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e650: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0002e660: 3220 3a20 4c69 7374 2020 2020 2020 2020  2 : List        
│ │ │ │ -0002e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e690: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0002e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e6c0: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6265  -----+.|i13 : be
│ │ │ │ -0002e6d0: 7474 6920 6672 6565 5265 736f 6c75 7469  tti freeResoluti
│ │ │ │ -0002e6e0: 6f6e 2028 4d2c 204c 656e 6774 684c 696d  on (M, LengthLim
│ │ │ │ -0002e6f0: 6974 203d 3e20 3529 2020 7c0a 7c20 2020  it => 5)  |.|   
│ │ │ │ -0002e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e730: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -0002e740: 2031 2032 2033 2034 2035 2020 2020 2020   1 2 3 4 5      
│ │ │ │ -0002e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e760: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74      |.|o13 = tot
│ │ │ │ -0002e770: 616c 3a20 3320 3420 3520 3620 3720 3820  al: 3 4 5 6 7 8 
│ │ │ │ -0002e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e790: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0002e7a0: 2020 2020 2020 323a 2033 2034 2035 2036        2: 3 4 5 6
│ │ │ │ -0002e7b0: 2037 2038 2020 2020 2020 2020 2020 2020   7 8            
│ │ │ │ -0002e7c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002e7d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e800: 2020 207c 0a7c 6f31 3320 3a20 4265 7474     |.|o13 : Bett
│ │ │ │ -0002e810: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ -0002e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e830: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -0002e870: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ -0002e880: 3d0a 0a20 202a 202a 6e6f 7465 206d 6174  =..  * *note mat
│ │ │ │ -0002e890: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0002e8a0: 3a20 6d61 7472 6978 4661 6374 6f72 697a  : matrixFactoriz
│ │ │ │ -0002e8b0: 6174 696f 6e2c 202d 2d20 4d61 7073 2069  ation, -- Maps i
│ │ │ │ -0002e8c0: 6e20 6120 6869 6768 6572 0a20 2020 2063  n a higher.    c
│ │ │ │ -0002e8d0: 6f64 696d 656e 7369 6f6e 206d 6174 7269  odimension matri
│ │ │ │ -0002e8e0: 7820 6661 6374 6f72 697a 6174 696f 6e0a  x factorization.
│ │ │ │ -0002e8f0: 2020 2a20 2a6e 6f74 6520 6669 6e69 7465    * *note finite
│ │ │ │ -0002e900: 4265 7474 694e 756d 6265 7273 3a20 6669  BettiNumbers: fi
│ │ │ │ -0002e910: 6e69 7465 4265 7474 694e 756d 6265 7273  niteBettiNumbers
│ │ │ │ -0002e920: 2c20 2d2d 2062 6574 7469 206e 756d 6265  , -- betti numbe
│ │ │ │ -0002e930: 7273 206f 6620 6669 6e69 7465 0a20 2020  rs of finite.   
│ │ │ │ -0002e940: 2072 6573 6f6c 7574 696f 6e20 636f 6d70   resolution comp
│ │ │ │ -0002e950: 7574 6564 2066 726f 6d20 6120 6d61 7472  uted from a matr
│ │ │ │ -0002e960: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ -0002e970: 0a0a 5761 7973 2074 6f20 7573 6520 696e  ..Ways to use in
│ │ │ │ -0002e980: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ -0002e990: 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  rs:.============
│ │ │ │ -0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002e9b0: 3d3d 3d3d 3d0a 0a20 202a 2022 696e 6669  =====..  * "infi
│ │ │ │ -0002e9c0: 6e69 7465 4265 7474 694e 756d 6265 7273  niteBettiNumbers
│ │ │ │ -0002e9d0: 284c 6973 742c 5a5a 2922 0a0a 466f 7220  (List,ZZ)"..For 
│ │ │ │ -0002e9e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -0002e9f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002ea00: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0002ea10: 6f74 6520 696e 6669 6e69 7465 4265 7474  ote infiniteBett
│ │ │ │ -0002ea20: 694e 756d 6265 7273 3a20 696e 6669 6e69  iNumbers: infini
│ │ │ │ -0002ea30: 7465 4265 7474 694e 756d 6265 7273 2c20  teBettiNumbers, 
│ │ │ │ -0002ea40: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -0002ea50: 640a 6675 6e63 7469 6f6e 3a20 284d 6163  d.function: (Mac
│ │ │ │ -0002ea60: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -0002ea70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +0002e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e630: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ +0002e640: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +0002e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0002e670: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002e680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e6a0: 2d2d 2d2d 2b0a 7c69 3133 203a 2062 6574  ----+.|i13 : bet
│ │ │ │ +0002e6b0: 7469 2066 7265 6552 6573 6f6c 7574 696f  ti freeResolutio
│ │ │ │ +0002e6c0: 6e20 284d 2c20 4c65 6e67 7468 4c69 6d69  n (M, LengthLimi
│ │ │ │ +0002e6d0: 7420 3d3e 2035 2920 207c 0a7c 2020 2020  t => 5)  |.|    
│ │ │ │ +0002e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e700: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0002e710: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +0002e720: 3120 3220 3320 3420 3520 2020 2020 2020  1 2 3 4 5       
│ │ │ │ +0002e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e740: 2020 207c 0a7c 6f31 3320 3d20 746f 7461     |.|o13 = tota
│ │ │ │ +0002e750: 6c3a 2033 2034 2035 2036 2037 2038 2020  l: 3 4 5 6 7 8  
│ │ │ │ +0002e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e770: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0002e780: 2020 2020 2032 3a20 3320 3420 3520 3620       2: 3 4 5 6 
│ │ │ │ +0002e790: 3720 3820 2020 2020 2020 2020 2020 2020  7 8             
│ │ │ │ +0002e7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e7e0: 2020 7c0a 7c6f 3133 203a 2042 6574 7469    |.|o13 : Betti
│ │ │ │ +0002e7f0: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +0002e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e810: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0002e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ +0002e850: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ +0002e860: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 7472  ..  * *note matr
│ │ │ │ +0002e870: 6978 4661 6374 6f72 697a 6174 696f 6e3a  ixFactorization:
│ │ │ │ +0002e880: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0002e890: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e  tion, -- Maps in
│ │ │ │ +0002e8a0: 2061 2068 6967 6865 720a 2020 2020 636f   a higher.    co
│ │ │ │ +0002e8b0: 6469 6d65 6e73 696f 6e20 6d61 7472 6978  dimension matrix
│ │ │ │ +0002e8c0: 2066 6163 746f 7269 7a61 7469 6f6e 0a20   factorization. 
│ │ │ │ +0002e8d0: 202a 202a 6e6f 7465 2066 696e 6974 6542   * *note finiteB
│ │ │ │ +0002e8e0: 6574 7469 4e75 6d62 6572 733a 2066 696e  ettiNumbers: fin
│ │ │ │ +0002e8f0: 6974 6542 6574 7469 4e75 6d62 6572 732c  iteBettiNumbers,
│ │ │ │ +0002e900: 202d 2d20 6265 7474 6920 6e75 6d62 6572   -- betti number
│ │ │ │ +0002e910: 7320 6f66 2066 696e 6974 650a 2020 2020  s of finite.    
│ │ │ │ +0002e920: 7265 736f 6c75 7469 6f6e 2063 6f6d 7075  resolution compu
│ │ │ │ +0002e930: 7465 6420 6672 6f6d 2061 206d 6174 7269  ted from a matri
│ │ │ │ +0002e940: 7820 6661 6374 6f72 697a 6174 696f 6e0a  x factorization.
│ │ │ │ +0002e950: 0a57 6179 7320 746f 2075 7365 2069 6e66  .Ways to use inf
│ │ │ │ +0002e960: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ +0002e970: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  s:.=============
│ │ │ │ +0002e980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002e990: 3d3d 3d3d 0a0a 2020 2a20 2269 6e66 696e  ====..  * "infin
│ │ │ │ +0002e9a0: 6974 6542 6574 7469 4e75 6d62 6572 7328  iteBettiNumbers(
│ │ │ │ +0002e9b0: 4c69 7374 2c5a 5a29 220a 0a46 6f72 2074  List,ZZ)"..For t
│ │ │ │ +0002e9c0: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +0002e9d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002e9e0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +0002e9f0: 7465 2069 6e66 696e 6974 6542 6574 7469  te infiniteBetti
│ │ │ │ +0002ea00: 4e75 6d62 6572 733a 2069 6e66 696e 6974  Numbers: infinit
│ │ │ │ +0002ea10: 6542 6574 7469 4e75 6d62 6572 732c 2069  eBettiNumbers, i
│ │ │ │ +0002ea20: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +0002ea30: 0a66 756e 6374 696f 6e3a 2028 4d61 6361  .function: (Maca
│ │ │ │ +0002ea40: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +0002ea50: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d  unction,...-----
│ │ │ │ +0002ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ea70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -0002ead0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ -0002eae0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ -0002eaf0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ -0002eb00: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -0002eb10: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ -0002eb20: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -0002eb30: 6573 2f0a 436f 6d70 6c65 7465 496e 7465  es/.CompleteInte
│ │ │ │ -0002eb40: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -0002eb50: 6f6e 732e 6d32 3a34 3131 333a 302e 0a1f  ons.m2:4113:0...
│ │ │ │ -0002eb60: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ -0002eb70: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -0002eb80: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ -0002eb90: 653a 2069 734c 696e 6561 722c 204e 6578  e: isLinear, Nex
│ │ │ │ -0002eba0: 743a 2069 7351 7561 7369 5265 6775 6c61  t: isQuasiRegula
│ │ │ │ -0002ebb0: 722c 2050 7265 763a 2069 6e66 696e 6974  r, Prev: infinit
│ │ │ │ -0002ebc0: 6542 6574 7469 4e75 6d62 6572 732c 2055  eBettiNumbers, U
│ │ │ │ -0002ebd0: 703a 2054 6f70 0a0a 6973 4c69 6e65 6172  p: Top..isLinear
│ │ │ │ -0002ebe0: 202d 2d20 6368 6563 6b20 7768 6574 6865   -- check whethe
│ │ │ │ -0002ebf0: 7220 6d61 7472 6978 2065 6e74 7269 6573  r matrix entries
│ │ │ │ -0002ec00: 2068 6176 6520 6465 6772 6565 2031 0a2a   have degree 1.*
│ │ │ │ +0002eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ +0002eab0: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ +0002eac0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ +0002ead0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +0002eae0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +0002eaf0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ +0002eb00: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +0002eb10: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ +0002eb20: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +0002eb30: 6e73 2e6d 323a 3431 3133 3a30 2e0a 1f0a  ns.m2:4113:0....
│ │ │ │ +0002eb40: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ +0002eb50: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +0002eb60: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ +0002eb70: 3a20 6973 4c69 6e65 6172 2c20 4e65 7874  : isLinear, Next
│ │ │ │ +0002eb80: 3a20 6973 5175 6173 6952 6567 756c 6172  : isQuasiRegular
│ │ │ │ +0002eb90: 2c20 5072 6576 3a20 696e 6669 6e69 7465  , Prev: infinite
│ │ │ │ +0002eba0: 4265 7474 694e 756d 6265 7273 2c20 5570  BettiNumbers, Up
│ │ │ │ +0002ebb0: 3a20 546f 700a 0a69 734c 696e 6561 7220  : Top..isLinear 
│ │ │ │ +0002ebc0: 2d2d 2063 6865 636b 2077 6865 7468 6572  -- check whether
│ │ │ │ +0002ebd0: 206d 6174 7269 7820 656e 7472 6965 7320   matrix entries 
│ │ │ │ +0002ebe0: 6861 7665 2064 6567 7265 6520 310a 2a2a  have degree 1.**
│ │ │ │ +0002ebf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002ec00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002ec10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ec20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ec30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002ec40: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ -0002ec50: 3a20 0a20 2020 2020 2020 2062 203d 2069  : .        b = i
│ │ │ │ -0002ec60: 734c 696e 6561 7220 4d0a 2020 2a20 496e  sLinear M.  * In
│ │ │ │ -0002ec70: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c  puts:.      * M,
│ │ │ │ -0002ec80: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ -0002ec90: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -0002eca0: 6174 7269 782c 2c20 0a20 202a 204f 7574  atrix,, .  * Out
│ │ │ │ -0002ecb0: 7075 7473 3a0a 2020 2020 2020 2a20 622c  puts:.      * b,
│ │ │ │ -0002ecc0: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e   a *note Boolean
│ │ │ │ -0002ecd0: 2076 616c 7565 3a20 284d 6163 6175 6c61   value: (Macaula
│ │ │ │ -0002ece0: 7932 446f 6329 426f 6f6c 6561 6e2c 2c20  y2Doc)Boolean,, 
│ │ │ │ -0002ecf0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -0002ed00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a4e 6f74 6520  =========..Note 
│ │ │ │ -0002ed10: 7468 6174 2061 206c 696e 6561 7220 6d61  that a linear ma
│ │ │ │ -0002ed20: 7472 6978 2c20 696e 2074 6869 7320 7365  trix, in this se
│ │ │ │ -0002ed30: 6e73 652c 2063 616e 2073 7469 6c6c 2068  nse, can still h
│ │ │ │ -0002ed40: 6176 6520 6469 6666 6572 656e 7420 7461  ave different ta
│ │ │ │ -0002ed50: 7267 6574 0a64 6567 7265 6573 2028 696e  rget.degrees (in
│ │ │ │ -0002ed60: 2077 6869 6368 2063 6173 6520 7468 6520   which case the 
│ │ │ │ -0002ed70: 636f 6b65 726e 656c 2064 6563 6f6d 706f  cokernel decompo
│ │ │ │ -0002ed80: 7365 7320 696e 746f 2061 2064 6972 6563  ses into a direc
│ │ │ │ -0002ed90: 7420 7375 6d20 6279 2067 656e 6572 6174  t sum by generat
│ │ │ │ -0002eda0: 6f72 0a64 6567 7265 652e 290a 0a57 6179  or.degree.)..Way
│ │ │ │ -0002edb0: 7320 746f 2075 7365 2069 734c 696e 6561  s to use isLinea
│ │ │ │ -0002edc0: 723a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r:.=============
│ │ │ │ -0002edd0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269  ========..  * "i
│ │ │ │ -0002ede0: 734c 696e 6561 7228 4d61 7472 6978 2922  sLinear(Matrix)"
│ │ │ │ -0002edf0: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -0002ee00: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -0002ee10: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -0002ee20: 6563 7420 2a6e 6f74 6520 6973 4c69 6e65  ect *note isLine
│ │ │ │ -0002ee30: 6172 3a20 6973 4c69 6e65 6172 2c20 6973  ar: isLinear, is
│ │ │ │ -0002ee40: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ -0002ee50: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ -0002ee60: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ -0002ee70: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ +0002ec20: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ +0002ec30: 200a 2020 2020 2020 2020 6220 3d20 6973   .        b = is
│ │ │ │ +0002ec40: 4c69 6e65 6172 204d 0a20 202a 2049 6e70  Linear M.  * Inp
│ │ │ │ +0002ec50: 7574 733a 0a20 2020 2020 202a 204d 2c20  uts:.      * M, 
│ │ │ │ +0002ec60: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ +0002ec70: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ +0002ec80: 7472 6978 2c2c 200a 2020 2a20 4f75 7470  trix,, .  * Outp
│ │ │ │ +0002ec90: 7574 733a 0a20 2020 2020 202a 2062 2c20  uts:.      * b, 
│ │ │ │ +0002eca0: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20  a *note Boolean 
│ │ │ │ +0002ecb0: 7661 6c75 653a 2028 4d61 6361 756c 6179  value: (Macaulay
│ │ │ │ +0002ecc0: 3244 6f63 2942 6f6f 6c65 616e 2c2c 200a  2Doc)Boolean,, .
│ │ │ │ +0002ecd0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0002ece0: 3d3d 3d3d 3d3d 3d3d 0a0a 4e6f 7465 2074  ========..Note t
│ │ │ │ +0002ecf0: 6861 7420 6120 6c69 6e65 6172 206d 6174  hat a linear mat
│ │ │ │ +0002ed00: 7269 782c 2069 6e20 7468 6973 2073 656e  rix, in this sen
│ │ │ │ +0002ed10: 7365 2c20 6361 6e20 7374 696c 6c20 6861  se, can still ha
│ │ │ │ +0002ed20: 7665 2064 6966 6665 7265 6e74 2074 6172  ve different tar
│ │ │ │ +0002ed30: 6765 740a 6465 6772 6565 7320 2869 6e20  get.degrees (in 
│ │ │ │ +0002ed40: 7768 6963 6820 6361 7365 2074 6865 2063  which case the c
│ │ │ │ +0002ed50: 6f6b 6572 6e65 6c20 6465 636f 6d70 6f73  okernel decompos
│ │ │ │ +0002ed60: 6573 2069 6e74 6f20 6120 6469 7265 6374  es into a direct
│ │ │ │ +0002ed70: 2073 756d 2062 7920 6765 6e65 7261 746f   sum by generato
│ │ │ │ +0002ed80: 720a 6465 6772 6565 2e29 0a0a 5761 7973  r.degree.)..Ways
│ │ │ │ +0002ed90: 2074 6f20 7573 6520 6973 4c69 6e65 6172   to use isLinear
│ │ │ │ +0002eda0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +0002edb0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6973  =======..  * "is
│ │ │ │ +0002edc0: 4c69 6e65 6172 284d 6174 7269 7829 220a  Linear(Matrix)".
│ │ │ │ +0002edd0: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +0002ede0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +0002edf0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +0002ee00: 6374 202a 6e6f 7465 2069 734c 696e 6561  ct *note isLinea
│ │ │ │ +0002ee10: 723a 2069 734c 696e 6561 722c 2069 7320  r: isLinear, is 
│ │ │ │ +0002ee20: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ +0002ee30: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ +0002ee40: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +0002ee50: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +0002ee60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ee90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eec0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ -0002eed0: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ -0002eee0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -0002eef0: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -0002ef00: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -0002ef10: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ -0002ef20: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -0002ef30: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ -0002ef40: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -0002ef50: 732e 6d32 3a33 3436 333a 302e 0a1f 0a46  s.m2:3463:0....F
│ │ │ │ -0002ef60: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -0002ef70: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -0002ef80: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -0002ef90: 2069 7351 7561 7369 5265 6775 6c61 722c   isQuasiRegular,
│ │ │ │ -0002efa0: 204e 6578 743a 2069 7353 7461 626c 7954   Next: isStablyT
│ │ │ │ -0002efb0: 7269 7669 616c 2c20 5072 6576 3a20 6973  rivial, Prev: is
│ │ │ │ -0002efc0: 4c69 6e65 6172 2c20 5570 3a20 546f 700a  Linear, Up: Top.
│ │ │ │ -0002efd0: 0a69 7351 7561 7369 5265 6775 6c61 7220  .isQuasiRegular 
│ │ │ │ -0002efe0: 2d2d 2074 6573 7473 2061 206d 6174 7269  -- tests a matri
│ │ │ │ -0002eff0: 7820 6f72 2073 6571 7565 6e63 6520 6f72  x or sequence or
│ │ │ │ -0002f000: 206c 6973 7420 666f 7220 7175 6173 692d   list for quasi-
│ │ │ │ -0002f010: 7265 6775 6c61 7269 7479 206f 6e20 6120  regularity on a 
│ │ │ │ -0002f020: 6d6f 6475 6c65 0a2a 2a2a 2a2a 2a2a 2a2a  module.*********
│ │ │ │ +0002eea0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +0002eeb0: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +0002eec0: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +0002eed0: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +0002eee0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +0002eef0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ +0002ef00: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +0002ef10: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ +0002ef20: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +0002ef30: 2e6d 323a 3334 3633 3a30 2e0a 1f0a 4669  .m2:3463:0....Fi
│ │ │ │ +0002ef40: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +0002ef50: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +0002ef60: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +0002ef70: 6973 5175 6173 6952 6567 756c 6172 2c20  isQuasiRegular, 
│ │ │ │ +0002ef80: 4e65 7874 3a20 6973 5374 6162 6c79 5472  Next: isStablyTr
│ │ │ │ +0002ef90: 6976 6961 6c2c 2050 7265 763a 2069 734c  ivial, Prev: isL
│ │ │ │ +0002efa0: 696e 6561 722c 2055 703a 2054 6f70 0a0a  inear, Up: Top..
│ │ │ │ +0002efb0: 6973 5175 6173 6952 6567 756c 6172 202d  isQuasiRegular -
│ │ │ │ +0002efc0: 2d20 7465 7374 7320 6120 6d61 7472 6978  - tests a matrix
│ │ │ │ +0002efd0: 206f 7220 7365 7175 656e 6365 206f 7220   or sequence or 
│ │ │ │ +0002efe0: 6c69 7374 2066 6f72 2071 7561 7369 2d72  list for quasi-r
│ │ │ │ +0002eff0: 6567 756c 6172 6974 7920 6f6e 2061 206d  egularity on a m
│ │ │ │ +0002f000: 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a 2a2a  odule.**********
│ │ │ │ +0002f010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002f020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002f030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002f040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002f050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002f060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002f070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ -0002f080: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0002f090: 2020 7420 3d20 6973 5175 6173 6952 6567    t = isQuasiReg
│ │ │ │ -0002f0a0: 756c 6172 2866 662c 4d29 0a20 202a 2049  ular(ff,M).  * I
│ │ │ │ -0002f0b0: 6e70 7574 733a 0a20 2020 2020 202a 2066  nputs:.      * f
│ │ │ │ -0002f0c0: 662c 2061 202a 6e6f 7465 206d 6174 7269  f, a *note matri
│ │ │ │ -0002f0d0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -0002f0e0: 294d 6174 7269 782c 2c20 0a20 2020 2020  )Matrix,, .     
│ │ │ │ -0002f0f0: 202a 2066 662c 2061 202a 6e6f 7465 206c   * ff, a *note l
│ │ │ │ -0002f100: 6973 743a 2028 4d61 6361 756c 6179 3244  ist: (Macaulay2D
│ │ │ │ -0002f110: 6f63 294c 6973 742c 2c20 0a20 2020 2020  oc)List,, .     
│ │ │ │ -0002f120: 202a 2066 662c 2061 202a 6e6f 7465 2073   * ff, a *note s
│ │ │ │ -0002f130: 6571 7565 6e63 653a 2028 4d61 6361 756c  equence: (Macaul
│ │ │ │ -0002f140: 6179 3244 6f63 2953 6571 7565 6e63 652c  ay2Doc)Sequence,
│ │ │ │ -0002f150: 2c20 0a20 2020 2020 202a 204d 2c20 6120  , .      * M, a 
│ │ │ │ -0002f160: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ -0002f170: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ -0002f180: 6c65 2c2c 200a 2020 2a20 4f75 7470 7574  le,, .  * Output
│ │ │ │ -0002f190: 733a 0a20 2020 2020 202a 2074 2c20 6120  s:.      * t, a 
│ │ │ │ -0002f1a0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661  *note Boolean va
│ │ │ │ -0002f1b0: 6c75 653a 2028 4d61 6361 756c 6179 3244  lue: (Macaulay2D
│ │ │ │ -0002f1c0: 6f63 2942 6f6f 6c65 616e 2c2c 200a 0a44  oc)Boolean,, ..D
│ │ │ │ -0002f1d0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -0002f1e0: 3d3d 3d3d 3d3d 0a0a 6666 2069 7320 7175  ======..ff is qu
│ │ │ │ -0002f1f0: 6173 692d 7265 6775 6c61 7220 6966 2074  asi-regular if t
│ │ │ │ -0002f200: 6865 206c 656e 6774 6820 6f66 2066 6620  he length of ff 
│ │ │ │ -0002f210: 6973 203c 3d20 6469 6d20 4d20 616e 6420  is <= dim M and 
│ │ │ │ -0002f220: 7468 6520 616e 6e69 6869 6c61 746f 7220  the annihilator 
│ │ │ │ -0002f230: 6f66 2066 665f 690a 6f6e 204d 2f28 6666  of ff_i.on M/(ff
│ │ │ │ -0002f240: 5f30 2e2e 6666 5f7b 2869 2d31 2929 7d4d  _0..ff_{(i-1))}M
│ │ │ │ -0002f250: 2068 6173 2066 696e 6974 6520 6c65 6e67   has finite leng
│ │ │ │ -0002f260: 7468 2066 6f72 2061 6c6c 2069 3d30 2e2e  th for all i=0..
│ │ │ │ -0002f270: 286c 656e 6774 6820 6666 292d 312e 0a0a  (length ff)-1...
│ │ │ │ -0002f280: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -0002f2b0: 3a20 6b6b 3d5a 5a2f 3130 313b 2020 2020  : kk=ZZ/101;    
│ │ │ │ -0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0002f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f300: 2d2d 2b0a 7c69 3220 3a20 5320 3d20 6b6b  --+.|i2 : S = kk
│ │ │ │ -0002f310: 5b61 2c62 2c63 5d3b 2020 2020 2020 2020  [a,b,c];        
│ │ │ │ -0002f320: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f330: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -0002f360: 3a20 4520 3d20 535e 312f 6964 6561 6c22  : E = S^1/ideal"
│ │ │ │ -0002f370: 6162 222b 2b53 5e31 2f69 6465 616c 2076  ab"++S^1/ideal v
│ │ │ │ -0002f380: 6172 7320 533b 7c0a 2b2d 2d2d 2d2d 2d2d  ars S;|.+-------
│ │ │ │ -0002f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f3b0: 2d2d 2b0a 7c69 3420 3a20 6631 203d 6d61  --+.|i4 : f1 =ma
│ │ │ │ -0002f3c0: 7472 6978 2261 223b 2020 2020 2020 2020  trix"a";        
│ │ │ │ -0002f3d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f3e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f400: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f410: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -0002f420: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -0002f430: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d61        |.|o4 : Ma
│ │ │ │ -0002f440: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
│ │ │ │ -0002f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f460: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0002f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0002f490: 7c69 3520 3a20 6632 203d 6d61 7472 6978  |i5 : f2 =matrix
│ │ │ │ -0002f4a0: 2261 2b62 2c63 223b 2020 2020 2020 2020  "a+b,c";        
│ │ │ │ -0002f4b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f4e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002f4f0: 2020 2020 2020 3120 2020 2020 2032 2020        1      2  
│ │ │ │ -0002f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f510: 2020 7c0a 7c6f 3520 3a20 4d61 7472 6978    |.|o5 : Matrix
│ │ │ │ -0002f520: 2053 2020 3c2d 2d20 5320 2020 2020 2020   S  <-- S       
│ │ │ │ -0002f530: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f540: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0002f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -0002f570: 3a20 6633 203d 206d 6174 7269 7822 612b  : f3 = matrix"a+
│ │ │ │ -0002f580: 6222 3b20 2020 2020 2020 2020 2020 2020  b";             
│ │ │ │ -0002f590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0002f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002f5d0: 2020 3120 2020 2020 2031 2020 2020 2020    1      1      
│ │ │ │ -0002f5e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f5f0: 7c6f 3620 3a20 4d61 7472 6978 2053 2020  |o6 : Matrix S  
│ │ │ │ -0002f600: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ -0002f610: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0002f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f640: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6634  ------+.|i7 : f4
│ │ │ │ -0002f650: 203d 206d 6174 7269 7822 612b 622c 2061   = matrix"a+b, a
│ │ │ │ -0002f660: 322b 6222 3b20 2020 2020 2020 2020 2020  2+b";           
│ │ │ │ -0002f670: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0002f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f6a0: 7c20 2020 2020 2020 2020 2020 2020 3120  |             1 
│ │ │ │ -0002f6b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0002f6c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ -0002f6d0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ -0002f6e0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -0002f6f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0002f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f720: 2d2d 2b0a 7c69 3820 3a20 6973 5175 6173  --+.|i8 : isQuas
│ │ │ │ -0002f730: 6952 6567 756c 6172 2866 312c 4529 2020  iRegular(f1,E)  
│ │ │ │ -0002f740: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f750: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f770: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ -0002f780: 3d20 6661 6c73 6520 2020 2020 2020 2020  = false         
│ │ │ │ -0002f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f7a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0002f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f7d0: 2d2d 2b0a 7c69 3920 3a20 6973 5175 6173  --+.|i9 : isQuas
│ │ │ │ -0002f7e0: 6952 6567 756c 6172 2866 322c 4529 2020  iRegular(f2,E)  
│ │ │ │ -0002f7f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f800: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f820: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ -0002f830: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │ -0002f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f850: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0002f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f880: 2d2d 2b0a 7c69 3130 203a 2069 7351 7561  --+.|i10 : isQua
│ │ │ │ -0002f890: 7369 5265 6775 6c61 7228 6633 2c45 2920  siRegular(f3,E) 
│ │ │ │ -0002f8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f8b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f8d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130            |.|o10
│ │ │ │ -0002f8e0: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ -0002f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f900: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0002f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f930: 2d2d 2b0a 7c69 3131 203a 2069 7351 7561  --+.|i11 : isQua
│ │ │ │ -0002f940: 7369 5265 6775 6c61 7228 6634 2c45 2920  siRegular(f4,E) 
│ │ │ │ -0002f950: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0002f960: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0002f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f980: 2020 2020 2020 2020 2020 7c0a 7c6f 3131            |.|o11
│ │ │ │ -0002f990: 203d 2066 616c 7365 2020 2020 2020 2020   = false        
│ │ │ │ -0002f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f9b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002f9e0: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365  --+..Ways to use
│ │ │ │ -0002f9f0: 2069 7351 7561 7369 5265 6775 6c61 723a   isQuasiRegular:
│ │ │ │ -0002fa00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0002fa10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -0002fa20: 2a20 2269 7351 7561 7369 5265 6775 6c61  * "isQuasiRegula
│ │ │ │ -0002fa30: 7228 4c69 7374 2c4d 6f64 756c 6529 220a  r(List,Module)".
│ │ │ │ -0002fa40: 2020 2a20 2269 7351 7561 7369 5265 6775    * "isQuasiRegu
│ │ │ │ -0002fa50: 6c61 7228 4d61 7472 6978 2c4d 6f64 756c  lar(Matrix,Modul
│ │ │ │ -0002fa60: 6529 220a 2020 2a20 2269 7351 7561 7369  e)".  * "isQuasi
│ │ │ │ -0002fa70: 5265 6775 6c61 7228 5365 7175 656e 6365  Regular(Sequence
│ │ │ │ -0002fa80: 2c4d 6f64 756c 6529 220a 0a46 6f72 2074  ,Module)"..For t
│ │ │ │ -0002fa90: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ -0002faa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0002fab0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -0002fac0: 7465 2069 7351 7561 7369 5265 6775 6c61  te isQuasiRegula
│ │ │ │ -0002fad0: 723a 2069 7351 7561 7369 5265 6775 6c61  r: isQuasiRegula
│ │ │ │ -0002fae0: 722c 2069 7320 6120 2a6e 6f74 6520 6d65  r, is a *note me
│ │ │ │ -0002faf0: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ -0002fb00: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ -0002fb10: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │ +0002f050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +0002f060: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0002f070: 2074 203d 2069 7351 7561 7369 5265 6775   t = isQuasiRegu
│ │ │ │ +0002f080: 6c61 7228 6666 2c4d 290a 2020 2a20 496e  lar(ff,M).  * In
│ │ │ │ +0002f090: 7075 7473 3a0a 2020 2020 2020 2a20 6666  puts:.      * ff
│ │ │ │ +0002f0a0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ +0002f0b0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0002f0c0: 4d61 7472 6978 2c2c 200a 2020 2020 2020  Matrix,, .      
│ │ │ │ +0002f0d0: 2a20 6666 2c20 6120 2a6e 6f74 6520 6c69  * ff, a *note li
│ │ │ │ +0002f0e0: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ +0002f0f0: 6329 4c69 7374 2c2c 200a 2020 2020 2020  c)List,, .      
│ │ │ │ +0002f100: 2a20 6666 2c20 6120 2a6e 6f74 6520 7365  * ff, a *note se
│ │ │ │ +0002f110: 7175 656e 6365 3a20 284d 6163 6175 6c61  quence: (Macaula
│ │ │ │ +0002f120: 7932 446f 6329 5365 7175 656e 6365 2c2c  y2Doc)Sequence,,
│ │ │ │ +0002f130: 200a 2020 2020 2020 2a20 4d2c 2061 202a   .      * M, a *
│ │ │ │ +0002f140: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ +0002f150: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ +0002f160: 652c 2c20 0a20 202a 204f 7574 7075 7473  e,, .  * Outputs
│ │ │ │ +0002f170: 3a0a 2020 2020 2020 2a20 742c 2061 202a  :.      * t, a *
│ │ │ │ +0002f180: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c  note Boolean val
│ │ │ │ +0002f190: 7565 3a20 284d 6163 6175 6c61 7932 446f  ue: (Macaulay2Do
│ │ │ │ +0002f1a0: 6329 426f 6f6c 6561 6e2c 2c20 0a0a 4465  c)Boolean,, ..De
│ │ │ │ +0002f1b0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +0002f1c0: 3d3d 3d3d 3d0a 0a66 6620 6973 2071 7561  =====..ff is qua
│ │ │ │ +0002f1d0: 7369 2d72 6567 756c 6172 2069 6620 7468  si-regular if th
│ │ │ │ +0002f1e0: 6520 6c65 6e67 7468 206f 6620 6666 2069  e length of ff i
│ │ │ │ +0002f1f0: 7320 3c3d 2064 696d 204d 2061 6e64 2074  s <= dim M and t
│ │ │ │ +0002f200: 6865 2061 6e6e 6968 696c 6174 6f72 206f  he annihilator o
│ │ │ │ +0002f210: 6620 6666 5f69 0a6f 6e20 4d2f 2866 665f  f ff_i.on M/(ff_
│ │ │ │ +0002f220: 302e 2e66 665f 7b28 692d 3129 297d 4d20  0..ff_{(i-1))}M 
│ │ │ │ +0002f230: 6861 7320 6669 6e69 7465 206c 656e 6774  has finite lengt
│ │ │ │ +0002f240: 6820 666f 7220 616c 6c20 693d 302e 2e28  h for all i=0..(
│ │ │ │ +0002f250: 6c65 6e67 7468 2066 6629 2d31 2e0a 0a2b  length ff)-1...+
│ │ │ │ +0002f260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f280: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ +0002f290: 206b 6b3d 5a5a 2f31 3031 3b20 2020 2020   kk=ZZ/101;     
│ │ │ │ +0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f2b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f2e0: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b  -+.|i2 : S = kk[
│ │ │ │ +0002f2f0: 612c 622c 635d 3b20 2020 2020 2020 2020  a,b,c];         
│ │ │ │ +0002f300: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002f310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f330: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ +0002f340: 2045 203d 2053 5e31 2f69 6465 616c 2261   E = S^1/ideal"a
│ │ │ │ +0002f350: 6222 2b2b 535e 312f 6964 6561 6c20 7661  b"++S^1/ideal va
│ │ │ │ +0002f360: 7273 2053 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d  rs S;|.+--------
│ │ │ │ +0002f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f390: 2d2b 0a7c 6934 203a 2066 3120 3d6d 6174  -+.|i4 : f1 =mat
│ │ │ │ +0002f3a0: 7269 7822 6122 3b20 2020 2020 2020 2020  rix"a";         
│ │ │ │ +0002f3b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002f3f0: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ +0002f400: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0002f410: 2020 2020 207c 0a7c 6f34 203a 204d 6174       |.|o4 : Mat
│ │ │ │ +0002f420: 7269 7820 5320 203c 2d2d 2053 2020 2020  rix S  <-- S    
│ │ │ │ +0002f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f440: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0002f450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0002f470: 6935 203a 2066 3220 3d6d 6174 7269 7822  i5 : f2 =matrix"
│ │ │ │ +0002f480: 612b 622c 6322 3b20 2020 2020 2020 2020  a+b,c";         
│ │ │ │ +0002f490: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0002f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f4c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f4d0: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +0002f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f4f0: 207c 0a7c 6f35 203a 204d 6174 7269 7820   |.|o5 : Matrix 
│ │ │ │ +0002f500: 5320 203c 2d2d 2053 2020 2020 2020 2020  S  <-- S        
│ │ │ │ +0002f510: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0002f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f540: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0002f550: 2066 3320 3d20 6d61 7472 6978 2261 2b62   f3 = matrix"a+b
│ │ │ │ +0002f560: 223b 2020 2020 2020 2020 2020 2020 2020  ";              
│ │ │ │ +0002f570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f5a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f5b0: 2031 2020 2020 2020 3120 2020 2020 2020   1      1       
│ │ │ │ +0002f5c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002f5d0: 6f36 203a 204d 6174 7269 7820 5320 203c  o6 : Matrix S  <
│ │ │ │ +0002f5e0: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ +0002f5f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0002f600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f620: 2d2d 2d2d 2d2b 0a7c 6937 203a 2066 3420  -----+.|i7 : f4 
│ │ │ │ +0002f630: 3d20 6d61 7472 6978 2261 2b62 2c20 6132  = matrix"a+b, a2
│ │ │ │ +0002f640: 2b62 223b 2020 2020 2020 2020 2020 2020  +b";            
│ │ │ │ +0002f650: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0002f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f670: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002f680: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +0002f690: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002f6a0: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ +0002f6b0: 204d 6174 7269 7820 5320 203c 2d2d 2053   Matrix S  <-- S
│ │ │ │ +0002f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f6d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f700: 2d2b 0a7c 6938 203a 2069 7351 7561 7369  -+.|i8 : isQuasi
│ │ │ │ +0002f710: 5265 6775 6c61 7228 6631 2c45 2920 2020  Regular(f1,E)   
│ │ │ │ +0002f720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f750: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ +0002f760: 2066 616c 7365 2020 2020 2020 2020 2020   false          
│ │ │ │ +0002f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f780: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002f790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f7b0: 2d2b 0a7c 6939 203a 2069 7351 7561 7369  -+.|i9 : isQuasi
│ │ │ │ +0002f7c0: 5265 6775 6c61 7228 6632 2c45 2920 2020  Regular(f2,E)   
│ │ │ │ +0002f7d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f800: 2020 2020 2020 2020 207c 0a7c 6f39 203d           |.|o9 =
│ │ │ │ +0002f810: 2074 7275 6520 2020 2020 2020 2020 2020   true           
│ │ │ │ +0002f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f830: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f860: 2d2b 0a7c 6931 3020 3a20 6973 5175 6173  -+.|i10 : isQuas
│ │ │ │ +0002f870: 6952 6567 756c 6172 2866 332c 4529 2020  iRegular(f3,E)  
│ │ │ │ +0002f880: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f8b0: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ +0002f8c0: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │ +0002f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f8e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002f8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f910: 2d2b 0a7c 6931 3120 3a20 6973 5175 6173  -+.|i11 : isQuas
│ │ │ │ +0002f920: 6952 6567 756c 6172 2866 342c 4529 2020  iRegular(f4,E)  
│ │ │ │ +0002f930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0002f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f960: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ +0002f970: 3d20 6661 6c73 6520 2020 2020 2020 2020  = false         
│ │ │ │ +0002f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f990: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0002f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002f9c0: 2d2b 0a0a 5761 7973 2074 6f20 7573 6520  -+..Ways to use 
│ │ │ │ +0002f9d0: 6973 5175 6173 6952 6567 756c 6172 3a0a  isQuasiRegular:.
│ │ │ │ +0002f9e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0002f9f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +0002fa00: 2022 6973 5175 6173 6952 6567 756c 6172   "isQuasiRegular
│ │ │ │ +0002fa10: 284c 6973 742c 4d6f 6475 6c65 2922 0a20  (List,Module)". 
│ │ │ │ +0002fa20: 202a 2022 6973 5175 6173 6952 6567 756c   * "isQuasiRegul
│ │ │ │ +0002fa30: 6172 284d 6174 7269 782c 4d6f 6475 6c65  ar(Matrix,Module
│ │ │ │ +0002fa40: 2922 0a20 202a 2022 6973 5175 6173 6952  )".  * "isQuasiR
│ │ │ │ +0002fa50: 6567 756c 6172 2853 6571 7565 6e63 652c  egular(Sequence,
│ │ │ │ +0002fa60: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ +0002fa70: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +0002fa80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0002fa90: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +0002faa0: 6520 6973 5175 6173 6952 6567 756c 6172  e isQuasiRegular
│ │ │ │ +0002fab0: 3a20 6973 5175 6173 6952 6567 756c 6172  : isQuasiRegular
│ │ │ │ +0002fac0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +0002fad0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ +0002fae0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +0002faf0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ +0002fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -0002fb70: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -0002fb80: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -0002fb90: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -0002fba0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -0002fbb0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ -0002fbc0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -0002fbd0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ -0002fbe0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -0002fbf0: 7574 696f 6e73 2e6d 323a 3436 3237 3a30  utions.m2:4627:0
│ │ │ │ -0002fc00: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ -0002fc10: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ -0002fc20: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ -0002fc30: 4e6f 6465 3a20 6973 5374 6162 6c79 5472  Node: isStablyTr
│ │ │ │ -0002fc40: 6976 6961 6c2c 204e 6578 743a 206b 6f73  ivial, Next: kos
│ │ │ │ -0002fc50: 7a75 6c45 7874 656e 7369 6f6e 2c20 5072  zulExtension, Pr
│ │ │ │ -0002fc60: 6576 3a20 6973 5175 6173 6952 6567 756c  ev: isQuasiRegul
│ │ │ │ -0002fc70: 6172 2c20 5570 3a20 546f 700a 0a69 7353  ar, Up: Top..isS
│ │ │ │ -0002fc80: 7461 626c 7954 7269 7669 616c 202d 2d20  tablyTrivial -- 
│ │ │ │ -0002fc90: 7265 7475 726e 7320 7472 7565 2069 6620  returns true if 
│ │ │ │ -0002fca0: 7468 6520 6d61 7020 676f 6573 2074 6f20  the map goes to 
│ │ │ │ -0002fcb0: 3020 756e 6465 7220 7374 6162 6c65 486f  0 under stableHo
│ │ │ │ -0002fcc0: 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  m.**************
│ │ │ │ +0002fb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +0002fb50: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +0002fb60: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +0002fb70: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +0002fb80: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +0002fb90: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ +0002fba0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +0002fbb0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ +0002fbc0: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +0002fbd0: 7469 6f6e 732e 6d32 3a34 3632 373a 302e  tions.m2:4627:0.
│ │ │ │ +0002fbe0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ +0002fbf0: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +0002fc00: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ +0002fc10: 6f64 653a 2069 7353 7461 626c 7954 7269  ode: isStablyTri
│ │ │ │ +0002fc20: 7669 616c 2c20 4e65 7874 3a20 6b6f 737a  vial, Next: kosz
│ │ │ │ +0002fc30: 756c 4578 7465 6e73 696f 6e2c 2050 7265  ulExtension, Pre
│ │ │ │ +0002fc40: 763a 2069 7351 7561 7369 5265 6775 6c61  v: isQuasiRegula
│ │ │ │ +0002fc50: 722c 2055 703a 2054 6f70 0a0a 6973 5374  r, Up: Top..isSt
│ │ │ │ +0002fc60: 6162 6c79 5472 6976 6961 6c20 2d2d 2072  ablyTrivial -- r
│ │ │ │ +0002fc70: 6574 7572 6e73 2074 7275 6520 6966 2074  eturns true if t
│ │ │ │ +0002fc80: 6865 206d 6170 2067 6f65 7320 746f 2030  he map goes to 0
│ │ │ │ +0002fc90: 2075 6e64 6572 2073 7461 626c 6548 6f6d   under stableHom
│ │ │ │ +0002fca0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +0002fcb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0002fcc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0002fcd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002fce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002fcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0002fd00: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -0002fd10: 653a 200a 2020 2020 2020 2020 6220 3d20  e: .        b = 
│ │ │ │ -0002fd20: 6973 5374 6162 6c79 5472 6976 6961 6c20  isStablyTrivial 
│ │ │ │ -0002fd30: 660a 2020 2a20 496e 7075 7473 3a0a 2020  f.  * Inputs:.  
│ │ │ │ -0002fd40: 2020 2020 2a20 662c 2061 202a 6e6f 7465      * f, a *note
│ │ │ │ -0002fd50: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ -0002fd60: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ -0002fd70: 6d61 7020 4d20 746f 204e 0a20 202a 204f  map M to N.  * O
│ │ │ │ -0002fd80: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ -0002fd90: 622c 2061 202a 6e6f 7465 2042 6f6f 6c65  b, a *note Boole
│ │ │ │ -0002fda0: 616e 2076 616c 7565 3a20 284d 6163 6175  an value: (Macau
│ │ │ │ -0002fdb0: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c  lay2Doc)Boolean,
│ │ │ │ -0002fdc0: 2c20 7472 7565 2069 6666 2066 2066 6163  , true iff f fac
│ │ │ │ -0002fdd0: 746f 7273 0a20 2020 2020 2020 2074 6872  tors.        thr
│ │ │ │ -0002fde0: 6f75 6768 2061 2070 726f 6a65 6374 6976  ough a projectiv
│ │ │ │ -0002fdf0: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d  e..Description.=
│ │ │ │ -0002fe00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120 706f  ==========..A po
│ │ │ │ -0002fe10: 7373 6962 6c65 206f 6273 7472 7563 7469  ssible obstructi
│ │ │ │ -0002fe20: 6f6e 2074 6f20 7468 6520 636f 6d6d 7574  on to the commut
│ │ │ │ -0002fe30: 6174 6976 6974 7920 6f66 2074 6865 2043  ativity of the C
│ │ │ │ -0002fe40: 4920 6f70 6572 6174 6f72 7320 696e 2063  I operators in c
│ │ │ │ -0002fe50: 6f64 696d 2063 2c0a 6576 656e 2061 7379  odim c,.even asy
│ │ │ │ -0002fe60: 6d70 746f 7469 6361 6c6c 792c 2077 6f75  mptotically, wou
│ │ │ │ -0002fe70: 6c64 2062 6520 7468 6520 6e6f 6e2d 7472  ld be the non-tr
│ │ │ │ -0002fe80: 6976 6961 6c69 7479 206f 6620 7468 6520  iviality of the 
│ │ │ │ -0002fe90: 6d61 7020 4d5f 7b28 6b2b 3429 7d20 2d2d  map M_{(k+4)} --
│ │ │ │ -0002fea0: 3e20 4d5f 6b0a 5c6f 7469 6d65 7320 5c77  > M_k.\otimes \w
│ │ │ │ -0002feb0: 6564 6765 5e32 2853 5e63 2920 696e 2074  edge^2(S^c) in t
│ │ │ │ -0002fec0: 6865 2073 7461 626c 6520 6361 7465 676f  he stable catego
│ │ │ │ -0002fed0: 7279 206f 6620 6d61 7869 6d61 6c20 436f  ry of maximal Co
│ │ │ │ -0002fee0: 6865 6e2d 4d61 6361 756c 6179 206d 6f64  hen-Macaulay mod
│ │ │ │ -0002fef0: 756c 6573 2e0a 0a49 6e20 7468 6520 666f  ules...In the fo
│ │ │ │ -0002ff00: 6c6c 6f77 696e 6720 6578 616d 706c 652c  llowing example,
│ │ │ │ -0002ff10: 2073 7475 6469 6564 2069 6e20 7468 6520   studied in the 
│ │ │ │ -0002ff20: 7061 7065 7220 2254 6f72 2061 7320 6120  paper "Tor as a 
│ │ │ │ -0002ff30: 6d6f 6475 6c65 206f 7665 7220 616e 0a65  module over an.e
│ │ │ │ -0002ff40: 7874 6572 696f 7220 616c 6765 6272 6122  xterior algebra"
│ │ │ │ -0002ff50: 206f 6620 4569 7365 6e62 7564 2c20 5065   of Eisenbud, Pe
│ │ │ │ -0002ff60: 6576 6120 616e 6420 5363 6872 6579 6572  eva and Schreyer
│ │ │ │ -0002ff70: 2c20 7468 6520 6d61 7020 6973 206e 6f6e  , the map is non
│ │ │ │ -0002ff80: 2d74 7269 7669 616c 2e2e 2e62 7574 0a69  -trivial...but.i
│ │ │ │ -0002ff90: 7420 6973 2073 7461 626c 7920 7472 6976  t is stably triv
│ │ │ │ -0002ffa0: 6961 6c2e 2054 6865 2073 616d 6520 676f  ial. The same go
│ │ │ │ -0002ffb0: 6573 2066 6f72 2068 6967 6865 7220 7661  es for higher va
│ │ │ │ -0002ffc0: 6c75 6573 206f 6620 6b20 2877 6869 6368  lues of k (which
│ │ │ │ -0002ffd0: 2074 616b 6520 6c6f 6e67 6572 0a74 6f20   take longer.to 
│ │ │ │ -0002ffe0: 636f 6d70 7574 6529 2e20 286e 6f74 6520  compute). (note 
│ │ │ │ -0002fff0: 7468 6174 2069 6e20 7468 6973 2063 6173  that in this cas
│ │ │ │ -00030000: 652c 2077 6974 6820 6320 3d20 332c 2074  e, with c = 3, t
│ │ │ │ -00030010: 776f 206f 6620 7468 6520 7468 7265 6520  wo of the three 
│ │ │ │ -00030020: 616c 7465 726e 6174 696e 670a 7072 6f64  alternating.prod
│ │ │ │ -00030030: 7563 7473 2061 7265 2061 6374 7561 6c6c  ucts are actuall
│ │ │ │ -00030040: 7920 6571 7561 6c20 746f 2030 2c20 736f  y equal to 0, so
│ │ │ │ -00030050: 2077 6520 7465 7374 206f 6e6c 7920 7468   we test only th
│ │ │ │ -00030060: 6520 7468 6972 642e 290a 0a4e 6f74 6520  e third.)..Note 
│ │ │ │ -00030070: 7468 6174 2054 2069 7320 7765 6c6c 2d64  that T is well-d
│ │ │ │ -00030080: 6566 696e 6564 2075 7020 746f 2068 6f6d  efined up to hom
│ │ │ │ -00030090: 6f74 6f70 793b 2073 6f20 545e 3220 6973  otopy; so T^2 is
│ │ │ │ -000300a0: 2077 656c 6c2d 6465 6669 6e65 6420 6d6f   well-defined mo
│ │ │ │ -000300b0: 6420 6d6d 5e32 2e0a 0a2b 2d2d 2d2d 2d2d  d mm^2...+------
│ │ │ │ +0002fce0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +0002fcf0: 3a20 0a20 2020 2020 2020 2062 203d 2069  : .        b = i
│ │ │ │ +0002fd00: 7353 7461 626c 7954 7269 7669 616c 2066  sStablyTrivial f
│ │ │ │ +0002fd10: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +0002fd20: 2020 202a 2066 2c20 6120 2a6e 6f74 6520     * f, a *note 
│ │ │ │ +0002fd30: 6d61 7472 6978 3a20 284d 6163 6175 6c61  matrix: (Macaula
│ │ │ │ +0002fd40: 7932 446f 6329 4d61 7472 6978 2c2c 206d  y2Doc)Matrix,, m
│ │ │ │ +0002fd50: 6170 204d 2074 6f20 4e0a 2020 2a20 4f75  ap M to N.  * Ou
│ │ │ │ +0002fd60: 7470 7574 733a 0a20 2020 2020 202a 2062  tputs:.      * b
│ │ │ │ +0002fd70: 2c20 6120 2a6e 6f74 6520 426f 6f6c 6561  , a *note Boolea
│ │ │ │ +0002fd80: 6e20 7661 6c75 653a 2028 4d61 6361 756c  n value: (Macaul
│ │ │ │ +0002fd90: 6179 3244 6f63 2942 6f6f 6c65 616e 2c2c  ay2Doc)Boolean,,
│ │ │ │ +0002fda0: 2074 7275 6520 6966 6620 6620 6661 6374   true iff f fact
│ │ │ │ +0002fdb0: 6f72 730a 2020 2020 2020 2020 7468 726f  ors.        thro
│ │ │ │ +0002fdc0: 7567 6820 6120 7072 6f6a 6563 7469 7665  ugh a projective
│ │ │ │ +0002fdd0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +0002fde0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 2070 6f73  =========..A pos
│ │ │ │ +0002fdf0: 7369 626c 6520 6f62 7374 7275 6374 696f  sible obstructio
│ │ │ │ +0002fe00: 6e20 746f 2074 6865 2063 6f6d 6d75 7461  n to the commuta
│ │ │ │ +0002fe10: 7469 7669 7479 206f 6620 7468 6520 4349  tivity of the CI
│ │ │ │ +0002fe20: 206f 7065 7261 746f 7273 2069 6e20 636f   operators in co
│ │ │ │ +0002fe30: 6469 6d20 632c 0a65 7665 6e20 6173 796d  dim c,.even asym
│ │ │ │ +0002fe40: 7074 6f74 6963 616c 6c79 2c20 776f 756c  ptotically, woul
│ │ │ │ +0002fe50: 6420 6265 2074 6865 206e 6f6e 2d74 7269  d be the non-tri
│ │ │ │ +0002fe60: 7669 616c 6974 7920 6f66 2074 6865 206d  viality of the m
│ │ │ │ +0002fe70: 6170 204d 5f7b 286b 2b34 297d 202d 2d3e  ap M_{(k+4)} -->
│ │ │ │ +0002fe80: 204d 5f6b 0a5c 6f74 696d 6573 205c 7765   M_k.\otimes \we
│ │ │ │ +0002fe90: 6467 655e 3228 535e 6329 2069 6e20 7468  dge^2(S^c) in th
│ │ │ │ +0002fea0: 6520 7374 6162 6c65 2063 6174 6567 6f72  e stable categor
│ │ │ │ +0002feb0: 7920 6f66 206d 6178 696d 616c 2043 6f68  y of maximal Coh
│ │ │ │ +0002fec0: 656e 2d4d 6163 6175 6c61 7920 6d6f 6475  en-Macaulay modu
│ │ │ │ +0002fed0: 6c65 732e 0a0a 496e 2074 6865 2066 6f6c  les...In the fol
│ │ │ │ +0002fee0: 6c6f 7769 6e67 2065 7861 6d70 6c65 2c20  lowing example, 
│ │ │ │ +0002fef0: 7374 7564 6965 6420 696e 2074 6865 2070  studied in the p
│ │ │ │ +0002ff00: 6170 6572 2022 546f 7220 6173 2061 206d  aper "Tor as a m
│ │ │ │ +0002ff10: 6f64 756c 6520 6f76 6572 2061 6e0a 6578  odule over an.ex
│ │ │ │ +0002ff20: 7465 7269 6f72 2061 6c67 6562 7261 2220  terior algebra" 
│ │ │ │ +0002ff30: 6f66 2045 6973 656e 6275 642c 2050 6565  of Eisenbud, Pee
│ │ │ │ +0002ff40: 7661 2061 6e64 2053 6368 7265 7965 722c  va and Schreyer,
│ │ │ │ +0002ff50: 2074 6865 206d 6170 2069 7320 6e6f 6e2d   the map is non-
│ │ │ │ +0002ff60: 7472 6976 6961 6c2e 2e2e 6275 740a 6974  trivial...but.it
│ │ │ │ +0002ff70: 2069 7320 7374 6162 6c79 2074 7269 7669   is stably trivi
│ │ │ │ +0002ff80: 616c 2e20 5468 6520 7361 6d65 2067 6f65  al. The same goe
│ │ │ │ +0002ff90: 7320 666f 7220 6869 6768 6572 2076 616c  s for higher val
│ │ │ │ +0002ffa0: 7565 7320 6f66 206b 2028 7768 6963 6820  ues of k (which 
│ │ │ │ +0002ffb0: 7461 6b65 206c 6f6e 6765 720a 746f 2063  take longer.to c
│ │ │ │ +0002ffc0: 6f6d 7075 7465 292e 2028 6e6f 7465 2074  ompute). (note t
│ │ │ │ +0002ffd0: 6861 7420 696e 2074 6869 7320 6361 7365  hat in this case
│ │ │ │ +0002ffe0: 2c20 7769 7468 2063 203d 2033 2c20 7477  , with c = 3, tw
│ │ │ │ +0002fff0: 6f20 6f66 2074 6865 2074 6872 6565 2061  o of the three a
│ │ │ │ +00030000: 6c74 6572 6e61 7469 6e67 0a70 726f 6475  lternating.produ
│ │ │ │ +00030010: 6374 7320 6172 6520 6163 7475 616c 6c79  cts are actually
│ │ │ │ +00030020: 2065 7175 616c 2074 6f20 302c 2073 6f20   equal to 0, so 
│ │ │ │ +00030030: 7765 2074 6573 7420 6f6e 6c79 2074 6865  we test only the
│ │ │ │ +00030040: 2074 6869 7264 2e29 0a0a 4e6f 7465 2074   third.)..Note t
│ │ │ │ +00030050: 6861 7420 5420 6973 2077 656c 6c2d 6465  hat T is well-de
│ │ │ │ +00030060: 6669 6e65 6420 7570 2074 6f20 686f 6d6f  fined up to homo
│ │ │ │ +00030070: 746f 7079 3b20 736f 2054 5e32 2069 7320  topy; so T^2 is 
│ │ │ │ +00030080: 7765 6c6c 2d64 6566 696e 6564 206d 6f64  well-defined mod
│ │ │ │ +00030090: 206d 6d5e 322e 0a0a 2b2d 2d2d 2d2d 2d2d   mm^2...+-------
│ │ │ │ +000300a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000300b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000300c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000300d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000300e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000300f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030100: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b  -------+.|i1 : k
│ │ │ │ -00030110: 6b20 3d20 5a5a 2f31 3031 2020 2020 2020  k = ZZ/101      
│ │ │ │ +000300e0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b  ------+.|i1 : kk
│ │ │ │ +000300f0: 203d 205a 5a2f 3130 3120 2020 2020 2020   = ZZ/101       
│ │ │ │ +00030100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030130: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030150: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030180: 2020 2020 2020 7c0a 7c6f 3120 3d20 6b6b        |.|o1 = kk
│ │ │ │  00030190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000301a0: 2020 2020 2020 207c 0a7c 6f31 203d 206b         |.|o1 = k
│ │ │ │ -000301b0: 6b20 2020 2020 2020 2020 2020 2020 2020  k               
│ │ │ │ +000301a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000301b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000301c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000301d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000301d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000301e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000301f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000301f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030240: 2020 2020 2020 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ -00030250: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00030220: 2020 2020 2020 7c0a 7c6f 3120 3a20 5175        |.|o1 : Qu
│ │ │ │ +00030230: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +00030240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030290: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030270: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00030280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000302a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000302b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000302c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000302d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000302e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053  -------+.|i2 : S
│ │ │ │ -000302f0: 203d 206b 6b5b 612c 622c 635d 2020 2020   = kk[a,b,c]    
│ │ │ │ +000302c0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320  ------+.|i2 : S 
│ │ │ │ +000302d0: 3d20 6b6b 5b61 2c62 2c63 5d20 2020 2020  = kk[a,b,c]     
│ │ │ │ +000302e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000302f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030310: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030330: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030360: 2020 2020 2020 7c0a 7c6f 3220 3d20 5320        |.|o2 = S 
│ │ │ │  00030370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030380: 2020 2020 2020 207c 0a7c 6f32 203d 2053         |.|o2 = S
│ │ │ │ +00030380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000303b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000303b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000303c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000303d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000303d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030420: 2020 2020 2020 207c 0a7c 6f32 203a 2050         |.|o2 : P
│ │ │ │ -00030430: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00030400: 2020 2020 2020 7c0a 7c6f 3220 3a20 506f        |.|o2 : Po
│ │ │ │ +00030410: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +00030420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030470: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030450: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00030460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000304a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000304c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066  -------+.|i3 : f
│ │ │ │ -000304d0: 6620 3d20 6d61 7472 6978 2261 322c 6232  f = matrix"a2,b2
│ │ │ │ -000304e0: 2c63 3222 2020 2020 2020 2020 2020 2020  ,c2"            
│ │ │ │ -000304f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000304a0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 6666  ------+.|i3 : ff
│ │ │ │ +000304b0: 203d 206d 6174 7269 7822 6132 2c62 322c   = matrix"a2,b2,
│ │ │ │ +000304c0: 6332 2220 2020 2020 2020 2020 2020 2020  c2"             
│ │ │ │ +000304d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000304e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000304f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030510: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030560: 2020 2020 2020 207c 0a7c 6f33 203d 207c         |.|o3 = |
│ │ │ │ -00030570: 2061 3220 6232 2063 3220 7c20 2020 2020   a2 b2 c2 |     
│ │ │ │ +00030540: 2020 2020 2020 7c0a 7c6f 3320 3d20 7c20        |.|o3 = | 
│ │ │ │ +00030550: 6132 2062 3220 6332 207c 2020 2020 2020  a2 b2 c2 |      
│ │ │ │ +00030560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000305a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000305b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030600: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00030610: 2020 2020 2020 2031 2020 2020 2020 3320         1      3 
│ │ │ │ +000305e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000305f0: 2020 2020 2020 3120 2020 2020 2033 2020        1      3  
│ │ │ │ +00030600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030650: 2020 2020 2020 207c 0a7c 6f33 203a 204d         |.|o3 : M
│ │ │ │ -00030660: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +00030630: 2020 2020 2020 7c0a 7c6f 3320 3a20 4d61        |.|o3 : Ma
│ │ │ │ +00030640: 7472 6978 2053 2020 3c2d 2d20 5320 2020  trix S  <-- S   
│ │ │ │ +00030650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000306a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030680: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00030690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000306a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000306b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000306c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000306d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000306e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000306f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ -00030700: 203d 2053 2f69 6465 616c 2066 6620 2020   = S/ideal ff   
│ │ │ │ +000306d0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 5220  ------+.|i4 : R 
│ │ │ │ +000306e0: 3d20 532f 6964 6561 6c20 6666 2020 2020  = S/ideal ff    
│ │ │ │ +000306f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030720: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030740: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030770: 2020 2020 2020 7c0a 7c6f 3420 3d20 5220        |.|o4 = R 
│ │ │ │  00030780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030790: 2020 2020 2020 207c 0a7c 6f34 203d 2052         |.|o4 = R
│ │ │ │ +00030790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000307c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000307c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000307d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000307e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000307e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000307f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030830: 2020 2020 2020 207c 0a7c 6f34 203a 2051         |.|o4 : Q
│ │ │ │ -00030840: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00030810: 2020 2020 2020 7c0a 7c6f 3420 3a20 5175        |.|o4 : Qu
│ │ │ │ +00030820: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +00030830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030880: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030860: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00030870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000308a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000308b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000308c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000308d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 204d  -------+.|i5 : M
│ │ │ │ -000308e0: 203d 2052 5e31 2f69 6465 616c 2261 2c62   = R^1/ideal"a,b
│ │ │ │ -000308f0: 6322 2020 2020 2020 2020 2020 2020 2020  c"              
│ │ │ │ -00030900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000308b0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 4d20  ------+.|i5 : M 
│ │ │ │ +000308c0: 3d20 525e 312f 6964 6561 6c22 612c 6263  = R^1/ideal"a,bc
│ │ │ │ +000308d0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ +000308e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000308f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030900: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030920: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030970: 2020 2020 2020 207c 0a7c 6f35 203d 2063         |.|o5 = c
│ │ │ │ -00030980: 6f6b 6572 6e65 6c20 7c20 6120 6263 207c  okernel | a bc |
│ │ │ │ +00030950: 2020 2020 2020 7c0a 7c6f 3520 3d20 636f        |.|o5 = co
│ │ │ │ +00030960: 6b65 726e 656c 207c 2061 2062 6320 7c20  kernel | a bc | 
│ │ │ │ +00030970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000309a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000309a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000309b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000309c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000309c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000309f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000309f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030a10: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │  00030a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a30: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -00030a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a60: 2020 2020 2020 207c 0a7c 6f35 203a 2052         |.|o5 : R
│ │ │ │ -00030a70: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00030a80: 7420 6f66 2052 2020 2020 2020 2020 2020  t of R          
│ │ │ │ -00030a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ab0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030a40: 2020 2020 2020 7c0a 7c6f 3520 3a20 522d        |.|o5 : R-
│ │ │ │ +00030a50: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +00030a60: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +00030a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030a90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00030aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030b00: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 206b  -------+.|i6 : k
│ │ │ │ -00030b10: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +00030ae0: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 6b20  ------+.|i6 : k 
│ │ │ │ +00030af0: 3d20 3120 2020 2020 2020 2020 2020 2020  = 1             
│ │ │ │ +00030b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030b30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030b80: 2020 2020 2020 7c0a 7c6f 3620 3d20 3120        |.|o6 = 1 
│ │ │ │  00030b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ba0: 2020 2020 2020 207c 0a7c 6f36 203d 2031         |.|o6 = 1
│ │ │ │ +00030ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030bf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030bd0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00030be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030c40: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 206d  -------+.|i7 : m
│ │ │ │ -00030c50: 203d 206b 2b35 2020 2020 2020 2020 2020   = k+5          
│ │ │ │ +00030c20: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6d20  ------+.|i7 : m 
│ │ │ │ +00030c30: 3d20 6b2b 3520 2020 2020 2020 2020 2020  = k+5           
│ │ │ │ +00030c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030c70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030c90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030cc0: 2020 2020 2020 7c0a 7c6f 3720 3d20 3620        |.|o7 = 6 
│ │ │ │  00030cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ce0: 2020 2020 2020 207c 0a7c 6f37 203d 2036         |.|o7 = 6
│ │ │ │ +00030ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030d10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00030d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2046  -------+.|i8 : F
│ │ │ │ -00030d90: 203d 2066 7265 6552 6573 6f6c 7574 696f   = freeResolutio
│ │ │ │ -00030da0: 6e28 4d2c 204c 656e 6774 684c 696d 6974  n(M, LengthLimit
│ │ │ │ -00030db0: 203d 3e20 6d29 2020 2020 2020 2020 2020   => m)          
│ │ │ │ +00030d60: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 4620  ------+.|i8 : F 
│ │ │ │ +00030d70: 3d20 6672 6565 5265 736f 6c75 7469 6f6e  = freeResolution
│ │ │ │ +00030d80: 284d 2c20 4c65 6e67 7468 4c69 6d69 7420  (M, LengthLimit 
│ │ │ │ +00030d90: 3d3e 206d 2920 2020 2020 2020 2020 2020  => m)           
│ │ │ │ +00030da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030db0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030dd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00030e30: 3120 2020 2020 2032 2020 2020 2020 3420  1      2      4 
│ │ │ │ -00030e40: 2020 2020 2037 2020 2020 2020 3131 2020       7      11  
│ │ │ │ -00030e50: 2020 2020 3136 2020 2020 2020 3232 2020      16      22  
│ │ │ │ -00030e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e70: 2020 2020 2020 207c 0a7c 6f38 203d 2052         |.|o8 = R
│ │ │ │ -00030e80: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020    <-- R  <-- R  
│ │ │ │ -00030e90: 3c2d 2d20 5220 203c 2d2d 2052 2020 203c  <-- R  <-- R   <
│ │ │ │ -00030ea0: 2d2d 2052 2020 203c 2d2d 2052 2020 2020  -- R   <-- R    
│ │ │ │ +00030e00: 2020 2020 2020 7c0a 7c20 2020 2020 2031        |.|      1
│ │ │ │ +00030e10: 2020 2020 2020 3220 2020 2020 2034 2020        2      4  
│ │ │ │ +00030e20: 2020 2020 3720 2020 2020 2031 3120 2020      7      11   
│ │ │ │ +00030e30: 2020 2031 3620 2020 2020 2032 3220 2020     16      22   
│ │ │ │ +00030e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030e50: 2020 2020 2020 7c0a 7c6f 3820 3d20 5220        |.|o8 = R 
│ │ │ │ +00030e60: 203c 2d2d 2052 2020 3c2d 2d20 5220 203c   <-- R  <-- R  <
│ │ │ │ +00030e70: 2d2d 2052 2020 3c2d 2d20 5220 2020 3c2d  -- R  <-- R   <-
│ │ │ │ +00030e80: 2d20 5220 2020 3c2d 2d20 5220 2020 2020  - R   <-- R     
│ │ │ │ +00030e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030ea0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f10: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ -00030f20: 2020 2020 2020 3120 2020 2020 2032 2020        1      2  
│ │ │ │ -00030f30: 2020 2020 3320 2020 2020 2034 2020 2020      3      4    
│ │ │ │ -00030f40: 2020 2035 2020 2020 2020 2036 2020 2020     5       6    
│ │ │ │ +00030ef0: 2020 2020 2020 7c0a 7c20 2020 2020 3020        |.|     0 
│ │ │ │ +00030f00: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +00030f10: 2020 2033 2020 2020 2020 3420 2020 2020     3      4     
│ │ │ │ +00030f20: 2020 3520 2020 2020 2020 3620 2020 2020    5       6     
│ │ │ │ +00030f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030f40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00030f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00030f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fb0: 2020 2020 2020 207c 0a7c 6f38 203a 2043         |.|o8 : C
│ │ │ │ -00030fc0: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +00030f90: 2020 2020 2020 7c0a 7c6f 3820 3a20 436f        |.|o8 : Co
│ │ │ │ +00030fa0: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │ +00030fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031000: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00030fe0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00030ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031050: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2073  -------+.|i9 : s
│ │ │ │ -00031060: 797a 7967 6965 7320 3d20 6170 706c 7928  yzygies = apply(
│ │ │ │ -00031070: 312e 2e6d 2c20 692d 3e63 6f6b 6572 2046  1..m, i->coker F
│ │ │ │ -00031080: 2e64 645f 6929 3b20 2020 2020 2020 2020  .dd_i);         
│ │ │ │ -00031090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000310a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00031030: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 7379  ------+.|i9 : sy
│ │ │ │ +00031040: 7a79 6769 6573 203d 2061 7070 6c79 2831  zygies = apply(1
│ │ │ │ +00031050: 2e2e 6d2c 2069 2d3e 636f 6b65 7220 462e  ..m, i->coker F.
│ │ │ │ +00031060: 6464 5f69 293b 2020 2020 2020 2020 2020  dd_i);          
│ │ │ │ +00031070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031080: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00031090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000310a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000310b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000310c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000310d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000310e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000310f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ -00031100: 7431 203d 206d 616b 6554 2866 662c 462c  t1 = makeT(ff,F,
│ │ │ │ -00031110: 6b2b 3429 3b20 2020 2020 2020 2020 2020  k+4);           
│ │ │ │ -00031120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000310d0: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2074  ------+.|i10 : t
│ │ │ │ +000310e0: 3120 3d20 6d61 6b65 5428 6666 2c46 2c6b  1 = makeT(ff,F,k
│ │ │ │ +000310f0: 2b34 293b 2020 2020 2020 2020 2020 2020  +4);            
│ │ │ │ +00031100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031120: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00031130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031190: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ -000311a0: 7432 203d 206d 616b 6554 2866 662c 462c  t2 = makeT(ff,F,
│ │ │ │ -000311b0: 6b2b 3229 3b20 2020 2020 2020 2020 2020  k+2);           
│ │ │ │ -000311c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000311d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000311e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00031170: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2074  ------+.|i11 : t
│ │ │ │ +00031180: 3220 3d20 6d61 6b65 5428 6666 2c46 2c6b  2 = makeT(ff,F,k
│ │ │ │ +00031190: 2b32 293b 2020 2020 2020 2020 2020 2020  +2);            
│ │ │ │ +000311a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000311b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000311c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000311d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000311e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000311f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031230: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -00031240: 5432 436f 6d70 6f6e 656e 7473 203d 2066  T2Components = f
│ │ │ │ -00031250: 6c61 7474 656e 2066 6f72 2069 2066 726f  latten for i fro
│ │ │ │ -00031260: 6d20 3020 746f 2031 206c 6973 7428 666f  m 0 to 1 list(fo
│ │ │ │ -00031270: 7220 6a20 6672 6f6d 2069 2b31 2074 6f20  r j from i+1 to 
│ │ │ │ -00031280: 3220 6c69 7374 207c 0a7c 2d2d 2d2d 2d2d  2 list |.|------
│ │ │ │ +00031210: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2054  ------+.|i12 : T
│ │ │ │ +00031220: 3243 6f6d 706f 6e65 6e74 7320 3d20 666c  2Components = fl
│ │ │ │ +00031230: 6174 7465 6e20 666f 7220 6920 6672 6f6d  atten for i from
│ │ │ │ +00031240: 2030 2074 6f20 3120 6c69 7374 2866 6f72   0 to 1 list(for
│ │ │ │ +00031250: 206a 2066 726f 6d20 692b 3120 746f 2032   j from i+1 to 2
│ │ │ │ +00031260: 206c 6973 7420 7c0a 7c2d 2d2d 2d2d 2d2d   list |.|-------
│ │ │ │ +00031270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000312a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000312b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000312c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000312d0: 2d2d 2d2d 2d2d 2d7c 0a7c 6d61 7028 465f  -------|.|map(F_
│ │ │ │ -000312e0: 6b2c 2046 5f28 6b2b 3429 2c20 7432 5f69  k, F_(k+4), t2_i
│ │ │ │ -000312f0: 2a74 315f 6a2d 7432 5f6a 2a74 315f 6929  *t1_j-t2_j*t1_i)
│ │ │ │ -00031300: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -00031310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031320: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000312b0: 2d2d 2d2d 2d2d 7c0a 7c6d 6170 2846 5f6b  ------|.|map(F_k
│ │ │ │ +000312c0: 2c20 465f 286b 2b34 292c 2074 325f 692a  , F_(k+4), t2_i*
│ │ │ │ +000312d0: 7431 5f6a 2d74 325f 6a2a 7431 5f69 2929  t1_j-t2_j*t1_i))
│ │ │ │ +000312e0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +000312f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031300: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00031310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031370: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20  -------+.|i13 : 
│ │ │ │ -00031380: 6720 3d20 6d61 7028 7379 7a79 6769 6573  g = map(syzygies
│ │ │ │ -00031390: 5f6b 2c20 7379 7a79 6769 6573 5f28 6b2b  _k, syzygies_(k+
│ │ │ │ -000313a0: 3429 2c20 5432 436f 6d70 6f6e 656e 7473  4), T2Components
│ │ │ │ -000313b0: 5f32 2920 2020 2020 2020 2020 2020 2020  _2)             
│ │ │ │ -000313c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00031350: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2067  ------+.|i13 : g
│ │ │ │ +00031360: 203d 206d 6170 2873 797a 7967 6965 735f   = map(syzygies_
│ │ │ │ +00031370: 6b2c 2073 797a 7967 6965 735f 286b 2b34  k, syzygies_(k+4
│ │ │ │ +00031380: 292c 2054 3243 6f6d 706f 6e65 6e74 735f  ), T2Components_
│ │ │ │ +00031390: 3229 2020 2020 2020 2020 2020 2020 2020  2)              
│ │ │ │ +000313a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000313b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000313c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000313f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031410: 2020 2020 2020 207c 0a7c 6f31 3320 3d20         |.|o13 = 
│ │ │ │ -00031420: 7b31 7d20 7c20 3020 3020 3020 3020 3020  {1} | 0 0 0 0 0 
│ │ │ │ -00031430: 2d63 2030 2030 2062 2030 2030 2030 2030  -c 0 0 b 0 0 0 0
│ │ │ │ -00031440: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -00031450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031460: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00031470: 7b32 7d20 7c20 3020 3020 3020 3020 3020  {2} | 0 0 0 0 0 
│ │ │ │ -00031480: 3020 2030 2030 2030 2030 2030 2030 2030  0  0 0 0 0 0 0 0
│ │ │ │ -00031490: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ +000313f0: 2020 2020 2020 7c0a 7c6f 3133 203d 207b        |.|o13 = {
│ │ │ │ +00031400: 317d 207c 2030 2030 2030 2030 2030 202d  1} | 0 0 0 0 0 -
│ │ │ │ +00031410: 6320 3020 3020 6220 3020 3020 3020 3020  c 0 0 b 0 0 0 0 
│ │ │ │ +00031420: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ +00031430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031440: 2020 2020 2020 7c0a 7c20 2020 2020 207b        |.|      {
│ │ │ │ +00031450: 327d 207c 2030 2030 2030 2030 2030 2030  2} | 0 0 0 0 0 0
│ │ │ │ +00031460: 2020 3020 3020 3020 3020 3020 3020 3020    0 0 0 0 0 0 0 
│ │ │ │ +00031470: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ +00031480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031490: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000314a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000314b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000314b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000314e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000314f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031500: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ -00031510: 4d61 7472 6978 2020 2020 2020 2020 2020  Matrix          
│ │ │ │ +000314e0: 2020 2020 2020 7c0a 7c6f 3133 203a 204d        |.|o13 : M
│ │ │ │ +000314f0: 6174 7269 7820 2020 2020 2020 2020 2020  atrix           
│ │ │ │ +00031500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031550: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00031530: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00031540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000315a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20  -------+.|i14 : 
│ │ │ │ -000315b0: 6973 5374 6162 6c79 5472 6976 6961 6c20  isStablyTrivial 
│ │ │ │ -000315c0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -000315d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031580: 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a 2069  ------+.|i14 : i
│ │ │ │ +00031590: 7353 7461 626c 7954 7269 7669 616c 2067  sStablyTrivial g
│ │ │ │ +000315a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000315b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000315c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000315d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000315e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000315f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000315f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031640: 2020 2020 2020 207c 0a7c 6f31 3420 3d20         |.|o14 = 
│ │ │ │ -00031650: 7472 7565 2020 2020 2020 2020 2020 2020  true            
│ │ │ │ +00031620: 2020 2020 2020 7c0a 7c6f 3134 203d 2074        |.|o14 = t
│ │ │ │ +00031630: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00031640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031690: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00031670: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00031680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000316a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000316b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000316c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000316d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000316e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -000316f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -00031700: 202a 6e6f 7465 2073 7461 626c 6548 6f6d   *note stableHom
│ │ │ │ -00031710: 3a20 7374 6162 6c65 486f 6d2c 202d 2d20  : stableHom, -- 
│ │ │ │ -00031720: 6d61 7020 6672 6f6d 2048 6f6d 284d 2c4e  map from Hom(M,N
│ │ │ │ -00031730: 2920 746f 2074 6865 2073 7461 626c 6520  ) to the stable 
│ │ │ │ -00031740: 486f 6d20 6d6f 6475 6c65 0a0a 5761 7973  Hom module..Ways
│ │ │ │ -00031750: 2074 6f20 7573 6520 6973 5374 6162 6c79   to use isStably
│ │ │ │ -00031760: 5472 6976 6961 6c3a 0a3d 3d3d 3d3d 3d3d  Trivial:.=======
│ │ │ │ -00031770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00031780: 3d3d 3d3d 3d0a 0a20 202a 2022 6973 5374  =====..  * "isSt
│ │ │ │ -00031790: 6162 6c79 5472 6976 6961 6c28 4d61 7472  ablyTrivial(Matr
│ │ │ │ -000317a0: 6978 2922 0a0a 466f 7220 7468 6520 7072  ix)"..For the pr
│ │ │ │ -000317b0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -000317c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -000317d0: 206f 626a 6563 7420 2a6e 6f74 6520 6973   object *note is
│ │ │ │ -000317e0: 5374 6162 6c79 5472 6976 6961 6c3a 2069  StablyTrivial: i
│ │ │ │ -000317f0: 7353 7461 626c 7954 7269 7669 616c 2c20  sStablyTrivial, 
│ │ │ │ -00031800: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -00031810: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -00031820: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -00031830: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +000316c0: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ +000316d0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +000316e0: 2a6e 6f74 6520 7374 6162 6c65 486f 6d3a  *note stableHom:
│ │ │ │ +000316f0: 2073 7461 626c 6548 6f6d 2c20 2d2d 206d   stableHom, -- m
│ │ │ │ +00031700: 6170 2066 726f 6d20 486f 6d28 4d2c 4e29  ap from Hom(M,N)
│ │ │ │ +00031710: 2074 6f20 7468 6520 7374 6162 6c65 2048   to the stable H
│ │ │ │ +00031720: 6f6d 206d 6f64 756c 650a 0a57 6179 7320  om module..Ways 
│ │ │ │ +00031730: 746f 2075 7365 2069 7353 7461 626c 7954  to use isStablyT
│ │ │ │ +00031740: 7269 7669 616c 3a0a 3d3d 3d3d 3d3d 3d3d  rivial:.========
│ │ │ │ +00031750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00031760: 3d3d 3d3d 0a0a 2020 2a20 2269 7353 7461  ====..  * "isSta
│ │ │ │ +00031770: 626c 7954 7269 7669 616c 284d 6174 7269  blyTrivial(Matri
│ │ │ │ +00031780: 7829 220a 0a46 6f72 2074 6865 2070 726f  x)"..For the pro
│ │ │ │ +00031790: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ +000317a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +000317b0: 6f62 6a65 6374 202a 6e6f 7465 2069 7353  object *note isS
│ │ │ │ +000317c0: 7461 626c 7954 7269 7669 616c 3a20 6973  tablyTrivial: is
│ │ │ │ +000317d0: 5374 6162 6c79 5472 6976 6961 6c2c 2069  StablyTrivial, i
│ │ │ │ +000317e0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +000317f0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ +00031800: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +00031810: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d  unction,...-----
│ │ │ │ +00031820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -00031890: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ -000318a0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ -000318b0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ -000318c0: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -000318d0: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ -000318e0: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -000318f0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465  es/.CompleteInte
│ │ │ │ -00031900: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -00031910: 6f6e 732e 6d32 3a34 3639 393a 302e 0a1f  ons.m2:4699:0...
│ │ │ │ -00031920: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ -00031930: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -00031940: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ -00031950: 653a 206b 6f73 7a75 6c45 7874 656e 7369  e: koszulExtensi
│ │ │ │ -00031960: 6f6e 2c20 4e65 7874 3a20 4c61 7965 7265  on, Next: Layere
│ │ │ │ -00031970: 642c 2050 7265 763a 2069 7353 7461 626c  d, Prev: isStabl
│ │ │ │ -00031980: 7954 7269 7669 616c 2c20 5570 3a20 546f  yTrivial, Up: To
│ │ │ │ -00031990: 700a 0a6b 6f73 7a75 6c45 7874 656e 7369  p..koszulExtensi
│ │ │ │ -000319a0: 6f6e 202d 2d20 6372 6561 7465 7320 7468  on -- creates th
│ │ │ │ -000319b0: 6520 4b6f 737a 756c 2065 7874 656e 7369  e Koszul extensi
│ │ │ │ -000319c0: 6f6e 2063 6f6d 706c 6578 206f 6620 6120  on complex of a 
│ │ │ │ -000319d0: 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  map.************
│ │ │ │ +00031860: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ +00031870: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ +00031880: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ +00031890: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +000318a0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +000318b0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ +000318c0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +000318d0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ +000318e0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +000318f0: 6e73 2e6d 323a 3436 3939 3a30 2e0a 1f0a  ns.m2:4699:0....
│ │ │ │ +00031900: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ +00031910: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +00031920: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ +00031930: 3a20 6b6f 737a 756c 4578 7465 6e73 696f  : koszulExtensio
│ │ │ │ +00031940: 6e2c 204e 6578 743a 204c 6179 6572 6564  n, Next: Layered
│ │ │ │ +00031950: 2c20 5072 6576 3a20 6973 5374 6162 6c79  , Prev: isStably
│ │ │ │ +00031960: 5472 6976 6961 6c2c 2055 703a 2054 6f70  Trivial, Up: Top
│ │ │ │ +00031970: 0a0a 6b6f 737a 756c 4578 7465 6e73 696f  ..koszulExtensio
│ │ │ │ +00031980: 6e20 2d2d 2063 7265 6174 6573 2074 6865  n -- creates the
│ │ │ │ +00031990: 204b 6f73 7a75 6c20 6578 7465 6e73 696f   Koszul extensio
│ │ │ │ +000319a0: 6e20 636f 6d70 6c65 7820 6f66 2061 206d  n complex of a m
│ │ │ │ +000319b0: 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ap.*************
│ │ │ │ +000319c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000319d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000319e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000319f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00031a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00031a10: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -00031a20: 200a 2020 2020 2020 2020 4d4d 203d 206b   .        MM = k
│ │ │ │ -00031a30: 6f73 7a75 6c45 7874 656e 7369 6f6e 2846  oszulExtension(F
│ │ │ │ -00031a40: 462c 4242 2c70 7369 312c 6666 290a 2020  F,BB,psi1,ff).  
│ │ │ │ -00031a50: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00031a60: 2a20 4646 2c20 6120 2a6e 6f74 6520 636f  * FF, a *note co
│ │ │ │ -00031a70: 6d70 6c65 783a 2028 436f 6d70 6c65 7865  mplex: (Complexe
│ │ │ │ -00031a80: 7329 436f 6d70 6c65 782c 2c20 7265 736f  s)Complex,, reso
│ │ │ │ -00031a90: 6c75 7469 6f6e 206f 7665 7220 530a 2020  lution over S.  
│ │ │ │ -00031aa0: 2020 2020 2a20 4242 2c20 6120 2a6e 6f74      * BB, a *not
│ │ │ │ -00031ab0: 6520 636f 6d70 6c65 783a 2028 436f 6d70  e complex: (Comp
│ │ │ │ -00031ac0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20  lexes)Complex,, 
│ │ │ │ -00031ad0: 7477 6f2d 7465 726d 2063 6f6d 706c 6578  two-term complex
│ │ │ │ -00031ae0: 2042 425f 312d 2d3e 4242 5f30 0a20 2020   BB_1-->BB_0.   
│ │ │ │ -00031af0: 2020 202a 2070 7369 312c 2061 202a 6e6f     * psi1, a *no
│ │ │ │ -00031b00: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ -00031b10: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ -00031b20: 2c20 6672 6f6d 2042 425f 3120 746f 2046  , from BB_1 to F
│ │ │ │ -00031b30: 465f 300a 2020 2020 2020 2a20 6666 2c20  F_0.      * ff, 
│ │ │ │ -00031b40: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ -00031b50: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ -00031b60: 7472 6978 2c2c 2072 6567 756c 6172 2073  trix,, regular s
│ │ │ │ -00031b70: 6571 7565 6e63 650a 2020 2020 2020 2020  equence.        
│ │ │ │ -00031b80: 616e 6e69 6869 6c61 7469 6e67 2074 6865  annihilating the
│ │ │ │ -00031b90: 206d 6f64 756c 6520 7265 736f 6c76 6564   module resolved
│ │ │ │ -00031ba0: 2062 7920 4646 0a20 202a 204f 7574 7075   by FF.  * Outpu
│ │ │ │ -00031bb0: 7473 3a0a 2020 2020 2020 2a20 4d4d 2c20  ts:.      * MM, 
│ │ │ │ -00031bc0: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a  a *note complex:
│ │ │ │ -00031bd0: 2028 436f 6d70 6c65 7865 7329 436f 6d70   (Complexes)Comp
│ │ │ │ -00031be0: 6c65 782c 2c20 7468 6520 6d61 7070 696e  lex,, the mappin
│ │ │ │ -00031bf0: 6720 636f 6e65 206f 6620 7468 650a 2020  g cone of the.  
│ │ │ │ -00031c00: 2020 2020 2020 696e 6475 6365 6420 6d61        induced ma
│ │ │ │ -00031c10: 7020 425b 2d31 5d5c 6f74 696d 6573 204b  p B[-1]\otimes K
│ │ │ │ -00031c20: 4b28 6666 2920 746f 2057 2065 7874 656e  K(ff) to W exten
│ │ │ │ -00031c30: 6469 6e67 2070 7369 0a0a 4465 7363 7269  ding psi..Descri
│ │ │ │ -00031c40: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -00031c50: 3d0a 0a49 6d70 6c65 6d65 6e74 7320 7468  =..Implements th
│ │ │ │ -00031c60: 6520 636f 6e73 7472 7563 7469 6f6e 2069  e construction i
│ │ │ │ -00031c70: 6e20 7468 6520 7061 7065 7220 224d 6174  n the paper "Mat
│ │ │ │ -00031c80: 7269 7820 4661 6374 6f72 697a 6174 696f  rix Factorizatio
│ │ │ │ -00031c90: 6e73 2069 6e20 4869 6768 6572 0a43 6f64  ns in Higher.Cod
│ │ │ │ -00031ca0: 696d 656e 7369 6f6e 2220 6279 2045 6973  imension" by Eis
│ │ │ │ -00031cb0: 656e 6275 6420 616e 6420 5065 6576 612e  enbud and Peeva.
│ │ │ │ -00031cc0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00031cd0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ -00031ce0: 616b 6546 696e 6974 6552 6573 6f6c 7574  akeFiniteResolut
│ │ │ │ -00031cf0: 696f 6e3a 206d 616b 6546 696e 6974 6552  ion: makeFiniteR
│ │ │ │ -00031d00: 6573 6f6c 7574 696f 6e2c 202d 2d20 6669  esolution, -- fi
│ │ │ │ -00031d10: 6e69 7465 2072 6573 6f6c 7574 696f 6e20  nite resolution 
│ │ │ │ -00031d20: 6f66 2061 0a20 2020 206d 6174 7269 7820  of a.    matrix 
│ │ │ │ -00031d30: 6661 6374 6f72 697a 6174 696f 6e20 6d6f  factorization mo
│ │ │ │ -00031d40: 6475 6c65 204d 0a0a 5761 7973 2074 6f20  dule M..Ways to 
│ │ │ │ -00031d50: 7573 6520 6b6f 737a 756c 4578 7465 6e73  use koszulExtens
│ │ │ │ -00031d60: 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  ion:.===========
│ │ │ │ -00031d70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00031d80: 3d0a 0a20 202a 2022 6b6f 737a 756c 4578  =..  * "koszulEx
│ │ │ │ -00031d90: 7465 6e73 696f 6e28 436f 6d70 6c65 782c  tension(Complex,
│ │ │ │ -00031da0: 436f 6d70 6c65 782c 4d61 7472 6978 2c4d  Complex,Matrix,M
│ │ │ │ -00031db0: 6174 7269 7829 220a 0a46 6f72 2074 6865  atrix)"..For the
│ │ │ │ -00031dc0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00031dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00031de0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00031df0: 206b 6f73 7a75 6c45 7874 656e 7369 6f6e   koszulExtension
│ │ │ │ -00031e00: 3a20 6b6f 737a 756c 4578 7465 6e73 696f  : koszulExtensio
│ │ │ │ -00031e10: 6e2c 2069 7320 6120 2a6e 6f74 6520 6d65  n, is a *note me
│ │ │ │ -00031e20: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ -00031e30: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ -00031e40: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │ +000319f0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +00031a00: 0a20 2020 2020 2020 204d 4d20 3d20 6b6f  .        MM = ko
│ │ │ │ +00031a10: 737a 756c 4578 7465 6e73 696f 6e28 4646  szulExtension(FF
│ │ │ │ +00031a20: 2c42 422c 7073 6931 2c66 6629 0a20 202a  ,BB,psi1,ff).  *
│ │ │ │ +00031a30: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00031a40: 2046 462c 2061 202a 6e6f 7465 2063 6f6d   FF, a *note com
│ │ │ │ +00031a50: 706c 6578 3a20 2843 6f6d 706c 6578 6573  plex: (Complexes
│ │ │ │ +00031a60: 2943 6f6d 706c 6578 2c2c 2072 6573 6f6c  )Complex,, resol
│ │ │ │ +00031a70: 7574 696f 6e20 6f76 6572 2053 0a20 2020  ution over S.   
│ │ │ │ +00031a80: 2020 202a 2042 422c 2061 202a 6e6f 7465     * BB, a *note
│ │ │ │ +00031a90: 2063 6f6d 706c 6578 3a20 2843 6f6d 706c   complex: (Compl
│ │ │ │ +00031aa0: 6578 6573 2943 6f6d 706c 6578 2c2c 2074  exes)Complex,, t
│ │ │ │ +00031ab0: 776f 2d74 6572 6d20 636f 6d70 6c65 7820  wo-term complex 
│ │ │ │ +00031ac0: 4242 5f31 2d2d 3e42 425f 300a 2020 2020  BB_1-->BB_0.    
│ │ │ │ +00031ad0: 2020 2a20 7073 6931 2c20 6120 2a6e 6f74    * psi1, a *not
│ │ │ │ +00031ae0: 6520 6d61 7472 6978 3a20 284d 6163 6175  e matrix: (Macau
│ │ │ │ +00031af0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c  lay2Doc)Matrix,,
│ │ │ │ +00031b00: 2066 726f 6d20 4242 5f31 2074 6f20 4646   from BB_1 to FF
│ │ │ │ +00031b10: 5f30 0a20 2020 2020 202a 2066 662c 2061  _0.      * ff, a
│ │ │ │ +00031b20: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ +00031b30: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ +00031b40: 7269 782c 2c20 7265 6775 6c61 7220 7365  rix,, regular se
│ │ │ │ +00031b50: 7175 656e 6365 0a20 2020 2020 2020 2061  quence.        a
│ │ │ │ +00031b60: 6e6e 6968 696c 6174 696e 6720 7468 6520  nnihilating the 
│ │ │ │ +00031b70: 6d6f 6475 6c65 2072 6573 6f6c 7665 6420  module resolved 
│ │ │ │ +00031b80: 6279 2046 460a 2020 2a20 4f75 7470 7574  by FF.  * Output
│ │ │ │ +00031b90: 733a 0a20 2020 2020 202a 204d 4d2c 2061  s:.      * MM, a
│ │ │ │ +00031ba0: 202a 6e6f 7465 2063 6f6d 706c 6578 3a20   *note complex: 
│ │ │ │ +00031bb0: 2843 6f6d 706c 6578 6573 2943 6f6d 706c  (Complexes)Compl
│ │ │ │ +00031bc0: 6578 2c2c 2074 6865 206d 6170 7069 6e67  ex,, the mapping
│ │ │ │ +00031bd0: 2063 6f6e 6520 6f66 2074 6865 0a20 2020   cone of the.   
│ │ │ │ +00031be0: 2020 2020 2069 6e64 7563 6564 206d 6170       induced map
│ │ │ │ +00031bf0: 2042 5b2d 315d 5c6f 7469 6d65 7320 4b4b   B[-1]\otimes KK
│ │ │ │ +00031c00: 2866 6629 2074 6f20 5720 6578 7465 6e64  (ff) to W extend
│ │ │ │ +00031c10: 696e 6720 7073 690a 0a44 6573 6372 6970  ing psi..Descrip
│ │ │ │ +00031c20: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +00031c30: 0a0a 496d 706c 656d 656e 7473 2074 6865  ..Implements the
│ │ │ │ +00031c40: 2063 6f6e 7374 7275 6374 696f 6e20 696e   construction in
│ │ │ │ +00031c50: 2074 6865 2070 6170 6572 2022 4d61 7472   the paper "Matr
│ │ │ │ +00031c60: 6978 2046 6163 746f 7269 7a61 7469 6f6e  ix Factorization
│ │ │ │ +00031c70: 7320 696e 2048 6967 6865 720a 436f 6469  s in Higher.Codi
│ │ │ │ +00031c80: 6d65 6e73 696f 6e22 2062 7920 4569 7365  mension" by Eise
│ │ │ │ +00031c90: 6e62 7564 2061 6e64 2050 6565 7661 2e0a  nbud and Peeva..
│ │ │ │ +00031ca0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00031cb0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6d61  ==..  * *note ma
│ │ │ │ +00031cc0: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ +00031cd0: 6f6e 3a20 6d61 6b65 4669 6e69 7465 5265  on: makeFiniteRe
│ │ │ │ +00031ce0: 736f 6c75 7469 6f6e 2c20 2d2d 2066 696e  solution, -- fin
│ │ │ │ +00031cf0: 6974 6520 7265 736f 6c75 7469 6f6e 206f  ite resolution o
│ │ │ │ +00031d00: 6620 610a 2020 2020 6d61 7472 6978 2066  f a.    matrix f
│ │ │ │ +00031d10: 6163 746f 7269 7a61 7469 6f6e 206d 6f64  actorization mod
│ │ │ │ +00031d20: 756c 6520 4d0a 0a57 6179 7320 746f 2075  ule M..Ways to u
│ │ │ │ +00031d30: 7365 206b 6f73 7a75 6c45 7874 656e 7369  se koszulExtensi
│ │ │ │ +00031d40: 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  on:.============
│ │ │ │ +00031d50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00031d60: 0a0a 2020 2a20 226b 6f73 7a75 6c45 7874  ..  * "koszulExt
│ │ │ │ +00031d70: 656e 7369 6f6e 2843 6f6d 706c 6578 2c43  ension(Complex,C
│ │ │ │ +00031d80: 6f6d 706c 6578 2c4d 6174 7269 782c 4d61  omplex,Matrix,Ma
│ │ │ │ +00031d90: 7472 6978 2922 0a0a 466f 7220 7468 6520  trix)"..For the 
│ │ │ │ +00031da0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00031db0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00031dc0: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +00031dd0: 6b6f 737a 756c 4578 7465 6e73 696f 6e3a  koszulExtension:
│ │ │ │ +00031de0: 206b 6f73 7a75 6c45 7874 656e 7369 6f6e   koszulExtension
│ │ │ │ +00031df0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ +00031e00: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ +00031e10: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ +00031e20: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ +00031e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00031e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ -00031ea0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ -00031eb0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ -00031ec0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ -00031ed0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ -00031ee0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ -00031ef0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ -00031f00: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ -00031f10: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -00031f20: 7574 696f 6e73 2e6d 323a 3330 3038 3a30  utions.m2:3008:0
│ │ │ │ -00031f30: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ -00031f40: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ -00031f50: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ -00031f60: 4e6f 6465 3a20 4c61 7965 7265 642c 204e  Node: Layered, N
│ │ │ │ -00031f70: 6578 743a 206c 6179 6572 6564 5265 736f  ext: layeredReso
│ │ │ │ -00031f80: 6c75 7469 6f6e 2c20 5072 6576 3a20 6b6f  lution, Prev: ko
│ │ │ │ -00031f90: 737a 756c 4578 7465 6e73 696f 6e2c 2055  szulExtension, U
│ │ │ │ -00031fa0: 703a 2054 6f70 0a0a 4c61 7965 7265 6420  p: Top..Layered 
│ │ │ │ -00031fb0: 2d2d 204f 7074 696f 6e20 666f 7220 6d61  -- Option for ma
│ │ │ │ -00031fc0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -00031fd0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │ -00031fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00031ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -00032000: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00032010: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -00032020: 7469 6f6e 2866 662c 6d2c 4c61 7965 7265  tion(ff,m,Layere
│ │ │ │ -00032030: 6420 3d3e 2074 7275 6529 0a20 202a 2049  d => true).  * I
│ │ │ │ -00032040: 6e70 7574 733a 0a20 2020 2020 202a 2043  nputs:.      * C
│ │ │ │ -00032050: 6865 636b 2c20 6120 2a6e 6f74 6520 426f  heck, a *note Bo
│ │ │ │ -00032060: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61  olean value: (Ma
│ │ │ │ -00032070: 6361 756c 6179 3244 6f63 2942 6f6f 6c65  caulay2Doc)Boole
│ │ │ │ -00032080: 616e 2c2c 200a 0a44 6573 6372 6970 7469  an,, ..Descripti
│ │ │ │ -00032090: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -000320a0: 4d61 6b65 7320 6d61 7472 6978 4661 6374  Makes matrixFact
│ │ │ │ -000320b0: 6f72 697a 6174 696f 6e20 7573 6520 7468  orization use th
│ │ │ │ -000320c0: 6520 226c 6179 6572 6564 2220 616c 676f  e "layered" algo
│ │ │ │ -000320d0: 7269 7468 6d2c 2077 6869 6368 2077 6f72  rithm, which wor
│ │ │ │ -000320e0: 6b73 2066 6f72 2061 6e79 204d 434d 0a6d  ks for any MCM.m
│ │ │ │ -000320f0: 6f64 756c 652c 2062 7574 2072 6574 7572  odule, but retur
│ │ │ │ -00032100: 6e73 2073 6f6d 6574 6869 6e67 206e 6f6e  ns something non
│ │ │ │ -00032110: 2d6d 696e 696d 616c 2069 6620 7468 6520  -minimal if the 
│ │ │ │ -00032120: 6d6f 6475 6c65 2069 7320 6e6f 7420 6120  module is not a 
│ │ │ │ -00032130: 2268 6967 6820 7379 7a79 6779 220a 696e  "high syzygy".in
│ │ │ │ -00032140: 2061 2073 7569 7461 626c 6520 7365 6e73   a suitable sens
│ │ │ │ -00032150: 652e 2044 6566 6175 6c74 2069 7320 2274  e. Default is "t
│ │ │ │ -00032160: 7275 6522 2e20 4e6f 7465 2074 6861 7420  rue". Note that 
│ │ │ │ -00032170: 7768 656e 2074 6865 206d 6f64 756c 6520  when the module 
│ │ │ │ -00032180: 6973 2061 2068 6967 680a 7379 7a79 6779  is a high.syzygy
│ │ │ │ -00032190: 2c20 4c61 7965 7265 643d 3e20 6661 6c73  , Layered=> fals
│ │ │ │ -000321a0: 6520 6973 206d 7563 6820 6661 7374 6572  e is much faster
│ │ │ │ -000321b0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  ...See also.====
│ │ │ │ -000321c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -000321d0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ -000321e0: 696f 6e3a 206d 6174 7269 7846 6163 746f  ion: matrixFacto
│ │ │ │ -000321f0: 7269 7a61 7469 6f6e 2c20 2d2d 204d 6170  rization, -- Map
│ │ │ │ -00032200: 7320 696e 2061 2068 6967 6865 720a 2020  s in a higher.  
│ │ │ │ -00032210: 2020 636f 6469 6d65 6e73 696f 6e20 6d61    codimension ma
│ │ │ │ -00032220: 7472 6978 2066 6163 746f 7269 7a61 7469  trix factorizati
│ │ │ │ -00032230: 6f6e 0a0a 4675 6e63 7469 6f6e 7320 7769  on..Functions wi
│ │ │ │ -00032240: 7468 206f 7074 696f 6e61 6c20 6172 6775  th optional argu
│ │ │ │ -00032250: 6d65 6e74 206e 616d 6564 204c 6179 6572  ment named Layer
│ │ │ │ -00032260: 6564 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ed:.============
│ │ │ │ -00032270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00032280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00032290: 3d3d 3d0a 0a20 202a 2022 6d61 7472 6978  ===..  * "matrix
│ │ │ │ -000322a0: 4661 6374 6f72 697a 6174 696f 6e28 2e2e  Factorization(..
│ │ │ │ -000322b0: 2e2c 4c61 7965 7265 643d 3e2e 2e2e 2922  .,Layered=>...)"
│ │ │ │ -000322c0: 202d 2d20 7365 6520 2a6e 6f74 6520 6d61   -- see *note ma
│ │ │ │ -000322d0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -000322e0: 6e3a 0a20 2020 206d 6174 7269 7846 6163  n:.    matrixFac
│ │ │ │ -000322f0: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d  torization, -- M
│ │ │ │ -00032300: 6170 7320 696e 2061 2068 6967 6865 7220  aps in a higher 
│ │ │ │ -00032310: 636f 6469 6d65 6e73 696f 6e20 6d61 7472  codimension matr
│ │ │ │ -00032320: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ -00032330: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ -00032340: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ -00032350: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ -00032360: 6563 7420 2a6e 6f74 6520 4c61 7965 7265  ect *note Layere
│ │ │ │ -00032370: 643a 204c 6179 6572 6564 2c20 6973 2061  d: Layered, is a
│ │ │ │ -00032380: 202a 6e6f 7465 2073 796d 626f 6c3a 2028   *note symbol: (
│ │ │ │ -00032390: 4d61 6361 756c 6179 3244 6f63 2953 796d  Macaulay2Doc)Sym
│ │ │ │ -000323a0: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  bol,...---------
│ │ │ │ +00031e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +00031e80: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +00031e90: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +00031ea0: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +00031eb0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +00031ec0: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ +00031ed0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +00031ee0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ +00031ef0: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +00031f00: 7469 6f6e 732e 6d32 3a33 3030 383a 302e  tions.m2:3008:0.
│ │ │ │ +00031f10: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ +00031f20: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ +00031f30: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ +00031f40: 6f64 653a 204c 6179 6572 6564 2c20 4e65  ode: Layered, Ne
│ │ │ │ +00031f50: 7874 3a20 6c61 7965 7265 6452 6573 6f6c  xt: layeredResol
│ │ │ │ +00031f60: 7574 696f 6e2c 2050 7265 763a 206b 6f73  ution, Prev: kos
│ │ │ │ +00031f70: 7a75 6c45 7874 656e 7369 6f6e 2c20 5570  zulExtension, Up
│ │ │ │ +00031f80: 3a20 546f 700a 0a4c 6179 6572 6564 202d  : Top..Layered -
│ │ │ │ +00031f90: 2d20 4f70 7469 6f6e 2066 6f72 206d 6174  - Option for mat
│ │ │ │ +00031fa0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +00031fb0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +00031fc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00031fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00031fe0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00031ff0: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +00032000: 696f 6e28 6666 2c6d 2c4c 6179 6572 6564  ion(ff,m,Layered
│ │ │ │ +00032010: 203d 3e20 7472 7565 290a 2020 2a20 496e   => true).  * In
│ │ │ │ +00032020: 7075 7473 3a0a 2020 2020 2020 2a20 4368  puts:.      * Ch
│ │ │ │ +00032030: 6563 6b2c 2061 202a 6e6f 7465 2042 6f6f  eck, a *note Boo
│ │ │ │ +00032040: 6c65 616e 2076 616c 7565 3a20 284d 6163  lean value: (Mac
│ │ │ │ +00032050: 6175 6c61 7932 446f 6329 426f 6f6c 6561  aulay2Doc)Boolea
│ │ │ │ +00032060: 6e2c 2c20 0a0a 4465 7363 7269 7074 696f  n,, ..Descriptio
│ │ │ │ +00032070: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4d  n.===========..M
│ │ │ │ +00032080: 616b 6573 206d 6174 7269 7846 6163 746f  akes matrixFacto
│ │ │ │ +00032090: 7269 7a61 7469 6f6e 2075 7365 2074 6865  rization use the
│ │ │ │ +000320a0: 2022 6c61 7965 7265 6422 2061 6c67 6f72   "layered" algor
│ │ │ │ +000320b0: 6974 686d 2c20 7768 6963 6820 776f 726b  ithm, which work
│ │ │ │ +000320c0: 7320 666f 7220 616e 7920 4d43 4d0a 6d6f  s for any MCM.mo
│ │ │ │ +000320d0: 6475 6c65 2c20 6275 7420 7265 7475 726e  dule, but return
│ │ │ │ +000320e0: 7320 736f 6d65 7468 696e 6720 6e6f 6e2d  s something non-
│ │ │ │ +000320f0: 6d69 6e69 6d61 6c20 6966 2074 6865 206d  minimal if the m
│ │ │ │ +00032100: 6f64 756c 6520 6973 206e 6f74 2061 2022  odule is not a "
│ │ │ │ +00032110: 6869 6768 2073 797a 7967 7922 0a69 6e20  high syzygy".in 
│ │ │ │ +00032120: 6120 7375 6974 6162 6c65 2073 656e 7365  a suitable sense
│ │ │ │ +00032130: 2e20 4465 6661 756c 7420 6973 2022 7472  . Default is "tr
│ │ │ │ +00032140: 7565 222e 204e 6f74 6520 7468 6174 2077  ue". Note that w
│ │ │ │ +00032150: 6865 6e20 7468 6520 6d6f 6475 6c65 2069  hen the module i
│ │ │ │ +00032160: 7320 6120 6869 6768 0a73 797a 7967 792c  s a high.syzygy,
│ │ │ │ +00032170: 204c 6179 6572 6564 3d3e 2066 616c 7365   Layered=> false
│ │ │ │ +00032180: 2069 7320 6d75 6368 2066 6173 7465 722e   is much faster.
│ │ │ │ +00032190: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +000321a0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ +000321b0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ +000321c0: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72  on: matrixFactor
│ │ │ │ +000321d0: 697a 6174 696f 6e2c 202d 2d20 4d61 7073  ization, -- Maps
│ │ │ │ +000321e0: 2069 6e20 6120 6869 6768 6572 0a20 2020   in a higher.   
│ │ │ │ +000321f0: 2063 6f64 696d 656e 7369 6f6e 206d 6174   codimension mat
│ │ │ │ +00032200: 7269 7820 6661 6374 6f72 697a 6174 696f  rix factorizatio
│ │ │ │ +00032210: 6e0a 0a46 756e 6374 696f 6e73 2077 6974  n..Functions wit
│ │ │ │ +00032220: 6820 6f70 7469 6f6e 616c 2061 7267 756d  h optional argum
│ │ │ │ +00032230: 656e 7420 6e61 6d65 6420 4c61 7965 7265  ent named Layere
│ │ │ │ +00032240: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ +00032250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00032260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00032270: 3d3d 0a0a 2020 2a20 226d 6174 7269 7846  ==..  * "matrixF
│ │ │ │ +00032280: 6163 746f 7269 7a61 7469 6f6e 282e 2e2e  actorization(...
│ │ │ │ +00032290: 2c4c 6179 6572 6564 3d3e 2e2e 2e29 2220  ,Layered=>...)" 
│ │ │ │ +000322a0: 2d2d 2073 6565 202a 6e6f 7465 206d 6174  -- see *note mat
│ │ │ │ +000322b0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +000322c0: 3a0a 2020 2020 6d61 7472 6978 4661 6374  :.    matrixFact
│ │ │ │ +000322d0: 6f72 697a 6174 696f 6e2c 202d 2d20 4d61  orization, -- Ma
│ │ │ │ +000322e0: 7073 2069 6e20 6120 6869 6768 6572 2063  ps in a higher c
│ │ │ │ +000322f0: 6f64 696d 656e 7369 6f6e 206d 6174 7269  odimension matri
│ │ │ │ +00032300: 7820 6661 6374 6f72 697a 6174 696f 6e0a  x factorization.
│ │ │ │ +00032310: 0a46 6f72 2074 6865 2070 726f 6772 616d  .For the program
│ │ │ │ +00032320: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  mer.============
│ │ │ │ +00032330: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65  ======..The obje
│ │ │ │ +00032340: 6374 202a 6e6f 7465 204c 6179 6572 6564  ct *note Layered
│ │ │ │ +00032350: 3a20 4c61 7965 7265 642c 2069 7320 6120  : Layered, is a 
│ │ │ │ +00032360: 2a6e 6f74 6520 7379 6d62 6f6c 3a20 284d  *note symbol: (M
│ │ │ │ +00032370: 6163 6175 6c61 7932 446f 6329 5379 6d62  acaulay2Doc)Symb
│ │ │ │ +00032380: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  ol,...----------
│ │ │ │ +00032390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000323a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000323b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000323c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000323d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000323e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000323f0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ -00032400: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ -00032410: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ -00032420: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ -00032430: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ -00032440: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ -00032450: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ -00032460: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -00032470: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ -00032480: 323a 3332 3531 3a30 2e0a 1f0a 4669 6c65  2:3251:0....File
│ │ │ │ -00032490: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ -000324a0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -000324b0: 732e 696e 666f 2c20 4e6f 6465 3a20 6c61  s.info, Node: la
│ │ │ │ -000324c0: 7965 7265 6452 6573 6f6c 7574 696f 6e2c  yeredResolution,
│ │ │ │ -000324d0: 204e 6578 743a 204c 6966 742c 2050 7265   Next: Lift, Pre
│ │ │ │ -000324e0: 763a 204c 6179 6572 6564 2c20 5570 3a20  v: Layered, Up: 
│ │ │ │ -000324f0: 546f 700a 0a6c 6179 6572 6564 5265 736f  Top..layeredReso
│ │ │ │ -00032500: 6c75 7469 6f6e 202d 2d20 6c61 7965 7265  lution -- layere
│ │ │ │ -00032510: 6420 6669 6e69 7465 2061 6e64 2069 6e66  d finite and inf
│ │ │ │ -00032520: 696e 6974 6520 6c61 7965 7265 6420 7265  inite layered re
│ │ │ │ -00032530: 736f 6c75 7469 6f6e 7320 6f66 2043 4d20  solutions of CM 
│ │ │ │ -00032540: 6d6f 6475 6c65 730a 2a2a 2a2a 2a2a 2a2a  modules.********
│ │ │ │ +000323d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ +000323e0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ +000323f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ +00032400: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ +00032410: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ +00032420: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ +00032430: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ +00032440: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +00032450: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
│ │ │ │ +00032460: 3a33 3235 313a 302e 0a1f 0a46 696c 653a  :3251:0....File:
│ │ │ │ +00032470: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ +00032480: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +00032490: 2e69 6e66 6f2c 204e 6f64 653a 206c 6179  .info, Node: lay
│ │ │ │ +000324a0: 6572 6564 5265 736f 6c75 7469 6f6e 2c20  eredResolution, 
│ │ │ │ +000324b0: 4e65 7874 3a20 4c69 6674 2c20 5072 6576  Next: Lift, Prev
│ │ │ │ +000324c0: 3a20 4c61 7965 7265 642c 2055 703a 2054  : Layered, Up: T
│ │ │ │ +000324d0: 6f70 0a0a 6c61 7965 7265 6452 6573 6f6c  op..layeredResol
│ │ │ │ +000324e0: 7574 696f 6e20 2d2d 206c 6179 6572 6564  ution -- layered
│ │ │ │ +000324f0: 2066 696e 6974 6520 616e 6420 696e 6669   finite and infi
│ │ │ │ +00032500: 6e69 7465 206c 6179 6572 6564 2072 6573  nite layered res
│ │ │ │ +00032510: 6f6c 7574 696f 6e73 206f 6620 434d 206d  olutions of CM m
│ │ │ │ +00032520: 6f64 756c 6573 0a2a 2a2a 2a2a 2a2a 2a2a  odules.*********
│ │ │ │ +00032530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00032540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00032550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00032560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00032570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00032580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00032590: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -000325a0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -000325b0: 2846 462c 2061 7567 2920 3d20 6c61 7965  (FF, aug) = laye
│ │ │ │ -000325c0: 7265 6452 6573 6f6c 7574 696f 6e28 6666  redResolution(ff
│ │ │ │ -000325d0: 2c4d 290a 2020 2020 2020 2020 2846 462c  ,M).        (FF,
│ │ │ │ -000325e0: 2061 7567 2920 3d20 6c61 7965 7265 6452   aug) = layeredR
│ │ │ │ -000325f0: 6573 6f6c 7574 696f 6e28 6666 2c4d 2c6c  esolution(ff,M,l
│ │ │ │ -00032600: 656e 290a 2020 2a20 496e 7075 7473 3a0a  en).  * Inputs:.
│ │ │ │ -00032610: 2020 2020 2020 2a20 6666 2c20 6120 2a6e        * ff, a *n
│ │ │ │ -00032620: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ -00032630: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ -00032640: 2c2c 2031 2078 2063 206d 6174 7269 7820  ,, 1 x c matrix 
│ │ │ │ -00032650: 7768 6f73 6520 656e 7472 6965 730a 2020  whose entries.  
│ │ │ │ -00032660: 2020 2020 2020 6172 6520 6120 7265 6775        are a regu
│ │ │ │ -00032670: 6c61 7220 7365 7175 656e 6365 2069 6e20  lar sequence in 
│ │ │ │ -00032680: 7468 6520 476f 7265 6e73 7465 696e 2072  the Gorenstein r
│ │ │ │ -00032690: 696e 6720 530a 2020 2020 2020 2a20 4d2c  ing S.      * M,
│ │ │ │ -000326a0: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ -000326b0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -000326c0: 6f64 756c 652c 2c20 4d43 4d20 6d6f 6475  odule,, MCM modu
│ │ │ │ -000326d0: 6c65 206f 7665 7220 522c 0a20 2020 2020  le over R,.     
│ │ │ │ -000326e0: 2020 2072 6570 7265 7365 6e74 6564 2061     represented a
│ │ │ │ -000326f0: 7320 616e 2053 2d6d 6f64 756c 6520 696e  s an S-module in
│ │ │ │ -00032700: 2074 6865 2066 6972 7374 2063 6173 6520   the first case 
│ │ │ │ -00032710: 616e 6420 6173 2061 6e20 522d 6d6f 6475  and as an R-modu
│ │ │ │ -00032720: 6c65 2069 6e20 7468 650a 2020 2020 2020  le in the.      
│ │ │ │ -00032730: 2020 7365 636f 6e64 0a20 2020 2020 202a    second.      *
│ │ │ │ -00032740: 206c 656e 2c20 616e 202a 6e6f 7465 2069   len, an *note i
│ │ │ │ -00032750: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ -00032760: 7932 446f 6329 5a5a 2c2c 206c 656e 6774  y2Doc)ZZ,, lengt
│ │ │ │ -00032770: 6820 6f66 2074 6865 2073 6567 6d65 6e74  h of the segment
│ │ │ │ -00032780: 206f 6620 7468 650a 2020 2020 2020 2020   of the.        
│ │ │ │ -00032790: 7265 736f 6c75 7469 6f6e 2074 6f20 6265  resolution to be
│ │ │ │ -000327a0: 2063 6f6d 7075 7465 6420 6f76 6572 2052   computed over R
│ │ │ │ -000327b0: 2c20 696e 2074 6865 2073 6563 6f6e 6420  , in the second 
│ │ │ │ -000327c0: 666f 726d 2e0a 2020 2a20 2a6e 6f74 6520  form..  * *note 
│ │ │ │ -000327d0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a  Optional inputs:
│ │ │ │ -000327e0: 2028 4d61 6361 756c 6179 3244 6f63 2975   (Macaulay2Doc)u
│ │ │ │ -000327f0: 7369 6e67 2066 756e 6374 696f 6e73 2077  sing functions w
│ │ │ │ -00032800: 6974 6820 6f70 7469 6f6e 616c 2069 6e70  ith optional inp
│ │ │ │ -00032810: 7574 732c 3a0a 2020 2020 2020 2a20 4368  uts,:.      * Ch
│ │ │ │ -00032820: 6563 6b20 3d3e 202e 2e2e 2c20 6465 6661  eck => ..., defa
│ │ │ │ -00032830: 756c 7420 7661 6c75 6520 6661 6c73 650a  ult value false.
│ │ │ │ -00032840: 2020 2020 2020 2a20 5665 7262 6f73 6520        * Verbose 
│ │ │ │ -00032850: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -00032860: 7661 6c75 6520 6661 6c73 650a 2020 2a20  value false.  * 
│ │ │ │ -00032870: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00032880: 2046 462c 2061 202a 6e6f 7465 2063 6f6d   FF, a *note com
│ │ │ │ -00032890: 706c 6578 3a20 2843 6f6d 706c 6578 6573  plex: (Complexes
│ │ │ │ -000328a0: 2943 6f6d 706c 6578 2c2c 2072 6573 6f6c  )Complex,, resol
│ │ │ │ -000328b0: 7574 696f 6e20 6f66 204d 206f 7665 7220  ution of M over 
│ │ │ │ -000328c0: 5320 696e 2074 6865 0a20 2020 2020 2020  S in the.       
│ │ │ │ -000328d0: 2066 6972 7374 2063 6173 653b 206c 656e   first case; len
│ │ │ │ -000328e0: 6774 6820 6c65 6e20 7365 676d 656e 7420  gth len segment 
│ │ │ │ -000328f0: 6f66 2074 6865 2072 6573 6f6c 7574 696f  of the resolutio
│ │ │ │ -00032900: 6e20 6f76 6572 2052 2069 6e20 7468 6520  n over R in the 
│ │ │ │ -00032910: 7365 636f 6e64 2e0a 0a44 6573 6372 6970  second...Descrip
│ │ │ │ -00032920: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00032930: 0a0a 5468 6520 7265 736f 6c75 7469 6f6e  ..The resolution
│ │ │ │ -00032940: 7320 636f 6d70 7574 6564 2061 7265 2074  s computed are t
│ │ │ │ -00032950: 686f 7365 2064 6573 6372 6962 6564 2069  hose described i
│ │ │ │ -00032960: 6e20 7468 6520 7061 7065 7220 224c 6179  n the paper "Lay
│ │ │ │ -00032970: 6572 6564 2052 6573 6f6c 7574 696f 6e73  ered Resolutions
│ │ │ │ -00032980: 0a6f 6620 436f 6865 6e2d 4d61 6361 756c  .of Cohen-Macaul
│ │ │ │ -00032990: 6179 206d 6f64 756c 6573 2220 6279 2045  ay modules" by E
│ │ │ │ -000329a0: 6973 656e 6275 6420 616e 6420 5065 6576  isenbud and Peev
│ │ │ │ -000329b0: 612e 2054 6865 7920 6172 6520 626f 7468  a. They are both
│ │ │ │ -000329c0: 206d 696e 696d 616c 2077 6865 6e20 4d0a   minimal when M.
│ │ │ │ -000329d0: 6973 2061 2073 7566 6669 6369 656e 746c  is a sufficientl
│ │ │ │ -000329e0: 7920 6869 6768 2073 797a 7967 7920 6f66  y high syzygy of
│ │ │ │ -000329f0: 2061 206d 6f64 756c 6520 4e2e 2049 6620   a module N. If 
│ │ │ │ -00032a00: 7468 6520 6f70 7469 6f6e 2056 6572 626f  the option Verbo
│ │ │ │ -00032a10: 7365 3d3e 7472 7565 2069 730a 7365 742c  se=>true is.set,
│ │ │ │ -00032a20: 2074 6865 6e20 2869 6e20 7468 6520 6361   then (in the ca
│ │ │ │ -00032a30: 7365 206f 6620 7468 6520 7265 736f 6c75  se of the resolu
│ │ │ │ -00032a40: 7469 6f6e 206f 7665 7220 5329 2074 6865  tion over S) the
│ │ │ │ -00032a50: 2072 616e 6b73 206f 6620 7468 6520 6d6f   ranks of the mo
│ │ │ │ -00032a60: 6475 6c65 7320 425f 730a 696e 2074 6865  dules B_s.in the
│ │ │ │ -00032a70: 2072 6573 6f6c 7574 696f 6e20 6172 6520   resolution are 
│ │ │ │ -00032a80: 6f75 7470 7574 2e0a 0a48 6572 6520 6973  output...Here is
│ │ │ │ -00032a90: 2061 6e20 6578 616d 706c 6520 636f 6d70   an example comp
│ │ │ │ -00032aa0: 7574 696e 6720 3520 7465 726d 7320 6f66  uting 5 terms of
│ │ │ │ -00032ab0: 2061 6e20 696e 6669 6e69 7465 2072 6573   an infinite res
│ │ │ │ -00032ac0: 6f6c 7574 696f 6e3a 0a0a 2b2d 2d2d 2d2d  olution:..+-----
│ │ │ │ +00032570: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ +00032580: 7361 6765 3a20 0a20 2020 2020 2020 2028  sage: .        (
│ │ │ │ +00032590: 4646 2c20 6175 6729 203d 206c 6179 6572  FF, aug) = layer
│ │ │ │ +000325a0: 6564 5265 736f 6c75 7469 6f6e 2866 662c  edResolution(ff,
│ │ │ │ +000325b0: 4d29 0a20 2020 2020 2020 2028 4646 2c20  M).        (FF, 
│ │ │ │ +000325c0: 6175 6729 203d 206c 6179 6572 6564 5265  aug) = layeredRe
│ │ │ │ +000325d0: 736f 6c75 7469 6f6e 2866 662c 4d2c 6c65  solution(ff,M,le
│ │ │ │ +000325e0: 6e29 0a20 202a 2049 6e70 7574 733a 0a20  n).  * Inputs:. 
│ │ │ │ +000325f0: 2020 2020 202a 2066 662c 2061 202a 6e6f       * ff, a *no
│ │ │ │ +00032600: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ +00032610: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ +00032620: 2c20 3120 7820 6320 6d61 7472 6978 2077  , 1 x c matrix w
│ │ │ │ +00032630: 686f 7365 2065 6e74 7269 6573 0a20 2020  hose entries.   
│ │ │ │ +00032640: 2020 2020 2061 7265 2061 2072 6567 756c       are a regul
│ │ │ │ +00032650: 6172 2073 6571 7565 6e63 6520 696e 2074  ar sequence in t
│ │ │ │ +00032660: 6865 2047 6f72 656e 7374 6569 6e20 7269  he Gorenstein ri
│ │ │ │ +00032670: 6e67 2053 0a20 2020 2020 202a 204d 2c20  ng S.      * M, 
│ │ │ │ +00032680: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20  a *note module: 
│ │ │ │ +00032690: 284d 6163 6175 6c61 7932 446f 6329 4d6f  (Macaulay2Doc)Mo
│ │ │ │ +000326a0: 6475 6c65 2c2c 204d 434d 206d 6f64 756c  dule,, MCM modul
│ │ │ │ +000326b0: 6520 6f76 6572 2052 2c0a 2020 2020 2020  e over R,.      
│ │ │ │ +000326c0: 2020 7265 7072 6573 656e 7465 6420 6173    represented as
│ │ │ │ +000326d0: 2061 6e20 532d 6d6f 6475 6c65 2069 6e20   an S-module in 
│ │ │ │ +000326e0: 7468 6520 6669 7273 7420 6361 7365 2061  the first case a
│ │ │ │ +000326f0: 6e64 2061 7320 616e 2052 2d6d 6f64 756c  nd as an R-modul
│ │ │ │ +00032700: 6520 696e 2074 6865 0a20 2020 2020 2020  e in the.       
│ │ │ │ +00032710: 2073 6563 6f6e 640a 2020 2020 2020 2a20   second.      * 
│ │ │ │ +00032720: 6c65 6e2c 2061 6e20 2a6e 6f74 6520 696e  len, an *note in
│ │ │ │ +00032730: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
│ │ │ │ +00032740: 3244 6f63 295a 5a2c 2c20 6c65 6e67 7468  2Doc)ZZ,, length
│ │ │ │ +00032750: 206f 6620 7468 6520 7365 676d 656e 7420   of the segment 
│ │ │ │ +00032760: 6f66 2074 6865 0a20 2020 2020 2020 2072  of the.        r
│ │ │ │ +00032770: 6573 6f6c 7574 696f 6e20 746f 2062 6520  esolution to be 
│ │ │ │ +00032780: 636f 6d70 7574 6564 206f 7665 7220 522c  computed over R,
│ │ │ │ +00032790: 2069 6e20 7468 6520 7365 636f 6e64 2066   in the second f
│ │ │ │ +000327a0: 6f72 6d2e 0a20 202a 202a 6e6f 7465 204f  orm..  * *note O
│ │ │ │ +000327b0: 7074 696f 6e61 6c20 696e 7075 7473 3a20  ptional inputs: 
│ │ │ │ +000327c0: 284d 6163 6175 6c61 7932 446f 6329 7573  (Macaulay2Doc)us
│ │ │ │ +000327d0: 696e 6720 6675 6e63 7469 6f6e 7320 7769  ing functions wi
│ │ │ │ +000327e0: 7468 206f 7074 696f 6e61 6c20 696e 7075  th optional inpu
│ │ │ │ +000327f0: 7473 2c3a 0a20 2020 2020 202a 2043 6865  ts,:.      * Che
│ │ │ │ +00032800: 636b 203d 3e20 2e2e 2e2c 2064 6566 6175  ck => ..., defau
│ │ │ │ +00032810: 6c74 2076 616c 7565 2066 616c 7365 0a20  lt value false. 
│ │ │ │ +00032820: 2020 2020 202a 2056 6572 626f 7365 203d       * Verbose =
│ │ │ │ +00032830: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ +00032840: 616c 7565 2066 616c 7365 0a20 202a 204f  alue false.  * O
│ │ │ │ +00032850: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +00032860: 4646 2c20 6120 2a6e 6f74 6520 636f 6d70  FF, a *note comp
│ │ │ │ +00032870: 6c65 783a 2028 436f 6d70 6c65 7865 7329  lex: (Complexes)
│ │ │ │ +00032880: 436f 6d70 6c65 782c 2c20 7265 736f 6c75  Complex,, resolu
│ │ │ │ +00032890: 7469 6f6e 206f 6620 4d20 6f76 6572 2053  tion of M over S
│ │ │ │ +000328a0: 2069 6e20 7468 650a 2020 2020 2020 2020   in the.        
│ │ │ │ +000328b0: 6669 7273 7420 6361 7365 3b20 6c65 6e67  first case; leng
│ │ │ │ +000328c0: 7468 206c 656e 2073 6567 6d65 6e74 206f  th len segment o
│ │ │ │ +000328d0: 6620 7468 6520 7265 736f 6c75 7469 6f6e  f the resolution
│ │ │ │ +000328e0: 206f 7665 7220 5220 696e 2074 6865 2073   over R in the s
│ │ │ │ +000328f0: 6563 6f6e 642e 0a0a 4465 7363 7269 7074  econd...Descript
│ │ │ │ +00032900: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00032910: 0a54 6865 2072 6573 6f6c 7574 696f 6e73  .The resolutions
│ │ │ │ +00032920: 2063 6f6d 7075 7465 6420 6172 6520 7468   computed are th
│ │ │ │ +00032930: 6f73 6520 6465 7363 7269 6265 6420 696e  ose described in
│ │ │ │ +00032940: 2074 6865 2070 6170 6572 2022 4c61 7965   the paper "Laye
│ │ │ │ +00032950: 7265 6420 5265 736f 6c75 7469 6f6e 730a  red Resolutions.
│ │ │ │ +00032960: 6f66 2043 6f68 656e 2d4d 6163 6175 6c61  of Cohen-Macaula
│ │ │ │ +00032970: 7920 6d6f 6475 6c65 7322 2062 7920 4569  y modules" by Ei
│ │ │ │ +00032980: 7365 6e62 7564 2061 6e64 2050 6565 7661  senbud and Peeva
│ │ │ │ +00032990: 2e20 5468 6579 2061 7265 2062 6f74 6820  . They are both 
│ │ │ │ +000329a0: 6d69 6e69 6d61 6c20 7768 656e 204d 0a69  minimal when M.i
│ │ │ │ +000329b0: 7320 6120 7375 6666 6963 6965 6e74 6c79  s a sufficiently
│ │ │ │ +000329c0: 2068 6967 6820 7379 7a79 6779 206f 6620   high syzygy of 
│ │ │ │ +000329d0: 6120 6d6f 6475 6c65 204e 2e20 4966 2074  a module N. If t
│ │ │ │ +000329e0: 6865 206f 7074 696f 6e20 5665 7262 6f73  he option Verbos
│ │ │ │ +000329f0: 653d 3e74 7275 6520 6973 0a73 6574 2c20  e=>true is.set, 
│ │ │ │ +00032a00: 7468 656e 2028 696e 2074 6865 2063 6173  then (in the cas
│ │ │ │ +00032a10: 6520 6f66 2074 6865 2072 6573 6f6c 7574  e of the resolut
│ │ │ │ +00032a20: 696f 6e20 6f76 6572 2053 2920 7468 6520  ion over S) the 
│ │ │ │ +00032a30: 7261 6e6b 7320 6f66 2074 6865 206d 6f64  ranks of the mod
│ │ │ │ +00032a40: 756c 6573 2042 5f73 0a69 6e20 7468 6520  ules B_s.in the 
│ │ │ │ +00032a50: 7265 736f 6c75 7469 6f6e 2061 7265 206f  resolution are o
│ │ │ │ +00032a60: 7574 7075 742e 0a0a 4865 7265 2069 7320  utput...Here is 
│ │ │ │ +00032a70: 616e 2065 7861 6d70 6c65 2063 6f6d 7075  an example compu
│ │ │ │ +00032a80: 7469 6e67 2035 2074 6572 6d73 206f 6620  ting 5 terms of 
│ │ │ │ +00032a90: 616e 2069 6e66 696e 6974 6520 7265 736f  an infinite reso
│ │ │ │ +00032aa0: 6c75 7469 6f6e 3a0a 0a2b 2d2d 2d2d 2d2d  lution:..+------
│ │ │ │ +00032ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032b10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ -00032b20: 5320 3d20 5a5a 2f31 3031 5b61 2c62 2c63  S = ZZ/101[a,b,c
│ │ │ │ -00032b30: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -00032b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032af0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ +00032b00: 203d 205a 5a2f 3130 315b 612c 622c 635d   = ZZ/101[a,b,c]
│ │ │ │ +00032b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032b40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00032b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032b60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032b90: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │  00032ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032bb0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20          |.|o1 = 
│ │ │ │ -00032bc0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00032bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032be0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00032bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c50: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ -00032c60: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +00032c30: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ +00032c40: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00032c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ca0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00032c80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00032c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032cf0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -00032d00: 6666 203d 206d 6174 7269 7822 6133 2c20  ff = matrix"a3, 
│ │ │ │ -00032d10: 6233 2c20 6333 2220 2020 2020 2020 2020  b3, c3"         
│ │ │ │ -00032d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032cd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2066  -------+.|i2 : f
│ │ │ │ +00032ce0: 6620 3d20 6d61 7472 6978 2261 332c 2062  f = matrix"a3, b
│ │ │ │ +00032cf0: 332c 2063 3322 2020 2020 2020 2020 2020  3, c3"          
│ │ │ │ +00032d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032d20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00032d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032d90: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20          |.|o2 = 
│ │ │ │ -00032da0: 7c20 6133 2062 3320 6333 207c 2020 2020  | a3 b3 c3 |    
│ │ │ │ +00032d70: 2020 2020 2020 207c 0a7c 6f32 203d 207c         |.|o2 = |
│ │ │ │ +00032d80: 2061 3320 6233 2063 3320 7c20 2020 2020   a3 b3 c3 |     
│ │ │ │ +00032d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032dc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00032dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00032e40: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ +00032e10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00032e20: 2020 2020 2020 2031 2020 2020 2020 3320         1      3 
│ │ │ │ +00032e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032e80: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20          |.|o2 : 
│ │ │ │ -00032e90: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +00032e60: 2020 2020 2020 207c 0a7c 6f32 203a 204d         |.|o2 : M
│ │ │ │ +00032e70: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +00032e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ed0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00032eb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00032ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032f20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -00032f30: 5220 3d20 532f 6964 6561 6c20 6666 2020  R = S/ideal ff  
│ │ │ │ +00032f00: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2052  -------+.|i3 : R
│ │ │ │ +00032f10: 203d 2053 2f69 6465 616c 2066 6620 2020   = S/ideal ff   
│ │ │ │ +00032f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032f50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00032f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00032f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032fa0: 2020 2020 2020 207c 0a7c 6f33 203d 2052         |.|o3 = R
│ │ │ │  00032fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032fc0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ -00032fd0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +00032fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032ff0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00033000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033010: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033060: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -00033070: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +00033040: 2020 2020 2020 207c 0a7c 6f33 203a 2051         |.|o3 : Q
│ │ │ │ +00033050: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00033060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000330a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000330b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033090: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000330a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000330b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000330c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000330d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000330e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033100: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00033110: 4d20 3d20 7379 7a79 6779 4d6f 6475 6c65  M = syzygyModule
│ │ │ │ -00033120: 2832 2c63 6f6b 6572 2076 6172 7320 5229  (2,coker vars R)
│ │ │ │ -00033130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000330e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 204d  -------+.|i4 : M
│ │ │ │ +000330f0: 203d 2073 797a 7967 794d 6f64 756c 6528   = syzygyModule(
│ │ │ │ +00033100: 322c 636f 6b65 7220 7661 7273 2052 2920  2,coker vars R) 
│ │ │ │ +00033110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033130: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00033140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033150: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000331a0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ -000331b0: 636f 6b65 726e 656c 207b 327d 207c 2061  cokernel {2} | a
│ │ │ │ -000331c0: 2020 3020 2d63 3220 3020 2020 6232 2030    0 -c2 0   b2 0
│ │ │ │ -000331d0: 2030 2020 2030 2020 3020 2030 207c 2020   0   0  0  0 |  
│ │ │ │ -000331e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000331f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033200: 2020 2020 2020 2020 207b 327d 207c 202d           {2} | -
│ │ │ │ -00033210: 6220 3020 3020 2020 2d63 3220 3020 2030  b 0 0   -c2 0  0
│ │ │ │ -00033220: 2030 2020 2061 3220 3020 2030 207c 2020   0   a2 0  0 |  
│ │ │ │ -00033230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033240: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033250: 2020 2020 2020 2020 207b 327d 207c 2063           {2} | c
│ │ │ │ -00033260: 2020 3020 3020 2020 3020 2020 3020 2030    0 0   0   0  0
│ │ │ │ -00033270: 202d 6232 2030 2020 6132 2030 207c 2020   -b2 0  a2 0 |  
│ │ │ │ -00033280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000332a0: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -000332b0: 2020 6320 6220 2020 6120 2020 3020 2030    c b   a   0  0
│ │ │ │ -000332c0: 2030 2020 2030 2020 3020 2030 207c 2020   0   0  0  0 |  
│ │ │ │ -000332d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000332e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000332f0: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -00033300: 2020 3020 3020 2020 3020 2020 6320 2062    0 0   0   c  b
│ │ │ │ -00033310: 2061 2020 2030 2020 3020 2030 207c 2020   a   0  0  0 |  
│ │ │ │ -00033320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033330: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033340: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ -00033350: 2020 3020 3020 2020 3020 2020 3020 2030    0 0   0   0  0
│ │ │ │ -00033360: 2030 2020 2063 2020 6220 2061 207c 2020   0   c  b  a |  
│ │ │ │ +00033180: 2020 2020 2020 207c 0a7c 6f34 203d 2063         |.|o4 = c
│ │ │ │ +00033190: 6f6b 6572 6e65 6c20 7b32 7d20 7c20 6120  okernel {2} | a 
│ │ │ │ +000331a0: 2030 202d 6332 2030 2020 2062 3220 3020   0 -c2 0   b2 0 
│ │ │ │ +000331b0: 3020 2020 3020 2030 2020 3020 7c20 2020  0   0  0  0 |   
│ │ │ │ +000331c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000331d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000331e0: 2020 2020 2020 2020 7b32 7d20 7c20 2d62          {2} | -b
│ │ │ │ +000331f0: 2030 2030 2020 202d 6332 2030 2020 3020   0 0   -c2 0  0 
│ │ │ │ +00033200: 3020 2020 6132 2030 2020 3020 7c20 2020  0   a2 0  0 |   
│ │ │ │ +00033210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033220: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033230: 2020 2020 2020 2020 7b32 7d20 7c20 6320          {2} | c 
│ │ │ │ +00033240: 2030 2030 2020 2030 2020 2030 2020 3020   0 0   0   0  0 
│ │ │ │ +00033250: 2d62 3220 3020 2061 3220 3020 7c20 2020  -b2 0  a2 0 |   
│ │ │ │ +00033260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033270: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033280: 2020 2020 2020 2020 7b33 7d20 7c20 3020          {3} | 0 
│ │ │ │ +00033290: 2063 2062 2020 2061 2020 2030 2020 3020   c b   a   0  0 
│ │ │ │ +000332a0: 3020 2020 3020 2030 2020 3020 7c20 2020  0   0  0  0 |   
│ │ │ │ +000332b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000332c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000332d0: 2020 2020 2020 2020 7b33 7d20 7c20 3020          {3} | 0 
│ │ │ │ +000332e0: 2030 2030 2020 2030 2020 2063 2020 6220   0 0   0   c  b 
│ │ │ │ +000332f0: 6120 2020 3020 2030 2020 3020 7c20 2020  a   0  0  0 |   
│ │ │ │ +00033300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033310: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033320: 2020 2020 2020 2020 7b33 7d20 7c20 3020          {3} | 0 
│ │ │ │ +00033330: 2030 2030 2020 2030 2020 2030 2020 3020   0 0   0   0  0 
│ │ │ │ +00033340: 3020 2020 6320 2062 2020 6120 7c20 2020  0   c  b  a |   
│ │ │ │ +00033350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033360: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00033370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033380: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000333a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000333b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000333b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000333c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000333d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000333d0: 2020 2020 2020 3620 2020 2020 2020 2020        6         
│ │ │ │  000333e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000333f0: 2020 2020 2020 2036 2020 2020 2020 2020         6        
│ │ │ │ -00033400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033420: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -00033430: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965  R-module, quotie
│ │ │ │ -00033440: 6e74 206f 6620 5220 2020 2020 2020 2020  nt of R         
│ │ │ │ -00033450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033470: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000333f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033400: 2020 2020 2020 207c 0a7c 6f34 203a 2052         |.|o4 : R
│ │ │ │ +00033410: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +00033420: 7420 6f66 2052 2020 2020 2020 2020 2020  t of R          
│ │ │ │ +00033430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033450: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00033460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000334c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ -000334d0: 2846 462c 2061 7567 2920 3d20 6c61 7965  (FF, aug) = laye
│ │ │ │ -000334e0: 7265 6452 6573 6f6c 7574 696f 6e28 6666  redResolution(ff
│ │ │ │ -000334f0: 2c4d 2c35 2920 2020 2020 2020 2020 2020  ,M,5)           
│ │ │ │ -00033500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033520: 2020 3020 3120 3220 3320 3420 3520 2020    0 1 2 3 4 5   
│ │ │ │ +000334a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2028  -------+.|i5 : (
│ │ │ │ +000334b0: 4646 2c20 6175 6729 203d 206c 6179 6572  FF, aug) = layer
│ │ │ │ +000334c0: 6564 5265 736f 6c75 7469 6f6e 2866 662c  edResolution(ff,
│ │ │ │ +000334d0: 4d2c 3529 2020 2020 2020 2020 2020 2020  M,5)            
│ │ │ │ +000334e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000334f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033500: 2030 2031 2032 2033 2034 2035 2020 2020   0 1 2 3 4 5    
│ │ │ │ +00033510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033560: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -00033570: 3a20 3420 3420 3420 3420 3420 3420 2020  : 4 4 4 4 4 4   
│ │ │ │ +00033540: 2020 2020 2020 207c 0a7c 746f 7461 6c3a         |.|total:
│ │ │ │ +00033550: 2034 2034 2034 2034 2034 2034 2020 2020   4 4 4 4 4 4    
│ │ │ │ +00033560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335b0: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -000335c0: 3a20 3320 3120 2e20 2e20 2e20 2e20 2020  : 3 1 . . . .   
│ │ │ │ +00033590: 2020 2020 2020 207c 0a7c 2020 2020 323a         |.|    2:
│ │ │ │ +000335a0: 2033 2031 202e 202e 202e 202e 2020 2020   3 1 . . . .    
│ │ │ │ +000335b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000335c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000335d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000335f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033600: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -00033610: 3a20 3120 3320 3320 3120 2e20 2e20 2020  : 1 3 3 1 . .   
│ │ │ │ +000335e0: 2020 2020 2020 207c 0a7c 2020 2020 333a         |.|    3:
│ │ │ │ +000335f0: 2031 2033 2033 2031 202e 202e 2020 2020   1 3 3 1 . .    
│ │ │ │ +00033600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033650: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00033660: 3a20 2e20 2e20 3120 3320 3320 3120 2020  : . . 1 3 3 1   
│ │ │ │ +00033630: 2020 2020 2020 207c 0a7c 2020 2020 343a         |.|    4:
│ │ │ │ +00033640: 202e 202e 2031 2033 2033 2031 2020 2020   . . 1 3 3 1    
│ │ │ │ +00033650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000336a0: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -000336b0: 3a20 2e20 2e20 2e20 2e20 3120 3320 2020  : . . . . 1 3   
│ │ │ │ +00033680: 2020 2020 2020 207c 0a7c 2020 2020 353a         |.|    5:
│ │ │ │ +00033690: 202e 202e 202e 202e 2031 2033 2020 2020   . . . . 1 3    
│ │ │ │ +000336a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000336b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000336c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000336d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000336e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000336f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033700: 2020 3020 3120 3220 2033 2020 3420 2035    0 1 2  3  4  5
│ │ │ │ +000336d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000336e0: 2030 2031 2032 2020 3320 2034 2020 3520   0 1 2  3  4  5 
│ │ │ │ +000336f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033740: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -00033750: 3a20 3520 3720 3920 3131 2031 3320 3135  : 5 7 9 11 13 15
│ │ │ │ +00033720: 2020 2020 2020 207c 0a7c 746f 7461 6c3a         |.|total:
│ │ │ │ +00033730: 2035 2037 2039 2031 3120 3133 2031 3520   5 7 9 11 13 15 
│ │ │ │ +00033740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033790: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -000337a0: 3a20 3320 3120 2e20 202e 2020 2e20 202e  : 3 1 .  .  .  .
│ │ │ │ +00033770: 2020 2020 2020 207c 0a7c 2020 2020 323a         |.|    2:
│ │ │ │ +00033780: 2033 2031 202e 2020 2e20 202e 2020 2e20   3 1 .  .  .  . 
│ │ │ │ +00033790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000337a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000337b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000337c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000337d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000337e0: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -000337f0: 3a20 3220 3620 3620 2032 2020 2e20 202e  : 2 6 6  2  .  .
│ │ │ │ +000337c0: 2020 2020 2020 207c 0a7c 2020 2020 333a         |.|    3:
│ │ │ │ +000337d0: 2032 2036 2036 2020 3220 202e 2020 2e20   2 6 6  2  .  . 
│ │ │ │ +000337e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000337f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033830: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00033840: 3a20 2e20 2e20 3320 2039 2020 3920 2033  : . . 3  9  9  3
│ │ │ │ +00033810: 2020 2020 2020 207c 0a7c 2020 2020 343a         |.|    4:
│ │ │ │ +00033820: 202e 202e 2033 2020 3920 2039 2020 3320   . . 3  9  9  3 
│ │ │ │ +00033830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033880: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -00033890: 3a20 2e20 2e20 2e20 202e 2020 3420 3132  : . . .  .  4 12
│ │ │ │ +00033860: 2020 2020 2020 207c 0a7c 2020 2020 353a         |.|    5:
│ │ │ │ +00033870: 202e 202e 202e 2020 2e20 2034 2031 3220   . . .  .  4 12 
│ │ │ │ +00033880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000338a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000338d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000338e0: 2020 3020 2031 2020 3220 2033 2020 3420    0  1  2  3  4 
│ │ │ │ -000338f0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ -00033900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033920: 2020 2020 2020 2020 7c0a 7c74 6f74 616c          |.|total
│ │ │ │ -00033930: 3a20 3620 3130 2031 3520 3231 2032 3820  : 6 10 15 21 28 
│ │ │ │ -00033940: 3336 2020 2020 2020 2020 2020 2020 2020  36              
│ │ │ │ -00033950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033970: 2020 2020 2020 2020 7c0a 7c20 2020 2032          |.|    2
│ │ │ │ -00033980: 3a20 3320 2031 2020 2e20 202e 2020 2e20  : 3  1  .  .  . 
│ │ │ │ -00033990: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -000339a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000339b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000339c0: 2020 2020 2020 2020 7c0a 7c20 2020 2033          |.|    3
│ │ │ │ -000339d0: 3a20 3320 2039 2020 3920 2033 2020 2e20  : 3  9  9  3  . 
│ │ │ │ -000339e0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -000339f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a10: 2020 2020 2020 2020 7c0a 7c20 2020 2034          |.|    4
│ │ │ │ -00033a20: 3a20 2e20 202e 2020 3620 3138 2031 3820  : .  .  6 18 18 
│ │ │ │ -00033a30: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ -00033a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033a60: 2020 2020 2020 2020 7c0a 7c20 2020 2035          |.|    5
│ │ │ │ -00033a70: 3a20 2e20 202e 2020 2e20 202e 2031 3020  : .  .  .  . 10 
│ │ │ │ -00033a80: 3330 2020 2020 2020 2020 2020 2020 2020  30              
│ │ │ │ -00033a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000338b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000338c0: 2030 2020 3120 2032 2020 3320 2034 2020   0  1  2  3  4  
│ │ │ │ +000338d0: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ +000338e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000338f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033900: 2020 2020 2020 207c 0a7c 746f 7461 6c3a         |.|total:
│ │ │ │ +00033910: 2036 2031 3020 3135 2032 3120 3238 2033   6 10 15 21 28 3
│ │ │ │ +00033920: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +00033930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033950: 2020 2020 2020 207c 0a7c 2020 2020 323a         |.|    2:
│ │ │ │ +00033960: 2033 2020 3120 202e 2020 2e20 202e 2020   3  1  .  .  .  
│ │ │ │ +00033970: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +00033980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000339a0: 2020 2020 2020 207c 0a7c 2020 2020 333a         |.|    3:
│ │ │ │ +000339b0: 2033 2020 3920 2039 2020 3320 202e 2020   3  9  9  3  .  
│ │ │ │ +000339c0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +000339d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000339e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000339f0: 2020 2020 2020 207c 0a7c 2020 2020 343a         |.|    4:
│ │ │ │ +00033a00: 202e 2020 2e20 2036 2031 3820 3138 2020   .  .  6 18 18  
│ │ │ │ +00033a10: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +00033a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a40: 2020 2020 2020 207c 0a7c 2020 2020 353a         |.|    5:
│ │ │ │ +00033a50: 202e 2020 2e20 202e 2020 2e20 3130 2033   .  .  .  . 10 3
│ │ │ │ +00033a60: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00033a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033a90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00033aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ab0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033b00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033b10: 2020 3620 2020 2020 2031 3020 2020 2020    6      10     
│ │ │ │ -00033b20: 2031 3520 2020 2020 2032 3120 2020 2020   15      21     
│ │ │ │ -00033b30: 2032 3820 2020 2020 2033 3620 2020 2020   28      36     
│ │ │ │ -00033b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033b50: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -00033b60: 2852 2020 3c2d 2d20 5220 2020 3c2d 2d20  (R  <-- R   <-- 
│ │ │ │ -00033b70: 5220 2020 3c2d 2d20 5220 2020 3c2d 2d20  R   <-- R   <-- 
│ │ │ │ -00033b80: 5220 2020 3c2d 2d20 5220 202c 207b 327d  R   <-- R  , {2}
│ │ │ │ -00033b90: 207c 2030 2030 2020 3020 2031 2030 2030   | 0 0  0  1 0 0
│ │ │ │ -00033ba0: 207c 2920 2020 2020 7c0a 7c20 2020 2020   |)     |.|     
│ │ │ │ -00033bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033bd0: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ -00033be0: 207c 2030 202d 3120 3020 2030 2030 2030   | 0 -1 0  0 0 0
│ │ │ │ -00033bf0: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ -00033c00: 2030 2020 2020 2020 3120 2020 2020 2020   0      1       
│ │ │ │ -00033c10: 3220 2020 2020 2020 3320 2020 2020 2020  2       3       
│ │ │ │ -00033c20: 3420 2020 2020 2020 3520 2020 207b 327d  4       5    {2}
│ │ │ │ -00033c30: 207c 2030 2030 2020 2d31 2030 2030 2030   | 0 0  -1 0 0 0
│ │ │ │ -00033c40: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ -00033c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033c70: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00033c80: 207c 2030 2030 2020 3020 2030 2030 2031   | 0 0  0  0 0 1
│ │ │ │ -00033c90: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ -00033ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033cc0: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00033cd0: 207c 2030 2030 2020 3020 2030 2031 2030   | 0 0  0  0 1 0
│ │ │ │ -00033ce0: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ -00033cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d10: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ -00033d20: 207c 2031 2030 2020 3020 2030 2030 2030   | 1 0  0  0 0 0
│ │ │ │ -00033d30: 207c 2020 2020 2020 7c0a 7c20 2020 2020   |      |.|     
│ │ │ │ +00033ae0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033af0: 2036 2020 2020 2020 3130 2020 2020 2020   6      10      
│ │ │ │ +00033b00: 3135 2020 2020 2020 3231 2020 2020 2020  15      21      
│ │ │ │ +00033b10: 3238 2020 2020 2020 3336 2020 2020 2020  28      36      
│ │ │ │ +00033b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033b30: 2020 2020 2020 207c 0a7c 6f35 203d 2028         |.|o5 = (
│ │ │ │ +00033b40: 5220 203c 2d2d 2052 2020 203c 2d2d 2052  R  <-- R   <-- R
│ │ │ │ +00033b50: 2020 203c 2d2d 2052 2020 203c 2d2d 2052     <-- R   <-- R
│ │ │ │ +00033b60: 2020 203c 2d2d 2052 2020 2c20 7b32 7d20     <-- R  , {2} 
│ │ │ │ +00033b70: 7c20 3020 3020 2030 2020 3120 3020 3020  | 0 0  0  1 0 0 
│ │ │ │ +00033b80: 7c29 2020 2020 207c 0a7c 2020 2020 2020  |)     |.|      
│ │ │ │ +00033b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033bb0: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ +00033bc0: 7c20 3020 2d31 2030 2020 3020 3020 3020  | 0 -1 0  0 0 0 
│ │ │ │ +00033bd0: 7c20 2020 2020 207c 0a7c 2020 2020 2020  |      |.|      
│ │ │ │ +00033be0: 3020 2020 2020 2031 2020 2020 2020 2032  0      1       2
│ │ │ │ +00033bf0: 2020 2020 2020 2033 2020 2020 2020 2034         3       4
│ │ │ │ +00033c00: 2020 2020 2020 2035 2020 2020 7b32 7d20         5    {2} 
│ │ │ │ +00033c10: 7c20 3020 3020 202d 3120 3020 3020 3020  | 0 0  -1 0 0 0 
│ │ │ │ +00033c20: 7c20 2020 2020 207c 0a7c 2020 2020 2020  |      |.|      
│ │ │ │ +00033c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033c50: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
│ │ │ │ +00033c60: 7c20 3020 3020 2030 2020 3020 3020 3120  | 0 0  0  0 0 1 
│ │ │ │ +00033c70: 7c20 2020 2020 207c 0a7c 2020 2020 2020  |      |.|      
│ │ │ │ +00033c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ca0: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
│ │ │ │ +00033cb0: 7c20 3020 3020 2030 2020 3020 3120 3020  | 0 0  0  0 1 0 
│ │ │ │ +00033cc0: 7c20 2020 2020 207c 0a7c 2020 2020 2020  |      |.|      
│ │ │ │ +00033cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033cf0: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
│ │ │ │ +00033d00: 7c20 3120 3020 2030 2020 3020 3020 3020  | 1 0  0  0 0 0 
│ │ │ │ +00033d10: 7c20 2020 2020 207c 0a7c 2020 2020 2020  |      |.|      
│ │ │ │ +00033d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033d80: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ -00033d90: 5365 7175 656e 6365 2020 2020 2020 2020  Sequence        
│ │ │ │ +00033d60: 2020 2020 2020 207c 0a7c 6f35 203a 2053         |.|o5 : S
│ │ │ │ +00033d70: 6571 7565 6e63 6520 2020 2020 2020 2020  equence         
│ │ │ │ +00033d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033dd0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00033db0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00033dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033e20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00033e30: 6265 7474 6920 4646 2020 2020 2020 2020  betti FF        
│ │ │ │ +00033e00: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2062  -------+.|i6 : b
│ │ │ │ +00033e10: 6574 7469 2046 4620 2020 2020 2020 2020  etti FF         
│ │ │ │ +00033e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033e50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00033e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033e70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00033e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00033e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ec0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033ed0: 2020 2020 2020 2030 2020 3120 2032 2020         0  1  2  
│ │ │ │ -00033ee0: 3320 2034 2020 3520 2020 2020 2020 2020  3  4  5         
│ │ │ │ -00033ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f10: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ -00033f20: 746f 7461 6c3a 2036 2031 3020 3135 2032  total: 6 10 15 2
│ │ │ │ -00033f30: 3120 3238 2033 3620 2020 2020 2020 2020  1 28 36         
│ │ │ │ -00033f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033f60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033f70: 2020 2020 323a 2033 2020 3120 202e 2020      2: 3  1  .  
│ │ │ │ -00033f80: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ -00033f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033fb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00033fc0: 2020 2020 333a 2033 2020 3920 2039 2020      3: 3  9  9  
│ │ │ │ -00033fd0: 3320 202e 2020 2e20 2020 2020 2020 2020  3  .  .         
│ │ │ │ -00033fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00033ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034000: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034010: 2020 2020 343a 202e 2020 2e20 2036 2031      4: .  .  6 1
│ │ │ │ -00034020: 3820 3138 2020 3620 2020 2020 2020 2020  8 18  6         
│ │ │ │ -00034030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034050: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034060: 2020 2020 353a 202e 2020 2e20 202e 2020      5: .  .  .  
│ │ │ │ -00034070: 2e20 3130 2033 3020 2020 2020 2020 2020  . 10 30         
│ │ │ │ -00034080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ea0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033eb0: 2020 2020 2020 3020 2031 2020 3220 2033        0  1  2  3
│ │ │ │ +00033ec0: 2020 3420 2035 2020 2020 2020 2020 2020    4  5          
│ │ │ │ +00033ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033ef0: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ +00033f00: 6f74 616c 3a20 3620 3130 2031 3520 3231  otal: 6 10 15 21
│ │ │ │ +00033f10: 2032 3820 3336 2020 2020 2020 2020 2020   28 36          
│ │ │ │ +00033f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033f40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033f50: 2020 2032 3a20 3320 2031 2020 2e20 202e     2: 3  1  .  .
│ │ │ │ +00033f60: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +00033f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033f90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033fa0: 2020 2033 3a20 3320 2039 2020 3920 2033     3: 3  9  9  3
│ │ │ │ +00033fb0: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +00033fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00033fe0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00033ff0: 2020 2034 3a20 2e20 202e 2020 3620 3138     4: .  .  6 18
│ │ │ │ +00034000: 2031 3820 2036 2020 2020 2020 2020 2020   18  6          
│ │ │ │ +00034010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00034040: 2020 2035 3a20 2e20 202e 2020 2e20 202e     5: .  .  .  .
│ │ │ │ +00034050: 2031 3020 3330 2020 2020 2020 2020 2020   10 30          
│ │ │ │ +00034060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034080: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000340a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000340b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000340c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340f0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ -00034100: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +000340d0: 2020 2020 2020 207c 0a7c 6f36 203a 2042         |.|o6 : B
│ │ │ │ +000340e0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +000340f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034140: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00034120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00034130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ -000341a0: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ -000341b0: 7469 6f6e 284d 2c20 4c65 6e67 7468 4c69  tion(M, LengthLi
│ │ │ │ -000341c0: 6d69 743d 3e35 2920 2020 2020 2020 2020  mit=>5)         
│ │ │ │ +00034170: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2062  -------+.|i7 : b
│ │ │ │ +00034180: 6574 7469 2066 7265 6552 6573 6f6c 7574  etti freeResolut
│ │ │ │ +00034190: 696f 6e28 4d2c 204c 656e 6774 684c 696d  ion(M, LengthLim
│ │ │ │ +000341a0: 6974 3d3e 3529 2020 2020 2020 2020 2020  it=>5)          
│ │ │ │ +000341b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000341c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000341d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000341e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000341e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000341f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034230: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034240: 2020 2020 2020 2030 2020 3120 2032 2020         0  1  2  
│ │ │ │ -00034250: 3320 2034 2020 3520 2020 2020 2020 2020  3  4  5         
│ │ │ │ -00034260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034280: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ -00034290: 746f 7461 6c3a 2036 2031 3020 3135 2032  total: 6 10 15 2
│ │ │ │ -000342a0: 3120 3238 2033 3620 2020 2020 2020 2020  1 28 36         
│ │ │ │ -000342b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000342e0: 2020 2020 323a 2033 2020 3120 202e 2020      2: 3  1  .  
│ │ │ │ -000342f0: 2e20 202e 2020 2e20 2020 2020 2020 2020  .  .  .         
│ │ │ │ -00034300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034320: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034330: 2020 2020 333a 2033 2020 3920 2039 2020      3: 3  9  9  
│ │ │ │ -00034340: 3320 202e 2020 2e20 2020 2020 2020 2020  3  .  .         
│ │ │ │ -00034350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034370: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034380: 2020 2020 343a 202e 2020 2e20 2036 2031      4: .  .  6 1
│ │ │ │ -00034390: 3820 3138 2020 3620 2020 2020 2020 2020  8 18  6         
│ │ │ │ -000343a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000343b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000343c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000343d0: 2020 2020 353a 202e 2020 2e20 202e 2020      5: .  .  .  
│ │ │ │ -000343e0: 2e20 3130 2033 3020 2020 2020 2020 2020  . 10 30         
│ │ │ │ -000343f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034210: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00034220: 2020 2020 2020 3020 2031 2020 3220 2033        0  1  2  3
│ │ │ │ +00034230: 2020 3420 2035 2020 2020 2020 2020 2020    4  5          
│ │ │ │ +00034240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034260: 2020 2020 2020 207c 0a7c 6f37 203d 2074         |.|o7 = t
│ │ │ │ +00034270: 6f74 616c 3a20 3620 3130 2031 3520 3231  otal: 6 10 15 21
│ │ │ │ +00034280: 2032 3820 3336 2020 2020 2020 2020 2020   28 36          
│ │ │ │ +00034290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000342a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000342b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000342c0: 2020 2032 3a20 3320 2031 2020 2e20 202e     2: 3  1  .  .
│ │ │ │ +000342d0: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +000342e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000342f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034300: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00034310: 2020 2033 3a20 3320 2039 2020 3920 2033     3: 3  9  9  3
│ │ │ │ +00034320: 2020 2e20 202e 2020 2020 2020 2020 2020    .  .          
│ │ │ │ +00034330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034350: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00034360: 2020 2034 3a20 2e20 202e 2020 3620 3138     4: .  .  6 18
│ │ │ │ +00034370: 2031 3820 2036 2020 2020 2020 2020 2020   18  6          
│ │ │ │ +00034380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000343a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000343b0: 2020 2035 3a20 2e20 202e 2020 2e20 202e     5: .  .  .  .
│ │ │ │ +000343c0: 2031 3020 3330 2020 2020 2020 2020 2020   10 30          
│ │ │ │ +000343d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000343e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000343f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034410: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034460: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ -00034470: 4265 7474 6954 616c 6c79 2020 2020 2020  BettiTally      
│ │ │ │ +00034440: 2020 2020 2020 207c 0a7c 6f37 203a 2042         |.|o7 : B
│ │ │ │ +00034450: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00034460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000344a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000344b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00034490: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000344a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000344b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000344c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000344d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000344e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000344f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -00034510: 4320 3d20 636f 6d70 6c65 7820 666c 6174  C = complex flat
│ │ │ │ -00034520: 7465 6e20 7b7b 6175 677d 207c 6170 706c  ten {{aug} |appl
│ │ │ │ -00034530: 7928 342c 2069 2d3e 2046 462e 6464 5f28  y(4, i-> FF.dd_(
│ │ │ │ -00034540: 692b 3129 297d 2020 2020 2020 2020 2020  i+1))}          
│ │ │ │ -00034550: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000344e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2043  -------+.|i8 : C
│ │ │ │ +000344f0: 203d 2063 6f6d 706c 6578 2066 6c61 7474   = complex flatt
│ │ │ │ +00034500: 656e 207b 7b61 7567 7d20 7c61 7070 6c79  en {{aug} |apply
│ │ │ │ +00034510: 2834 2c20 692d 3e20 4646 2e64 645f 2869  (4, i-> FF.dd_(i
│ │ │ │ +00034520: 2b31 2929 7d20 2020 2020 2020 2020 2020  +1))}           
│ │ │ │ +00034530: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00034540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000345b0: 2020 2020 2020 2036 2020 2020 2020 3130         6      10
│ │ │ │ -000345c0: 2020 2020 2020 3135 2020 2020 2020 3231        15      21
│ │ │ │ -000345d0: 2020 2020 2020 3238 2020 2020 2020 2020        28        
│ │ │ │ -000345e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345f0: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ -00034600: 4d20 3c2d 2d20 5220 203c 2d2d 2052 2020  M <-- R  <-- R  
│ │ │ │ -00034610: 203c 2d2d 2052 2020 203c 2d2d 2052 2020   <-- R   <-- R  
│ │ │ │ -00034620: 203c 2d2d 2052 2020 2020 2020 2020 2020   <-- R          
│ │ │ │ +00034580: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00034590: 2020 2020 2020 3620 2020 2020 2031 3020        6      10 
│ │ │ │ +000345a0: 2020 2020 2031 3520 2020 2020 2032 3120       15      21 
│ │ │ │ +000345b0: 2020 2020 2032 3820 2020 2020 2020 2020       28         
│ │ │ │ +000345c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000345d0: 2020 2020 2020 207c 0a7c 6f38 203d 204d         |.|o8 = M
│ │ │ │ +000345e0: 203c 2d2d 2052 2020 3c2d 2d20 5220 2020   <-- R  <-- R   
│ │ │ │ +000345f0: 3c2d 2d20 5220 2020 3c2d 2d20 5220 2020  <-- R   <-- R   
│ │ │ │ +00034600: 3c2d 2d20 5220 2020 2020 2020 2020 2020  <-- R           
│ │ │ │ +00034610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034620: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034640: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034690: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000346a0: 3020 2020 2020 3120 2020 2020 2032 2020  0     1      2  
│ │ │ │ -000346b0: 2020 2020 2033 2020 2020 2020 2034 2020       3       4  
│ │ │ │ -000346c0: 2020 2020 2035 2020 2020 2020 2020 2020       5          
│ │ │ │ +00034670: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00034680: 2020 2020 2031 2020 2020 2020 3220 2020       1      2   
│ │ │ │ +00034690: 2020 2020 3320 2020 2020 2020 3420 2020      3       4   
│ │ │ │ +000346a0: 2020 2020 3520 2020 2020 2020 2020 2020      5           
│ │ │ │ +000346b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000346c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000346d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000346e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000346e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000346f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034730: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ -00034740: 436f 6d70 6c65 7820 2020 2020 2020 2020  Complex         
│ │ │ │ +00034710: 2020 2020 2020 207c 0a7c 6f38 203a 2043         |.|o8 : C
│ │ │ │ +00034720: 6f6d 706c 6578 2020 2020 2020 2020 2020  omplex          
│ │ │ │ +00034730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00034760: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00034770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000347a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000347b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000347c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000347d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20  --------+.|i9 : 
│ │ │ │ -000347e0: 6170 706c 7928 342c 2069 202d 3e46 462e  apply(4, i ->FF.
│ │ │ │ -000347f0: 6464 5f28 692b 3129 2920 2020 2020 2020  dd_(i+1))       
│ │ │ │ -00034800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000347b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2061  -------+.|i9 : a
│ │ │ │ +000347c0: 7070 6c79 2834 2c20 6920 2d3e 4646 2e64  pply(4, i ->FF.d
│ │ │ │ +000347d0: 645f 2869 2b31 2929 2020 2020 2020 2020  d_(i+1))        
│ │ │ │ +000347e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000347f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034800: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034820: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034870: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -00034880: 7b7b 337d 207c 2063 202d 6220 6120 2030  {{3} | c -b a  0
│ │ │ │ -00034890: 2020 3020 3020 2030 2020 3020 3020 2030    0 0  0  0 0  0
│ │ │ │ -000348a0: 2020 7c2c 207b 347d 207c 2063 3220 3020    |, {4} | c2 0 
│ │ │ │ -000348b0: 2d61 2030 2062 2030 2030 2020 3020 2020  -a 0 b 0 0  0   
│ │ │ │ -000348c0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -000348d0: 207b 337d 207c 2030 2030 2020 3020 2063   {3} | 0 0  0  c
│ │ │ │ -000348e0: 2020 6220 6120 2030 2020 3020 3020 2030    b a  0  0 0  0
│ │ │ │ -000348f0: 2020 7c20 207b 347d 207c 2030 2020 3020    |  {4} | 0  0 
│ │ │ │ -00034900: 3020 2030 2063 2030 2030 2020 3020 2020  0  0 c 0 0  0   
│ │ │ │ -00034910: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00034920: 207b 337d 207c 2030 2030 2020 6332 2030   {3} | 0 0  c2 0
│ │ │ │ -00034930: 2020 3020 3020 2030 2020 6120 2d63 202d    0 0  0  a -c -
│ │ │ │ -00034940: 6220 7c20 207b 347d 207c 2030 2020 3020  b |  {4} | 0  0 
│ │ │ │ -00034950: 6320 2030 2030 2030 2030 2020 3020 2020  c  0 0 0 0  0   
│ │ │ │ -00034960: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00034970: 207b 327d 207c 2030 2030 2020 3020 2030   {2} | 0 0  0  0
│ │ │ │ -00034980: 2020 3020 3020 2062 2020 3020 6132 2030    0 0  b  0 a2 0
│ │ │ │ -00034990: 2020 7c20 207b 347d 207c 2030 2020 3020    |  {4} | 0  0 
│ │ │ │ -000349a0: 3020 2030 2030 2030 2030 2020 3020 2020  0  0 0 0 0  0   
│ │ │ │ -000349b0: 2d61 2030 2020 6220 7c0a 7c20 2020 2020  -a 0  b |.|     
│ │ │ │ -000349c0: 207b 327d 207c 2030 2063 3220 3020 2030   {2} | 0 c2 0  0
│ │ │ │ -000349d0: 2020 3020 6232 202d 6320 3020 3020 2061    0 b2 -c 0 0  a
│ │ │ │ -000349e0: 3220 7c20 207b 347d 207c 2030 2020 3020  2 |  {4} | 0  0 
│ │ │ │ -000349f0: 3020 2030 2030 2030 2062 3220 3020 2020  0  0 0 0 b2 0   
│ │ │ │ -00034a00: 3020 202d 6120 2d63 7c0a 7c20 2020 2020  0  -a -c|.|     
│ │ │ │ -00034a10: 207b 327d 207c 2030 2030 2020 3020 2062   {2} | 0 0  0  b
│ │ │ │ -00034a20: 3220 3020 3020 2061 2020 3020 3020 2030  2 0 0  a  0 0  0
│ │ │ │ -00034a30: 2020 7c20 207b 347d 207c 2030 2020 3020    |  {4} | 0  0 
│ │ │ │ -00034a40: 3020 2030 2030 2030 2030 2020 6132 2020  0  0 0 0 0  a2  
│ │ │ │ -00034a50: 6320 2062 2020 3020 7c0a 7c20 2020 2020  c  b  0 |.|     
│ │ │ │ -00034a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034a80: 2020 2020 207b 337d 207c 2030 2020 3020       {3} | 0  0 
│ │ │ │ -00034a90: 3020 2030 2030 2030 2030 2020 3020 2020  0  0 0 0 0  0   
│ │ │ │ -00034aa0: 6232 2030 2020 3020 7c0a 7c20 2020 2020  b2 0  0 |.|     
│ │ │ │ -00034ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ad0: 2020 2020 207b 347d 207c 2030 2020 3020       {4} | 0  0 
│ │ │ │ -00034ae0: 3020 2030 2030 2030 2030 2020 3020 2020  0  0 0 0 0  0   
│ │ │ │ -00034af0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00034b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b20: 2020 2020 207b 347d 207c 2030 2020 3020       {4} | 0  0 
│ │ │ │ -00034b30: 3020 2030 2030 2030 2030 2020 3020 2020  0  0 0 0 0  0   
│ │ │ │ -00034b40: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00034b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034b70: 2020 2020 207b 347d 207c 2030 2020 3020       {4} | 0  0 
│ │ │ │ -00034b80: 3020 2030 2030 2030 2030 2020 2d62 3220  0  0 0 0 0  -b2 
│ │ │ │ -00034b90: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ +00034850: 2020 2020 2020 207c 0a7c 6f39 203d 207b         |.|o9 = {
│ │ │ │ +00034860: 7b33 7d20 7c20 6320 2d62 2061 2020 3020  {3} | c -b a  0 
│ │ │ │ +00034870: 2030 2030 2020 3020 2030 2030 2020 3020   0 0  0  0 0  0 
│ │ │ │ +00034880: 207c 2c20 7b34 7d20 7c20 6332 2030 202d   |, {4} | c2 0 -
│ │ │ │ +00034890: 6120 3020 6220 3020 3020 2030 2020 2030  a 0 b 0 0  0   0
│ │ │ │ +000348a0: 2020 3020 2030 207c 0a7c 2020 2020 2020    0  0 |.|      
│ │ │ │ +000348b0: 7b33 7d20 7c20 3020 3020 2030 2020 6320  {3} | 0 0  0  c 
│ │ │ │ +000348c0: 2062 2061 2020 3020 2030 2030 2020 3020   b a  0  0 0  0 
│ │ │ │ +000348d0: 207c 2020 7b34 7d20 7c20 3020 2030 2030   |  {4} | 0  0 0
│ │ │ │ +000348e0: 2020 3020 6320 3020 3020 2030 2020 2030    0 c 0 0  0   0
│ │ │ │ +000348f0: 2020 3020 2030 207c 0a7c 2020 2020 2020    0  0 |.|      
│ │ │ │ +00034900: 7b33 7d20 7c20 3020 3020 2063 3220 3020  {3} | 0 0  c2 0 
│ │ │ │ +00034910: 2030 2030 2020 3020 2061 202d 6320 2d62   0 0  0  a -c -b
│ │ │ │ +00034920: 207c 2020 7b34 7d20 7c20 3020 2030 2063   |  {4} | 0  0 c
│ │ │ │ +00034930: 2020 3020 3020 3020 3020 2030 2020 2030    0 0 0 0  0   0
│ │ │ │ +00034940: 2020 3020 2030 207c 0a7c 2020 2020 2020    0  0 |.|      
│ │ │ │ +00034950: 7b32 7d20 7c20 3020 3020 2030 2020 3020  {2} | 0 0  0  0 
│ │ │ │ +00034960: 2030 2030 2020 6220 2030 2061 3220 3020   0 0  b  0 a2 0 
│ │ │ │ +00034970: 207c 2020 7b34 7d20 7c20 3020 2030 2030   |  {4} | 0  0 0
│ │ │ │ +00034980: 2020 3020 3020 3020 3020 2030 2020 202d    0 0 0 0  0   -
│ │ │ │ +00034990: 6120 3020 2062 207c 0a7c 2020 2020 2020  a 0  b |.|      
│ │ │ │ +000349a0: 7b32 7d20 7c20 3020 6332 2030 2020 3020  {2} | 0 c2 0  0 
│ │ │ │ +000349b0: 2030 2062 3220 2d63 2030 2030 2020 6132   0 b2 -c 0 0  a2
│ │ │ │ +000349c0: 207c 2020 7b34 7d20 7c20 3020 2030 2030   |  {4} | 0  0 0
│ │ │ │ +000349d0: 2020 3020 3020 3020 6232 2030 2020 2030    0 0 0 b2 0   0
│ │ │ │ +000349e0: 2020 2d61 202d 637c 0a7c 2020 2020 2020    -a -c|.|      
│ │ │ │ +000349f0: 7b32 7d20 7c20 3020 3020 2030 2020 6232  {2} | 0 0  0  b2
│ │ │ │ +00034a00: 2030 2030 2020 6120 2030 2030 2020 3020   0 0  a  0 0  0 
│ │ │ │ +00034a10: 207c 2020 7b34 7d20 7c20 3020 2030 2030   |  {4} | 0  0 0
│ │ │ │ +00034a20: 2020 3020 3020 3020 3020 2061 3220 2063    0 0 0 0  a2  c
│ │ │ │ +00034a30: 2020 6220 2030 207c 0a7c 2020 2020 2020    b  0 |.|      
│ │ │ │ +00034a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034a60: 2020 2020 7b33 7d20 7c20 3020 2030 2030      {3} | 0  0 0
│ │ │ │ +00034a70: 2020 3020 3020 3020 3020 2030 2020 2062    0 0 0 0  0   b
│ │ │ │ +00034a80: 3220 3020 2030 207c 0a7c 2020 2020 2020  2 0  0 |.|      
│ │ │ │ +00034a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034ab0: 2020 2020 7b34 7d20 7c20 3020 2030 2030      {4} | 0  0 0
│ │ │ │ +00034ac0: 2020 3020 3020 3020 3020 2030 2020 2030    0 0 0 0  0   0
│ │ │ │ +00034ad0: 2020 3020 2030 207c 0a7c 2020 2020 2020    0  0 |.|      
│ │ │ │ +00034ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034b00: 2020 2020 7b34 7d20 7c20 3020 2030 2030      {4} | 0  0 0
│ │ │ │ +00034b10: 2020 3020 3020 3020 3020 2030 2020 2030    0 0 0 0  0   0
│ │ │ │ +00034b20: 2020 3020 2030 207c 0a7c 2020 2020 2020    0  0 |.|      
│ │ │ │ +00034b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034b50: 2020 2020 7b34 7d20 7c20 3020 2030 2030      {4} | 0  0 0
│ │ │ │ +00034b60: 2020 3020 3020 3020 3020 202d 6232 2030    0 0 0 0  -b2 0
│ │ │ │ +00034b70: 2020 3020 2030 207c 0a7c 2020 2020 2020    0  0 |.|      
│ │ │ │ +00034b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034bc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034be0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034c10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034c60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034cb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034cd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034d00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034d50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034da0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034dc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034df0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034e40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034e90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00034ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034eb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034f00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034ee0: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ +00034ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034f50: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00034f60: 3020 2030 2030 2030 2020 7c2c 207b 367d  0  0 0 0  |, {6}
│ │ │ │ -00034f70: 207c 2063 202d 6220 6120 2030 2020 3020   | c -b a  0  0 
│ │ │ │ -00034f80: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00034f90: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00034fa0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00034fb0: 3020 2030 2030 2030 2020 7c20 207b 367d  0  0 0 0  |  {6}
│ │ │ │ -00034fc0: 207c 2030 2030 2020 3020 2063 2020 6220   | 0 0  0  c  b 
│ │ │ │ -00034fd0: 6120 2030 2020 3020 3020 2030 2020 3020  a  0  0 0  0  0 
│ │ │ │ -00034fe0: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00034ff0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035000: 3020 2030 2030 2030 2020 7c20 207b 367d  0  0 0 0  |  {6}
│ │ │ │ -00035010: 207c 2030 2030 2020 6332 2030 2020 3020   | 0 0  c2 0  0 
│ │ │ │ -00035020: 3020 2030 2020 6120 2d63 202d 6220 3020  0  0  a -c -b 0 
│ │ │ │ -00035030: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035040: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035050: 3020 2030 2030 2030 2020 7c20 207b 357d  0  0 0 0  |  {5}
│ │ │ │ -00035060: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035070: 3020 2062 2020 3020 6132 2030 2020 3020  0  b  0 a2 0  0 
│ │ │ │ -00035080: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035090: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -000350a0: 3020 2030 2030 2030 2020 7c20 207b 357d  0  0 0 0  |  {5}
│ │ │ │ -000350b0: 207c 2030 2063 3220 3020 2030 2020 3020   | 0 c2 0  0  0 
│ │ │ │ -000350c0: 6232 202d 6320 3020 3020 2061 3220 3020  b2 -c 0 0  a2 0 
│ │ │ │ -000350d0: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -000350e0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -000350f0: 3020 2030 2030 2030 2020 7c20 207b 357d  0  0 0 0  |  {5}
│ │ │ │ -00035100: 207c 2030 2030 2020 3020 2062 3220 3020   | 0 0  0  b2 0 
│ │ │ │ -00035110: 3020 2061 2020 3020 3020 2030 2020 3020  0  a  0 0  0  0 
│ │ │ │ -00035120: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035130: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035140: 3020 2030 2030 2061 3220 7c20 207b 367d  0  0 0 a2 |  {6}
│ │ │ │ -00035150: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035160: 3020 2030 2020 3020 3020 2030 2020 6320  0  0  0 0  0  c 
│ │ │ │ -00035170: 2062 2061 2020 3020 2030 2020 3020 2020   b a  0  0  0   
│ │ │ │ -00035180: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035190: 6132 2063 2062 2030 2020 7c20 207b 367d  a2 c b 0  |  {6}
│ │ │ │ -000351a0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -000351b0: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -000351c0: 2030 2030 2020 3020 2061 2020 2d63 2020   0 0  0  a  -c  
│ │ │ │ -000351d0: 2d62 2030 2020 3020 7c0a 7c20 2020 2020  -b 0  0 |.|     
│ │ │ │ -000351e0: 3020 2061 2030 202d 6220 7c20 207b 357d  0  a 0 -b |  {5}
│ │ │ │ -000351f0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035200: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00035210: 2030 2030 2020 6220 2030 2020 6132 2020   0 0  b  0  a2  
│ │ │ │ -00035220: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -00035230: 3020 2030 2061 2063 2020 7c20 207b 357d  0  0 a c  |  {5}
│ │ │ │ -00035240: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035250: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00035260: 2030 2062 3220 2d63 2030 2020 3020 2020   0 b2 -c 0  0   
│ │ │ │ -00035270: 6132 2030 2020 3020 7c0a 7c20 2020 2020  a2 0  0 |.|     
│ │ │ │ -00035280: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -00035290: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -000352a0: 3020 2030 2020 3020 3020 2030 2020 6232  0  0  0 0  0  b2
│ │ │ │ -000352b0: 2030 2030 2020 6120 2030 2020 3020 2020   0 0  a  0  0   
│ │ │ │ -000352c0: 3020 2030 2020 3020 7c0a 7c20 2020 2020  0  0  0 |.|     
│ │ │ │ -000352d0: 2020 2020 2020 2020 2020 2020 207b 367d               {6}
│ │ │ │ -000352e0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -000352f0: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00035300: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035310: 3020 2030 2020 6120 7c0a 7c20 2020 2020  0  0  a |.|     
│ │ │ │ -00035320: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -00035330: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035340: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -00035350: 2030 2030 2020 3020 2030 2020 3020 2020   0 0  0  0  0   
│ │ │ │ -00035360: 3020 2062 2020 3020 7c0a 7c20 2020 2020  0  b  0 |.|     
│ │ │ │ -00035370: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -00035380: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00035390: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -000353a0: 2030 2030 2020 3020 2062 3220 3020 2020   0 0  0  b2 0   
│ │ │ │ -000353b0: 3020 202d 6320 3020 7c0a 7c20 2020 2020  0  -c 0 |.|     
│ │ │ │ -000353c0: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -000353d0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -000353e0: 3020 2030 2020 3020 3020 2030 2020 3020  0  0  0 0  0  0 
│ │ │ │ -000353f0: 2030 2030 2020 3020 2030 2020 2d62 3220   0 0  0  0  -b2 
│ │ │ │ -00035400: 3020 2061 2020 3020 7c0a 7c20 2020 2020  0  a  0 |.|     
│ │ │ │ +00034f30: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030  -------|.|     0
│ │ │ │ +00034f40: 2020 3020 3020 3020 207c 2c20 7b36 7d20    0 0 0  |, {6} 
│ │ │ │ +00034f50: 7c20 6320 2d62 2061 2020 3020 2030 2030  | c -b a  0  0 0
│ │ │ │ +00034f60: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +00034f70: 3020 3020 2030 2020 3020 2030 2020 2030  0 0  0  0  0   0
│ │ │ │ +00034f80: 2020 3020 2030 207c 0a7c 2020 2020 2030    0  0 |.|     0
│ │ │ │ +00034f90: 2020 3020 3020 3020 207c 2020 7b36 7d20    0 0 0  |  {6} 
│ │ │ │ +00034fa0: 7c20 3020 3020 2030 2020 6320 2062 2061  | 0 0  0  c  b a
│ │ │ │ +00034fb0: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +00034fc0: 3020 3020 2030 2020 3020 2030 2020 2030  0 0  0  0  0   0
│ │ │ │ +00034fd0: 2020 3020 2030 207c 0a7c 2020 2020 2030    0  0 |.|     0
│ │ │ │ +00034fe0: 2020 3020 3020 3020 207c 2020 7b36 7d20    0 0 0  |  {6} 
│ │ │ │ +00034ff0: 7c20 3020 3020 2063 3220 3020 2030 2030  | 0 0  c2 0  0 0
│ │ │ │ +00035000: 2020 3020 2061 202d 6320 2d62 2030 2020    0  a -c -b 0  
│ │ │ │ +00035010: 3020 3020 2030 2020 3020 2030 2020 2030  0 0  0  0  0   0
│ │ │ │ +00035020: 2020 3020 2030 207c 0a7c 2020 2020 2030    0  0 |.|     0
│ │ │ │ +00035030: 2020 3020 3020 3020 207c 2020 7b35 7d20    0 0 0  |  {5} 
│ │ │ │ +00035040: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +00035050: 2020 6220 2030 2061 3220 3020 2030 2020    b  0 a2 0  0  
│ │ │ │ +00035060: 3020 3020 2030 2020 3020 2030 2020 2030  0 0  0  0  0   0
│ │ │ │ +00035070: 2020 3020 2030 207c 0a7c 2020 2020 2030    0  0 |.|     0
│ │ │ │ +00035080: 2020 3020 3020 3020 207c 2020 7b35 7d20    0 0 0  |  {5} 
│ │ │ │ +00035090: 7c20 3020 6332 2030 2020 3020 2030 2062  | 0 c2 0  0  0 b
│ │ │ │ +000350a0: 3220 2d63 2030 2030 2020 6132 2030 2020  2 -c 0 0  a2 0  
│ │ │ │ +000350b0: 3020 3020 2030 2020 3020 2030 2020 2030  0 0  0  0  0   0
│ │ │ │ +000350c0: 2020 3020 2030 207c 0a7c 2020 2020 2030    0  0 |.|     0
│ │ │ │ +000350d0: 2020 3020 3020 3020 207c 2020 7b35 7d20    0 0 0  |  {5} 
│ │ │ │ +000350e0: 7c20 3020 3020 2030 2020 6232 2030 2030  | 0 0  0  b2 0 0
│ │ │ │ +000350f0: 2020 6120 2030 2030 2020 3020 2030 2020    a  0 0  0  0  
│ │ │ │ +00035100: 3020 3020 2030 2020 3020 2030 2020 2030  0 0  0  0  0   0
│ │ │ │ +00035110: 2020 3020 2030 207c 0a7c 2020 2020 2030    0  0 |.|     0
│ │ │ │ +00035120: 2020 3020 3020 6132 207c 2020 7b36 7d20    0 0 a2 |  {6} 
│ │ │ │ +00035130: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +00035140: 2020 3020 2030 2030 2020 3020 2063 2020    0  0 0  0  c  
│ │ │ │ +00035150: 6220 6120 2030 2020 3020 2030 2020 2030  b a  0  0  0   0
│ │ │ │ +00035160: 2020 3020 2030 207c 0a7c 2020 2020 2061    0  0 |.|     a
│ │ │ │ +00035170: 3220 6320 6220 3020 207c 2020 7b36 7d20  2 c b 0  |  {6} 
│ │ │ │ +00035180: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +00035190: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +000351a0: 3020 3020 2030 2020 6120 202d 6320 202d  0 0  0  a  -c  -
│ │ │ │ +000351b0: 6220 3020 2030 207c 0a7c 2020 2020 2030  b 0  0 |.|     0
│ │ │ │ +000351c0: 2020 6120 3020 2d62 207c 2020 7b35 7d20    a 0 -b |  {5} 
│ │ │ │ +000351d0: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +000351e0: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +000351f0: 3020 3020 2062 2020 3020 2061 3220 2030  0 0  b  0  a2  0
│ │ │ │ +00035200: 2020 3020 2030 207c 0a7c 2020 2020 2030    0  0 |.|     0
│ │ │ │ +00035210: 2020 3020 6120 6320 207c 2020 7b35 7d20    0 a c  |  {5} 
│ │ │ │ +00035220: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +00035230: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +00035240: 3020 6232 202d 6320 3020 2030 2020 2061  0 b2 -c 0  0   a
│ │ │ │ +00035250: 3220 3020 2030 207c 0a7c 2020 2020 2020  2 0  0 |.|      
│ │ │ │ +00035260: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ +00035270: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +00035280: 2020 3020 2030 2030 2020 3020 2062 3220    0  0 0  0  b2 
│ │ │ │ +00035290: 3020 3020 2061 2020 3020 2030 2020 2030  0 0  a  0  0   0
│ │ │ │ +000352a0: 2020 3020 2030 207c 0a7c 2020 2020 2020    0  0 |.|      
│ │ │ │ +000352b0: 2020 2020 2020 2020 2020 2020 7b36 7d20              {6} 
│ │ │ │ +000352c0: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +000352d0: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +000352e0: 3020 3020 2030 2020 3020 2030 2020 2030  0 0  0  0  0   0
│ │ │ │ +000352f0: 2020 3020 2061 207c 0a7c 2020 2020 2020    0  a |.|      
│ │ │ │ +00035300: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ +00035310: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +00035320: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +00035330: 3020 3020 2030 2020 3020 2030 2020 2030  0 0  0  0  0   0
│ │ │ │ +00035340: 2020 6220 2030 207c 0a7c 2020 2020 2020    b  0 |.|      
│ │ │ │ +00035350: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ +00035360: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +00035370: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +00035380: 3020 3020 2030 2020 6232 2030 2020 2030  0 0  0  b2 0   0
│ │ │ │ +00035390: 2020 2d63 2030 207c 0a7c 2020 2020 2020    -c 0 |.|      
│ │ │ │ +000353a0: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ +000353b0: 7c20 3020 3020 2030 2020 3020 2030 2030  | 0 0  0  0  0 0
│ │ │ │ +000353c0: 2020 3020 2030 2030 2020 3020 2030 2020    0  0 0  0  0  
│ │ │ │ +000353d0: 3020 3020 2030 2020 3020 202d 6232 2030  0 0  0  0  -b2 0
│ │ │ │ +000353e0: 2020 6120 2030 207c 0a7c 2020 2020 2020    a  0 |.|      
│ │ │ │ +000353f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035430: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00035440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035450: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00035450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035480: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00035490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000354a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000354a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000354d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000354d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000354e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000354f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000354f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035520: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00035530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00035540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035570: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00035580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035590: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00035590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000355c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000355d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000355e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000355c0: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ +000355d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000355e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000355f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035630: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00035640: 3020 2030 2020 7c2c 207b 377d 207c 2063  0  0  |, {7} | c
│ │ │ │ -00035650: 3220 3020 2d61 2030 2062 2030 2030 2020  2 0 -a 0 b 0 0  
│ │ │ │ -00035660: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035670: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035680: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035690: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -000356a0: 2020 3020 3020 2030 2063 2030 2030 2020    0 0  0 c 0 0  
│ │ │ │ -000356b0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -000356c0: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -000356d0: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -000356e0: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -000356f0: 2020 3020 6320 2030 2030 2030 2030 2020    0 c  0 0 0 0  
│ │ │ │ -00035700: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035710: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035720: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035730: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -00035740: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035750: 3020 2020 2d61 2030 2020 6220 2030 2020  0   -a 0  b  0  
│ │ │ │ -00035760: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035770: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035780: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -00035790: 2020 3020 3020 2030 2030 2030 2062 3220    0 0  0 0 0 b2 
│ │ │ │ -000357a0: 3020 2020 3020 202d 6120 2d63 2030 2020  0   0  -a -c 0  
│ │ │ │ -000357b0: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -000357c0: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -000357d0: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -000357e0: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -000357f0: 6132 2020 6320 2062 2020 3020 2030 2020  a2  c  b  0  0  
│ │ │ │ -00035800: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035810: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035820: 3020 2030 2020 7c20 207b 367d 207c 2030  0  0  |  {6} | 0
│ │ │ │ -00035830: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035840: 3020 2020 6232 2030 2020 3020 2030 2020  0   b2 0  0  0  
│ │ │ │ -00035850: 3020 3020 6132 2030 2020 3020 2020 3020  0 0 a2 0  0   0 
│ │ │ │ -00035860: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035870: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -00035880: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035890: 3020 2020 3020 2030 2020 3020 2061 3220  0   0  0  0  a2 
│ │ │ │ -000358a0: 6320 6220 3020 2030 2020 3020 2020 3020  c b 0  0  0   0 
│ │ │ │ -000358b0: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -000358c0: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -000358d0: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -000358e0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -000358f0: 6120 3020 2d62 2030 2020 3020 2020 3020  a 0 -b 0  0   0 
│ │ │ │ -00035900: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035910: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -00035920: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035930: 2d62 3220 3020 2030 2020 3020 2030 2020  -b2 0  0  0  0  
│ │ │ │ -00035940: 3020 6120 6320 2030 2020 3020 2020 3020  0 a c  0  0   0 
│ │ │ │ -00035950: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035960: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -00035970: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035980: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035990: 3020 3020 3020 2030 2020 3020 2020 2d61  0 0 0  0  0   -a
│ │ │ │ -000359a0: 2030 2020 6220 2020 7c0a 7c20 2020 2020   0  b   |.|     
│ │ │ │ -000359b0: 2d63 202d 6220 7c20 207b 377d 207c 2030  -c -b |  {7} | 0
│ │ │ │ -000359c0: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -000359d0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -000359e0: 3020 3020 3020 2062 3220 3020 2020 3020  0 0 0  b2 0   0 
│ │ │ │ -000359f0: 202d 6120 2d63 2020 7c0a 7c20 2020 2020   -a -c  |.|     
│ │ │ │ -00035a00: 6132 2030 2020 7c20 207b 377d 207c 2030  a2 0  |  {7} | 0
│ │ │ │ -00035a10: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035a20: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035a30: 3020 3020 3020 2030 2020 6132 2020 6320  0 0 0  0  a2  c 
│ │ │ │ -00035a40: 2062 2020 3020 2020 7c0a 7c20 2020 2020   b  0   |.|     
│ │ │ │ -00035a50: 3020 2061 3220 7c20 207b 367d 207c 2030  0  a2 |  {6} | 0
│ │ │ │ -00035a60: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035a70: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035a80: 3020 3020 3020 2030 2020 3020 2020 6232  0 0 0  0  0   b2
│ │ │ │ -00035a90: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035aa0: 3020 2030 2020 7c20 207b 377d 207c 2030  0  0  |  {7} | 0
│ │ │ │ -00035ab0: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035ac0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035ad0: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035ae0: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035af0: 2020 2020 2020 2020 207b 377d 207c 2030           {7} | 0
│ │ │ │ -00035b00: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035b10: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035b20: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035b30: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035b40: 2020 2020 2020 2020 207b 377d 207c 2030           {7} | 0
│ │ │ │ -00035b50: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035b60: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035b70: 3020 3020 3020 2030 2020 2d62 3220 3020  0 0 0  0  -b2 0 
│ │ │ │ -00035b80: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035b90: 2020 2020 2020 2020 207b 367d 207c 2030           {6} | 0
│ │ │ │ -00035ba0: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035bb0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035bc0: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035bd0: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035be0: 2020 2020 2020 2020 207b 377d 207c 2030           {7} | 0
│ │ │ │ -00035bf0: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035c00: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035c10: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035c20: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035c30: 2020 2020 2020 2020 207b 377d 207c 2030           {7} | 0
│ │ │ │ -00035c40: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035c50: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035c60: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035c70: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ -00035c80: 2020 2020 2020 2020 207b 377d 207c 2030           {7} | 0
│ │ │ │ -00035c90: 2020 3020 3020 2030 2030 2030 2030 2020    0 0  0 0 0 0  
│ │ │ │ -00035ca0: 3020 2020 3020 2030 2020 3020 2030 2020  0   0  0  0  0  
│ │ │ │ -00035cb0: 3020 3020 3020 2030 2020 3020 2020 3020  0 0 0  0  0   0 
│ │ │ │ -00035cc0: 2030 2020 3020 2020 7c0a 7c20 2020 2020   0  0   |.|     
│ │ │ │ +00035610: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030  -------|.|     0
│ │ │ │ +00035620: 2020 3020 207c 2c20 7b37 7d20 7c20 6332    0  |, {7} | c2
│ │ │ │ +00035630: 2030 202d 6120 3020 6220 3020 3020 2030   0 -a 0 b 0 0  0
│ │ │ │ +00035640: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035650: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035660: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +00035670: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +00035680: 2030 2030 2020 3020 6320 3020 3020 2030   0 0  0 c 0 0  0
│ │ │ │ +00035690: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +000356a0: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +000356b0: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +000356c0: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +000356d0: 2030 2063 2020 3020 3020 3020 3020 2030   0 c  0 0 0 0  0
│ │ │ │ +000356e0: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +000356f0: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035700: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +00035710: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +00035720: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035730: 2020 202d 6120 3020 2062 2020 3020 2030     -a 0  b  0  0
│ │ │ │ +00035740: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035750: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +00035760: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +00035770: 2030 2030 2020 3020 3020 3020 6232 2030   0 0  0 0 0 b2 0
│ │ │ │ +00035780: 2020 2030 2020 2d61 202d 6320 3020 2030     0  -a -c 0  0
│ │ │ │ +00035790: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +000357a0: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +000357b0: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +000357c0: 2030 2030 2020 3020 3020 3020 3020 2061   0 0  0 0 0 0  a
│ │ │ │ +000357d0: 3220 2063 2020 6220 2030 2020 3020 2030  2  c  b  0  0  0
│ │ │ │ +000357e0: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +000357f0: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +00035800: 2020 3020 207c 2020 7b36 7d20 7c20 3020    0  |  {6} | 0 
│ │ │ │ +00035810: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035820: 2020 2062 3220 3020 2030 2020 3020 2030     b2 0  0  0  0
│ │ │ │ +00035830: 2030 2061 3220 3020 2030 2020 2030 2020   0 a2 0  0   0  
│ │ │ │ +00035840: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +00035850: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +00035860: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035870: 2020 2030 2020 3020 2030 2020 6132 2063     0  0  0  a2 c
│ │ │ │ +00035880: 2062 2030 2020 3020 2030 2020 2030 2020   b 0  0  0   0  
│ │ │ │ +00035890: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +000358a0: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +000358b0: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +000358c0: 2020 2030 2020 3020 2030 2020 3020 2061     0  0  0  0  a
│ │ │ │ +000358d0: 2030 202d 6220 3020 2030 2020 2030 2020   0 -b 0  0   0  
│ │ │ │ +000358e0: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +000358f0: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +00035900: 2030 2030 2020 3020 3020 3020 3020 202d   0 0  0 0 0 0  -
│ │ │ │ +00035910: 6232 2030 2020 3020 2030 2020 3020 2030  b2 0  0  0  0  0
│ │ │ │ +00035920: 2061 2063 2020 3020 2030 2020 2030 2020   a c  0  0   0  
│ │ │ │ +00035930: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +00035940: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +00035950: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035960: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035970: 2030 2030 2020 3020 2030 2020 202d 6120   0 0  0  0   -a 
│ │ │ │ +00035980: 3020 2062 2020 207c 0a7c 2020 2020 202d  0  b   |.|     -
│ │ │ │ +00035990: 6320 2d62 207c 2020 7b37 7d20 7c20 3020  c -b |  {7} | 0 
│ │ │ │ +000359a0: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +000359b0: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +000359c0: 2030 2030 2020 6232 2030 2020 2030 2020   0 0  b2 0   0  
│ │ │ │ +000359d0: 2d61 202d 6320 207c 0a7c 2020 2020 2061  -a -c  |.|     a
│ │ │ │ +000359e0: 3220 3020 207c 2020 7b37 7d20 7c20 3020  2 0  |  {7} | 0 
│ │ │ │ +000359f0: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035a00: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035a10: 2030 2030 2020 3020 2061 3220 2063 2020   0 0  0  a2  c  
│ │ │ │ +00035a20: 6220 2030 2020 207c 0a7c 2020 2020 2030  b  0   |.|     0
│ │ │ │ +00035a30: 2020 6132 207c 2020 7b36 7d20 7c20 3020    a2 |  {6} | 0 
│ │ │ │ +00035a40: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035a50: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035a60: 2030 2030 2020 3020 2030 2020 2062 3220   0 0  0  0   b2 
│ │ │ │ +00035a70: 3020 2030 2020 207c 0a7c 2020 2020 2030  0  0   |.|     0
│ │ │ │ +00035a80: 2020 3020 207c 2020 7b37 7d20 7c20 3020    0  |  {7} | 0 
│ │ │ │ +00035a90: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035aa0: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035ab0: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035ac0: 3020 2030 2020 207c 0a7c 2020 2020 2020  0  0   |.|      
│ │ │ │ +00035ad0: 2020 2020 2020 2020 7b37 7d20 7c20 3020          {7} | 0 
│ │ │ │ +00035ae0: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035af0: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035b00: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035b10: 3020 2030 2020 207c 0a7c 2020 2020 2020  0  0   |.|      
│ │ │ │ +00035b20: 2020 2020 2020 2020 7b37 7d20 7c20 3020          {7} | 0 
│ │ │ │ +00035b30: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035b40: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035b50: 2030 2030 2020 3020 202d 6232 2030 2020   0 0  0  -b2 0  
│ │ │ │ +00035b60: 3020 2030 2020 207c 0a7c 2020 2020 2020  0  0   |.|      
│ │ │ │ +00035b70: 2020 2020 2020 2020 7b36 7d20 7c20 3020          {6} | 0 
│ │ │ │ +00035b80: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035b90: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035ba0: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035bb0: 3020 2030 2020 207c 0a7c 2020 2020 2020  0  0   |.|      
│ │ │ │ +00035bc0: 2020 2020 2020 2020 7b37 7d20 7c20 3020          {7} | 0 
│ │ │ │ +00035bd0: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035be0: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035bf0: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035c00: 3020 2030 2020 207c 0a7c 2020 2020 2020  0  0   |.|      
│ │ │ │ +00035c10: 2020 2020 2020 2020 7b37 7d20 7c20 3020          {7} | 0 
│ │ │ │ +00035c20: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035c30: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035c40: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035c50: 3020 2030 2020 207c 0a7c 2020 2020 2020  0  0   |.|      
│ │ │ │ +00035c60: 2020 2020 2020 2020 7b37 7d20 7c20 3020          {7} | 0 
│ │ │ │ +00035c70: 2030 2030 2020 3020 3020 3020 3020 2030   0 0  0 0 0 0  0
│ │ │ │ +00035c80: 2020 2030 2020 3020 2030 2020 3020 2030     0  0  0  0  0
│ │ │ │ +00035c90: 2030 2030 2020 3020 2030 2020 2030 2020   0 0  0  0   0  
│ │ │ │ +00035ca0: 3020 2030 2020 207c 0a7c 2020 2020 202d  0  0   |.|     -
│ │ │ │ +00035cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00035cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035d10: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00035d20: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035d30: 2030 2030 2020 7c7d 2020 2020 2020 2020   0 0  |}        
│ │ │ │ -00035d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035d60: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035d70: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035d80: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00035d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035db0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035dc0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035dd0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00035de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035e10: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035e20: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00035e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e50: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035e60: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035e70: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00035e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ea0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035eb0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035ec0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00035ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035f00: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035f10: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00035f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035f50: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035f60: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00035f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035fa0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00035fb0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00035fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035fe0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00035ff0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00036000: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00036010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036030: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036040: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -00036050: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00036060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036080: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036090: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -000360a0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -000360b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000360d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000360e0: 3020 2020 3020 2030 2030 2020 3020 2030  0   0  0 0  0  0
│ │ │ │ -000360f0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00036100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036120: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036130: 3020 2020 3020 2030 2061 3220 3020 2030  0   0  0 a2 0  0
│ │ │ │ -00036140: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00036150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036170: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036180: 6132 2020 6320 2062 2030 2020 3020 2030  a2  c  b 0  0  0
│ │ │ │ -00036190: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -000361a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000361d0: 3020 2020 6120 2030 202d 6220 3020 2030  0   a  0 -b 0  0
│ │ │ │ -000361e0: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -000361f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036210: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036220: 3020 2020 3020 2061 2063 2020 3020 2030  0   0  a c  0  0
│ │ │ │ -00036230: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ -00036240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036260: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036270: 3020 2020 6232 2030 2030 2020 3020 2030  0   b2 0 0  0  0
│ │ │ │ -00036280: 2030 2061 3220 7c20 2020 2020 2020 2020   0 a2 |         
│ │ │ │ -00036290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000362a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000362b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000362c0: 3020 2020 3020 2030 2030 2020 6132 2063  0   0  0 0  a2 c
│ │ │ │ -000362d0: 2062 2030 2020 7c20 2020 2020 2020 2020   b 0  |         
│ │ │ │ -000362e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000362f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036310: 3020 2020 3020 2030 2030 2020 3020 2061  0   0  0 0  0  a
│ │ │ │ -00036320: 2030 202d 6220 7c20 2020 2020 2020 2020   0 -b |         
│ │ │ │ -00036330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036350: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00036360: 2d62 3220 3020 2030 2030 2020 3020 2030  -b2 0  0 0  0  0
│ │ │ │ -00036370: 2061 2063 2020 7c20 2020 2020 2020 2020   a c  |         
│ │ │ │ -00036380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035cf0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030  -------|.|     0
│ │ │ │ +00035d00: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035d10: 3020 3020 207c 7d20 2020 2020 2020 2020  0 0  |}         
│ │ │ │ +00035d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035d40: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035d50: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035d60: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035d90: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035da0: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035db0: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035de0: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035df0: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035e00: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035e30: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035e40: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035e50: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035e80: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035e90: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035ea0: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035ed0: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035ee0: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035ef0: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035f20: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035f30: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035f40: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035f70: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035f80: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035f90: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035fc0: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00035fd0: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00035fe0: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00035ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036010: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00036020: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00036030: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00036040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036060: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00036070: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +00036080: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00036090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000360a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000360b0: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +000360c0: 2020 2030 2020 3020 3020 2030 2020 3020     0  0 0  0  0 
│ │ │ │ +000360d0: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +000360e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000360f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036100: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00036110: 2020 2030 2020 3020 6132 2030 2020 3020     0  0 a2 0  0 
│ │ │ │ +00036120: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00036130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036150: 2020 2020 2020 207c 0a7c 2020 2020 2061         |.|     a
│ │ │ │ +00036160: 3220 2063 2020 6220 3020 2030 2020 3020  2  c  b 0  0  0 
│ │ │ │ +00036170: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00036180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000361a0: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +000361b0: 2020 2061 2020 3020 2d62 2030 2020 3020     a  0 -b 0  0 
│ │ │ │ +000361c0: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +000361d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000361e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000361f0: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00036200: 2020 2030 2020 6120 6320 2030 2020 3020     0  a c  0  0 
│ │ │ │ +00036210: 3020 3020 207c 2020 2020 2020 2020 2020  0 0  |          
│ │ │ │ +00036220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036240: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +00036250: 2020 2062 3220 3020 3020 2030 2020 3020     b2 0 0  0  0 
│ │ │ │ +00036260: 3020 6132 207c 2020 2020 2020 2020 2020  0 a2 |          
│ │ │ │ +00036270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036290: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +000362a0: 2020 2030 2020 3020 3020 2061 3220 6320     0  0 0  a2 c 
│ │ │ │ +000362b0: 6220 3020 207c 2020 2020 2020 2020 2020  b 0  |          
│ │ │ │ +000362c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000362d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000362e0: 2020 2020 2020 207c 0a7c 2020 2020 2030         |.|     0
│ │ │ │ +000362f0: 2020 2030 2020 3020 3020 2030 2020 6120     0  0 0  0  a 
│ │ │ │ +00036300: 3020 2d62 207c 2020 2020 2020 2020 2020  0 -b |          
│ │ │ │ +00036310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036330: 2020 2020 2020 207c 0a7c 2020 2020 202d         |.|     -
│ │ │ │ +00036340: 6232 2030 2020 3020 3020 2030 2020 3020  b2 0  0 0  0  0 
│ │ │ │ +00036350: 6120 6320 207c 2020 2020 2020 2020 2020  a c  |          
│ │ │ │ +00036360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036380: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00036390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000363a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000363b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000363c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000363f0: 2020 2020 2020 2020 7c0a 7c6f 3920 3a20          |.|o9 : 
│ │ │ │ -00036400: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +000363d0: 2020 2020 2020 207c 0a7c 6f39 203a 204c         |.|o9 : L
│ │ │ │ +000363e0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +000363f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036440: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00036420: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00036430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036490: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a  --------+.|i10 :
│ │ │ │ -000364a0: 2061 7070 6c79 2835 2c20 6a2d 3e20 7072   apply(5, j-> pr
│ │ │ │ -000364b0: 756e 6520 4848 5f6a 2043 203d 3d20 3029  une HH_j C == 0)
│ │ │ │ -000364c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036470: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ +00036480: 6170 706c 7928 352c 206a 2d3e 2070 7275  apply(5, j-> pru
│ │ │ │ +00036490: 6e65 2048 485f 6a20 4320 3d3d 2030 2920  ne HH_j C == 0) 
│ │ │ │ +000364a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000364b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000364c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000364d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000364e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000364e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000364f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036530: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ -00036540: 207b 7472 7565 2c20 6661 6c73 652c 2066   {true, false, f
│ │ │ │ -00036550: 616c 7365 2c20 6661 6c73 652c 2066 616c  alse, false, fal
│ │ │ │ -00036560: 7365 7d20 2020 2020 2020 2020 2020 2020  se}             
│ │ │ │ +00036510: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ +00036520: 7b74 7275 652c 2066 616c 7365 2c20 6661  {true, false, fa
│ │ │ │ +00036530: 6c73 652c 2066 616c 7365 2c20 6661 6c73  lse, false, fals
│ │ │ │ +00036540: 657d 2020 2020 2020 2020 2020 2020 2020  e}              
│ │ │ │ +00036550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036560: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00036570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036580: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000365a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365d0: 2020 2020 2020 2020 7c0a 7c6f 3130 203a          |.|o10 :
│ │ │ │ -000365e0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +000365b0: 2020 2020 2020 207c 0a7c 6f31 3020 3a20         |.|o10 : 
│ │ │ │ +000365c0: 4c69 7374 2020 2020 2020 2020 2020 2020  List            
│ │ │ │ +000365d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000365e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000365f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036620: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00036600: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00036610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036670: 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6e64 206f  --------+..And o
│ │ │ │ -00036680: 6e65 2063 6f6d 7075 7469 6e67 2074 6865  ne computing the
│ │ │ │ -00036690: 2077 686f 6c65 2066 696e 6974 6520 7265   whole finite re
│ │ │ │ -000366a0: 736f 6c75 7469 6f6e 3a0a 0a2b 2d2d 2d2d  solution:..+----
│ │ │ │ +00036650: 2d2d 2d2d 2d2d 2d2b 0a0a 416e 6420 6f6e  -------+..And on
│ │ │ │ +00036660: 6520 636f 6d70 7574 696e 6720 7468 6520  e computing the 
│ │ │ │ +00036670: 7768 6f6c 6520 6669 6e69 7465 2072 6573  whole finite res
│ │ │ │ +00036680: 6f6c 7574 696f 6e3a 0a0a 2b2d 2d2d 2d2d  olution:..+-----
│ │ │ │ +00036690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000366a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000366b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000366c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000366d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000366e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000366f0: 2d2d 2d2b 0a7c 6931 3120 3a20 4d53 203d  ---+.|i11 : MS =
│ │ │ │ -00036700: 2070 7573 6846 6f72 7761 7264 286d 6170   pushForward(map
│ │ │ │ -00036710: 2852 2c53 292c 204d 293b 2020 2020 2020  (R,S), M);      
│ │ │ │ -00036720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036730: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000366d0: 2d2d 2b0a 7c69 3131 203a 204d 5320 3d20  --+.|i11 : MS = 
│ │ │ │ +000366e0: 7075 7368 466f 7277 6172 6428 6d61 7028  pushForward(map(
│ │ │ │ +000366f0: 522c 5329 2c20 4d29 3b20 2020 2020 2020  R,S), M);       
│ │ │ │ +00036700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036710: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00036720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036780: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20  -------+.|i12 : 
│ │ │ │ -00036790: 2847 472c 2061 7567 2920 3d20 6c61 7965  (GG, aug) = laye
│ │ │ │ -000367a0: 7265 6452 6573 6f6c 7574 696f 6e28 6666  redResolution(ff
│ │ │ │ -000367b0: 2c4d 5329 2020 2020 2020 2020 2020 2020  ,MS)            
│ │ │ │ +00036760: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2028  ------+.|i12 : (
│ │ │ │ +00036770: 4747 2c20 6175 6729 203d 206c 6179 6572  GG, aug) = layer
│ │ │ │ +00036780: 6564 5265 736f 6c75 7469 6f6e 2866 662c  edResolution(ff,
│ │ │ │ +00036790: 4d53 2920 2020 2020 2020 2020 2020 2020  MS)             
│ │ │ │ +000367a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000367b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000367c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000367d0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000367d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000367e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000367f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036810: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00036820: 2020 2020 2020 3620 2020 2020 2031 3320        6      13 
│ │ │ │ -00036830: 2020 2020 2031 3020 2020 2020 2033 2020       10      3  
│ │ │ │ -00036840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036860: 2020 2020 207c 0a7c 6f31 3220 3d20 2853       |.|o12 = (S
│ │ │ │ -00036870: 2020 3c2d 2d20 5320 2020 3c2d 2d20 5320    <-- S   <-- S 
│ │ │ │ -00036880: 2020 3c2d 2d20 5320 2c20 7b32 7d20 7c20    <-- S , {2} | 
│ │ │ │ -00036890: 3020 3020 3020 3020 2030 2020 3120 7c29  0 0 0 0  0  1 |)
│ │ │ │ -000368a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000368b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000368c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000368d0: 2020 2020 7b32 7d20 7c20 3020 3020 3020      {2} | 0 0 0 
│ │ │ │ -000368e0: 2d31 2030 2020 3020 7c20 2020 2020 2020  -1 0  0 |       
│ │ │ │ -000368f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00036900: 2020 2030 2020 2020 2020 3120 2020 2020     0      1     
│ │ │ │ -00036910: 2020 3220 2020 2020 2020 3320 2020 7b32    2       3   {2
│ │ │ │ -00036920: 7d20 7c20 3020 3020 3020 3020 202d 3120  } | 0 0 0 0  -1 
│ │ │ │ -00036930: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ -00036940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00036950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036960: 2020 2020 2020 2020 7b33 7d20 7c20 3120          {3} | 1 
│ │ │ │ -00036970: 3020 3020 3020 2030 2020 3020 7c20 2020  0 0 0  0  0 |   
│ │ │ │ -00036980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00036990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369b0: 2020 7b33 7d20 7c20 3020 3120 3020 3020    {3} | 0 1 0 0 
│ │ │ │ -000369c0: 2030 2020 3020 7c20 2020 2020 2020 2020   0  0 |         
│ │ │ │ -000369d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000369e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000369f0: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
│ │ │ │ -00036a00: 7c20 3020 3020 3120 3020 2030 2020 3020  | 0 0 1 0  0  0 
│ │ │ │ -00036a10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00036a20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000367f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036800: 2020 2020 2036 2020 2020 2020 3133 2020       6      13  
│ │ │ │ +00036810: 2020 2020 3130 2020 2020 2020 3320 2020      10      3   
│ │ │ │ +00036820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036840: 2020 2020 7c0a 7c6f 3132 203d 2028 5320      |.|o12 = (S 
│ │ │ │ +00036850: 203c 2d2d 2053 2020 203c 2d2d 2053 2020   <-- S   <-- S  
│ │ │ │ +00036860: 203c 2d2d 2053 202c 207b 327d 207c 2030   <-- S , {2} | 0
│ │ │ │ +00036870: 2030 2030 2030 2020 3020 2031 207c 2920   0 0 0  0  1 |) 
│ │ │ │ +00036880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00036890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000368a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000368b0: 2020 207b 327d 207c 2030 2030 2030 202d     {2} | 0 0 0 -
│ │ │ │ +000368c0: 3120 3020 2030 207c 2020 2020 2020 2020  1 0  0 |        
│ │ │ │ +000368d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000368e0: 2020 3020 2020 2020 2031 2020 2020 2020    0      1      
│ │ │ │ +000368f0: 2032 2020 2020 2020 2033 2020 207b 327d   2       3   {2}
│ │ │ │ +00036900: 207c 2030 2030 2030 2030 2020 2d31 2030   | 0 0 0 0  -1 0
│ │ │ │ +00036910: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00036920: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00036930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036940: 2020 2020 2020 207b 337d 207c 2031 2030         {3} | 1 0
│ │ │ │ +00036950: 2030 2030 2020 3020 2030 207c 2020 2020   0 0  0  0 |    
│ │ │ │ +00036960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00036970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036990: 207b 337d 207c 2030 2031 2030 2030 2020   {3} | 0 1 0 0  
│ │ │ │ +000369a0: 3020 2030 207c 2020 2020 2020 2020 2020  0  0 |          
│ │ │ │ +000369b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000369c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000369d0: 2020 2020 2020 2020 2020 207b 337d 207c             {3} |
│ │ │ │ +000369e0: 2030 2030 2031 2030 2020 3020 2030 207c   0 0 1 0  0  0 |
│ │ │ │ +000369f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036a00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00036a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a60: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00036a70: 3220 3a20 5365 7175 656e 6365 2020 2020  2 : Sequence    
│ │ │ │ +00036a40: 2020 2020 2020 2020 2020 7c0a 7c6f 3132            |.|o12
│ │ │ │ +00036a50: 203a 2053 6571 7565 6e63 6520 2020 2020   : Sequence     
│ │ │ │ +00036a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ab0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00036a90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00036aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00036b00: 0a7c 6931 3320 3a20 2847 472c 2061 7567  .|i13 : (GG, aug
│ │ │ │ -00036b10: 2920 3d20 6c61 7965 7265 6452 6573 6f6c  ) = layeredResol
│ │ │ │ -00036b20: 7574 696f 6e28 6666 2c4d 532c 2056 6572  ution(ff,MS, Ver
│ │ │ │ -00036b30: 626f 7365 203d 3e74 7275 6529 2020 2020  bose =>true)    
│ │ │ │ -00036b40: 2020 2020 2020 2020 207c 0a7c 7b33 2c20           |.|{3, 
│ │ │ │ -00036b50: 317d 2069 6e20 636f 6469 6d65 6e73 696f  1} in codimensio
│ │ │ │ -00036b60: 6e20 3320 2020 2020 2020 2020 2020 2020  n 3             
│ │ │ │ -00036b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036b90: 2020 207c 0a7c 7b33 2c20 317d 2069 6e20     |.|{3, 1} in 
│ │ │ │ -00036ba0: 636f 6469 6d65 6e73 696f 6e20 3220 2020  codimension 2   
│ │ │ │ -00036bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00036ae0: 7c69 3133 203a 2028 4747 2c20 6175 6729  |i13 : (GG, aug)
│ │ │ │ +00036af0: 203d 206c 6179 6572 6564 5265 736f 6c75   = layeredResolu
│ │ │ │ +00036b00: 7469 6f6e 2866 662c 4d53 2c20 5665 7262  tion(ff,MS, Verb
│ │ │ │ +00036b10: 6f73 6520 3d3e 7472 7565 2920 2020 2020  ose =>true)     
│ │ │ │ +00036b20: 2020 2020 2020 2020 7c0a 7c7b 332c 2031          |.|{3, 1
│ │ │ │ +00036b30: 7d20 696e 2063 6f64 696d 656e 7369 6f6e  } in codimension
│ │ │ │ +00036b40: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00036b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036b70: 2020 7c0a 7c7b 332c 2031 7d20 696e 2063    |.|{3, 1} in c
│ │ │ │ +00036b80: 6f64 696d 656e 7369 6f6e 2032 2020 2020  odimension 2    
│ │ │ │ +00036b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00036bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00036c30: 2020 3620 2020 2020 2031 3320 2020 2020    6      13     
│ │ │ │ -00036c40: 2031 3020 2020 2020 2033 2020 2020 2020   10      3      
│ │ │ │ -00036c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036c70: 207c 0a7c 6f31 3320 3d20 2853 2020 3c2d   |.|o13 = (S  <-
│ │ │ │ -00036c80: 2d20 5320 2020 3c2d 2d20 5320 2020 3c2d  - S   <-- S   <-
│ │ │ │ -00036c90: 2d20 5320 2c20 7b32 7d20 7c20 3020 3020  - S , {2} | 0 0 
│ │ │ │ -00036ca0: 3020 3020 2030 2020 3120 7c29 2020 2020  0 0  0  1 |)    
│ │ │ │ -00036cb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00036cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ce0: 7b32 7d20 7c20 3020 3020 3020 2d31 2030  {2} | 0 0 0 -1 0
│ │ │ │ -00036cf0: 2020 3020 7c20 2020 2020 2020 2020 2020    0 |           
│ │ │ │ -00036d00: 2020 2020 207c 0a7c 2020 2020 2020 2030       |.|       0
│ │ │ │ -00036d10: 2020 2020 2020 3120 2020 2020 2020 3220        1       2 
│ │ │ │ -00036d20: 2020 2020 2020 3320 2020 7b32 7d20 7c20        3   {2} | 
│ │ │ │ -00036d30: 3020 3020 3020 3020 202d 3120 3020 7c20  0 0 0 0  -1 0 | 
│ │ │ │ -00036d40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00036d50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00036d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036d70: 2020 2020 7b33 7d20 7c20 3120 3020 3020      {3} | 1 0 0 
│ │ │ │ -00036d80: 3020 2030 2020 3020 7c20 2020 2020 2020  0  0  0 |       
│ │ │ │ -00036d90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00036da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036db0: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ -00036dc0: 7d20 7c20 3020 3120 3020 3020 2030 2020  } | 0 1 0 0  0  
│ │ │ │ -00036dd0: 3020 7c20 2020 2020 2020 2020 2020 2020  0 |             
│ │ │ │ -00036de0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00036df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e00: 2020 2020 2020 2020 7b33 7d20 7c20 3020          {3} | 0 
│ │ │ │ -00036e10: 3020 3120 3020 2030 2020 3020 7c20 2020  0 1 0  0  0 |   
│ │ │ │ -00036e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00036c00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00036c10: 2036 2020 2020 2020 3133 2020 2020 2020   6      13      
│ │ │ │ +00036c20: 3130 2020 2020 2020 3320 2020 2020 2020  10      3       
│ │ │ │ +00036c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036c50: 7c0a 7c6f 3133 203d 2028 5320 203c 2d2d  |.|o13 = (S  <--
│ │ │ │ +00036c60: 2053 2020 203c 2d2d 2053 2020 203c 2d2d   S   <-- S   <--
│ │ │ │ +00036c70: 2053 202c 207b 327d 207c 2030 2030 2030   S , {2} | 0 0 0
│ │ │ │ +00036c80: 2030 2020 3020 2031 207c 2920 2020 2020   0  0  1 |)     
│ │ │ │ +00036c90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00036ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036cb0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00036cc0: 327d 207c 2030 2030 2030 202d 3120 3020  2} | 0 0 0 -1 0 
│ │ │ │ +00036cd0: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +00036ce0: 2020 2020 7c0a 7c20 2020 2020 2020 3020      |.|       0 
│ │ │ │ +00036cf0: 2020 2020 2031 2020 2020 2020 2032 2020       1       2  
│ │ │ │ +00036d00: 2020 2020 2033 2020 207b 327d 207c 2030       3   {2} | 0
│ │ │ │ +00036d10: 2030 2030 2030 2020 2d31 2030 207c 2020   0 0 0  -1 0 |  
│ │ │ │ +00036d20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00036d30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00036d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036d50: 2020 207b 337d 207c 2031 2030 2030 2030     {3} | 1 0 0 0
│ │ │ │ +00036d60: 2020 3020 2030 207c 2020 2020 2020 2020    0  0 |        
│ │ │ │ +00036d70: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00036d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036d90: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ +00036da0: 207c 2030 2031 2030 2030 2020 3020 2030   | 0 1 0 0  0  0
│ │ │ │ +00036db0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00036dc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00036dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036de0: 2020 2020 2020 207b 337d 207c 2030 2030         {3} | 0 0
│ │ │ │ +00036df0: 2031 2030 2020 3020 2030 207c 2020 2020   1 0  0  0 |    
│ │ │ │ +00036e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00036e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036e70: 2020 2020 2020 207c 0a7c 6f31 3320 3a20         |.|o13 : 
│ │ │ │ -00036e80: 5365 7175 656e 6365 2020 2020 2020 2020  Sequence        
│ │ │ │ +00036e50: 2020 2020 2020 7c0a 7c6f 3133 203a 2053        |.|o13 : S
│ │ │ │ +00036e60: 6571 7565 6e63 6520 2020 2020 2020 2020  equence         
│ │ │ │ +00036e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036ec0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00036ea0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00036eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00036f10: 3420 3a20 6265 7474 6920 4747 2020 2020  4 : betti GG    
│ │ │ │ +00036ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134  ----------+.|i14
│ │ │ │ +00036ef0: 203a 2062 6574 7469 2047 4720 2020 2020   : betti GG     
│ │ │ │ +00036f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036f30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00036f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00036f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00036fa0: 0a7c 2020 2020 2020 2020 2020 2020 2030  .|             0
│ │ │ │ -00036fb0: 2020 3120 2032 2033 2020 2020 2020 2020    1  2 3        
│ │ │ │ -00036fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036fe0: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ -00036ff0: 3d20 746f 7461 6c3a 2036 2031 3320 3130  = total: 6 13 10
│ │ │ │ -00037000: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ -00037010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037030: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00037040: 323a 2033 2020 3120 202e 202e 2020 2020  2: 3  1  . .    
│ │ │ │ -00037050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00037080: 2020 2020 2020 2020 2020 333a 2033 2020            3: 3  
│ │ │ │ -00037090: 3920 202e 202e 2020 2020 2020 2020 2020  9  . .          
│ │ │ │ -000370a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000370b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000370c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000370d0: 2020 2020 343a 202e 2020 2e20 202e 202e      4: .  .  . .
│ │ │ │ +00036f70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00036f80: 7c20 2020 2020 2020 2020 2020 2020 3020  |             0 
│ │ │ │ +00036f90: 2031 2020 3220 3320 2020 2020 2020 2020   1  2 3         
│ │ │ │ +00036fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036fc0: 2020 2020 2020 2020 7c0a 7c6f 3134 203d          |.|o14 =
│ │ │ │ +00036fd0: 2074 6f74 616c 3a20 3620 3133 2031 3020   total: 6 13 10 
│ │ │ │ +00036fe0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00036ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037010: 2020 7c0a 7c20 2020 2020 2020 2020 2032    |.|          2
│ │ │ │ +00037020: 3a20 3320 2031 2020 2e20 2e20 2020 2020  : 3  1  . .     
│ │ │ │ +00037030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00037060: 2020 2020 2020 2020 2033 3a20 3320 2039           3: 3  9
│ │ │ │ +00037070: 2020 2e20 2e20 2020 2020 2020 2020 2020    . .           
│ │ │ │ +00037080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000370a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000370b0: 2020 2034 3a20 2e20 202e 2020 2e20 2e20     4: .  .  . . 
│ │ │ │ +000370c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000370d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000370e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000370f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037110: 207c 0a7c 2020 2020 2020 2020 2020 353a   |.|          5:
│ │ │ │ -00037120: 202e 2020 3320 2039 202e 2020 2020 2020   .  3  9 .      
│ │ │ │ -00037130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037150: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00037160: 2020 2020 2020 2020 363a 202e 2020 2e20          6: .  . 
│ │ │ │ -00037170: 202e 202e 2020 2020 2020 2020 2020 2020   . .            
│ │ │ │ -00037180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000371b0: 2020 373a 202e 2020 2e20 2031 2033 2020    7: .  .  1 3  
│ │ │ │ -000371c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000371f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000370f0: 7c0a 7c20 2020 2020 2020 2020 2035 3a20  |.|          5: 
│ │ │ │ +00037100: 2e20 2033 2020 3920 2e20 2020 2020 2020  .  3  9 .       
│ │ │ │ +00037110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037130: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00037140: 2020 2020 2020 2036 3a20 2e20 202e 2020         6: .  .  
│ │ │ │ +00037150: 2e20 2e20 2020 2020 2020 2020 2020 2020  . .             
│ │ │ │ +00037160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037180: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00037190: 2037 3a20 2e20 202e 2020 3120 3320 2020   7: .  .  1 3   
│ │ │ │ +000371a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000371b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000371c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000371d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000371e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000371f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037230: 2020 2020 2020 2020 207c 0a7c 6f31 3420           |.|o14 
│ │ │ │ -00037240: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ +00037210: 2020 2020 2020 2020 7c0a 7c6f 3134 203a          |.|o14 :
│ │ │ │ +00037220: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +00037230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037280: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00037260: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00037270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00037280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000372a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000372b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000372c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000372d0: 6931 3520 3a20 6265 7474 6920 6672 6565  i15 : betti free
│ │ │ │ -000372e0: 5265 736f 6c75 7469 6f6e 204d 5320 2020  Resolution MS   
│ │ │ │ -000372f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000372a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000372b0: 3135 203a 2062 6574 7469 2066 7265 6552  15 : betti freeR
│ │ │ │ +000372c0: 6573 6f6c 7574 696f 6e20 4d53 2020 2020  esolution MS    
│ │ │ │ +000372d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000372e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000372f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00037300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037310: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00037310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037360: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00037370: 2030 2020 3120 2032 2033 2020 2020 2020   0  1  2 3      
│ │ │ │ -00037380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000373a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -000373b0: 3520 3d20 746f 7461 6c3a 2036 2031 3320  5 = total: 6 13 
│ │ │ │ -000373c0: 3130 2033 2020 2020 2020 2020 2020 2020  10 3            
│ │ │ │ -000373d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000373e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000373f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00037400: 2020 323a 2033 2020 3120 202e 202e 2020    2: 3  1  . .  
│ │ │ │ -00037410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00037440: 0a7c 2020 2020 2020 2020 2020 333a 2033  .|          3: 3
│ │ │ │ -00037450: 2020 3920 202e 202e 2020 2020 2020 2020    9  . .        
│ │ │ │ -00037460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037480: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00037490: 2020 2020 2020 343a 202e 2020 2e20 202e        4: .  .  .
│ │ │ │ -000374a0: 202e 2020 2020 2020 2020 2020 2020 2020   .              
│ │ │ │ -000374b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -000374e0: 353a 202e 2020 3320 2039 202e 2020 2020  5: .  3  9 .    
│ │ │ │ -000374f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00037520: 2020 2020 2020 2020 2020 363a 202e 2020            6: .  
│ │ │ │ -00037530: 2e20 202e 202e 2020 2020 2020 2020 2020  .  . .          
│ │ │ │ -00037540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037560: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00037570: 2020 2020 373a 202e 2020 2e20 2031 2033      7: .  .  1 3
│ │ │ │ +00037340: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00037350: 3020 2031 2020 3220 3320 2020 2020 2020  0  1  2 3       
│ │ │ │ +00037360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037380: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ +00037390: 203d 2074 6f74 616c 3a20 3620 3133 2031   = total: 6 13 1
│ │ │ │ +000373a0: 3020 3320 2020 2020 2020 2020 2020 2020  0 3             
│ │ │ │ +000373b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000373c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000373d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000373e0: 2032 3a20 3320 2031 2020 2e20 2e20 2020   2: 3  1  . .   
│ │ │ │ +000373f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00037420: 7c20 2020 2020 2020 2020 2033 3a20 3320  |          3: 3 
│ │ │ │ +00037430: 2039 2020 2e20 2e20 2020 2020 2020 2020   9  . .         
│ │ │ │ +00037440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037460: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00037470: 2020 2020 2034 3a20 2e20 202e 2020 2e20       4: .  .  . 
│ │ │ │ +00037480: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +00037490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000374a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000374b0: 2020 7c0a 7c20 2020 2020 2020 2020 2035    |.|          5
│ │ │ │ +000374c0: 3a20 2e20 2033 2020 3920 2e20 2020 2020  : .  3  9 .     
│ │ │ │ +000374d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000374e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000374f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00037500: 2020 2020 2020 2020 2036 3a20 2e20 202e           6: .  .
│ │ │ │ +00037510: 2020 2e20 2e20 2020 2020 2020 2020 2020    . .           
│ │ │ │ +00037520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037540: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00037550: 2020 2037 3a20 2e20 202e 2020 3120 3320     7: .  .  1 3 
│ │ │ │ +00037560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037590: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000375a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000375b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375f0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00037600: 3520 3a20 4265 7474 6954 616c 6c79 2020  5 : BettiTally  
│ │ │ │ +000375d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3135            |.|o15
│ │ │ │ +000375e0: 203a 2042 6574 7469 5461 6c6c 7920 2020   : BettiTally   
│ │ │ │ +000375f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00037620: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00037630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00037640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00037690: 0a7c 6931 3620 3a20 4320 3d20 636f 6d70  .|i16 : C = comp
│ │ │ │ -000376a0: 6c65 7820 666c 6174 7465 6e20 7b7b 6175  lex flatten {{au
│ │ │ │ -000376b0: 677d 207c 6170 706c 7928 6c65 6e67 7468  g} |apply(length
│ │ │ │ -000376c0: 2047 4720 2d31 2c20 692d 3e20 4747 2e64   GG -1, i-> GG.d
│ │ │ │ -000376d0: 645f 2869 2b31 2929 7d7c 0a7c 2020 2020  d_(i+1))}|.|    
│ │ │ │ +00037660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00037670: 7c69 3136 203a 2043 203d 2063 6f6d 706c  |i16 : C = compl
│ │ │ │ +00037680: 6578 2066 6c61 7474 656e 207b 7b61 7567  ex flatten {{aug
│ │ │ │ +00037690: 7d20 7c61 7070 6c79 286c 656e 6774 6820  } |apply(length 
│ │ │ │ +000376a0: 4747 202d 312c 2069 2d3e 2047 472e 6464  GG -1, i-> GG.dd
│ │ │ │ +000376b0: 5f28 692b 3129 297d 7c0a 7c20 2020 2020  _(i+1))}|.|     
│ │ │ │ +000376c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000376d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037720: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00037730: 2020 2020 3620 2020 2020 2031 3320 2020      6      13   
│ │ │ │ -00037740: 2020 2031 3020 2020 2020 2020 2020 2020     10           
│ │ │ │ -00037750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00037770: 6f31 3620 3d20 4d53 203c 2d2d 2053 2020  o16 = MS <-- S  
│ │ │ │ -00037780: 3c2d 2d20 5320 2020 3c2d 2d20 5320 2020  <-- S   <-- S   
│ │ │ │ -00037790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037700: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00037710: 2020 2036 2020 2020 2020 3133 2020 2020     6      13    
│ │ │ │ +00037720: 2020 3130 2020 2020 2020 2020 2020 2020    10            
│ │ │ │ +00037730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037740: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00037750: 3136 203d 204d 5320 3c2d 2d20 5320 203c  16 = MS <-- S  <
│ │ │ │ +00037760: 2d2d 2053 2020 203c 2d2d 2053 2020 2020  -- S   <-- S    
│ │ │ │ +00037770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037790: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000377a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000377b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000377c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000377d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000377f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037800: 207c 0a7c 2020 2020 2020 3020 2020 2020   |.|      0     
│ │ │ │ -00037810: 2031 2020 2020 2020 3220 2020 2020 2020   1      2       
│ │ │ │ -00037820: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +000377e0: 7c0a 7c20 2020 2020 2030 2020 2020 2020  |.|      0      
│ │ │ │ +000377f0: 3120 2020 2020 2032 2020 2020 2020 2033  1      2       3
│ │ │ │ +00037800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037820: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00037830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037840: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00037840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037890: 2020 2020 207c 0a7c 6f31 3620 3a20 436f       |.|o16 : Co
│ │ │ │ -000378a0: 6d70 6c65 7820 2020 2020 2020 2020 2020  mplex           
│ │ │ │ -000378b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000378c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000378d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000378e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00037870: 2020 2020 7c0a 7c6f 3136 203a 2043 6f6d      |.|o16 : Com
│ │ │ │ +00037880: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │ +00037890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000378a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000378b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000378c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000378d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000378e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000378f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037920: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720  ---------+.|i17 
│ │ │ │ -00037930: 3a20 6170 706c 7928 6c65 6e67 7468 2047  : apply(length G
│ │ │ │ -00037940: 4720 2b31 202c 206a 2d3e 2070 7275 6e65  G +1 , j-> prune
│ │ │ │ -00037950: 2048 485f 6a20 4320 3d3d 2030 2920 2020   HH_j C == 0)   
│ │ │ │ +00037900: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a  --------+.|i17 :
│ │ │ │ +00037910: 2061 7070 6c79 286c 656e 6774 6820 4747   apply(length GG
│ │ │ │ +00037920: 202b 3120 2c20 6a2d 3e20 7072 756e 6520   +1 , j-> prune 
│ │ │ │ +00037930: 4848 5f6a 2043 203d 3d20 3029 2020 2020  HH_j C == 0)    
│ │ │ │ +00037940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037950: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00037960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037970: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00037970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000379c0: 6f31 3720 3d20 7b74 7275 652c 2074 7275  o17 = {true, tru
│ │ │ │ -000379d0: 652c 2074 7275 652c 2066 616c 7365 7d20  e, true, false} 
│ │ │ │ -000379e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037990: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000379a0: 3137 203d 207b 7472 7565 2c20 7472 7565  17 = {true, true
│ │ │ │ +000379b0: 2c20 7472 7565 2c20 6661 6c73 657d 2020  , true, false}  
│ │ │ │ +000379c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000379d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000379e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000379f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00037a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037a30: 7c0a 7c6f 3137 203a 204c 6973 7420 2020  |.|o17 : List   
│ │ │ │  00037a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a50: 207c 0a7c 6f31 3720 3a20 4c69 7374 2020   |.|o17 : List  
│ │ │ │ +00037a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00037a70: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00037a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00037a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037ae0: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20  -----+..Ways to 
│ │ │ │ -00037af0: 7573 6520 6c61 7965 7265 6452 6573 6f6c  use layeredResol
│ │ │ │ -00037b00: 7574 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d  ution:.=========
│ │ │ │ -00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00037b20: 3d3d 3d3d 3d0a 0a20 202a 2022 6c61 7965  =====..  * "laye
│ │ │ │ -00037b30: 7265 6452 6573 6f6c 7574 696f 6e28 4d61  redResolution(Ma
│ │ │ │ -00037b40: 7472 6978 2c4d 6f64 756c 6529 220a 2020  trix,Module)".  
│ │ │ │ -00037b50: 2a20 226c 6179 6572 6564 5265 736f 6c75  * "layeredResolu
│ │ │ │ -00037b60: 7469 6f6e 284d 6174 7269 782c 4d6f 6475  tion(Matrix,Modu
│ │ │ │ -00037b70: 6c65 2c5a 5a29 220a 0a46 6f72 2074 6865  le,ZZ)"..For the
│ │ │ │ -00037b80: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -00037b90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -00037ba0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -00037bb0: 206c 6179 6572 6564 5265 736f 6c75 7469   layeredResoluti
│ │ │ │ -00037bc0: 6f6e 3a20 6c61 7965 7265 6452 6573 6f6c  on: layeredResol
│ │ │ │ -00037bd0: 7574 696f 6e2c 2069 7320 6120 2a6e 6f74  ution, is a *not
│ │ │ │ -00037be0: 6520 6d65 7468 6f64 0a66 756e 6374 696f  e method.functio
│ │ │ │ -00037bf0: 6e20 7769 7468 206f 7074 696f 6e73 3a20  n with options: 
│ │ │ │ -00037c00: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -00037c10: 7468 6f64 4675 6e63 7469 6f6e 5769 7468  thodFunctionWith
│ │ │ │ -00037c20: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d  Options,...-----
│ │ │ │ +00037ac0: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075  ----+..Ways to u
│ │ │ │ +00037ad0: 7365 206c 6179 6572 6564 5265 736f 6c75  se layeredResolu
│ │ │ │ +00037ae0: 7469 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  tion:.==========
│ │ │ │ +00037af0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00037b00: 3d3d 3d3d 0a0a 2020 2a20 226c 6179 6572  ====..  * "layer
│ │ │ │ +00037b10: 6564 5265 736f 6c75 7469 6f6e 284d 6174  edResolution(Mat
│ │ │ │ +00037b20: 7269 782c 4d6f 6475 6c65 2922 0a20 202a  rix,Module)".  *
│ │ │ │ +00037b30: 2022 6c61 7965 7265 6452 6573 6f6c 7574   "layeredResolut
│ │ │ │ +00037b40: 696f 6e28 4d61 7472 6978 2c4d 6f64 756c  ion(Matrix,Modul
│ │ │ │ +00037b50: 652c 5a5a 2922 0a0a 466f 7220 7468 6520  e,ZZ)"..For the 
│ │ │ │ +00037b60: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00037b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00037b80: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +00037b90: 6c61 7965 7265 6452 6573 6f6c 7574 696f  layeredResolutio
│ │ │ │ +00037ba0: 6e3a 206c 6179 6572 6564 5265 736f 6c75  n: layeredResolu
│ │ │ │ +00037bb0: 7469 6f6e 2c20 6973 2061 202a 6e6f 7465  tion, is a *note
│ │ │ │ +00037bc0: 206d 6574 686f 640a 6675 6e63 7469 6f6e   method.function
│ │ │ │ +00037bd0: 2077 6974 6820 6f70 7469 6f6e 733a 2028   with options: (
│ │ │ │ +00037be0: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +00037bf0: 686f 6446 756e 6374 696f 6e57 6974 684f  hodFunctionWithO
│ │ │ │ +00037c00: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d  ptions,...------
│ │ │ │ +00037c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00037c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037c70: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ -00037c80: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ -00037c90: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ -00037ca0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -00037cb0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -00037cc0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ -00037cd0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -00037ce0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ -00037cf0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00037d00: 6e73 2e6d 323a 3438 3935 3a30 2e0a 1f0a  ns.m2:4895:0....
│ │ │ │ -00037d10: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ -00037d20: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -00037d30: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ -00037d40: 3a20 4c69 6674 2c20 4e65 7874 3a20 6d61  : Lift, Next: ma
│ │ │ │ -00037d50: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ -00037d60: 6f6e 2c20 5072 6576 3a20 6c61 7965 7265  on, Prev: layere
│ │ │ │ -00037d70: 6452 6573 6f6c 7574 696f 6e2c 2055 703a  dResolution, Up:
│ │ │ │ -00037d80: 2054 6f70 0a0a 4c69 6674 202d 2d20 4f70   Top..Lift -- Op
│ │ │ │ -00037d90: 7469 6f6e 2066 6f72 206e 6577 4578 740a  tion for newExt.
│ │ │ │ -00037da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00037db0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -00037dc0: 7361 6765 3a20 0a20 2020 2020 2020 206e  sage: .        n
│ │ │ │ -00037dd0: 6577 4578 7428 4d2c 4e2c 4368 6563 6b20  ewExt(M,N,Check 
│ │ │ │ -00037de0: 3d3e 7472 7565 290a 2020 2a20 496e 7075  =>true).  * Inpu
│ │ │ │ -00037df0: 7473 3a0a 2020 2020 2020 2a20 4368 6563  ts:.      * Chec
│ │ │ │ -00037e00: 6b2c 2061 202a 6e6f 7465 2042 6f6f 6c65  k, a *note Boole
│ │ │ │ -00037e10: 616e 2076 616c 7565 3a20 284d 6163 6175  an value: (Macau
│ │ │ │ -00037e20: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c  lay2Doc)Boolean,
│ │ │ │ -00037e30: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a  , ..Description.
│ │ │ │ -00037e40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4d 616b  ===========..Mak
│ │ │ │ -00037e50: 6573 206e 6577 4578 7420 7065 7266 6f72  es newExt perfor
│ │ │ │ -00037e60: 6d20 7661 7269 6f75 7320 6368 6563 6b73  m various checks
│ │ │ │ -00037e70: 2061 7320 6974 2063 6f6d 7075 7465 732e   as it computes.
│ │ │ │ -00037e80: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00037e90: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206e  ===..  * *note n
│ │ │ │ -00037ea0: 6577 4578 743a 206e 6577 4578 742c 202d  ewExt: newExt, -
│ │ │ │ -00037eb0: 2d20 476c 6f62 616c 2045 7874 2066 6f72  - Global Ext for
│ │ │ │ -00037ec0: 206d 6f64 756c 6573 206f 7665 7220 6120   modules over a 
│ │ │ │ -00037ed0: 636f 6d70 6c65 7465 0a20 2020 2049 6e74  complete.    Int
│ │ │ │ -00037ee0: 6572 7365 6374 696f 6e0a 0a46 756e 6374  ersection..Funct
│ │ │ │ -00037ef0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ -00037f00: 616c 2061 7267 756d 656e 7420 6e61 6d65  al argument name
│ │ │ │ -00037f10: 6420 4c69 6674 3a0a 3d3d 3d3d 3d3d 3d3d  d Lift:.========
│ │ │ │ -00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00037f30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00037f40: 3d3d 3d3d 0a0a 2020 2a20 226e 6577 4578  ====..  * "newEx
│ │ │ │ -00037f50: 7428 2e2e 2e2c 4c69 6674 3d3e 2e2e 2e29  t(...,Lift=>...)
│ │ │ │ -00037f60: 2220 2d2d 2073 6565 202a 6e6f 7465 206e  " -- see *note n
│ │ │ │ -00037f70: 6577 4578 743a 206e 6577 4578 742c 202d  ewExt: newExt, -
│ │ │ │ -00037f80: 2d20 476c 6f62 616c 2045 7874 2066 6f72  - Global Ext for
│ │ │ │ -00037f90: 0a20 2020 206d 6f64 756c 6573 206f 7665  .    modules ove
│ │ │ │ -00037fa0: 7220 6120 636f 6d70 6c65 7465 2049 6e74  r a complete Int
│ │ │ │ -00037fb0: 6572 7365 6374 696f 6e0a 0a46 6f72 2074  ersection..For t
│ │ │ │ -00037fc0: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ -00037fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00037fe0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -00037ff0: 7465 204c 6966 743a 204c 6966 742c 2069  te Lift: Lift, i
│ │ │ │ -00038000: 7320 6120 2a6e 6f74 6520 7379 6d62 6f6c  s a *note symbol
│ │ │ │ -00038010: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00038020: 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d  Symbol,...------
│ │ │ │ +00037c50: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00037c60: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00037c70: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +00037c80: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +00037c90: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +00037ca0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ +00037cb0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +00037cc0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +00037cd0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00037ce0: 732e 6d32 3a34 3839 353a 302e 0a1f 0a46  s.m2:4895:0....F
│ │ │ │ +00037cf0: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +00037d00: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00037d10: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +00037d20: 204c 6966 742c 204e 6578 743a 206d 616b   Lift, Next: mak
│ │ │ │ +00037d30: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ +00037d40: 6e2c 2050 7265 763a 206c 6179 6572 6564  n, Prev: layered
│ │ │ │ +00037d50: 5265 736f 6c75 7469 6f6e 2c20 5570 3a20  Resolution, Up: 
│ │ │ │ +00037d60: 546f 700a 0a4c 6966 7420 2d2d 204f 7074  Top..Lift -- Opt
│ │ │ │ +00037d70: 696f 6e20 666f 7220 6e65 7745 7874 0a2a  ion for newExt.*
│ │ │ │ +00037d80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00037d90: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +00037da0: 6167 653a 200a 2020 2020 2020 2020 6e65  age: .        ne
│ │ │ │ +00037db0: 7745 7874 284d 2c4e 2c43 6865 636b 203d  wExt(M,N,Check =
│ │ │ │ +00037dc0: 3e74 7275 6529 0a20 202a 2049 6e70 7574  >true).  * Input
│ │ │ │ +00037dd0: 733a 0a20 2020 2020 202a 2043 6865 636b  s:.      * Check
│ │ │ │ +00037de0: 2c20 6120 2a6e 6f74 6520 426f 6f6c 6561  , a *note Boolea
│ │ │ │ +00037df0: 6e20 7661 6c75 653a 2028 4d61 6361 756c  n value: (Macaul
│ │ │ │ +00037e00: 6179 3244 6f63 2942 6f6f 6c65 616e 2c2c  ay2Doc)Boolean,,
│ │ │ │ +00037e10: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d   ..Description.=
│ │ │ │ +00037e20: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4d61 6b65  ==========..Make
│ │ │ │ +00037e30: 7320 6e65 7745 7874 2070 6572 666f 726d  s newExt perform
│ │ │ │ +00037e40: 2076 6172 696f 7573 2063 6865 636b 7320   various checks 
│ │ │ │ +00037e50: 6173 2069 7420 636f 6d70 7574 6573 2e0a  as it computes..
│ │ │ │ +00037e60: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00037e70: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6e65  ==..  * *note ne
│ │ │ │ +00037e80: 7745 7874 3a20 6e65 7745 7874 2c20 2d2d  wExt: newExt, --
│ │ │ │ +00037e90: 2047 6c6f 6261 6c20 4578 7420 666f 7220   Global Ext for 
│ │ │ │ +00037ea0: 6d6f 6475 6c65 7320 6f76 6572 2061 2063  modules over a c
│ │ │ │ +00037eb0: 6f6d 706c 6574 650a 2020 2020 496e 7465  omplete.    Inte
│ │ │ │ +00037ec0: 7273 6563 7469 6f6e 0a0a 4675 6e63 7469  rsection..Functi
│ │ │ │ +00037ed0: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ +00037ee0: 6c20 6172 6775 6d65 6e74 206e 616d 6564  l argument named
│ │ │ │ +00037ef0: 204c 6966 743a 0a3d 3d3d 3d3d 3d3d 3d3d   Lift:.=========
│ │ │ │ +00037f00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00037f10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00037f20: 3d3d 3d0a 0a20 202a 2022 6e65 7745 7874  ===..  * "newExt
│ │ │ │ +00037f30: 282e 2e2e 2c4c 6966 743d 3e2e 2e2e 2922  (...,Lift=>...)"
│ │ │ │ +00037f40: 202d 2d20 7365 6520 2a6e 6f74 6520 6e65   -- see *note ne
│ │ │ │ +00037f50: 7745 7874 3a20 6e65 7745 7874 2c20 2d2d  wExt: newExt, --
│ │ │ │ +00037f60: 2047 6c6f 6261 6c20 4578 7420 666f 720a   Global Ext for.
│ │ │ │ +00037f70: 2020 2020 6d6f 6475 6c65 7320 6f76 6572      modules over
│ │ │ │ +00037f80: 2061 2063 6f6d 706c 6574 6520 496e 7465   a complete Inte
│ │ │ │ +00037f90: 7273 6563 7469 6f6e 0a0a 466f 7220 7468  rsection..For th
│ │ │ │ +00037fa0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +00037fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +00037fc0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +00037fd0: 6520 4c69 6674 3a20 4c69 6674 2c20 6973  e Lift: Lift, is
│ │ │ │ +00037fe0: 2061 202a 6e6f 7465 2073 796d 626f 6c3a   a *note symbol:
│ │ │ │ +00037ff0: 2028 4d61 6361 756c 6179 3244 6f63 2953   (Macaulay2Doc)S
│ │ │ │ +00038000: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d  ymbol,...-------
│ │ │ │ +00038010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038070: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ -00038080: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ -00038090: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -000380a0: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -000380b0: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -000380c0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ -000380d0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -000380e0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ -000380f0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -00038100: 732e 6d32 3a33 3139 383a 302e 0a1f 0a46  s.m2:3198:0....F
│ │ │ │ -00038110: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -00038120: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -00038130: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -00038140: 206d 616b 6546 696e 6974 6552 6573 6f6c   makeFiniteResol
│ │ │ │ -00038150: 7574 696f 6e2c 204e 6578 743a 206d 616b  ution, Next: mak
│ │ │ │ -00038160: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ -00038170: 6e43 6f64 696d 322c 2050 7265 763a 204c  nCodim2, Prev: L
│ │ │ │ -00038180: 6966 742c 2055 703a 2054 6f70 0a0a 6d61  ift, Up: Top..ma
│ │ │ │ -00038190: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ -000381a0: 6f6e 202d 2d20 6669 6e69 7465 2072 6573  on -- finite res
│ │ │ │ -000381b0: 6f6c 7574 696f 6e20 6f66 2061 206d 6174  olution of a mat
│ │ │ │ -000381c0: 7269 7820 6661 6374 6f72 697a 6174 696f  rix factorizatio
│ │ │ │ -000381d0: 6e20 6d6f 6475 6c65 204d 0a2a 2a2a 2a2a  n module M.*****
│ │ │ │ +00038050: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +00038060: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +00038070: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +00038080: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +00038090: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +000380a0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ +000380b0: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +000380c0: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ +000380d0: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +000380e0: 2e6d 323a 3331 3938 3a30 2e0a 1f0a 4669  .m2:3198:0....Fi
│ │ │ │ +000380f0: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +00038100: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +00038110: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +00038120: 6d61 6b65 4669 6e69 7465 5265 736f 6c75  makeFiniteResolu
│ │ │ │ +00038130: 7469 6f6e 2c20 4e65 7874 3a20 6d61 6b65  tion, Next: make
│ │ │ │ +00038140: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ +00038150: 436f 6469 6d32 2c20 5072 6576 3a20 4c69  Codim2, Prev: Li
│ │ │ │ +00038160: 6674 2c20 5570 3a20 546f 700a 0a6d 616b  ft, Up: Top..mak
│ │ │ │ +00038170: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ +00038180: 6e20 2d2d 2066 696e 6974 6520 7265 736f  n -- finite reso
│ │ │ │ +00038190: 6c75 7469 6f6e 206f 6620 6120 6d61 7472  lution of a matr
│ │ │ │ +000381a0: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ +000381b0: 206d 6f64 756c 6520 4d0a 2a2a 2a2a 2a2a   module M.******
│ │ │ │ +000381c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000381d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000381e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000381f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00038200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00038210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00038220: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -00038230: 6765 3a20 0a20 2020 2020 2020 2041 203d  ge: .        A =
│ │ │ │ -00038240: 206d 616b 6546 696e 6974 6552 6573 6f6c   makeFiniteResol
│ │ │ │ -00038250: 7574 696f 6e28 6666 2c6d 6629 0a20 202a  ution(ff,mf).  *
│ │ │ │ -00038260: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -00038270: 206d 662c 2061 202a 6e6f 7465 206c 6973   mf, a *note lis
│ │ │ │ -00038280: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -00038290: 294c 6973 742c 2c20 6f75 7470 7574 206f  )List,, output o
│ │ │ │ -000382a0: 6620 6d61 7472 6978 4661 6374 6f72 697a  f matrixFactoriz
│ │ │ │ -000382b0: 6174 696f 6e0a 2020 2020 2020 2a20 6666  ation.      * ff
│ │ │ │ -000382c0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ -000382d0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -000382e0: 4d61 7472 6978 2c2c 2074 6865 2072 6567  Matrix,, the reg
│ │ │ │ -000382f0: 756c 6172 2073 6571 7565 6e63 6520 7573  ular sequence us
│ │ │ │ -00038300: 6564 0a20 2020 2020 2020 2066 6f72 2074  ed.        for t
│ │ │ │ -00038310: 6865 206d 6174 7269 7846 6163 746f 7269  he matrixFactori
│ │ │ │ -00038320: 7a61 7469 6f6e 2063 6f6d 7075 7461 7469  zation computati
│ │ │ │ -00038330: 6f6e 0a20 202a 204f 7574 7075 7473 3a0a  on.  * Outputs:.
│ │ │ │ -00038340: 2020 2020 2020 2a20 412c 2061 202a 6e6f        * A, a *no
│ │ │ │ -00038350: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d  te complex: (Com
│ │ │ │ -00038360: 706c 6578 6573 2943 6f6d 706c 6578 2c2c  plexes)Complex,,
│ │ │ │ -00038370: 2041 2069 7320 7468 6520 6d69 6e69 6d61   A is the minima
│ │ │ │ -00038380: 6c20 6669 6e69 7465 0a20 2020 2020 2020  l finite.       
│ │ │ │ -00038390: 2072 6573 6f6c 7574 696f 6e20 6f66 204d   resolution of M
│ │ │ │ -000383a0: 206f 7665 7220 522e 0a0a 4465 7363 7269   over R...Descri
│ │ │ │ -000383b0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -000383c0: 3d0a 0a53 7570 706f 7365 2074 6861 7420  =..Suppose that 
│ │ │ │ -000383d0: 665f 312e 2e66 5f63 2069 7320 6120 686f  f_1..f_c is a ho
│ │ │ │ -000383e0: 6d6f 6765 6e65 6f75 7320 7265 6775 6c61  mogeneous regula
│ │ │ │ -000383f0: 7220 7365 7175 656e 6365 206f 6620 666f  r sequence of fo
│ │ │ │ -00038400: 726d 7320 6f66 2074 6865 2073 616d 650a  rms of the same.
│ │ │ │ -00038410: 6465 6772 6565 2069 6e20 6120 706f 6c79  degree in a poly
│ │ │ │ -00038420: 6e6f 6d69 616c 2072 696e 6720 5320 616e  nomial ring S an
│ │ │ │ -00038430: 6420 4d20 6973 2061 2068 6967 6820 7379  d M is a high sy
│ │ │ │ -00038440: 7a79 6779 206d 6f64 756c 6520 6f76 6572  zygy module over
│ │ │ │ -00038450: 2053 2f28 665f 312c 2e2e 2c66 5f63 290a   S/(f_1,..,f_c).
│ │ │ │ -00038460: 3d20 5228 6329 2c20 616e 6420 6d66 203d  = R(c), and mf =
│ │ │ │ -00038470: 2028 642c 6829 2069 7320 7468 6520 6f75   (d,h) is the ou
│ │ │ │ -00038480: 7470 7574 206f 6620 6d61 7472 6978 4661  tput of matrixFa
│ │ │ │ -00038490: 6374 6f72 697a 6174 696f 6e28 4d2c 6666  ctorization(M,ff
│ │ │ │ -000384a0: 292e 2049 6620 7468 650a 636f 6d70 6c65  ). If the.comple
│ │ │ │ -000384b0: 7869 7479 206f 6620 4d20 6973 2063 272c  xity of M is c',
│ │ │ │ -000384c0: 2074 6865 6e20 4d20 6861 7320 6120 6669   then M has a fi
│ │ │ │ -000384d0: 6e69 7465 2066 7265 6520 7265 736f 6c75  nite free resolu
│ │ │ │ -000384e0: 7469 6f6e 206f 7665 7220 5220 3d0a 532f  tion over R =.S/
│ │ │ │ -000384f0: 2866 5f31 2c2e 2e2c 665f 7b28 632d 6327  (f_1,..,f_{(c-c'
│ │ │ │ -00038500: 297d 2920 2861 6e64 2c20 6d6f 7265 2067  )}) (and, more g
│ │ │ │ -00038510: 656e 6572 616c 6c79 2c20 6861 7320 636f  enerally, has co
│ │ │ │ -00038520: 6d70 6c65 7869 7479 2063 2d64 206f 7665  mplexity c-d ove
│ │ │ │ -00038530: 720a 532f 2866 5f31 2c2e 2e2c 665f 7b28  r.S/(f_1,..,f_{(
│ │ │ │ -00038540: 632d 6429 7d29 2066 6f72 2064 3e3d 6327  c-d)}) for d>=c'
│ │ │ │ -00038550: 292e 0a0a 5468 6520 636f 6d70 6c65 7820  )...The complex 
│ │ │ │ -00038560: 4120 6973 2074 6865 206d 696e 696d 616c  A is the minimal
│ │ │ │ -00038570: 2066 696e 6974 6520 6672 6565 2072 6573   finite free res
│ │ │ │ -00038580: 6f6c 7574 696f 6e20 6f66 204d 206f 7665  olution of M ove
│ │ │ │ -00038590: 7220 412c 2063 6f6e 7374 7275 6374 6564  r A, constructed
│ │ │ │ -000385a0: 2061 730a 616e 2069 7465 7261 7465 6420   as.an iterated 
│ │ │ │ -000385b0: 4b6f 737a 756c 2065 7874 656e 7369 6f6e  Koszul extension
│ │ │ │ -000385c0: 2c20 6d61 6465 2066 726f 6d20 7468 6520  , made from the 
│ │ │ │ -000385d0: 6d61 7073 2069 6e20 624d 6170 7320 6d66  maps in bMaps mf
│ │ │ │ -000385e0: 2061 6e64 2070 7369 4d61 7073 206d 662c   and psiMaps mf,
│ │ │ │ -000385f0: 2061 730a 6465 7363 7269 6265 6420 696e   as.described in
│ │ │ │ -00038600: 2045 6973 656e 6275 642d 5065 6576 612e   Eisenbud-Peeva.
│ │ │ │ -00038610: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +00038200: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ +00038210: 653a 200a 2020 2020 2020 2020 4120 3d20  e: .        A = 
│ │ │ │ +00038220: 6d61 6b65 4669 6e69 7465 5265 736f 6c75  makeFiniteResolu
│ │ │ │ +00038230: 7469 6f6e 2866 662c 6d66 290a 2020 2a20  tion(ff,mf).  * 
│ │ │ │ +00038240: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00038250: 6d66 2c20 6120 2a6e 6f74 6520 6c69 7374  mf, a *note list
│ │ │ │ +00038260: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00038270: 4c69 7374 2c2c 206f 7574 7075 7420 6f66  List,, output of
│ │ │ │ +00038280: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +00038290: 7469 6f6e 0a20 2020 2020 202a 2066 662c  tion.      * ff,
│ │ │ │ +000382a0: 2061 202a 6e6f 7465 206d 6174 7269 783a   a *note matrix:
│ │ │ │ +000382b0: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +000382c0: 6174 7269 782c 2c20 7468 6520 7265 6775  atrix,, the regu
│ │ │ │ +000382d0: 6c61 7220 7365 7175 656e 6365 2075 7365  lar sequence use
│ │ │ │ +000382e0: 640a 2020 2020 2020 2020 666f 7220 7468  d.        for th
│ │ │ │ +000382f0: 6520 6d61 7472 6978 4661 6374 6f72 697a  e matrixFactoriz
│ │ │ │ +00038300: 6174 696f 6e20 636f 6d70 7574 6174 696f  ation computatio
│ │ │ │ +00038310: 6e0a 2020 2a20 4f75 7470 7574 733a 0a20  n.  * Outputs:. 
│ │ │ │ +00038320: 2020 2020 202a 2041 2c20 6120 2a6e 6f74       * A, a *not
│ │ │ │ +00038330: 6520 636f 6d70 6c65 783a 2028 436f 6d70  e complex: (Comp
│ │ │ │ +00038340: 6c65 7865 7329 436f 6d70 6c65 782c 2c20  lexes)Complex,, 
│ │ │ │ +00038350: 4120 6973 2074 6865 206d 696e 696d 616c  A is the minimal
│ │ │ │ +00038360: 2066 696e 6974 650a 2020 2020 2020 2020   finite.        
│ │ │ │ +00038370: 7265 736f 6c75 7469 6f6e 206f 6620 4d20  resolution of M 
│ │ │ │ +00038380: 6f76 6572 2052 2e0a 0a44 6573 6372 6970  over R...Descrip
│ │ │ │ +00038390: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +000383a0: 0a0a 5375 7070 6f73 6520 7468 6174 2066  ..Suppose that f
│ │ │ │ +000383b0: 5f31 2e2e 665f 6320 6973 2061 2068 6f6d  _1..f_c is a hom
│ │ │ │ +000383c0: 6f67 656e 656f 7573 2072 6567 756c 6172  ogeneous regular
│ │ │ │ +000383d0: 2073 6571 7565 6e63 6520 6f66 2066 6f72   sequence of for
│ │ │ │ +000383e0: 6d73 206f 6620 7468 6520 7361 6d65 0a64  ms of the same.d
│ │ │ │ +000383f0: 6567 7265 6520 696e 2061 2070 6f6c 796e  egree in a polyn
│ │ │ │ +00038400: 6f6d 6961 6c20 7269 6e67 2053 2061 6e64  omial ring S and
│ │ │ │ +00038410: 204d 2069 7320 6120 6869 6768 2073 797a   M is a high syz
│ │ │ │ +00038420: 7967 7920 6d6f 6475 6c65 206f 7665 7220  ygy module over 
│ │ │ │ +00038430: 532f 2866 5f31 2c2e 2e2c 665f 6329 0a3d  S/(f_1,..,f_c).=
│ │ │ │ +00038440: 2052 2863 292c 2061 6e64 206d 6620 3d20   R(c), and mf = 
│ │ │ │ +00038450: 2864 2c68 2920 6973 2074 6865 206f 7574  (d,h) is the out
│ │ │ │ +00038460: 7075 7420 6f66 206d 6174 7269 7846 6163  put of matrixFac
│ │ │ │ +00038470: 746f 7269 7a61 7469 6f6e 284d 2c66 6629  torization(M,ff)
│ │ │ │ +00038480: 2e20 4966 2074 6865 0a63 6f6d 706c 6578  . If the.complex
│ │ │ │ +00038490: 6974 7920 6f66 204d 2069 7320 6327 2c20  ity of M is c', 
│ │ │ │ +000384a0: 7468 656e 204d 2068 6173 2061 2066 696e  then M has a fin
│ │ │ │ +000384b0: 6974 6520 6672 6565 2072 6573 6f6c 7574  ite free resolut
│ │ │ │ +000384c0: 696f 6e20 6f76 6572 2052 203d 0a53 2f28  ion over R =.S/(
│ │ │ │ +000384d0: 665f 312c 2e2e 2c66 5f7b 2863 2d63 2729  f_1,..,f_{(c-c')
│ │ │ │ +000384e0: 7d29 2028 616e 642c 206d 6f72 6520 6765  }) (and, more ge
│ │ │ │ +000384f0: 6e65 7261 6c6c 792c 2068 6173 2063 6f6d  nerally, has com
│ │ │ │ +00038500: 706c 6578 6974 7920 632d 6420 6f76 6572  plexity c-d over
│ │ │ │ +00038510: 0a53 2f28 665f 312c 2e2e 2c66 5f7b 2863  .S/(f_1,..,f_{(c
│ │ │ │ +00038520: 2d64 297d 2920 666f 7220 643e 3d63 2729  -d)}) for d>=c')
│ │ │ │ +00038530: 2e0a 0a54 6865 2063 6f6d 706c 6578 2041  ...The complex A
│ │ │ │ +00038540: 2069 7320 7468 6520 6d69 6e69 6d61 6c20   is the minimal 
│ │ │ │ +00038550: 6669 6e69 7465 2066 7265 6520 7265 736f  finite free reso
│ │ │ │ +00038560: 6c75 7469 6f6e 206f 6620 4d20 6f76 6572  lution of M over
│ │ │ │ +00038570: 2041 2c20 636f 6e73 7472 7563 7465 6420   A, constructed 
│ │ │ │ +00038580: 6173 0a61 6e20 6974 6572 6174 6564 204b  as.an iterated K
│ │ │ │ +00038590: 6f73 7a75 6c20 6578 7465 6e73 696f 6e2c  oszul extension,
│ │ │ │ +000385a0: 206d 6164 6520 6672 6f6d 2074 6865 206d   made from the m
│ │ │ │ +000385b0: 6170 7320 696e 2062 4d61 7073 206d 6620  aps in bMaps mf 
│ │ │ │ +000385c0: 616e 6420 7073 694d 6170 7320 6d66 2c20  and psiMaps mf, 
│ │ │ │ +000385d0: 6173 0a64 6573 6372 6962 6564 2069 6e20  as.described in 
│ │ │ │ +000385e0: 4569 7365 6e62 7564 2d50 6565 7661 2e0a  Eisenbud-Peeva..
│ │ │ │ +000385f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00038600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038660: 2b0a 7c69 3120 3a20 7365 7452 616e 646f  +.|i1 : setRando
│ │ │ │ -00038670: 6d53 6565 6420 3020 2020 2020 2020 2020  mSeed 0         
│ │ │ │ -00038680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386b0: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072  |.| -- setting r
│ │ │ │ -000386c0: 616e 646f 6d20 7365 6564 2074 6f20 3020  andom seed to 0 
│ │ │ │ -000386d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000386e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00038640: 0a7c 6931 203a 2073 6574 5261 6e64 6f6d  .|i1 : setRandom
│ │ │ │ +00038650: 5365 6564 2030 2020 2020 2020 2020 2020  Seed 0          
│ │ │ │ +00038660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038680: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038690: 0a7c 202d 2d20 7365 7474 696e 6720 7261  .| -- setting ra
│ │ │ │ +000386a0: 6e64 6f6d 2073 6565 6420 746f 2030 2020  ndom seed to 0  
│ │ │ │ +000386b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000386c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000386d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000386e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000386f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038700: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00038700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038730: 0a7c 6f31 203d 2030 2020 2020 2020 2020  .|o1 = 0        
│ │ │ │  00038740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038750: 7c0a 7c6f 3120 3d20 3020 2020 2020 2020  |.|o1 = 0       
│ │ │ │ +00038750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000387a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00038770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038780: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00038790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000387a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000387b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000387c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000387d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000387e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000387f0: 2b0a 7c69 3220 3a20 5320 3d20 5a5a 2f31  +.|i2 : S = ZZ/1
│ │ │ │ -00038800: 3031 5b61 2c62 2c63 5d3b 2020 2020 2020  01[a,b,c];      
│ │ │ │ -00038810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038840: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000387c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000387d0: 0a7c 6932 203a 2053 203d 205a 5a2f 3130  .|i2 : S = ZZ/10
│ │ │ │ +000387e0: 315b 612c 622c 635d 3b20 2020 2020 2020  1[a,b,c];       
│ │ │ │ +000387f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038820: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00038830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038890: 2b0a 7c69 3320 3a20 6666 203d 206d 6174  +.|i3 : ff = mat
│ │ │ │ -000388a0: 7269 7822 6133 2c62 3322 3b20 2020 2020  rix"a3,b3";     
│ │ │ │ -000388b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00038870: 0a7c 6933 203a 2066 6620 3d20 6d61 7472  .|i3 : ff = matr
│ │ │ │ +00038880: 6978 2261 332c 6233 223b 2020 2020 2020  ix"a3,b3";      
│ │ │ │ +00038890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000388a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000388b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000388c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000388d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000388e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038930: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00038940: 3120 2020 2020 2032 2020 2020 2020 2020  1      2        
│ │ │ │ -00038950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038980: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053  |.|o3 : Matrix S
│ │ │ │ -00038990: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -000389a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00038900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038910: 0a7c 2020 2020 2020 2020 2020 2020 2031  .|             1
│ │ │ │ +00038920: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00038930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038960: 0a7c 6f33 203a 204d 6174 7269 7820 5320  .|o3 : Matrix S 
│ │ │ │ +00038970: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +00038980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000389a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000389b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000389c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000389d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000389e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000389f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038a20: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964  +.|i4 : R = S/id
│ │ │ │ -00038a30: 6561 6c20 6666 3b20 2020 2020 2020 2020  eal ff;         
│ │ │ │ -00038a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038a70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +000389f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00038a00: 0a7c 6934 203a 2052 203d 2053 2f69 6465  .|i4 : R = S/ide
│ │ │ │ +00038a10: 616c 2066 663b 2020 2020 2020 2020 2020  al ff;          
│ │ │ │ +00038a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038a50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00038a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038ac0: 2b0a 7c69 3520 3a20 4d20 3d20 6869 6768  +.|i5 : M = high
│ │ │ │ -00038ad0: 5379 7a79 6779 2028 525e 312f 6964 6561  Syzygy (R^1/idea
│ │ │ │ -00038ae0: 6c20 7661 7273 2052 293b 2020 2020 2020  l vars R);      
│ │ │ │ -00038af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038b10: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00038a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00038aa0: 0a7c 6935 203a 204d 203d 2068 6967 6853  .|i5 : M = highS
│ │ │ │ +00038ab0: 797a 7967 7920 2852 5e31 2f69 6465 616c  yzygy (R^1/ideal
│ │ │ │ +00038ac0: 2076 6172 7320 5229 3b20 2020 2020 2020   vars R);       
│ │ │ │ +00038ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038af0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00038b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038b60: 2b0a 7c69 3620 3a20 6d66 203d 206d 6174  +.|i6 : mf = mat
│ │ │ │ -00038b70: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00038b80: 2028 6666 2c20 4d29 2020 2020 2020 2020   (ff, M)        
│ │ │ │ -00038b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00038b40: 0a7c 6936 203a 206d 6620 3d20 6d61 7472  .|i6 : mf = matr
│ │ │ │ +00038b50: 6978 4661 6374 6f72 697a 6174 696f 6e20  ixFactorization 
│ │ │ │ +00038b60: 2866 662c 204d 2920 2020 2020 2020 2020  (ff, M)         
│ │ │ │ +00038b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038b90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00038ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038bb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00038bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c00: 7c0a 7c6f 3620 3d20 7b7b 347d 207c 202d  |.|o6 = {{4} | -
│ │ │ │ -00038c10: 6320 6220 3020 2061 3220 3020 2030 2020  c b 0  a2 0  0  
│ │ │ │ -00038c20: 3020 2030 2020 3020 2020 7c2c 207b 357d  0  0  0   |, {5}
│ │ │ │ -00038c30: 207c 2030 2061 3220 3020 202d 6220 3020   | 0 a2 0  -b 0 
│ │ │ │ -00038c40: 2030 2020 2030 2020 2030 2030 2020 3020   0   0   0 0  0 
│ │ │ │ -00038c50: 7c0a 7c20 2020 2020 207b 347d 207c 2061  |.|      {4} | a
│ │ │ │ -00038c60: 2020 3020 6220 2030 2020 3020 2030 2020    0 b  0  0  0  
│ │ │ │ -00038c70: 3020 2030 2020 3020 2020 7c20 207b 357d  0  0  0   |  {5}
│ │ │ │ -00038c80: 207c 2030 2030 2020 6132 202d 6320 6232   | 0 0  a2 -c b2
│ │ │ │ -00038c90: 2030 2020 2030 2020 2030 2030 2020 3020   0   0   0 0  0 
│ │ │ │ -00038ca0: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00038cb0: 2020 6120 6320 2030 2020 3020 2030 2020    a c  0  0  0  
│ │ │ │ -00038cc0: 3020 2030 2020 2d62 3220 7c20 207b 357d  0  0  -b2 |  {5}
│ │ │ │ -00038cd0: 207c 2030 2030 2020 3020 2061 2020 3020   | 0 0  0  a  0 
│ │ │ │ -00038ce0: 2062 3220 2030 2020 2030 2030 2020 3020   b2  0   0 0  0 
│ │ │ │ -00038cf0: 7c0a 7c20 2020 2020 207b 337d 207c 2030  |.|      {3} | 0
│ │ │ │ -00038d00: 2020 3020 6132 2030 2020 3020 2062 3220    0 a2 0  0  b2 
│ │ │ │ -00038d10: 3020 2030 2020 3020 2020 7c20 207b 367d  0  0  0   |  {6}
│ │ │ │ -00038d20: 207c 2061 2063 2020 2d62 2030 2020 3020   | a c  -b 0  0 
│ │ │ │ -00038d30: 2030 2020 2030 2020 2030 2030 2020 3020   0   0   0 0  0 
│ │ │ │ -00038d40: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00038d50: 2020 3020 3020 2030 2020 6220 202d 6120    0 0  0  b  -a 
│ │ │ │ -00038d60: 3020 2030 2020 3020 2020 7c20 207b 357d  0  0  0   |  {5}
│ │ │ │ -00038d70: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00038d80: 2030 2020 2030 2020 2061 2062 3220 3020   0   0   a b2 0 
│ │ │ │ -00038d90: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00038da0: 2020 3020 3020 2030 2020 2d63 2030 2020    0 0  0  -c 0  
│ │ │ │ -00038db0: 6120 2062 3220 3020 2020 7c20 207b 357d  a  b2 0   |  {5}
│ │ │ │ -00038dc0: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00038dd0: 202d 6132 2030 2020 2062 2030 2020 3020   -a2 0   b 0  0 
│ │ │ │ -00038de0: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00038df0: 2020 3020 3020 2030 2020 3020 2063 2020    0 0  0  0  c  
│ │ │ │ -00038e00: 2d62 2030 2020 6132 2020 7c20 207b 357d  -b 0  a2  |  {5}
│ │ │ │ -00038e10: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00038e20: 2030 2020 202d 6132 2063 2030 2020 3020   0   -a2 c 0  0 
│ │ │ │ -00038e30: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00038e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038e50: 2020 2020 2020 2020 2020 2020 207b 367d               {6}
│ │ │ │ -00038e60: 207c 2030 2030 2020 3020 2030 2020 3020   | 0 0  0  0  0 
│ │ │ │ -00038e70: 2030 2020 2030 2020 2030 2063 2020 6220   0   0   0 c  b 
│ │ │ │ -00038e80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00038e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ea0: 2020 2020 2020 2020 2020 2020 207b 367d               {6}
│ │ │ │ -00038eb0: 207c 2030 2030 2020 3020 2030 2020 6120   | 0 0  0  0  a 
│ │ │ │ -00038ec0: 2063 2020 202d 6220 2030 2030 2020 3020   c   -b  0 0  0 
│ │ │ │ -00038ed0: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d  |.|     --------
│ │ │ │ +00038bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038be0: 0a7c 6f36 203d 207b 7b34 7d20 7c20 2d63  .|o6 = {{4} | -c
│ │ │ │ +00038bf0: 2062 2030 2020 6132 2030 2020 3020 2030   b 0  a2 0  0  0
│ │ │ │ +00038c00: 2020 3020 2030 2020 207c 2c20 7b35 7d20    0  0   |, {5} 
│ │ │ │ +00038c10: 7c20 3020 6132 2030 2020 2d62 2030 2020  | 0 a2 0  -b 0  
│ │ │ │ +00038c20: 3020 2020 3020 2020 3020 3020 2030 207c  0   0   0 0  0 |
│ │ │ │ +00038c30: 0a7c 2020 2020 2020 7b34 7d20 7c20 6120  .|      {4} | a 
│ │ │ │ +00038c40: 2030 2062 2020 3020 2030 2020 3020 2030   0 b  0  0  0  0
│ │ │ │ +00038c50: 2020 3020 2030 2020 207c 2020 7b35 7d20    0  0   |  {5} 
│ │ │ │ +00038c60: 7c20 3020 3020 2061 3220 2d63 2062 3220  | 0 0  a2 -c b2 
│ │ │ │ +00038c70: 3020 2020 3020 2020 3020 3020 2030 207c  0   0   0 0  0 |
│ │ │ │ +00038c80: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00038c90: 2061 2063 2020 3020 2030 2020 3020 2030   a c  0  0  0  0
│ │ │ │ +00038ca0: 2020 3020 202d 6232 207c 2020 7b35 7d20    0  -b2 |  {5} 
│ │ │ │ +00038cb0: 7c20 3020 3020 2030 2020 6120 2030 2020  | 0 0  0  a  0  
│ │ │ │ +00038cc0: 6232 2020 3020 2020 3020 3020 2030 207c  b2  0   0 0  0 |
│ │ │ │ +00038cd0: 0a7c 2020 2020 2020 7b33 7d20 7c20 3020  .|      {3} | 0 
│ │ │ │ +00038ce0: 2030 2061 3220 3020 2030 2020 6232 2030   0 a2 0  0  b2 0
│ │ │ │ +00038cf0: 2020 3020 2030 2020 207c 2020 7b36 7d20    0  0   |  {6} 
│ │ │ │ +00038d00: 7c20 6120 6320 202d 6220 3020 2030 2020  | a c  -b 0  0  
│ │ │ │ +00038d10: 3020 2020 3020 2020 3020 3020 2030 207c  0   0   0 0  0 |
│ │ │ │ +00038d20: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00038d30: 2030 2030 2020 3020 2062 2020 2d61 2030   0 0  0  b  -a 0
│ │ │ │ +00038d40: 2020 3020 2030 2020 207c 2020 7b35 7d20    0  0   |  {5} 
│ │ │ │ +00038d50: 7c20 3020 3020 2030 2020 3020 2030 2020  | 0 0  0  0  0  
│ │ │ │ +00038d60: 3020 2020 3020 2020 6120 6232 2030 207c  0   0   a b2 0 |
│ │ │ │ +00038d70: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00038d80: 2030 2030 2020 3020 202d 6320 3020 2061   0 0  0  -c 0  a
│ │ │ │ +00038d90: 2020 6232 2030 2020 207c 2020 7b35 7d20    b2 0   |  {5} 
│ │ │ │ +00038da0: 7c20 3020 3020 2030 2020 3020 2030 2020  | 0 0  0  0  0  
│ │ │ │ +00038db0: 2d61 3220 3020 2020 6220 3020 2030 207c  -a2 0   b 0  0 |
│ │ │ │ +00038dc0: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00038dd0: 2030 2030 2020 3020 2030 2020 6320 202d   0 0  0  0  c  -
│ │ │ │ +00038de0: 6220 3020 2061 3220 207c 2020 7b35 7d20  b 0  a2  |  {5} 
│ │ │ │ +00038df0: 7c20 3020 3020 2030 2020 3020 2030 2020  | 0 0  0  0  0  
│ │ │ │ +00038e00: 3020 2020 2d61 3220 6320 3020 2030 207c  0   -a2 c 0  0 |
│ │ │ │ +00038e10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00038e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038e30: 2020 2020 2020 2020 2020 2020 7b36 7d20              {6} 
│ │ │ │ +00038e40: 7c20 3020 3020 2030 2020 3020 2030 2020  | 0 0  0  0  0  
│ │ │ │ +00038e50: 3020 2020 3020 2020 3020 6320 2062 207c  0   0   0 c  b |
│ │ │ │ +00038e60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00038e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038e80: 2020 2020 2020 2020 2020 2020 7b36 7d20              {6} 
│ │ │ │ +00038e90: 7c20 3020 3020 2030 2020 3020 2061 2020  | 0 0  0  0  a  
│ │ │ │ +00038ea0: 6320 2020 2d62 2020 3020 3020 2030 207c  c   -b  0 0  0 |
│ │ │ │ +00038eb0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d  .|     ---------
│ │ │ │ +00038ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038f20: 7c0a 7c20 2020 2020 3020 2020 7c2c 207b  |.|     0   |, {
│ │ │ │ -00038f30: 337d 207c 2030 2030 2030 2020 3120 3020  3} | 0 0 0  1 0 
│ │ │ │ -00038f40: 3020 3020 7c7d 2020 2020 2020 2020 2020  0 0 |}          
│ │ │ │ -00038f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f70: 7c0a 7c20 2020 2020 3020 2020 7c20 207b  |.|     0   |  {
│ │ │ │ -00038f80: 347d 207c 2030 2030 2030 2020 3020 3120  4} | 0 0 0  0 1 
│ │ │ │ -00038f90: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │ -00038fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038fc0: 7c0a 7c20 2020 2020 3020 2020 7c20 207b  |.|     0   |  {
│ │ │ │ -00038fd0: 347d 207c 2030 2030 2030 2020 3020 3020  4} | 0 0 0  0 0 
│ │ │ │ -00038fe0: 3120 3020 7c20 2020 2020 2020 2020 2020  1 0 |           
│ │ │ │ -00038ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039010: 7c0a 7c20 2020 2020 3020 2020 7c20 207b  |.|     0   |  {
│ │ │ │ -00039020: 347d 207c 2030 2030 2030 2020 3020 3020  4} | 0 0 0  0 0 
│ │ │ │ -00039030: 3020 3120 7c20 2020 2020 2020 2020 2020  0 1 |           
│ │ │ │ -00039040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039060: 7c0a 7c20 2020 2020 3020 2020 7c20 207b  |.|     0   |  {
│ │ │ │ -00039070: 347d 207c 2030 2031 2030 2020 3020 3020  4} | 0 1 0  0 0 
│ │ │ │ -00039080: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │ -00039090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000390a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000390b0: 7c0a 7c20 2020 2020 3020 2020 7c20 207b  |.|     0   |  {
│ │ │ │ -000390c0: 347d 207c 2030 2030 202d 3120 3020 3020  4} | 0 0 -1 0 0 
│ │ │ │ -000390d0: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │ -000390e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000390f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039100: 7c0a 7c20 2020 2020 2d62 3220 7c20 207b  |.|     -b2 |  {
│ │ │ │ -00039110: 347d 207c 2031 2030 2030 2020 3020 3020  4} | 1 0 0  0 0 
│ │ │ │ -00039120: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │ -00039130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ +00038f00: 0a7c 2020 2020 2030 2020 207c 2c20 7b33  .|     0   |, {3
│ │ │ │ +00038f10: 7d20 7c20 3020 3020 3020 2031 2030 2030  } | 0 0 0  1 0 0
│ │ │ │ +00038f20: 2030 207c 7d20 2020 2020 2020 2020 2020   0 |}           
│ │ │ │ +00038f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038f50: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +00038f60: 7d20 7c20 3020 3020 3020 2030 2031 2030  } | 0 0 0  0 1 0
│ │ │ │ +00038f70: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +00038f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038fa0: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +00038fb0: 7d20 7c20 3020 3020 3020 2030 2030 2031  } | 0 0 0  0 0 1
│ │ │ │ +00038fc0: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +00038fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00038ff0: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +00039000: 7d20 7c20 3020 3020 3020 2030 2030 2030  } | 0 0 0  0 0 0
│ │ │ │ +00039010: 2031 207c 2020 2020 2020 2020 2020 2020   1 |            
│ │ │ │ +00039020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039040: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +00039050: 7d20 7c20 3020 3120 3020 2030 2030 2030  } | 0 1 0  0 0 0
│ │ │ │ +00039060: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +00039070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039090: 0a7c 2020 2020 2030 2020 207c 2020 7b34  .|     0   |  {4
│ │ │ │ +000390a0: 7d20 7c20 3020 3020 2d31 2030 2030 2030  } | 0 0 -1 0 0 0
│ │ │ │ +000390b0: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +000390c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000390d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000390e0: 0a7c 2020 2020 202d 6232 207c 2020 7b34  .|     -b2 |  {4
│ │ │ │ +000390f0: 7d20 7c20 3120 3020 3020 2030 2030 2030  } | 1 0 0  0 0 0
│ │ │ │ +00039100: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +00039110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039130: 0a7c 2020 2020 2061 2020 207c 2020 2020  .|     a   |    
│ │ │ │  00039140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039150: 7c0a 7c20 2020 2020 6120 2020 7c20 2020  |.|     a   |   
│ │ │ │ +00039150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039180: 0a7c 2020 2020 2030 2020 207c 2020 2020  .|     0   |    
│ │ │ │  00039190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391a0: 7c0a 7c20 2020 2020 3020 2020 7c20 2020  |.|     0   |   
│ │ │ │ +000391a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000391b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000391c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000391d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000391e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000391f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039220: 0a7c 6f36 203a 204c 6973 7420 2020 2020  .|o6 : List     
│ │ │ │  00039230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039240: 7c0a 7c6f 3620 3a20 4c69 7374 2020 2020  |.|o6 : List    
│ │ │ │ +00039240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039290: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00039260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039270: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00039280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000392a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000392b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000392c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000392d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000392e0: 2b0a 7c69 3720 3a20 4720 3d20 6d61 6b65  +.|i7 : G = make
│ │ │ │ -000392f0: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ -00039300: 2866 662c 6d66 2920 2020 2020 2020 2020  (ff,mf)         
│ │ │ │ -00039310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000392b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000392c0: 0a7c 6937 203a 2047 203d 206d 616b 6546  .|i7 : G = makeF
│ │ │ │ +000392d0: 696e 6974 6552 6573 6f6c 7574 696f 6e28  initeResolution(
│ │ │ │ +000392e0: 6666 2c6d 6629 2020 2020 2020 2020 2020  ff,mf)          
│ │ │ │ +000392f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00039320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039330: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039380: 7c0a 7c20 2020 2020 2037 2020 2020 2020  |.|      7      
│ │ │ │ -00039390: 3132 2020 2020 2020 3520 2020 2020 2020  12      5       
│ │ │ │ -000393a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000393b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000393c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000393d0: 7c0a 7c6f 3720 3d20 5320 203c 2d2d 2053  |.|o7 = S  <-- S
│ │ │ │ -000393e0: 2020 203c 2d2d 2053 2020 2020 2020 2020     <-- S        
│ │ │ │ -000393f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039360: 0a7c 2020 2020 2020 3720 2020 2020 2031  .|      7      1
│ │ │ │ +00039370: 3220 2020 2020 2035 2020 2020 2020 2020  2      5        
│ │ │ │ +00039380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000393a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000393b0: 0a7c 6f37 203d 2053 2020 3c2d 2d20 5320  .|o7 = S  <-- S 
│ │ │ │ +000393c0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ +000393d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000393e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000393f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039400: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00039410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039420: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039470: 7c0a 7c20 2020 2020 3020 2020 2020 2031  |.|     0      1
│ │ │ │ -00039480: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00039490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039450: 0a7c 2020 2020 2030 2020 2020 2020 3120  .|     0      1 
│ │ │ │ +00039460: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00039470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000394a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000394b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000394c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000394d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000394f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000394e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000394f0: 0a7c 6f37 203a 2043 6f6d 706c 6578 2020  .|o7 : Complex  
│ │ │ │  00039500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039510: 7c0a 7c6f 3720 3a20 436f 6d70 6c65 7820  |.|o7 : Complex 
│ │ │ │ +00039510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039560: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00039530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039540: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00039550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000395a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000395b0: 2b0a 7c69 3820 3a20 4620 3d20 6672 6565  +.|i8 : F = free
│ │ │ │ -000395c0: 5265 736f 6c75 7469 6f6e 2070 7573 6846  Resolution pushF
│ │ │ │ -000395d0: 6f72 7761 7264 286d 6170 2852 2c53 292c  orward(map(R,S),
│ │ │ │ -000395e0: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ +00039580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00039590: 0a7c 6938 203a 2046 203d 2066 7265 6552  .|i8 : F = freeR
│ │ │ │ +000395a0: 6573 6f6c 7574 696f 6e20 7075 7368 466f  esolution pushFo
│ │ │ │ +000395b0: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ +000395c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +000395d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000395e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000395f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039600: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039650: 7c0a 7c20 2020 2020 2037 2020 2020 2020  |.|      7      
│ │ │ │ -00039660: 3132 2020 2020 2020 3520 2020 2020 2020  12      5       
│ │ │ │ -00039670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396a0: 7c0a 7c6f 3820 3d20 5320 203c 2d2d 2053  |.|o8 = S  <-- S
│ │ │ │ -000396b0: 2020 203c 2d2d 2053 2020 2020 2020 2020     <-- S        
│ │ │ │ -000396c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039630: 0a7c 2020 2020 2020 3720 2020 2020 2031  .|      7      1
│ │ │ │ +00039640: 3220 2020 2020 2035 2020 2020 2020 2020  2      5        
│ │ │ │ +00039650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039680: 0a7c 6f38 203d 2053 2020 3c2d 2d20 5320  .|o8 = S  <-- S 
│ │ │ │ +00039690: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ +000396a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000396b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000396c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000396d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000396e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000396f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039740: 7c0a 7c20 2020 2020 3020 2020 2020 2031  |.|     0      1
│ │ │ │ -00039750: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00039760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039720: 0a7c 2020 2020 2030 2020 2020 2020 3120  .|     0      1 
│ │ │ │ +00039730: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00039740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039770: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00039780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039790: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000397b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000397c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000397b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000397c0: 0a7c 6f38 203a 2043 6f6d 706c 6578 2020  .|o8 : Complex  
│ │ │ │  000397d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000397e0: 7c0a 7c6f 3820 3a20 436f 6d70 6c65 7820  |.|o8 : Complex 
│ │ │ │ +000397e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039830: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00039800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039810: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00039820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039880: 2b0a 7c69 3920 3a20 472e 6464 5f31 2020  +.|i9 : G.dd_1  
│ │ │ │ +00039850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00039860: 0a7c 6939 203a 2047 2e64 645f 3120 2020  .|i9 : G.dd_1   
│ │ │ │ +00039870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000398a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000398b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000398c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000398d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000398e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000398f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039920: 7c0a 7c6f 3920 3d20 7b34 7d20 7c20 6220  |.|o9 = {4} | b 
│ │ │ │ -00039930: 202d 6120 3020 2030 2020 3020 2020 2d61   -a 0  0  0   -a
│ │ │ │ -00039940: 3320 3020 2020 3020 2020 3020 2030 2030  3 0   0   0  0 0
│ │ │ │ -00039950: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -00039960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039970: 7c0a 7c20 2020 2020 7b34 7d20 7c20 2d63  |.|     {4} | -c
│ │ │ │ -00039980: 2030 2020 6120 2062 3220 3020 2020 3020   0  a  b2 0   0 
│ │ │ │ -00039990: 2020 2d61 3320 3020 2020 3020 2030 2030    -a3 0   0  0 0
│ │ │ │ -000399a0: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -000399b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000399c0: 7c0a 7c20 2020 2020 7b34 7d20 7c20 3020  |.|     {4} | 0 
│ │ │ │ -000399d0: 2063 2020 2d62 2030 2020 6132 2020 3020   c  -b 0  a2  0 
│ │ │ │ -000399e0: 2020 3020 2020 2d61 3320 3020 2030 2030    0   -a3 0  0 0
│ │ │ │ -000399f0: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -00039a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039a10: 7c0a 7c20 2020 2020 7b34 7d20 7c20 3020  |.|     {4} | 0 
│ │ │ │ -00039a20: 2030 2020 3020 2030 2020 3020 2020 3020   0  0  0  0   0 
│ │ │ │ -00039a30: 2020 3020 2020 3020 2020 2d63 2062 2030    0   0   -c b 0
│ │ │ │ -00039a40: 2020 6132 207c 2020 2020 2020 2020 2020    a2 |          
│ │ │ │ -00039a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039a60: 7c0a 7c20 2020 2020 7b34 7d20 7c20 3020  |.|     {4} | 0 
│ │ │ │ -00039a70: 2030 2020 3020 2030 2020 3020 2020 3020   0  0  0  0   0 
│ │ │ │ -00039a80: 2020 3020 2020 3020 2020 6120 2030 2062    0   0   a  0 b
│ │ │ │ -00039a90: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -00039aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ab0: 7c0a 7c20 2020 2020 7b34 7d20 7c20 3020  |.|     {4} | 0 
│ │ │ │ -00039ac0: 2030 2020 3020 2030 2020 2d62 3220 3020   0  0  0  -b2 0 
│ │ │ │ -00039ad0: 2020 3020 2020 3020 2020 3020 2061 2063    0   0   0  a c
│ │ │ │ -00039ae0: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ -00039af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b00: 7c0a 7c20 2020 2020 7b33 7d20 7c20 3020  |.|     {3} | 0 
│ │ │ │ -00039b10: 2062 3220 3020 2030 2020 3020 2020 3020   b2 0  0  0   0 
│ │ │ │ -00039b20: 2020 3020 2020 3020 2020 3020 2030 2061    0   0   0  0 a
│ │ │ │ -00039b30: 3220 3020 207c 2020 2020 2020 2020 2020  2 0  |          
│ │ │ │ +000398f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039900: 0a7c 6f39 203d 207b 347d 207c 2062 2020  .|o9 = {4} | b  
│ │ │ │ +00039910: 2d61 2030 2020 3020 2030 2020 202d 6133  -a 0  0  0   -a3
│ │ │ │ +00039920: 2030 2020 2030 2020 2030 2020 3020 3020   0   0   0  0 0 
│ │ │ │ +00039930: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ +00039940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039950: 0a7c 2020 2020 207b 347d 207c 202d 6320  .|     {4} | -c 
│ │ │ │ +00039960: 3020 2061 2020 6232 2030 2020 2030 2020  0  a  b2 0   0  
│ │ │ │ +00039970: 202d 6133 2030 2020 2030 2020 3020 3020   -a3 0   0  0 0 
│ │ │ │ +00039980: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ +00039990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000399a0: 0a7c 2020 2020 207b 347d 207c 2030 2020  .|     {4} | 0  
│ │ │ │ +000399b0: 6320 202d 6220 3020 2061 3220 2030 2020  c  -b 0  a2  0  
│ │ │ │ +000399c0: 2030 2020 202d 6133 2030 2020 3020 3020   0   -a3 0  0 0 
│ │ │ │ +000399d0: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ +000399e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000399f0: 0a7c 2020 2020 207b 347d 207c 2030 2020  .|     {4} | 0  
│ │ │ │ +00039a00: 3020 2030 2020 3020 2030 2020 2030 2020  0  0  0  0   0  
│ │ │ │ +00039a10: 2030 2020 2030 2020 202d 6320 6220 3020   0   0   -c b 0 
│ │ │ │ +00039a20: 2061 3220 7c20 2020 2020 2020 2020 2020   a2 |           
│ │ │ │ +00039a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039a40: 0a7c 2020 2020 207b 347d 207c 2030 2020  .|     {4} | 0  
│ │ │ │ +00039a50: 3020 2030 2020 3020 2030 2020 2030 2020  0  0  0  0   0  
│ │ │ │ +00039a60: 2030 2020 2030 2020 2061 2020 3020 6220   0   0   a  0 b 
│ │ │ │ +00039a70: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ +00039a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039a90: 0a7c 2020 2020 207b 347d 207c 2030 2020  .|     {4} | 0  
│ │ │ │ +00039aa0: 3020 2030 2020 3020 202d 6232 2030 2020  0  0  0  -b2 0  
│ │ │ │ +00039ab0: 2030 2020 2030 2020 2030 2020 6120 6320   0   0   0  a c 
│ │ │ │ +00039ac0: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ +00039ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039ae0: 0a7c 2020 2020 207b 337d 207c 2030 2020  .|     {3} | 0  
│ │ │ │ +00039af0: 6232 2030 2020 3020 2030 2020 2030 2020  b2 0  0  0   0  
│ │ │ │ +00039b00: 2030 2020 2030 2020 2030 2020 3020 6132   0   0   0  0 a2
│ │ │ │ +00039b10: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ +00039b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039b30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00039b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b50: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00039bb0: 3720 2020 2020 2031 3220 2020 2020 2020  7      12       
│ │ │ │ -00039bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039bf0: 7c0a 7c6f 3920 3a20 4d61 7472 6978 2053  |.|o9 : Matrix S
│ │ │ │ -00039c00: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -00039c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039c40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00039b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039b80: 0a7c 2020 2020 2020 2020 2020 2020 2037  .|             7
│ │ │ │ +00039b90: 2020 2020 2020 3132 2020 2020 2020 2020        12        
│ │ │ │ +00039ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039bd0: 0a7c 6f39 203a 204d 6174 7269 7820 5320  .|o9 : Matrix S 
│ │ │ │ +00039be0: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +00039bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039c20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00039c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039c90: 2b0a 7c69 3130 203a 2046 2e64 645f 3120  +.|i10 : F.dd_1 
│ │ │ │ +00039c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00039c70: 0a7c 6931 3020 3a20 462e 6464 5f31 2020  .|i10 : F.dd_1  
│ │ │ │ +00039c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00039cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d30: 7c0a 7c6f 3130 203d 207b 337d 207c 2061  |.|o10 = {3} | a
│ │ │ │ -00039d40: 3220 6232 2030 2020 3020 2030 2020 3020  2 b2 0  0  0  0 
│ │ │ │ -00039d50: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00039d60: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00039d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d80: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00039d90: 2020 2d61 2030 2020 3020 2030 2020 6220    -a 0  0  0  b 
│ │ │ │ -00039da0: 2030 2020 3020 2030 2020 3020 2030 2020   0  0  0  0  0  
│ │ │ │ -00039db0: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00039dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039dd0: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00039de0: 2020 3020 2061 2020 3020 2030 2020 2d63    0  a  0  0  -c
│ │ │ │ -00039df0: 2030 2020 3020 2062 3220 3020 2030 2020   0  0  b2 0  0  
│ │ │ │ -00039e00: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00039e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039e20: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00039e30: 2020 6320 202d 6220 3020 2030 2020 3020    c  -b 0  0  0 
│ │ │ │ -00039e40: 2061 3220 3020 2030 2020 3020 2030 2020   a2 0  0  0  0  
│ │ │ │ -00039e50: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00039e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039e70: 7c0a 7c20 2020 2020 207b 347d 207c 2062  |.|      {4} | b
│ │ │ │ -00039e80: 2020 3020 2030 2020 6120 2030 2020 3020    0  0  a  0  0 
│ │ │ │ -00039e90: 2030 2020 3020 2030 2020 6233 2030 2020   0  0  0  b3 0  
│ │ │ │ -00039ea0: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00039eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ec0: 7c0a 7c20 2020 2020 207b 347d 207c 202d  |.|      {4} | -
│ │ │ │ -00039ed0: 6320 3020 2030 2020 3020 2061 2020 3020  c 0  0  0  a  0 
│ │ │ │ -00039ee0: 2062 3220 3020 2030 2020 3020 2062 3320   b2 0  0  0  b3 
│ │ │ │ -00039ef0: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00039f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f10: 7c0a 7c20 2020 2020 207b 347d 207c 2030  |.|      {4} | 0
│ │ │ │ -00039f20: 2020 3020 2030 2020 2d63 202d 6220 3020    0  0  -c -b 0 
│ │ │ │ -00039f30: 2030 2020 6132 2030 2020 3020 2030 2020   0  a2 0  0  0  
│ │ │ │ -00039f40: 6233 207c 2020 2020 2020 2020 2020 2020  b3 |            
│ │ │ │ +00039d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039d10: 0a7c 6f31 3020 3d20 7b33 7d20 7c20 6132  .|o10 = {3} | a2
│ │ │ │ +00039d20: 2062 3220 3020 2030 2020 3020 2030 2020   b2 0  0  0  0  
│ │ │ │ +00039d30: 3020 2030 2020 3020 2030 2020 3020 2030  0  0  0  0  0  0
│ │ │ │ +00039d40: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00039d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039d60: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00039d70: 202d 6120 3020 2030 2020 3020 2062 2020   -a 0  0  0  b  
│ │ │ │ +00039d80: 3020 2030 2020 3020 2030 2020 3020 2030  0  0  0  0  0  0
│ │ │ │ +00039d90: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00039da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039db0: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00039dc0: 2030 2020 6120 2030 2020 3020 202d 6320   0  a  0  0  -c 
│ │ │ │ +00039dd0: 3020 2030 2020 6232 2030 2020 3020 2030  0  0  b2 0  0  0
│ │ │ │ +00039de0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00039df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039e00: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00039e10: 2063 2020 2d62 2030 2020 3020 2030 2020   c  -b 0  0  0  
│ │ │ │ +00039e20: 6132 2030 2020 3020 2030 2020 3020 2030  a2 0  0  0  0  0
│ │ │ │ +00039e30: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00039e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039e50: 0a7c 2020 2020 2020 7b34 7d20 7c20 6220  .|      {4} | b 
│ │ │ │ +00039e60: 2030 2020 3020 2061 2020 3020 2030 2020   0  0  a  0  0  
│ │ │ │ +00039e70: 3020 2030 2020 3020 2062 3320 3020 2030  0  0  0  b3 0  0
│ │ │ │ +00039e80: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00039e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039ea0: 0a7c 2020 2020 2020 7b34 7d20 7c20 2d63  .|      {4} | -c
│ │ │ │ +00039eb0: 2030 2020 3020 2030 2020 6120 2030 2020   0  0  0  a  0  
│ │ │ │ +00039ec0: 6232 2030 2020 3020 2030 2020 6233 2030  b2 0  0  0  b3 0
│ │ │ │ +00039ed0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00039ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039ef0: 0a7c 2020 2020 2020 7b34 7d20 7c20 3020  .|      {4} | 0 
│ │ │ │ +00039f00: 2030 2020 3020 202d 6320 2d62 2030 2020   0  0  -c -b 0  
│ │ │ │ +00039f10: 3020 2061 3220 3020 2030 2020 3020 2062  0  a2 0  0  0  b
│ │ │ │ +00039f20: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │ +00039f30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039f40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00039f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f60: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00039f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039fb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00039fc0: 2037 2020 2020 2020 3132 2020 2020 2020   7      12      
│ │ │ │ -00039fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a000: 7c0a 7c6f 3130 203a 204d 6174 7269 7820  |.|o10 : Matrix 
│ │ │ │ -0003a010: 5320 203c 2d2d 2053 2020 2020 2020 2020  S  <-- S        
│ │ │ │ -0003a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a050: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00039f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039f90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00039fa0: 3720 2020 2020 2031 3220 2020 2020 2020  7      12       
│ │ │ │ +00039fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00039fe0: 0a7c 6f31 3020 3a20 4d61 7472 6978 2053  .|o10 : Matrix S
│ │ │ │ +00039ff0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ +0003a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a030: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a0a0: 2b0a 7c69 3131 203a 2047 2e64 645f 3220  +.|i11 : G.dd_2 
│ │ │ │ +0003a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0003a080: 0a7c 6931 3120 3a20 472e 6464 5f32 2020  .|i11 : G.dd_2  
│ │ │ │ +0003a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a0c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a0f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a140: 7c0a 7c6f 3131 203d 207b 357d 207c 202d  |.|o11 = {5} | -
│ │ │ │ -0003a150: 6133 2030 2020 2020 3020 2020 3020 2020  a3 0    0   0   
│ │ │ │ -0003a160: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ -0003a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a190: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a1a0: 2020 202d 6133 2020 3020 2020 3020 2020     -a3  0   0   
│ │ │ │ -0003a1b0: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ -0003a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a1e0: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a1f0: 2020 2030 2020 2020 2d61 3320 3020 2020     0    -a3 0   
│ │ │ │ -0003a200: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ -0003a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a230: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a240: 2020 2030 2020 2020 3020 2020 2d61 3320     0    0   -a3 
│ │ │ │ -0003a250: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ -0003a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a280: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a290: 2020 2030 2020 2020 3020 2020 3020 2020     0    0   0   
│ │ │ │ -0003a2a0: 2d61 3320 2020 7c20 2020 2020 2020 2020  -a3   |         
│ │ │ │ -0003a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a2d0: 7c0a 7c20 2020 2020 207b 377d 207c 202d  |.|      {7} | -
│ │ │ │ -0003a2e0: 6220 2061 2020 2020 3020 2020 3020 2020  b  a    0   0   
│ │ │ │ -0003a2f0: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ -0003a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a320: 7c0a 7c20 2020 2020 207b 377d 207c 2063  |.|      {7} | c
│ │ │ │ -0003a330: 2020 2030 2020 2020 2d61 2020 2d62 3220     0    -a  -b2 
│ │ │ │ -0003a340: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ -0003a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a370: 7c0a 7c20 2020 2020 207b 377d 207c 2030  |.|      {7} | 0
│ │ │ │ -0003a380: 2020 202d 6320 2020 6220 2020 3020 2020     -c   b   0   
│ │ │ │ -0003a390: 2d61 3220 2020 7c20 2020 2020 2020 2020  -a2   |         
│ │ │ │ -0003a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a3c0: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a3d0: 2020 202d 6233 2020 3020 2020 3020 2020     -b3  0   0   
│ │ │ │ -0003a3e0: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ -0003a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a410: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a420: 2020 202d 6232 6320 3020 2020 3020 2020     -b2c 0   0   
│ │ │ │ -0003a430: 2d61 3262 3220 7c20 2020 2020 2020 2020  -a2b2 |         
│ │ │ │ -0003a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a460: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a470: 2020 2061 6232 2020 3020 2020 3020 2020     ab2  0   0   
│ │ │ │ -0003a480: 3020 2020 2020 7c20 2020 2020 2020 2020  0     |         
│ │ │ │ -0003a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a4b0: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a4c0: 2020 2030 2020 2020 3020 2020 3020 2020     0    0   0   
│ │ │ │ -0003a4d0: 6233 2020 2020 7c20 2020 2020 2020 2020  b3    |         
│ │ │ │ -0003a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a120: 0a7c 6f31 3120 3d20 7b35 7d20 7c20 2d61  .|o11 = {5} | -a
│ │ │ │ +0003a130: 3320 3020 2020 2030 2020 2030 2020 2030  3 0    0   0   0
│ │ │ │ +0003a140: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +0003a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a170: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
│ │ │ │ +0003a180: 2020 2d61 3320 2030 2020 2030 2020 2030    -a3  0   0   0
│ │ │ │ +0003a190: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +0003a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a1b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a1c0: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
│ │ │ │ +0003a1d0: 2020 3020 2020 202d 6133 2030 2020 2030    0    -a3 0   0
│ │ │ │ +0003a1e0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +0003a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a210: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
│ │ │ │ +0003a220: 2020 3020 2020 2030 2020 202d 6133 2030    0    0   -a3 0
│ │ │ │ +0003a230: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +0003a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a260: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
│ │ │ │ +0003a270: 2020 3020 2020 2030 2020 2030 2020 202d    0    0   0   -
│ │ │ │ +0003a280: 6133 2020 207c 2020 2020 2020 2020 2020  a3   |          
│ │ │ │ +0003a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a2b0: 0a7c 2020 2020 2020 7b37 7d20 7c20 2d62  .|      {7} | -b
│ │ │ │ +0003a2c0: 2020 6120 2020 2030 2020 2030 2020 2030    a    0   0   0
│ │ │ │ +0003a2d0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +0003a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a2f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a300: 0a7c 2020 2020 2020 7b37 7d20 7c20 6320  .|      {7} | c 
│ │ │ │ +0003a310: 2020 3020 2020 202d 6120 202d 6232 2030    0    -a  -b2 0
│ │ │ │ +0003a320: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +0003a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a350: 0a7c 2020 2020 2020 7b37 7d20 7c20 3020  .|      {7} | 0 
│ │ │ │ +0003a360: 2020 2d63 2020 2062 2020 2030 2020 202d    -c   b   0   -
│ │ │ │ +0003a370: 6132 2020 207c 2020 2020 2020 2020 2020  a2   |          
│ │ │ │ +0003a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a3a0: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
│ │ │ │ +0003a3b0: 2020 2d62 3320 2030 2020 2030 2020 2030    -b3  0   0   0
│ │ │ │ +0003a3c0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +0003a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a3e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a3f0: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
│ │ │ │ +0003a400: 2020 2d62 3263 2030 2020 2030 2020 202d    -b2c 0   0   -
│ │ │ │ +0003a410: 6132 6232 207c 2020 2020 2020 2020 2020  a2b2 |          
│ │ │ │ +0003a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a440: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
│ │ │ │ +0003a450: 2020 6162 3220 2030 2020 2030 2020 2030    ab2  0   0   0
│ │ │ │ +0003a460: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +0003a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a490: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
│ │ │ │ +0003a4a0: 2020 3020 2020 2030 2020 2030 2020 2062    0    0   0   b
│ │ │ │ +0003a4b0: 3320 2020 207c 2020 2020 2020 2020 2020  3    |          
│ │ │ │ +0003a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a4d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a4e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003a4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a500: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a550: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003a560: 2031 3220 2020 2020 2035 2020 2020 2020   12      5      
│ │ │ │ -0003a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5a0: 7c0a 7c6f 3131 203a 204d 6174 7269 7820  |.|o11 : Matrix 
│ │ │ │ -0003a5b0: 5320 2020 3c2d 2d20 5320 2020 2020 2020  S   <-- S       
│ │ │ │ -0003a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5f0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003a520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a530: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003a540: 3132 2020 2020 2020 3520 2020 2020 2020  12      5       
│ │ │ │ +0003a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a580: 0a7c 6f31 3120 3a20 4d61 7472 6978 2053  .|o11 : Matrix S
│ │ │ │ +0003a590: 2020 203c 2d2d 2053 2020 2020 2020 2020     <-- S        
│ │ │ │ +0003a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a5c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a5d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a640: 2b0a 7c69 3132 203a 2046 2e64 645f 3220  +.|i12 : F.dd_2 
│ │ │ │ +0003a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0003a620: 0a7c 6931 3220 3a20 462e 6464 5f32 2020  .|i12 : F.dd_2  
│ │ │ │ +0003a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003a680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a690: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6e0: 7c0a 7c6f 3132 203d 207b 357d 207c 2062  |.|o12 = {5} | b
│ │ │ │ -0003a6f0: 3320 2020 3020 2020 3020 2020 3020 2020  3   0   0   0   
│ │ │ │ -0003a700: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a730: 7c0a 7c20 2020 2020 207b 357d 207c 202d  |.|      {5} | -
│ │ │ │ -0003a740: 6132 6220 3020 2020 3020 2020 3020 2020  a2b 0   0   0   
│ │ │ │ -0003a750: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a780: 7c0a 7c20 2020 2020 207b 357d 207c 202d  |.|      {5} | -
│ │ │ │ -0003a790: 6132 6320 3020 2020 3020 2020 2d61 3262  a2c 0   0   -a2b
│ │ │ │ -0003a7a0: 3220 3020 2020 7c20 2020 2020 2020 2020  2 0   |         
│ │ │ │ -0003a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a7d0: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a7e0: 2020 2020 2d62 3320 3020 2020 3020 2020      -b3 0   0   
│ │ │ │ -0003a7f0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a820: 7c0a 7c20 2020 2020 207b 357d 207c 2030  |.|      {5} | 0
│ │ │ │ -0003a830: 2020 2020 3020 2020 2d62 3320 3020 2020      0   -b3 0   
│ │ │ │ -0003a840: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a870: 7c0a 7c20 2020 2020 207b 357d 207c 202d  |.|      {5} | -
│ │ │ │ -0003a880: 6133 2020 3020 2020 3020 2020 3020 2020  a3  0   0   0   
│ │ │ │ -0003a890: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a8c0: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a8d0: 2020 2020 3020 2020 3020 2020 2d62 3320      0   0   -b3 
│ │ │ │ -0003a8e0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a910: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a920: 2020 2020 3020 2020 3020 2020 3020 2020      0   0   0   
│ │ │ │ -0003a930: 2020 2d62 3320 7c20 2020 2020 2020 2020    -b3 |         
│ │ │ │ -0003a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a960: 7c0a 7c20 2020 2020 207b 367d 207c 2030  |.|      {6} | 0
│ │ │ │ -0003a970: 2020 2020 3020 2020 3020 2020 6133 2020      0   0   a3  
│ │ │ │ -0003a980: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9b0: 7c0a 7c20 2020 2020 207b 377d 207c 202d  |.|      {7} | -
│ │ │ │ -0003a9c0: 6220 2020 6120 2020 3020 2020 3020 2020  b   a   0   0   
│ │ │ │ -0003a9d0: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa00: 7c0a 7c20 2020 2020 207b 377d 207c 2063  |.|      {7} | c
│ │ │ │ -0003aa10: 2020 2020 3020 2020 6120 2020 6232 2020      0   a   b2  
│ │ │ │ -0003aa20: 2020 3020 2020 7c20 2020 2020 2020 2020    0   |         
│ │ │ │ -0003aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa50: 7c0a 7c20 2020 2020 207b 377d 207c 2030  |.|      {7} | 0
│ │ │ │ -0003aa60: 2020 2020 2d63 2020 2d62 2020 3020 2020      -c  -b  0   
│ │ │ │ -0003aa70: 2020 6132 2020 7c20 2020 2020 2020 2020    a2  |         
│ │ │ │ -0003aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a6b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a6c0: 0a7c 6f31 3220 3d20 7b35 7d20 7c20 6233  .|o12 = {5} | b3
│ │ │ │ +0003a6d0: 2020 2030 2020 2030 2020 2030 2020 2020     0   0   0    
│ │ │ │ +0003a6e0: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a710: 0a7c 2020 2020 2020 7b35 7d20 7c20 2d61  .|      {5} | -a
│ │ │ │ +0003a720: 3262 2030 2020 2030 2020 2030 2020 2020  2b 0   0   0    
│ │ │ │ +0003a730: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a760: 0a7c 2020 2020 2020 7b35 7d20 7c20 2d61  .|      {5} | -a
│ │ │ │ +0003a770: 3263 2030 2020 2030 2020 202d 6132 6232  2c 0   0   -a2b2
│ │ │ │ +0003a780: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a7a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a7b0: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
│ │ │ │ +0003a7c0: 2020 202d 6233 2030 2020 2030 2020 2020     -b3 0   0    
│ │ │ │ +0003a7d0: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a7f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a800: 0a7c 2020 2020 2020 7b35 7d20 7c20 3020  .|      {5} | 0 
│ │ │ │ +0003a810: 2020 2030 2020 202d 6233 2030 2020 2020     0   -b3 0    
│ │ │ │ +0003a820: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a850: 0a7c 2020 2020 2020 7b35 7d20 7c20 2d61  .|      {5} | -a
│ │ │ │ +0003a860: 3320 2030 2020 2030 2020 2030 2020 2020  3  0   0   0    
│ │ │ │ +0003a870: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a8a0: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
│ │ │ │ +0003a8b0: 2020 2030 2020 2030 2020 202d 6233 2020     0   0   -b3  
│ │ │ │ +0003a8c0: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a8e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a8f0: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
│ │ │ │ +0003a900: 2020 2030 2020 2030 2020 2030 2020 2020     0   0   0    
│ │ │ │ +0003a910: 202d 6233 207c 2020 2020 2020 2020 2020   -b3 |          
│ │ │ │ +0003a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a940: 0a7c 2020 2020 2020 7b36 7d20 7c20 3020  .|      {6} | 0 
│ │ │ │ +0003a950: 2020 2030 2020 2030 2020 2061 3320 2020     0   0   a3   
│ │ │ │ +0003a960: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a990: 0a7c 2020 2020 2020 7b37 7d20 7c20 2d62  .|      {7} | -b
│ │ │ │ +0003a9a0: 2020 2061 2020 2030 2020 2030 2020 2020     a   0   0    
│ │ │ │ +0003a9b0: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a9d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003a9e0: 0a7c 2020 2020 2020 7b37 7d20 7c20 6320  .|      {7} | c 
│ │ │ │ +0003a9f0: 2020 2030 2020 2061 2020 2062 3220 2020     0   a   b2   
│ │ │ │ +0003aa00: 2030 2020 207c 2020 2020 2020 2020 2020   0   |          
│ │ │ │ +0003aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aa20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003aa30: 0a7c 2020 2020 2020 7b37 7d20 7c20 3020  .|      {7} | 0 
│ │ │ │ +0003aa40: 2020 202d 6320 202d 6220 2030 2020 2020     -c  -b  0    
│ │ │ │ +0003aa50: 2061 3220 207c 2020 2020 2020 2020 2020   a2  |          
│ │ │ │ +0003aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aa70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003aa80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aaa0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aaf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003ab00: 2031 3220 2020 2020 2035 2020 2020 2020   12      5      
│ │ │ │ -0003ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab40: 7c0a 7c6f 3132 203a 204d 6174 7269 7820  |.|o12 : Matrix 
│ │ │ │ -0003ab50: 5320 2020 3c2d 2d20 5320 2020 2020 2020  S   <-- S       
│ │ │ │ -0003ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ab90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003aac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003aad0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003aae0: 3132 2020 2020 2020 3520 2020 2020 2020  12      5       
│ │ │ │ +0003aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ab10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003ab20: 0a7c 6f31 3220 3a20 4d61 7472 6978 2053  .|o12 : Matrix S
│ │ │ │ +0003ab30: 2020 203c 2d2d 2053 2020 2020 2020 2020     <-- S        
│ │ │ │ +0003ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ab60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003ab70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003abe0: 2b0a 0a49 6620 7468 6520 636f 6d70 6c65  +..If the comple
│ │ │ │ -0003abf0: 7869 7479 206f 6620 4d20 6973 206e 6f74  xity of M is not
│ │ │ │ -0003ac00: 206d 6178 696d 616c 2c20 7468 656e 2074   maximal, then t
│ │ │ │ -0003ac10: 6865 2066 696e 6974 6520 7265 736f 6c75  he finite resolu
│ │ │ │ -0003ac20: 7469 6f6e 2074 616b 6573 2070 6c61 6365  tion takes place
│ │ │ │ -0003ac30: 0a6f 7665 7220 616e 2069 6e74 6572 6d65  .over an interme
│ │ │ │ -0003ac40: 6469 6174 6520 636f 6d70 6c65 7465 2069  diate complete i
│ │ │ │ -0003ac50: 6e74 6572 7365 6374 696f 6e3a 0a0a 2b2d  ntersection:..+-
│ │ │ │ +0003abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0003abc0: 0a0a 4966 2074 6865 2063 6f6d 706c 6578  ..If the complex
│ │ │ │ +0003abd0: 6974 7920 6f66 204d 2069 7320 6e6f 7420  ity of M is not 
│ │ │ │ +0003abe0: 6d61 7869 6d61 6c2c 2074 6865 6e20 7468  maximal, then th
│ │ │ │ +0003abf0: 6520 6669 6e69 7465 2072 6573 6f6c 7574  e finite resolut
│ │ │ │ +0003ac00: 696f 6e20 7461 6b65 7320 706c 6163 650a  ion takes place.
│ │ │ │ +0003ac10: 6f76 6572 2061 6e20 696e 7465 726d 6564  over an intermed
│ │ │ │ +0003ac20: 6961 7465 2063 6f6d 706c 6574 6520 696e  iate complete in
│ │ │ │ +0003ac30: 7465 7273 6563 7469 6f6e 3a0a 0a2b 2d2d  tersection:..+--
│ │ │ │ +0003ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ac50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003acb0: 3133 203a 2053 203d 205a 5a2f 3130 315b  13 : S = ZZ/101[
│ │ │ │ -0003acc0: 612c 622c 632c 645d 2020 2020 2020 2020  a,b,c,d]        
│ │ │ │ -0003acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0003ac90: 3320 3a20 5320 3d20 5a5a 2f31 3031 5b61  3 : S = ZZ/101[a
│ │ │ │ +0003aca0: 2c62 2c63 2c64 5d20 2020 2020 2020 2020  ,b,c,d]         
│ │ │ │ +0003acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003acd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003acf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003ad50: 3133 203d 2053 2020 2020 2020 2020 2020  13 = S          
│ │ │ │ +0003ad20: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003ad30: 3320 3d20 5320 2020 2020 2020 2020 2020  3 = S           
│ │ │ │ +0003ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ad70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ade0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003adf0: 3133 203a 2050 6f6c 796e 6f6d 6961 6c52  13 : PolynomialR
│ │ │ │ -0003ae00: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0003ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003adc0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003add0: 3320 3a20 506f 6c79 6e6f 6d69 616c 5269  3 : PolynomialRi
│ │ │ │ +0003ade0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0003adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ae10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003ae20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ae40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003ae90: 3134 203a 2066 6631 203d 206d 6174 7269  14 : ff1 = matri
│ │ │ │ -0003aea0: 7822 6133 2c62 332c 6333 2c64 3322 2020  x"a3,b3,c3,d3"  
│ │ │ │ -0003aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0003ae70: 3420 3a20 6666 3120 3d20 6d61 7472 6978  4 : ff1 = matrix
│ │ │ │ +0003ae80: 2261 332c 6233 2c63 332c 6433 2220 2020  "a3,b3,c3,d3"   
│ │ │ │ +0003ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aeb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003af30: 3134 203d 207c 2061 3320 6233 2063 3320  14 = | a3 b3 c3 
│ │ │ │ -0003af40: 6433 207c 2020 2020 2020 2020 2020 2020  d3 |            
│ │ │ │ -0003af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003af00: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003af10: 3420 3d20 7c20 6133 2062 3320 6333 2064  4 = | a3 b3 c3 d
│ │ │ │ +0003af20: 3320 7c20 2020 2020 2020 2020 2020 2020  3 |             
│ │ │ │ +0003af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003af50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003afc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003afd0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0003afe0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ -0003aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b010: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b020: 3134 203a 204d 6174 7269 7820 5320 203c  14 : Matrix S  <
│ │ │ │ -0003b030: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ -0003b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b060: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003afa0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003afb0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +0003afc0: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +0003afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aff0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003b000: 3420 3a20 4d61 7472 6978 2053 2020 3c2d  4 : Matrix S  <-
│ │ │ │ +0003b010: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │ +0003b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b040: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b0c0: 3135 203a 2066 6620 3d66 6631 2a72 616e  15 : ff =ff1*ran
│ │ │ │ -0003b0d0: 646f 6d28 736f 7572 6365 2066 6631 2c20  dom(source ff1, 
│ │ │ │ -0003b0e0: 736f 7572 6365 2066 6631 2920 2020 2020  source ff1)     
│ │ │ │ +0003b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0003b0a0: 3520 3a20 6666 203d 6666 312a 7261 6e64  5 : ff =ff1*rand
│ │ │ │ +0003b0b0: 6f6d 2873 6f75 7263 6520 6666 312c 2073  om(source ff1, s
│ │ │ │ +0003b0c0: 6f75 7263 6520 6666 3129 2020 2020 2020  ource ff1)      
│ │ │ │ +0003b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b0e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b100: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b150: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b160: 3135 203d 207c 2032 3461 332d 3336 6233  15 = | 24a3-36b3
│ │ │ │ -0003b170: 2d33 3063 332d 3239 6433 2031 3961 332b  -30c3-29d3 19a3+
│ │ │ │ -0003b180: 3139 6233 2d31 3063 332d 3239 6433 202d  19b3-10c3-29d3 -
│ │ │ │ -0003b190: 3861 332d 3232 6233 2d32 3963 332d 3234  8a3-22b3-29c3-24
│ │ │ │ -0003b1a0: 6433 2020 2020 2020 2020 2020 7c0a 7c20  d3          |.| 
│ │ │ │ -0003b1b0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │ +0003b130: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003b140: 3520 3d20 7c20 3234 6133 2d33 3662 332d  5 = | 24a3-36b3-
│ │ │ │ +0003b150: 3330 6333 2d32 3964 3320 3139 6133 2b31  30c3-29d3 19a3+1
│ │ │ │ +0003b160: 3962 332d 3130 6333 2d32 3964 3320 2d38  9b3-10c3-29d3 -8
│ │ │ │ +0003b170: 6133 2d32 3262 332d 3239 6333 2d32 3464  a3-22b3-29c3-24d
│ │ │ │ +0003b180: 3320 2020 2020 2020 2020 207c 0a7c 2020  3          |.|  
│ │ │ │ +0003b190: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │ +0003b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0003b200: 2020 2020 202d 3338 6133 2d31 3662 332b       -38a3-16b3+
│ │ │ │ -0003b210: 3339 6333 2b32 3164 3320 7c20 2020 2020  39c3+21d3 |     
│ │ │ │ -0003b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0003b1e0: 2020 2020 2d33 3861 332d 3136 6233 2b33      -38a3-16b3+3
│ │ │ │ +0003b1f0: 3963 332b 3231 6433 207c 2020 2020 2020  9c3+21d3 |      
│ │ │ │ +0003b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b220: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b290: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b2a0: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0003b2b0: 2020 2020 3420 2020 2020 2020 2020 2020      4           
│ │ │ │ -0003b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b2f0: 3135 203a 204d 6174 7269 7820 5320 203c  15 : Matrix S  <
│ │ │ │ -0003b300: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ -0003b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b330: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b270: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003b280: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +0003b290: 2020 2034 2020 2020 2020 2020 2020 2020     4            
│ │ │ │ +0003b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b2c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003b2d0: 3520 3a20 4d61 7472 6978 2053 2020 3c2d  5 : Matrix S  <-
│ │ │ │ +0003b2e0: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │ +0003b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b310: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b390: 3136 203a 2052 203d 2053 2f69 6465 616c  16 : R = S/ideal
│ │ │ │ -0003b3a0: 2066 6620 2020 2020 2020 2020 2020 2020   ff             
│ │ │ │ -0003b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0003b370: 3620 3a20 5220 3d20 532f 6964 6561 6c20  6 : R = S/ideal 
│ │ │ │ +0003b380: 6666 2020 2020 2020 2020 2020 2020 2020  ff              
│ │ │ │ +0003b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b3b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b420: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b430: 3136 203d 2052 2020 2020 2020 2020 2020  16 = R          
│ │ │ │ +0003b400: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003b410: 3620 3d20 5220 2020 2020 2020 2020 2020  6 = R           
│ │ │ │ +0003b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b450: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b4d0: 3136 203a 2051 756f 7469 656e 7452 696e  16 : QuotientRin
│ │ │ │ -0003b4e0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -0003b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b510: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b4a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003b4b0: 3620 3a20 5175 6f74 6965 6e74 5269 6e67  6 : QuotientRing
│ │ │ │ +0003b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b4f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b570: 3137 203a 204d 203d 2068 6967 6853 797a  17 : M = highSyz
│ │ │ │ -0003b580: 7967 7920 2852 5e31 2f69 6465 616c 2261  ygy (R^1/ideal"a
│ │ │ │ -0003b590: 3262 3222 2920 2020 2020 2020 2020 2020  2b2")           
│ │ │ │ +0003b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0003b550: 3720 3a20 4d20 3d20 6869 6768 5379 7a79  7 : M = highSyzy
│ │ │ │ +0003b560: 6779 2028 525e 312f 6964 6561 6c22 6132  gy (R^1/ideal"a2
│ │ │ │ +0003b570: 6232 2229 2020 2020 2020 2020 2020 2020  b2")            
│ │ │ │ +0003b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b600: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b610: 3137 203d 2063 6f6b 6572 6e65 6c20 7b36  17 = cokernel {6
│ │ │ │ -0003b620: 7d20 7c20 6232 2030 202d 6132 2030 207c  } | b2 0 -a2 0 |
│ │ │ │ -0003b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b650: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b660: 2020 2020 2020 2020 2020 2020 2020 7b37                {7
│ │ │ │ -0003b670: 7d20 7c20 6120 2062 2030 2020 2030 207c  } | a  b 0   0 |
│ │ │ │ -0003b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003b6b0: 2020 2020 2020 2020 2020 2020 2020 7b37                {7
│ │ │ │ -0003b6c0: 7d20 7c20 3020 2030 2062 2020 2061 207c  } | 0  0 b   a |
│ │ │ │ -0003b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b5e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003b5f0: 3720 3d20 636f 6b65 726e 656c 207b 367d  7 = cokernel {6}
│ │ │ │ +0003b600: 207c 2062 3220 3020 2d61 3220 3020 7c20   | b2 0 -a2 0 | 
│ │ │ │ +0003b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b630: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003b640: 2020 2020 2020 2020 2020 2020 207b 377d               {7}
│ │ │ │ +0003b650: 207c 2061 2020 6220 3020 2020 3020 7c20   | a  b 0   0 | 
│ │ │ │ +0003b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b680: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003b690: 2020 2020 2020 2020 2020 2020 207b 377d               {7}
│ │ │ │ +0003b6a0: 207c 2030 2020 3020 6220 2020 6120 7c20   | 0  0 b   a | 
│ │ │ │ +0003b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b6d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b740: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b750: 3137 203a 2052 2d6d 6f64 756c 652c 2071  17 : R-module, q
│ │ │ │ -0003b760: 756f 7469 6520 2020 2020 2020 3320 2020  uotie       3   
│ │ │ │ -0003b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b790: 2020 2020 2020 2020 2020 2020 7c0a 7c6e              |.|n
│ │ │ │ -0003b7a0: 7420 6f66 2052 2020 2020 2020 2020 2020  t of R          
│ │ │ │ +0003b720: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003b730: 3720 3a20 522d 6d6f 6475 6c65 2c20 7175  7 : R-module, qu
│ │ │ │ +0003b740: 6f74 6965 2020 2020 2020 2033 2020 2020  otie       3    
│ │ │ │ +0003b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b770: 2020 2020 2020 2020 2020 207c 0a7c 6e74             |.|nt
│ │ │ │ +0003b780: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ +0003b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b7e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b7c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003b7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b840: 3138 203a 2063 6f6d 706c 6578 6974 7920  18 : complexity 
│ │ │ │ -0003b850: 4d20 2020 2020 2020 2020 2020 2020 2020  M               
│ │ │ │ -0003b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0003b820: 3820 3a20 636f 6d70 6c65 7869 7479 204d  8 : complexity M
│ │ │ │ +0003b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b860: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003b8e0: 3138 203d 2032 2020 2020 2020 2020 2020  18 = 2          
│ │ │ │ +0003b8b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003b8c0: 3820 3d20 3220 2020 2020 2020 2020 2020  8 = 2           
│ │ │ │ +0003b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b920: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003b900: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003b910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003b980: 3139 203a 206d 6620 3d20 6d61 7472 6978  19 : mf = matrix
│ │ │ │ -0003b990: 4661 6374 6f72 697a 6174 696f 6e20 2866  Factorization (f
│ │ │ │ -0003b9a0: 662c 204d 2920 2020 2020 2020 2020 2020  f, M)           
│ │ │ │ +0003b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +0003b960: 3920 3a20 6d66 203d 206d 6174 7269 7846  9 : mf = matrixF
│ │ │ │ +0003b970: 6163 746f 7269 7a61 7469 6f6e 2028 6666  actorization (ff
│ │ │ │ +0003b980: 2c20 4d29 2020 2020 2020 2020 2020 2020  , M)            
│ │ │ │ +0003b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b9a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ba10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003ba20: 3139 203d 207b 7b37 7d20 7c20 2d61 202d  19 = {{7} | -a -
│ │ │ │ -0003ba30: 3336 6220 3020 6120 7c2c 207b 387d 207c  36b 0 a |, {8} |
│ │ │ │ -0003ba40: 2033 3561 3220 2034 3862 2020 3020 2020   35a2  48b  0   
│ │ │ │ -0003ba50: 2020 2d33 3362 2030 2020 2020 207c 2c20    -33b 0     |, 
│ │ │ │ -0003ba60: 7b36 7d20 7c20 3020 2020 3336 7c0a 7c20  {6} | 0   36|.| 
│ │ │ │ -0003ba70: 2020 2020 2020 7b36 7d20 7c20 6232 2061        {6} | b2 a
│ │ │ │ -0003ba80: 3220 2020 3020 3020 7c20 207b 387d 207c  2   0 0 |  {8} |
│ │ │ │ -0003ba90: 202d 3335 6232 202d 3335 6120 3020 2020   -35b2 -35a 0   
│ │ │ │ -0003baa0: 2020 3020 2020 2030 2020 2020 207c 2020    0    0     |  
│ │ │ │ -0003bab0: 7b37 7d20 7c20 2d33 3620 3020 7c0a 7c20  {7} | -36 0 |.| 
│ │ │ │ -0003bac0: 2020 2020 2020 7b37 7d20 7c20 3020 2030        {7} | 0  0
│ │ │ │ -0003bad0: 2020 2020 6220 6120 7c20 207b 387d 207c      b a |  {8} |
│ │ │ │ -0003bae0: 2030 2020 2020 2030 2020 2020 3333 6232   0     0    33b2
│ │ │ │ -0003baf0: 2020 3333 6120 202d 3333 6232 207c 2020    33a  -33b2 |  
│ │ │ │ -0003bb00: 7b37 7d20 7c20 3120 2020 3020 7c0a 7c20  {7} | 1   0 |.| 
│ │ │ │ -0003bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bb20: 2020 2020 2020 2020 2020 207b 387d 207c             {8} |
│ │ │ │ -0003bb30: 2030 2020 2020 2030 2020 2020 2d34 3361   0     0    -43a
│ │ │ │ -0003bb40: 3220 2d33 3362 2030 2020 2020 207c 2020  2 -33b 0     |  
│ │ │ │ -0003bb50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003bb60: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d       -----------
│ │ │ │ +0003b9f0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003ba00: 3920 3d20 7b7b 377d 207c 202d 6120 2d33  9 = {{7} | -a -3
│ │ │ │ +0003ba10: 3662 2030 2061 207c 2c20 7b38 7d20 7c20  6b 0 a |, {8} | 
│ │ │ │ +0003ba20: 3335 6132 2020 3438 6220 2030 2020 2020  35a2  48b  0    
│ │ │ │ +0003ba30: 202d 3333 6220 3020 2020 2020 7c2c 207b   -33b 0     |, {
│ │ │ │ +0003ba40: 367d 207c 2030 2020 2033 367c 0a7c 2020  6} | 0   36|.|  
│ │ │ │ +0003ba50: 2020 2020 207b 367d 207c 2062 3220 6132       {6} | b2 a2
│ │ │ │ +0003ba60: 2020 2030 2030 207c 2020 7b38 7d20 7c20     0 0 |  {8} | 
│ │ │ │ +0003ba70: 2d33 3562 3220 2d33 3561 2030 2020 2020  -35b2 -35a 0    
│ │ │ │ +0003ba80: 2030 2020 2020 3020 2020 2020 7c20 207b   0    0     |  {
│ │ │ │ +0003ba90: 377d 207c 202d 3336 2030 207c 0a7c 2020  7} | -36 0 |.|  
│ │ │ │ +0003baa0: 2020 2020 207b 377d 207c 2030 2020 3020       {7} | 0  0 
│ │ │ │ +0003bab0: 2020 2062 2061 207c 2020 7b38 7d20 7c20     b a |  {8} | 
│ │ │ │ +0003bac0: 3020 2020 2020 3020 2020 2033 3362 3220  0     0    33b2 
│ │ │ │ +0003bad0: 2033 3361 2020 2d33 3362 3220 7c20 207b   33a  -33b2 |  {
│ │ │ │ +0003bae0: 377d 207c 2031 2020 2030 207c 0a7c 2020  7} | 1   0 |.|  
│ │ │ │ +0003baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bb00: 2020 2020 2020 2020 2020 7b38 7d20 7c20            {8} | 
│ │ │ │ +0003bb10: 3020 2020 2020 3020 2020 202d 3433 6132  0     0    -43a2
│ │ │ │ +0003bb20: 202d 3333 6220 3020 2020 2020 7c20 2020   -33b 0     |   
│ │ │ │ +0003bb30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003bb40: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d      ------------
│ │ │ │ +0003bb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ -0003bbb0: 2020 2020 2030 2020 7c7d 2020 2020 2020       0  |}      
│ │ │ │ +0003bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +0003bb90: 2020 2020 3020 207c 7d20 2020 2020 2020      0  |}       
│ │ │ │ +0003bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bbf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003bc00: 2020 2020 2033 3620 7c20 2020 2020 2020       36 |       
│ │ │ │ +0003bbd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003bbe0: 2020 2020 3336 207c 2020 2020 2020 2020      36 |        
│ │ │ │ +0003bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003bc50: 2020 2020 2030 2020 7c20 2020 2020 2020       0  |       
│ │ │ │ +0003bc20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003bc30: 2020 2020 3020 207c 2020 2020 2020 2020      0  |        
│ │ │ │ +0003bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bc70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bce0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003bcf0: 3139 203a 204c 6973 7420 2020 2020 2020  19 : List       
│ │ │ │ +0003bcc0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003bcd0: 3920 3a20 4c69 7374 2020 2020 2020 2020  9 : List        
│ │ │ │ +0003bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003bd10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003bd90: 3230 203a 2063 6f6d 706c 6578 6974 7920  20 : complexity 
│ │ │ │ -0003bda0: 6d66 2020 2020 2020 2020 2020 2020 2020  mf              
│ │ │ │ -0003bdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0003bd70: 3020 3a20 636f 6d70 6c65 7869 7479 206d  0 : complexity m
│ │ │ │ +0003bd80: 6620 2020 2020 2020 2020 2020 2020 2020  f               
│ │ │ │ +0003bd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bdb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bdd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003be30: 3230 203d 2032 2020 2020 2020 2020 2020  20 = 2          
│ │ │ │ +0003be00: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003be10: 3020 3d20 3220 2020 2020 2020 2020 2020  0 = 2           
│ │ │ │ +0003be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003be70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003be50: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003be70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003be80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003be90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003bed0: 3231 203a 2042 5261 6e6b 7320 6d66 2020  21 : BRanks mf  
│ │ │ │ +0003bea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0003beb0: 3120 3a20 4252 616e 6b73 206d 6620 2020  1 : BRanks mf   
│ │ │ │ +0003bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bef0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bf10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bf60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003bf70: 3231 203d 207b 7b32 2c20 327d 2c20 7b31  21 = {{2, 2}, {1
│ │ │ │ -0003bf80: 2c20 327d 7d20 2020 2020 2020 2020 2020  , 2}}           
│ │ │ │ -0003bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bf40: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003bf50: 3120 3d20 7b7b 322c 2032 7d2c 207b 312c  1 = {{2, 2}, {1,
│ │ │ │ +0003bf60: 2032 7d7d 2020 2020 2020 2020 2020 2020   2}}            
│ │ │ │ +0003bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bf90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bfb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c000: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c010: 3231 203a 204c 6973 7420 2020 2020 2020  21 : List       
│ │ │ │ +0003bfe0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003bff0: 3120 3a20 4c69 7374 2020 2020 2020 2020  1 : List        
│ │ │ │ +0003c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c050: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c030: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003c040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c0b0: 3232 203a 2047 203d 206d 616b 6546 696e  22 : G = makeFin
│ │ │ │ -0003c0c0: 6974 6552 6573 6f6c 7574 696f 6e28 6666  iteResolution(ff
│ │ │ │ -0003c0d0: 2c6d 6629 3b20 2020 2020 2020 2020 2020  ,mf);           
│ │ │ │ -0003c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c0f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0003c090: 3220 3a20 4720 3d20 6d61 6b65 4669 6e69  2 : G = makeFini
│ │ │ │ +0003c0a0: 7465 5265 736f 6c75 7469 6f6e 2866 662c  teResolution(ff,
│ │ │ │ +0003c0b0: 6d66 293b 2020 2020 2020 2020 2020 2020  mf);            
│ │ │ │ +0003c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c0d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c150: 3233 203a 2063 6f64 696d 2072 696e 6720  23 : codim ring 
│ │ │ │ -0003c160: 4720 2020 2020 2020 2020 2020 2020 2020  G               
│ │ │ │ -0003c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0003c130: 3320 3a20 636f 6469 6d20 7269 6e67 2047  3 : codim ring G
│ │ │ │ +0003c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c170: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c1e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c1f0: 3233 203d 2032 2020 2020 2020 2020 2020  23 = 2          
│ │ │ │ +0003c1c0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003c1d0: 3320 3d20 3220 2020 2020 2020 2020 2020  3 = 2           
│ │ │ │ +0003c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c230: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c210: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003c220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c290: 3234 203a 2052 3120 3d20 7269 6e67 2047  24 : R1 = ring G
│ │ │ │ +0003c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0003c270: 3420 3a20 5231 203d 2072 696e 6720 4720  4 : R1 = ring G 
│ │ │ │ +0003c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c2b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c2d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c320: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c330: 3234 203d 2052 3120 2020 2020 2020 2020  24 = R1         
│ │ │ │ +0003c300: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003c310: 3420 3d20 5231 2020 2020 2020 2020 2020  4 = R1          
│ │ │ │ +0003c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c350: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c370: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c3d0: 3234 203a 2051 756f 7469 656e 7452 696e  24 : QuotientRin
│ │ │ │ -0003c3e0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -0003c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c410: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c3a0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003c3b0: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ +0003c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c3f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c470: 3235 203a 2046 203d 2066 7265 6552 6573  25 : F = freeRes
│ │ │ │ -0003c480: 6f6c 7574 696f 6e28 7072 756e 6520 7075  olution(prune pu
│ │ │ │ -0003c490: 7368 466f 7277 6172 6428 6d61 7028 522c  shForward(map(R,
│ │ │ │ -0003c4a0: 5231 292c 4d29 2c20 4c65 6e67 7468 4c69  R1),M), LengthLi
│ │ │ │ -0003c4b0: 6d69 7420 3d3e 2034 2920 2020 7c0a 7c20  mit => 4)   |.| 
│ │ │ │ +0003c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0003c450: 3520 3a20 4620 3d20 6672 6565 5265 736f  5 : F = freeReso
│ │ │ │ +0003c460: 6c75 7469 6f6e 2870 7275 6e65 2070 7573  lution(prune pus
│ │ │ │ +0003c470: 6846 6f72 7761 7264 286d 6170 2852 2c52  hForward(map(R,R
│ │ │ │ +0003c480: 3129 2c4d 292c 204c 656e 6774 684c 696d  1),M), LengthLim
│ │ │ │ +0003c490: 6974 203d 3e20 3429 2020 207c 0a7c 2020  it => 4)   |.|  
│ │ │ │ +0003c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c510: 2020 2020 2020 2033 2020 2020 2020 2035         3       5
│ │ │ │ -0003c520: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0003c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c560: 3235 203d 2052 3120 203c 2d2d 2052 3120  25 = R1  <-- R1 
│ │ │ │ -0003c570: 203c 2d2d 2052 3120 2020 2020 2020 2020   <-- R1         
│ │ │ │ -0003c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c4e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003c4f0: 2020 2020 2020 3320 2020 2020 2020 3520        3       5 
│ │ │ │ +0003c500: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0003c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c530: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003c540: 3520 3d20 5231 2020 3c2d 2d20 5231 2020  5 = R1  <-- R1  
│ │ │ │ +0003c550: 3c2d 2d20 5231 2020 2020 2020 2020 2020  <-- R1          
│ │ │ │ +0003c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c580: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c5a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c600: 2020 2020 2030 2020 2020 2020 2031 2020       0       1  
│ │ │ │ -0003c610: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0003c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c5d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003c5e0: 2020 2020 3020 2020 2020 2020 3120 2020      0       1   
│ │ │ │ +0003c5f0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0003c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c620: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c690: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c6a0: 3235 203a 2043 6f6d 706c 6578 2020 2020  25 : Complex    
│ │ │ │ +0003c670: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003c680: 3520 3a20 436f 6d70 6c65 7820 2020 2020  5 : Complex     
│ │ │ │ +0003c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c6e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003c6c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003c6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003c740: 3236 203a 2062 6574 7469 2046 2020 2020  26 : betti F    
│ │ │ │ +0003c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0003c720: 3620 3a20 6265 7474 6920 4620 2020 2020  6 : betti F     
│ │ │ │ +0003c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c760: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c780: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c7d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c7e0: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -0003c7f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0003c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c820: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003c830: 3236 203d 2074 6f74 616c 3a20 3320 3520  26 = total: 3 5 
│ │ │ │ -0003c840: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0003c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c880: 2020 2020 2020 2020 2036 3a20 3120 2e20           6: 1 . 
│ │ │ │ -0003c890: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0003c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c8c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c8d0: 2020 2020 2020 2020 2037 3a20 3220 3420           7: 2 4 
│ │ │ │ -0003c8e0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0003c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c910: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c920: 2020 2020 2020 2020 2038 3a20 2e20 2e20           8: . . 
│ │ │ │ -0003c930: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0003c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003c970: 2020 2020 2020 2020 2039 3a20 2e20 3120           9: . 1 
│ │ │ │ -0003c980: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0003c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c7b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003c7c0: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +0003c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c800: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003c810: 3620 3d20 746f 7461 6c3a 2033 2035 2032  6 = total: 3 5 2
│ │ │ │ +0003c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c850: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003c860: 2020 2020 2020 2020 363a 2031 202e 202e          6: 1 . .
│ │ │ │ +0003c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c8a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003c8b0: 2020 2020 2020 2020 373a 2032 2034 202e          7: 2 4 .
│ │ │ │ +0003c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003c900: 2020 2020 2020 2020 383a 202e 202e 202e          8: . . .
│ │ │ │ +0003c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c940: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003c950: 2020 2020 2020 2020 393a 202e 2031 2032          9: . 1 2
│ │ │ │ +0003c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c990: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c9b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003ca10: 3236 203a 2042 6574 7469 5461 6c6c 7920  26 : BettiTally 
│ │ │ │ +0003c9e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003c9f0: 3620 3a20 4265 7474 6954 616c 6c79 2020  6 : BettiTally  
│ │ │ │ +0003ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003ca30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003ca40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003cab0: 3237 203a 2062 6574 7469 2047 2020 2020  27 : betti G    
│ │ │ │ +0003ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +0003ca90: 3720 3a20 6265 7474 6920 4720 2020 2020  7 : betti G     
│ │ │ │ +0003caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003caf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cb50: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -0003cb60: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0003cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003cba0: 3237 203d 2074 6f74 616c 3a20 3320 3520  27 = total: 3 5 
│ │ │ │ -0003cbb0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0003cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cbe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cbf0: 2020 2020 2020 2020 2036 3a20 3120 2e20           6: 1 . 
│ │ │ │ -0003cc00: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0003cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cc40: 2020 2020 2020 2020 2037 3a20 3220 3420           7: 2 4 
│ │ │ │ -0003cc50: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0003cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cc90: 2020 2020 2020 2020 2038 3a20 2e20 2e20           8: . . 
│ │ │ │ -0003cca0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ -0003ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ccd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003cce0: 2020 2020 2020 2020 2039 3a20 2e20 3120           9: . 1 
│ │ │ │ -0003ccf0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0003cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cb20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003cb30: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +0003cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cb70: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003cb80: 3720 3d20 746f 7461 6c3a 2033 2035 2032  7 = total: 3 5 2
│ │ │ │ +0003cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cbc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003cbd0: 2020 2020 2020 2020 363a 2031 202e 202e          6: 1 . .
│ │ │ │ +0003cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cc10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003cc20: 2020 2020 2020 2020 373a 2032 2034 202e          7: 2 4 .
│ │ │ │ +0003cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cc60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003cc70: 2020 2020 2020 2020 383a 202e 202e 202e          8: . . .
│ │ │ │ +0003cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ccb0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003ccc0: 2020 2020 2020 2020 393a 202e 2031 2032          9: . 1 2
│ │ │ │ +0003ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cd00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cd20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cd70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003cd80: 3237 203a 2042 6574 7469 5461 6c6c 7920  27 : BettiTally 
│ │ │ │ +0003cd50: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +0003cd60: 3720 3a20 4265 7474 6954 616c 6c79 2020  7 : BettiTally  
│ │ │ │ +0003cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cdc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003cda0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003cdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53  ------------+..S
│ │ │ │ -0003ce20: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -0003ce30: 0a0a 2020 2a20 2a6e 6f74 6520 6d61 7472  ..  * *note matr
│ │ │ │ -0003ce40: 6978 4661 6374 6f72 697a 6174 696f 6e3a  ixFactorization:
│ │ │ │ -0003ce50: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -0003ce60: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e  tion, -- Maps in
│ │ │ │ -0003ce70: 2061 2068 6967 6865 720a 2020 2020 636f   a higher.    co
│ │ │ │ -0003ce80: 6469 6d65 6e73 696f 6e20 6d61 7472 6978  dimension matrix
│ │ │ │ -0003ce90: 2066 6163 746f 7269 7a61 7469 6f6e 0a20   factorization. 
│ │ │ │ -0003cea0: 202a 202a 6e6f 7465 2062 4d61 7073 3a20   * *note bMaps: 
│ │ │ │ -0003ceb0: 624d 6170 732c 202d 2d20 6c69 7374 2074  bMaps, -- list t
│ │ │ │ -0003cec0: 6865 206d 6170 7320 2064 5f70 3a42 5f31  he maps  d_p:B_1
│ │ │ │ -0003ced0: 2870 292d 2d3e 425f 3028 7029 2069 6e20  (p)-->B_0(p) in 
│ │ │ │ -0003cee0: 610a 2020 2020 6d61 7472 6978 4661 6374  a.    matrixFact
│ │ │ │ -0003cef0: 6f72 697a 6174 696f 6e0a 2020 2a20 2a6e  orization.  * *n
│ │ │ │ -0003cf00: 6f74 6520 7073 694d 6170 733a 2070 7369  ote psiMaps: psi
│ │ │ │ -0003cf10: 4d61 7073 2c20 2d2d 206c 6973 7420 7468  Maps, -- list th
│ │ │ │ -0003cf20: 6520 6d61 7073 2020 7073 6928 7029 3a20  e maps  psi(p): 
│ │ │ │ -0003cf30: 425f 3128 7029 202d 2d3e 2041 5f30 2870  B_1(p) --> A_0(p
│ │ │ │ -0003cf40: 2d31 2920 696e 2061 0a20 2020 206d 6174  -1) in a.    mat
│ │ │ │ -0003cf50: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0003cf60: 0a20 202a 202a 6e6f 7465 2068 4d61 7073  .  * *note hMaps
│ │ │ │ -0003cf70: 3a20 684d 6170 732c 202d 2d20 6c69 7374  : hMaps, -- list
│ │ │ │ -0003cf80: 2074 6865 206d 6170 7320 2068 2870 293a   the maps  h(p):
│ │ │ │ -0003cf90: 2041 5f30 2870 292d 2d3e 2041 5f31 2870   A_0(p)--> A_1(p
│ │ │ │ -0003cfa0: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ -0003cfb0: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ -0003cfc0: 202a 202a 6e6f 7465 2063 6f6d 706c 6578   * *note complex
│ │ │ │ -0003cfd0: 6974 793a 2063 6f6d 706c 6578 6974 792c  ity: complexity,
│ │ │ │ -0003cfe0: 202d 2d20 636f 6d70 6c65 7869 7479 206f   -- complexity o
│ │ │ │ -0003cff0: 6620 6120 6d6f 6475 6c65 206f 7665 7220  f a module over 
│ │ │ │ -0003d000: 6120 636f 6d70 6c65 7465 0a20 2020 2069  a complete.    i
│ │ │ │ -0003d010: 6e74 6572 7365 6374 696f 6e0a 0a57 6179  ntersection..Way
│ │ │ │ -0003d020: 7320 746f 2075 7365 206d 616b 6546 696e  s to use makeFin
│ │ │ │ -0003d030: 6974 6552 6573 6f6c 7574 696f 6e3a 0a3d  iteResolution:.=
│ │ │ │ -0003d040: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003d050: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003d060: 0a0a 2020 2a20 226d 616b 6546 696e 6974  ..  * "makeFinit
│ │ │ │ -0003d070: 6552 6573 6f6c 7574 696f 6e28 4d61 7472  eResolution(Matr
│ │ │ │ -0003d080: 6978 2c4c 6973 7429 220a 0a46 6f72 2074  ix,List)"..For t
│ │ │ │ -0003d090: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ -0003d0a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003d0b0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ -0003d0c0: 7465 206d 616b 6546 696e 6974 6552 6573  te makeFiniteRes
│ │ │ │ -0003d0d0: 6f6c 7574 696f 6e3a 206d 616b 6546 696e  olution: makeFin
│ │ │ │ -0003d0e0: 6974 6552 6573 6f6c 7574 696f 6e2c 2069  iteResolution, i
│ │ │ │ -0003d0f0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ -0003d100: 0a66 756e 6374 696f 6e3a 2028 4d61 6361  .function: (Maca
│ │ │ │ -0003d110: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ -0003d120: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d  unction,...-----
│ │ │ │ +0003cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +0003ce00: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +0003ce10: 0a20 202a 202a 6e6f 7465 206d 6174 7269  .  * *note matri
│ │ │ │ +0003ce20: 7846 6163 746f 7269 7a61 7469 6f6e 3a20  xFactorization: 
│ │ │ │ +0003ce30: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +0003ce40: 696f 6e2c 202d 2d20 4d61 7073 2069 6e20  ion, -- Maps in 
│ │ │ │ +0003ce50: 6120 6869 6768 6572 0a20 2020 2063 6f64  a higher.    cod
│ │ │ │ +0003ce60: 696d 656e 7369 6f6e 206d 6174 7269 7820  imension matrix 
│ │ │ │ +0003ce70: 6661 6374 6f72 697a 6174 696f 6e0a 2020  factorization.  
│ │ │ │ +0003ce80: 2a20 2a6e 6f74 6520 624d 6170 733a 2062  * *note bMaps: b
│ │ │ │ +0003ce90: 4d61 7073 2c20 2d2d 206c 6973 7420 7468  Maps, -- list th
│ │ │ │ +0003cea0: 6520 6d61 7073 2020 645f 703a 425f 3128  e maps  d_p:B_1(
│ │ │ │ +0003ceb0: 7029 2d2d 3e42 5f30 2870 2920 696e 2061  p)-->B_0(p) in a
│ │ │ │ +0003cec0: 0a20 2020 206d 6174 7269 7846 6163 746f  .    matrixFacto
│ │ │ │ +0003ced0: 7269 7a61 7469 6f6e 0a20 202a 202a 6e6f  rization.  * *no
│ │ │ │ +0003cee0: 7465 2070 7369 4d61 7073 3a20 7073 694d  te psiMaps: psiM
│ │ │ │ +0003cef0: 6170 732c 202d 2d20 6c69 7374 2074 6865  aps, -- list the
│ │ │ │ +0003cf00: 206d 6170 7320 2070 7369 2870 293a 2042   maps  psi(p): B
│ │ │ │ +0003cf10: 5f31 2870 2920 2d2d 3e20 415f 3028 702d  _1(p) --> A_0(p-
│ │ │ │ +0003cf20: 3129 2069 6e20 610a 2020 2020 6d61 7472  1) in a.    matr
│ │ │ │ +0003cf30: 6978 4661 6374 6f72 697a 6174 696f 6e0a  ixFactorization.
│ │ │ │ +0003cf40: 2020 2a20 2a6e 6f74 6520 684d 6170 733a    * *note hMaps:
│ │ │ │ +0003cf50: 2068 4d61 7073 2c20 2d2d 206c 6973 7420   hMaps, -- list 
│ │ │ │ +0003cf60: 7468 6520 6d61 7073 2020 6828 7029 3a20  the maps  h(p): 
│ │ │ │ +0003cf70: 415f 3028 7029 2d2d 3e20 415f 3128 7029  A_0(p)--> A_1(p)
│ │ │ │ +0003cf80: 2069 6e20 610a 2020 2020 6d61 7472 6978   in a.    matrix
│ │ │ │ +0003cf90: 4661 6374 6f72 697a 6174 696f 6e0a 2020  Factorization.  
│ │ │ │ +0003cfa0: 2a20 2a6e 6f74 6520 636f 6d70 6c65 7869  * *note complexi
│ │ │ │ +0003cfb0: 7479 3a20 636f 6d70 6c65 7869 7479 2c20  ty: complexity, 
│ │ │ │ +0003cfc0: 2d2d 2063 6f6d 706c 6578 6974 7920 6f66  -- complexity of
│ │ │ │ +0003cfd0: 2061 206d 6f64 756c 6520 6f76 6572 2061   a module over a
│ │ │ │ +0003cfe0: 2063 6f6d 706c 6574 650a 2020 2020 696e   complete.    in
│ │ │ │ +0003cff0: 7465 7273 6563 7469 6f6e 0a0a 5761 7973  tersection..Ways
│ │ │ │ +0003d000: 2074 6f20 7573 6520 6d61 6b65 4669 6e69   to use makeFini
│ │ │ │ +0003d010: 7465 5265 736f 6c75 7469 6f6e 3a0a 3d3d  teResolution:.==
│ │ │ │ +0003d020: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0003d030: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0003d040: 0a20 202a 2022 6d61 6b65 4669 6e69 7465  .  * "makeFinite
│ │ │ │ +0003d050: 5265 736f 6c75 7469 6f6e 284d 6174 7269  Resolution(Matri
│ │ │ │ +0003d060: 782c 4c69 7374 2922 0a0a 466f 7220 7468  x,List)"..For th
│ │ │ │ +0003d070: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ +0003d080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0003d090: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ +0003d0a0: 6520 6d61 6b65 4669 6e69 7465 5265 736f  e makeFiniteReso
│ │ │ │ +0003d0b0: 6c75 7469 6f6e 3a20 6d61 6b65 4669 6e69  lution: makeFini
│ │ │ │ +0003d0c0: 7465 5265 736f 6c75 7469 6f6e 2c20 6973  teResolution, is
│ │ │ │ +0003d0d0: 2061 202a 6e6f 7465 206d 6574 686f 640a   a *note method.
│ │ │ │ +0003d0e0: 6675 6e63 7469 6f6e 3a20 284d 6163 6175  function: (Macau
│ │ │ │ +0003d0f0: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +0003d100: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ +0003d110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d170: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ -0003d180: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ -0003d190: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ -0003d1a0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -0003d1b0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -0003d1c0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ -0003d1d0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -0003d1e0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ -0003d1f0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -0003d200: 6e73 2e6d 323a 3238 3939 3a30 2e0a 1f0a  ns.m2:2899:0....
│ │ │ │ -0003d210: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ -0003d220: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -0003d230: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ -0003d240: 3a20 6d61 6b65 4669 6e69 7465 5265 736f  : makeFiniteReso
│ │ │ │ -0003d250: 6c75 7469 6f6e 436f 6469 6d32 2c20 4e65  lutionCodim2, Ne
│ │ │ │ -0003d260: 7874 3a20 6d61 6b65 486f 6d6f 746f 7069  xt: makeHomotopi
│ │ │ │ -0003d270: 6573 2c20 5072 6576 3a20 6d61 6b65 4669  es, Prev: makeFi
│ │ │ │ -0003d280: 6e69 7465 5265 736f 6c75 7469 6f6e 2c20  niteResolution, 
│ │ │ │ -0003d290: 5570 3a20 546f 700a 0a6d 616b 6546 696e  Up: Top..makeFin
│ │ │ │ -0003d2a0: 6974 6552 6573 6f6c 7574 696f 6e43 6f64  iteResolutionCod
│ │ │ │ -0003d2b0: 696d 3220 2d2d 204d 6170 7320 6173 736f  im2 -- Maps asso
│ │ │ │ -0003d2c0: 6369 6174 6564 2074 6f20 7468 6520 6669  ciated to the fi
│ │ │ │ -0003d2d0: 6e69 7465 2072 6573 6f6c 7574 696f 6e20  nite resolution 
│ │ │ │ -0003d2e0: 6f66 2061 2068 6967 6820 7379 7a79 6779  of a high syzygy
│ │ │ │ -0003d2f0: 206d 6f64 756c 6520 696e 2063 6f64 696d   module in codim
│ │ │ │ -0003d300: 2032 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a   2.*************
│ │ │ │ +0003d150: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +0003d160: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +0003d170: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +0003d180: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +0003d190: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +0003d1a0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ +0003d1b0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +0003d1c0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +0003d1d0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +0003d1e0: 732e 6d32 3a32 3839 393a 302e 0a1f 0a46  s.m2:2899:0....F
│ │ │ │ +0003d1f0: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +0003d200: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +0003d210: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +0003d220: 206d 616b 6546 696e 6974 6552 6573 6f6c   makeFiniteResol
│ │ │ │ +0003d230: 7574 696f 6e43 6f64 696d 322c 204e 6578  utionCodim2, Nex
│ │ │ │ +0003d240: 743a 206d 616b 6548 6f6d 6f74 6f70 6965  t: makeHomotopie
│ │ │ │ +0003d250: 732c 2050 7265 763a 206d 616b 6546 696e  s, Prev: makeFin
│ │ │ │ +0003d260: 6974 6552 6573 6f6c 7574 696f 6e2c 2055  iteResolution, U
│ │ │ │ +0003d270: 703a 2054 6f70 0a0a 6d61 6b65 4669 6e69  p: Top..makeFini
│ │ │ │ +0003d280: 7465 5265 736f 6c75 7469 6f6e 436f 6469  teResolutionCodi
│ │ │ │ +0003d290: 6d32 202d 2d20 4d61 7073 2061 7373 6f63  m2 -- Maps assoc
│ │ │ │ +0003d2a0: 6961 7465 6420 746f 2074 6865 2066 696e  iated to the fin
│ │ │ │ +0003d2b0: 6974 6520 7265 736f 6c75 7469 6f6e 206f  ite resolution o
│ │ │ │ +0003d2c0: 6620 6120 6869 6768 2073 797a 7967 7920  f a high syzygy 
│ │ │ │ +0003d2d0: 6d6f 6475 6c65 2069 6e20 636f 6469 6d20  module in codim 
│ │ │ │ +0003d2e0: 320a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  2.**************
│ │ │ │ +0003d2f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003d300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003d310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003d320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003d330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003d340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003d350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003d360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ -0003d370: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -0003d380: 2020 6d61 7073 203d 206d 616b 6546 696e    maps = makeFin
│ │ │ │ -0003d390: 6974 6552 6573 6f6c 7574 696f 6e43 6f64  iteResolutionCod
│ │ │ │ -0003d3a0: 696d 3228 6666 2c6d 6629 0a20 202a 2049  im2(ff,mf).  * I
│ │ │ │ -0003d3b0: 6e70 7574 733a 0a20 2020 2020 202a 206d  nputs:.      * m
│ │ │ │ -0003d3c0: 662c 2061 202a 6e6f 7465 206c 6973 743a  f, a *note list:
│ │ │ │ -0003d3d0: 2028 4d61 6361 756c 6179 3244 6f63 294c   (Macaulay2Doc)L
│ │ │ │ -0003d3e0: 6973 742c 2c20 6d61 7472 6978 2066 6163  ist,, matrix fac
│ │ │ │ -0003d3f0: 746f 7269 7a61 7469 6f6e 0a20 2020 2020  torization.     
│ │ │ │ -0003d400: 202a 2066 662c 2061 202a 6e6f 7465 206d   * ff, a *note m
│ │ │ │ -0003d410: 6174 7269 783a 2028 4d61 6361 756c 6179  atrix: (Macaulay
│ │ │ │ -0003d420: 3244 6f63 294d 6174 7269 782c 2c20 7265  2Doc)Matrix,, re
│ │ │ │ -0003d430: 6775 6c61 7220 7365 7175 656e 6365 0a20  gular sequence. 
│ │ │ │ -0003d440: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -0003d450: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ -0003d460: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ -0003d470: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -0003d480: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ -0003d490: 2020 2020 202a 2043 6865 636b 203d 3e20       * Check => 
│ │ │ │ -0003d4a0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -0003d4b0: 7565 2066 616c 7365 0a20 202a 204f 7574  ue false.  * Out
│ │ │ │ -0003d4c0: 7075 7473 3a0a 2020 2020 2020 2a20 6d61  puts:.      * ma
│ │ │ │ -0003d4d0: 7073 2c20 6120 2a6e 6f74 6520 6861 7368  ps, a *note hash
│ │ │ │ -0003d4e0: 2074 6162 6c65 3a20 284d 6163 6175 6c61   table: (Macaula
│ │ │ │ -0003d4f0: 7932 446f 6329 4861 7368 5461 626c 652c  y2Doc)HashTable,
│ │ │ │ -0003d500: 2c20 6d61 6e79 206d 6170 730a 0a44 6573  , many maps..Des
│ │ │ │ -0003d510: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -0003d520: 3d3d 3d3d 0a0a 4769 7665 6e20 6120 636f  ====..Given a co
│ │ │ │ -0003d530: 6469 6d20 3220 6d61 7472 6978 2066 6163  dim 2 matrix fac
│ │ │ │ -0003d540: 746f 7269 7a61 7469 6f6e 2c20 6d61 6b65  torization, make
│ │ │ │ -0003d550: 7320 616c 6c20 7468 6520 636f 6d70 6f6e  s all the compon
│ │ │ │ -0003d560: 656e 7473 206f 6620 7468 650a 6469 6666  ents of the.diff
│ │ │ │ -0003d570: 6572 656e 7469 616c 2061 6e64 206f 6620  erential and of 
│ │ │ │ -0003d580: 7468 6520 686f 6d6f 746f 7069 6573 2074  the homotopies t
│ │ │ │ -0003d590: 6861 7420 6172 6520 7265 6c65 7661 6e74  hat are relevant
│ │ │ │ -0003d5a0: 2074 6f20 7468 6520 6669 6e69 7465 2072   to the finite r
│ │ │ │ -0003d5b0: 6573 6f6c 7574 696f 6e2c 0a61 7320 696e  esolution,.as in
│ │ │ │ -0003d5c0: 2034 2e32 2e33 206f 6620 4569 7365 6e62   4.2.3 of Eisenb
│ │ │ │ -0003d5d0: 7564 2d50 6565 7661 2022 4d69 6e69 6d61  ud-Peeva "Minima
│ │ │ │ -0003d5e0: 6c20 4672 6565 2052 6573 6f6c 7574 696f  l Free Resolutio
│ │ │ │ -0003d5f0: 6e73 2061 6e64 2048 6967 6865 7220 4d61  ns and Higher Ma
│ │ │ │ -0003d600: 7472 6978 0a46 6163 746f 7269 7a61 7469  trix.Factorizati
│ │ │ │ -0003d610: 6f6e 7322 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ons"..+---------
│ │ │ │ +0003d340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +0003d350: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +0003d360: 206d 6170 7320 3d20 6d61 6b65 4669 6e69   maps = makeFini
│ │ │ │ +0003d370: 7465 5265 736f 6c75 7469 6f6e 436f 6469  teResolutionCodi
│ │ │ │ +0003d380: 6d32 2866 662c 6d66 290a 2020 2a20 496e  m2(ff,mf).  * In
│ │ │ │ +0003d390: 7075 7473 3a0a 2020 2020 2020 2a20 6d66  puts:.      * mf
│ │ │ │ +0003d3a0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ +0003d3b0: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ +0003d3c0: 7374 2c2c 206d 6174 7269 7820 6661 6374  st,, matrix fact
│ │ │ │ +0003d3d0: 6f72 697a 6174 696f 6e0a 2020 2020 2020  orization.      
│ │ │ │ +0003d3e0: 2a20 6666 2c20 6120 2a6e 6f74 6520 6d61  * ff, a *note ma
│ │ │ │ +0003d3f0: 7472 6978 3a20 284d 6163 6175 6c61 7932  trix: (Macaulay2
│ │ │ │ +0003d400: 446f 6329 4d61 7472 6978 2c2c 2072 6567  Doc)Matrix,, reg
│ │ │ │ +0003d410: 756c 6172 2073 6571 7565 6e63 650a 2020  ular sequence.  
│ │ │ │ +0003d420: 2a20 2a6e 6f74 6520 4f70 7469 6f6e 616c  * *note Optional
│ │ │ │ +0003d430: 2069 6e70 7574 733a 2028 4d61 6361 756c   inputs: (Macaul
│ │ │ │ +0003d440: 6179 3244 6f63 2975 7369 6e67 2066 756e  ay2Doc)using fun
│ │ │ │ +0003d450: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ +0003d460: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020  onal inputs,:.  
│ │ │ │ +0003d470: 2020 2020 2a20 4368 6563 6b20 3d3e 202e      * Check => .
│ │ │ │ +0003d480: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +0003d490: 6520 6661 6c73 650a 2020 2a20 4f75 7470  e false.  * Outp
│ │ │ │ +0003d4a0: 7574 733a 0a20 2020 2020 202a 206d 6170  uts:.      * map
│ │ │ │ +0003d4b0: 732c 2061 202a 6e6f 7465 2068 6173 6820  s, a *note hash 
│ │ │ │ +0003d4c0: 7461 626c 653a 2028 4d61 6361 756c 6179  table: (Macaulay
│ │ │ │ +0003d4d0: 3244 6f63 2948 6173 6854 6162 6c65 2c2c  2Doc)HashTable,,
│ │ │ │ +0003d4e0: 206d 616e 7920 6d61 7073 0a0a 4465 7363   many maps..Desc
│ │ │ │ +0003d4f0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +0003d500: 3d3d 3d0a 0a47 6976 656e 2061 2063 6f64  ===..Given a cod
│ │ │ │ +0003d510: 696d 2032 206d 6174 7269 7820 6661 6374  im 2 matrix fact
│ │ │ │ +0003d520: 6f72 697a 6174 696f 6e2c 206d 616b 6573  orization, makes
│ │ │ │ +0003d530: 2061 6c6c 2074 6865 2063 6f6d 706f 6e65   all the compone
│ │ │ │ +0003d540: 6e74 7320 6f66 2074 6865 0a64 6966 6665  nts of the.diffe
│ │ │ │ +0003d550: 7265 6e74 6961 6c20 616e 6420 6f66 2074  rential and of t
│ │ │ │ +0003d560: 6865 2068 6f6d 6f74 6f70 6965 7320 7468  he homotopies th
│ │ │ │ +0003d570: 6174 2061 7265 2072 656c 6576 616e 7420  at are relevant 
│ │ │ │ +0003d580: 746f 2074 6865 2066 696e 6974 6520 7265  to the finite re
│ │ │ │ +0003d590: 736f 6c75 7469 6f6e 2c0a 6173 2069 6e20  solution,.as in 
│ │ │ │ +0003d5a0: 342e 322e 3320 6f66 2045 6973 656e 6275  4.2.3 of Eisenbu
│ │ │ │ +0003d5b0: 642d 5065 6576 6120 224d 696e 696d 616c  d-Peeva "Minimal
│ │ │ │ +0003d5c0: 2046 7265 6520 5265 736f 6c75 7469 6f6e   Free Resolution
│ │ │ │ +0003d5d0: 7320 616e 6420 4869 6768 6572 204d 6174  s and Higher Mat
│ │ │ │ +0003d5e0: 7269 780a 4661 6374 6f72 697a 6174 696f  rix.Factorizatio
│ │ │ │ +0003d5f0: 6e73 220a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  ns"..+----------
│ │ │ │ +0003d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d650: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b  -------+.|i1 : k
│ │ │ │ -0003d660: 6b3d 5a5a 2f31 3031 2020 2020 2020 2020  k=ZZ/101        
│ │ │ │ -0003d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d630: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b  ------+.|i1 : kk
│ │ │ │ +0003d640: 3d5a 5a2f 3130 3120 2020 2020 2020 2020  =ZZ/101         
│ │ │ │ +0003d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d670: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0003d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d690: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003d6e0: 6f31 203d 206b 6b20 2020 2020 2020 2020  o1 = kk         
│ │ │ │ -0003d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0003d6c0: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +0003d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d6f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003d700: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d720: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d760: 2020 207c 0a7c 6f31 203a 2051 756f 7469     |.|o1 : Quoti
│ │ │ │ -0003d770: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ -0003d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d7a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0003d740: 2020 7c0a 7c6f 3120 3a20 5175 6f74 6965    |.|o1 : Quotie
│ │ │ │ +0003d750: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +0003d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d780: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0003d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d7e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -0003d7f0: 2053 203d 206b 6b5b 612c 625d 2020 2020   S = kk[a,b]    
│ │ │ │ -0003d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +0003d7d0: 5320 3d20 6b6b 5b61 2c62 5d20 2020 2020  S = kk[a,b]     
│ │ │ │ +0003d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d800: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0003d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003d870: 0a7c 6f32 203d 2053 2020 2020 2020 2020  .|o2 = S        
│ │ │ │ +0003d840: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003d850: 7c6f 3220 3d20 5320 2020 2020 2020 2020  |o2 = S         
│ │ │ │ +0003d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d890: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d8b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0003d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d8f0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ -0003d900: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ -0003d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d930: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003d8d0: 2020 2020 7c0a 7c6f 3220 3a20 506f 6c79      |.|o2 : Poly
│ │ │ │ +0003d8e0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +0003d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d910: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0003d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933  -----------+.|i3
│ │ │ │ -0003d980: 203a 2066 6620 3d20 6d61 7472 6978 2261   : ff = matrix"a
│ │ │ │ -0003d990: 342c 6234 2220 2020 2020 2020 2020 2020  4,b4"           
│ │ │ │ +0003d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ +0003d960: 3a20 6666 203d 206d 6174 7269 7822 6134  : ff = matrix"a4
│ │ │ │ +0003d970: 2c62 3422 2020 2020 2020 2020 2020 2020  ,b4"            
│ │ │ │ +0003d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d990: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d9b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003d9c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da00: 207c 0a7c 6f33 203d 207c 2061 3420 6234   |.|o3 = | a4 b4
│ │ │ │ -0003da10: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0003da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d9e0: 7c0a 7c6f 3320 3d20 7c20 6134 2062 3420  |.|o3 = | a4 b4 
│ │ │ │ +0003d9f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003da20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0003da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da80: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003da90: 2020 2020 2020 2031 2020 2020 2020 3220         1      2 
│ │ │ │ -0003daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dac0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320            |.|o3 
│ │ │ │ -0003dad0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20  : Matrix S  <-- 
│ │ │ │ -0003dae0: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -0003daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003db00: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0003da60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003da70: 2020 2020 2020 3120 2020 2020 2032 2020        1      2  
│ │ │ │ +0003da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003daa0: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ +0003dab0: 204d 6174 7269 7820 5320 203c 2d2d 2053   Matrix S  <-- S
│ │ │ │ +0003dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dae0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0003daf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003db00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db50: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964  +.|i4 : R = S/id
│ │ │ │ -0003db60: 6561 6c20 6666 2020 2020 2020 2020 2020  eal ff          
│ │ │ │ -0003db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0003db30: 0a7c 6934 203a 2052 203d 2053 2f69 6465  .|i4 : R = S/ide
│ │ │ │ +0003db40: 616c 2066 6620 2020 2020 2020 2020 2020  al ff           
│ │ │ │ +0003db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003db70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0003db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003db90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dbb0: 2020 2020 207c 0a7c 6f34 203d 2052 2020       |.|o4 = R  
│ │ │ │  0003dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dbd0: 2020 2020 2020 7c0a 7c6f 3420 3d20 5220        |.|o4 = R 
│ │ │ │ +0003dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dbf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0003dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003dc60: 3420 3a20 5175 6f74 6965 6e74 5269 6e67  4 : QuotientRing
│ │ │ │ -0003dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dc90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003dca0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003dc30: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +0003dc40: 203a 2051 756f 7469 656e 7452 696e 6720   : QuotientRing 
│ │ │ │ +0003dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dc70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003dc80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0003dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dce0: 2d2d 2b0a 7c69 3520 3a20 4e20 3d20 525e  --+.|i5 : N = R^
│ │ │ │ -0003dcf0: 312f 6964 6561 6c22 6132 2c20 6162 2c20  1/ideal"a2, ab, 
│ │ │ │ -0003dd00: 6233 2220 2020 2020 2020 2020 2020 2020  b3"             
│ │ │ │ +0003dcc0: 2d2b 0a7c 6935 203a 204e 203d 2052 5e31  -+.|i5 : N = R^1
│ │ │ │ +0003dcd0: 2f69 6465 616c 2261 322c 2061 622c 2062  /ideal"a2, ab, b
│ │ │ │ +0003dce0: 3322 2020 2020 2020 2020 2020 2020 2020  3"              
│ │ │ │ +0003dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dd00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0003dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dd60: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ -0003dd70: 636f 6b65 726e 656c 207c 2061 3220 6162  cokernel | a2 ab
│ │ │ │ -0003dd80: 2062 3320 7c20 2020 2020 2020 2020 2020   b3 |           
│ │ │ │ +0003dd40: 2020 2020 2020 207c 0a7c 6f35 203d 2063         |.|o5 = c
│ │ │ │ +0003dd50: 6f6b 6572 6e65 6c20 7c20 6132 2061 6220  okernel | a2 ab 
│ │ │ │ +0003dd60: 6233 207c 2020 2020 2020 2020 2020 2020  b3 |            
│ │ │ │ +0003dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dd80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0003dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dda0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ddc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dde0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003ddf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003de00: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0003de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de30: 207c 0a7c 6f35 203a 2052 2d6d 6f64 756c   |.|o5 : R-modul
│ │ │ │ -0003de40: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -0003de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0003dde0: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +0003ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003de10: 7c0a 7c6f 3520 3a20 522d 6d6f 6475 6c65  |.|o5 : R-module
│ │ │ │ +0003de20: 2c20 7175 6f74 6965 6e74 206f 6620 5220  , quotient of R 
│ │ │ │ +0003de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003de50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003deb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204e  -------+.|i6 : N
│ │ │ │ -0003dec0: 203d 2063 6f6b 6572 2076 6172 7320 5220   = coker vars R 
│ │ │ │ -0003ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003de90: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4e20  ------+.|i6 : N 
│ │ │ │ +0003dea0: 3d20 636f 6b65 7220 7661 7273 2052 2020  = coker vars R  
│ │ │ │ +0003deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ded0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0003dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003def0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003df40: 6f36 203d 2063 6f6b 6572 6e65 6c20 7c20  o6 = cokernel | 
│ │ │ │ -0003df50: 6120 6220 7c20 2020 2020 2020 2020 2020  a b |           
│ │ │ │ -0003df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003df10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0003df20: 3620 3d20 636f 6b65 726e 656c 207c 2061  6 = cokernel | a
│ │ │ │ +0003df30: 2062 207c 2020 2020 2020 2020 2020 2020   b |            
│ │ │ │ +0003df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003df50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003df60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dfa0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0003dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dfc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003dfc0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  0003dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dfe0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ -0003dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e000: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d        |.|o6 : R-
│ │ │ │ -0003e010: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ -0003e020: 206f 6620 5220 2020 2020 2020 2020 2020   of R           
│ │ │ │ -0003e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e040: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0003dfe0: 2020 2020 207c 0a7c 6f36 203a 2052 2d6d       |.|o6 : R-m
│ │ │ │ +0003dff0: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ +0003e000: 6f66 2052 2020 2020 2020 2020 2020 2020  of R            
│ │ │ │ +0003e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e020: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0003e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0003e090: 3720 3a20 4d20 3d20 6869 6768 5379 7a79  7 : M = highSyzy
│ │ │ │ -0003e0a0: 6779 204e 2020 2020 2020 2020 2020 2020  gy N            
│ │ │ │ -0003e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e0c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003e0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +0003e070: 203a 204d 203d 2068 6967 6853 797a 7967   : M = highSyzyg
│ │ │ │ +0003e080: 7920 4e20 2020 2020 2020 2020 2020 2020  y N             
│ │ │ │ +0003e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003e0b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e110: 2020 7c0a 7c6f 3720 3d20 636f 6b65 726e    |.|o7 = cokern
│ │ │ │ -0003e120: 656c 207b 327d 207c 2030 202d 6233 2061  el {2} | 0 -b3 a
│ │ │ │ -0003e130: 3320 3020 7c20 2020 2020 2020 2020 2020  3 0 |           
│ │ │ │ -0003e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e150: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0003e160: 2020 2020 2020 7b34 7d20 7c20 6220 6120        {4} | b a 
│ │ │ │ -0003e170: 2020 3020 2030 207c 2020 2020 2020 2020    0  0 |        
│ │ │ │ -0003e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e190: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003e1a0: 2020 2020 2020 2020 207b 347d 207c 2030           {4} | 0
│ │ │ │ -0003e1b0: 2030 2020 2062 2020 6120 7c20 2020 2020   0   b  a |     
│ │ │ │ +0003e0f0: 207c 0a7c 6f37 203d 2063 6f6b 6572 6e65   |.|o7 = cokerne
│ │ │ │ +0003e100: 6c20 7b32 7d20 7c20 3020 2d62 3320 6133  l {2} | 0 -b3 a3
│ │ │ │ +0003e110: 2030 207c 2020 2020 2020 2020 2020 2020   0 |            
│ │ │ │ +0003e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e130: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003e140: 2020 2020 207b 347d 207c 2062 2061 2020       {4} | b a  
│ │ │ │ +0003e150: 2030 2020 3020 7c20 2020 2020 2020 2020   0  0 |         
│ │ │ │ +0003e160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e170: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003e180: 2020 2020 2020 2020 7b34 7d20 7c20 3020          {4} | 0 
│ │ │ │ +0003e190: 3020 2020 6220 2061 207c 2020 2020 2020  0   b  a |      
│ │ │ │ +0003e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e1b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0003e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e1d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e1f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003e220: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003e230: 2020 2020 2020 2020 2020 2020 2033 2020               3  
│ │ │ │ -0003e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e260: 207c 0a7c 6f37 203a 2052 2d6d 6f64 756c   |.|o7 : R-modul
│ │ │ │ -0003e270: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -0003e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e2a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0003e210: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ +0003e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e240: 7c0a 7c6f 3720 3a20 522d 6d6f 6475 6c65  |.|o7 : R-module
│ │ │ │ +0003e250: 2c20 7175 6f74 6965 6e74 206f 6620 5220  , quotient of R 
│ │ │ │ +0003e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e280: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0003e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 204d  -------+.|i8 : M
│ │ │ │ -0003e2f0: 5320 3d20 7075 7368 466f 7277 6172 6428  S = pushForward(
│ │ │ │ -0003e300: 6d61 7028 522c 5329 2c4d 2920 2020 2020  map(R,S),M)     
│ │ │ │ +0003e2c0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53  ------+.|i8 : MS
│ │ │ │ +0003e2d0: 203d 2070 7573 6846 6f72 7761 7264 286d   = pushForward(m
│ │ │ │ +0003e2e0: 6170 2852 2c53 292c 4d29 2020 2020 2020  ap(R,S),M)      
│ │ │ │ +0003e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e300: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0003e310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e320: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003e370: 6f38 203d 2063 6f6b 6572 6e65 6c20 7b32  o8 = cokernel {2
│ │ │ │ -0003e380: 7d20 7c20 3020 6233 2061 3320 3020 3020  } | 0 b3 a3 0 0 
│ │ │ │ -0003e390: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0003e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e3b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003e3c0: 207b 347d 207c 2062 202d 6120 3020 2030   {4} | b -a 0  0
│ │ │ │ -0003e3d0: 2030 2020 7c20 2020 2020 2020 2020 2020   0  |           
│ │ │ │ -0003e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e3f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003e400: 2020 2020 7b34 7d20 7c20 3020 3020 2062      {4} | 0 0  b
│ │ │ │ -0003e410: 2020 6120 6234 207c 2020 2020 2020 2020    a b4 |        
│ │ │ │ +0003e340: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0003e350: 3820 3d20 636f 6b65 726e 656c 207b 327d  8 = cokernel {2}
│ │ │ │ +0003e360: 207c 2030 2062 3320 6133 2030 2030 2020   | 0 b3 a3 0 0  
│ │ │ │ +0003e370: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003e380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003e390: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003e3a0: 7b34 7d20 7c20 6220 2d61 2030 2020 3020  {4} | b -a 0  0 
│ │ │ │ +0003e3b0: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ +0003e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e3d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0003e3e0: 2020 207b 347d 207c 2030 2030 2020 6220     {4} | 0 0  b 
│ │ │ │ +0003e3f0: 2061 2062 3420 7c20 2020 2020 2020 2020   a b4 |         
│ │ │ │ +0003e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e410: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0003e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e430: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e450: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0003e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e470: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003e470: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │  0003e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e490: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -0003e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003e4c0: 3820 3a20 532d 6d6f 6475 6c65 2c20 7175  8 : S-module, qu
│ │ │ │ -0003e4d0: 6f74 6965 6e74 206f 6620 5320 2020 2020  otient of S     
│ │ │ │ -0003e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e4f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003e500: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003e490: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +0003e4a0: 203a 2053 2d6d 6f64 756c 652c 2071 756f   : S-module, quo
│ │ │ │ +0003e4b0: 7469 656e 7420 6f66 2053 2020 2020 2020  tient of S      
│ │ │ │ +0003e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e4d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003e4e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0003e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e540: 2d2d 2b0a 7c69 3920 3a20 6d66 203d 206d  --+.|i9 : mf = m
│ │ │ │ -0003e550: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -0003e560: 6f6e 2866 662c 204d 2920 2020 2020 2020  on(ff, M)       
│ │ │ │ +0003e520: 2d2b 0a7c 6939 203a 206d 6620 3d20 6d61  -+.|i9 : mf = ma
│ │ │ │ +0003e530: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +0003e540: 6e28 6666 2c20 4d29 2020 2020 2020 2020  n(ff, M)        
│ │ │ │ +0003e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e560: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0003e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e580: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e5c0: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -0003e5d0: 7b7b 347d 207c 2061 202d 6220 3020 3020  {{4} | a -b 0 0 
│ │ │ │ -0003e5e0: 207c 2c20 7b35 7d20 7c20 6133 2062 2030   |, {5} | a3 b 0
│ │ │ │ -0003e5f0: 2020 2030 2020 3020 207c 2c20 7b32 7d20     0  0  |, {2} 
│ │ │ │ -0003e600: 7c20 3020 2d31 2030 207c 7d7c 0a7c 2020  | 0 -1 0 |}|.|  
│ │ │ │ -0003e610: 2020 2020 7b32 7d20 7c20 3020 6133 2030      {2} | 0 a3 0
│ │ │ │ -0003e620: 2062 3320 7c20 207b 357d 207c 2030 2020   b3 |  {5} | 0  
│ │ │ │ -0003e630: 6120 2d62 3320 3020 2030 2020 7c20 207b  a -b3 0  0  |  {
│ │ │ │ -0003e640: 347d 207c 2030 2030 2020 3120 7c20 7c0a  4} | 0 0  1 | |.
│ │ │ │ -0003e650: 7c20 2020 2020 207b 347d 207c 2030 2030  |      {4} | 0 0
│ │ │ │ -0003e660: 2020 6220 6120 207c 2020 7b35 7d20 7c20    b a  |  {5} | 
│ │ │ │ -0003e670: 3020 2030 2030 2020 202d 6120 6233 207c  0  0 0   -a b3 |
│ │ │ │ -0003e680: 2020 7b34 7d20 7c20 3120 3020 2030 207c    {4} | 1 0  0 |
│ │ │ │ -0003e690: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003e6a0: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ -0003e6b0: 207c 2030 2020 3020 6133 2020 6220 2030   | 0  0 a3  b  0
│ │ │ │ -0003e6c0: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ -0003e6d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0003e5a0: 2020 2020 2020 207c 0a7c 6f39 203d 207b         |.|o9 = {
│ │ │ │ +0003e5b0: 7b34 7d20 7c20 6120 2d62 2030 2030 2020  {4} | a -b 0 0  
│ │ │ │ +0003e5c0: 7c2c 207b 357d 207c 2061 3320 6220 3020  |, {5} | a3 b 0 
│ │ │ │ +0003e5d0: 2020 3020 2030 2020 7c2c 207b 327d 207c    0  0  |, {2} |
│ │ │ │ +0003e5e0: 2030 202d 3120 3020 7c7d 7c0a 7c20 2020   0 -1 0 |}|.|   
│ │ │ │ +0003e5f0: 2020 207b 327d 207c 2030 2061 3320 3020     {2} | 0 a3 0 
│ │ │ │ +0003e600: 6233 207c 2020 7b35 7d20 7c20 3020 2061  b3 |  {5} | 0  a
│ │ │ │ +0003e610: 202d 6233 2030 2020 3020 207c 2020 7b34   -b3 0  0  |  {4
│ │ │ │ +0003e620: 7d20 7c20 3020 3020 2031 207c 207c 0a7c  } | 0 0  1 | |.|
│ │ │ │ +0003e630: 2020 2020 2020 7b34 7d20 7c20 3020 3020        {4} | 0 0 
│ │ │ │ +0003e640: 2062 2061 2020 7c20 207b 357d 207c 2030   b a  |  {5} | 0
│ │ │ │ +0003e650: 2020 3020 3020 2020 2d61 2062 3320 7c20    0 0   -a b3 | 
│ │ │ │ +0003e660: 207b 347d 207c 2031 2030 2020 3020 7c20   {4} | 1 0  0 | 
│ │ │ │ +0003e670: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003e680: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ +0003e690: 7c20 3020 2030 2061 3320 2062 2020 3020  | 0  0 a3  b  0 
│ │ │ │ +0003e6a0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0003e6b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e710: 2020 2020 2020 207c 0a7c 6f39 203a 204c         |.|o9 : L
│ │ │ │ -0003e720: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ -0003e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e750: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0003e6f0: 2020 2020 2020 7c0a 7c6f 3920 3a20 4c69        |.|o9 : Li
│ │ │ │ +0003e700: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0003e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e730: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0003e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0003e7a0: 6931 3020 3a20 4720 3d20 6d61 6b65 4669  i10 : G = makeFi
│ │ │ │ -0003e7b0: 6e69 7465 5265 736f 6c75 7469 6f6e 436f  niteResolutionCo
│ │ │ │ -0003e7c0: 6469 6d32 2866 662c 6d66 2920 2020 2020  dim2(ff,mf)     
│ │ │ │ +0003e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0003e780: 3130 203a 2047 203d 206d 616b 6546 696e  10 : G = makeFin
│ │ │ │ +0003e790: 6974 6552 6573 6f6c 7574 696f 6e43 6f64  iteResolutionCod
│ │ │ │ +0003e7a0: 696d 3228 6666 2c6d 6629 2020 2020 2020  im2(ff,mf)      
│ │ │ │ +0003e7b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003e7c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e7e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e820: 2020 207c 0a7c 6f31 3020 3d20 4861 7368     |.|o10 = Hash
│ │ │ │ -0003e830: 5461 626c 657b 2261 6c70 6861 2220 3d3e  Table{"alpha" =>
│ │ │ │ -0003e840: 207b 357d 207c 2030 2020 2030 207c 2020   {5} | 0   0 |  
│ │ │ │ -0003e850: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │ -0003e860: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003e800: 2020 7c0a 7c6f 3130 203d 2048 6173 6854    |.|o10 = HashT
│ │ │ │ +0003e810: 6162 6c65 7b22 616c 7068 6122 203d 3e20  able{"alpha" => 
│ │ │ │ +0003e820: 7b35 7d20 7c20 3020 2020 3020 7c20 2020  {5} | 0   0 |   
│ │ │ │ +0003e830: 2020 2020 2020 2020 2020 207d 2020 2020             }    
│ │ │ │ +0003e840: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e860: 2020 207b 357d 207c 202d 6233 2030 207c     {5} | -b3 0 |
│ │ │ │  0003e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e880: 2020 2020 7b35 7d20 7c20 2d62 3320 3020      {5} | -b3 0 
│ │ │ │ -0003e890: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003e8a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003e8b0: 2020 2020 2020 2020 2020 2020 2262 2220              "b" 
│ │ │ │ -0003e8c0: 3d3e 207b 347d 207c 2062 2061 207c 2020  => {4} | b a |  
│ │ │ │ -0003e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2022                 "
│ │ │ │ -0003e900: 6831 2722 203d 3e20 7b35 7d20 7c20 3020  h1'" => {5} | 0 
│ │ │ │ -0003e910: 2020 3020 2030 2020 7c20 2020 2020 2020    0  0  |       
│ │ │ │ -0003e920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003e930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0003e940: 2020 2020 2020 2020 2020 207b 357d 207c             {5} |
│ │ │ │ -0003e950: 202d 6233 2030 2020 3020 207c 2020 2020   -b3 0  0  |    
│ │ │ │ -0003e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e970: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003e980: 2020 2020 2020 2020 2020 2020 2020 7b35                {5
│ │ │ │ -0003e990: 7d20 7c20 3020 2020 2d61 2062 3320 7c20  } | 0   -a b3 | 
│ │ │ │ +0003e880: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003e890: 2020 2020 2020 2020 2020 2022 6222 203d             "b" =
│ │ │ │ +0003e8a0: 3e20 7b34 7d20 7c20 6220 6120 7c20 2020  > {4} | b a |   
│ │ │ │ +0003e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e8c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0003e8d0: 2020 2020 2020 2020 2020 2020 2020 2268                "h
│ │ │ │ +0003e8e0: 3127 2220 3d3e 207b 357d 207c 2030 2020  1'" => {5} | 0  
│ │ │ │ +0003e8f0: 2030 2020 3020 207c 2020 2020 2020 2020   0  0  |        
│ │ │ │ +0003e900: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003e910: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003e920: 2020 2020 2020 2020 2020 7b35 7d20 7c20            {5} | 
│ │ │ │ +0003e930: 2d62 3320 3020 2030 2020 7c20 2020 2020  -b3 0  0  |     
│ │ │ │ +0003e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e950: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003e960: 2020 2020 2020 2020 2020 2020 207b 357d               {5}
│ │ │ │ +0003e970: 207c 2030 2020 202d 6120 6233 207c 2020   | 0   -a b3 |  
│ │ │ │ +0003e980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0003e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e9b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0003e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e9d0: 207b 357d 207c 2061 3320 2062 2020 3020   {5} | a3  b  0 
│ │ │ │ -0003e9e0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -0003e9f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003ea00: 2020 2020 2020 2020 2020 2022 6831 2220             "h1" 
│ │ │ │ -0003ea10: 3d3e 207b 357d 207c 2030 2020 2030 2020  => {5} | 0   0  
│ │ │ │ -0003ea20: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -0003ea30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ea50: 2020 2020 2020 7b35 7d20 7c20 2d62 3320        {5} | -b3 
│ │ │ │ -0003ea60: 3020 2030 2020 7c20 2020 2020 2020 2020  0  0  |         
│ │ │ │ -0003ea70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003ea80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003ea90: 2020 2020 2020 2020 207b 357d 207c 2030           {5} | 0
│ │ │ │ -0003eaa0: 2020 202d 6120 6233 207c 2020 2020 2020     -a b3 |      
│ │ │ │ -0003eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eac0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003ead0: 2020 2020 2020 2020 2020 2020 7b35 7d20              {5} 
│ │ │ │ -0003eae0: 7c20 6133 2020 6220 2030 2020 7c20 2020  | a3  b  0  |   
│ │ │ │ -0003eaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb00: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003eb10: 2020 2020 2020 2022 6d75 2220 3d3e 207b         "mu" => {
│ │ │ │ -0003eb20: 357d 207c 2061 3320 6220 7c20 2020 2020  5} | a3 b |     
│ │ │ │ +0003e9b0: 7b35 7d20 7c20 6133 2020 6220 2030 2020  {5} | a3  b  0  
│ │ │ │ +0003e9c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003e9d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003e9e0: 2020 2020 2020 2020 2020 2268 3122 203d            "h1" =
│ │ │ │ +0003e9f0: 3e20 7b35 7d20 7c20 3020 2020 3020 2030  > {5} | 0   0  0
│ │ │ │ +0003ea00: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +0003ea10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003ea20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea30: 2020 2020 207b 357d 207c 202d 6233 2030       {5} | -b3 0
│ │ │ │ +0003ea40: 2020 3020 207c 2020 2020 2020 2020 2020    0  |          
│ │ │ │ +0003ea50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0003ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea70: 2020 2020 2020 2020 7b35 7d20 7c20 3020          {5} | 0 
│ │ │ │ +0003ea80: 2020 2d61 2062 3320 7c20 2020 2020 2020    -a b3 |       
│ │ │ │ +0003ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eaa0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003eab0: 2020 2020 2020 2020 2020 207b 357d 207c             {5} |
│ │ │ │ +0003eac0: 2061 3320 2062 2020 3020 207c 2020 2020   a3  b  0  |    
│ │ │ │ +0003ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eae0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003eaf0: 2020 2020 2020 226d 7522 203d 3e20 7b35        "mu" => {5
│ │ │ │ +0003eb00: 7d20 7c20 6133 2062 207c 2020 2020 2020  } | a3 b |      
│ │ │ │ +0003eb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eb20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0003eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003eb40: 207b 357d 207c 2030 2020 6120 7c20 2020   {5} | 0  a |   
│ │ │ │  0003eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb60: 2020 7b35 7d20 7c20 3020 2061 207c 2020    {5} | 0  a |  
│ │ │ │ -0003eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eb80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0003eb90: 2020 2020 2020 2020 2020 2020 2022 7061               "pa
│ │ │ │ -0003eba0: 7274 6961 6c22 203d 3e20 7b34 7d20 7c20  rtial" => {4} | 
│ │ │ │ -0003ebb0: 6120 2d62 207c 2020 2020 2020 2020 2020  a -b |          
│ │ │ │ -0003ebc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ebe0: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ -0003ebf0: 207c 2030 2061 3320 7c20 2020 2020 2020   | 0 a3 |       
│ │ │ │ -0003ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0003ec20: 2020 2022 7073 6922 203d 3e20 7b34 7d20     "psi" => {4} 
│ │ │ │ -0003ec30: 7c20 3020 3020 207c 2020 2020 2020 2020  | 0 0  |        
│ │ │ │ -0003ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec50: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0003ec60: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -0003ec70: 327d 207c 2030 2062 3320 7c20 2020 2020  2} | 0 b3 |     
│ │ │ │ +0003eb60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0003eb70: 2020 2020 2020 2020 2020 2020 2270 6172              "par
│ │ │ │ +0003eb80: 7469 616c 2220 3d3e 207b 347d 207c 2061  tial" => {4} | a
│ │ │ │ +0003eb90: 202d 6220 7c20 2020 2020 2020 2020 2020   -b |           
│ │ │ │ +0003eba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0003ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ebc0: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ +0003ebd0: 7c20 3020 6133 207c 2020 2020 2020 2020  | 0 a3 |        
│ │ │ │ +0003ebe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003ebf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003ec00: 2020 2270 7369 2220 3d3e 207b 347d 207c    "psi" => {4} |
│ │ │ │ +0003ec10: 2030 2030 2020 7c20 2020 2020 2020 2020   0 0  |         
│ │ │ │ +0003ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec30: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0003ec40: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +0003ec50: 7d20 7c20 3020 6233 207c 2020 2020 2020  } | 0 b3 |      
│ │ │ │ +0003ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ec70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0003ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ec90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0003eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ecb0: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ -0003ecc0: 2035 2020 2020 2020 3220 2020 2020 2020   5      2       
│ │ │ │ -0003ecd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003ece0: 2020 2020 2020 2020 2020 2020 2272 6573              "res
│ │ │ │ -0003ecf0: 6f6c 7574 696f 6e22 203d 3e20 5320 203c  olution" => S  <
│ │ │ │ -0003ed00: 2d2d 2053 2020 3c2d 2d20 5320 203c 2d2d  -- S  <-- S  <--
│ │ │ │ -0003ed10: 2030 2020 2020 2020 2020 2020 7c0a 7c20   0          |.| 
│ │ │ │ +0003ec90: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ +0003eca0: 3520 2020 2020 2032 2020 2020 2020 2020  5      2        
│ │ │ │ +0003ecb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003ecc0: 2020 2020 2020 2020 2020 2022 7265 736f             "reso
│ │ │ │ +0003ecd0: 6c75 7469 6f6e 2220 3d3e 2053 2020 3c2d  lution" => S  <-
│ │ │ │ +0003ece0: 2d20 5320 203c 2d2d 2053 2020 3c2d 2d20  - S  <-- S  <-- 
│ │ │ │ +0003ecf0: 3020 2020 2020 2020 2020 207c 0a7c 2020  0          |.|  
│ │ │ │ +0003ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003ed60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0003ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed80: 2020 3020 2020 2020 2031 2020 2020 2020    0      1      
│ │ │ │ -0003ed90: 3220 2020 2020 2033 2020 2020 2020 2020  2      3        
│ │ │ │ -0003eda0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0003edb0: 2020 2020 2022 7369 676d 6122 203d 3e20       "sigma" => 
│ │ │ │ -0003edc0: 7b35 7d20 7c20 6233 207c 2020 2020 2020  {5} | b3 |      
│ │ │ │ +0003ed30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003ed40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ed60: 2030 2020 2020 2020 3120 2020 2020 2032   0      1      2
│ │ │ │ +0003ed70: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +0003ed80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003ed90: 2020 2020 2273 6967 6d61 2220 3d3e 207b      "sigma" => {
│ │ │ │ +0003eda0: 357d 207c 2062 3320 7c20 2020 2020 2020  5} | b3 |       
│ │ │ │ +0003edb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003edc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0003edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ede0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003ede0: 2020 7b35 7d20 7c20 3020 207c 2020 2020    {5} | 0  |    
│ │ │ │  0003edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee00: 2020 207b 357d 207c 2030 2020 7c20 2020     {5} | 0  |   
│ │ │ │ -0003ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee20: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0003ee30: 2020 2020 2020 2020 2020 2022 7461 7522             "tau"
│ │ │ │ -0003ee40: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ -0003ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee60: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003ee70: 2020 2020 2020 2020 2020 2020 2020 2275                "u
│ │ │ │ -0003ee80: 2220 3d3e 207b 387d 207c 2031 2030 207c  " => {8} | 1 0 |
│ │ │ │ +0003ee00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0003ee10: 2020 2020 2020 2020 2020 2274 6175 2220            "tau" 
│ │ │ │ +0003ee20: 3d3e 2030 2020 2020 2020 2020 2020 2020  => 0            
│ │ │ │ +0003ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ee40: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003ee50: 2020 2020 2020 2020 2020 2020 2022 7522               "u"
│ │ │ │ +0003ee60: 203d 3e20 7b38 7d20 7c20 3120 3020 7c20   => {8} | 1 0 | 
│ │ │ │ +0003ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ee80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0003ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eea0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0003eeb0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0003eec0: 2022 7622 203d 3e20 7b39 7d20 7c20 3020   "v" => {9} | 0 
│ │ │ │ -0003eed0: 2d31 207c 2020 2020 2020 2020 2020 2020  -1 |            
│ │ │ │ -0003eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eef0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0003ef00: 2020 2020 2020 2020 2020 207b 397d 207c             {9} |
│ │ │ │ -0003ef10: 2031 2030 2020 7c20 2020 2020 2020 2020   1 0  |         
│ │ │ │ -0003ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0003ef40: 2020 2020 2020 2022 5822 203d 3e20 3020         "X" => 0 
│ │ │ │ -0003ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0003ef80: 2020 2020 2020 2020 2020 2259 2220 3d3e            "Y" =>
│ │ │ │ -0003ef90: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +0003eea0: 2276 2220 3d3e 207b 397d 207c 2030 202d  "v" => {9} | 0 -
│ │ │ │ +0003eeb0: 3120 7c20 2020 2020 2020 2020 2020 2020  1 |             
│ │ │ │ +0003eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003eed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0003eee0: 2020 2020 2020 2020 2020 7b39 7d20 7c20            {9} | 
│ │ │ │ +0003eef0: 3120 3020 207c 2020 2020 2020 2020 2020  1 0  |          
│ │ │ │ +0003ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0003ef20: 2020 2020 2020 2258 2220 3d3e 2030 2020        "X" => 0  
│ │ │ │ +0003ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003ef60: 2020 2020 2020 2020 2022 5922 203d 3e20           "Y" => 
│ │ │ │ +0003ef70: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +0003ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0003efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003efb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0003efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003eff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0003f000: 6f31 3020 3a20 4861 7368 5461 626c 6520  o10 : HashTable 
│ │ │ │ -0003f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f040: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0003efd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0003efe0: 3130 203a 2048 6173 6854 6162 6c65 2020  10 : HashTable  
│ │ │ │ +0003eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0003f020: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0003f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f080: 2d2d 2d2b 0a7c 6931 3120 3a20 4620 3d20  ---+.|i11 : F = 
│ │ │ │ -0003f090: 4723 2272 6573 6f6c 7574 696f 6e22 2020  G#"resolution"  
│ │ │ │ -0003f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f060: 2d2d 2b0a 7c69 3131 203a 2046 203d 2047  --+.|i11 : F = G
│ │ │ │ +0003f070: 2322 7265 736f 6c75 7469 6f6e 2220 2020  #"resolution"   
│ │ │ │ +0003f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f0a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0003f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0003f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0003f110: 2020 2033 2020 2020 2020 3520 2020 2020     3      5     
│ │ │ │ -0003f120: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0003f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f140: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0003f150: 3131 203d 2053 2020 3c2d 2d20 5320 203c  11 = S  <-- S  <
│ │ │ │ -0003f160: 2d2d 2053 2020 3c2d 2d20 3020 2020 2020  -- S  <-- 0     
│ │ │ │ -0003f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0003f190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0003f0e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0003f0f0: 2020 3320 2020 2020 2035 2020 2020 2020    3      5      
│ │ │ │ +0003f100: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f120: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0003f130: 3120 3d20 5320 203c 2d2d 2053 2020 3c2d  1 = S  <-- S  <-
│ │ │ │ +0003f140: 2d20 5320 203c 2d2d 2030 2020 2020 2020  - S  <-- 0      
│ │ │ │ +0003f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0003f170: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0003f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f1d0: 2020 7c0a 7c20 2020 2020 2030 2020 2020    |.|      0    
│ │ │ │ -0003f1e0: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ -0003f1f0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0003f1b0: 207c 0a7c 2020 2020 2020 3020 2020 2020   |.|      0     
│ │ │ │ +0003f1c0: 2031 2020 2020 2020 3220 2020 2020 2033   1      2      3
│ │ │ │ +0003f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f1f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0003f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f210: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0003f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f250: 2020 2020 2020 2020 7c0a 7c6f 3131 203a          |.|o11 :
│ │ │ │ -0003f260: 2043 6f6d 706c 6578 2020 2020 2020 2020   Complex        
│ │ │ │ -0003f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f290: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0003f230: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ +0003f240: 436f 6d70 6c65 7820 2020 2020 2020 2020  Complex         
│ │ │ │ +0003f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f270: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0003f280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0003f2e0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -0003f2f0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6d61  ==..  * *note ma
│ │ │ │ -0003f300: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ -0003f310: 6f6e 3a20 6d61 6b65 4669 6e69 7465 5265  on: makeFiniteRe
│ │ │ │ -0003f320: 736f 6c75 7469 6f6e 2c20 2d2d 2066 696e  solution, -- fin
│ │ │ │ -0003f330: 6974 6520 7265 736f 6c75 7469 6f6e 206f  ite resolution o
│ │ │ │ -0003f340: 6620 610a 2020 2020 6d61 7472 6978 2066  f a.    matrix f
│ │ │ │ -0003f350: 6163 746f 7269 7a61 7469 6f6e 206d 6f64  actorization mod
│ │ │ │ -0003f360: 756c 6520 4d0a 0a57 6179 7320 746f 2075  ule M..Ways to u
│ │ │ │ -0003f370: 7365 206d 616b 6546 696e 6974 6552 6573  se makeFiniteRes
│ │ │ │ -0003f380: 6f6c 7574 696f 6e43 6f64 696d 323a 0a3d  olutionCodim2:.=
│ │ │ │ -0003f390: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003f3a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0003f3b0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b  ======..  * "mak
│ │ │ │ -0003f3c0: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ -0003f3d0: 6e43 6f64 696d 3228 4d61 7472 6978 2c4c  nCodim2(Matrix,L
│ │ │ │ -0003f3e0: 6973 7429 220a 0a46 6f72 2074 6865 2070  ist)"..For the p
│ │ │ │ -0003f3f0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -0003f400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -0003f410: 6520 6f62 6a65 6374 202a 6e6f 7465 206d  e object *note m
│ │ │ │ -0003f420: 616b 6546 696e 6974 6552 6573 6f6c 7574  akeFiniteResolut
│ │ │ │ -0003f430: 696f 6e43 6f64 696d 323a 206d 616b 6546  ionCodim2: makeF
│ │ │ │ -0003f440: 696e 6974 6552 6573 6f6c 7574 696f 6e43  initeResolutionC
│ │ │ │ -0003f450: 6f64 696d 322c 2069 7320 610a 2a6e 6f74  odim2, is a.*not
│ │ │ │ -0003f460: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ -0003f470: 6e20 7769 7468 206f 7074 696f 6e73 3a20  n with options: 
│ │ │ │ -0003f480: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -0003f490: 7468 6f64 4675 6e63 7469 6f6e 5769 7468  thodFunctionWith
│ │ │ │ -0003f4a0: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d  Options,...-----
│ │ │ │ +0003f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +0003f2c0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +0003f2d0: 3d0a 0a20 202a 202a 6e6f 7465 206d 616b  =..  * *note mak
│ │ │ │ +0003f2e0: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ +0003f2f0: 6e3a 206d 616b 6546 696e 6974 6552 6573  n: makeFiniteRes
│ │ │ │ +0003f300: 6f6c 7574 696f 6e2c 202d 2d20 6669 6e69  olution, -- fini
│ │ │ │ +0003f310: 7465 2072 6573 6f6c 7574 696f 6e20 6f66  te resolution of
│ │ │ │ +0003f320: 2061 0a20 2020 206d 6174 7269 7820 6661   a.    matrix fa
│ │ │ │ +0003f330: 6374 6f72 697a 6174 696f 6e20 6d6f 6475  ctorization modu
│ │ │ │ +0003f340: 6c65 204d 0a0a 5761 7973 2074 6f20 7573  le M..Ways to us
│ │ │ │ +0003f350: 6520 6d61 6b65 4669 6e69 7465 5265 736f  e makeFiniteReso
│ │ │ │ +0003f360: 6c75 7469 6f6e 436f 6469 6d32 3a0a 3d3d  lutionCodim2:.==
│ │ │ │ +0003f370: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0003f380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0003f390: 3d3d 3d3d 3d0a 0a20 202a 2022 6d61 6b65  =====..  * "make
│ │ │ │ +0003f3a0: 4669 6e69 7465 5265 736f 6c75 7469 6f6e  FiniteResolution
│ │ │ │ +0003f3b0: 436f 6469 6d32 284d 6174 7269 782c 4c69  Codim2(Matrix,Li
│ │ │ │ +0003f3c0: 7374 2922 0a0a 466f 7220 7468 6520 7072  st)"..For the pr
│ │ │ │ +0003f3d0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +0003f3e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +0003f3f0: 206f 626a 6563 7420 2a6e 6f74 6520 6d61   object *note ma
│ │ │ │ +0003f400: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ +0003f410: 6f6e 436f 6469 6d32 3a20 6d61 6b65 4669  onCodim2: makeFi
│ │ │ │ +0003f420: 6e69 7465 5265 736f 6c75 7469 6f6e 436f  niteResolutionCo
│ │ │ │ +0003f430: 6469 6d32 2c20 6973 2061 0a2a 6e6f 7465  dim2, is a.*note
│ │ │ │ +0003f440: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +0003f450: 2077 6974 6820 6f70 7469 6f6e 733a 2028   with options: (
│ │ │ │ +0003f460: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +0003f470: 686f 6446 756e 6374 696f 6e57 6974 684f  hodFunctionWithO
│ │ │ │ +0003f480: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d  ptions,...------
│ │ │ │ +0003f490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ -0003f500: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ -0003f510: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ -0003f520: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -0003f530: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -0003f540: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ -0003f550: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -0003f560: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ -0003f570: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -0003f580: 6e73 2e6d 323a 3239 3339 3a30 2e0a 1f0a  ns.m2:2939:0....
│ │ │ │ -0003f590: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ -0003f5a0: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -0003f5b0: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ -0003f5c0: 3a20 6d61 6b65 486f 6d6f 746f 7069 6573  : makeHomotopies
│ │ │ │ -0003f5d0: 2c20 4e65 7874 3a20 6d61 6b65 486f 6d6f  , Next: makeHomo
│ │ │ │ -0003f5e0: 746f 7069 6573 312c 2050 7265 763a 206d  topies1, Prev: m
│ │ │ │ -0003f5f0: 616b 6546 696e 6974 6552 6573 6f6c 7574  akeFiniteResolut
│ │ │ │ -0003f600: 696f 6e43 6f64 696d 322c 2055 703a 2054  ionCodim2, Up: T
│ │ │ │ -0003f610: 6f70 0a0a 6d61 6b65 486f 6d6f 746f 7069  op..makeHomotopi
│ │ │ │ -0003f620: 6573 202d 2d20 7265 7475 726e 7320 6120  es -- returns a 
│ │ │ │ -0003f630: 7379 7374 656d 206f 6620 6869 6768 6572  system of higher
│ │ │ │ -0003f640: 2068 6f6d 6f74 6f70 6965 730a 2a2a 2a2a   homotopies.****
│ │ │ │ +0003f4d0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +0003f4e0: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +0003f4f0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +0003f500: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +0003f510: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +0003f520: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ +0003f530: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +0003f540: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +0003f550: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +0003f560: 732e 6d32 3a32 3933 393a 302e 0a1f 0a46  s.m2:2939:0....F
│ │ │ │ +0003f570: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +0003f580: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +0003f590: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +0003f5a0: 206d 616b 6548 6f6d 6f74 6f70 6965 732c   makeHomotopies,
│ │ │ │ +0003f5b0: 204e 6578 743a 206d 616b 6548 6f6d 6f74   Next: makeHomot
│ │ │ │ +0003f5c0: 6f70 6965 7331 2c20 5072 6576 3a20 6d61  opies1, Prev: ma
│ │ │ │ +0003f5d0: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ +0003f5e0: 6f6e 436f 6469 6d32 2c20 5570 3a20 546f  onCodim2, Up: To
│ │ │ │ +0003f5f0: 700a 0a6d 616b 6548 6f6d 6f74 6f70 6965  p..makeHomotopie
│ │ │ │ +0003f600: 7320 2d2d 2072 6574 7572 6e73 2061 2073  s -- returns a s
│ │ │ │ +0003f610: 7973 7465 6d20 6f66 2068 6967 6865 7220  ystem of higher 
│ │ │ │ +0003f620: 686f 6d6f 746f 7069 6573 0a2a 2a2a 2a2a  homotopies.*****
│ │ │ │ +0003f630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0003f640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0003f650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003f660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003f670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0003f680: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -0003f690: 0a20 2020 2020 2020 2048 203d 206d 616b  .        H = mak
│ │ │ │ -0003f6a0: 6548 6f6d 6f74 6f70 6965 7328 662c 462c  eHomotopies(f,F,
│ │ │ │ -0003f6b0: 6229 0a20 202a 2049 6e70 7574 733a 0a20  b).  * Inputs:. 
│ │ │ │ -0003f6c0: 2020 2020 202a 2066 2c20 6120 2a6e 6f74       * f, a *not
│ │ │ │ -0003f6d0: 6520 6d61 7472 6978 3a20 284d 6163 6175  e matrix: (Macau
│ │ │ │ -0003f6e0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c  lay2Doc)Matrix,,
│ │ │ │ -0003f6f0: 2031 786e 206d 6174 7269 7820 6f66 2065   1xn matrix of e
│ │ │ │ -0003f700: 6c65 6d65 6e74 7320 6f66 2053 0a20 2020  lements of S.   
│ │ │ │ -0003f710: 2020 202a 2046 2c20 6120 2a6e 6f74 6520     * F, a *note 
│ │ │ │ -0003f720: 636f 6d70 6c65 783a 2028 436f 6d70 6c65  complex: (Comple
│ │ │ │ -0003f730: 7865 7329 436f 6d70 6c65 782c 2c20 6164  xes)Complex,, ad
│ │ │ │ -0003f740: 6d69 7474 696e 6720 686f 6d6f 746f 7069  mitting homotopi
│ │ │ │ -0003f750: 6573 2066 6f72 2074 6865 0a20 2020 2020  es for the.     
│ │ │ │ -0003f760: 2020 2065 6e74 7269 6573 206f 6620 660a     entries of f.
│ │ │ │ -0003f770: 2020 2020 2020 2a20 622c 2061 6e20 2a6e        * b, an *n
│ │ │ │ -0003f780: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -0003f790: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -0003f7a0: 686f 7720 6661 7220 6261 636b 2074 6f20  how far back to 
│ │ │ │ -0003f7b0: 636f 6d70 7574 6520 7468 650a 2020 2020  compute the.    
│ │ │ │ -0003f7c0: 2020 2020 686f 6d6f 746f 7069 6573 2028      homotopies (
│ │ │ │ -0003f7d0: 6465 6661 756c 7473 2074 6f20 6c65 6e67  defaults to leng
│ │ │ │ -0003f7e0: 7468 206f 6620 4629 0a20 202a 204f 7574  th of F).  * Out
│ │ │ │ -0003f7f0: 7075 7473 3a0a 2020 2020 2020 2a20 482c  puts:.      * H,
│ │ │ │ -0003f800: 2061 202a 6e6f 7465 2068 6173 6820 7461   a *note hash ta
│ │ │ │ -0003f810: 626c 653a 2028 4d61 6361 756c 6179 3244  ble: (Macaulay2D
│ │ │ │ -0003f820: 6f63 2948 6173 6854 6162 6c65 2c2c 2067  oc)HashTable,, g
│ │ │ │ -0003f830: 6976 6573 2074 6865 2068 6967 6865 720a  ives the higher.
│ │ │ │ -0003f840: 2020 2020 2020 2020 686f 6d6f 746f 7079          homotopy
│ │ │ │ -0003f850: 2066 726f 6d20 465f 6920 636f 7272 6573   from F_i corres
│ │ │ │ -0003f860: 706f 6e64 696e 6720 746f 2061 206d 6f6e  ponding to a mon
│ │ │ │ -0003f870: 6f6d 6961 6c20 7769 7468 2065 7870 6f6e  omial with expon
│ │ │ │ -0003f880: 656e 7420 7665 6374 6f72 204c 2061 730a  ent vector L as.
│ │ │ │ -0003f890: 2020 2020 2020 2020 7468 6520 7661 6c75          the valu
│ │ │ │ -0003f8a0: 6520 2448 235c 7b4c 2c69 5c7d 240a 0a44  e $H#\{L,i\}$..D
│ │ │ │ -0003f8b0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -0003f8c0: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120  ======..Given a 
│ │ │ │ -0003f8d0: 2431 5c74 696d 6573 206e 2420 6d61 7472  $1\times n$ matr
│ │ │ │ -0003f8e0: 6978 2066 2c20 616e 6420 6120 6368 6169  ix f, and a chai
│ │ │ │ -0003f8f0: 6e20 636f 6d70 6c65 7820 462c 2074 6865  n complex F, the
│ │ │ │ -0003f900: 2073 6372 6970 7420 6174 7465 6d70 7473   script attempts
│ │ │ │ -0003f910: 2074 6f0a 6d61 6b65 2061 2066 616d 696c   to.make a famil
│ │ │ │ -0003f920: 7920 6f66 2068 6967 6865 7220 686f 6d6f  y of higher homo
│ │ │ │ -0003f930: 746f 7069 6573 206f 6e20 4620 666f 7220  topies on F for 
│ │ │ │ -0003f940: 7468 6520 656c 656d 656e 7473 206f 6620  the elements of 
│ │ │ │ -0003f950: 662c 2069 6e20 7468 6520 7365 6e73 650a  f, in the sense.
│ │ │ │ -0003f960: 6465 7363 7269 6265 642c 2066 6f72 2065  described, for e
│ │ │ │ -0003f970: 7861 6d70 6c65 2c20 696e 2045 6973 656e  xample, in Eisen
│ │ │ │ -0003f980: 6275 6420 2245 6e72 6963 6865 6420 4672  bud "Enriched Fr
│ │ │ │ -0003f990: 6565 2052 6573 6f6c 7574 696f 6e73 2061  ee Resolutions a
│ │ │ │ -0003f9a0: 6e64 2043 6861 6e67 6520 6f66 0a52 696e  nd Change of.Rin
│ │ │ │ -0003f9b0: 6773 222e 0a0a 5468 6520 6f75 7470 7574  gs"...The output
│ │ │ │ -0003f9c0: 2069 7320 6120 6861 7368 2074 6162 6c65   is a hash table
│ │ │ │ -0003f9d0: 2077 6974 6820 656e 7472 6965 7320 6f66   with entries of
│ │ │ │ -0003f9e0: 2074 6865 2066 6f72 6d20 245c 7b4a 2c69   the form $\{J,i
│ │ │ │ -0003f9f0: 5c7d 3d3e 7324 2c20 7768 6572 6520 4a20  \}=>s$, where J 
│ │ │ │ -0003fa00: 6973 2061 0a6c 6973 7420 6f66 206e 6f6e  is a.list of non
│ │ │ │ -0003fa10: 2d6e 6567 6174 6976 6520 696e 7465 6765  -negative intege
│ │ │ │ -0003fa20: 7273 2c20 6f66 206c 656e 6774 6820 6e20  rs, of length n 
│ │ │ │ -0003fa30: 616e 6420 2448 5c23 5c7b 4a2c 695c 7d3a  and $H\#\{J,i\}:
│ │ │ │ -0003fa40: 2046 5f69 2d3e 465f 7b69 2b32 7c4a 7c2d   F_i->F_{i+2|J|-
│ │ │ │ -0003fa50: 317d 240a 6172 6520 6d61 7073 2073 6174  1}$.are maps sat
│ │ │ │ -0003fa60: 6973 6679 696e 6720 7468 6520 636f 6e64  isfying the cond
│ │ │ │ -0003fa70: 6974 696f 6e73 2024 2420 485c 235c 7b65  itions $$ H\#\{e
│ │ │ │ -0003fa80: 302c 695c 7d20 3d20 643b 2024 2420 616e  0,i\} = d; $$ an
│ │ │ │ -0003fa90: 6420 2424 0a48 235c 7b65 302c 692b 315c  d $$.H#\{e0,i+1\
│ │ │ │ -0003faa0: 7d2a 4823 5c7b 652c 695c 7d2b 4823 5c7b  }*H#\{e,i\}+H#\{
│ │ │ │ -0003fab0: 652c 692d 315c 7d48 235c 7b65 302c 695c  e,i-1\}H#\{e0,i\
│ │ │ │ -0003fac0: 7d20 3d20 665f 692c 2024 2420 7768 6572  } = f_i, $$ wher
│ │ │ │ -0003fad0: 6520 2465 3020 3d0a 5c7b 302c 5c64 6f74  e $e0 =.\{0,\dot
│ │ │ │ -0003fae0: 732c 305c 7d24 2061 6e64 2024 6524 2069  s,0\}$ and $e$ i
│ │ │ │ -0003faf0: 7320 7468 6520 696e 6465 7820 6f66 2064  s the index of d
│ │ │ │ -0003fb00: 6567 7265 6520 3120 7769 7468 2061 2031  egree 1 with a 1
│ │ │ │ -0003fb10: 2069 6e20 7468 6520 2469 242d 7468 2070   in the $i$-th p
│ │ │ │ -0003fb20: 6c61 6365 3b0a 616e 642c 2066 6f72 2065  lace;.and, for e
│ │ │ │ -0003fb30: 6163 6820 696e 6465 7820 6c69 7374 2049  ach index list I
│ │ │ │ -0003fb40: 2077 6974 6820 7c49 7c3c 3d64 2c20 2424   with |I|<=d, $$
│ │ │ │ -0003fb50: 2073 756d 5f7b 4a3c 497d 2048 235c 7b49   sum_{Js$, where J i
│ │ │ │ +0003f9e0: 7320 610a 6c69 7374 206f 6620 6e6f 6e2d  s a.list of non-
│ │ │ │ +0003f9f0: 6e65 6761 7469 7665 2069 6e74 6567 6572  negative integer
│ │ │ │ +0003fa00: 732c 206f 6620 6c65 6e67 7468 206e 2061  s, of length n a
│ │ │ │ +0003fa10: 6e64 2024 485c 235c 7b4a 2c69 5c7d 3a20  nd $H\#\{J,i\}: 
│ │ │ │ +0003fa20: 465f 692d 3e46 5f7b 692b 327c 4a7c 2d31  F_i->F_{i+2|J|-1
│ │ │ │ +0003fa30: 7d24 0a61 7265 206d 6170 7320 7361 7469  }$.are maps sati
│ │ │ │ +0003fa40: 7366 7969 6e67 2074 6865 2063 6f6e 6469  sfying the condi
│ │ │ │ +0003fa50: 7469 6f6e 7320 2424 2048 5c23 5c7b 6530  tions $$ H\#\{e0
│ │ │ │ +0003fa60: 2c69 5c7d 203d 2064 3b20 2424 2061 6e64  ,i\} = d; $$ and
│ │ │ │ +0003fa70: 2024 240a 4823 5c7b 6530 2c69 2b31 5c7d   $$.H#\{e0,i+1\}
│ │ │ │ +0003fa80: 2a48 235c 7b65 2c69 5c7d 2b48 235c 7b65  *H#\{e,i\}+H#\{e
│ │ │ │ +0003fa90: 2c69 2d31 5c7d 4823 5c7b 6530 2c69 5c7d  ,i-1\}H#\{e0,i\}
│ │ │ │ +0003faa0: 203d 2066 5f69 2c20 2424 2077 6865 7265   = f_i, $$ where
│ │ │ │ +0003fab0: 2024 6530 203d 0a5c 7b30 2c5c 646f 7473   $e0 =.\{0,\dots
│ │ │ │ +0003fac0: 2c30 5c7d 2420 616e 6420 2465 2420 6973  ,0\}$ and $e$ is
│ │ │ │ +0003fad0: 2074 6865 2069 6e64 6578 206f 6620 6465   the index of de
│ │ │ │ +0003fae0: 6772 6565 2031 2077 6974 6820 6120 3120  gree 1 with a 1 
│ │ │ │ +0003faf0: 696e 2074 6865 2024 6924 2d74 6820 706c  in the $i$-th pl
│ │ │ │ +0003fb00: 6163 653b 0a61 6e64 2c20 666f 7220 6561  ace;.and, for ea
│ │ │ │ +0003fb10: 6368 2069 6e64 6578 206c 6973 7420 4920  ch index list I 
│ │ │ │ +0003fb20: 7769 7468 207c 497c 3c3d 642c 2024 2420  with |I|<=d, $$ 
│ │ │ │ +0003fb30: 7375 6d5f 7b4a 3c49 7d20 4823 5c7b 495c  sum_{J 
│ │ │ │ -000403a0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -000403b0: 2020 2020 2020 2020 207d 7c0a 7c20 2020           }|.|   
│ │ │ │ -000403c0: 2020 2020 2020 2020 2020 2020 7b7b 302c              {{0,
│ │ │ │ -000403d0: 2030 2c20 307d 2c20 317d 203d 3e20 7c20   0, 0}, 1} => | 
│ │ │ │ -000403e0: 6120 6220 6320 6420 7c20 2020 2020 2020  a b c d |       
│ │ │ │ -000403f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00040400: 2020 2020 2020 2020 2020 7b7b 302c 2030            {{0, 0
│ │ │ │ -00040410: 2c20 307d 2c20 327d 203d 3e20 7b31 7d20  , 0}, 2} => {1} 
│ │ │ │ -00040420: 7c20 2d62 202d 6320 3020 202d 6420 3020  | -b -c 0  -d 0 
│ │ │ │ -00040430: 2030 2020 7c20 7c0a 7c20 2020 2020 2020   0  | |.|       
│ │ │ │ -00040440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040450: 2020 2020 2020 2020 2020 7b31 7d20 7c20            {1} | 
│ │ │ │ -00040460: 6120 2030 2020 2d63 2030 2020 2d64 2030  a  0  -c 0  -d 0
│ │ │ │ -00040470: 2020 7c20 7c0a 7c20 2020 2020 2020 2020    | |.|         
│ │ │ │ -00040480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040490: 2020 2020 2020 2020 7b31 7d20 7c20 3020          {1} | 0 
│ │ │ │ -000404a0: 2061 2020 6220 2030 2020 3020 202d 6420   a  b  0  0  -d 
│ │ │ │ -000404b0: 7c20 7c0a 7c20 2020 2020 2020 2020 2020  | |.|           
│ │ │ │ -000404c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000404d0: 2020 2020 2020 7b31 7d20 7c20 3020 2030        {1} | 0  0
│ │ │ │ -000404e0: 2020 3020 2061 2020 6220 2063 2020 7c20    0  a  b  c  | 
│ │ │ │ -000404f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00040500: 2020 7b7b 302c 2030 2c20 307d 2c20 337d    {{0, 0, 0}, 3}
│ │ │ │ -00040510: 203d 3e20 7b32 7d20 7c20 6320 2064 2020   => {2} | c  d  
│ │ │ │ -00040520: 3020 2030 2020 7c20 2020 2020 2020 7c0a  0  0  |       |.
│ │ │ │ -00040530: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00040540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040550: 2020 7b32 7d20 7c20 2d62 2030 2020 6420    {2} | -b 0  d 
│ │ │ │ -00040560: 2030 2020 7c20 2020 2020 2020 7c0a 7c20   0  |       |.| 
│ │ │ │ -00040570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040590: 7b32 7d20 7c20 6120 2030 2020 3020 2064  {2} | a  0  0  d
│ │ │ │ -000405a0: 2020 7c20 2020 2020 2020 7c0a 7c20 2020    |       |.|   
│ │ │ │ -000405b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000405c0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ -000405d0: 7d20 7c20 3020 202d 6220 2d63 2030 2020  } | 0  -b -c 0  
│ │ │ │ -000405e0: 7c20 2020 2020 2020 7c0a 7c20 2020 2020  |       |.|     
│ │ │ │ -000405f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040600: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ -00040610: 7c20 3020 2061 2020 3020 202d 6320 7c20  | 0  a  0  -c | 
│ │ │ │ -00040620: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00040630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040640: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ -00040650: 3020 2030 2020 6120 2062 2020 7c20 2020  0  0  a  b  |   
│ │ │ │ -00040660: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00040670: 2020 2020 2020 7b7b 302c 2030 2c20 317d        {{0, 0, 1}
│ │ │ │ -00040680: 2c20 2d31 7d20 3d3e 2030 2020 2020 2020  , -1} => 0      
│ │ │ │ -00040690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000406a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000406b0: 2020 2020 7b7b 302c 2030 2c20 317d 2c20      {{0, 0, 1}, 
│ │ │ │ -000406c0: 307d 203d 3e20 7b31 7d20 7c20 3020 7c20  0} => {1} | 0 | 
│ │ │ │ +00040220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00040230: 2020 2020 2031 2020 2020 2020 3320 2020       1      3   
│ │ │ │ +00040240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040260: 2020 207c 0a7c 6f34 203a 204d 6174 7269     |.|o4 : Matri
│ │ │ │ +00040270: 7820 5320 203c 2d2d 2053 2020 2020 2020  x S  <-- S      
│ │ │ │ +00040280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000402a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000402b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000402c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000402d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000402e0: 0a7c 6935 203a 2068 6f6d 6f74 203d 206d  .|i5 : homot = m
│ │ │ │ +000402f0: 616b 6548 6f6d 6f74 6f70 6965 7328 662c  akeHomotopies(f,
│ │ │ │ +00040300: 462c 3229 2020 2020 2020 2020 2020 2020  F,2)            
│ │ │ │ +00040310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00040320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040350: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +00040360: 203d 2048 6173 6854 6162 6c65 7b7b 7b30   = HashTable{{{0
│ │ │ │ +00040370: 2c20 302c 2030 7d2c 2030 7d20 3d3e 2030  , 0, 0}, 0} => 0
│ │ │ │ +00040380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040390: 2020 2020 2020 2020 7d7c 0a7c 2020 2020          }|.|    
│ │ │ │ +000403a0: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +000403b0: 302c 2030 7d2c 2031 7d20 3d3e 207c 2061  0, 0}, 1} => | a
│ │ │ │ +000403c0: 2062 2063 2064 207c 2020 2020 2020 2020   b c d |        
│ │ │ │ +000403d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000403e0: 2020 2020 2020 2020 207b 7b30 2c20 302c           {{0, 0,
│ │ │ │ +000403f0: 2030 7d2c 2032 7d20 3d3e 207b 317d 207c   0}, 2} => {1} |
│ │ │ │ +00040400: 202d 6220 2d63 2030 2020 2d64 2030 2020   -b -c 0  -d 0  
│ │ │ │ +00040410: 3020 207c 207c 0a7c 2020 2020 2020 2020  0  | |.|        
│ │ │ │ +00040420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040430: 2020 2020 2020 2020 207b 317d 207c 2061           {1} | a
│ │ │ │ +00040440: 2020 3020 202d 6320 3020 202d 6420 3020    0  -c 0  -d 0 
│ │ │ │ +00040450: 207c 207c 0a7c 2020 2020 2020 2020 2020   | |.|          
│ │ │ │ +00040460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040470: 2020 2020 2020 207b 317d 207c 2030 2020         {1} | 0  
│ │ │ │ +00040480: 6120 2062 2020 3020 2030 2020 2d64 207c  a  b  0  0  -d |
│ │ │ │ +00040490: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000404a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000404b0: 2020 2020 207b 317d 207c 2030 2020 3020       {1} | 0  0 
│ │ │ │ +000404c0: 2030 2020 6120 2062 2020 6320 207c 207c   0  a  b  c  | |
│ │ │ │ +000404d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000404e0: 207b 7b30 2c20 302c 2030 7d2c 2033 7d20   {{0, 0, 0}, 3} 
│ │ │ │ +000404f0: 3d3e 207b 327d 207c 2063 2020 6420 2030  => {2} | c  d  0
│ │ │ │ +00040500: 2020 3020 207c 2020 2020 2020 207c 0a7c    0  |       |.|
│ │ │ │ +00040510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040530: 207b 327d 207c 202d 6220 3020 2064 2020   {2} | -b 0  d  
│ │ │ │ +00040540: 3020 207c 2020 2020 2020 207c 0a7c 2020  0  |       |.|  
│ │ │ │ +00040550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040560: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00040570: 327d 207c 2061 2020 3020 2030 2020 6420  2} | a  0  0  d 
│ │ │ │ +00040580: 207c 2020 2020 2020 207c 0a7c 2020 2020   |       |.|    
│ │ │ │ +00040590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000405a0: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ +000405b0: 207c 2030 2020 2d62 202d 6320 3020 207c   | 0  -b -c 0  |
│ │ │ │ +000405c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000405d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000405e0: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ +000405f0: 2030 2020 6120 2030 2020 2d63 207c 2020   0  a  0  -c |  
│ │ │ │ +00040600: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00040610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040620: 2020 2020 2020 2020 207b 327d 207c 2030           {2} | 0
│ │ │ │ +00040630: 2020 3020 2061 2020 6220 207c 2020 2020    0  a  b  |    
│ │ │ │ +00040640: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00040650: 2020 2020 207b 7b30 2c20 302c 2031 7d2c       {{0, 0, 1},
│ │ │ │ +00040660: 202d 317d 203d 3e20 3020 2020 2020 2020   -1} => 0       
│ │ │ │ +00040670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040680: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00040690: 2020 207b 7b30 2c20 302c 2031 7d2c 2030     {{0, 0, 1}, 0
│ │ │ │ +000406a0: 7d20 3d3e 207b 317d 207c 2030 207c 2020  } => {1} | 0 |  
│ │ │ │ +000406b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000406c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000406d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000406e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000406f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040700: 2020 2020 7b31 7d20 7c20 3020 7c20 2020      {1} | 0 |   
│ │ │ │ -00040710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00040720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00040730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040740: 2020 7b31 7d20 7c20 3120 7c20 2020 2020    {1} | 1 |     
│ │ │ │ -00040750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00040760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040780: 7b31 7d20 7c20 3020 7c20 2020 2020 2020  {1} | 0 |       
│ │ │ │ -00040790: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000407a0: 2020 2020 2020 2020 2020 2020 7b7b 302c              {{0,
│ │ │ │ -000407b0: 2030 2c20 317d 2c20 317d 203d 3e20 7b32   0, 1}, 1} => {2
│ │ │ │ -000407c0: 7d20 7c20 3020 2030 2020 3020 3020 7c20  } | 0  0  0 0 | 
│ │ │ │ -000407d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -000407e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000407f0: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ -00040800: 7c20 2d31 2030 2020 3020 3020 7c20 2020  | -1 0  0 0 |   
│ │ │ │ -00040810: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00040820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040830: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ -00040840: 3020 202d 3120 3020 3020 7c20 2020 2020  0  -1 0 0 |     
│ │ │ │ -00040850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00040860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040870: 2020 2020 2020 2020 7b32 7d20 7c20 3020          {2} | 0 
│ │ │ │ -00040880: 2030 2020 3020 3020 7c20 2020 2020 2020   0  0 0 |       
│ │ │ │ -00040890: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -000408a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000408b0: 2020 2020 2020 7b32 7d20 7c20 3020 2030        {2} | 0  0
│ │ │ │ -000408c0: 2020 3020 3020 7c20 2020 2020 2020 2020    0 0 |         
│ │ │ │ -000408d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000408e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000408f0: 2020 2020 7b32 7d20 7c20 3020 2030 2020      {2} | 0  0  
│ │ │ │ -00040900: 3020 3120 7c20 2020 2020 2020 2020 7c0a  0 1 |         |.
│ │ │ │ -00040910: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00040920: 7b7b 302c 2030 2c20 317d 2c20 327d 203d  {{0, 0, 1}, 2} =
│ │ │ │ -00040930: 3e20 7b33 7d20 7c20 3120 3020 3020 3020  > {3} | 1 0 0 0 
│ │ │ │ -00040940: 2030 2020 3020 7c20 2020 2020 7c0a 7c20   0  0 |     |.| 
│ │ │ │ -00040950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040970: 7b33 7d20 7c20 3020 3020 3020 3020 2030  {3} | 0 0 0 0  0
│ │ │ │ -00040980: 2020 3020 7c20 2020 2020 7c0a 7c20 2020    0 |     |.|   
│ │ │ │ -00040990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000409a0: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ -000409b0: 7d20 7c20 3020 3020 3020 2d31 2030 2020  } | 0 0 0 -1 0  
│ │ │ │ -000409c0: 3020 7c20 2020 2020 7c0a 7c20 2020 2020  0 |     |.|     
│ │ │ │ -000409d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000409e0: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
│ │ │ │ -000409f0: 7c20 3020 3020 3020 3020 202d 3120 3020  | 0 0 0 0  -1 0 
│ │ │ │ -00040a00: 7c20 2020 2020 7c0a 7c20 2020 2020 2020  |     |.|       
│ │ │ │ -00040a10: 2020 2020 2020 2020 7b7b 302c 2030 2c20          {{0, 0, 
│ │ │ │ -00040a20: 327d 2c20 2d31 7d20 3d3e 2030 2020 2020  2}, -1} => 0    
│ │ │ │ -00040a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040a40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00040a50: 2020 2020 2020 7b7b 302c 2031 2c20 307d        {{0, 1, 0}
│ │ │ │ -00040a60: 2c20 2d31 7d20 3d3e 2030 2020 2020 2020  , -1} => 0      
│ │ │ │ -00040a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040a80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00040a90: 2020 2020 7b7b 302c 2031 2c20 307d 2c20      {{0, 1, 0}, 
│ │ │ │ -00040aa0: 307d 203d 3e20 7b31 7d20 7c20 3020 7c20  0} => {1} | 0 | 
│ │ │ │ +000406e0: 2020 207b 317d 207c 2030 207c 2020 2020     {1} | 0 |    
│ │ │ │ +000406f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00040700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040720: 207b 317d 207c 2031 207c 2020 2020 2020   {1} | 1 |      
│ │ │ │ +00040730: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00040740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040750: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00040760: 317d 207c 2030 207c 2020 2020 2020 2020  1} | 0 |        
│ │ │ │ +00040770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00040780: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +00040790: 302c 2031 7d2c 2031 7d20 3d3e 207b 327d  0, 1}, 1} => {2}
│ │ │ │ +000407a0: 207c 2030 2020 3020 2030 2030 207c 2020   | 0  0  0 0 |  
│ │ │ │ +000407b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000407c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000407d0: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ +000407e0: 202d 3120 3020 2030 2030 207c 2020 2020   -1 0  0 0 |    
│ │ │ │ +000407f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00040800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040810: 2020 2020 2020 2020 207b 327d 207c 2030           {2} | 0
│ │ │ │ +00040820: 2020 2d31 2030 2030 207c 2020 2020 2020    -1 0 0 |      
│ │ │ │ +00040830: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00040840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040850: 2020 2020 2020 207b 327d 207c 2030 2020         {2} | 0  
│ │ │ │ +00040860: 3020 2030 2030 207c 2020 2020 2020 2020  0  0 0 |        
│ │ │ │ +00040870: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00040880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040890: 2020 2020 207b 327d 207c 2030 2020 3020       {2} | 0  0 
│ │ │ │ +000408a0: 2030 2030 207c 2020 2020 2020 2020 207c   0 0 |         |
│ │ │ │ +000408b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000408c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000408d0: 2020 207b 327d 207c 2030 2020 3020 2030     {2} | 0  0  0
│ │ │ │ +000408e0: 2031 207c 2020 2020 2020 2020 207c 0a7c   1 |         |.|
│ │ │ │ +000408f0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00040900: 7b30 2c20 302c 2031 7d2c 2032 7d20 3d3e  {0, 0, 1}, 2} =>
│ │ │ │ +00040910: 207b 337d 207c 2031 2030 2030 2030 2020   {3} | 1 0 0 0  
│ │ │ │ +00040920: 3020 2030 207c 2020 2020 207c 0a7c 2020  0  0 |     |.|  
│ │ │ │ +00040930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040940: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00040950: 337d 207c 2030 2030 2030 2030 2020 3020  3} | 0 0 0 0  0 
│ │ │ │ +00040960: 2030 207c 2020 2020 207c 0a7c 2020 2020   0 |     |.|    
│ │ │ │ +00040970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040980: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ +00040990: 207c 2030 2030 2030 202d 3120 3020 2030   | 0 0 0 -1 0  0
│ │ │ │ +000409a0: 207c 2020 2020 207c 0a7c 2020 2020 2020   |     |.|      
│ │ │ │ +000409b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000409c0: 2020 2020 2020 2020 2020 207b 337d 207c             {3} |
│ │ │ │ +000409d0: 2030 2030 2030 2030 2020 2d31 2030 207c   0 0 0 0  -1 0 |
│ │ │ │ +000409e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000409f0: 2020 2020 2020 207b 7b30 2c20 302c 2032         {{0, 0, 2
│ │ │ │ +00040a00: 7d2c 202d 317d 203d 3e20 3020 2020 2020  }, -1} => 0     
│ │ │ │ +00040a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040a20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00040a30: 2020 2020 207b 7b30 2c20 312c 2030 7d2c       {{0, 1, 0},
│ │ │ │ +00040a40: 202d 317d 203d 3e20 3020 2020 2020 2020   -1} => 0       
│ │ │ │ +00040a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040a60: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00040a70: 2020 207b 7b30 2c20 312c 2030 7d2c 2030     {{0, 1, 0}, 0
│ │ │ │ +00040a80: 7d20 3d3e 207b 317d 207c 2030 207c 2020  } => {1} | 0 |  
│ │ │ │ +00040a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00040aa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00040ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040ac0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00040ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040ae0: 2020 2020 7b31 7d20 7c20 3120 7c20 2020      {1} | 1 |   
│ │ │ │ -00040af0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00040b00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00040b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040b20: 2020 7b31 7d20 7c20 3020 7c20 2020 2020    {1} | 0 |     
│ │ │ │ -00040b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00040b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040b60: 7b31 7d20 7c20 3020 7c20 2020 2020 2020  {1} | 0 |       
│ │ │ │ -00040b70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00040b80: 2020 2020 2020 2020 2020 2020 7b7b 302c              {{0,
│ │ │ │ -00040b90: 2031 2c20 307d 2c20 317d 203d 3e20 7b32   1, 0}, 1} => {2
│ │ │ │ -00040ba0: 7d20 7c20 2d31 2030 2030 2030 207c 2020  } | -1 0 0 0 |  
│ │ │ │ -00040bb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00040bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040bd0: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ -00040be0: 7c20 3020 2030 2030 2030 207c 2020 2020  | 0  0 0 0 |    
│ │ │ │ -00040bf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00040c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040c10: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ -00040c20: 3020 2030 2031 2030 207c 2020 2020 2020  0  0 1 0 |      
│ │ │ │ -00040c30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00040c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040c50: 2020 2020 2020 2020 7b32 7d20 7c20 3020          {2} | 0 
│ │ │ │ -00040c60: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -00040c70: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00040c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040c90: 2020 2020 2020 7b32 7d20 7c20 3020 2030        {2} | 0  0
│ │ │ │ -00040ca0: 2030 2031 207c 2020 2020 2020 2020 2020   0 1 |          
│ │ │ │ -00040cb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00040cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040cd0: 2020 2020 7b32 7d20 7c20 3020 2030 2030      {2} | 0  0 0
│ │ │ │ -00040ce0: 2030 207c 2020 2020 2020 2020 2020 7c0a   0 |          |.
│ │ │ │ -00040cf0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00040d00: 7b7b 302c 2031 2c20 307d 2c20 327d 203d  {{0, 1, 0}, 2} =
│ │ │ │ -00040d10: 3e20 7b33 7d20 7c20 3020 2d31 2030 2030  > {3} | 0 -1 0 0
│ │ │ │ -00040d20: 2020 3020 3020 7c20 2020 2020 7c0a 7c20    0 0 |     |.| 
│ │ │ │ -00040d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040d50: 7b33 7d20 7c20 3020 3020 2030 202d 3120  {3} | 0 0  0 -1 
│ │ │ │ -00040d60: 3020 3020 7c20 2020 2020 7c0a 7c20 2020  0 0 |     |.|   
│ │ │ │ -00040d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040d80: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ -00040d90: 7d20 7c20 3020 3020 2030 2030 2020 3020  } | 0 0  0 0  0 
│ │ │ │ -00040da0: 3020 7c20 2020 2020 7c0a 7c20 2020 2020  0 |     |.|     
│ │ │ │ -00040db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040dc0: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
│ │ │ │ -00040dd0: 7c20 3020 3020 2030 2030 2020 3020 3120  | 0 0  0 0  0 1 
│ │ │ │ -00040de0: 7c20 2020 2020 7c0a 7c20 2020 2020 2020  |     |.|       
│ │ │ │ -00040df0: 2020 2020 2020 2020 7b7b 302c 2031 2c20          {{0, 1, 
│ │ │ │ -00040e00: 317d 2c20 2d31 7d20 3d3e 2030 2020 2020  1}, -1} => 0    
│ │ │ │ -00040e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040e20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00040e30: 2020 2020 2020 7b7b 302c 2032 2c20 307d        {{0, 2, 0}
│ │ │ │ -00040e40: 2c20 2d31 7d20 3d3e 2030 2020 2020 2020  , -1} => 0      
│ │ │ │ -00040e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040e60: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00040e70: 2020 2020 7b7b 312c 2030 2c20 307d 2c20      {{1, 0, 0}, 
│ │ │ │ -00040e80: 2d31 7d20 3d3e 2030 2020 2020 2020 2020  -1} => 0        
│ │ │ │ -00040e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040ea0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00040eb0: 2020 7b7b 312c 2030 2c20 307d 2c20 307d    {{1, 0, 0}, 0}
│ │ │ │ -00040ec0: 203d 3e20 7b31 7d20 7c20 3120 7c20 2020   => {1} | 1 |   
│ │ │ │ -00040ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00040ee0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00040ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040f00: 2020 7b31 7d20 7c20 3020 7c20 2020 2020    {1} | 0 |     
│ │ │ │ -00040f10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00040f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040f40: 7b31 7d20 7c20 3020 7c20 2020 2020 2020  {1} | 0 |       
│ │ │ │ -00040f50: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00040f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040f70: 2020 2020 2020 2020 2020 2020 2020 7b31                {1
│ │ │ │ -00040f80: 7d20 7c20 3020 7c20 2020 2020 2020 2020  } | 0 |         
│ │ │ │ -00040f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00040fa0: 2020 2020 2020 2020 2020 7b7b 312c 2030            {{1, 0
│ │ │ │ -00040fb0: 2c20 307d 2c20 317d 203d 3e20 7b32 7d20  , 0}, 1} => {2} 
│ │ │ │ -00040fc0: 7c20 3020 3120 3020 3020 7c20 2020 2020  | 0 1 0 0 |     
│ │ │ │ -00040fd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00040fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040ff0: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ -00041000: 3020 3020 3120 3020 7c20 2020 2020 2020  0 0 1 0 |       
│ │ │ │ -00041010: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00041020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041030: 2020 2020 2020 2020 7b32 7d20 7c20 3020          {2} | 0 
│ │ │ │ -00041040: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ -00041050: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00041060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041070: 2020 2020 2020 7b32 7d20 7c20 3020 3020        {2} | 0 0 
│ │ │ │ -00041080: 3020 3120 7c20 2020 2020 2020 2020 2020  0 1 |           
│ │ │ │ -00041090: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -000410a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000410b0: 2020 2020 7b32 7d20 7c20 3020 3020 3020      {2} | 0 0 0 
│ │ │ │ -000410c0: 3020 7c20 2020 2020 2020 2020 2020 7c0a  0 |           |.
│ │ │ │ -000410d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000410e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000410f0: 2020 7b32 7d20 7c20 3020 3020 3020 3020    {2} | 0 0 0 0 
│ │ │ │ -00041100: 7c20 2020 2020 2020 2020 2020 7c0a 7c20  |           |.| 
│ │ │ │ -00041110: 2020 2020 2020 2020 2020 2020 2020 7b7b                {{
│ │ │ │ -00041120: 312c 2030 2c20 307d 2c20 327d 203d 3e20  1, 0, 0}, 2} => 
│ │ │ │ -00041130: 7b33 7d20 7c20 3020 3020 3120 3020 3020  {3} | 0 0 1 0 0 
│ │ │ │ -00041140: 3020 7c20 2020 2020 2020 7c0a 7c20 2020  0 |       |.|   
│ │ │ │ -00041150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041160: 2020 2020 2020 2020 2020 2020 2020 7b33                {3
│ │ │ │ -00041170: 7d20 7c20 3020 3020 3020 3020 3120 3020  } | 0 0 0 0 1 0 
│ │ │ │ -00041180: 7c20 2020 2020 2020 7c0a 7c20 2020 2020  |       |.|     
│ │ │ │ -00041190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000411a0: 2020 2020 2020 2020 2020 2020 7b33 7d20              {3} 
│ │ │ │ -000411b0: 7c20 3020 3020 3020 3020 3020 3120 7c20  | 0 0 0 0 0 1 | 
│ │ │ │ -000411c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000411d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000411e0: 2020 2020 2020 2020 2020 7b33 7d20 7c20            {3} | 
│ │ │ │ -000411f0: 3020 3020 3020 3020 3020 3020 7c20 2020  0 0 0 0 0 0 |   
│ │ │ │ -00041200: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00041210: 2020 2020 2020 7b7b 312c 2030 2c20 317d        {{1, 0, 1}
│ │ │ │ -00041220: 2c20 2d31 7d20 3d3e 2030 2020 2020 2020  , -1} => 0      
│ │ │ │ -00041230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041240: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -00041250: 2020 2020 7b7b 312c 2031 2c20 307d 2c20      {{1, 1, 0}, 
│ │ │ │ -00041260: 2d31 7d20 3d3e 2030 2020 2020 2020 2020  -1} => 0        
│ │ │ │ -00041270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041280: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00041290: 2020 7b7b 322c 2030 2c20 307d 2c20 2d31    {{2, 0, 0}, -1
│ │ │ │ -000412a0: 7d20 3d3e 2030 2020 2020 2020 2020 2020  } => 0          
│ │ │ │ -000412b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000412c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000412d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000412e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000412f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00041300: 3520 3a20 4861 7368 5461 626c 6520 2020  5 : HashTable   
│ │ │ │ -00041310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041330: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00040ac0: 2020 207b 317d 207c 2031 207c 2020 2020     {1} | 1 |    
│ │ │ │ +00040ad0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00040ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040b00: 207b 317d 207c 2030 207c 2020 2020 2020   {1} | 0 |      
│ │ │ │ +00040b10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00040b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040b30: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00040b40: 317d 207c 2030 207c 2020 2020 2020 2020  1} | 0 |        
│ │ │ │ +00040b50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00040b60: 2020 2020 2020 2020 2020 207b 7b30 2c20             {{0, 
│ │ │ │ +00040b70: 312c 2030 7d2c 2031 7d20 3d3e 207b 327d  1, 0}, 1} => {2}
│ │ │ │ +00040b80: 207c 202d 3120 3020 3020 3020 7c20 2020   | -1 0 0 0 |   
│ │ │ │ +00040b90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00040ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040bb0: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ +00040bc0: 2030 2020 3020 3020 3020 7c20 2020 2020   0  0 0 0 |     
│ │ │ │ +00040bd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00040be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040bf0: 2020 2020 2020 2020 207b 327d 207c 2030           {2} | 0
│ │ │ │ +00040c00: 2020 3020 3120 3020 7c20 2020 2020 2020    0 1 0 |       
│ │ │ │ +00040c10: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00040c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040c30: 2020 2020 2020 207b 327d 207c 2030 2020         {2} | 0  
│ │ │ │ +00040c40: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ +00040c50: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00040c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040c70: 2020 2020 207b 327d 207c 2030 2020 3020       {2} | 0  0 
│ │ │ │ +00040c80: 3020 3120 7c20 2020 2020 2020 2020 207c  0 1 |          |
│ │ │ │ +00040c90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00040ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040cb0: 2020 207b 327d 207c 2030 2020 3020 3020     {2} | 0  0 0 
│ │ │ │ +00040cc0: 3020 7c20 2020 2020 2020 2020 207c 0a7c  0 |          |.|
│ │ │ │ +00040cd0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00040ce0: 7b30 2c20 312c 2030 7d2c 2032 7d20 3d3e  {0, 1, 0}, 2} =>
│ │ │ │ +00040cf0: 207b 337d 207c 2030 202d 3120 3020 3020   {3} | 0 -1 0 0 
│ │ │ │ +00040d00: 2030 2030 207c 2020 2020 207c 0a7c 2020   0 0 |     |.|  
│ │ │ │ +00040d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d20: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00040d30: 337d 207c 2030 2030 2020 3020 2d31 2030  3} | 0 0  0 -1 0
│ │ │ │ +00040d40: 2030 207c 2020 2020 207c 0a7c 2020 2020   0 |     |.|    
│ │ │ │ +00040d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d60: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ +00040d70: 207c 2030 2030 2020 3020 3020 2030 2030   | 0 0  0 0  0 0
│ │ │ │ +00040d80: 207c 2020 2020 207c 0a7c 2020 2020 2020   |     |.|      
│ │ │ │ +00040d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040da0: 2020 2020 2020 2020 2020 207b 337d 207c             {3} |
│ │ │ │ +00040db0: 2030 2030 2020 3020 3020 2030 2031 207c   0 0  0 0  0 1 |
│ │ │ │ +00040dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00040dd0: 2020 2020 2020 207b 7b30 2c20 312c 2031         {{0, 1, 1
│ │ │ │ +00040de0: 7d2c 202d 317d 203d 3e20 3020 2020 2020  }, -1} => 0     
│ │ │ │ +00040df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040e00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00040e10: 2020 2020 207b 7b30 2c20 322c 2030 7d2c       {{0, 2, 0},
│ │ │ │ +00040e20: 202d 317d 203d 3e20 3020 2020 2020 2020   -1} => 0       
│ │ │ │ +00040e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040e40: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00040e50: 2020 207b 7b31 2c20 302c 2030 7d2c 202d     {{1, 0, 0}, -
│ │ │ │ +00040e60: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +00040e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00040e80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00040e90: 207b 7b31 2c20 302c 2030 7d2c 2030 7d20   {{1, 0, 0}, 0} 
│ │ │ │ +00040ea0: 3d3e 207b 317d 207c 2031 207c 2020 2020  => {1} | 1 |    
│ │ │ │ +00040eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00040ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040ee0: 207b 317d 207c 2030 207c 2020 2020 2020   {1} | 0 |      
│ │ │ │ +00040ef0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00040f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040f10: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00040f20: 317d 207c 2030 207c 2020 2020 2020 2020  1} | 0 |        
│ │ │ │ +00040f30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00040f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040f50: 2020 2020 2020 2020 2020 2020 207b 317d               {1}
│ │ │ │ +00040f60: 207c 2030 207c 2020 2020 2020 2020 2020   | 0 |          
│ │ │ │ +00040f70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00040f80: 2020 2020 2020 2020 207b 7b31 2c20 302c           {{1, 0,
│ │ │ │ +00040f90: 2030 7d2c 2031 7d20 3d3e 207b 327d 207c   0}, 1} => {2} |
│ │ │ │ +00040fa0: 2030 2031 2030 2030 207c 2020 2020 2020   0 1 0 0 |      
│ │ │ │ +00040fb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00040fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040fd0: 2020 2020 2020 2020 207b 327d 207c 2030           {2} | 0
│ │ │ │ +00040fe0: 2030 2031 2030 207c 2020 2020 2020 2020   0 1 0 |        
│ │ │ │ +00040ff0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00041000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041010: 2020 2020 2020 207b 327d 207c 2030 2030         {2} | 0 0
│ │ │ │ +00041020: 2030 2030 207c 2020 2020 2020 2020 2020   0 0 |          
│ │ │ │ +00041030: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00041040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041050: 2020 2020 207b 327d 207c 2030 2030 2030       {2} | 0 0 0
│ │ │ │ +00041060: 2031 207c 2020 2020 2020 2020 2020 207c   1 |           |
│ │ │ │ +00041070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00041080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041090: 2020 207b 327d 207c 2030 2030 2030 2030     {2} | 0 0 0 0
│ │ │ │ +000410a0: 207c 2020 2020 2020 2020 2020 207c 0a7c   |           |.|
│ │ │ │ +000410b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000410c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000410d0: 207b 327d 207c 2030 2030 2030 2030 207c   {2} | 0 0 0 0 |
│ │ │ │ +000410e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000410f0: 2020 2020 2020 2020 2020 2020 207b 7b31               {{1
│ │ │ │ +00041100: 2c20 302c 2030 7d2c 2032 7d20 3d3e 207b  , 0, 0}, 2} => {
│ │ │ │ +00041110: 337d 207c 2030 2030 2031 2030 2030 2030  3} | 0 0 1 0 0 0
│ │ │ │ +00041120: 207c 2020 2020 2020 207c 0a7c 2020 2020   |       |.|    
│ │ │ │ +00041130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041140: 2020 2020 2020 2020 2020 2020 207b 337d               {3}
│ │ │ │ +00041150: 207c 2030 2030 2030 2030 2031 2030 207c   | 0 0 0 0 1 0 |
│ │ │ │ +00041160: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00041170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041180: 2020 2020 2020 2020 2020 207b 337d 207c             {3} |
│ │ │ │ +00041190: 2030 2030 2030 2030 2030 2031 207c 2020   0 0 0 0 0 1 |  
│ │ │ │ +000411a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000411b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000411c0: 2020 2020 2020 2020 207b 337d 207c 2030           {3} | 0
│ │ │ │ +000411d0: 2030 2030 2030 2030 2030 207c 2020 2020   0 0 0 0 0 |    
│ │ │ │ +000411e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000411f0: 2020 2020 207b 7b31 2c20 302c 2031 7d2c       {{1, 0, 1},
│ │ │ │ +00041200: 202d 317d 203d 3e20 3020 2020 2020 2020   -1} => 0       
│ │ │ │ +00041210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041220: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00041230: 2020 207b 7b31 2c20 312c 2030 7d2c 202d     {{1, 1, 0}, -
│ │ │ │ +00041240: 317d 203d 3e20 3020 2020 2020 2020 2020  1} => 0         
│ │ │ │ +00041250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00041260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00041270: 207b 7b32 2c20 302c 2030 7d2c 202d 317d   {{2, 0, 0}, -1}
│ │ │ │ +00041280: 203d 3e20 3020 2020 2020 2020 2020 2020   => 0           
│ │ │ │ +00041290: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000412a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000412b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000412c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000412d0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +000412e0: 203a 2048 6173 6854 6162 6c65 2020 2020   : HashTable    
│ │ │ │ +000412f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041310: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00041320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041370: 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 7468  --------+..In th
│ │ │ │ -00041380: 6973 2063 6173 6520 7468 6520 6869 6768  is case the high
│ │ │ │ -00041390: 6572 2068 6f6d 6f74 6f70 6965 7320 6172  er homotopies ar
│ │ │ │ -000413a0: 6520 303a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  e 0:..+---------
│ │ │ │ +00041350: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6869  -------+..In thi
│ │ │ │ +00041360: 7320 6361 7365 2074 6865 2068 6967 6865  s case the highe
│ │ │ │ +00041370: 7220 686f 6d6f 746f 7069 6573 2061 7265  r homotopies are
│ │ │ │ +00041380: 2030 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d   0:..+----------
│ │ │ │ +00041390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000413a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000413b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000413c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000413d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000413e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204c  -------+.|i6 : L
│ │ │ │ -000413f0: 203d 2073 6f72 7420 7365 6c65 6374 286b   = sort select(k
│ │ │ │ -00041400: 6579 7320 686f 6d6f 742c 206b 2d3e 2868  eys homot, k->(h
│ │ │ │ -00041410: 6f6d 6f74 236b 213d 3020 616e 6420 7375  omot#k!=0 and su
│ │ │ │ -00041420: 6d28 6b5f 3029 3e31 2929 7c0a 7c20 2020  m(k_0)>1))|.|   
│ │ │ │ +000413c0: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4c20  ------+.|i6 : L 
│ │ │ │ +000413d0: 3d20 736f 7274 2073 656c 6563 7428 6b65  = sort select(ke
│ │ │ │ +000413e0: 7973 2068 6f6d 6f74 2c20 6b2d 3e28 686f  ys homot, k->(ho
│ │ │ │ +000413f0: 6d6f 7423 6b21 3d30 2061 6e64 2073 756d  mot#k!=0 and sum
│ │ │ │ +00041400: 286b 5f30 293e 3129 297c 0a7c 2020 2020  (k_0)>1))|.|    
│ │ │ │ +00041410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041470: 6f36 203d 207b 7d20 2020 2020 2020 2020  o6 = {}         
│ │ │ │ -00041480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041440: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00041450: 3620 3d20 7b7d 2020 2020 2020 2020 2020  6 = {}          
│ │ │ │ +00041460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00041490: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  000414a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000414b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +000414b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000414c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000414d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000414d0: 2020 7c0a 7c6f 3620 3a20 4c69 7374 2020    |.|o6 : List  
│ │ │ │  000414e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000414f0: 2020 207c 0a7c 6f36 203a 204c 6973 7420     |.|o6 : List 
│ │ │ │ +000414f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041530: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00041510: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00041520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041570: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4f6e 2074  ---------+..On t
│ │ │ │ -00041580: 6865 206f 7468 6572 2068 616e 642c 2069  he other hand, i
│ │ │ │ -00041590: 6620 7765 2074 616b 6520 6120 636f 6d70  f we take a comp
│ │ │ │ -000415a0: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ -000415b0: 6e20 616e 6420 736f 6d65 7468 696e 6720  n and something 
│ │ │ │ -000415c0: 636f 6e74 6169 6e65 640a 696e 2069 7420  contained.in it 
│ │ │ │ -000415d0: 696e 2061 206d 6f72 6520 636f 6d70 6c69  in a more compli
│ │ │ │ -000415e0: 6361 7465 6420 7369 7475 6174 696f 6e2c  cated situation,
│ │ │ │ -000415f0: 2074 6865 2070 726f 6772 616d 2067 6976   the program giv
│ │ │ │ -00041600: 6573 206e 6f6e 7a65 726f 2068 6967 6865  es nonzero highe
│ │ │ │ -00041610: 720a 686f 6d6f 746f 7069 6573 3a0a 0a2b  r.homotopies:..+
│ │ │ │ +00041550: 2d2d 2d2d 2d2d 2d2d 2b0a 0a4f 6e20 7468  --------+..On th
│ │ │ │ +00041560: 6520 6f74 6865 7220 6861 6e64 2c20 6966  e other hand, if
│ │ │ │ +00041570: 2077 6520 7461 6b65 2061 2063 6f6d 706c   we take a compl
│ │ │ │ +00041580: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ +00041590: 2061 6e64 2073 6f6d 6574 6869 6e67 2063   and something c
│ │ │ │ +000415a0: 6f6e 7461 696e 6564 0a69 6e20 6974 2069  ontained.in it i
│ │ │ │ +000415b0: 6e20 6120 6d6f 7265 2063 6f6d 706c 6963  n a more complic
│ │ │ │ +000415c0: 6174 6564 2073 6974 7561 7469 6f6e 2c20  ated situation, 
│ │ │ │ +000415d0: 7468 6520 7072 6f67 7261 6d20 6769 7665  the program give
│ │ │ │ +000415e0: 7320 6e6f 6e7a 6572 6f20 6869 6768 6572  s nonzero higher
│ │ │ │ +000415f0: 0a68 6f6d 6f74 6f70 6965 733a 0a0a 2b2d  .homotopies:..+-
│ │ │ │ +00041600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041670: 6937 203a 206b 6b3d 205a 5a2f 3332 3030  i7 : kk= ZZ/3200
│ │ │ │ -00041680: 333b 2020 2020 2020 2020 2020 2020 2020  3;              
│ │ │ │ -00041690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000416a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000416b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00041650: 3720 3a20 6b6b 3d20 5a5a 2f33 3230 3033  7 : kk= ZZ/32003
│ │ │ │ +00041660: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00041670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041690: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000416a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000416b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000416c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000416d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000416e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000416f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041710: 6938 203a 2053 203d 206b 6b5b 612c 622c  i8 : S = kk[a,b,
│ │ │ │ -00041720: 632c 645d 3b20 2020 2020 2020 2020 2020  c,d];           
│ │ │ │ -00041730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041750: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000416e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000416f0: 3820 3a20 5320 3d20 6b6b 5b61 2c62 2c63  8 : S = kk[a,b,c
│ │ │ │ +00041700: 2c64 5d3b 2020 2020 2020 2020 2020 2020  ,d];            
│ │ │ │ +00041710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041730: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00041740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000417a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000417b0: 6939 203a 204d 203d 2053 5e31 2f28 6964  i9 : M = S^1/(id
│ │ │ │ -000417c0: 6561 6c22 6132 2c62 322c 6332 2c64 3222  eal"a2,b2,c2,d2"
│ │ │ │ -000417d0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -000417e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000417f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00041790: 3920 3a20 4d20 3d20 535e 312f 2869 6465  9 : M = S^1/(ide
│ │ │ │ +000417a0: 616c 2261 322c 6232 2c63 322c 6432 2229  al"a2,b2,c2,d2")
│ │ │ │ +000417b0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +000417c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000417d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +000417e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000417f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041850: 6931 3020 3a20 4620 3d20 6672 6565 5265  i10 : F = freeRe
│ │ │ │ -00041860: 736f 6c75 7469 6f6e 204d 2020 2020 2020  solution M      
│ │ │ │ -00041870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00041830: 3130 203a 2046 203d 2066 7265 6552 6573  10 : F = freeRes
│ │ │ │ +00041840: 6f6c 7574 696f 6e20 4d20 2020 2020 2020  olution M       
│ │ │ │ +00041850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00041880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000418a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000418b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000418c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000418d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000418e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000418f0: 2020 2020 2020 2031 2020 2020 2020 3420         1      4 
│ │ │ │ -00041900: 2020 2020 2036 2020 2020 2020 3420 2020       6      4   
│ │ │ │ -00041910: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -00041920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041940: 6f31 3020 3d20 5320 203c 2d2d 2053 2020  o10 = S  <-- S  
│ │ │ │ -00041950: 3c2d 2d20 5320 203c 2d2d 2053 2020 3c2d  <-- S  <-- S  <-
│ │ │ │ -00041960: 2d20 5320 2020 2020 2020 2020 2020 2020  - S             
│ │ │ │ +000418c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000418d0: 2020 2020 2020 3120 2020 2020 2034 2020        1      4  
│ │ │ │ +000418e0: 2020 2020 3620 2020 2020 2034 2020 2020      6      4    
│ │ │ │ +000418f0: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00041900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041910: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00041920: 3130 203d 2053 2020 3c2d 2d20 5320 203c  10 = S  <-- S  <
│ │ │ │ +00041930: 2d2d 2053 2020 3c2d 2d20 5320 203c 2d2d  -- S  <-- S  <--
│ │ │ │ +00041940: 2053 2020 2020 2020 2020 2020 2020 2020   S              
│ │ │ │ +00041950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00041970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000419a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000419b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000419c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000419d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000419e0: 2020 2020 2020 3020 2020 2020 2031 2020        0      1  
│ │ │ │ -000419f0: 2020 2020 3220 2020 2020 2033 2020 2020      2      3    
│ │ │ │ -00041a00: 2020 3420 2020 2020 2020 2020 2020 2020    4             
│ │ │ │ +000419b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000419c0: 2020 2020 2030 2020 2020 2020 3120 2020       0      1   
│ │ │ │ +000419d0: 2020 2032 2020 2020 2020 3320 2020 2020     2      3     
│ │ │ │ +000419e0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +000419f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00041a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041a70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041a80: 6f31 3020 3a20 436f 6d70 6c65 7820 2020  o10 : Complex   
│ │ │ │ +00041a50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00041a60: 3130 203a 2043 6f6d 706c 6578 2020 2020  10 : Complex    
│ │ │ │ +00041a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041ac0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041aa0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00041ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041b20: 6931 3120 3a20 7365 7452 616e 646f 6d53  i11 : setRandomS
│ │ │ │ -00041b30: 6565 6420 3020 2020 2020 2020 2020 2020  eed 0           
│ │ │ │ -00041b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041b70: 202d 2d20 7365 7474 696e 6720 7261 6e64   -- setting rand
│ │ │ │ -00041b80: 6f6d 2073 6565 6420 746f 2030 2020 2020  om seed to 0    
│ │ │ │ -00041b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00041b00: 3131 203a 2073 6574 5261 6e64 6f6d 5365  11 : setRandomSe
│ │ │ │ +00041b10: 6564 2030 2020 2020 2020 2020 2020 2020  ed 0            
│ │ │ │ +00041b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00041b50: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ +00041b60: 6d20 7365 6564 2074 6f20 3020 2020 2020  m seed to 0     
│ │ │ │ +00041b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00041ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041c10: 6f31 3120 3d20 3020 2020 2020 2020 2020  o11 = 0         
│ │ │ │ +00041be0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00041bf0: 3131 203d 2030 2020 2020 2020 2020 2020  11 = 0          
│ │ │ │ +00041c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041c50: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041c30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00041c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041cb0: 6931 3220 3a20 6620 3d20 7261 6e64 6f6d  i12 : f = random
│ │ │ │ -00041cc0: 2853 5e31 2c53 5e7b 323a 2d35 7d29 3b20  (S^1,S^{2:-5}); 
│ │ │ │ -00041cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00041c90: 3132 203a 2066 203d 2072 616e 646f 6d28  12 : f = random(
│ │ │ │ +00041ca0: 535e 312c 535e 7b32 3a2d 357d 293b 2020  S^1,S^{2:-5});  
│ │ │ │ +00041cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00041ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041d50: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -00041d60: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00041d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041da0: 6f31 3220 3a20 4d61 7472 6978 2053 2020  o12 : Matrix S  
│ │ │ │ -00041db0: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ -00041dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041de0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00041d20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00041d30: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ +00041d40: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00041d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041d70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00041d80: 3132 203a 204d 6174 7269 7820 5320 203c  12 : Matrix S  <
│ │ │ │ +00041d90: 2d2d 2053 2020 2020 2020 2020 2020 2020  -- S            
│ │ │ │ +00041da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041dc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00041dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00041e40: 6931 3320 3a20 686f 6d6f 7420 3d20 6d61  i13 : homot = ma
│ │ │ │ -00041e50: 6b65 486f 6d6f 746f 7069 6573 2866 2c46  keHomotopies(f,F
│ │ │ │ -00041e60: 2c35 2920 2020 2020 2020 2020 2020 2020  ,5)             
│ │ │ │ +00041e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00041e20: 3133 203a 2068 6f6d 6f74 203d 206d 616b  13 : homot = mak
│ │ │ │ +00041e30: 6548 6f6d 6f74 6f70 6965 7328 662c 462c  eHomotopies(f,F,
│ │ │ │ +00041e40: 3529 2020 2020 2020 2020 2020 2020 2020  5)              
│ │ │ │ +00041e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00041e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041e80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00041e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041ed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041ee0: 6f31 3320 3d20 4861 7368 5461 626c 657b  o13 = HashTable{
│ │ │ │ -00041ef0: 7b7b 302c 2030 7d2c 2030 7d20 3d3e 2030  {{0, 0}, 0} => 0
│ │ │ │ -00041f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f40: 7b7b 302c 2030 7d2c 2031 7d20 3d3e 207c  {{0, 0}, 1} => |
│ │ │ │ -00041f50: 2061 3220 2020 2020 2020 2020 2020 2020   a2             
│ │ │ │ -00041f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041f90: 7b7b 302c 2030 7d2c 2032 7d20 3d3e 207b  {{0, 0}, 2} => {
│ │ │ │ -00041fa0: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00041eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00041ec0: 3133 203d 2048 6173 6854 6162 6c65 7b7b  13 = HashTable{{
│ │ │ │ +00041ed0: 7b30 2c20 307d 2c20 307d 203d 3e20 3020  {0, 0}, 0} => 0 
│ │ │ │ +00041ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00041f10: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00041f20: 7b30 2c20 307d 2c20 317d 203d 3e20 7c20  {0, 0}, 1} => | 
│ │ │ │ +00041f30: 6132 2020 2020 2020 2020 2020 2020 2020  a2              
│ │ │ │ +00041f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00041f60: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00041f70: 7b30 2c20 307d 2c20 327d 203d 3e20 7b32  {0, 0}, 2} => {2
│ │ │ │ +00041f80: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00041f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00041fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00041fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041fe0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00041ff0: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00041fc0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00041fd0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00041fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042030: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042040: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00042010: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00042020: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042080: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042090: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ -000420a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000420b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000420c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000420d0: 7b7b 302c 2030 7d2c 2033 7d20 3d3e 207b  {{0, 0}, 3} => {
│ │ │ │ -000420e0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042060: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00042070: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000420a0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000420b0: 7b30 2c20 307d 2c20 337d 203d 3e20 7b34  {0, 0}, 3} => {4
│ │ │ │ +000420c0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000420d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000420e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000420f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042120: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042130: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042100: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042110: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042130: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042170: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042180: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042150: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042160: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042180: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000421a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000421b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000421c0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000421d0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +000421a0: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +000421b0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000421c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000421d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000421e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000421f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042210: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042220: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +000421f0: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042200: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042260: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042270: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ -00042280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042290: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000422a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000422b0: 7b7b 302c 2030 7d2c 2034 7d20 3d3e 207b  {{0, 0}, 4} => {
│ │ │ │ -000422c0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042240: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042250: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042270: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042280: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042290: 7b30 2c20 307d 2c20 347d 203d 3e20 7b36  {0, 0}, 4} => {6
│ │ │ │ +000422a0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000422b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000422c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000422d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000422e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000422f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042300: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042310: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +000422e0: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +000422f0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042350: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042360: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042330: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00042340: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042380: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000423a0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000423b0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ -000423c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000423d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042380: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00042390: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000423a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000423b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000423c0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000423d0: 7b30 2c20 307d 2c20 357d 203d 3e20 3020  {0, 0}, 5} => 0 
│ │ │ │  000423e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000423f0: 7b7b 302c 2030 7d2c 2035 7d20 3d3e 2030  {{0, 0}, 5} => 0
│ │ │ │ -00042400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000423f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042410: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042420: 7b30 2c20 317d 2c20 2d31 7d20 3d3e 2030  {0, 1}, -1} => 0
│ │ │ │  00042430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042440: 7b7b 302c 2031 7d2c 202d 317d 203d 3e20  {{0, 1}, -1} => 
│ │ │ │ -00042450: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00042460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042490: 7b7b 302c 2031 7d2c 2030 7d20 3d3e 207b  {{0, 1}, 0} => {
│ │ │ │ -000424a0: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00042440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042450: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042460: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042470: 7b30 2c20 317d 2c20 307d 203d 3e20 7b32  {0, 1}, 0} => {2
│ │ │ │ +00042480: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000424a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000424b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000424c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000424d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000424e0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000424f0: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +000424c0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +000424d0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000424e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000424f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042530: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042540: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00042510: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00042520: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042560: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042580: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042590: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ -000425a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000425b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000425c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000425d0: 7b7b 302c 2031 7d2c 2031 7d20 3d3e 207b  {{0, 1}, 1} => {
│ │ │ │ -000425e0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042560: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00042570: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042590: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000425a0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000425b0: 7b30 2c20 317d 2c20 317d 203d 3e20 7b34  {0, 1}, 1} => {4
│ │ │ │ +000425c0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000425d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000425e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000425f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042600: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042620: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042630: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042600: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042610: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042630: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042650: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042670: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042680: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042650: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042660: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000426a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000426b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000426c0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000426d0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +000426a0: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +000426b0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000426c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000426d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000426e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000426f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042710: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042720: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +000426f0: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042700: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042720: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042760: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042770: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ -00042780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042790: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000427a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000427b0: 7b7b 302c 2031 7d2c 2032 7d20 3d3e 207b  {{0, 1}, 2} => {
│ │ │ │ -000427c0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042740: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042750: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042780: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042790: 7b30 2c20 317d 2c20 327d 203d 3e20 7b36  {0, 1}, 2} => {6
│ │ │ │ +000427a0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000427b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000427c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000427d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000427e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000427f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042800: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042810: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +000427e0: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +000427f0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042810: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042830: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042850: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042860: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042830: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00042840: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042880: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000428a0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000428b0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ -000428c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000428d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000428e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000428f0: 7b7b 302c 2031 7d2c 2033 7d20 3d3e 207b  {{0, 1}, 3} => {
│ │ │ │ -00042900: 387d 207c 2020 2020 2020 2020 2020 2020  8} |            
│ │ │ │ -00042910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042920: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042880: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00042890: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000428a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000428b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000428c0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000428d0: 7b30 2c20 317d 2c20 337d 203d 3e20 7b38  {0, 1}, 3} => {8
│ │ │ │ +000428e0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000428f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042900: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042910: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042920: 7b30 2c20 317d 2c20 347d 203d 3e20 3020  {0, 1}, 4} => 0 
│ │ │ │  00042930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042940: 7b7b 302c 2031 7d2c 2034 7d20 3d3e 2030  {{0, 1}, 4} => 0
│ │ │ │ -00042950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042970: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042950: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042960: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042970: 7b30 2c20 327d 2c20 2d31 7d20 3d3e 2030  {0, 2}, -1} => 0
│ │ │ │  00042980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042990: 7b7b 302c 2032 7d2c 202d 317d 203d 3e20  {{0, 2}, -1} => 
│ │ │ │ -000429a0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -000429b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000429c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000429d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000429e0: 7b7b 302c 2032 7d2c 2030 7d20 3d3e 207b  {{0, 2}, 0} => {
│ │ │ │ -000429f0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000429a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000429b0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000429c0: 7b30 2c20 327d 2c20 307d 203d 3e20 7b36  {0, 2}, 0} => {6
│ │ │ │ +000429d0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000429e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000429f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042a10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042a30: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042a40: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042a10: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00042a20: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042a60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042a80: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042a90: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042a60: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00042a70: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042a90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ab0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ad0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042ae0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ -00042af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042b00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042b20: 7b7b 302c 2032 7d2c 2031 7d20 3d3e 207b  {{0, 2}, 1} => {
│ │ │ │ -00042b30: 387d 207c 2020 2020 2020 2020 2020 2020  8} |            
│ │ │ │ -00042b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042b50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042ab0: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00042ac0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042ae0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042af0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042b00: 7b30 2c20 327d 2c20 317d 203d 3e20 7b38  {0, 2}, 1} => {8
│ │ │ │ +00042b10: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042b40: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042b50: 7b30 2c20 337d 2c20 2d31 7d20 3d3e 2030  {0, 3}, -1} => 0
│ │ │ │  00042b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042b70: 7b7b 302c 2033 7d2c 202d 317d 203d 3e20  {{0, 3}, -1} => 
│ │ │ │ -00042b80: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00042b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042b90: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042ba0: 7b30 2c20 347d 2c20 2d31 7d20 3d3e 2030  {0, 4}, -1} => 0
│ │ │ │  00042bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042bc0: 7b7b 302c 2034 7d2c 202d 317d 203d 3e20  {{0, 4}, -1} => 
│ │ │ │ -00042bd0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00042be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00042bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042bd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042be0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042bf0: 7b31 2c20 307d 2c20 2d31 7d20 3d3e 2030  {1, 0}, -1} => 0
│ │ │ │  00042c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042c10: 7b7b 312c 2030 7d2c 202d 317d 203d 3e20  {{1, 0}, -1} => 
│ │ │ │ -00042c20: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00042c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042c40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042c60: 7b7b 312c 2030 7d2c 2030 7d20 3d3e 207b  {{1, 0}, 0} => {
│ │ │ │ -00042c70: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00042c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042c20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042c30: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042c40: 7b31 2c20 307d 2c20 307d 203d 3e20 7b32  {1, 0}, 0} => {2
│ │ │ │ +00042c50: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042c90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042cb0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042cc0: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00042c90: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00042ca0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ce0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042d00: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042d10: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ +00042ce0: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00042cf0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042d30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042d50: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042d60: 327d 207c 2020 2020 2020 2020 2020 2020  2} |            
│ │ │ │ -00042d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042d80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042da0: 7b7b 312c 2030 7d2c 2031 7d20 3d3e 207b  {{1, 0}, 1} => {
│ │ │ │ -00042db0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042d30: 2020 2020 2020 2020 2020 2020 2020 7b32                {2
│ │ │ │ +00042d40: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042d60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042d70: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042d80: 7b31 2c20 307d 2c20 317d 203d 3e20 7b34  {1, 0}, 1} => {4
│ │ │ │ +00042d90: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042df0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042e00: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042dd0: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042de0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042e40: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042e50: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042e20: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042e30: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042e50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042e70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042e90: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042ea0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042e70: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042e80: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ee0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042ef0: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ +00042ec0: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042ed0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042f10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042f30: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042f40: 347d 207c 2020 2020 2020 2020 2020 2020  4} |            
│ │ │ │ -00042f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042f60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042f80: 7b7b 312c 2030 7d2c 2032 7d20 3d3e 207b  {{1, 0}, 2} => {
│ │ │ │ -00042f90: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042f10: 2020 2020 2020 2020 2020 2020 2020 7b34                {4
│ │ │ │ +00042f20: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00042f50: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00042f60: 7b31 2c20 307d 2c20 327d 203d 3e20 7b36  {1, 0}, 2} => {6
│ │ │ │ +00042f70: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00042fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042fd0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00042fe0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00042fb0: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00042fc0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00042fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00042ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043020: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00043030: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00043000: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00043010: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00043020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00043040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043070: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00043080: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ -00043090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000430a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000430b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000430c0: 7b7b 312c 2030 7d2c 2033 7d20 3d3e 207b  {{1, 0}, 3} => {
│ │ │ │ -000430d0: 387d 207c 2020 2020 2020 2020 2020 2020  8} |            
│ │ │ │ -000430e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000430f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043050: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00043060: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00043070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043080: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043090: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000430a0: 7b31 2c20 307d 2c20 337d 203d 3e20 7b38  {1, 0}, 3} => {8
│ │ │ │ +000430b0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000430c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000430d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000430e0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000430f0: 7b31 2c20 307d 2c20 347d 203d 3e20 3020  {1, 0}, 4} => 0 
│ │ │ │  00043100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043110: 7b7b 312c 2030 7d2c 2034 7d20 3d3e 2030  {{1, 0}, 4} => 0
│ │ │ │ -00043120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043120: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043130: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00043140: 7b31 2c20 317d 2c20 2d31 7d20 3d3e 2030  {1, 1}, -1} => 0
│ │ │ │  00043150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043160: 7b7b 312c 2031 7d2c 202d 317d 203d 3e20  {{1, 1}, -1} => 
│ │ │ │ -00043170: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00043180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000431a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000431b0: 7b7b 312c 2031 7d2c 2030 7d20 3d3e 207b  {{1, 1}, 0} => {
│ │ │ │ -000431c0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00043160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043170: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043180: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00043190: 7b31 2c20 317d 2c20 307d 203d 3e20 7b36  {1, 1}, 0} => {6
│ │ │ │ +000431a0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000431b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000431c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000431d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000431e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000431f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043200: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00043210: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +000431e0: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +000431f0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00043200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043210: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00043220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043230: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043250: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00043260: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00043230: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00043240: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00043250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043260: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00043270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000432a0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000432b0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ -000432c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000432d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000432e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000432f0: 7b7b 312c 2031 7d2c 2031 7d20 3d3e 207b  {{1, 1}, 1} => {
│ │ │ │ -00043300: 387d 207c 2020 2020 2020 2020 2020 2020  8} |            
│ │ │ │ -00043310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043280: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00043290: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000432a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000432b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000432c0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000432d0: 7b31 2c20 317d 2c20 317d 203d 3e20 7b38  {1, 1}, 1} => {8
│ │ │ │ +000432e0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000432f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043300: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043310: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00043320: 7b31 2c20 327d 2c20 2d31 7d20 3d3e 2030  {1, 2}, -1} => 0
│ │ │ │  00043330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043340: 7b7b 312c 2032 7d2c 202d 317d 203d 3e20  {{1, 2}, -1} => 
│ │ │ │ -00043350: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00043360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043370: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043350: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043360: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00043370: 7b31 2c20 337d 2c20 2d31 7d20 3d3e 2030  {1, 3}, -1} => 0
│ │ │ │  00043380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043390: 7b7b 312c 2033 7d2c 202d 317d 203d 3e20  {{1, 3}, -1} => 
│ │ │ │ -000433a0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -000433b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000433c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000433a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000433b0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000433c0: 7b32 2c20 307d 2c20 2d31 7d20 3d3e 2030  {2, 0}, -1} => 0
│ │ │ │  000433d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000433e0: 7b7b 322c 2030 7d2c 202d 317d 203d 3e20  {{2, 0}, -1} => 
│ │ │ │ -000433f0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00043400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043410: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043430: 7b7b 322c 2030 7d2c 2030 7d20 3d3e 207b  {{2, 0}, 0} => {
│ │ │ │ -00043440: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +000433e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000433f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043400: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00043410: 7b32 2c20 307d 2c20 307d 203d 3e20 7b36  {2, 0}, 0} => {6
│ │ │ │ +00043420: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00043430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00043450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043480: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00043490: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +00043460: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00043470: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00043480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043490: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000434a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000434b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000434c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000434d0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -000434e0: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ +000434b0: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +000434c0: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +000434d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000434e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000434f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043520: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ -00043530: 367d 207c 2020 2020 2020 2020 2020 2020  6} |            
│ │ │ │ -00043540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043570: 7b7b 322c 2030 7d2c 2031 7d20 3d3e 207b  {{2, 0}, 1} => {
│ │ │ │ -00043580: 387d 207c 2020 2020 2020 2020 2020 2020  8} |            
│ │ │ │ -00043590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000435a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043500: 2020 2020 2020 2020 2020 2020 2020 7b36                {6
│ │ │ │ +00043510: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00043520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043530: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043540: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00043550: 7b32 2c20 307d 2c20 317d 203d 3e20 7b38  {2, 0}, 1} => {8
│ │ │ │ +00043560: 7d20 7c20 2020 2020 2020 2020 2020 2020  } |             
│ │ │ │ +00043570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043580: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043590: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000435a0: 7b32 2c20 317d 2c20 2d31 7d20 3d3e 2030  {2, 1}, -1} => 0
│ │ │ │  000435b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000435c0: 7b7b 322c 2031 7d2c 202d 317d 203d 3e20  {{2, 1}, -1} => 
│ │ │ │ -000435d0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -000435e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000435f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000435c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000435d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000435e0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000435f0: 7b32 2c20 327d 2c20 2d31 7d20 3d3e 2030  {2, 2}, -1} => 0
│ │ │ │  00043600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043610: 7b7b 322c 2032 7d2c 202d 317d 203d 3e20  {{2, 2}, -1} => 
│ │ │ │ -00043620: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00043630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043640: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043620: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043630: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00043640: 7b33 2c20 307d 2c20 2d31 7d20 3d3e 2030  {3, 0}, -1} => 0
│ │ │ │  00043650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043660: 7b7b 332c 2030 7d2c 202d 317d 203d 3e20  {{3, 0}, -1} => 
│ │ │ │ -00043670: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -00043680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043670: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043680: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +00043690: 7b33 2c20 317d 2c20 2d31 7d20 3d3e 2030  {3, 1}, -1} => 0
│ │ │ │  000436a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000436b0: 7b7b 332c 2031 7d2c 202d 317d 203d 3e20  {{3, 1}, -1} => 
│ │ │ │ -000436c0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ -000436d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000436e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000436b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000436c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000436d0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +000436e0: 7b34 2c20 307d 2c20 2d31 7d20 3d3e 2030  {4, 0}, -1} => 0
│ │ │ │  000436f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043700: 7b7b 342c 2030 7d2c 202d 317d 203d 3e20  {{4, 0}, -1} => 
│ │ │ │ -00043710: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00043700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00043720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043780: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043790: 6f31 3320 3a20 4861 7368 5461 626c 6520  o13 : HashTable 
│ │ │ │ +00043760: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00043770: 3133 203a 2048 6173 6854 6162 6c65 2020  13 : HashTable  
│ │ │ │ +00043780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000437a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000437b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000437c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000437d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000437b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +000437c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000437d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000437e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000437f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00043800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00043810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043880: 6232 2063 3220 6432 207c 2020 2020 2020  b2 c2 d2 |      
│ │ │ │ +00043850: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +00043860: 3220 6332 2064 3220 7c20 2020 2020 2020  2 c2 d2 |       
│ │ │ │ +00043870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000438a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000438b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000438c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000438d0: 202d 6232 202d 6332 2030 2020 202d 6432   -b2 -c2 0   -d2
│ │ │ │ -000438e0: 2030 2020 2030 2020 207c 2020 2020 2020   0   0   |      
│ │ │ │ -000438f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043910: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043920: 2061 3220 2030 2020 202d 6332 2030 2020   a2  0   -c2 0  
│ │ │ │ -00043930: 202d 6432 2030 2020 207c 2020 2020 2020   -d2 0   |      
│ │ │ │ -00043940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043970: 2030 2020 2061 3220 2062 3220 2030 2020   0   a2  b2  0  
│ │ │ │ -00043980: 2030 2020 202d 6432 207c 2020 2020 2020   0   -d2 |      
│ │ │ │ -00043990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000439a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000439b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000439c0: 2030 2020 2030 2020 2030 2020 2061 3220   0   0   0   a2 
│ │ │ │ -000439d0: 2062 3220 2063 3220 207c 2020 2020 2020   b2  c2  |      
│ │ │ │ -000439e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000439f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043a00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043a10: 2063 3220 2064 3220 2030 2020 2030 2020   c2  d2  0   0  
│ │ │ │ -00043a20: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00043a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043a60: 202d 6232 2030 2020 2064 3220 2030 2020   -b2 0   d2  0  
│ │ │ │ -00043a70: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00043a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043ab0: 2061 3220 2030 2020 2030 2020 2064 3220   a2  0   0   d2 
│ │ │ │ -00043ac0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00043ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043af0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043b00: 2030 2020 202d 6232 202d 6332 2030 2020   0   -b2 -c2 0  
│ │ │ │ -00043b10: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00043b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043b40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043b50: 2030 2020 2061 3220 2030 2020 202d 6332   0   a2  0   -c2
│ │ │ │ -00043b60: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00043b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043b90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043ba0: 2030 2020 2030 2020 2061 3220 2062 3220   0   0   a2  b2 
│ │ │ │ -00043bb0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00043bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043be0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043bf0: 202d 6432 207c 2020 2020 2020 2020 2020   -d2 |          
│ │ │ │ +000438a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000438b0: 2d62 3220 2d63 3220 3020 2020 2d64 3220  -b2 -c2 0   -d2 
│ │ │ │ +000438c0: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ +000438d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000438e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000438f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043900: 6132 2020 3020 2020 2d63 3220 3020 2020  a2  0   -c2 0   
│ │ │ │ +00043910: 2d64 3220 3020 2020 7c20 2020 2020 2020  -d2 0   |       
│ │ │ │ +00043920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043950: 3020 2020 6132 2020 6232 2020 3020 2020  0   a2  b2  0   
│ │ │ │ +00043960: 3020 2020 2d64 3220 7c20 2020 2020 2020  0   -d2 |       
│ │ │ │ +00043970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043990: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000439a0: 3020 2020 3020 2020 3020 2020 6132 2020  0   0   0   a2  
│ │ │ │ +000439b0: 6232 2020 6332 2020 7c20 2020 2020 2020  b2  c2  |       
│ │ │ │ +000439c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000439d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000439e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000439f0: 6332 2020 6432 2020 3020 2020 3020 2020  c2  d2  0   0   
│ │ │ │ +00043a00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00043a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043a40: 2d62 3220 3020 2020 6432 2020 3020 2020  -b2 0   d2  0   
│ │ │ │ +00043a50: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00043a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043a90: 6132 2020 3020 2020 3020 2020 6432 2020  a2  0   0   d2  
│ │ │ │ +00043aa0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00043ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043ae0: 3020 2020 2d62 3220 2d63 3220 3020 2020  0   -b2 -c2 0   
│ │ │ │ +00043af0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00043b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043b30: 3020 2020 6132 2020 3020 2020 2d63 3220  0   a2  0   -c2 
│ │ │ │ +00043b40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00043b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043b70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043b80: 3020 2020 3020 2020 6132 2020 6232 2020  0   0   a2  b2  
│ │ │ │ +00043b90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00043ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043bd0: 2d64 3220 7c20 2020 2020 2020 2020 2020  -d2 |           
│ │ │ │ +00043be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043c30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043c40: 2063 3220 207c 2020 2020 2020 2020 2020   c2  |          
│ │ │ │ +00043c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043c20: 6332 2020 7c20 2020 2020 2020 2020 2020  c2  |           
│ │ │ │ +00043c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043c80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043c90: 202d 6232 207c 2020 2020 2020 2020 2020   -b2 |          
│ │ │ │ +00043c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043c70: 2d62 3220 7c20 2020 2020 2020 2020 2020  -b2 |           
│ │ │ │ +00043c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043cd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043ce0: 2061 3220 207c 2020 2020 2020 2020 2020   a2  |          
│ │ │ │ +00043cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043cc0: 6132 2020 7c20 2020 2020 2020 2020 2020  a2  |           
│ │ │ │ +00043cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00043d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043d20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00043d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00043d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043dd0: 2031 3233 3661 332b 3839 3232 6132 622d   1236a3+8922a2b-
│ │ │ │ -00043de0: 3335 3839 6162 322b 3839 3731 6233 2d35  3589ab2+8971b3-5
│ │ │ │ -00043df0: 3030 3661 3263 2d35 3539 3961 6263 2d31  006a2c-5599abc-1
│ │ │ │ -00043e00: 3431 3635 6232 632b 3838 3830 6132 642b  4165b2c+8880a2d+
│ │ │ │ -00043e10: 3432 3539 6162 642d 3330 3032 627c 0a7c  4259abd-3002b|.|
│ │ │ │ -00043e20: 2031 3033 3730 6162 322d 3730 3932 6233   10370ab2-7092b3
│ │ │ │ -00043e30: 2d39 3730 3261 6263 2d36 3632 3762 3263  -9702abc-6627b2c
│ │ │ │ -00043e40: 2b38 3838 3661 6264 2b34 3730 3062 3264  +8886abd+4700b2d
│ │ │ │ -00043e50: 2d31 3561 6364 2b35 3936 3962 6364 2020  -15acd+5969bcd  
│ │ │ │ -00043e60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00043e70: 2031 3337 3037 6133 2b32 3137 3761 3262   13707a3+2177a2b
│ │ │ │ -00043e80: 2d37 3032 3861 6232 2b39 3739 3762 332d  -7028ab2+9797b3-
│ │ │ │ -00043e90: 3730 3231 6132 632b 3633 3737 6162 632b  7021a2c+6377abc+
│ │ │ │ -00043ea0: 3538 3734 6232 632d 3736 3030 6163 322d  5874b2c-7600ac2-
│ │ │ │ -00043eb0: 3131 3732 3662 6332 2b31 3134 307c 0a7c  11726bc2+1140|.|
│ │ │ │ -00043ec0: 202d 3939 3461 332d 3139 3436 6132 622d   -994a3-1946a2b-
│ │ │ │ -00043ed0: 3637 3233 6162 322d 3534 3833 6233 2d36  6723ab2-5483b3-6
│ │ │ │ -00043ee0: 3435 3361 3263 2d31 3139 3261 6263 2d31  453a2c-1192abc-1
│ │ │ │ -00043ef0: 3532 3530 6232 632d 3331 3634 6163 322b  5250b2c-3164ac2+
│ │ │ │ -00043f00: 3239 3562 6332 2d37 3635 3063 337c 0a7c  295bc2-7650c3|.|
│ │ │ │ -00043f10: 202d 3130 3337 3061 6232 2b37 3039 3262   -10370ab2+7092b
│ │ │ │ -00043f20: 332b 3937 3032 6162 632b 3636 3237 6232  3+9702abc+6627b2
│ │ │ │ -00043f30: 632b 3730 3238 6163 322d 3937 3937 6263  c+7028ac2-9797bc
│ │ │ │ -00043f40: 322d 3538 3734 6333 2d38 3838 3661 6264  2-5874c3-8886abd
│ │ │ │ -00043f50: 2d34 3730 3062 3264 2b31 3561 637c 0a7c  -4700b2d+15ac|.|
│ │ │ │ -00043f60: 202d 3133 3730 3761 332d 3231 3737 6132   -13707a3-2177a2
│ │ │ │ -00043f70: 622b 3730 3231 6132 632d 3633 3737 6162  b+7021a2c-6377ab
│ │ │ │ -00043f80: 632b 3736 3030 6163 322b 3131 3732 3662  c+7600ac2+11726b
│ │ │ │ -00043f90: 6332 2d31 3134 3063 332d 3132 3036 6132  c2-1140c3-1206a2
│ │ │ │ -00043fa0: 642b 3131 3433 3561 6264 2b36 307c 0a7c  d+11435abd+60|.|
│ │ │ │ -00043fb0: 2037 3032 3861 332d 3937 3937 6132 622d   7028a3-9797a2b-
│ │ │ │ -00043fc0: 3538 3734 6132 632d 3930 3734 6132 6420  5874a2c-9074a2d 
│ │ │ │ -00043fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00044000: 2039 3934 6133 2b31 3934 3661 3262 2b36   994a3+1946a2b+6
│ │ │ │ -00044010: 3435 3361 3263 2b31 3139 3261 6263 2b31  453a2c+1192abc+1
│ │ │ │ -00044020: 3130 3435 6132 642b 3135 3333 3361 6264  1045a2d+15333abd
│ │ │ │ -00044030: 2b33 3536 3061 6364 2d32 3239 3262 6364  +3560acd-2292bcd
│ │ │ │ -00044040: 2b37 3139 3461 6432 2d36 3234 357c 0a7c  +7194ad2-6245|.|
│ │ │ │ -00044050: 2036 3732 3361 332b 3534 3833 6132 622b   6723a3+5483a2b+
│ │ │ │ -00044060: 3135 3235 3061 3263 2d31 3035 3637 6132  15250a2c-10567a2
│ │ │ │ -00044070: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ -00044080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000440a0: 2033 3136 3461 332d 3239 3561 3262 2b37   3164a3-295a2b+7
│ │ │ │ -000440b0: 3635 3061 3263 2d31 3433 3838 6132 6420  650a2c-14388a2d 
│ │ │ │ -000440c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000440d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000440e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000440f0: 2031 3337 3037 6133 2b32 3137 3761 3262   13707a3+2177a2b
│ │ │ │ -00044100: 2d37 3032 3161 3263 2b36 3337 3761 6263  -7021a2c+6377abc
│ │ │ │ -00044110: 2d37 3630 3061 6332 2d31 3137 3236 6263  -7600ac2-11726bc
│ │ │ │ -00044120: 322b 3131 3430 6333 2b31 3230 3661 3264  2+1140c3+1206a2d
│ │ │ │ -00044130: 2d31 3134 3335 6162 642d 3630 347c 0a7c  -11435abd-604|.|
│ │ │ │ -00044140: 202d 3939 3461 332d 3139 3436 6132 622d   -994a3-1946a2b-
│ │ │ │ -00044150: 3634 3533 6132 632d 3131 3932 6162 632d  6453a2c-1192abc-
│ │ │ │ -00044160: 3131 3034 3561 3264 2d31 3533 3333 6162  11045a2d-15333ab
│ │ │ │ -00044170: 642d 3335 3630 6163 642b 3232 3932 6263  d-3560acd+2292bc
│ │ │ │ -00044180: 642d 3731 3934 6164 322b 3632 347c 0a7c  d-7194ad2+624|.|
│ │ │ │ -00044190: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00043da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043db0: 3132 3336 6133 2b38 3932 3261 3262 2d33  1236a3+8922a2b-3
│ │ │ │ +00043dc0: 3538 3961 6232 2b38 3937 3162 332d 3530  589ab2+8971b3-50
│ │ │ │ +00043dd0: 3036 6132 632d 3535 3939 6162 632d 3134  06a2c-5599abc-14
│ │ │ │ +00043de0: 3136 3562 3263 2b38 3838 3061 3264 2b34  165b2c+8880a2d+4
│ │ │ │ +00043df0: 3235 3961 6264 2d33 3030 3262 7c0a 7c20  259abd-3002b|.| 
│ │ │ │ +00043e00: 3130 3337 3061 6232 2d37 3039 3262 332d  10370ab2-7092b3-
│ │ │ │ +00043e10: 3937 3032 6162 632d 3636 3237 6232 632b  9702abc-6627b2c+
│ │ │ │ +00043e20: 3838 3836 6162 642b 3437 3030 6232 642d  8886abd+4700b2d-
│ │ │ │ +00043e30: 3135 6163 642b 3539 3639 6263 6420 2020  15acd+5969bcd   
│ │ │ │ +00043e40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043e50: 3133 3730 3761 332b 3231 3737 6132 622d  13707a3+2177a2b-
│ │ │ │ +00043e60: 3730 3238 6162 322b 3937 3937 6233 2d37  7028ab2+9797b3-7
│ │ │ │ +00043e70: 3032 3161 3263 2b36 3337 3761 6263 2b35  021a2c+6377abc+5
│ │ │ │ +00043e80: 3837 3462 3263 2d37 3630 3061 6332 2d31  874b2c-7600ac2-1
│ │ │ │ +00043e90: 3137 3236 6263 322b 3131 3430 7c0a 7c20  1726bc2+1140|.| 
│ │ │ │ +00043ea0: 2d39 3934 6133 2d31 3934 3661 3262 2d36  -994a3-1946a2b-6
│ │ │ │ +00043eb0: 3732 3361 6232 2d35 3438 3362 332d 3634  723ab2-5483b3-64
│ │ │ │ +00043ec0: 3533 6132 632d 3131 3932 6162 632d 3135  53a2c-1192abc-15
│ │ │ │ +00043ed0: 3235 3062 3263 2d33 3136 3461 6332 2b32  250b2c-3164ac2+2
│ │ │ │ +00043ee0: 3935 6263 322d 3736 3530 6333 7c0a 7c20  95bc2-7650c3|.| 
│ │ │ │ +00043ef0: 2d31 3033 3730 6162 322b 3730 3932 6233  -10370ab2+7092b3
│ │ │ │ +00043f00: 2b39 3730 3261 6263 2b36 3632 3762 3263  +9702abc+6627b2c
│ │ │ │ +00043f10: 2b37 3032 3861 6332 2d39 3739 3762 6332  +7028ac2-9797bc2
│ │ │ │ +00043f20: 2d35 3837 3463 332d 3838 3836 6162 642d  -5874c3-8886abd-
│ │ │ │ +00043f30: 3437 3030 6232 642b 3135 6163 7c0a 7c20  4700b2d+15ac|.| 
│ │ │ │ +00043f40: 2d31 3337 3037 6133 2d32 3137 3761 3262  -13707a3-2177a2b
│ │ │ │ +00043f50: 2b37 3032 3161 3263 2d36 3337 3761 6263  +7021a2c-6377abc
│ │ │ │ +00043f60: 2b37 3630 3061 6332 2b31 3137 3236 6263  +7600ac2+11726bc
│ │ │ │ +00043f70: 322d 3131 3430 6333 2d31 3230 3661 3264  2-1140c3-1206a2d
│ │ │ │ +00043f80: 2b31 3134 3335 6162 642b 3630 7c0a 7c20  +11435abd+60|.| 
│ │ │ │ +00043f90: 3730 3238 6133 2d39 3739 3761 3262 2d35  7028a3-9797a2b-5
│ │ │ │ +00043fa0: 3837 3461 3263 2d39 3037 3461 3264 2020  874a2c-9074a2d  
│ │ │ │ +00043fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00043fe0: 3939 3461 332b 3139 3436 6132 622b 3634  994a3+1946a2b+64
│ │ │ │ +00043ff0: 3533 6132 632b 3131 3932 6162 632b 3131  53a2c+1192abc+11
│ │ │ │ +00044000: 3034 3561 3264 2b31 3533 3333 6162 642b  045a2d+15333abd+
│ │ │ │ +00044010: 3335 3630 6163 642d 3232 3932 6263 642b  3560acd-2292bcd+
│ │ │ │ +00044020: 3731 3934 6164 322d 3632 3435 7c0a 7c20  7194ad2-6245|.| 
│ │ │ │ +00044030: 3637 3233 6133 2b35 3438 3361 3262 2b31  6723a3+5483a2b+1
│ │ │ │ +00044040: 3532 3530 6132 632d 3130 3536 3761 3264  5250a2c-10567a2d
│ │ │ │ +00044050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044070: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044080: 3331 3634 6133 2d32 3935 6132 622b 3736  3164a3-295a2b+76
│ │ │ │ +00044090: 3530 6132 632d 3134 3338 3861 3264 2020  50a2c-14388a2d  
│ │ │ │ +000440a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000440b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000440c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000440d0: 3133 3730 3761 332b 3231 3737 6132 622d  13707a3+2177a2b-
│ │ │ │ +000440e0: 3730 3231 6132 632b 3633 3737 6162 632d  7021a2c+6377abc-
│ │ │ │ +000440f0: 3736 3030 6163 322d 3131 3732 3662 6332  7600ac2-11726bc2
│ │ │ │ +00044100: 2b31 3134 3063 332b 3132 3036 6132 642d  +1140c3+1206a2d-
│ │ │ │ +00044110: 3131 3433 3561 6264 2d36 3034 7c0a 7c20  11435abd-604|.| 
│ │ │ │ +00044120: 2d39 3934 6133 2d31 3934 3661 3262 2d36  -994a3-1946a2b-6
│ │ │ │ +00044130: 3435 3361 3263 2d31 3139 3261 6263 2d31  453a2c-1192abc-1
│ │ │ │ +00044140: 3130 3435 6132 642d 3135 3333 3361 6264  1045a2d-15333abd
│ │ │ │ +00044150: 2d33 3536 3061 6364 2b32 3239 3262 6364  -3560acd+2292bcd
│ │ │ │ +00044160: 2d37 3139 3461 6432 2b36 3234 7c0a 7c20  -7194ad2+624|.| 
│ │ │ │ +00044170: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00044180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000441a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000441b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000441c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000441d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000441e0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +000441b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000441c0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000441d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000441e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000441f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044220: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00044230: 2039 3934 6133 2b31 3934 3661 3262 2b36   994a3+1946a2b+6
│ │ │ │ -00044240: 3732 3361 6232 2b36 3435 3361 3263 2b31  723ab2+6453a2c+1
│ │ │ │ -00044250: 3139 3261 6263 2b31 3130 3435 6132 642b  192abc+11045a2d+
│ │ │ │ -00044260: 3135 3333 3361 6264 2b33 3536 3061 6364  15333abd+3560acd
│ │ │ │ -00044270: 2d32 3239 3262 6364 2b37 3139 347c 0a7c  -2292bcd+7194|.|
│ │ │ │ +00044200: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044210: 3939 3461 332b 3139 3436 6132 622b 3637  994a3+1946a2b+67
│ │ │ │ +00044220: 3233 6162 322b 3634 3533 6132 632b 3131  23ab2+6453a2c+11
│ │ │ │ +00044230: 3932 6162 632b 3131 3034 3561 3264 2b31  92abc+11045a2d+1
│ │ │ │ +00044240: 3533 3333 6162 642b 3335 3630 6163 642d  5333abd+3560acd-
│ │ │ │ +00044250: 3232 3932 6263 642b 3731 3934 7c0a 7c20  2292bcd+7194|.| 
│ │ │ │ +00044260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000442a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000442a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000442b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000442c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000442c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000442d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000442e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000442f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00044320: 202d 3133 3739 3561 342b 3230 3139 6133   -13795a4+2019a3
│ │ │ │ -00044330: 622b 3133 3736 3961 3262 322b 3735 3836  b+13769a2b2+7586
│ │ │ │ -00044340: 6162 332b 3836 3439 6234 2b36 3435 3461  ab3+8649b4+6454a
│ │ │ │ -00044350: 3363 2d31 3031 3837 6132 6263 2d31 3738  3c-10187a2bc-178
│ │ │ │ -00044360: 3361 6232 632b 3932 3139 6233 637c 0a7c  3ab2c+9219b3c|.|
│ │ │ │ -00044370: 2031 3131 3532 6134 2d31 3333 3661 3362   11152a4-1336a3b
│ │ │ │ -00044380: 2b31 3138 3436 6132 6232 2b31 3032 3634  +11846a2b2+10264
│ │ │ │ -00044390: 6162 332b 3631 3862 342d 3131 3035 3161  ab3+618b4-11051a
│ │ │ │ -000443a0: 3363 2b31 3231 3239 6132 6263 2b35 3932  3c+12129a2bc+592
│ │ │ │ -000443b0: 3761 6232 632b 3438 3962 3363 2d7c 0a7c  7ab2c+489b3c-|.|
│ │ │ │ -000443c0: 202d 3633 3338 6134 2b31 3030 3235 6133   -6338a4+10025a3
│ │ │ │ -000443d0: 622b 3134 3938 3761 3363 2d39 3935 3961  b+14987a3c-9959a
│ │ │ │ -000443e0: 3262 632d 3131 3639 3161 3263 322b 3132  2bc-11691a2c2+12
│ │ │ │ -000443f0: 3333 3661 6263 322d 3737 3836 6133 642d  336abc2-7786a3d-
│ │ │ │ -00044400: 3131 3536 6132 6264 2b34 3936 307c 0a7c  1156a2bd+4960|.|
│ │ │ │ -00044410: 2032 3237 3561 342d 3233 3961 3362 2b31   2275a4-239a3b+1
│ │ │ │ -00044420: 3435 3934 6132 6232 2d38 3135 3361 6233  4594a2b2-8153ab3
│ │ │ │ -00044430: 2d31 3139 3435 6234 2d38 3431 3661 3363  -11945b4-8416a3c
│ │ │ │ -00044440: 2b36 3235 3161 3262 632d 3330 3233 6162  +6251a2bc-3023ab
│ │ │ │ -00044450: 3263 2b35 3933 3362 3363 2b39 327c 0a7c  2c+5933b3c+92|.|
│ │ │ │ -00044460: 202d 3935 3736 6134 2d39 3935 3761 3362   -9576a4-9957a3b
│ │ │ │ -00044470: 2b39 3830 3461 3262 322b 3134 3039 3161  +9804a2b2+14091a
│ │ │ │ -00044480: 3363 2b31 3331 3434 6132 6263 2b31 3338  3c+13144a2bc+138
│ │ │ │ -00044490: 3939 6132 6332 2d31 3035 3039 6162 6332  99a2c2-10509abc2
│ │ │ │ -000444a0: 2b31 3535 3033 6163 332b 3132 317c 0a7c  +15503ac3+121|.|
│ │ │ │ +000442f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044300: 2d31 3337 3935 6134 2b32 3031 3961 3362  -13795a4+2019a3b
│ │ │ │ +00044310: 2b31 3337 3639 6132 6232 2b37 3538 3661  +13769a2b2+7586a
│ │ │ │ +00044320: 6233 2b38 3634 3962 342b 3634 3534 6133  b3+8649b4+6454a3
│ │ │ │ +00044330: 632d 3130 3138 3761 3262 632d 3137 3833  c-10187a2bc-1783
│ │ │ │ +00044340: 6162 3263 2b39 3231 3962 3363 7c0a 7c20  ab2c+9219b3c|.| 
│ │ │ │ +00044350: 3131 3135 3261 342d 3133 3336 6133 622b  11152a4-1336a3b+
│ │ │ │ +00044360: 3131 3834 3661 3262 322b 3130 3236 3461  11846a2b2+10264a
│ │ │ │ +00044370: 6233 2b36 3138 6234 2d31 3130 3531 6133  b3+618b4-11051a3
│ │ │ │ +00044380: 632b 3132 3132 3961 3262 632b 3539 3237  c+12129a2bc+5927
│ │ │ │ +00044390: 6162 3263 2b34 3839 6233 632d 7c0a 7c20  ab2c+489b3c-|.| 
│ │ │ │ +000443a0: 2d36 3333 3861 342b 3130 3032 3561 3362  -6338a4+10025a3b
│ │ │ │ +000443b0: 2b31 3439 3837 6133 632d 3939 3539 6132  +14987a3c-9959a2
│ │ │ │ +000443c0: 6263 2d31 3136 3931 6132 6332 2b31 3233  bc-11691a2c2+123
│ │ │ │ +000443d0: 3336 6162 6332 2d37 3738 3661 3364 2d31  36abc2-7786a3d-1
│ │ │ │ +000443e0: 3135 3661 3262 642b 3439 3630 7c0a 7c20  156a2bd+4960|.| 
│ │ │ │ +000443f0: 3232 3735 6134 2d32 3339 6133 622b 3134  2275a4-239a3b+14
│ │ │ │ +00044400: 3539 3461 3262 322d 3831 3533 6162 332d  594a2b2-8153ab3-
│ │ │ │ +00044410: 3131 3934 3562 342d 3834 3136 6133 632b  11945b4-8416a3c+
│ │ │ │ +00044420: 3632 3531 6132 6263 2d33 3032 3361 6232  6251a2bc-3023ab2
│ │ │ │ +00044430: 632b 3539 3333 6233 632b 3932 7c0a 7c20  c+5933b3c+92|.| 
│ │ │ │ +00044440: 2d39 3537 3661 342d 3939 3537 6133 622b  -9576a4-9957a3b+
│ │ │ │ +00044450: 3938 3034 6132 6232 2b31 3430 3931 6133  9804a2b2+14091a3
│ │ │ │ +00044460: 632b 3133 3134 3461 3262 632b 3133 3839  c+13144a2bc+1389
│ │ │ │ +00044470: 3961 3263 322d 3130 3530 3961 6263 322b  9a2c2-10509abc2+
│ │ │ │ +00044480: 3135 3530 3361 6333 2b31 3231 7c0a 7c20  15503ac3+121|.| 
│ │ │ │ +00044490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000444a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000444b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000444c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000444d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000444d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000444e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000444f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000444f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044520: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00044530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00044540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000445a0: 2031 3037 6133 2b34 3337 3661 3262 2b33   107a3+4376a2b+3
│ │ │ │ -000445b0: 3738 3361 6232 2b31 3033 3539 6233 2d35  783ab2+10359b3-5
│ │ │ │ -000445c0: 3537 3061 3263 2d35 3330 3761 6263 2d37  570a2c-5307abc-7
│ │ │ │ -000445d0: 3436 3462 3263 2b33 3138 3761 3264 2b38  464b2c+3187a2d+8
│ │ │ │ -000445e0: 3537 3061 6264 2d38 3235 3162 327c 0a7c  570abd-8251b2|.|
│ │ │ │ -000445f0: 2038 3233 3161 6232 2b31 3331 3737 6233   8231ab2+13177b3
│ │ │ │ -00044600: 2b35 3836 3461 6263 2b31 3339 3930 6232  +5864abc+13990b2
│ │ │ │ -00044610: 632b 3530 3236 6162 642d 3131 3532 3162  c+5026abd-11521b
│ │ │ │ -00044620: 3264 2d37 3530 3161 6364 2d31 3737 3962  2d-7501acd-1779b
│ │ │ │ -00044630: 6364 2020 2020 2020 2020 2020 207c 0a7c  cd           |.|
│ │ │ │ -00044640: 202d 3135 3334 3461 332b 3236 3533 6132   -15344a3+2653a2
│ │ │ │ -00044650: 622b 3130 3235 3961 6232 2d31 3330 3962  b+10259ab2-1309b
│ │ │ │ -00044660: 332b 3132 3336 3561 3263 2d37 3231 3661  3+12365a2c-7216a
│ │ │ │ -00044670: 6263 2b35 3339 3862 3263 2b36 3233 3061  bc+5398b2c+6230a
│ │ │ │ -00044680: 6332 2d35 3332 3662 6332 2b31 307c 0a7c  c2-5326bc2+10|.|
│ │ │ │ -00044690: 202d 3130 3438 3061 332d 3632 3033 6132   -10480a3-6203a2
│ │ │ │ -000446a0: 622b 3935 3334 6162 322b 3130 3836 3662  b+9534ab2+10866b
│ │ │ │ -000446b0: 332d 3934 3830 6132 632b 3732 3536 6162  3-9480a2c+7256ab
│ │ │ │ -000446c0: 632d 3730 3631 6232 632b 3531 3037 6163  c-7061b2c+5107ac
│ │ │ │ -000446d0: 322b 3536 3739 6263 322d 3633 327c 0a7c  2+5679bc2-632|.|
│ │ │ │ -000446e0: 202d 3832 3331 6162 322d 3133 3137 3762   -8231ab2-13177b
│ │ │ │ -000446f0: 332d 3538 3634 6162 632d 3133 3939 3062  3-5864abc-13990b
│ │ │ │ -00044700: 3263 2d31 3032 3539 6163 322b 3133 3039  2c-10259ac2+1309
│ │ │ │ -00044710: 6263 322d 3533 3938 6333 2d35 3032 3661  bc2-5398c3-5026a
│ │ │ │ -00044720: 6264 2b31 3135 3231 6232 642b 377c 0a7c  bd+11521b2d+7|.|
│ │ │ │ -00044730: 2031 3533 3434 6133 2d32 3635 3361 3262   15344a3-2653a2b
│ │ │ │ -00044740: 2d31 3233 3635 6132 632b 3732 3136 6162  -12365a2c+7216ab
│ │ │ │ -00044750: 632d 3632 3330 6163 322b 3533 3236 6263  c-6230ac2+5326bc
│ │ │ │ -00044760: 322d 3130 3331 6333 2b31 3335 3038 6132  2-1031c3+13508a2
│ │ │ │ -00044770: 642b 3130 3132 3561 6264 2d39 307c 0a7c  d+10125abd-90|.|
│ │ │ │ -00044780: 202d 3130 3235 3961 332b 3133 3039 6132   -10259a3+1309a2
│ │ │ │ -00044790: 622d 3533 3938 6132 632d 3535 3439 6132  b-5398a2c-5549a2
│ │ │ │ -000447a0: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ -000447b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000447c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000447d0: 2031 3034 3830 6133 2b36 3230 3361 3262   10480a3+6203a2b
│ │ │ │ -000447e0: 2b39 3438 3061 3263 2d37 3235 3661 6263  +9480a2c-7256abc
│ │ │ │ -000447f0: 2b31 3139 3530 6132 642b 3533 3231 6162  +11950a2d+5321ab
│ │ │ │ -00044800: 642b 3339 3936 6163 642b 3731 3532 6263  d+3996acd+7152bc
│ │ │ │ -00044810: 642d 3933 3938 6164 322d 3135 337c 0a7c  d-9398ad2-153|.|
│ │ │ │ -00044820: 202d 3935 3334 6133 2d31 3038 3636 6132   -9534a3-10866a2
│ │ │ │ -00044830: 622b 3730 3631 6132 632d 3236 3237 6132  b+7061a2c-2627a2
│ │ │ │ -00044840: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ -00044850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00044870: 202d 3531 3037 6133 2d35 3637 3961 3262   -5107a3-5679a2b
│ │ │ │ -00044880: 2b36 3332 3561 3263 2b31 3137 3430 6132  +6325a2c+11740a2
│ │ │ │ -00044890: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ -000448a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000448b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000448c0: 202d 3135 3334 3461 332b 3236 3533 6132   -15344a3+2653a2
│ │ │ │ -000448d0: 622b 3132 3336 3561 3263 2d37 3231 3661  b+12365a2c-7216a
│ │ │ │ -000448e0: 6263 2b36 3233 3061 6332 2d35 3332 3662  bc+6230ac2-5326b
│ │ │ │ -000448f0: 6332 2b31 3033 3163 332d 3133 3530 3861  c2+1031c3-13508a
│ │ │ │ -00044900: 3264 2d31 3031 3235 6162 642b 397c 0a7c  2d-10125abd+9|.|
│ │ │ │ -00044910: 202d 3130 3438 3061 332d 3632 3033 6132   -10480a3-6203a2
│ │ │ │ -00044920: 622d 3934 3830 6132 632b 3732 3536 6162  b-9480a2c+7256ab
│ │ │ │ -00044930: 632d 3131 3935 3061 3264 2d35 3332 3161  c-11950a2d-5321a
│ │ │ │ -00044940: 6264 2d33 3939 3661 6364 2d37 3135 3262  bd-3996acd-7152b
│ │ │ │ -00044950: 6364 2b39 3339 3861 6432 2b31 357c 0a7c  cd+9398ad2+15|.|
│ │ │ │ -00044960: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00044570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044580: 3130 3761 332b 3433 3736 6132 622b 3337  107a3+4376a2b+37
│ │ │ │ +00044590: 3833 6162 322b 3130 3335 3962 332d 3535  83ab2+10359b3-55
│ │ │ │ +000445a0: 3730 6132 632d 3533 3037 6162 632d 3734  70a2c-5307abc-74
│ │ │ │ +000445b0: 3634 6232 632b 3331 3837 6132 642b 3835  64b2c+3187a2d+85
│ │ │ │ +000445c0: 3730 6162 642d 3832 3531 6232 7c0a 7c20  70abd-8251b2|.| 
│ │ │ │ +000445d0: 3832 3331 6162 322b 3133 3137 3762 332b  8231ab2+13177b3+
│ │ │ │ +000445e0: 3538 3634 6162 632b 3133 3939 3062 3263  5864abc+13990b2c
│ │ │ │ +000445f0: 2b35 3032 3661 6264 2d31 3135 3231 6232  +5026abd-11521b2
│ │ │ │ +00044600: 642d 3735 3031 6163 642d 3137 3739 6263  d-7501acd-1779bc
│ │ │ │ +00044610: 6420 2020 2020 2020 2020 2020 7c0a 7c20  d           |.| 
│ │ │ │ +00044620: 2d31 3533 3434 6133 2b32 3635 3361 3262  -15344a3+2653a2b
│ │ │ │ +00044630: 2b31 3032 3539 6162 322d 3133 3039 6233  +10259ab2-1309b3
│ │ │ │ +00044640: 2b31 3233 3635 6132 632d 3732 3136 6162  +12365a2c-7216ab
│ │ │ │ +00044650: 632b 3533 3938 6232 632b 3632 3330 6163  c+5398b2c+6230ac
│ │ │ │ +00044660: 322d 3533 3236 6263 322b 3130 7c0a 7c20  2-5326bc2+10|.| 
│ │ │ │ +00044670: 2d31 3034 3830 6133 2d36 3230 3361 3262  -10480a3-6203a2b
│ │ │ │ +00044680: 2b39 3533 3461 6232 2b31 3038 3636 6233  +9534ab2+10866b3
│ │ │ │ +00044690: 2d39 3438 3061 3263 2b37 3235 3661 6263  -9480a2c+7256abc
│ │ │ │ +000446a0: 2d37 3036 3162 3263 2b35 3130 3761 6332  -7061b2c+5107ac2
│ │ │ │ +000446b0: 2b35 3637 3962 6332 2d36 3332 7c0a 7c20  +5679bc2-632|.| 
│ │ │ │ +000446c0: 2d38 3233 3161 6232 2d31 3331 3737 6233  -8231ab2-13177b3
│ │ │ │ +000446d0: 2d35 3836 3461 6263 2d31 3339 3930 6232  -5864abc-13990b2
│ │ │ │ +000446e0: 632d 3130 3235 3961 6332 2b31 3330 3962  c-10259ac2+1309b
│ │ │ │ +000446f0: 6332 2d35 3339 3863 332d 3530 3236 6162  c2-5398c3-5026ab
│ │ │ │ +00044700: 642b 3131 3532 3162 3264 2b37 7c0a 7c20  d+11521b2d+7|.| 
│ │ │ │ +00044710: 3135 3334 3461 332d 3236 3533 6132 622d  15344a3-2653a2b-
│ │ │ │ +00044720: 3132 3336 3561 3263 2b37 3231 3661 6263  12365a2c+7216abc
│ │ │ │ +00044730: 2d36 3233 3061 6332 2b35 3332 3662 6332  -6230ac2+5326bc2
│ │ │ │ +00044740: 2d31 3033 3163 332b 3133 3530 3861 3264  -1031c3+13508a2d
│ │ │ │ +00044750: 2b31 3031 3235 6162 642d 3930 7c0a 7c20  +10125abd-90|.| 
│ │ │ │ +00044760: 2d31 3032 3539 6133 2b31 3330 3961 3262  -10259a3+1309a2b
│ │ │ │ +00044770: 2d35 3339 3861 3263 2d35 3534 3961 3264  -5398a2c-5549a2d
│ │ │ │ +00044780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000447a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000447b0: 3130 3438 3061 332b 3632 3033 6132 622b  10480a3+6203a2b+
│ │ │ │ +000447c0: 3934 3830 6132 632d 3732 3536 6162 632b  9480a2c-7256abc+
│ │ │ │ +000447d0: 3131 3935 3061 3264 2b35 3332 3161 6264  11950a2d+5321abd
│ │ │ │ +000447e0: 2b33 3939 3661 6364 2b37 3135 3262 6364  +3996acd+7152bcd
│ │ │ │ +000447f0: 2d39 3339 3861 6432 2d31 3533 7c0a 7c20  -9398ad2-153|.| 
│ │ │ │ +00044800: 2d39 3533 3461 332d 3130 3836 3661 3262  -9534a3-10866a2b
│ │ │ │ +00044810: 2b37 3036 3161 3263 2d32 3632 3761 3264  +7061a2c-2627a2d
│ │ │ │ +00044820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044840: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044850: 2d35 3130 3761 332d 3536 3739 6132 622b  -5107a3-5679a2b+
│ │ │ │ +00044860: 3633 3235 6132 632b 3131 3734 3061 3264  6325a2c+11740a2d
│ │ │ │ +00044870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044890: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000448a0: 2d31 3533 3434 6133 2b32 3635 3361 3262  -15344a3+2653a2b
│ │ │ │ +000448b0: 2b31 3233 3635 6132 632d 3732 3136 6162  +12365a2c-7216ab
│ │ │ │ +000448c0: 632b 3632 3330 6163 322d 3533 3236 6263  c+6230ac2-5326bc
│ │ │ │ +000448d0: 322b 3130 3331 6333 2d31 3335 3038 6132  2+1031c3-13508a2
│ │ │ │ +000448e0: 642d 3130 3132 3561 6264 2b39 7c0a 7c20  d-10125abd+9|.| 
│ │ │ │ +000448f0: 2d31 3034 3830 6133 2d36 3230 3361 3262  -10480a3-6203a2b
│ │ │ │ +00044900: 2d39 3438 3061 3263 2b37 3235 3661 6263  -9480a2c+7256abc
│ │ │ │ +00044910: 2d31 3139 3530 6132 642d 3533 3231 6162  -11950a2d-5321ab
│ │ │ │ +00044920: 642d 3339 3936 6163 642d 3731 3532 6263  d-3996acd-7152bc
│ │ │ │ +00044930: 642b 3933 3938 6164 322b 3135 7c0a 7c20  d+9398ad2+15|.| 
│ │ │ │ +00044940: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +00044950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000449a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000449b0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00044980: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044990: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000449a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000449b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000449c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000449d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000449e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000449f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00044a00: 2031 3034 3830 6133 2b36 3230 3361 3262   10480a3+6203a2b
│ │ │ │ -00044a10: 2d39 3533 3461 6232 2b39 3438 3061 3263  -9534ab2+9480a2c
│ │ │ │ -00044a20: 2d37 3235 3661 6263 2b31 3139 3530 6132  -7256abc+11950a2
│ │ │ │ -00044a30: 642b 3533 3231 6162 642b 3339 3936 6163  d+5321abd+3996ac
│ │ │ │ -00044a40: 642b 3731 3532 6263 642d 3933 397c 0a7c  d+7152bcd-939|.|
│ │ │ │ +000449d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000449e0: 3130 3438 3061 332b 3632 3033 6132 622d  10480a3+6203a2b-
│ │ │ │ +000449f0: 3935 3334 6162 322b 3934 3830 6132 632d  9534ab2+9480a2c-
│ │ │ │ +00044a00: 3732 3536 6162 632b 3131 3935 3061 3264  7256abc+11950a2d
│ │ │ │ +00044a10: 2b35 3332 3161 6264 2b33 3939 3661 6364  +5321abd+3996acd
│ │ │ │ +00044a20: 2b37 3135 3262 6364 2d39 3339 7c0a 7c20  +7152bcd-939|.| 
│ │ │ │ +00044a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044a70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00044a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044a90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00044a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044ae0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00044af0: 202d 3839 3931 6134 2b38 3938 3261 3362   -8991a4+8982a3b
│ │ │ │ -00044b00: 2b31 3333 3230 6132 6232 2d37 3232 3961  +13320a2b2-7229a
│ │ │ │ -00044b10: 6233 2b37 3637 3262 342d 3133 3638 3961  b3+7672b4-13689a
│ │ │ │ -00044b20: 3363 2b31 3030 3635 6132 6263 2d38 3738  3c+10065a2bc-878
│ │ │ │ -00044b30: 3761 6232 632d 3137 3236 6233 637c 0a7c  7ab2c-1726b3c|.|
│ │ │ │ -00044b40: 202d 3833 3735 6134 2d32 3235 6133 622d   -8375a4-225a3b-
│ │ │ │ -00044b50: 3131 3033 3961 3262 322d 3130 3139 3661  11039a2b2-10196a
│ │ │ │ -00044b60: 6233 2b34 3637 3662 342b 3236 3935 6133  b3+4676b4+2695a3
│ │ │ │ -00044b70: 632d 3839 3737 6132 6263 2d31 3539 3530  c-8977a2bc-15950
│ │ │ │ -00044b80: 6162 3263 2d37 3432 3162 3363 2d7c 0a7c  ab2c-7421b3c-|.|
│ │ │ │ -00044b90: 202d 3130 3835 3761 342b 3134 3136 3661   -10857a4+14166a
│ │ │ │ -00044ba0: 3362 2d38 3538 6133 632b 3533 3335 6132  3b-858a3c+5335a2
│ │ │ │ -00044bb0: 6263 2d35 3531 3361 3263 322b 3531 3030  bc-5513a2c2+5100
│ │ │ │ -00044bc0: 6162 6332 2d31 3037 3233 6133 642d 3138  abc2-10723a3d-18
│ │ │ │ -00044bd0: 3730 6132 6264 2d39 3832 3261 327c 0a7c  70a2bd-9822a2|.|
│ │ │ │ -00044be0: 202d 3535 3038 6134 2b31 3336 3630 6133   -5508a4+13660a3
│ │ │ │ -00044bf0: 622b 3135 3333 3661 3262 322d 3133 3934  b+15336a2b2-1394
│ │ │ │ -00044c00: 3361 6233 2d38 3432 6234 2b31 3931 3061  3ab3-842b4+1910a
│ │ │ │ -00044c10: 3363 2b31 3232 3330 6132 6263 2d39 3235  3c+12230a2bc-925
│ │ │ │ -00044c20: 3261 6232 632d 3634 3438 6233 637c 0a7c  2ab2c-6448b3c|.|
│ │ │ │ -00044c30: 2035 3232 3461 342d 3833 3330 6133 622d   5224a4-8330a3b-
│ │ │ │ -00044c40: 3133 3135 3361 3262 322b 3132 3230 3661  13153a2b2+12206a
│ │ │ │ -00044c50: 3363 2b31 3331 3338 6132 6263 2b34 3439  3c+13138a2bc+449
│ │ │ │ -00044c60: 3861 3263 322b 3133 3836 3461 6263 322d  8a2c2+13864abc2-
│ │ │ │ -00044c70: 3937 3861 6333 2d34 3036 3261 337c 0a7c  978ac3-4062a3|.|
│ │ │ │ +00044ac0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044ad0: 2d38 3939 3161 342b 3839 3832 6133 622b  -8991a4+8982a3b+
│ │ │ │ +00044ae0: 3133 3332 3061 3262 322d 3732 3239 6162  13320a2b2-7229ab
│ │ │ │ +00044af0: 332b 3736 3732 6234 2d31 3336 3839 6133  3+7672b4-13689a3
│ │ │ │ +00044b00: 632b 3130 3036 3561 3262 632d 3837 3837  c+10065a2bc-8787
│ │ │ │ +00044b10: 6162 3263 2d31 3732 3662 3363 7c0a 7c20  ab2c-1726b3c|.| 
│ │ │ │ +00044b20: 2d38 3337 3561 342d 3232 3561 3362 2d31  -8375a4-225a3b-1
│ │ │ │ +00044b30: 3130 3339 6132 6232 2d31 3031 3936 6162  1039a2b2-10196ab
│ │ │ │ +00044b40: 332b 3436 3736 6234 2b32 3639 3561 3363  3+4676b4+2695a3c
│ │ │ │ +00044b50: 2d38 3937 3761 3262 632d 3135 3935 3061  -8977a2bc-15950a
│ │ │ │ +00044b60: 6232 632d 3734 3231 6233 632d 7c0a 7c20  b2c-7421b3c-|.| 
│ │ │ │ +00044b70: 2d31 3038 3537 6134 2b31 3431 3636 6133  -10857a4+14166a3
│ │ │ │ +00044b80: 622d 3835 3861 3363 2b35 3333 3561 3262  b-858a3c+5335a2b
│ │ │ │ +00044b90: 632d 3535 3133 6132 6332 2b35 3130 3061  c-5513a2c2+5100a
│ │ │ │ +00044ba0: 6263 322d 3130 3732 3361 3364 2d31 3837  bc2-10723a3d-187
│ │ │ │ +00044bb0: 3061 3262 642d 3938 3232 6132 7c0a 7c20  0a2bd-9822a2|.| 
│ │ │ │ +00044bc0: 2d35 3530 3861 342b 3133 3636 3061 3362  -5508a4+13660a3b
│ │ │ │ +00044bd0: 2b31 3533 3336 6132 6232 2d31 3339 3433  +15336a2b2-13943
│ │ │ │ +00044be0: 6162 332d 3834 3262 342b 3139 3130 6133  ab3-842b4+1910a3
│ │ │ │ +00044bf0: 632b 3132 3233 3061 3262 632d 3932 3532  c+12230a2bc-9252
│ │ │ │ +00044c00: 6162 3263 2d36 3434 3862 3363 7c0a 7c20  ab2c-6448b3c|.| 
│ │ │ │ +00044c10: 3532 3234 6134 2d38 3333 3061 3362 2d31  5224a4-8330a3b-1
│ │ │ │ +00044c20: 3331 3533 6132 6232 2b31 3232 3036 6133  3153a2b2+12206a3
│ │ │ │ +00044c30: 632b 3133 3133 3861 3262 632b 3434 3938  c+13138a2bc+4498
│ │ │ │ +00044c40: 6132 6332 2b31 3338 3634 6162 6332 2d39  a2c2+13864abc2-9
│ │ │ │ +00044c50: 3738 6163 332d 3430 3632 6133 7c0a 7c20  78ac3-4062a3|.| 
│ │ │ │ +00044c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00044cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00044cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00044d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044d10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00044d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044d60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00044d70: 2039 3631 3161 342b 3133 3132 3761 3362   9611a4+13127a3b
│ │ │ │ -00044d80: 2d39 3438 3961 3262 322d 3436 3461 6233  -9489a2b2-464ab3
│ │ │ │ -00044d90: 2b39 3334 3162 342d 3135 3734 3361 3363  +9341b4-15743a3c
│ │ │ │ -00044da0: 2d39 3533 3061 3262 632b 3134 3933 3561  -9530a2bc+14935a
│ │ │ │ -00044db0: 6232 632d 3133 3930 3162 3363 2b7c 0a7c  b2c-13901b3c+|.|
│ │ │ │ -00044dc0: 202d 3339 3538 6134 2d35 3734 6133 622d   -3958a4-574a3b-
│ │ │ │ -00044dd0: 3731 3031 6132 6232 2b31 3536 3734 6162  7101a2b2+15674ab
│ │ │ │ -00044de0: 332b 3633 3433 6234 2b31 3134 3261 3363  3+6343b4+1142a3c
│ │ │ │ -00044df0: 2b39 3832 3061 3262 632d 3438 3231 6162  +9820a2bc-4821ab
│ │ │ │ -00044e00: 3263 2b35 3733 3762 3363 2b35 307c 0a7c  2c+5737b3c+50|.|
│ │ │ │ -00044e10: 2032 3339 3861 342b 3937 3334 6133 622d   2398a4+9734a3b-
│ │ │ │ -00044e20: 3335 6133 632d 3439 3739 6132 6263 2d35  35a3c-4979a2bc-5
│ │ │ │ -00044e30: 3035 3361 3263 322d 3433 3732 6162 6332  053a2c2-4372abc2
│ │ │ │ -00044e40: 2b31 3034 3232 6133 642d 3532 3631 6132  +10422a3d-5261a2
│ │ │ │ -00044e50: 6264 2d32 3735 3061 3263 642d 397c 0a7c  bd-2750a2cd-9|.|
│ │ │ │ -00044e60: 2036 3934 3561 342b 3132 3039 3861 3362   6945a4+12098a3b
│ │ │ │ -00044e70: 2d38 3937 3861 3262 322b 3132 3039 3961  -8978a2b2+12099a
│ │ │ │ -00044e80: 6233 2b39 3136 3962 342d 3135 3933 3661  b3+9169b4-15936a
│ │ │ │ -00044e90: 3363 2d31 3937 3561 3262 632b 3130 3537  3c-1975a2bc+1057
│ │ │ │ -00044ea0: 3361 6232 632b 3930 3531 6233 637c 0a7c  3ab2c+9051b3c|.|
│ │ │ │ -00044eb0: 2031 3237 3938 6134 2d36 3034 3061 3362   12798a4-6040a3b
│ │ │ │ -00044ec0: 2d37 3234 3761 3262 322d 3331 3238 6133  -7247a2b2-3128a3
│ │ │ │ -00044ed0: 632d 3134 3136 3561 3262 632d 3436 3237  c-14165a2bc-4627
│ │ │ │ -00044ee0: 6132 6332 2d31 3036 3737 6162 6332 2d34  a2c2-10677abc2-4
│ │ │ │ -00044ef0: 3633 3361 6333 2b33 3333 3861 337c 0a7c  633ac3+3338a3|.|
│ │ │ │ +00044d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00044d50: 3936 3131 6134 2b31 3331 3237 6133 622d  9611a4+13127a3b-
│ │ │ │ +00044d60: 3934 3839 6132 6232 2d34 3634 6162 332b  9489a2b2-464ab3+
│ │ │ │ +00044d70: 3933 3431 6234 2d31 3537 3433 6133 632d  9341b4-15743a3c-
│ │ │ │ +00044d80: 3935 3330 6132 6263 2b31 3439 3335 6162  9530a2bc+14935ab
│ │ │ │ +00044d90: 3263 2d31 3339 3031 6233 632b 7c0a 7c20  2c-13901b3c+|.| 
│ │ │ │ +00044da0: 2d33 3935 3861 342d 3537 3461 3362 2d37  -3958a4-574a3b-7
│ │ │ │ +00044db0: 3130 3161 3262 322b 3135 3637 3461 6233  101a2b2+15674ab3
│ │ │ │ +00044dc0: 2b36 3334 3362 342b 3131 3432 6133 632b  +6343b4+1142a3c+
│ │ │ │ +00044dd0: 3938 3230 6132 6263 2d34 3832 3161 6232  9820a2bc-4821ab2
│ │ │ │ +00044de0: 632b 3537 3337 6233 632b 3530 7c0a 7c20  c+5737b3c+50|.| 
│ │ │ │ +00044df0: 3233 3938 6134 2b39 3733 3461 3362 2d33  2398a4+9734a3b-3
│ │ │ │ +00044e00: 3561 3363 2d34 3937 3961 3262 632d 3530  5a3c-4979a2bc-50
│ │ │ │ +00044e10: 3533 6132 6332 2d34 3337 3261 6263 322b  53a2c2-4372abc2+
│ │ │ │ +00044e20: 3130 3432 3261 3364 2d35 3236 3161 3262  10422a3d-5261a2b
│ │ │ │ +00044e30: 642d 3237 3530 6132 6364 2d39 7c0a 7c20  d-2750a2cd-9|.| 
│ │ │ │ +00044e40: 3639 3435 6134 2b31 3230 3938 6133 622d  6945a4+12098a3b-
│ │ │ │ +00044e50: 3839 3738 6132 6232 2b31 3230 3939 6162  8978a2b2+12099ab
│ │ │ │ +00044e60: 332b 3931 3639 6234 2d31 3539 3336 6133  3+9169b4-15936a3
│ │ │ │ +00044e70: 632d 3139 3735 6132 6263 2b31 3035 3733  c-1975a2bc+10573
│ │ │ │ +00044e80: 6162 3263 2b39 3035 3162 3363 7c0a 7c20  ab2c+9051b3c|.| 
│ │ │ │ +00044e90: 3132 3739 3861 342d 3630 3430 6133 622d  12798a4-6040a3b-
│ │ │ │ +00044ea0: 3732 3437 6132 6232 2d33 3132 3861 3363  7247a2b2-3128a3c
│ │ │ │ +00044eb0: 2d31 3431 3635 6132 6263 2d34 3632 3761  -14165a2bc-4627a
│ │ │ │ +00044ec0: 3263 322d 3130 3637 3761 6263 322d 3436  2c2-10677abc2-46
│ │ │ │ +00044ed0: 3333 6163 332b 3333 3338 6133 7c0a 7c2d  33ac3+3338a3|.|-
│ │ │ │ +00044ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00044ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00044f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00044f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00044f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00044f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044f70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00044f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044f90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00044f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00044fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00044fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00044fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00044fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00044ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045060: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000450a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000450b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000450b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000450c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000450d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000450d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000450e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000450f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045100: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000451a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000451a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000451b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000451c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000451c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000451d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000451e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000451f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000451f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045290: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000452a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000452b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000452b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000452c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000452d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000452e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000452e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000452f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045350: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045380: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000453a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000453a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000453b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000453c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000453d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000453d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000453e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000453f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000453f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000454a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000454b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000454c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000454d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000454e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000454f0: 3264 2d31 3338 3932 6163 642d 3130 3532  2d-13892acd-1052
│ │ │ │ -00045500: 3162 6364 2020 2020 2020 2020 2020 2020  1bcd            
│ │ │ │ -00045510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000454c0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +000454d0: 642d 3133 3839 3261 6364 2d31 3035 3231  d-13892acd-10521
│ │ │ │ +000454e0: 6263 6420 2020 2020 2020 2020 2020 2020  bcd             
│ │ │ │ +000454f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045510: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00045590: 6333 2b31 3230 3661 3264 2d31 3134 3335  c3+1206a2d-11435
│ │ │ │ -000455a0: 6162 642b 3930 3734 6232 642d 3630 3430  abd+9074b2d-6040
│ │ │ │ -000455b0: 6163 642b 3830 3232 6263 642b 3339 3638  acd+8022bcd+3968
│ │ │ │ -000455c0: 6332 6420 2020 2020 2020 2020 2020 2020  c2d             
│ │ │ │ -000455d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000455e0: 2d31 3130 3435 6132 642d 3135 3333 3361  -11045a2d-15333a
│ │ │ │ -000455f0: 6264 2b31 3035 3637 6232 642d 3335 3630  bd+10567b2d-3560
│ │ │ │ -00045600: 6163 642b 3232 3932 6263 642b 3134 3338  acd+2292bcd+1438
│ │ │ │ -00045610: 3863 3264 2d37 3139 3461 6432 2b36 3234  8c2d-7194ad2+624
│ │ │ │ -00045620: 3562 6432 2d38 3633 3963 6432 2b7c 0a7c  5bd2-8639cd2+|.|
│ │ │ │ -00045630: 642d 3539 3639 6263 642d 3930 3734 6332  d-5969bcd-9074c2
│ │ │ │ -00045640: 642b 3637 3233 6164 322b 3534 3833 6264  d+6723ad2+5483bd
│ │ │ │ -00045650: 322b 3135 3235 3063 6432 2d31 3035 3637  2+15250cd2-10567
│ │ │ │ -00045660: 6433 2020 2031 3233 3661 332b 3839 3232  d3   1236a3+8922
│ │ │ │ -00045670: 6132 622d 3335 3839 6162 322b 387c 0a7c  a2b-3589ab2+8|.|
│ │ │ │ -00045680: 3430 6163 642d 3830 3232 6263 642d 3339  40acd-8022bcd-39
│ │ │ │ -00045690: 3638 6332 642b 3331 3634 6164 322d 3239  68c2d+3164ad2-29
│ │ │ │ -000456a0: 3562 6432 2b37 3635 3063 6432 2d31 3433  5bd2+7650cd2-143
│ │ │ │ -000456b0: 3838 6433 2030 2020 2020 2020 2020 2020  88d3 0          
│ │ │ │ -000456c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045560: 2020 2020 2020 2020 2020 2020 7c0a 7c63              |.|c
│ │ │ │ +00045570: 332b 3132 3036 6132 642d 3131 3433 3561  3+1206a2d-11435a
│ │ │ │ +00045580: 6264 2b39 3037 3462 3264 2d36 3034 3061  bd+9074b2d-6040a
│ │ │ │ +00045590: 6364 2b38 3032 3262 6364 2b33 3936 3863  cd+8022bcd+3968c
│ │ │ │ +000455a0: 3264 2020 2020 2020 2020 2020 2020 2020  2d              
│ │ │ │ +000455b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +000455c0: 3131 3034 3561 3264 2d31 3533 3333 6162  11045a2d-15333ab
│ │ │ │ +000455d0: 642b 3130 3536 3762 3264 2d33 3536 3061  d+10567b2d-3560a
│ │ │ │ +000455e0: 6364 2b32 3239 3262 6364 2b31 3433 3838  cd+2292bcd+14388
│ │ │ │ +000455f0: 6332 642d 3731 3934 6164 322b 3632 3435  c2d-7194ad2+6245
│ │ │ │ +00045600: 6264 322d 3836 3339 6364 322b 7c0a 7c64  bd2-8639cd2+|.|d
│ │ │ │ +00045610: 2d35 3936 3962 6364 2d39 3037 3463 3264  -5969bcd-9074c2d
│ │ │ │ +00045620: 2b36 3732 3361 6432 2b35 3438 3362 6432  +6723ad2+5483bd2
│ │ │ │ +00045630: 2b31 3532 3530 6364 322d 3130 3536 3764  +15250cd2-10567d
│ │ │ │ +00045640: 3320 2020 3132 3336 6133 2b38 3932 3261  3   1236a3+8922a
│ │ │ │ +00045650: 3262 2d33 3538 3961 6232 2b38 7c0a 7c34  2b-3589ab2+8|.|4
│ │ │ │ +00045660: 3061 6364 2d38 3032 3262 6364 2d33 3936  0acd-8022bcd-396
│ │ │ │ +00045670: 3863 3264 2b33 3136 3461 6432 2d32 3935  8c2d+3164ad2-295
│ │ │ │ +00045680: 6264 322b 3736 3530 6364 322d 3134 3338  bd2+7650cd2-1438
│ │ │ │ +00045690: 3864 3320 3020 2020 2020 2020 2020 2020  8d3 0           
│ │ │ │ +000456a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000456b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000456c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000456d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000456e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000456f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045700: 2020 2020 202d 3133 3730 3761 332d 3231       -13707a3-21
│ │ │ │ -00045710: 3737 6132 622b 3730 3238 6162 327c 0a7c  77a2b+7028ab2|.|
│ │ │ │ -00045720: 6264 322b 3836 3339 6364 322d 3934 3236  bd2+8639cd2-9426
│ │ │ │ -00045730: 6433 2020 2020 2020 2020 2020 2020 2020  d3              
│ │ │ │ -00045740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045750: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ -00045760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000456e0: 2020 2020 2d31 3337 3037 6133 2d32 3137      -13707a3-217
│ │ │ │ +000456f0: 3761 3262 2b37 3032 3861 6232 7c0a 7c62  7a2b+7028ab2|.|b
│ │ │ │ +00045700: 6432 2b38 3633 3963 6432 2d39 3432 3664  d2+8639cd2-9426d
│ │ │ │ +00045710: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00045720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045730: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ +00045740: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00045750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000457a0: 2020 2020 2039 3934 6133 2b31 3934 3661       994a3+1946a
│ │ │ │ -000457b0: 3262 2b36 3732 3361 6232 2b35 347c 0a7c  2b+6723ab2+54|.|
│ │ │ │ +00045780: 2020 2020 3939 3461 332b 3139 3436 6132      994a3+1946a2
│ │ │ │ +00045790: 622b 3637 3233 6162 322b 3534 7c0a 7c20  b+6723ab2+54|.| 
│ │ │ │ +000457a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000457b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000457c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000457d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000457e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000457f0: 2020 2020 2033 3136 3461 6232 2d32 3935       3164ab2-295
│ │ │ │ -00045800: 6233 2b37 3635 3062 3263 2d31 347c 0a7c  b3+7650b2c-14|.|
│ │ │ │ -00045810: 3061 6364 2b38 3032 3262 6364 2b33 3936  0acd+8022bcd+396
│ │ │ │ -00045820: 3863 3264 2d33 3136 3461 6432 2b32 3935  8c2d-3164ad2+295
│ │ │ │ -00045830: 6264 322d 3736 3530 6364 322b 3134 3338  bd2-7650cd2+1438
│ │ │ │ -00045840: 3864 3320 2d31 3033 3730 6162 322b 3730  8d3 -10370ab2+70
│ │ │ │ -00045850: 3932 6233 2b39 3730 3261 6263 2b7c 0a7c  92b3+9702abc+|.|
│ │ │ │ -00045860: 3562 6432 2d38 3633 3963 6432 2b39 3432  5bd2-8639cd2+942
│ │ │ │ -00045870: 3664 3320 2020 2020 2020 2020 2020 2020  6d3             
│ │ │ │ -00045880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045890: 2020 2020 3637 3233 6163 3220 2020 2020      6723ac2     
│ │ │ │ -000458a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000457d0: 2020 2020 3331 3634 6162 322d 3239 3562      3164ab2-295b
│ │ │ │ +000457e0: 332b 3736 3530 6232 632d 3134 7c0a 7c30  3+7650b2c-14|.|0
│ │ │ │ +000457f0: 6163 642b 3830 3232 6263 642b 3339 3638  acd+8022bcd+3968
│ │ │ │ +00045800: 6332 642d 3331 3634 6164 322b 3239 3562  c2d-3164ad2+295b
│ │ │ │ +00045810: 6432 2d37 3635 3063 6432 2b31 3433 3838  d2-7650cd2+14388
│ │ │ │ +00045820: 6433 202d 3130 3337 3061 6232 2b37 3039  d3 -10370ab2+709
│ │ │ │ +00045830: 3262 332b 3937 3032 6162 632b 7c0a 7c35  2b3+9702abc+|.|5
│ │ │ │ +00045840: 6264 322d 3836 3339 6364 322b 3934 3236  bd2-8639cd2+9426
│ │ │ │ +00045850: 6433 2020 2020 2020 2020 2020 2020 2020  d3              
│ │ │ │ +00045860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045870: 2020 2036 3732 3361 6332 2020 2020 2020     6723ac2      
│ │ │ │ +00045880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00045890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000458a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000458b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000458c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000458d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000458e0: 2020 2020 2d39 3934 6133 2d31 3934 3661      -994a3-1946a
│ │ │ │ -000458f0: 3262 2d36 3732 3361 6232 2d36 347c 0a7c  2b-6723ab2-64|.|
│ │ │ │ +000458c0: 2020 202d 3939 3461 332d 3139 3436 6132     -994a3-1946a2
│ │ │ │ +000458d0: 622d 3637 3233 6162 322d 3634 7c0a 7c20  b-6723ab2-64|.| 
│ │ │ │ +000458e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000458f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045930: 2020 2020 2d35 3438 3361 3262 2d31 3532      -5483a2b-152
│ │ │ │ -00045940: 3530 6132 632b 3130 3536 3761 327c 0a7c  50a2c+10567a2|.|
│ │ │ │ -00045950: 6164 322d 3632 3435 6264 322b 3836 3339  ad2-6245bd2+8639
│ │ │ │ -00045960: 6364 322d 3934 3236 6433 2031 3337 3037  cd2-9426d3 13707
│ │ │ │ -00045970: 6133 2b32 3137 3761 3262 2d37 3032 3861  a3+2177a2b-7028a
│ │ │ │ -00045980: 6232 2d37 3032 3161 3263 2b36 3337 3761  b2-7021a2c+6377a
│ │ │ │ -00045990: 6263 2d37 3630 3061 6332 2d31 317c 0a7c  bc-7600ac2-11|.|
│ │ │ │ +00045910: 2020 202d 3534 3833 6132 622d 3135 3235     -5483a2b-1525
│ │ │ │ +00045920: 3061 3263 2b31 3035 3637 6132 7c0a 7c61  0a2c+10567a2|.|a
│ │ │ │ +00045930: 6432 2d36 3234 3562 6432 2b38 3633 3963  d2-6245bd2+8639c
│ │ │ │ +00045940: 6432 2d39 3432 3664 3320 3133 3730 3761  d2-9426d3 13707a
│ │ │ │ +00045950: 332b 3231 3737 6132 622d 3730 3238 6162  3+2177a2b-7028ab
│ │ │ │ +00045960: 322d 3730 3231 6132 632b 3633 3737 6162  2-7021a2c+6377ab
│ │ │ │ +00045970: 632d 3736 3030 6163 322d 3131 7c0a 7c20  c-7600ac2-11|.| 
│ │ │ │ +00045980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000459a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000459b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000459c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000459c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000459d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000459e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000459e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000459f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045a30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00045a40: 2b35 3531 3361 3263 322b 3130 3535 3861  +5513a2c2+10558a
│ │ │ │ -00045a50: 6263 322b 3235 3930 6232 6332 2b31 3136  bc2+2590b2c2+116
│ │ │ │ -00045a60: 3234 6133 642d 3536 3033 6132 6264 2b31  24a3d-5603a2bd+1
│ │ │ │ -00045a70: 3430 3538 6162 3264 2d31 3236 3135 6233  4058ab2d-12615b3
│ │ │ │ -00045a80: 642b 3738 3639 6132 6364 2d32 307c 0a7c  d+7869a2cd-20|.|
│ │ │ │ -00045a90: 3135 3338 3361 3263 322b 3530 3761 6263  15383a2c2+507abc
│ │ │ │ -00045aa0: 322d 3133 3830 3462 3263 322d 3834 3136  2-13804b2c2-8416
│ │ │ │ -00045ab0: 6163 332b 3932 6334 2d31 3130 3537 6133  ac3+92c4-11057a3
│ │ │ │ -00045ac0: 642d 3531 3133 6132 6264 2d32 3736 3261  d-5113a2bd-2762a
│ │ │ │ -00045ad0: 6232 642b 3134 3039 3562 3364 2d7c 0a7c  b2d+14095b3d-|.|
│ │ │ │ -00045ae0: 6132 6364 2d35 3538 3961 6263 642d 3831  a2cd-5589abcd-81
│ │ │ │ -00045af0: 3633 6163 3264 2d31 3839 3562 6332 642b  63ac2d-1895bc2d+
│ │ │ │ -00045b00: 3934 3634 6132 6432 2d37 3235 3361 6264  9464a2d2-7253abd
│ │ │ │ -00045b10: 322b 3132 3634 3261 6364 322d 3139 3538  2+12642acd2-1958
│ │ │ │ -00045b20: 6263 6432 2020 2020 2020 2020 207c 0a7c  bcd2         |.|
│ │ │ │ -00045b30: 6132 6332 2b35 3334 3361 6263 322b 3337  a2c2+5343abc2+37
│ │ │ │ -00045b40: 3938 6232 6332 2d31 3539 3638 6133 642b  98b2c2-15968a3d+
│ │ │ │ -00045b50: 3437 3361 3262 642b 3133 3239 3361 6232  473a2bd+13293ab2
│ │ │ │ -00045b60: 642d 3337 3631 6233 642d 3737 3137 6132  d-3761b3d-7717a2
│ │ │ │ -00045b70: 6364 2d37 3338 3961 6263 642b 347c 0a7c  cd-7389abcd+4|.|
│ │ │ │ -00045b80: 3239 6133 642b 3134 3738 3461 3262 642d  29a3d+14784a2bd-
│ │ │ │ -00045b90: 3130 3630 3861 3263 642d 3338 3136 6162  10608a2cd-3816ab
│ │ │ │ -00045ba0: 6364 2d31 3336 3338 6163 3264 2d36 3135  cd-13638ac2d-615
│ │ │ │ -00045bb0: 3961 3264 322d 3134 3634 3861 6264 322b  9a2d2-14648abd2+
│ │ │ │ -00045bc0: 3330 3732 6163 6432 2d31 3432 367c 0a7c  3072acd2-1426|.|
│ │ │ │ +00045a10: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ +00045a20: 3535 3133 6132 6332 2b31 3035 3538 6162  5513a2c2+10558ab
│ │ │ │ +00045a30: 6332 2b32 3539 3062 3263 322b 3131 3632  c2+2590b2c2+1162
│ │ │ │ +00045a40: 3461 3364 2d35 3630 3361 3262 642b 3134  4a3d-5603a2bd+14
│ │ │ │ +00045a50: 3035 3861 6232 642d 3132 3631 3562 3364  058ab2d-12615b3d
│ │ │ │ +00045a60: 2b37 3836 3961 3263 642d 3230 7c0a 7c31  +7869a2cd-20|.|1
│ │ │ │ +00045a70: 3533 3833 6132 6332 2b35 3037 6162 6332  5383a2c2+507abc2
│ │ │ │ +00045a80: 2d31 3338 3034 6232 6332 2d38 3431 3661  -13804b2c2-8416a
│ │ │ │ +00045a90: 6333 2b39 3263 342d 3131 3035 3761 3364  c3+92c4-11057a3d
│ │ │ │ +00045aa0: 2d35 3131 3361 3262 642d 3237 3632 6162  -5113a2bd-2762ab
│ │ │ │ +00045ab0: 3264 2b31 3430 3935 6233 642d 7c0a 7c61  2d+14095b3d-|.|a
│ │ │ │ +00045ac0: 3263 642d 3535 3839 6162 6364 2d38 3136  2cd-5589abcd-816
│ │ │ │ +00045ad0: 3361 6332 642d 3138 3935 6263 3264 2b39  3ac2d-1895bc2d+9
│ │ │ │ +00045ae0: 3436 3461 3264 322d 3732 3533 6162 6432  464a2d2-7253abd2
│ │ │ │ +00045af0: 2b31 3236 3432 6163 6432 2d31 3935 3862  +12642acd2-1958b
│ │ │ │ +00045b00: 6364 3220 2020 2020 2020 2020 7c0a 7c61  cd2         |.|a
│ │ │ │ +00045b10: 3263 322b 3533 3433 6162 6332 2b33 3739  2c2+5343abc2+379
│ │ │ │ +00045b20: 3862 3263 322d 3135 3936 3861 3364 2b34  8b2c2-15968a3d+4
│ │ │ │ +00045b30: 3733 6132 6264 2b31 3332 3933 6162 3264  73a2bd+13293ab2d
│ │ │ │ +00045b40: 2d33 3736 3162 3364 2d37 3731 3761 3263  -3761b3d-7717a2c
│ │ │ │ +00045b50: 642d 3733 3839 6162 6364 2b34 7c0a 7c32  d-7389abcd+4|.|2
│ │ │ │ +00045b60: 3961 3364 2b31 3437 3834 6132 6264 2d31  9a3d+14784a2bd-1
│ │ │ │ +00045b70: 3036 3038 6132 6364 2d33 3831 3661 6263  0608a2cd-3816abc
│ │ │ │ +00045b80: 642d 3133 3633 3861 6332 642d 3631 3539  d-13638ac2d-6159
│ │ │ │ +00045b90: 6132 6432 2d31 3436 3438 6162 6432 2b33  a2d2-14648abd2+3
│ │ │ │ +00045ba0: 3037 3261 6364 322d 3134 3236 7c0a 7c20  072acd2-1426|.| 
│ │ │ │ +00045bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045c10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045c40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045c60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045cb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00045cc0: 642b 3834 3434 6163 642b 3530 3731 6263  d+8444acd+5071bc
│ │ │ │ -00045cd0: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ -00045ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045c90: 2020 2020 2020 2020 2020 2020 7c0a 7c64              |.|d
│ │ │ │ +00045ca0: 2b38 3434 3461 6364 2b35 3037 3162 6364  +8444acd+5071bcd
│ │ │ │ +00045cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045ce0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00045cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045d00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045d50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00045d60: 3331 6333 2d31 3335 3038 6132 642d 3130  31c3-13508a2d-10
│ │ │ │ -00045d70: 3132 3561 6264 2b35 3534 3962 3264 2b39  125abd+5549b2d+9
│ │ │ │ -00045d80: 3033 3361 6364 2b32 3939 3862 6364 2d32  033acd+2998bcd-2
│ │ │ │ -00045d90: 3033 3663 3264 2020 2020 2020 2020 2020  036c2d          
│ │ │ │ -00045da0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00045db0: 3563 332d 3131 3935 3061 3264 2d35 3332  5c3-11950a2d-532
│ │ │ │ -00045dc0: 3161 6264 2b32 3632 3762 3264 2d33 3939  1abd+2627b2d-399
│ │ │ │ -00045dd0: 3661 6364 2d37 3135 3262 6364 2d31 3137  6acd-7152bcd-117
│ │ │ │ -00045de0: 3430 6332 642b 3933 3938 6164 322b 3135  40c2d+9398ad2+15
│ │ │ │ -00045df0: 3331 3762 6432 2d36 3932 3263 647c 0a7c  317bd2-6922cd|.|
│ │ │ │ -00045e00: 3530 3161 6364 2b31 3737 3962 6364 2d35  501acd+1779bcd-5
│ │ │ │ -00045e10: 3534 3963 3264 2d39 3533 3461 6432 2d31  549c2d-9534ad2-1
│ │ │ │ -00045e20: 3038 3636 6264 322b 3730 3631 6364 322d  0866bd2+7061cd2-
│ │ │ │ -00045e30: 3236 3237 6433 2031 3037 6133 2b34 3337  2627d3 107a3+437
│ │ │ │ -00045e40: 3661 3262 2b33 3738 3361 6232 2b7c 0a7c  6a2b+3783ab2+|.|
│ │ │ │ -00045e50: 3333 6163 642d 3239 3938 6263 642b 3230  33acd-2998bcd+20
│ │ │ │ -00045e60: 3336 6332 642d 3531 3037 6164 322d 3536  36c2d-5107ad2-56
│ │ │ │ -00045e70: 3739 6264 322b 3633 3235 6364 322b 3131  79bd2+6325cd2+11
│ │ │ │ -00045e80: 3734 3064 3320 2030 2020 2020 2020 2020  740d3  0        
│ │ │ │ -00045e90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045d30: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +00045d40: 3163 332d 3133 3530 3861 3264 2d31 3031  1c3-13508a2d-101
│ │ │ │ +00045d50: 3235 6162 642b 3535 3439 6232 642b 3930  25abd+5549b2d+90
│ │ │ │ +00045d60: 3333 6163 642b 3239 3938 6263 642d 3230  33acd+2998bcd-20
│ │ │ │ +00045d70: 3336 6332 6420 2020 2020 2020 2020 2020  36c2d           
│ │ │ │ +00045d80: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +00045d90: 6333 2d31 3139 3530 6132 642d 3533 3231  c3-11950a2d-5321
│ │ │ │ +00045da0: 6162 642b 3236 3237 6232 642d 3339 3936  abd+2627b2d-3996
│ │ │ │ +00045db0: 6163 642d 3731 3532 6263 642d 3131 3734  acd-7152bcd-1174
│ │ │ │ +00045dc0: 3063 3264 2b39 3339 3861 6432 2b31 3533  0c2d+9398ad2+153
│ │ │ │ +00045dd0: 3137 6264 322d 3639 3232 6364 7c0a 7c35  17bd2-6922cd|.|5
│ │ │ │ +00045de0: 3031 6163 642b 3137 3739 6263 642d 3535  01acd+1779bcd-55
│ │ │ │ +00045df0: 3439 6332 642d 3935 3334 6164 322d 3130  49c2d-9534ad2-10
│ │ │ │ +00045e00: 3836 3662 6432 2b37 3036 3163 6432 2d32  866bd2+7061cd2-2
│ │ │ │ +00045e10: 3632 3764 3320 3130 3761 332b 3433 3736  627d3 107a3+4376
│ │ │ │ +00045e20: 6132 622b 3337 3833 6162 322b 7c0a 7c33  a2b+3783ab2+|.|3
│ │ │ │ +00045e30: 3361 6364 2d32 3939 3862 6364 2b32 3033  3acd-2998bcd+203
│ │ │ │ +00045e40: 3663 3264 2d35 3130 3761 6432 2d35 3637  6c2d-5107ad2-567
│ │ │ │ +00045e50: 3962 6432 2b36 3332 3563 6432 2b31 3137  9bd2+6325cd2+117
│ │ │ │ +00045e60: 3430 6433 2020 3020 2020 2020 2020 2020  40d3  0         
│ │ │ │ +00045e70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00045e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045ed0: 2020 2020 2020 2031 3533 3434 6133 2d32         15344a3-2
│ │ │ │ -00045ee0: 3635 3361 3262 2d31 3032 3539 617c 0a7c  653a2b-10259a|.|
│ │ │ │ -00045ef0: 3137 6264 322b 3639 3232 6364 322b 3530  17bd2+6922cd2+50
│ │ │ │ -00045f00: 3830 6433 2020 2020 2020 2020 2020 2020  80d3            
│ │ │ │ -00045f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045f20: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00045f30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045eb0: 2020 2020 2020 3135 3334 3461 332d 3236        15344a3-26
│ │ │ │ +00045ec0: 3533 6132 622d 3130 3235 3961 7c0a 7c31  53a2b-10259a|.|1
│ │ │ │ +00045ed0: 3762 6432 2b36 3932 3263 6432 2b35 3038  7bd2+6922cd2+508
│ │ │ │ +00045ee0: 3064 3320 2020 2020 2020 2020 2020 2020  0d3             
│ │ │ │ +00045ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045f00: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00045f10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00045f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045f70: 2020 2020 2020 2031 3034 3830 6133 2b36         10480a3+6
│ │ │ │ -00045f80: 3230 3361 3262 2d39 3533 3461 627c 0a7c  203a2b-9534ab|.|
│ │ │ │ +00045f50: 2020 2020 2020 3130 3438 3061 332b 3632        10480a3+62
│ │ │ │ +00045f60: 3033 6132 622d 3935 3334 6162 7c0a 7c20  03a2b-9534ab|.| 
│ │ │ │ +00045f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00045f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00045f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00045fc0: 2020 2020 2020 202d 3531 3037 6162 322d         -5107ab2-
│ │ │ │ -00045fd0: 3536 3739 6233 2b36 3332 3562 327c 0a7c  5679b3+6325b2|.|
│ │ │ │ -00045fe0: 3033 3361 6364 2b32 3939 3862 6364 2d32  033acd+2998bcd-2
│ │ │ │ -00045ff0: 3033 3663 3264 2b35 3130 3761 6432 2b35  036c2d+5107ad2+5
│ │ │ │ -00046000: 3637 3962 6432 2d36 3332 3563 6432 2d31  679bd2-6325cd2-1
│ │ │ │ -00046010: 3137 3430 6433 202d 3832 3331 6162 322d  1740d3 -8231ab2-
│ │ │ │ -00046020: 3133 3137 3762 332d 3538 3634 617c 0a7c  13177b3-5864a|.|
│ │ │ │ -00046030: 3331 3762 6432 2d36 3932 3263 6432 2d35  317bd2-6922cd2-5
│ │ │ │ -00046040: 3038 3064 3320 2020 2020 2020 2020 2020  080d3           
│ │ │ │ -00046050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046060: 2020 2020 2020 202d 3935 3334 6163 3220         -9534ac2 
│ │ │ │ -00046070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00045fa0: 2020 2020 2020 2d35 3130 3761 6232 2d35        -5107ab2-5
│ │ │ │ +00045fb0: 3637 3962 332b 3633 3235 6232 7c0a 7c30  679b3+6325b2|.|0
│ │ │ │ +00045fc0: 3333 6163 642b 3239 3938 6263 642d 3230  33acd+2998bcd-20
│ │ │ │ +00045fd0: 3336 6332 642b 3531 3037 6164 322b 3536  36c2d+5107ad2+56
│ │ │ │ +00045fe0: 3739 6264 322d 3633 3235 6364 322d 3131  79bd2-6325cd2-11
│ │ │ │ +00045ff0: 3734 3064 3320 2d38 3233 3161 6232 2d31  740d3 -8231ab2-1
│ │ │ │ +00046000: 3331 3737 6233 2d35 3836 3461 7c0a 7c33  3177b3-5864a|.|3
│ │ │ │ +00046010: 3137 6264 322d 3639 3232 6364 322d 3530  17bd2-6922cd2-50
│ │ │ │ +00046020: 3830 6433 2020 2020 2020 2020 2020 2020  80d3            
│ │ │ │ +00046030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046040: 2020 2020 2020 2d39 3533 3461 6332 2020        -9534ac2  
│ │ │ │ +00046050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00046060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000460a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000460b0: 2020 2020 2020 202d 3130 3438 3061 332d         -10480a3-
│ │ │ │ -000460c0: 3632 3033 6132 622b 3935 3334 617c 0a7c  6203a2b+9534a|.|
│ │ │ │ +00046090: 2020 2020 2020 2d31 3034 3830 6133 2d36        -10480a3-6
│ │ │ │ +000460a0: 3230 3361 3262 2b39 3533 3461 7c0a 7c20  203a2b+9534a|.| 
│ │ │ │ +000460b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000460c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000460d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000460e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000460f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046100: 2020 2020 2020 2031 3038 3636 6132 622d         10866a2b-
│ │ │ │ -00046110: 3730 3631 6132 632b 3236 3237 617c 0a7c  7061a2c+2627a|.|
│ │ │ │ -00046120: 3861 6432 2d31 3533 3137 6264 322b 3639  8ad2-15317bd2+69
│ │ │ │ -00046130: 3232 6364 322b 3530 3830 6433 202d 3135  22cd2+5080d3 -15
│ │ │ │ -00046140: 3334 3461 332b 3236 3533 6132 622b 3130  344a3+2653a2b+10
│ │ │ │ -00046150: 3235 3961 6232 2b31 3233 3635 6132 632d  259ab2+12365a2c-
│ │ │ │ -00046160: 3732 3136 6162 632b 3632 3330 617c 0a7c  7216abc+6230a|.|
│ │ │ │ +000460e0: 2020 2020 2020 3130 3836 3661 3262 2d37        10866a2b-7
│ │ │ │ +000460f0: 3036 3161 3263 2b32 3632 3761 7c0a 7c38  061a2c+2627a|.|8
│ │ │ │ +00046100: 6164 322d 3135 3331 3762 6432 2b36 3932  ad2-15317bd2+692
│ │ │ │ +00046110: 3263 6432 2b35 3038 3064 3320 2d31 3533  2cd2+5080d3 -153
│ │ │ │ +00046120: 3434 6133 2b32 3635 3361 3262 2b31 3032  44a3+2653a2b+102
│ │ │ │ +00046130: 3539 6162 322b 3132 3336 3561 3263 2d37  59ab2+12365a2c-7
│ │ │ │ +00046140: 3231 3661 6263 2b36 3233 3061 7c0a 7c20  216abc+6230a|.| 
│ │ │ │ +00046150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000461a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000461b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000461b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000461c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000461d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000461e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000461f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046210: 2b39 3033 3361 3263 322d 3134 3936 3661  +9033a2c2-14966a
│ │ │ │ -00046220: 6263 322d 3639 3239 6232 6332 2b31 3139  bc2-6929b2c2+119
│ │ │ │ -00046230: 3933 6133 642b 3133 3539 3361 3262 642b  93a3d+13593a2bd+
│ │ │ │ -00046240: 3831 3033 6162 3264 2d31 3430 3534 6233  8103ab2d-14054b3
│ │ │ │ -00046250: 642d 3432 3430 6132 6364 2d39 317c 0a7c  d-4240a2cd-91|.|
│ │ │ │ -00046260: 3134 3931 3961 3263 322d 3131 3334 3661  14919a2c2-11346a
│ │ │ │ -00046270: 6263 322b 3134 3137 3262 3263 322b 3139  bc2+14172b2c2+19
│ │ │ │ -00046280: 3130 6163 332b 3131 3635 3662 6333 2d38  10ac3+11656bc3-8
│ │ │ │ -00046290: 3532 3763 342d 3839 3739 6133 642d 3831  527c4-8979a3d-81
│ │ │ │ -000462a0: 3034 6132 6264 2b38 3733 3461 627c 0a7c  04a2bd+8734ab|.|
│ │ │ │ -000462b0: 6364 2b32 3736 3061 6263 642b 3431 3139  cd+2760abcd+4119
│ │ │ │ -000462c0: 6163 3264 2b35 3537 3462 6332 642b 3838  ac2d+5574bc2d+88
│ │ │ │ -000462d0: 3331 6132 6432 2d31 3433 3733 6162 6432  31a2d2-14373abd2
│ │ │ │ -000462e0: 2b31 3336 3736 6163 6432 2b31 3136 3731  +13676acd2+11671
│ │ │ │ -000462f0: 6263 6432 2020 2020 2020 2020 207c 0a7c  bcd2         |.|
│ │ │ │ -00046300: 2d38 3532 3761 3263 322d 3739 3938 6162  -8527a2c2-7998ab
│ │ │ │ -00046310: 6332 2d31 3336 3233 6232 6332 2d35 3934  c2-13623b2c2-594
│ │ │ │ -00046320: 3261 3364 2d31 3131 3530 6132 6264 2d31  2a3d-11150a2bd-1
│ │ │ │ -00046330: 3237 3931 6162 3264 2d31 3234 3031 6233  2791ab2d-12401b3
│ │ │ │ -00046340: 642b 3636 3338 6132 6364 2b31 337c 0a7c  d+6638a2cd+13|.|
│ │ │ │ -00046350: 642b 3131 3938 3261 3262 642b 3934 3732  d+11982a2bd+9472
│ │ │ │ -00046360: 6132 6364 2b31 3030 3535 6162 6364 2b35  a2cd+10055abcd+5
│ │ │ │ -00046370: 3839 3061 6332 642d 3933 3037 6132 6432  890ac2d-9307a2d2
│ │ │ │ -00046380: 2b31 3137 3337 6162 6432 2b31 3333 3761  +11737abd2+1337a
│ │ │ │ -00046390: 6364 322d 3131 3739 3361 6433 207c 0a7c  cd2-11793ad3 |.|
│ │ │ │ +000461e0: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ +000461f0: 3930 3333 6132 6332 2d31 3439 3636 6162  9033a2c2-14966ab
│ │ │ │ +00046200: 6332 2d36 3932 3962 3263 322b 3131 3939  c2-6929b2c2+1199
│ │ │ │ +00046210: 3361 3364 2b31 3335 3933 6132 6264 2b38  3a3d+13593a2bd+8
│ │ │ │ +00046220: 3130 3361 6232 642d 3134 3035 3462 3364  103ab2d-14054b3d
│ │ │ │ +00046230: 2d34 3234 3061 3263 642d 3931 7c0a 7c31  -4240a2cd-91|.|1
│ │ │ │ +00046240: 3439 3139 6132 6332 2d31 3133 3436 6162  4919a2c2-11346ab
│ │ │ │ +00046250: 6332 2b31 3431 3732 6232 6332 2b31 3931  c2+14172b2c2+191
│ │ │ │ +00046260: 3061 6333 2b31 3136 3536 6263 332d 3835  0ac3+11656bc3-85
│ │ │ │ +00046270: 3237 6334 2d38 3937 3961 3364 2d38 3130  27c4-8979a3d-810
│ │ │ │ +00046280: 3461 3262 642b 3837 3334 6162 7c0a 7c63  4a2bd+8734ab|.|c
│ │ │ │ +00046290: 642b 3237 3630 6162 6364 2b34 3131 3961  d+2760abcd+4119a
│ │ │ │ +000462a0: 6332 642b 3535 3734 6263 3264 2b38 3833  c2d+5574bc2d+883
│ │ │ │ +000462b0: 3161 3264 322d 3134 3337 3361 6264 322b  1a2d2-14373abd2+
│ │ │ │ +000462c0: 3133 3637 3661 6364 322b 3131 3637 3162  13676acd2+11671b
│ │ │ │ +000462d0: 6364 3220 2020 2020 2020 2020 7c0a 7c2d  cd2         |.|-
│ │ │ │ +000462e0: 3835 3237 6132 6332 2d37 3939 3861 6263  8527a2c2-7998abc
│ │ │ │ +000462f0: 322d 3133 3632 3362 3263 322d 3539 3432  2-13623b2c2-5942
│ │ │ │ +00046300: 6133 642d 3131 3135 3061 3262 642d 3132  a3d-11150a2bd-12
│ │ │ │ +00046310: 3739 3161 6232 642d 3132 3430 3162 3364  791ab2d-12401b3d
│ │ │ │ +00046320: 2b36 3633 3861 3263 642b 3133 7c0a 7c64  +6638a2cd+13|.|d
│ │ │ │ +00046330: 2b31 3139 3832 6132 6264 2b39 3437 3261  +11982a2bd+9472a
│ │ │ │ +00046340: 3263 642b 3130 3035 3561 6263 642b 3538  2cd+10055abcd+58
│ │ │ │ +00046350: 3930 6163 3264 2d39 3330 3761 3264 322b  90ac2d-9307a2d2+
│ │ │ │ +00046360: 3131 3733 3761 6264 322b 3133 3337 6163  11737abd2+1337ac
│ │ │ │ +00046370: 6432 2d31 3137 3933 6164 3320 7c0a 7c20  d2-11793ad3 |.| 
│ │ │ │ +00046380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000463a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000463b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000463c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000463c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000463d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000463e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000463e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000463f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046410: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046480: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046490: 3135 3936 3061 3263 322d 3435 3031 6162  15960a2c2-4501ab
│ │ │ │ -000464a0: 6332 2b31 3130 3562 3263 322b 3630 3536  c2+1105b2c2+6056
│ │ │ │ -000464b0: 6133 642d 3131 3835 3761 3262 642b 3133  a3d-11857a2bd+13
│ │ │ │ -000464c0: 3734 3861 6232 642d 3131 3735 3262 3364  748ab2d-11752b3d
│ │ │ │ -000464d0: 2d31 3233 3435 6132 6364 2d31 347c 0a7c  -12345a2cd-14|.|
│ │ │ │ -000464e0: 3238 6132 6332 2b39 3331 3261 6263 322b  28a2c2+9312abc2+
│ │ │ │ -000464f0: 3136 3130 6232 6332 2d31 3539 3336 6163  1610b2c2-15936ac
│ │ │ │ -00046500: 332b 3533 3835 6263 332b 3533 3735 6334  3+5385bc3+5375c4
│ │ │ │ -00046510: 2b37 3037 3361 3364 2b31 3230 3932 6132  +7073a3d+12092a2
│ │ │ │ -00046520: 6264 2b38 3234 3161 6232 642d 347c 0a7c  bd+8241ab2d-4|.|
│ │ │ │ -00046530: 3638 3061 6263 642b 3437 3037 6163 3264  680abcd+4707ac2d
│ │ │ │ -00046540: 2d37 3036 3962 6332 642d 3338 3733 6132  -7069bc2d-3873a2
│ │ │ │ -00046550: 6432 2b35 3633 3261 6264 322b 3132 3733  d2+5632abd2+1273
│ │ │ │ -00046560: 3461 6364 322d 3739 3630 6263 6432 2020  4acd2-7960bcd2  
│ │ │ │ -00046570: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046580: 2b35 3337 3561 3263 322b 3136 3737 6162  +5375a2c2+1677ab
│ │ │ │ -00046590: 6332 2b31 3534 3562 3263 322b 3130 3330  c2+1545b2c2+1030
│ │ │ │ -000465a0: 3561 3364 2d36 3932 3161 3262 642d 3331  5a3d-6921a2bd-31
│ │ │ │ -000465b0: 3335 6162 3264 2d39 3130 3562 3364 2d31  35ab2d-9105b3d-1
│ │ │ │ -000465c0: 3036 3739 6132 6364 2d31 3430 367c 0a7c  0679a2cd-1406|.|
│ │ │ │ -000465d0: 642d 3430 3235 6132 6264 2b31 3039 3933  d-4025a2bd+10993
│ │ │ │ -000465e0: 6132 6364 2b36 3235 3861 6263 642d 3134  a2cd+6258abcd-14
│ │ │ │ -000465f0: 3539 3761 6332 642b 3133 3831 3261 3264  597ac2d+13812a2d
│ │ │ │ -00046600: 322b 3738 3633 6162 6432 2b31 3037 3537  2+7863abd2+10757
│ │ │ │ -00046610: 6163 6432 2d31 3637 6164 3320 2d7c 0a7c  acd2-167ad3 -|.|
│ │ │ │ +00046460: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +00046470: 3539 3630 6132 6332 2d34 3530 3161 6263  5960a2c2-4501abc
│ │ │ │ +00046480: 322b 3131 3035 6232 6332 2b36 3035 3661  2+1105b2c2+6056a
│ │ │ │ +00046490: 3364 2d31 3138 3537 6132 6264 2b31 3337  3d-11857a2bd+137
│ │ │ │ +000464a0: 3438 6162 3264 2d31 3137 3532 6233 642d  48ab2d-11752b3d-
│ │ │ │ +000464b0: 3132 3334 3561 3263 642d 3134 7c0a 7c32  12345a2cd-14|.|2
│ │ │ │ +000464c0: 3861 3263 322b 3933 3132 6162 6332 2b31  8a2c2+9312abc2+1
│ │ │ │ +000464d0: 3631 3062 3263 322d 3135 3933 3661 6333  610b2c2-15936ac3
│ │ │ │ +000464e0: 2b35 3338 3562 6333 2b35 3337 3563 342b  +5385bc3+5375c4+
│ │ │ │ +000464f0: 3730 3733 6133 642b 3132 3039 3261 3262  7073a3d+12092a2b
│ │ │ │ +00046500: 642b 3832 3431 6162 3264 2d34 7c0a 7c36  d+8241ab2d-4|.|6
│ │ │ │ +00046510: 3830 6162 6364 2b34 3730 3761 6332 642d  80abcd+4707ac2d-
│ │ │ │ +00046520: 3730 3639 6263 3264 2d33 3837 3361 3264  7069bc2d-3873a2d
│ │ │ │ +00046530: 322b 3536 3332 6162 6432 2b31 3237 3334  2+5632abd2+12734
│ │ │ │ +00046540: 6163 6432 2d37 3936 3062 6364 3220 2020  acd2-7960bcd2   
│ │ │ │ +00046550: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ +00046560: 3533 3735 6132 6332 2b31 3637 3761 6263  5375a2c2+1677abc
│ │ │ │ +00046570: 322b 3135 3435 6232 6332 2b31 3033 3035  2+1545b2c2+10305
│ │ │ │ +00046580: 6133 642d 3639 3231 6132 6264 2d33 3133  a3d-6921a2bd-313
│ │ │ │ +00046590: 3561 6232 642d 3931 3035 6233 642d 3130  5ab2d-9105b3d-10
│ │ │ │ +000465a0: 3637 3961 3263 642d 3134 3036 7c0a 7c64  679a2cd-1406|.|d
│ │ │ │ +000465b0: 2d34 3032 3561 3262 642b 3130 3939 3361  -4025a2bd+10993a
│ │ │ │ +000465c0: 3263 642b 3632 3538 6162 6364 2d31 3435  2cd+6258abcd-145
│ │ │ │ +000465d0: 3937 6163 3264 2b31 3338 3132 6132 6432  97ac2d+13812a2d2
│ │ │ │ +000465e0: 2b37 3836 3361 6264 322b 3130 3735 3761  +7863abd2+10757a
│ │ │ │ +000465f0: 6364 322d 3136 3761 6433 202d 7c0a 7c2d  cd2-167ad3 -|.|-
│ │ │ │ +00046600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00046610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00046630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00046640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00046650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00046660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00046640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00046650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000466a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000466b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000466b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000466c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000466d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000466e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000466e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000466f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046780: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000467a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000467a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000467b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000467c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000467d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000467d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000467e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000467f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000467f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046840: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000468a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000468b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000468c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000468c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000468d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000468e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000468e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000468f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046910: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000469a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000469b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000469b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000469c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000469d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000469d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000469e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000469f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046a70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046b10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00046ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00046bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046c10: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +00046be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00046bf0: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ +00046c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046c50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046c60: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +00046c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00046c40: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ +00046c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046cb0: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +00046c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00046c90: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ +00046ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046d00: 3934 3236 6433 207c 2020 2020 2020 2020  9426d3 |        
│ │ │ │ +00046cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +00046ce0: 3432 3664 3320 7c20 2020 2020 2020 2020  426d3 |         
│ │ │ │ +00046cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046d50: 3937 3162 332d 3530 3036 6132 632d 3535  971b3-5006a2c-55
│ │ │ │ -00046d60: 3939 6162 632d 3134 3136 3562 3263 2b38  99abc-14165b2c+8
│ │ │ │ -00046d70: 3838 3061 3264 2b34 3235 3961 6264 2d33  880a2d+4259abd-3
│ │ │ │ -00046d80: 3030 3262 3264 2d31 3338 3932 6163 642d  002b2d-13892acd-
│ │ │ │ -00046d90: 3130 3532 3162 6364 2020 2020 207c 0a7c  10521bcd     |.|
│ │ │ │ +00046d20: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +00046d30: 3731 6233 2d35 3030 3661 3263 2d35 3539  71b3-5006a2c-559
│ │ │ │ +00046d40: 3961 6263 2d31 3431 3635 6232 632b 3838  9abc-14165b2c+88
│ │ │ │ +00046d50: 3830 6132 642b 3432 3539 6162 642d 3330  80a2d+4259abd-30
│ │ │ │ +00046d60: 3032 6232 642d 3133 3839 3261 6364 2d31  02b2d-13892acd-1
│ │ │ │ +00046d70: 3035 3231 6263 6420 2020 2020 7c0a 7c20  0521bcd     |.| 
│ │ │ │ +00046d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046de0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046df0: 2d39 3739 3762 332b 3730 3231 6132 632d  -9797b3+7021a2c-
│ │ │ │ -00046e00: 3633 3737 6162 632d 3538 3734 6232 632b  6377abc-5874b2c+
│ │ │ │ -00046e10: 3736 3030 6163 322b 3131 3732 3662 6332  7600ac2+11726bc2
│ │ │ │ -00046e20: 2d31 3134 3063 332d 3132 3036 6132 642b  -1140c3-1206a2d+
│ │ │ │ -00046e30: 3131 3433 3561 6264 2d39 3037 347c 0a7c  11435abd-9074|.|
│ │ │ │ +00046dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00046dd0: 3937 3937 6233 2b37 3032 3161 3263 2d36  9797b3+7021a2c-6
│ │ │ │ +00046de0: 3337 3761 6263 2d35 3837 3462 3263 2b37  377abc-5874b2c+7
│ │ │ │ +00046df0: 3630 3061 6332 2b31 3137 3236 6263 322d  600ac2+11726bc2-
│ │ │ │ +00046e00: 3131 3430 6333 2d31 3230 3661 3264 2b31  1140c3-1206a2d+1
│ │ │ │ +00046e10: 3134 3335 6162 642d 3930 3734 7c0a 7c20  1435abd-9074|.| 
│ │ │ │ +00046e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046e80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046e90: 3833 6233 2b36 3435 3361 3263 2b31 3139  83b3+6453a2c+119
│ │ │ │ -00046ea0: 3261 6263 2b31 3532 3530 6232 632b 3131  2abc+15250b2c+11
│ │ │ │ -00046eb0: 3034 3561 3264 2b31 3533 3333 6162 642d  045a2d+15333abd-
│ │ │ │ -00046ec0: 3130 3536 3762 3264 2b33 3536 3061 6364  10567b2d+3560acd
│ │ │ │ -00046ed0: 2d32 3239 3262 6364 2b37 3139 347c 0a7c  -2292bcd+7194|.|
│ │ │ │ -00046ee0: 3338 3862 3264 2020 2020 2020 2020 2020  388b2d          
│ │ │ │ +00046e60: 2020 2020 2020 2020 2020 2020 7c0a 7c38              |.|8
│ │ │ │ +00046e70: 3362 332b 3634 3533 6132 632b 3131 3932  3b3+6453a2c+1192
│ │ │ │ +00046e80: 6162 632b 3135 3235 3062 3263 2b31 3130  abc+15250b2c+110
│ │ │ │ +00046e90: 3435 6132 642b 3135 3333 3361 6264 2d31  45a2d+15333abd-1
│ │ │ │ +00046ea0: 3035 3637 6232 642b 3335 3630 6163 642d  0567b2d+3560acd-
│ │ │ │ +00046eb0: 3232 3932 6263 642b 3731 3934 7c0a 7c33  2292bcd+7194|.|3
│ │ │ │ +00046ec0: 3838 6232 6420 2020 2020 2020 2020 2020  88b2d           
│ │ │ │ +00046ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046f30: 3636 3237 6232 632b 3730 3238 6163 322d  6627b2c+7028ac2-
│ │ │ │ -00046f40: 3937 3937 6263 322d 3538 3734 6333 2d38  9797bc2-5874c3-8
│ │ │ │ -00046f50: 3838 3661 6264 2d34 3730 3062 3264 2b31  886abd-4700b2d+1
│ │ │ │ -00046f60: 3561 6364 2d35 3936 3962 6364 2d39 3037  5acd-5969bcd-907
│ │ │ │ -00046f70: 3463 3264 2b35 3438 3362 6432 2b7c 0a7c  4c2d+5483bd2+|.|
│ │ │ │ +00046f00: 2020 2020 2020 2020 2020 2020 7c0a 7c36              |.|6
│ │ │ │ +00046f10: 3632 3762 3263 2b37 3032 3861 6332 2d39  627b2c+7028ac2-9
│ │ │ │ +00046f20: 3739 3762 6332 2d35 3837 3463 332d 3838  797bc2-5874c3-88
│ │ │ │ +00046f30: 3836 6162 642d 3437 3030 6232 642b 3135  86abd-4700b2d+15
│ │ │ │ +00046f40: 6163 642d 3539 3639 6263 642d 3930 3734  acd-5969bcd-9074
│ │ │ │ +00046f50: 6332 642b 3534 3833 6264 322b 7c0a 7c20  c2d+5483bd2+|.| 
│ │ │ │ +00046f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00046f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00046f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00046fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00046fd0: 3533 6132 632d 3131 3932 6162 632d 3131  53a2c-1192abc-11
│ │ │ │ -00046fe0: 3034 3561 3264 2d31 3533 3333 6162 642d  045a2d-15333abd-
│ │ │ │ -00046ff0: 3335 3630 6163 642b 3232 3932 6263 642d  3560acd+2292bcd-
│ │ │ │ -00047000: 3731 3934 6164 322b 3632 3435 6264 322d  7194ad2+6245bd2-
│ │ │ │ -00047010: 3836 3339 6364 322b 3934 3236 647c 0a7c  8639cd2+9426d|.|
│ │ │ │ -00047020: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ +00046fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +00046fb0: 3361 3263 2d31 3139 3261 6263 2d31 3130  3a2c-1192abc-110
│ │ │ │ +00046fc0: 3435 6132 642d 3135 3333 3361 6264 2d33  45a2d-15333abd-3
│ │ │ │ +00046fd0: 3536 3061 6364 2b32 3239 3262 6364 2d37  560acd+2292bcd-7
│ │ │ │ +00046fe0: 3139 3461 6432 2b36 3234 3562 6432 2d38  194ad2+6245bd2-8
│ │ │ │ +00046ff0: 3633 3963 6432 2b39 3432 3664 7c0a 7c64  639cd2+9426d|.|d
│ │ │ │ +00047000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047070: 3732 3662 6332 2b31 3134 3063 332b 3132  726bc2+1140c3+12
│ │ │ │ -00047080: 3036 6132 642d 3131 3433 3561 6264 2d36  06a2d-11435abd-6
│ │ │ │ -00047090: 3034 3061 6364 2b38 3032 3262 6364 2b33  040acd+8022bcd+3
│ │ │ │ -000470a0: 3936 3863 3264 2d33 3136 3461 6432 2b32  968c2d-3164ad2+2
│ │ │ │ -000470b0: 3935 6264 322d 3736 3530 6364 327c 0a7c  95bd2-7650cd2|.|
│ │ │ │ +00047040: 2020 2020 2020 2020 2020 2020 7c0a 7c37              |.|7
│ │ │ │ +00047050: 3236 6263 322b 3131 3430 6333 2b31 3230  26bc2+1140c3+120
│ │ │ │ +00047060: 3661 3264 2d31 3134 3335 6162 642d 3630  6a2d-11435abd-60
│ │ │ │ +00047070: 3430 6163 642b 3830 3232 6263 642b 3339  40acd+8022bcd+39
│ │ │ │ +00047080: 3638 6332 642d 3331 3634 6164 322b 3239  68c2d-3164ad2+29
│ │ │ │ +00047090: 3562 6432 2d37 3635 3063 6432 7c0a 7c20  5bd2-7650cd2|.| 
│ │ │ │ +000470a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000470b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000470c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000470d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000470e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000470e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000470f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047160: 3532 6162 6364 2d31 3833 3162 3263 642b  52abcd-1831b2cd+
│ │ │ │ -00047170: 3630 3432 6163 3264 2d32 3536 3162 6332  6042ac2d-2561bc2
│ │ │ │ -00047180: 642d 3837 3039 6132 6432 2d31 3332 3139  d-8709a2d2-13219
│ │ │ │ -00047190: 6162 6432 2b34 3230 3962 3264 322b 3132  abd2+4209b2d2+12
│ │ │ │ -000471a0: 3232 3561 6364 322d 3236 3035 627c 0a7c  225acd2-2605b|.|
│ │ │ │ -000471b0: 3135 3838 6132 6364 2b32 3030 3061 6263  1588a2cd+2000abc
│ │ │ │ -000471c0: 642d 3230 3830 6232 6364 2b39 3137 3561  d-2080b2cd+9175a
│ │ │ │ -000471d0: 6332 642d 3634 3962 6332 642b 3838 3239  c2d-649bc2d+8829
│ │ │ │ -000471e0: 6333 642b 3231 3634 6132 6432 2b38 3633  c3d+2164a2d2+863
│ │ │ │ -000471f0: 3561 6264 322d 3731 3631 6232 647c 0a7c  5abd2-7161b2d|.|
│ │ │ │ +00047130: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +00047140: 3261 6263 642d 3138 3331 6232 6364 2b36  2abcd-1831b2cd+6
│ │ │ │ +00047150: 3034 3261 6332 642d 3235 3631 6263 3264  042ac2d-2561bc2d
│ │ │ │ +00047160: 2d38 3730 3961 3264 322d 3133 3231 3961  -8709a2d2-13219a
│ │ │ │ +00047170: 6264 322b 3432 3039 6232 6432 2b31 3232  bd2+4209b2d2+122
│ │ │ │ +00047180: 3235 6163 6432 2d32 3630 3562 7c0a 7c31  25acd2-2605b|.|1
│ │ │ │ +00047190: 3538 3861 3263 642b 3230 3030 6162 6364  588a2cd+2000abcd
│ │ │ │ +000471a0: 2d32 3038 3062 3263 642b 3931 3735 6163  -2080b2cd+9175ac
│ │ │ │ +000471b0: 3264 2d36 3439 6263 3264 2b38 3832 3963  2d-649bc2d+8829c
│ │ │ │ +000471c0: 3364 2b32 3136 3461 3264 322b 3836 3335  3d+2164a2d2+8635
│ │ │ │ +000471d0: 6162 6432 2d37 3136 3162 3264 7c0a 7c20  abd2-7161b2d|.| 
│ │ │ │ +000471e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000471f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047250: 3732 3362 3263 642d 3133 3236 3261 6332  723b2cd-13262ac2
│ │ │ │ -00047260: 642b 3534 3331 6263 3264 2b31 3132 3734  d+5431bc2d+11274
│ │ │ │ -00047270: 6132 6432 2d32 3137 6162 6432 2b31 3236  a2d2-217abd2+126
│ │ │ │ -00047280: 3162 3264 322b 3832 3031 6163 6432 2d31  1b2d2+8201acd2-1
│ │ │ │ -00047290: 3430 3830 6263 6432 2020 2020 207c 0a7c  4080bcd2     |.|
│ │ │ │ -000472a0: 3361 6433 202d 3232 3735 6132 6232 2b32  3ad3 -2275a2b2+2
│ │ │ │ -000472b0: 3339 6162 332b 3938 3034 6234 2b38 3431  39ab3+9804b4+841
│ │ │ │ -000472c0: 3661 6232 632d 3932 6232 6332 2b31 3539  6ab2c-92b2c2+159
│ │ │ │ -000472d0: 3638 6162 3264 2b31 3438 3630 6233 642d  68ab2d+14860b3d-
│ │ │ │ -000472e0: 3838 3239 6232 6364 2d31 3132 377c 0a7c  8829b2cd-1127|.|
│ │ │ │ +00047220: 2020 2020 2020 2020 2020 2020 7c0a 7c37              |.|7
│ │ │ │ +00047230: 3233 6232 6364 2d31 3332 3632 6163 3264  23b2cd-13262ac2d
│ │ │ │ +00047240: 2b35 3433 3162 6332 642b 3131 3237 3461  +5431bc2d+11274a
│ │ │ │ +00047250: 3264 322d 3231 3761 6264 322b 3132 3631  2d2-217abd2+1261
│ │ │ │ +00047260: 6232 6432 2b38 3230 3161 6364 322d 3134  b2d2+8201acd2-14
│ │ │ │ +00047270: 3038 3062 6364 3220 2020 2020 7c0a 7c33  080bcd2     |.|3
│ │ │ │ +00047280: 6164 3320 2d32 3237 3561 3262 322b 3233  ad3 -2275a2b2+23
│ │ │ │ +00047290: 3961 6233 2b39 3830 3462 342b 3834 3136  9ab3+9804b4+8416
│ │ │ │ +000472a0: 6162 3263 2d39 3262 3263 322b 3135 3936  ab2c-92b2c2+1596
│ │ │ │ +000472b0: 3861 6232 642b 3134 3836 3062 3364 2d38  8ab2d+14860b3d-8
│ │ │ │ +000472c0: 3832 3962 3263 642d 3131 3237 7c0a 7c20  829b2cd-1127|.| 
│ │ │ │ +000472d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000472e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000472f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047380: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000473a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000473b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000473c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000473d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000473e0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +000473b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000473c0: 2020 2020 2020 2020 7c20 2020 2020 2020          |       
│ │ │ │ +000473d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000473e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000473f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047430: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00047400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00047410: 2020 2020 2020 2020 7c20 2020 2020 2020          |       
│ │ │ │ +00047420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047480: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00047450: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00047460: 2020 2020 2020 2020 7c20 2020 2020 2020          |       
│ │ │ │ +00047470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000474a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000474b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000474c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000474d0: 322d 3530 3830 6433 207c 2020 2020 2020  2-5080d3 |      
│ │ │ │ +000474a0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +000474b0: 2d35 3038 3064 3320 7c20 2020 2020 2020  -5080d3 |       
│ │ │ │ +000474c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000474d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000474e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000474f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047520: 3130 3335 3962 332d 3535 3730 6132 632d  10359b3-5570a2c-
│ │ │ │ -00047530: 3533 3037 6162 632d 3734 3634 6232 632b  5307abc-7464b2c+
│ │ │ │ -00047540: 3331 3837 6132 642b 3835 3730 6162 642d  3187a2d+8570abd-
│ │ │ │ -00047550: 3832 3531 6232 642b 3834 3434 6163 642b  8251b2d+8444acd+
│ │ │ │ -00047560: 3530 3731 6263 6420 2020 2020 207c 0a7c  5071bcd      |.|
│ │ │ │ +000474f0: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +00047500: 3033 3539 6233 2d35 3537 3061 3263 2d35  0359b3-5570a2c-5
│ │ │ │ +00047510: 3330 3761 6263 2d37 3436 3462 3263 2b33  307abc-7464b2c+3
│ │ │ │ +00047520: 3138 3761 3264 2b38 3537 3061 6264 2d38  187a2d+8570abd-8
│ │ │ │ +00047530: 3235 3162 3264 2b38 3434 3461 6364 2b35  251b2d+8444acd+5
│ │ │ │ +00047540: 3037 3162 6364 2020 2020 2020 7c0a 7c20  071bcd      |.| 
│ │ │ │ +00047550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000475a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000475b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000475c0: 6232 2b31 3330 3962 332d 3132 3336 3561  b2+1309b3-12365a
│ │ │ │ -000475d0: 3263 2b37 3231 3661 6263 2d35 3339 3862  2c+7216abc-5398b
│ │ │ │ -000475e0: 3263 2d36 3233 3061 6332 2b35 3332 3662  2c-6230ac2+5326b
│ │ │ │ -000475f0: 6332 2d31 3033 3163 332b 3133 3530 3861  c2-1031c3+13508a
│ │ │ │ -00047600: 3264 2b31 3031 3235 6162 642d 357c 0a7c  2d+10125abd-5|.|
│ │ │ │ +00047590: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +000475a0: 322b 3133 3039 6233 2d31 3233 3635 6132  2+1309b3-12365a2
│ │ │ │ +000475b0: 632b 3732 3136 6162 632d 3533 3938 6232  c+7216abc-5398b2
│ │ │ │ +000475c0: 632d 3632 3330 6163 322b 3533 3236 6263  c-6230ac2+5326bc
│ │ │ │ +000475d0: 322d 3130 3331 6333 2b31 3335 3038 6132  2-1031c3+13508a2
│ │ │ │ +000475e0: 642b 3130 3132 3561 6264 2d35 7c0a 7c20  d+10125abd-5|.| 
│ │ │ │ +000475f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047650: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047660: 322d 3130 3836 3662 332b 3934 3830 6132  2-10866b3+9480a2
│ │ │ │ -00047670: 632d 3732 3536 6162 632b 3730 3631 6232  c-7256abc+7061b2
│ │ │ │ -00047680: 632b 3131 3935 3061 3264 2b35 3332 3161  c+11950a2d+5321a
│ │ │ │ -00047690: 6264 2d32 3632 3762 3264 2b33 3939 3661  bd-2627b2d+3996a
│ │ │ │ -000476a0: 6364 2b37 3135 3262 6364 2d39 337c 0a7c  cd+7152bcd-93|.|
│ │ │ │ -000476b0: 632b 3131 3734 3062 3264 2020 2020 2020  c+11740b2d      
│ │ │ │ +00047630: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +00047640: 2d31 3038 3636 6233 2b39 3438 3061 3263  -10866b3+9480a2c
│ │ │ │ +00047650: 2d37 3235 3661 6263 2b37 3036 3162 3263  -7256abc+7061b2c
│ │ │ │ +00047660: 2b31 3139 3530 6132 642b 3533 3231 6162  +11950a2d+5321ab
│ │ │ │ +00047670: 642d 3236 3237 6232 642b 3339 3936 6163  d-2627b2d+3996ac
│ │ │ │ +00047680: 642b 3731 3532 6263 642d 3933 7c0a 7c63  d+7152bcd-93|.|c
│ │ │ │ +00047690: 2b31 3137 3430 6232 6420 2020 2020 2020  +11740b2d       
│ │ │ │ +000476a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000476b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000476c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000476d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000476e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000476f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047700: 6263 2d31 3339 3930 6232 632d 3130 3235  bc-13990b2c-1025
│ │ │ │ -00047710: 3961 6332 2b31 3330 3962 6332 2d35 3339  9ac2+1309bc2-539
│ │ │ │ -00047720: 3863 332d 3530 3236 6162 642b 3131 3532  8c3-5026abd+1152
│ │ │ │ -00047730: 3162 3264 2b37 3530 3161 6364 2b31 3737  1b2d+7501acd+177
│ │ │ │ -00047740: 3962 6364 2d35 3534 3963 3264 2d7c 0a7c  9bcd-5549c2d-|.|
│ │ │ │ +000476d0: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +000476e0: 632d 3133 3939 3062 3263 2d31 3032 3539  c-13990b2c-10259
│ │ │ │ +000476f0: 6163 322b 3133 3039 6263 322d 3533 3938  ac2+1309bc2-5398
│ │ │ │ +00047700: 6333 2d35 3032 3661 6264 2b31 3135 3231  c3-5026abd+11521
│ │ │ │ +00047710: 6232 642b 3735 3031 6163 642b 3137 3739  b2d+7501acd+1779
│ │ │ │ +00047720: 6263 642d 3535 3439 6332 642d 7c0a 7c20  bcd-5549c2d-|.| 
│ │ │ │ +00047730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047790: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000477a0: 6232 2d39 3438 3061 3263 2b37 3235 3661  b2-9480a2c+7256a
│ │ │ │ -000477b0: 6263 2d31 3139 3530 6132 642d 3533 3231  bc-11950a2d-5321
│ │ │ │ -000477c0: 6162 642d 3339 3936 6163 642d 3731 3532  abd-3996acd-7152
│ │ │ │ -000477d0: 6263 642b 3933 3938 6164 322b 3135 3331  bcd+9398ad2+1531
│ │ │ │ -000477e0: 3762 6432 2d36 3932 3263 6432 2d7c 0a7c  7bd2-6922cd2-|.|
│ │ │ │ -000477f0: 3264 2020 2020 2020 2020 2020 2020 2020  2d              
│ │ │ │ +00047770: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +00047780: 322d 3934 3830 6132 632b 3732 3536 6162  2-9480a2c+7256ab
│ │ │ │ +00047790: 632d 3131 3935 3061 3264 2d35 3332 3161  c-11950a2d-5321a
│ │ │ │ +000477a0: 6264 2d33 3939 3661 6364 2d37 3135 3262  bd-3996acd-7152b
│ │ │ │ +000477b0: 6364 2b39 3339 3861 6432 2b31 3533 3137  cd+9398ad2+15317
│ │ │ │ +000477c0: 6264 322d 3639 3232 6364 322d 7c0a 7c32  bd2-6922cd2-|.|2
│ │ │ │ +000477d0: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ +000477e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000477f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047830: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047840: 6332 2d35 3332 3662 6332 2b31 3033 3163  c2-5326bc2+1031c
│ │ │ │ -00047850: 332d 3133 3530 3861 3264 2d31 3031 3235  3-13508a2d-10125
│ │ │ │ -00047860: 6162 642b 3930 3333 6163 642b 3239 3938  abd+9033acd+2998
│ │ │ │ -00047870: 6263 642d 3230 3336 6332 642b 3531 3037  bcd-2036c2d+5107
│ │ │ │ -00047880: 6164 322b 3536 3739 6264 322d 367c 0a7c  ad2+5679bd2-6|.|
│ │ │ │ +00047810: 2020 2020 2020 2020 2020 2020 7c0a 7c63              |.|c
│ │ │ │ +00047820: 322d 3533 3236 6263 322b 3130 3331 6333  2-5326bc2+1031c3
│ │ │ │ +00047830: 2d31 3335 3038 6132 642d 3130 3132 3561  -13508a2d-10125a
│ │ │ │ +00047840: 6264 2b39 3033 3361 6364 2b32 3939 3862  bd+9033acd+2998b
│ │ │ │ +00047850: 6364 2d32 3033 3663 3264 2b35 3130 3761  cd-2036c2d+5107a
│ │ │ │ +00047860: 6432 2b35 3637 3962 6432 2d36 7c0a 7c20  d2+5679bd2-6|.| 
│ │ │ │ +00047870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000478a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000478b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000478b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000478c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000478d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000478d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000478e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000478f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047920: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047930: 3837 6162 6364 2d31 3436 3835 6232 6364  87abcd-14685b2cd
│ │ │ │ -00047940: 2d31 3035 3831 6163 3264 2b35 3232 3862  -10581ac2d+5228b
│ │ │ │ -00047950: 6332 642d 3135 3836 3661 3264 322d 3137  c2d-15866a2d2-17
│ │ │ │ -00047960: 3933 6162 6432 2d31 3232 3435 6232 6432  93abd2-12245b2d2
│ │ │ │ -00047970: 2b31 3232 3833 6163 6432 2b36 377c 0a7c  +12283acd2+67|.|
│ │ │ │ -00047980: 3264 2b35 3231 3862 3364 2d31 3238 3032  2d+5218b3d-12802
│ │ │ │ -00047990: 6132 6364 2d31 3236 3138 6162 6364 2d35  a2cd-12618abcd-5
│ │ │ │ -000479a0: 3732 3862 3263 642b 3533 3437 6163 3264  728b2cd+5347ac2d
│ │ │ │ -000479b0: 2b31 3333 3934 6263 3264 2b32 3936 6333  +13394bc2d+296c3
│ │ │ │ -000479c0: 642b 3734 3436 6132 6432 2b31 307c 0a7c  d+7446a2d2+10|.|
│ │ │ │ +00047900: 2020 2020 2020 2020 2020 2020 7c0a 7c38              |.|8
│ │ │ │ +00047910: 3761 6263 642d 3134 3638 3562 3263 642d  7abcd-14685b2cd-
│ │ │ │ +00047920: 3130 3538 3161 6332 642b 3532 3238 6263  10581ac2d+5228bc
│ │ │ │ +00047930: 3264 2d31 3538 3636 6132 6432 2d31 3739  2d-15866a2d2-179
│ │ │ │ +00047940: 3361 6264 322d 3132 3234 3562 3264 322b  3abd2-12245b2d2+
│ │ │ │ +00047950: 3132 3238 3361 6364 322b 3637 7c0a 7c32  12283acd2+67|.|2
│ │ │ │ +00047960: 642b 3532 3138 6233 642d 3132 3830 3261  d+5218b3d-12802a
│ │ │ │ +00047970: 3263 642d 3132 3631 3861 6263 642d 3537  2cd-12618abcd-57
│ │ │ │ +00047980: 3238 6232 6364 2b35 3334 3761 6332 642b  28b2cd+5347ac2d+
│ │ │ │ +00047990: 3133 3339 3462 6332 642b 3239 3663 3364  13394bc2d+296c3d
│ │ │ │ +000479a0: 2b37 3434 3661 3264 322b 3130 7c0a 7c20  +7446a2d2+10|.| 
│ │ │ │ +000479b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000479c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000479d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000479e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000479f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047a10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047a20: 3433 3961 6263 642d 3132 3337 3162 3263  439abcd-12371b2c
│ │ │ │ -00047a30: 642b 3133 3761 6332 642d 3134 3331 3362  d+137ac2d-14313b
│ │ │ │ -00047a40: 6332 642d 3830 3834 6132 6432 2d34 3535  c2d-8084a2d2-455
│ │ │ │ -00047a50: 3261 6264 322b 3635 3634 6232 6432 2b35  2abd2+6564b2d2+5
│ │ │ │ -00047a60: 3831 3361 6364 322d 3135 3334 357c 0a7c  813acd2-15345|.|
│ │ │ │ -00047a70: 3535 3038 6132 6232 2d31 3336 3630 6162  5508a2b2-13660ab
│ │ │ │ -00047a80: 332d 3133 3135 3362 342d 3139 3130 6162  3-13153b4-1910ab
│ │ │ │ -00047a90: 3263 2d31 3136 3536 6233 632b 3835 3237  2c-11656b3c+8527
│ │ │ │ -00047aa0: 6232 6332 2b35 3934 3261 6232 642b 3134  b2c2+5942ab2d+14
│ │ │ │ -00047ab0: 3932 3562 3364 2d32 3936 6232 637c 0a7c  925b3d-296b2c|.|
│ │ │ │ +000479f0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +00047a00: 3339 6162 6364 2d31 3233 3731 6232 6364  39abcd-12371b2cd
│ │ │ │ +00047a10: 2b31 3337 6163 3264 2d31 3433 3133 6263  +137ac2d-14313bc
│ │ │ │ +00047a20: 3264 2d38 3038 3461 3264 322d 3435 3532  2d-8084a2d2-4552
│ │ │ │ +00047a30: 6162 6432 2b36 3536 3462 3264 322b 3538  abd2+6564b2d2+58
│ │ │ │ +00047a40: 3133 6163 6432 2d31 3533 3435 7c0a 7c35  13acd2-15345|.|5
│ │ │ │ +00047a50: 3530 3861 3262 322d 3133 3636 3061 6233  508a2b2-13660ab3
│ │ │ │ +00047a60: 2d31 3331 3533 6234 2d31 3931 3061 6232  -13153b4-1910ab2
│ │ │ │ +00047a70: 632d 3131 3635 3662 3363 2b38 3532 3762  c-11656b3c+8527b
│ │ │ │ +00047a80: 3263 322b 3539 3432 6162 3264 2b31 3439  2c2+5942ab2d+149
│ │ │ │ +00047a90: 3235 6233 642d 3239 3662 3263 7c0a 7c20  25b3d-296b2c|.| 
│ │ │ │ +00047aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047ae0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047b00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047b50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047bb0: 3930 3861 6263 642d 3938 3934 6232 6364  908abcd-9894b2cd
│ │ │ │ -00047bc0: 2b38 3434 3061 6332 642b 3130 3639 3362  +8440ac2d+10693b
│ │ │ │ -00047bd0: 6332 642b 3132 3036 3261 3264 322d 3133  c2d+12062a2d2-13
│ │ │ │ -00047be0: 3632 3661 6264 322b 3432 3437 6232 6432  626abd2+4247b2d2
│ │ │ │ -00047bf0: 2d31 3237 3033 6163 6432 2b32 397c 0a7c  -12703acd2+29|.|
│ │ │ │ -00047c00: 3432 3062 3364 2b35 3233 3461 3263 642b  420b3d+5234a2cd+
│ │ │ │ -00047c10: 3730 3036 6162 6364 2d31 3439 3231 6232  7006abcd-14921b2
│ │ │ │ -00047c20: 6364 2b38 3831 3061 6332 642d 3332 3731  cd+8810ac2d-3271
│ │ │ │ -00047c30: 6263 3264 2d31 3036 3633 6333 642d 3132  bc2d-10663c3d-12
│ │ │ │ -00047c40: 3533 3761 3264 322b 3135 3238 317c 0a7c  537a2d2+15281|.|
│ │ │ │ +00047b80: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +00047b90: 3038 6162 6364 2d39 3839 3462 3263 642b  08abcd-9894b2cd+
│ │ │ │ +00047ba0: 3834 3430 6163 3264 2b31 3036 3933 6263  8440ac2d+10693bc
│ │ │ │ +00047bb0: 3264 2b31 3230 3632 6132 6432 2d31 3336  2d+12062a2d2-136
│ │ │ │ +00047bc0: 3236 6162 6432 2b34 3234 3762 3264 322d  26abd2+4247b2d2-
│ │ │ │ +00047bd0: 3132 3730 3361 6364 322b 3239 7c0a 7c34  12703acd2+29|.|4
│ │ │ │ +00047be0: 3230 6233 642b 3532 3334 6132 6364 2b37  20b3d+5234a2cd+7
│ │ │ │ +00047bf0: 3030 3661 6263 642d 3134 3932 3162 3263  006abcd-14921b2c
│ │ │ │ +00047c00: 642b 3838 3130 6163 3264 2d33 3237 3162  d+8810ac2d-3271b
│ │ │ │ +00047c10: 6332 642d 3130 3636 3363 3364 2d31 3235  c2d-10663c3d-125
│ │ │ │ +00047c20: 3337 6132 6432 2b31 3532 3831 7c0a 7c20  37a2d2+15281|.| 
│ │ │ │ +00047c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047c90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00047ca0: 3561 6263 642b 3632 3937 6232 6364 2b31  5abcd+6297b2cd+1
│ │ │ │ -00047cb0: 3533 3739 6163 3264 2d31 3238 3831 6263  5379ac2d-12881bc
│ │ │ │ -00047cc0: 3264 2d36 3334 3161 3264 322b 3832 3932  2d-6341a2d2+8292
│ │ │ │ -00047cd0: 6162 6432 2b31 3138 3632 6232 6432 2d31  abd2+11862b2d2-1
│ │ │ │ -00047ce0: 3035 3136 6163 6432 2d31 3234 397c 0a7c  0516acd2-1249|.|
│ │ │ │ -00047cf0: 3639 3435 6132 6232 2d31 3230 3938 6162  6945a2b2-12098ab
│ │ │ │ -00047d00: 332d 3732 3437 6234 2b31 3539 3336 6162  3-7247b4+15936ab
│ │ │ │ -00047d10: 3263 2d35 3338 3562 3363 2d35 3337 3562  2c-5385b3c-5375b
│ │ │ │ -00047d20: 3263 322d 3130 3330 3561 6232 642b 3832  2c2-10305ab2d+82
│ │ │ │ -00047d30: 3937 6233 642b 3130 3636 3362 327c 0a7c  97b3d+10663b2|.|
│ │ │ │ +00047c70: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +00047c80: 6162 6364 2b36 3239 3762 3263 642b 3135  abcd+6297b2cd+15
│ │ │ │ +00047c90: 3337 3961 6332 642d 3132 3838 3162 6332  379ac2d-12881bc2
│ │ │ │ +00047ca0: 642d 3633 3431 6132 6432 2b38 3239 3261  d-6341a2d2+8292a
│ │ │ │ +00047cb0: 6264 322b 3131 3836 3262 3264 322d 3130  bd2+11862b2d2-10
│ │ │ │ +00047cc0: 3531 3661 6364 322d 3132 3439 7c0a 7c36  516acd2-1249|.|6
│ │ │ │ +00047cd0: 3934 3561 3262 322d 3132 3039 3861 6233  945a2b2-12098ab3
│ │ │ │ +00047ce0: 2d37 3234 3762 342b 3135 3933 3661 6232  -7247b4+15936ab2
│ │ │ │ +00047cf0: 632d 3533 3835 6233 632d 3533 3735 6232  c-5385b3c-5375b2
│ │ │ │ +00047d00: 6332 2d31 3033 3035 6162 3264 2b38 3239  c2-10305ab2d+829
│ │ │ │ +00047d10: 3762 3364 2b31 3036 3633 6232 7c0a 7c2d  7b3d+10663b2|.|-
│ │ │ │ +00047d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00047d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00047d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00047d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00047d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00047d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047e50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047e70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047ec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047ef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047f10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047f60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00047fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00047fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00047fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00047fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00047ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048080: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000480a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000480a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000480b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000480c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000480d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000480d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000480e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000480f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000480f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048120: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048170: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000481a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000481b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000481c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000481c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000481d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000481e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000481e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000481f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048210: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048230: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048260: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000482a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000482b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000482b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000482c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000482d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000482d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000482e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000482f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048300: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048350: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048370: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000483a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000483a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000483b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000483c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000483c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000483d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000483e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000483f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000483f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048410: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000484a0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -000484b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048480: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +00048490: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000484a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000484b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000484c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000484d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000484e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000484f0: 2020 2020 2020 2020 2020 2031 3233 3661             1236a
│ │ │ │ -00048500: 332b 3839 3232 6132 622d 3335 387c 0a7c  3+8922a2b-358|.|
│ │ │ │ -00048510: 6232 642b 3630 3430 6163 642d 3830 3232  b2d+6040acd-8022
│ │ │ │ -00048520: 6263 642d 3339 3638 6332 642b 3331 3634  bcd-3968c2d+3164
│ │ │ │ -00048530: 6164 322d 3239 3562 6432 2b37 3635 3063  ad2-295bd2+7650c
│ │ │ │ -00048540: 6432 2d31 3433 3838 6433 2031 3033 3730  d2-14388d3 10370
│ │ │ │ -00048550: 6162 322d 3730 3932 6233 2d39 377c 0a7c  ab2-7092b3-97|.|
│ │ │ │ +000484d0: 2020 2020 2020 2020 2020 3132 3336 6133            1236a3
│ │ │ │ +000484e0: 2b38 3932 3261 3262 2d33 3538 7c0a 7c62  +8922a2b-358|.|b
│ │ │ │ +000484f0: 3264 2b36 3034 3061 6364 2d38 3032 3262  2d+6040acd-8022b
│ │ │ │ +00048500: 6364 2d33 3936 3863 3264 2b33 3136 3461  cd-3968c2d+3164a
│ │ │ │ +00048510: 6432 2d32 3935 6264 322b 3736 3530 6364  d2-295bd2+7650cd
│ │ │ │ +00048520: 322d 3134 3338 3864 3320 3130 3337 3061  2-14388d3 10370a
│ │ │ │ +00048530: 6232 2d37 3039 3262 332d 3937 7c0a 7c20  b2-7092b3-97|.| 
│ │ │ │ +00048540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048590: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -000485a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000485b0: 6164 322d 3632 3435 6264 322b 3836 3339  ad2-6245bd2+8639
│ │ │ │ -000485c0: 6364 322d 3934 3236 6433 2020 2020 2020  cd2-9426d3      
│ │ │ │ -000485d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000485e0: 2020 2020 2020 2020 2020 2030 2020 2020             0    
│ │ │ │ -000485f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048570: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +00048580: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +00048590: 6432 2d36 3234 3562 6432 2b38 3633 3963  d2-6245bd2+8639c
│ │ │ │ +000485a0: 6432 2d39 3432 3664 3320 2020 2020 2020  d2-9426d3       
│ │ │ │ +000485b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000485c0: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ +000485d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000485e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000485f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048630: 2020 2020 2020 2020 2020 2039 3934 6133             994a3
│ │ │ │ -00048640: 2b31 3934 3661 3262 2b36 3732 337c 0a7c  +1946a2b+6723|.|
│ │ │ │ -00048650: 3135 3235 3063 6432 2d31 3035 3637 6433  15250cd2-10567d3
│ │ │ │ -00048660: 2031 3233 3661 332b 3839 3232 6132 622d   1236a3+8922a2b-
│ │ │ │ -00048670: 3335 3839 6162 322b 3839 3731 6233 2d35  3589ab2+8971b3-5
│ │ │ │ -00048680: 3030 3661 3263 2d35 3539 3961 6263 2d31  006a2c-5599abc-1
│ │ │ │ -00048690: 3431 3635 6232 632b 3838 3830 617c 0a7c  4165b2c+8880a|.|
│ │ │ │ +00048610: 2020 2020 2020 2020 2020 3939 3461 332b            994a3+
│ │ │ │ +00048620: 3139 3436 6132 622b 3637 3233 7c0a 7c31  1946a2b+6723|.|1
│ │ │ │ +00048630: 3532 3530 6364 322d 3130 3536 3764 3320  5250cd2-10567d3 
│ │ │ │ +00048640: 3132 3336 6133 2b38 3932 3261 3262 2d33  1236a3+8922a2b-3
│ │ │ │ +00048650: 3538 3961 6232 2b38 3937 3162 332d 3530  589ab2+8971b3-50
│ │ │ │ +00048660: 3036 6132 632d 3535 3939 6162 632d 3134  06a2c-5599abc-14
│ │ │ │ +00048670: 3136 3562 3263 2b38 3838 3061 7c0a 7c20  165b2c+8880a|.| 
│ │ │ │ +00048680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048690: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │  000486a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000486b0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -000486c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000486b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000486c0: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │  000486d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000486e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000486f0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -00048700: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -00048710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000486e0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000486f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048750: 202d 3939 3461 332d 3139 3436 6132 622d   -994a3-1946a2b-
│ │ │ │ -00048760: 3637 3233 6162 322d 3534 3833 6233 2d36  6723ab2-5483b3-6
│ │ │ │ -00048770: 3435 3361 3263 2d31 3139 3261 6263 2d31  453a2c-1192abc-1
│ │ │ │ -00048780: 3532 3530 6232 632d 3131 3034 357c 0a7c  5250b2c-11045|.|
│ │ │ │ -00048790: 2b31 3433 3838 6433 202d 3130 3337 3061  +14388d3 -10370a
│ │ │ │ -000487a0: 6232 2b37 3039 3262 332b 3937 3032 6162  b2+7092b3+9702ab
│ │ │ │ -000487b0: 632b 3636 3237 6232 632d 3937 3937 6263  c+6627b2c-9797bc
│ │ │ │ -000487c0: 322d 3538 3734 6333 2d38 3838 3661 6264  2-5874c3-8886abd
│ │ │ │ -000487d0: 2d34 3730 3062 3264 2b31 3561 637c 0a7c  -4700b2d+15ac|.|
│ │ │ │ +00048730: 2d39 3934 6133 2d31 3934 3661 3262 2d36  -994a3-1946a2b-6
│ │ │ │ +00048740: 3732 3361 6232 2d35 3438 3362 332d 3634  723ab2-5483b3-64
│ │ │ │ +00048750: 3533 6132 632d 3131 3932 6162 632d 3135  53a2c-1192abc-15
│ │ │ │ +00048760: 3235 3062 3263 2d31 3130 3435 7c0a 7c2b  250b2c-11045|.|+
│ │ │ │ +00048770: 3134 3338 3864 3320 2d31 3033 3730 6162  14388d3 -10370ab
│ │ │ │ +00048780: 322b 3730 3932 6233 2b39 3730 3261 6263  2+7092b3+9702abc
│ │ │ │ +00048790: 2b36 3632 3762 3263 2d39 3739 3762 6332  +6627b2c-9797bc2
│ │ │ │ +000487a0: 2d35 3837 3463 332d 3838 3836 6162 642d  -5874c3-8886abd-
│ │ │ │ +000487b0: 3437 3030 6232 642b 3135 6163 7c0a 7c20  4700b2d+15ac|.| 
│ │ │ │ +000487c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000487d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000487e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000487f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048880: 6364 322d 3932 6332 6432 2b31 3539 3638  cd2-92c2d2+15968
│ │ │ │ -00048890: 6164 332b 3134 3836 3062 6433 2d38 3832  ad3+14860bd3-882
│ │ │ │ -000488a0: 3963 6433 2d31 3132 3734 6434 207c 2020  9cd3-11274d4 |  
│ │ │ │ -000488b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000488c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000488d0: 322b 3939 3761 6364 322b 3330 3135 6263  2+997acd2+3015bc
│ │ │ │ -000488e0: 6432 2b31 3132 3734 6332 6432 2020 2020  d2+11274c2d2    
│ │ │ │ -000488f0: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │ +00048850: 2020 2020 2020 2020 2020 2020 7c0a 7c63              |.|c
│ │ │ │ +00048860: 6432 2d39 3263 3264 322b 3135 3936 3861  d2-92c2d2+15968a
│ │ │ │ +00048870: 6433 2b31 3438 3630 6264 332d 3838 3239  d3+14860bd3-8829
│ │ │ │ +00048880: 6364 332d 3131 3237 3464 3420 7c20 2020  cd3-11274d4 |   
│ │ │ │ +00048890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000488a0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +000488b0: 2b39 3937 6163 6432 2b33 3031 3562 6364  +997acd2+3015bcd
│ │ │ │ +000488c0: 322b 3131 3237 3463 3264 3220 2020 2020  2+11274c2d2     
│ │ │ │ +000488d0: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │ +000488e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000488f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048910: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048920: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │  00048930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048940: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │ +00048940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048970: 2020 2020 2020 2020 2020 2020 7c20 2020              |   
│ │ │ │  00048980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048990: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │ -000489a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000489b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000489c0: 3462 3264 3220 2d37 3634 3861 3362 2d31  4b2d2 -7648a3b-1
│ │ │ │ -000489d0: 3332 3631 6132 6232 2b33 3335 3861 6233  3261a2b2+3358ab3
│ │ │ │ -000489e0: 2d36 3138 6234 2d37 3637 6133 632d 3535  -618b4-767a3c-55
│ │ │ │ -000489f0: 3830 6132 6263 2d31 3535 3630 6162 3263  80a2bc-15560ab2c
│ │ │ │ -00048a00: 2d34 3839 6233 632b 3135 3338 337c 0a7c  -489b3c+15383|.|
│ │ │ │ +00048990: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +000489a0: 6232 6432 202d 3736 3438 6133 622d 3133  b2d2 -7648a3b-13
│ │ │ │ +000489b0: 3236 3161 3262 322b 3333 3538 6162 332d  261a2b2+3358ab3-
│ │ │ │ +000489c0: 3631 3862 342d 3736 3761 3363 2d35 3538  618b4-767a3c-558
│ │ │ │ +000489d0: 3061 3262 632d 3135 3536 3061 6232 632d  0a2bc-15560ab2c-
│ │ │ │ +000489e0: 3438 3962 3363 2b31 3533 3833 7c0a 7c20  489b3c+15383|.| 
│ │ │ │ +000489f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048af0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048b40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048b70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048b90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048be0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048c30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048c70: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00048c80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048c50: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00048c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00048c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048cc0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00048cd0: 3037 6133 2b34 3337 3661 3262 2b7c 0a7c  07a3+4376a2b+|.|
│ │ │ │ -00048ce0: 3534 3962 3264 2d39 3033 3361 6364 2d32  549b2d-9033acd-2
│ │ │ │ -00048cf0: 3939 3862 6364 2b32 3033 3663 3264 2d35  998bcd+2036c2d-5
│ │ │ │ -00048d00: 3130 3761 6432 2d35 3637 3962 6432 2b36  107ad2-5679bd2+6
│ │ │ │ -00048d10: 3332 3563 6432 2b31 3137 3430 6433 2038  325cd2+11740d3 8
│ │ │ │ -00048d20: 3233 3161 6232 2b31 3331 3737 627c 0a7c  231ab2+13177b|.|
│ │ │ │ +00048ca0: 2020 2020 2020 2020 2020 2020 2020 3130                10
│ │ │ │ +00048cb0: 3761 332b 3433 3736 6132 622b 7c0a 7c35  7a3+4376a2b+|.|5
│ │ │ │ +00048cc0: 3439 6232 642d 3930 3333 6163 642d 3239  49b2d-9033acd-29
│ │ │ │ +00048cd0: 3938 6263 642b 3230 3336 6332 642d 3531  98bcd+2036c2d-51
│ │ │ │ +00048ce0: 3037 6164 322d 3536 3739 6264 322b 3633  07ad2-5679bd2+63
│ │ │ │ +00048cf0: 3235 6364 322b 3131 3734 3064 3320 3832  25cd2+11740d3 82
│ │ │ │ +00048d00: 3331 6162 322b 3133 3137 3762 7c0a 7c20  31ab2+13177b|.| 
│ │ │ │ +00048d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048d60: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00048d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048d80: 3938 6164 322d 3135 3331 3762 6432 2b36  98ad2-15317bd2+6
│ │ │ │ -00048d90: 3932 3263 6432 2b35 3038 3064 3320 2020  922cd2+5080d3   
│ │ │ │ -00048da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048db0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00048dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048d40: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00048d50: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +00048d60: 3861 6432 2d31 3533 3137 6264 322b 3639  8ad2-15317bd2+69
│ │ │ │ +00048d70: 3232 6364 322b 3530 3830 6433 2020 2020  22cd2+5080d3    
│ │ │ │ +00048d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048d90: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ +00048da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00048db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048e00: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00048e10: 3034 3830 6133 2b36 3230 3361 327c 0a7c  0480a3+6203a2|.|
│ │ │ │ -00048e20: 3130 3836 3662 6432 2b37 3036 3163 6432  10866bd2+7061cd2
│ │ │ │ -00048e30: 2d32 3632 3764 3320 3130 3761 332b 3433  -2627d3 107a3+43
│ │ │ │ -00048e40: 3736 6132 622b 3337 3833 6162 322b 3130  76a2b+3783ab2+10
│ │ │ │ -00048e50: 3335 3962 332d 3535 3730 6132 632d 3533  359b3-5570a2c-53
│ │ │ │ -00048e60: 3037 6162 632d 3734 3634 6232 637c 0a7c  07abc-7464b2c|.|
│ │ │ │ +00048de0: 2020 2020 2020 2020 2020 2020 2020 3130                10
│ │ │ │ +00048df0: 3438 3061 332b 3632 3033 6132 7c0a 7c31  480a3+6203a2|.|1
│ │ │ │ +00048e00: 3038 3636 6264 322b 3730 3631 6364 322d  0866bd2+7061cd2-
│ │ │ │ +00048e10: 3236 3237 6433 2031 3037 6133 2b34 3337  2627d3 107a3+437
│ │ │ │ +00048e20: 3661 3262 2b33 3738 3361 6232 2b31 3033  6a2b+3783ab2+103
│ │ │ │ +00048e30: 3539 6233 2d35 3537 3061 3263 2d35 3330  59b3-5570a2c-530
│ │ │ │ +00048e40: 3761 6263 2d37 3436 3462 3263 7c0a 7c20  7abc-7464b2c|.| 
│ │ │ │ +00048e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048e60: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │  00048e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048e80: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -00048e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048ec0: 3530 3830 6433 2020 2020 2020 2020 2020  5080d3          
│ │ │ │ -00048ed0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ -00048ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048e90: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +00048ea0: 3038 3064 3320 2020 2020 2020 2020 2020  080d3           
│ │ │ │ +00048eb0: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +00048ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048f00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00048f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048f20: 2020 2020 2020 2020 2d31 3034 3830 6133          -10480a3
│ │ │ │ -00048f30: 2d36 3230 3361 3262 2b39 3533 3461 6232  -6203a2b+9534ab2
│ │ │ │ -00048f40: 2b31 3038 3636 6233 2d39 3438 3061 3263  +10866b3-9480a2c
│ │ │ │ -00048f50: 2b37 3235 3661 6263 2d37 3036 317c 0a7c  +7256abc-7061|.|
│ │ │ │ -00048f60: 3332 3563 6432 2d31 3137 3430 6433 202d  325cd2-11740d3 -
│ │ │ │ -00048f70: 3832 3331 6162 322d 3133 3137 3762 332d  8231ab2-13177b3-
│ │ │ │ -00048f80: 3538 3634 6162 632d 3133 3939 3062 3263  5864abc-13990b2c
│ │ │ │ -00048f90: 2b31 3330 3962 6332 2d35 3339 3863 332d  +1309bc2-5398c3-
│ │ │ │ -00048fa0: 3530 3236 6162 642b 3131 3532 317c 0a7c  5026abd+11521|.|
│ │ │ │ +00048f00: 2020 2020 2020 202d 3130 3438 3061 332d         -10480a3-
│ │ │ │ +00048f10: 3632 3033 6132 622b 3935 3334 6162 322b  6203a2b+9534ab2+
│ │ │ │ +00048f20: 3130 3836 3662 332d 3934 3830 6132 632b  10866b3-9480a2c+
│ │ │ │ +00048f30: 3732 3536 6162 632d 3730 3631 7c0a 7c33  7256abc-7061|.|3
│ │ │ │ +00048f40: 3235 6364 322d 3131 3734 3064 3320 2d38  25cd2-11740d3 -8
│ │ │ │ +00048f50: 3233 3161 6232 2d31 3331 3737 6233 2d35  231ab2-13177b3-5
│ │ │ │ +00048f60: 3836 3461 6263 2d31 3339 3930 6232 632b  864abc-13990b2c+
│ │ │ │ +00048f70: 3133 3039 6263 322d 3533 3938 6333 2d35  1309bc2-5398c3-5
│ │ │ │ +00048f80: 3032 3661 6264 2b31 3135 3231 7c0a 7c20  026abd+11521|.| 
│ │ │ │ +00048f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00048fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00048fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00048fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00048ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00048ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049040: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00049050: 3330 6263 6432 2b38 3532 3763 3264 322b  30bcd2+8527c2d2+
│ │ │ │ -00049060: 3539 3432 6164 332b 3134 3932 3562 6433  5942ad3+14925bd3
│ │ │ │ -00049070: 2d32 3936 6364 332b 3830 3834 6434 207c  -296cd3+8084d4 |
│ │ │ │ -00049080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000490a0: 3034 3661 6264 322b 3633 3432 6232 6432  046abd2+6342b2d2
│ │ │ │ -000490b0: 2d37 3438 3061 6364 322d 3738 3433 6263  -7480acd2-7843bc
│ │ │ │ -000490c0: 6432 2d38 3038 3463 3264 3220 2020 207c  d2-8084c2d2    |
│ │ │ │ +00049020: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +00049030: 3062 6364 322b 3835 3237 6332 6432 2b35  0bcd2+8527c2d2+5
│ │ │ │ +00049040: 3934 3261 6433 2b31 3439 3235 6264 332d  942ad3+14925bd3-
│ │ │ │ +00049050: 3239 3663 6433 2b38 3038 3464 3420 7c20  296cd3+8084d4 | 
│ │ │ │ +00049060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049070: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +00049080: 3436 6162 6432 2b36 3334 3262 3264 322d  46abd2+6342b2d2-
│ │ │ │ +00049090: 3734 3830 6163 6432 2d37 3834 3362 6364  7480acd2-7843bcd
│ │ │ │ +000490a0: 322d 3830 3834 6332 6432 2020 2020 7c20  2-8084c2d2    | 
│ │ │ │ +000490b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000490c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000490d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000490e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000490f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000490e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000490f0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  00049100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00049120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049130: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00049140: 6263 6432 2020 2020 2020 2020 2020 2020  bcd2            
│ │ │ │ +00049110: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +00049120: 6364 3220 2020 2020 2020 2020 2020 2020  cd2             
│ │ │ │ +00049130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049140: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  00049150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00049170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049180: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00049190: 642b 3830 3834 6232 6432 202d 3130 3439  d+8084b2d2 -1049
│ │ │ │ -000491a0: 3261 3362 2b31 3339 3936 6132 6232 2d39  2a3b+13996a2b2-9
│ │ │ │ -000491b0: 3333 3861 6233 2d34 3637 3662 342d 3938  338ab3-4676b4-98
│ │ │ │ -000491c0: 3236 6133 632b 3437 3331 6132 6263 2d31  26a3c+4731a2bc-1
│ │ │ │ -000491d0: 3237 3335 6162 3263 2b37 3432 317c 0a7c  2735ab2c+7421|.|
│ │ │ │ +00049160: 2020 2020 2020 2020 2020 2020 7c0a 7c64              |.|d
│ │ │ │ +00049170: 2b38 3038 3462 3264 3220 2d31 3034 3932  +8084b2d2 -10492
│ │ │ │ +00049180: 6133 622b 3133 3939 3661 3262 322d 3933  a3b+13996a2b2-93
│ │ │ │ +00049190: 3338 6162 332d 3436 3736 6234 2d39 3832  38ab3-4676b4-982
│ │ │ │ +000491a0: 3661 3363 2b34 3733 3161 3262 632d 3132  6a3c+4731a2bc-12
│ │ │ │ +000491b0: 3733 3561 6232 632b 3734 3231 7c0a 7c20  735ab2c+7421|.| 
│ │ │ │ +000491c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000491d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000491e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000491f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049200: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049220: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000492a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000492b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000492c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000492d0: 3537 6263 6432 2d35 3337 3563 3264 322d  57bcd2-5375c2d2-
│ │ │ │ -000492e0: 3130 3330 3561 6433 2b38 3239 3762 6433  10305ad3+8297bd3
│ │ │ │ -000492f0: 2b31 3036 3633 6364 332b 3633 3431 6434  +10663cd3+6341d4
│ │ │ │ -00049300: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00049310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00049320: 6162 6432 2d39 3334 3662 3264 322b 3433  abd2-9346b2d2+43
│ │ │ │ -00049330: 3039 6163 6432 2b38 3236 3962 6364 322d  09acd2+8269bcd2-
│ │ │ │ -00049340: 3633 3431 6332 6432 2020 2020 2020 2020  6341c2d2        
│ │ │ │ -00049350: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00049360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000492a0: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +000492b0: 3762 6364 322d 3533 3735 6332 6432 2d31  7bcd2-5375c2d2-1
│ │ │ │ +000492c0: 3033 3035 6164 332b 3832 3937 6264 332b  0305ad3+8297bd3+
│ │ │ │ +000492d0: 3130 3636 3363 6433 2b36 3334 3164 3420  10663cd3+6341d4 
│ │ │ │ +000492e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000492f0: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +00049300: 6264 322d 3933 3436 6232 6432 2b34 3330  bd2-9346b2d2+430
│ │ │ │ +00049310: 3961 6364 322b 3832 3639 6263 6432 2d36  9acd2+8269bcd2-6
│ │ │ │ +00049320: 3334 3163 3264 3220 2020 2020 2020 2020  341c2d2         
│ │ │ │ +00049330: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00049340: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00049350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000493a0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -000493b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000493c0: 3962 6364 3220 2020 2020 2020 2020 2020  9bcd2           
│ │ │ │ -000493d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000493e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000493f0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00049400: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00049410: 6364 2b36 3334 3162 3264 3220 2d31 3035  cd+6341b2d2 -105
│ │ │ │ -00049420: 3534 6133 622b 3638 3432 6132 6232 2d31  54a3b+6842a2b2-1
│ │ │ │ -00049430: 3432 3236 6162 332d 3633 3433 6234 2d31  4226ab3-6343b4-1
│ │ │ │ -00049440: 3235 3435 6133 632d 3130 3031 3561 3262  2545a3c-10015a2b
│ │ │ │ -00049450: 632d 3134 3238 3661 6232 632d 357c 0a7c  c-14286ab2c-5|.|
│ │ │ │ +00049380: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00049390: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +000493a0: 6263 6432 2020 2020 2020 2020 2020 2020  bcd2            
│ │ │ │ +000493b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000493c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000493d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000493e0: 2020 2020 2020 2020 2020 2020 7c0a 7c63              |.|c
│ │ │ │ +000493f0: 642b 3633 3431 6232 6432 202d 3130 3535  d+6341b2d2 -1055
│ │ │ │ +00049400: 3461 3362 2b36 3834 3261 3262 322d 3134  4a3b+6842a2b2-14
│ │ │ │ +00049410: 3232 3661 6233 2d36 3334 3362 342d 3132  226ab3-6343b4-12
│ │ │ │ +00049420: 3534 3561 3363 2d31 3030 3135 6132 6263  545a3c-10015a2bc
│ │ │ │ +00049430: 2d31 3432 3836 6162 3263 2d35 7c0a 7c2d  -14286ab2c-5|.|-
│ │ │ │ +00049440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00049450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00049470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00049490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000494a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00049480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00049490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000494a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000494b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000494c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000494d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000494d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000494e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000494f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000494f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049520: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000495a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000495b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000495c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000495c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000495d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000495e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000495e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000495f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049610: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049630: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049660: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049680: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000496a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000496b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000496b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000496c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000496d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000496d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000496e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000496f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049770: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000497a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000497a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000497b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000497c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000497c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000497d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000497e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000497f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000497f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049810: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049840: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049890: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000498a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000498b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000498b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000498c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000498d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000498e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000498e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000498f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049900: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049930: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049950: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049980: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000499a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000499a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000499b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000499c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000499d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000499d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000499e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000499f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000499f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049a20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049a40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049a70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049a90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049ac0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049ae0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049b10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049b30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049b60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049b80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00049be0: 3961 6232 2b38 3937 3162 332d 3530 3036  9ab2+8971b3-5006
│ │ │ │ -00049bf0: 6132 632d 3535 3939 6162 632d 3134 3136  a2c-5599abc-1416
│ │ │ │ -00049c00: 3562 3263 2b38 3838 3061 3264 2b34 3235  5b2c+8880a2d+425
│ │ │ │ -00049c10: 3961 6264 2d33 3030 3262 3264 2d31 3338  9abd-3002b2d-138
│ │ │ │ -00049c20: 3932 6163 642d 3130 3532 3162 637c 0a7c  92acd-10521bc|.|
│ │ │ │ -00049c30: 3032 6162 632d 3636 3237 6232 632b 3838  02abc-6627b2c+88
│ │ │ │ -00049c40: 3836 6162 642b 3437 3030 6232 642d 3135  86abd+4700b2d-15
│ │ │ │ -00049c50: 6163 642b 3539 3639 6263 6420 2020 2020  acd+5969bcd     
│ │ │ │ +00049bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +00049bc0: 6162 322b 3839 3731 6233 2d35 3030 3661  ab2+8971b3-5006a
│ │ │ │ +00049bd0: 3263 2d35 3539 3961 6263 2d31 3431 3635  2c-5599abc-14165
│ │ │ │ +00049be0: 6232 632b 3838 3830 6132 642b 3432 3539  b2c+8880a2d+4259
│ │ │ │ +00049bf0: 6162 642d 3330 3032 6232 642d 3133 3839  abd-3002b2d-1389
│ │ │ │ +00049c00: 3261 6364 2d31 3035 3231 6263 7c0a 7c30  2acd-10521bc|.|0
│ │ │ │ +00049c10: 3261 6263 2d36 3632 3762 3263 2b38 3838  2abc-6627b2c+888
│ │ │ │ +00049c20: 3661 6264 2b34 3730 3062 3264 2d31 3561  6abd+4700b2d-15a
│ │ │ │ +00049c30: 6364 2b35 3936 3962 6364 2020 2020 2020  cd+5969bcd      
│ │ │ │ +00049c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049cc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049d10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00049d20: 6162 322b 3534 3833 6233 2b36 3435 3361  ab2+5483b3+6453a
│ │ │ │ -00049d30: 3263 2b31 3139 3261 6263 2b31 3532 3530  2c+1192abc+15250
│ │ │ │ -00049d40: 6232 632b 3331 3634 6163 322d 3239 3562  b2c+3164ac2-295b
│ │ │ │ -00049d50: 6332 2b37 3635 3063 332b 3131 3034 3561  c2+7650c3+11045a
│ │ │ │ -00049d60: 3264 2b31 3533 3333 6162 642d 317c 0a7c  2d+15333abd-1|.|
│ │ │ │ -00049d70: 3264 2b34 3235 3961 6264 2d33 3030 3262  2d+4259abd-3002b
│ │ │ │ -00049d80: 3264 2d31 3338 3932 6163 642d 3130 3532  2d-13892acd-1052
│ │ │ │ -00049d90: 3162 6364 2020 2020 2020 2020 2020 2020  1bcd            
│ │ │ │ +00049cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +00049d00: 6232 2b35 3438 3362 332b 3634 3533 6132  b2+5483b3+6453a2
│ │ │ │ +00049d10: 632b 3131 3932 6162 632b 3135 3235 3062  c+1192abc+15250b
│ │ │ │ +00049d20: 3263 2b33 3136 3461 6332 2d32 3935 6263  2c+3164ac2-295bc
│ │ │ │ +00049d30: 322b 3736 3530 6333 2b31 3130 3435 6132  2+7650c3+11045a2
│ │ │ │ +00049d40: 642b 3135 3333 3361 6264 2d31 7c0a 7c32  d+15333abd-1|.|2
│ │ │ │ +00049d50: 642b 3432 3539 6162 642d 3330 3032 6232  d+4259abd-3002b2
│ │ │ │ +00049d60: 642d 3133 3839 3261 6364 2d31 3035 3231  d-13892acd-10521
│ │ │ │ +00049d70: 6263 6420 2020 2020 2020 2020 2020 2020  bcd             
│ │ │ │ +00049d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049d90: 2020 2020 3730 3238 6164 3220 7c0a 7c20      7028ad2 |.| 
│ │ │ │  00049da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049db0: 2020 2020 2037 3032 3861 6432 207c 0a7c       7028ad2 |.|
│ │ │ │ +00049db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049de0: 2020 2020 2d31 3033 3730 6162 7c0a 7c20      -10370ab|.| 
│ │ │ │  00049df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049e00: 2020 2020 202d 3130 3337 3061 627c 0a7c       -10370ab|.|
│ │ │ │ +00049e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049e50: 2020 2020 202d 3133 3730 3761 337c 0a7c       -13707a3|.|
│ │ │ │ -00049e60: 6132 642d 3135 3333 3361 6264 2b31 3035  a2d-15333abd+105
│ │ │ │ -00049e70: 3637 6232 642d 3335 3630 6163 642b 3232  67b2d-3560acd+22
│ │ │ │ -00049e80: 3932 6263 642d 3731 3934 6164 322b 3632  92bcd-7194ad2+62
│ │ │ │ -00049e90: 3435 6264 322d 3836 3339 6364 322b 3934  45bd2-8639cd2+94
│ │ │ │ -00049ea0: 3236 6433 202d 3937 3937 6132 627c 0a7c  26d3 -9797a2b|.|
│ │ │ │ -00049eb0: 642d 3539 3639 6263 642d 3930 3734 6332  d-5969bcd-9074c2
│ │ │ │ -00049ec0: 642b 3534 3833 6264 322b 3135 3235 3063  d+5483bd2+15250c
│ │ │ │ -00049ed0: 6432 2d31 3035 3637 6433 2031 3233 3661  d2-10567d3 1236a
│ │ │ │ -00049ee0: 332b 3839 3232 6132 622d 3335 3839 6162  3+8922a2b-3589ab
│ │ │ │ -00049ef0: 322b 3839 3731 6233 2d35 3030 367c 0a7c  2+8971b3-5006|.|
│ │ │ │ +00049e30: 2020 2020 2d31 3337 3037 6133 7c0a 7c61      -13707a3|.|a
│ │ │ │ +00049e40: 3264 2d31 3533 3333 6162 642b 3130 3536  2d-15333abd+1056
│ │ │ │ +00049e50: 3762 3264 2d33 3536 3061 6364 2b32 3239  7b2d-3560acd+229
│ │ │ │ +00049e60: 3262 6364 2d37 3139 3461 6432 2b36 3234  2bcd-7194ad2+624
│ │ │ │ +00049e70: 3562 6432 2d38 3633 3963 6432 2b39 3432  5bd2-8639cd2+942
│ │ │ │ +00049e80: 3664 3320 2d39 3739 3761 3262 7c0a 7c64  6d3 -9797a2b|.|d
│ │ │ │ +00049e90: 2d35 3936 3962 6364 2d39 3037 3463 3264  -5969bcd-9074c2d
│ │ │ │ +00049ea0: 2b35 3438 3362 6432 2b31 3532 3530 6364  +5483bd2+15250cd
│ │ │ │ +00049eb0: 322d 3130 3536 3764 3320 3132 3336 6133  2-10567d3 1236a3
│ │ │ │ +00049ec0: 2b38 3932 3261 3262 2d33 3538 3961 6232  +8922a2b-3589ab2
│ │ │ │ +00049ed0: 2b38 3937 3162 332d 3530 3036 7c0a 7c20  +8971b3-5006|.| 
│ │ │ │ +00049ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049f20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049f40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049f70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049f90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00049fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00049fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00049fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00049fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00049ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a060: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a0d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a0e0: 6132 6332 2d35 3037 6162 6332 2b31 3338  a2c2-507abc2+138
│ │ │ │ -0004a0f0: 3034 6232 6332 2b38 3431 3661 6333 2d39  04b2c2+8416ac3-9
│ │ │ │ -0004a100: 3263 342d 3335 3838 6133 642b 3134 3534  2c4-3588a3d+1454
│ │ │ │ -0004a110: 3161 3262 642d 3836 3835 6162 3264 2d31  1a2bd-8685ab2d-1
│ │ │ │ -0004a120: 3430 3935 6233 642b 3132 3735 307c 0a7c  4095b3d+12750|.|
│ │ │ │ +0004a0b0: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +0004a0c0: 3263 322d 3530 3761 6263 322b 3133 3830  2c2-507abc2+1380
│ │ │ │ +0004a0d0: 3462 3263 322b 3834 3136 6163 332d 3932  4b2c2+8416ac3-92
│ │ │ │ +0004a0e0: 6334 2d33 3538 3861 3364 2b31 3435 3431  c4-3588a3d+14541
│ │ │ │ +0004a0f0: 6132 6264 2d38 3638 3561 6232 642d 3134  a2bd-8685ab2d-14
│ │ │ │ +0004a100: 3039 3562 3364 2b31 3237 3530 7c0a 7c20  095b3d+12750|.| 
│ │ │ │ +0004a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a1c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a290: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a2b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a350: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a3a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a3b0: 3337 3833 6162 322b 3130 3335 3962 332d  3783ab2+10359b3-
│ │ │ │ -0004a3c0: 3535 3730 6132 632d 3533 3037 6162 632d  5570a2c-5307abc-
│ │ │ │ -0004a3d0: 3734 3634 6232 632b 3331 3837 6132 642b  7464b2c+3187a2d+
│ │ │ │ -0004a3e0: 3835 3730 6162 642d 3832 3531 6232 642b  8570abd-8251b2d+
│ │ │ │ -0004a3f0: 3834 3434 6163 642b 3530 3731 627c 0a7c  8444acd+5071b|.|
│ │ │ │ -0004a400: 332b 3538 3634 6162 632b 3133 3939 3062  3+5864abc+13990b
│ │ │ │ -0004a410: 3263 2b35 3032 3661 6264 2d31 3135 3231  2c+5026abd-11521
│ │ │ │ -0004a420: 6232 642d 3735 3031 6163 642d 3137 3739  b2d-7501acd-1779
│ │ │ │ -0004a430: 6263 6420 2020 2020 2020 2020 2020 2020  bcd             
│ │ │ │ -0004a440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a380: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +0004a390: 3738 3361 6232 2b31 3033 3539 6233 2d35  783ab2+10359b3-5
│ │ │ │ +0004a3a0: 3537 3061 3263 2d35 3330 3761 6263 2d37  570a2c-5307abc-7
│ │ │ │ +0004a3b0: 3436 3462 3263 2b33 3138 3761 3264 2b38  464b2c+3187a2d+8
│ │ │ │ +0004a3c0: 3537 3061 6264 2d38 3235 3162 3264 2b38  570abd-8251b2d+8
│ │ │ │ +0004a3d0: 3434 3461 6364 2b35 3037 3162 7c0a 7c33  444acd+5071b|.|3
│ │ │ │ +0004a3e0: 2b35 3836 3461 6263 2b31 3339 3930 6232  +5864abc+13990b2
│ │ │ │ +0004a3f0: 632b 3530 3236 6162 642d 3131 3532 3162  c+5026abd-11521b
│ │ │ │ +0004a400: 3264 2d37 3530 3161 6364 2d31 3737 3962  2d-7501acd-1779b
│ │ │ │ +0004a410: 6364 2020 2020 2020 2020 2020 2020 2020  cd              
│ │ │ │ +0004a420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0004a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a4e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a4f0: 622d 3935 3334 6162 322d 3130 3836 3662  b-9534ab2-10866b
│ │ │ │ -0004a500: 332b 3934 3830 6132 632d 3732 3536 6162  3+9480a2c-7256ab
│ │ │ │ -0004a510: 632b 3730 3631 6232 632d 3531 3037 6163  c+7061b2c-5107ac
│ │ │ │ -0004a520: 322d 3536 3739 6263 322b 3633 3235 6333  2-5679bc2+6325c3
│ │ │ │ -0004a530: 2b31 3139 3530 6132 642b 3533 327c 0a7c  +11950a2d+532|.|
│ │ │ │ -0004a540: 2b33 3138 3761 3264 2b38 3537 3061 6264  +3187a2d+8570abd
│ │ │ │ -0004a550: 2d38 3235 3162 3264 2b38 3434 3461 6364  -8251b2d+8444acd
│ │ │ │ -0004a560: 2b35 3037 3162 6364 2020 2020 2020 2020  +5071bcd        
│ │ │ │ +0004a4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +0004a4d0: 2d39 3533 3461 6232 2d31 3038 3636 6233  -9534ab2-10866b3
│ │ │ │ +0004a4e0: 2b39 3438 3061 3263 2d37 3235 3661 6263  +9480a2c-7256abc
│ │ │ │ +0004a4f0: 2b37 3036 3162 3263 2d35 3130 3761 6332  +7061b2c-5107ac2
│ │ │ │ +0004a500: 2d35 3637 3962 6332 2b36 3332 3563 332b  -5679bc2+6325c3+
│ │ │ │ +0004a510: 3131 3935 3061 3264 2b35 3332 7c0a 7c2b  11950a2d+532|.|+
│ │ │ │ +0004a520: 3331 3837 6132 642b 3835 3730 6162 642d  3187a2d+8570abd-
│ │ │ │ +0004a530: 3832 3531 6232 642b 3834 3434 6163 642b  8251b2d+8444acd+
│ │ │ │ +0004a540: 3530 3731 6263 6420 2020 2020 2020 2020  5071bcd         
│ │ │ │ +0004a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a560: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a5d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a620: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a630: 6232 632d 3131 3935 3061 3264 2d35 3332  b2c-11950a2d-532
│ │ │ │ -0004a640: 3161 6264 2b32 3632 3762 3264 2d33 3939  1abd+2627b2d-399
│ │ │ │ -0004a650: 3661 6364 2d37 3135 3262 6364 2b39 3339  6acd-7152bcd+939
│ │ │ │ -0004a660: 3861 6432 2b31 3533 3137 6264 322d 3639  8ad2+15317bd2-69
│ │ │ │ -0004a670: 3232 6364 322d 3530 3830 6433 207c 0a7c  22cd2-5080d3 |.|
│ │ │ │ -0004a680: 6232 642b 3735 3031 6163 642b 3137 3739  b2d+7501acd+1779
│ │ │ │ -0004a690: 6263 642d 3535 3439 6332 642d 3130 3836  bcd-5549c2d-1086
│ │ │ │ -0004a6a0: 3662 6432 2b37 3036 3163 6432 2d32 3632  6bd2+7061cd2-262
│ │ │ │ -0004a6b0: 3764 3320 3130 3761 332b 3433 3736 6132  7d3 107a3+4376a2
│ │ │ │ -0004a6c0: 622b 3337 3833 6162 322b 3130 337c 0a7c  b+3783ab2+103|.|
│ │ │ │ +0004a600: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +0004a610: 3263 2d31 3139 3530 6132 642d 3533 3231  2c-11950a2d-5321
│ │ │ │ +0004a620: 6162 642b 3236 3237 6232 642d 3339 3936  abd+2627b2d-3996
│ │ │ │ +0004a630: 6163 642d 3731 3532 6263 642b 3933 3938  acd-7152bcd+9398
│ │ │ │ +0004a640: 6164 322b 3135 3331 3762 6432 2d36 3932  ad2+15317bd2-692
│ │ │ │ +0004a650: 3263 6432 2d35 3038 3064 3320 7c0a 7c62  2cd2-5080d3 |.|b
│ │ │ │ +0004a660: 3264 2b37 3530 3161 6364 2b31 3737 3962  2d+7501acd+1779b
│ │ │ │ +0004a670: 6364 2d35 3534 3963 3264 2d31 3038 3636  cd-5549c2d-10866
│ │ │ │ +0004a680: 6264 322b 3730 3631 6364 322d 3236 3237  bd2+7061cd2-2627
│ │ │ │ +0004a690: 6433 2031 3037 6133 2b34 3337 3661 3262  d3 107a3+4376a2b
│ │ │ │ +0004a6a0: 2b33 3738 3361 6232 2b31 3033 7c0a 7c20  +3783ab2+103|.| 
│ │ │ │ +0004a6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a710: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a740: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a790: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a7b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a7e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a800: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a830: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a8a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004a8b0: 6233 632b 3134 3931 3961 3263 322b 3131  b3c+14919a2c2+11
│ │ │ │ -0004a8c0: 3334 3661 6263 322d 3134 3137 3262 3263  346abc2-14172b2c
│ │ │ │ -0004a8d0: 322d 3139 3130 6163 332d 3131 3635 3662  2-1910ac3-11656b
│ │ │ │ -0004a8e0: 6333 2b38 3532 3763 342b 3637 3730 6133  c3+8527c4+6770a3
│ │ │ │ -0004a8f0: 642d 3931 3037 6132 6264 2d38 387c 0a7c  d-9107a2bd-88|.|
│ │ │ │ +0004a880: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +0004a890: 3363 2b31 3439 3139 6132 6332 2b31 3133  3c+14919a2c2+113
│ │ │ │ +0004a8a0: 3436 6162 6332 2d31 3431 3732 6232 6332  46abc2-14172b2c2
│ │ │ │ +0004a8b0: 2d31 3931 3061 6333 2d31 3136 3536 6263  -1910ac3-11656bc
│ │ │ │ +0004a8c0: 332b 3835 3237 6334 2b36 3737 3061 3364  3+8527c4+6770a3d
│ │ │ │ +0004a8d0: 2d39 3130 3761 3262 642d 3838 7c0a 7c20  -9107a2bd-88|.| 
│ │ │ │ +0004a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a920: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a970: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a990: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004a9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004a9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004a9e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004aa10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aa30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004aa60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aa80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004aab0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004aac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aad0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ab20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004ab30: 3733 3762 3363 2d35 3032 3861 3263 322d  737b3c-5028a2c2-
│ │ │ │ -0004ab40: 3933 3132 6162 6332 2d31 3631 3062 3263  9312abc2-1610b2c
│ │ │ │ -0004ab50: 322b 3135 3933 3661 6333 2d35 3338 3562  2+15936ac3-5385b
│ │ │ │ -0004ab60: 6333 2d35 3337 3563 342b 3639 3338 6133  c3-5375c4+6938a3
│ │ │ │ -0004ab70: 642d 3933 3731 6132 6264 2d39 397c 0a7c  d-9371a2bd-99|.|
│ │ │ │ +0004ab00: 2020 2020 2020 2020 2020 2020 7c0a 7c37              |.|7
│ │ │ │ +0004ab10: 3337 6233 632d 3530 3238 6132 6332 2d39  37b3c-5028a2c2-9
│ │ │ │ +0004ab20: 3331 3261 6263 322d 3136 3130 6232 6332  312abc2-1610b2c2
│ │ │ │ +0004ab30: 2b31 3539 3336 6163 332d 3533 3835 6263  +15936ac3-5385bc
│ │ │ │ +0004ab40: 332d 3533 3735 6334 2b36 3933 3861 3364  3-5375c4+6938a3d
│ │ │ │ +0004ab50: 2d39 3337 3161 3262 642d 3939 7c0a 7c2d  -9371a2bd-99|.|-
│ │ │ │ +0004ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0004ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +0004aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0004abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004abf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ac10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ac40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ac60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ac90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004acb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ace0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ad00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ad30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ad50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ad80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ada0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004add0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004adf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ae20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ae40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ae70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ae90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004aec0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004aee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004af10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004af30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004af60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004af80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004afb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004afd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b000: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b0a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b0c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b140: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b1b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b1e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b200: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b230: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b250: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b280: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b2a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b2c0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +0004b2d0: 2020 2020 2020 2020 2020 2020 7c0a 7c64              |.|d
│ │ │ │  0004b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b2f0: 3020 2020 2020 2020 2020 2020 207c 0a7c  0            |.|
│ │ │ │ -0004b300: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ -0004b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b310: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +0004b320: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b340: 3020 2020 2020 2020 2020 2020 207c 0a7c  0            |.|
│ │ │ │ +0004b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b360: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +0004b370: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b390: 3020 2020 2020 2020 2020 2020 207c 0a7c  0            |.|
│ │ │ │ +0004b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b3b0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0004b3c0: 3233 3661 332b 3839 3232 6132 7c0a 7c20  236a3+8922a2|.| 
│ │ │ │  0004b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b3e0: 3132 3336 6133 2b38 3932 3261 327c 0a7c  1236a3+8922a2|.|
│ │ │ │ +0004b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b430: 3130 3337 3061 6232 2d37 3039 327c 0a7c  10370ab2-7092|.|
│ │ │ │ -0004b440: 3035 3637 6232 642b 3335 3630 6163 642d  0567b2d+3560acd-
│ │ │ │ -0004b450: 3232 3932 6263 642d 3134 3338 3863 3264  2292bcd-14388c2d
│ │ │ │ -0004b460: 2b37 3139 3461 6432 2d36 3234 3562 6432  +7194ad2-6245bd2
│ │ │ │ -0004b470: 2b38 3633 3963 6432 2d39 3432 3664 3320  +8639cd2-9426d3 
│ │ │ │ -0004b480: 3133 3730 3761 332b 3231 3737 617c 0a7c  13707a3+2177a|.|
│ │ │ │ +0004b400: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0004b410: 3033 3730 6162 322d 3730 3932 7c0a 7c30  0370ab2-7092|.|0
│ │ │ │ +0004b420: 3536 3762 3264 2b33 3536 3061 6364 2d32  567b2d+3560acd-2
│ │ │ │ +0004b430: 3239 3262 6364 2d31 3433 3838 6332 642b  292bcd-14388c2d+
│ │ │ │ +0004b440: 3731 3934 6164 322d 3632 3435 6264 322b  7194ad2-6245bd2+
│ │ │ │ +0004b450: 3836 3339 6364 322d 3934 3236 6433 2031  8639cd2-9426d3 1
│ │ │ │ +0004b460: 3337 3037 6133 2b32 3137 3761 7c0a 7c20  3707a3+2177a|.| 
│ │ │ │ +0004b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b4d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004b4e0: 322b 3730 3932 6233 2b39 3730 3261 6263  2+7092b3+9702abc
│ │ │ │ -0004b4f0: 2b36 3632 3762 3263 2d39 3739 3762 6332  +6627b2c-9797bc2
│ │ │ │ -0004b500: 2d35 3837 3463 332d 3838 3836 6162 642d  -5874c3-8886abd-
│ │ │ │ -0004b510: 3437 3030 6232 642b 3135 6163 642d 3539  4700b2d+15acd-59
│ │ │ │ -0004b520: 3639 6263 642d 3930 3734 6332 647c 0a7c  69bcd-9074c2d|.|
│ │ │ │ -0004b530: 2d32 3137 3761 3262 2b37 3032 3861 6232  -2177a2b+7028ab2
│ │ │ │ -0004b540: 2b37 3032 3161 3263 2d36 3337 3761 6263  +7021a2c-6377abc
│ │ │ │ -0004b550: 2b37 3630 3061 6332 2b31 3137 3236 6263  +7600ac2+11726bc
│ │ │ │ -0004b560: 322d 3131 3430 6333 2d31 3230 3661 3264  2-1140c3-1206a2d
│ │ │ │ -0004b570: 2b31 3134 3335 6162 642b 3630 347c 0a7c  +11435abd+604|.|
│ │ │ │ -0004b580: 2d35 3837 3461 3263 2d39 3037 3461 3264  -5874a2c-9074a2d
│ │ │ │ +0004b4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +0004b4c0: 2b37 3039 3262 332b 3937 3032 6162 632b  +7092b3+9702abc+
│ │ │ │ +0004b4d0: 3636 3237 6232 632d 3937 3937 6263 322d  6627b2c-9797bc2-
│ │ │ │ +0004b4e0: 3538 3734 6333 2d38 3838 3661 6264 2d34  5874c3-8886abd-4
│ │ │ │ +0004b4f0: 3730 3062 3264 2b31 3561 6364 2d35 3936  700b2d+15acd-596
│ │ │ │ +0004b500: 3962 6364 2d39 3037 3463 3264 7c0a 7c2d  9bcd-9074c2d|.|-
│ │ │ │ +0004b510: 3231 3737 6132 622b 3730 3238 6162 322b  2177a2b+7028ab2+
│ │ │ │ +0004b520: 3730 3231 6132 632d 3633 3737 6162 632b  7021a2c-6377abc+
│ │ │ │ +0004b530: 3736 3030 6163 322b 3131 3732 3662 6332  7600ac2+11726bc2
│ │ │ │ +0004b540: 2d31 3134 3063 332d 3132 3036 6132 642b  -1140c3-1206a2d+
│ │ │ │ +0004b550: 3131 3433 3561 6264 2b36 3034 7c0a 7c2d  11435abd+604|.|-
│ │ │ │ +0004b560: 3538 3734 6132 632d 3930 3734 6132 6420  5874a2c-9074a2d 
│ │ │ │ +0004b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b5c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004b5d0: 6132 632d 3535 3939 6162 632d 3134 3136  a2c-5599abc-1416
│ │ │ │ -0004b5e0: 3562 3263 2b38 3838 3061 3264 2b34 3235  5b2c+8880a2d+425
│ │ │ │ -0004b5f0: 3961 6264 2d33 3030 3262 3264 2d31 3338  9abd-3002b2d-138
│ │ │ │ -0004b600: 3932 6163 642d 3130 3532 3162 6364 207c  92acd-10521bcd |
│ │ │ │ -0004b610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b5a0: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +0004b5b0: 3263 2d35 3539 3961 6263 2d31 3431 3635  2c-5599abc-14165
│ │ │ │ +0004b5c0: 6232 632b 3838 3830 6132 642b 3432 3539  b2c+8880a2d+4259
│ │ │ │ +0004b5d0: 6162 642d 3330 3032 6232 642d 3133 3839  abd-3002b2d-1389
│ │ │ │ +0004b5e0: 3261 6364 2d31 3035 3231 6263 6420 7c20  2acd-10521bcd | 
│ │ │ │ +0004b5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0004b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b660: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b6b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b6e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b780: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004b800: 6132 6364 2b34 3035 3361 6263 642b 3230  a2cd+4053abcd+20
│ │ │ │ -0004b810: 3830 6232 6364 2d39 3137 3561 6332 642b  80b2cd-9175ac2d+
│ │ │ │ -0004b820: 3634 3962 6332 642d 3838 3239 6333 642d  649bc2d-8829c3d-
│ │ │ │ -0004b830: 3231 3634 6132 6432 2d38 3633 3561 6264  2164a2d2-8635abd
│ │ │ │ -0004b840: 322b 3731 3631 6232 6432 2d39 397c 0a7c  2+7161b2d2-99|.|
│ │ │ │ +0004b7d0: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +0004b7e0: 3263 642b 3430 3533 6162 6364 2b32 3038  2cd+4053abcd+208
│ │ │ │ +0004b7f0: 3062 3263 642d 3931 3735 6163 3264 2b36  0b2cd-9175ac2d+6
│ │ │ │ +0004b800: 3439 6263 3264 2d38 3832 3963 3364 2d32  49bc2d-8829c3d-2
│ │ │ │ +0004b810: 3136 3461 3264 322d 3836 3335 6162 6432  164a2d2-8635abd2
│ │ │ │ +0004b820: 2b37 3136 3162 3264 322d 3939 7c0a 7c20  +7161b2d2-99|.| 
│ │ │ │ +0004b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b8c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b8e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b910: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004b9b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004b9d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ba00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ba50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ba70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bac0: 2020 2020 2020 3020 2020 2020 207c 0a7c        0      |.|
│ │ │ │ -0004bad0: 6364 2020 2020 2020 2020 2020 2020 2020  cd              
│ │ │ │ +0004baa0: 2020 2020 2030 2020 2020 2020 7c0a 7c63       0      |.|c
│ │ │ │ +0004bab0: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ +0004bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004baf0: 2020 2020 2030 2020 2020 2020 7c0a 7c20       0      |.| 
│ │ │ │  0004bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb10: 2020 2020 2020 3020 2020 2020 207c 0a7c        0      |.|
│ │ │ │ +0004bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bb40: 2020 2020 2030 2020 2020 2020 7c0a 7c20       0      |.| 
│ │ │ │  0004bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb60: 2020 2020 2020 3020 2020 2020 207c 0a7c        0      |.|
│ │ │ │ +0004bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bb90: 2020 2020 2031 3037 6133 2b34 7c0a 7c20       107a3+4|.| 
│ │ │ │  0004bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bbb0: 2020 2020 2020 3130 3761 332b 347c 0a7c        107a3+4|.|
│ │ │ │ +0004bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc00: 2020 2020 2020 3832 3331 6162 327c 0a7c        8231ab2|.|
│ │ │ │ -0004bc10: 3161 6264 2d32 3632 3762 3264 2b33 3939  1abd-2627b2d+399
│ │ │ │ -0004bc20: 3661 6364 2b37 3135 3262 6364 2b31 3137  6acd+7152bcd+117
│ │ │ │ -0004bc30: 3430 6332 642d 3933 3938 6164 322d 3135  40c2d-9398ad2-15
│ │ │ │ -0004bc40: 3331 3762 6432 2b36 3932 3263 6432 2b35  317bd2+6922cd2+5
│ │ │ │ -0004bc50: 3038 3064 3320 2d31 3533 3434 617c 0a7c  080d3 -15344a|.|
│ │ │ │ -0004bc60: 2d31 3032 3539 6164 3220 2020 2020 2020  -10259ad2       
│ │ │ │ +0004bbe0: 2020 2020 2038 3233 3161 6232 7c0a 7c31       8231ab2|.|1
│ │ │ │ +0004bbf0: 6162 642d 3236 3237 6232 642b 3339 3936  abd-2627b2d+3996
│ │ │ │ +0004bc00: 6163 642b 3731 3532 6263 642b 3131 3734  acd+7152bcd+1174
│ │ │ │ +0004bc10: 3063 3264 2d39 3339 3861 6432 2d31 3533  0c2d-9398ad2-153
│ │ │ │ +0004bc20: 3137 6264 322b 3639 3232 6364 322b 3530  17bd2+6922cd2+50
│ │ │ │ +0004bc30: 3830 6433 202d 3135 3334 3461 7c0a 7c2d  80d3 -15344a|.|-
│ │ │ │ +0004bc40: 3130 3235 3961 6432 2020 2020 2020 2020  10259ad2        
│ │ │ │ +0004bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004bcb0: 2d38 3233 3161 6232 2d31 3331 3737 6233  -8231ab2-13177b3
│ │ │ │ -0004bcc0: 2d35 3836 3461 6263 2d31 3339 3930 6232  -5864abc-13990b2
│ │ │ │ -0004bcd0: 632b 3133 3039 6263 322d 3533 3938 6333  c+1309bc2-5398c3
│ │ │ │ -0004bce0: 2d35 3032 3661 6264 2b31 3135 3231 6232  -5026abd+11521b2
│ │ │ │ -0004bcf0: 642b 3735 3031 6163 642b 3137 377c 0a7c  d+7501acd+177|.|
│ │ │ │ -0004bd00: 3135 3334 3461 332d 3236 3533 6132 622d  15344a3-2653a2b-
│ │ │ │ -0004bd10: 3130 3235 3961 6232 2d31 3233 3635 6132  10259ab2-12365a2
│ │ │ │ -0004bd20: 632b 3732 3136 6162 632d 3632 3330 6163  c+7216abc-6230ac
│ │ │ │ -0004bd30: 322b 3533 3236 6263 322d 3130 3331 6333  2+5326bc2-1031c3
│ │ │ │ -0004bd40: 2b31 3335 3038 6132 642b 3130 317c 0a7c  +13508a2d+101|.|
│ │ │ │ -0004bd50: 3133 3039 6132 622d 3533 3938 6132 632d  1309a2b-5398a2c-
│ │ │ │ -0004bd60: 3535 3439 6132 6420 2020 2020 2020 2020  5549a2d         
│ │ │ │ -0004bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bd90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004bda0: 3539 6233 2d35 3537 3061 3263 2d35 3330  59b3-5570a2c-530
│ │ │ │ -0004bdb0: 3761 6263 2d37 3436 3462 3263 2b33 3138  7abc-7464b2c+318
│ │ │ │ -0004bdc0: 3761 3264 2b38 3537 3061 6264 2d38 3235  7a2d+8570abd-825
│ │ │ │ -0004bdd0: 3162 3264 2b38 3434 3461 6364 2b35 3037  1b2d+8444acd+507
│ │ │ │ -0004bde0: 3162 6364 207c 2020 2020 2020 207c 0a7c  1bcd |       |.|
│ │ │ │ +0004bc80: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0004bc90: 3832 3331 6162 322d 3133 3137 3762 332d  8231ab2-13177b3-
│ │ │ │ +0004bca0: 3538 3634 6162 632d 3133 3939 3062 3263  5864abc-13990b2c
│ │ │ │ +0004bcb0: 2b31 3330 3962 6332 2d35 3339 3863 332d  +1309bc2-5398c3-
│ │ │ │ +0004bcc0: 3530 3236 6162 642b 3131 3532 3162 3264  5026abd+11521b2d
│ │ │ │ +0004bcd0: 2b37 3530 3161 6364 2b31 3737 7c0a 7c31  +7501acd+177|.|1
│ │ │ │ +0004bce0: 3533 3434 6133 2d32 3635 3361 3262 2d31  5344a3-2653a2b-1
│ │ │ │ +0004bcf0: 3032 3539 6162 322d 3132 3336 3561 3263  0259ab2-12365a2c
│ │ │ │ +0004bd00: 2b37 3231 3661 6263 2d36 3233 3061 6332  +7216abc-6230ac2
│ │ │ │ +0004bd10: 2b35 3332 3662 6332 2d31 3033 3163 332b  +5326bc2-1031c3+
│ │ │ │ +0004bd20: 3133 3530 3861 3264 2b31 3031 7c0a 7c31  13508a2d+101|.|1
│ │ │ │ +0004bd30: 3330 3961 3262 2d35 3339 3861 3263 2d35  309a2b-5398a2c-5
│ │ │ │ +0004bd40: 3534 3961 3264 2020 2020 2020 2020 2020  549a2d          
│ │ │ │ +0004bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bd70: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +0004bd80: 3962 332d 3535 3730 6132 632d 3533 3037  9b3-5570a2c-5307
│ │ │ │ +0004bd90: 6162 632d 3734 3634 6232 632b 3331 3837  abc-7464b2c+3187
│ │ │ │ +0004bda0: 6132 642b 3835 3730 6162 642d 3832 3531  a2d+8570abd-8251
│ │ │ │ +0004bdb0: 6232 642b 3834 3434 6163 642b 3530 3731  b2d+8444acd+5071
│ │ │ │ +0004bdc0: 6263 6420 7c20 2020 2020 2020 7c0a 7c20  bcd |       |.| 
│ │ │ │ +0004bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004be00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004be10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004be10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004be20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004be30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004be60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004be80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004beb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bf00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bf20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004bf50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004bf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004bfc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004bfd0: 3436 6162 3264 2d35 3231 3862 3364 2d38  46ab2d-5218b3d-8
│ │ │ │ -0004bfe0: 3032 6132 6364 2d36 3333 3261 6263 642b  02a2cd-6332abcd+
│ │ │ │ -0004bff0: 3537 3238 6232 6364 2d35 3334 3761 6332  5728b2cd-5347ac2
│ │ │ │ -0004c000: 642d 3133 3339 3462 6332 642d 3239 3663  d-13394bc2d-296c
│ │ │ │ -0004c010: 3364 2d37 3434 3661 3264 322d 317c 0a7c  3d-7446a2d2-1|.|
│ │ │ │ +0004bfa0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +0004bfb0: 3661 6232 642d 3532 3138 6233 642d 3830  6ab2d-5218b3d-80
│ │ │ │ +0004bfc0: 3261 3263 642d 3633 3332 6162 6364 2b35  2a2cd-6332abcd+5
│ │ │ │ +0004bfd0: 3732 3862 3263 642d 3533 3437 6163 3264  728b2cd-5347ac2d
│ │ │ │ +0004bfe0: 2d31 3333 3934 6263 3264 2d32 3936 6333  -13394bc2d-296c3
│ │ │ │ +0004bff0: 642d 3734 3436 6132 6432 2d31 7c0a 7c20  d-7446a2d2-1|.| 
│ │ │ │ +0004c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c0b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c0e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c130: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c180: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c1a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c1d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c1f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004c250: 3031 6162 3264 2b34 3432 3062 3364 2b31  01ab2d+4420b3d+1
│ │ │ │ -0004c260: 3233 3137 6132 6364 2b31 3533 3738 6162  2317a2cd+15378ab
│ │ │ │ -0004c270: 6364 2b31 3439 3231 6232 6364 2d38 3831  cd+14921b2cd-881
│ │ │ │ -0004c280: 3061 6332 642b 3332 3731 6263 3264 2b31  0ac2d+3271bc2d+1
│ │ │ │ -0004c290: 3036 3633 6333 642b 3132 3533 377c 0a7c  0663c3d+12537|.|
│ │ │ │ +0004c220: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0004c230: 3161 6232 642b 3434 3230 6233 642b 3132  1ab2d+4420b3d+12
│ │ │ │ +0004c240: 3331 3761 3263 642b 3135 3337 3861 6263  317a2cd+15378abc
│ │ │ │ +0004c250: 642b 3134 3932 3162 3263 642d 3838 3130  d+14921b2cd-8810
│ │ │ │ +0004c260: 6163 3264 2b33 3237 3162 6332 642b 3130  ac2d+3271bc2d+10
│ │ │ │ +0004c270: 3636 3363 3364 2b31 3235 3337 7c0a 7c2d  663c3d+12537|.|-
│ │ │ │ +0004c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0004c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004c2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +0004c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0004c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c380: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c3b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c3d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c450: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c4a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c4c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c4f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c560: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c590: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c5b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c600: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c630: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c650: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c6a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c6d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c6f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c720: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c790: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c7c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c7e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c810: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c830: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c880: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c8b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c900: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c920: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c950: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c970: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004c9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c9c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004c9f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ca10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ca40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ca60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cab0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004cac0: 622d 3335 3839 6162 322b 3839 3731 6233  b-3589ab2+8971b3
│ │ │ │ -0004cad0: 2d35 3030 3661 3263 2d35 3539 3961 6263  -5006a2c-5599abc
│ │ │ │ -0004cae0: 2d31 3431 3635 6232 632b 3838 3830 6132  -14165b2c+8880a2
│ │ │ │ -0004caf0: 642b 3432 3539 6162 642d 3330 3032 6232  d+4259abd-3002b2
│ │ │ │ -0004cb00: 642d 3133 3839 3261 6364 2d31 307c 0a7c  d-13892acd-10|.|
│ │ │ │ -0004cb10: 6233 2d39 3730 3261 6263 2d36 3632 3762  b3-9702abc-6627b
│ │ │ │ -0004cb20: 3263 2b38 3838 3661 6264 2b34 3730 3062  2c+8886abd+4700b
│ │ │ │ -0004cb30: 3264 2d31 3561 6364 2b35 3936 3962 6364  2d-15acd+5969bcd
│ │ │ │ -0004cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cb50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004cb60: 3262 2d37 3032 3861 6232 2b39 3739 3762  2b-7028ab2+9797b
│ │ │ │ -0004cb70: 332d 3730 3231 6132 632b 3633 3737 6162  3-7021a2c+6377ab
│ │ │ │ -0004cb80: 632b 3538 3734 6232 632d 3736 3030 6163  c+5874b2c-7600ac
│ │ │ │ -0004cb90: 322d 3131 3732 3662 6332 2b31 3134 3063  2-11726bc2+1140c
│ │ │ │ -0004cba0: 332b 3132 3036 6132 642d 3131 347c 0a7c  3+1206a2d-114|.|
│ │ │ │ +0004ca90: 2020 2020 2020 2020 2020 2020 7c0a 7c62              |.|b
│ │ │ │ +0004caa0: 2d33 3538 3961 6232 2b38 3937 3162 332d  -3589ab2+8971b3-
│ │ │ │ +0004cab0: 3530 3036 6132 632d 3535 3939 6162 632d  5006a2c-5599abc-
│ │ │ │ +0004cac0: 3134 3136 3562 3263 2b38 3838 3061 3264  14165b2c+8880a2d
│ │ │ │ +0004cad0: 2b34 3235 3961 6264 2d33 3030 3262 3264  +4259abd-3002b2d
│ │ │ │ +0004cae0: 2d31 3338 3932 6163 642d 3130 7c0a 7c62  -13892acd-10|.|b
│ │ │ │ +0004caf0: 332d 3937 3032 6162 632d 3636 3237 6232  3-9702abc-6627b2
│ │ │ │ +0004cb00: 632b 3838 3836 6162 642b 3437 3030 6232  c+8886abd+4700b2
│ │ │ │ +0004cb10: 642d 3135 6163 642b 3539 3639 6263 6420  d-15acd+5969bcd 
│ │ │ │ +0004cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cb30: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +0004cb40: 622d 3730 3238 6162 322b 3937 3937 6233  b-7028ab2+9797b3
│ │ │ │ +0004cb50: 2d37 3032 3161 3263 2b36 3337 3761 6263  -7021a2c+6377abc
│ │ │ │ +0004cb60: 2b35 3837 3462 3263 2d37 3630 3061 6332  +5874b2c-7600ac2
│ │ │ │ +0004cb70: 2d31 3137 3236 6263 322b 3131 3430 6333  -11726bc2+1140c3
│ │ │ │ +0004cb80: 2b31 3230 3661 3264 2d31 3134 7c0a 7c20  +1206a2d-114|.| 
│ │ │ │ +0004cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cbe0: 2020 2020 3020 2020 2020 2020 2020 2020      0           
│ │ │ │ -0004cbf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004cc00: 2b36 3732 3361 6432 2b35 3438 3362 6432  +6723ad2+5483bd2
│ │ │ │ -0004cc10: 2b31 3532 3530 6364 322d 3130 3536 3764  +15250cd2-10567d
│ │ │ │ -0004cc20: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ -0004cc30: 2020 2020 3132 3336 6133 2b38 3932 3261      1236a3+8922a
│ │ │ │ -0004cc40: 3262 2d33 3538 3961 6232 2b38 397c 0a7c  2b-3589ab2+89|.|
│ │ │ │ -0004cc50: 3061 6364 2d38 3032 3262 6364 2d33 3936  0acd-8022bcd-396
│ │ │ │ -0004cc60: 3863 3264 2b33 3136 3461 6432 2d32 3935  8c2d+3164ad2-295
│ │ │ │ -0004cc70: 6264 322b 3736 3530 6364 322d 3134 3338  bd2+7650cd2-1438
│ │ │ │ -0004cc80: 3864 3320 3020 2020 2020 2020 2020 2020  8d3 0           
│ │ │ │ -0004cc90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004cbc0: 2020 2030 2020 2020 2020 2020 2020 2020     0            
│ │ │ │ +0004cbd0: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ +0004cbe0: 3637 3233 6164 322b 3534 3833 6264 322b  6723ad2+5483bd2+
│ │ │ │ +0004cbf0: 3135 3235 3063 6432 2d31 3035 3637 6433  15250cd2-10567d3
│ │ │ │ +0004cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cc10: 2020 2031 3233 3661 332b 3839 3232 6132     1236a3+8922a2
│ │ │ │ +0004cc20: 622d 3335 3839 6162 322b 3839 7c0a 7c30  b-3589ab2+89|.|0
│ │ │ │ +0004cc30: 6163 642d 3830 3232 6263 642d 3339 3638  acd-8022bcd-3968
│ │ │ │ +0004cc40: 6332 642b 3331 3634 6164 322d 3239 3562  c2d+3164ad2-295b
│ │ │ │ +0004cc50: 6432 2b37 3635 3063 6432 2d31 3433 3838  d2+7650cd2-14388
│ │ │ │ +0004cc60: 6433 2030 2020 2020 2020 2020 2020 2020  d3 0            
│ │ │ │ +0004cc70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0004cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ccd0: 2020 2020 2d31 3337 3037 6133 2d32 3137      -13707a3-217
│ │ │ │ -0004cce0: 3761 3262 2b37 3032 3861 6232 2d7c 0a7c  7a2b+7028ab2-|.|
│ │ │ │ +0004ccb0: 2020 202d 3133 3730 3761 332d 3231 3737     -13707a3-2177
│ │ │ │ +0004ccc0: 6132 622b 3730 3238 6162 322d 7c0a 7c20  a2b+7028ab2-|.| 
│ │ │ │ +0004ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cd10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cd30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cd60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cd80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cdb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cdd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ce00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ce20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ce50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ce70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cf10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004cf20: 3761 6364 322d 3330 3135 6263 6432 2d31  7acd2-3015bcd2-1
│ │ │ │ -0004cf30: 3132 3734 6332 6432 2031 3139 3538 6133  1274c2d2 11958a3
│ │ │ │ -0004cf40: 622b 3836 3431 6132 6232 2b39 3836 3461  b+8641a2b2+9864a
│ │ │ │ -0004cf50: 6233 2b38 3634 3962 342d 3434 3137 6133  b3+8649b4-4417a3
│ │ │ │ -0004cf60: 632b 3337 3331 6132 6263 2b37 397c 0a7c  c+3731a2bc+79|.|
│ │ │ │ +0004cef0: 2020 2020 2020 2020 2020 2020 7c0a 7c37              |.|7
│ │ │ │ +0004cf00: 6163 6432 2d33 3031 3562 6364 322d 3131  acd2-3015bcd2-11
│ │ │ │ +0004cf10: 3237 3463 3264 3220 3131 3935 3861 3362  274c2d2 11958a3b
│ │ │ │ +0004cf20: 2b38 3634 3161 3262 322b 3938 3634 6162  +8641a2b2+9864ab
│ │ │ │ +0004cf30: 332b 3836 3439 6234 2d34 3431 3761 3363  3+8649b4-4417a3c
│ │ │ │ +0004cf40: 2b33 3733 3161 3262 632b 3739 7c0a 7c20  +3731a2bc+79|.| 
│ │ │ │ +0004cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cf90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cfb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004cfe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d080: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d0a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d0d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d0f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d120: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d170: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d1c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d210: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d230: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004d290: 3337 3661 3262 2b33 3738 3361 6232 2b31  376a2b+3783ab2+1
│ │ │ │ -0004d2a0: 3033 3539 6233 2d35 3537 3061 3263 2d35  0359b3-5570a2c-5
│ │ │ │ -0004d2b0: 3330 3761 6263 2d37 3436 3462 3263 2b33  307abc-7464b2c+3
│ │ │ │ -0004d2c0: 3138 3761 3264 2b38 3537 3061 6264 2d38  187a2d+8570abd-8
│ │ │ │ -0004d2d0: 3235 3162 3264 2b38 3434 3461 637c 0a7c  251b2d+8444ac|.|
│ │ │ │ -0004d2e0: 2b31 3331 3737 6233 2b35 3836 3461 6263  +13177b3+5864abc
│ │ │ │ -0004d2f0: 2b31 3339 3930 6232 632b 3530 3236 6162  +13990b2c+5026ab
│ │ │ │ -0004d300: 642d 3131 3532 3162 3264 2d37 3530 3161  d-11521b2d-7501a
│ │ │ │ -0004d310: 6364 2d31 3737 3962 6364 2020 2020 2020  cd-1779bcd      
│ │ │ │ -0004d320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004d330: 332b 3236 3533 6132 622b 3130 3235 3961  3+2653a2b+10259a
│ │ │ │ -0004d340: 6232 2d31 3330 3962 332b 3132 3336 3561  b2-1309b3+12365a
│ │ │ │ -0004d350: 3263 2d37 3231 3661 6263 2b35 3339 3862  2c-7216abc+5398b
│ │ │ │ -0004d360: 3263 2b36 3233 3061 6332 2d35 3332 3662  2c+6230ac2-5326b
│ │ │ │ -0004d370: 6332 2b31 3033 3163 332d 3133 357c 0a7c  c2+1031c3-135|.|
│ │ │ │ +0004d260: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +0004d270: 3736 6132 622b 3337 3833 6162 322b 3130  76a2b+3783ab2+10
│ │ │ │ +0004d280: 3335 3962 332d 3535 3730 6132 632d 3533  359b3-5570a2c-53
│ │ │ │ +0004d290: 3037 6162 632d 3734 3634 6232 632b 3331  07abc-7464b2c+31
│ │ │ │ +0004d2a0: 3837 6132 642b 3835 3730 6162 642d 3832  87a2d+8570abd-82
│ │ │ │ +0004d2b0: 3531 6232 642b 3834 3434 6163 7c0a 7c2b  51b2d+8444ac|.|+
│ │ │ │ +0004d2c0: 3133 3137 3762 332b 3538 3634 6162 632b  13177b3+5864abc+
│ │ │ │ +0004d2d0: 3133 3939 3062 3263 2b35 3032 3661 6264  13990b2c+5026abd
│ │ │ │ +0004d2e0: 2d31 3135 3231 6232 642d 3735 3031 6163  -11521b2d-7501ac
│ │ │ │ +0004d2f0: 642d 3137 3739 6263 6420 2020 2020 2020  d-1779bcd       
│ │ │ │ +0004d300: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +0004d310: 2b32 3635 3361 3262 2b31 3032 3539 6162  +2653a2b+10259ab
│ │ │ │ +0004d320: 322d 3133 3039 6233 2b31 3233 3635 6132  2-1309b3+12365a2
│ │ │ │ +0004d330: 632d 3732 3136 6162 632b 3533 3938 6232  c-7216abc+5398b2
│ │ │ │ +0004d340: 632b 3632 3330 6163 322d 3533 3236 6263  c+6230ac2-5326bc
│ │ │ │ +0004d350: 322b 3130 3331 6333 2d31 3335 7c0a 7c20  2+1031c3-135|.| 
│ │ │ │ +0004d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d3b0: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0004d3c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004d3d0: 3962 6364 2d35 3534 3963 3264 2d39 3533  9bcd-5549c2d-953
│ │ │ │ -0004d3e0: 3461 6432 2d31 3038 3636 6264 322b 3730  4ad2-10866bd2+70
│ │ │ │ -0004d3f0: 3631 6364 322d 3236 3237 6433 2020 2020  61cd2-2627d3    
│ │ │ │ -0004d400: 2020 2020 2020 2020 2020 2020 2020 3130                10
│ │ │ │ -0004d410: 3761 332b 3433 3736 6132 622b 337c 0a7c  7a3+4376a2b+3|.|
│ │ │ │ -0004d420: 3235 6162 642d 3930 3333 6163 642d 3239  25abd-9033acd-29
│ │ │ │ -0004d430: 3938 6263 642b 3230 3336 6332 642d 3531  98bcd+2036c2d-51
│ │ │ │ -0004d440: 3037 6164 322d 3536 3739 6264 322b 3633  07ad2-5679bd2+63
│ │ │ │ -0004d450: 3235 6364 322b 3131 3734 3064 3320 3020  25cd2+11740d3 0 
│ │ │ │ -0004d460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d390: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +0004d3a0: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +0004d3b0: 6263 642d 3535 3439 6332 642d 3935 3334  bcd-5549c2d-9534
│ │ │ │ +0004d3c0: 6164 322d 3130 3836 3662 6432 2b37 3036  ad2-10866bd2+706
│ │ │ │ +0004d3d0: 3163 6432 2d32 3632 3764 3320 2020 2020  1cd2-2627d3     
│ │ │ │ +0004d3e0: 2020 2020 2020 2020 2020 2020 2031 3037               107
│ │ │ │ +0004d3f0: 6133 2b34 3337 3661 3262 2b33 7c0a 7c32  a3+4376a2b+3|.|2
│ │ │ │ +0004d400: 3561 6264 2d39 3033 3361 6364 2d32 3939  5abd-9033acd-299
│ │ │ │ +0004d410: 3862 6364 2b32 3033 3663 3264 2d35 3130  8bcd+2036c2d-510
│ │ │ │ +0004d420: 3761 6432 2d35 3637 3962 6432 2b36 3332  7ad2-5679bd2+632
│ │ │ │ +0004d430: 3563 6432 2b31 3137 3430 6433 2030 2020  5cd2+11740d3 0  
│ │ │ │ +0004d440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0004d450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d4a0: 2020 2020 2020 2020 2020 2020 2020 3135                15
│ │ │ │ -0004d4b0: 3334 3461 332d 3236 3533 6132 627c 0a7c  344a3-2653a2b|.|
│ │ │ │ +0004d480: 2020 2020 2020 2020 2020 2020 2031 3533               153
│ │ │ │ +0004d490: 3434 6133 2d32 3635 3361 3262 7c0a 7c20  44a3-2653a2b|.| 
│ │ │ │ +0004d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d4e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d530: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d580: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d5a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d5d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d5f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d620: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d640: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d670: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d6e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004d6f0: 3030 3436 6162 6432 2d36 3334 3262 3264  0046abd2-6342b2d
│ │ │ │ -0004d700: 322b 3734 3830 6163 6432 2b37 3834 3362  2+7480acd2+7843b
│ │ │ │ -0004d710: 6364 322b 3830 3834 6332 6432 2036 3430  cd2+8084c2d2 640
│ │ │ │ -0004d720: 3961 3362 2d31 3031 3531 6132 6232 2d33  9a3b-10151a2b2-3
│ │ │ │ -0004d730: 3836 3361 6233 2b37 3637 3262 347c 0a7c  863ab3+7672b4|.|
│ │ │ │ +0004d6c0: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0004d6d0: 3034 3661 6264 322d 3633 3432 6232 6432  046abd2-6342b2d2
│ │ │ │ +0004d6e0: 2b37 3438 3061 6364 322b 3738 3433 6263  +7480acd2+7843bc
│ │ │ │ +0004d6f0: 6432 2b38 3038 3463 3264 3220 3634 3039  d2+8084c2d2 6409
│ │ │ │ +0004d700: 6133 622d 3130 3135 3161 3262 322d 3338  a3b-10151a2b2-38
│ │ │ │ +0004d710: 3633 6162 332b 3736 3732 6234 7c0a 7c20  63ab3+7672b4|.| 
│ │ │ │ +0004d720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d760: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d780: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d7d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d8a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d8c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004d8f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d910: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004d960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004d970: 6132 6432 2d31 3532 3831 6162 6432 2b39  a2d2-15281abd2+9
│ │ │ │ -0004d980: 3334 3662 3264 322d 3433 3039 6163 6432  346b2d2-4309acd2
│ │ │ │ -0004d990: 2d38 3236 3962 6364 322b 3633 3431 6332  -8269bcd2+6341c2
│ │ │ │ -0004d9a0: 6432 202d 3132 3035 3161 3362 2b33 3533  d2 -12051a3b+353
│ │ │ │ -0004d9b0: 6132 6232 2b38 3531 3861 6233 2b7c 0a7c  a2b2+8518ab3+|.|
│ │ │ │ +0004d940: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +0004d950: 3264 322d 3135 3238 3161 6264 322b 3933  2d2-15281abd2+93
│ │ │ │ +0004d960: 3436 6232 6432 2d34 3330 3961 6364 322d  46b2d2-4309acd2-
│ │ │ │ +0004d970: 3832 3639 6263 6432 2b36 3334 3163 3264  8269bcd2+6341c2d
│ │ │ │ +0004d980: 3220 2d31 3230 3531 6133 622b 3335 3361  2 -12051a3b+353a
│ │ │ │ +0004d990: 3262 322b 3835 3138 6162 332b 7c0a 7c2d  2b2+8518ab3+|.|-
│ │ │ │ +0004d9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0004d9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004d9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004d9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004d9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004da00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +0004d9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0004d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004da30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004da40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004da50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004da80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004da90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004daa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004daf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004db20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004db40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004db70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004db90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dbe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dc10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dc30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dc60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dc80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dcb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dcd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dd00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dd20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dd50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dd70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dda0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ddc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ddf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004de00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004de10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004de40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004de60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004de90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004deb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004df00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004df30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004df50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004df80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dfa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004dfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004dfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004dfd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004dff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004dff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e020: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e040: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e070: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e0e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e0f0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │  0004e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e110: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ +0004e110: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e130: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e140: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │  0004e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e160: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ +0004e160: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e180: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e190: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │  0004e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e1b0: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ -0004e1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e1d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e1e0: 3532 3162 6364 2020 2020 2020 2020 2020  521bcd          
│ │ │ │ +0004e1b0: 2020 2020 2020 2020 2020 2020 7c0a 7c35              |.|5
│ │ │ │ +0004e1c0: 3231 6263 6420 2020 2020 2020 2020 2020  21bcd           
│ │ │ │ +0004e1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e1e0: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │  0004e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e200: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ +0004e200: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e220: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e230: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │  0004e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e250: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │ -0004e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e280: 3335 6162 642b 3930 3734 6232 642d 3630  35abd+9074b2d-60
│ │ │ │ -0004e290: 3430 6163 642b 3830 3232 6263 642b 3339  40acd+8022bcd+39
│ │ │ │ -0004e2a0: 3638 6332 6420 7c20 2020 2020 2020 2020  68c2d |         
│ │ │ │ +0004e250: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +0004e260: 3561 6264 2b39 3037 3462 3264 2d36 3034  5abd+9074b2d-604
│ │ │ │ +0004e270: 3061 6364 2b38 3032 3262 6364 2b33 3936  0acd+8022bcd+396
│ │ │ │ +0004e280: 3863 3264 207c 2020 2020 2020 2020 2020  8c2d |          
│ │ │ │ +0004e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e2a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e2c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e320: 3731 6233 2d35 3030 3661 3263 2d35 3539  71b3-5006a2c-559
│ │ │ │ -0004e330: 3961 6263 2d31 3431 3635 6232 632b 3838  9abc-14165b2c+88
│ │ │ │ -0004e340: 3830 6132 642b 3432 3539 6162 642d 3330  80a2d+4259abd-30
│ │ │ │ -0004e350: 3032 6232 642d 3133 3839 3261 6364 2d31  02b2d-13892acd-1
│ │ │ │ -0004e360: 3035 3231 6263 6420 2020 2020 207c 0a7c  0521bcd      |.|
│ │ │ │ +0004e2f0: 2020 2020 2020 2020 2020 2020 7c0a 7c37              |.|7
│ │ │ │ +0004e300: 3162 332d 3530 3036 6132 632d 3535 3939  1b3-5006a2c-5599
│ │ │ │ +0004e310: 6162 632d 3134 3136 3562 3263 2b38 3838  abc-14165b2c+888
│ │ │ │ +0004e320: 3061 3264 2b34 3235 3961 6264 2d33 3030  0a2d+4259abd-300
│ │ │ │ +0004e330: 3262 3264 2d31 3338 3932 6163 642d 3130  2b2d-13892acd-10
│ │ │ │ +0004e340: 3532 3162 6364 2020 2020 2020 7c0a 7c20  521bcd      |.| 
│ │ │ │ +0004e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e3b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e3c0: 3937 3937 6233 2b37 3032 3161 3263 2d36  9797b3+7021a2c-6
│ │ │ │ -0004e3d0: 3337 3761 6263 2d35 3837 3462 3263 2b37  377abc-5874b2c+7
│ │ │ │ -0004e3e0: 3630 3061 6332 2b31 3137 3236 6263 322d  600ac2+11726bc2-
│ │ │ │ -0004e3f0: 3131 3430 6333 2d31 3230 3661 3264 2b31  1140c3-1206a2d+1
│ │ │ │ -0004e400: 3134 3335 6162 642d 3930 3734 627c 0a7c  1435abd-9074b|.|
│ │ │ │ +0004e390: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +0004e3a0: 3739 3762 332b 3730 3231 6132 632d 3633  797b3+7021a2c-63
│ │ │ │ +0004e3b0: 3737 6162 632d 3538 3734 6232 632b 3736  77abc-5874b2c+76
│ │ │ │ +0004e3c0: 3030 6163 322b 3131 3732 3662 6332 2d31  00ac2+11726bc2-1
│ │ │ │ +0004e3d0: 3134 3063 332d 3132 3036 6132 642b 3131  140c3-1206a2d+11
│ │ │ │ +0004e3e0: 3433 3561 6264 2d39 3037 3462 7c0a 7c20  435abd-9074b|.| 
│ │ │ │ +0004e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e430: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e4a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e4d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e4f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e520: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e5c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e5e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e630: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e640: 3330 6162 3263 2b39 3231 3962 3363 2b35  30ab2c+9219b3c+5
│ │ │ │ -0004e650: 3531 3361 3263 322b 3130 3535 3861 6263  513a2c2+10558abc
│ │ │ │ -0004e660: 322b 3235 3930 6232 6332 2b31 3434 3134  2+2590b2c2+14414
│ │ │ │ -0004e670: 6133 642b 3338 3434 6132 6264 2b35 3937  a3d+3844a2bd+597
│ │ │ │ -0004e680: 3961 6232 642d 3132 3631 3562 337c 0a7c  9ab2d-12615b3|.|
│ │ │ │ +0004e610: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +0004e620: 3061 6232 632b 3932 3139 6233 632b 3535  0ab2c+9219b3c+55
│ │ │ │ +0004e630: 3133 6132 6332 2b31 3035 3538 6162 6332  13a2c2+10558abc2
│ │ │ │ +0004e640: 2b32 3539 3062 3263 322b 3134 3431 3461  +2590b2c2+14414a
│ │ │ │ +0004e650: 3364 2b33 3834 3461 3262 642b 3539 3739  3d+3844a2bd+5979
│ │ │ │ +0004e660: 6162 3264 2d31 3236 3135 6233 7c0a 7c20  ab2d-12615b3|.| 
│ │ │ │ +0004e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e720: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e770: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e7c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e810: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e840: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e860: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004e860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004e880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e890: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e8c0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  0004e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e8e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0004e8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e900: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e910: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  0004e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0004e930: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e950: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e960: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  0004e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0004e990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e9a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004e9b0: 642b 3530 3731 6263 6420 2020 2020 2020  d+5071bcd       
│ │ │ │ +0004e980: 2020 2020 2020 2020 2020 2020 7c0a 7c64              |.|d
│ │ │ │ +0004e990: 2b35 3037 3162 6364 2020 2020 2020 2020  +5071bcd        
│ │ │ │ +0004e9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e9b0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  0004e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e9d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0004e9d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004e9f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004ea00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004e9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ea00: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │  0004ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ea20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0004ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ea40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004ea50: 3038 6132 642d 3130 3132 3561 6264 2b35  08a2d-10125abd+5
│ │ │ │ -0004ea60: 3534 3962 3264 2b39 3033 3361 6364 2b32  549b2d+9033acd+2
│ │ │ │ -0004ea70: 3939 3862 6364 2d32 3033 3663 3264 207c  998bcd-2036c2d |
│ │ │ │ +0004ea20: 2020 2020 2020 2020 2020 2020 7c0a 7c30              |.|0
│ │ │ │ +0004ea30: 3861 3264 2d31 3031 3235 6162 642b 3535  8a2d-10125abd+55
│ │ │ │ +0004ea40: 3439 6232 642b 3930 3333 6163 642b 3239  49b2d+9033acd+29
│ │ │ │ +0004ea50: 3938 6263 642d 3230 3336 6332 6420 7c20  98bcd-2036c2d | 
│ │ │ │ +0004ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ea70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ea90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ea90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eae0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004eaf0: 3738 3361 6232 2b31 3033 3539 6233 2d35  783ab2+10359b3-5
│ │ │ │ -0004eb00: 3537 3061 3263 2d35 3330 3761 6263 2d37  570a2c-5307abc-7
│ │ │ │ -0004eb10: 3436 3462 3263 2b33 3138 3761 3264 2b38  464b2c+3187a2d+8
│ │ │ │ -0004eb20: 3537 3061 6264 2d38 3235 3162 3264 2b38  570abd-8251b2d+8
│ │ │ │ -0004eb30: 3434 3461 6364 2b35 3037 3162 637c 0a7c  444acd+5071bc|.|
│ │ │ │ +0004eac0: 2020 2020 2020 2020 2020 2020 7c0a 7c37              |.|7
│ │ │ │ +0004ead0: 3833 6162 322b 3130 3335 3962 332d 3535  83ab2+10359b3-55
│ │ │ │ +0004eae0: 3730 6132 632d 3533 3037 6162 632d 3734  70a2c-5307abc-74
│ │ │ │ +0004eaf0: 3634 6232 632b 3331 3837 6132 642b 3835  64b2c+3187a2d+85
│ │ │ │ +0004eb00: 3730 6162 642d 3832 3531 6232 642b 3834  70abd-8251b2d+84
│ │ │ │ +0004eb10: 3434 6163 642b 3530 3731 6263 7c0a 7c20  44acd+5071bc|.| 
│ │ │ │ +0004eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004eb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eb80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004eb90: 2d31 3032 3539 6162 322b 3133 3039 6233  -10259ab2+1309b3
│ │ │ │ -0004eba0: 2d31 3233 3635 6132 632b 3732 3136 6162  -12365a2c+7216ab
│ │ │ │ -0004ebb0: 632d 3533 3938 6232 632d 3632 3330 6163  c-5398b2c-6230ac
│ │ │ │ -0004ebc0: 322b 3533 3236 6263 322d 3130 3331 6333  2+5326bc2-1031c3
│ │ │ │ -0004ebd0: 2b31 3335 3038 6132 642b 3130 317c 0a7c  +13508a2d+101|.|
│ │ │ │ +0004eb60: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0004eb70: 3130 3235 3961 6232 2b31 3330 3962 332d  10259ab2+1309b3-
│ │ │ │ +0004eb80: 3132 3336 3561 3263 2b37 3231 3661 6263  12365a2c+7216abc
│ │ │ │ +0004eb90: 2d35 3339 3862 3263 2d36 3233 3061 6332  -5398b2c-6230ac2
│ │ │ │ +0004eba0: 2b35 3332 3662 6332 2d31 3033 3163 332b  +5326bc2-1031c3+
│ │ │ │ +0004ebb0: 3133 3530 3861 3264 2b31 3031 7c0a 7c20  13508a2d+101|.| 
│ │ │ │ +0004ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ebd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ebe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ec00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ec00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ec20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ec20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ec50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ec50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ec70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ec70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ec80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004eca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ecc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ecf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ed10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ed40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ed60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ed90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004edb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004edb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ee00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004ee10: 2b33 3736 3261 3363 2d39 3238 3061 3262  +3762a3c-9280a2b
│ │ │ │ -0004ee20: 632b 3132 3235 3361 6232 632d 3137 3236  c+12253ab2c-1726
│ │ │ │ -0004ee30: 6233 632b 3930 3333 6132 6332 2d31 3439  b3c+9033a2c2-149
│ │ │ │ -0004ee40: 3636 6162 6332 2d36 3932 3962 3263 322d  66abc2-6929b2c2-
│ │ │ │ -0004ee50: 3131 3235 3361 3364 2b34 3932 317c 0a7c  11253a3d+4921|.|
│ │ │ │ +0004ede0: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ +0004edf0: 3337 3632 6133 632d 3932 3830 6132 6263  3762a3c-9280a2bc
│ │ │ │ +0004ee00: 2b31 3232 3533 6162 3263 2d31 3732 3662  +12253ab2c-1726b
│ │ │ │ +0004ee10: 3363 2b39 3033 3361 3263 322d 3134 3936  3c+9033a2c2-1496
│ │ │ │ +0004ee20: 3661 6263 322d 3639 3239 6232 6332 2d31  6abc2-6929b2c2-1
│ │ │ │ +0004ee30: 3132 3533 6133 642b 3439 3231 7c0a 7c20  1253a3d+4921|.| 
│ │ │ │ +0004ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ee80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eea0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004eed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004eef0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004eef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ef20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ef40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ef70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ef90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004efc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004efe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f030: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f080: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004f090: 3933 3431 6234 2b31 3533 3261 3363 2d32  9341b4+1532a3c-2
│ │ │ │ -0004f0a0: 3132 3061 3262 632b 3439 3332 6162 3263  120a2bc+4932ab2c
│ │ │ │ -0004f0b0: 2d31 3339 3031 6233 632b 3135 3936 3061  -13901b3c+15960a
│ │ │ │ -0004f0c0: 3263 322d 3435 3031 6162 6332 2b31 3130  2c2-4501abc2+110
│ │ │ │ -0004f0d0: 3562 3263 322d 3134 3331 3461 337c 0a7c  5b2c2-14314a3|.|
│ │ │ │ +0004f060: 2020 2020 2020 2020 2020 2020 7c0a 7c39              |.|9
│ │ │ │ +0004f070: 3334 3162 342b 3135 3332 6133 632d 3231  341b4+1532a3c-21
│ │ │ │ +0004f080: 3230 6132 6263 2b34 3933 3261 6232 632d  20a2bc+4932ab2c-
│ │ │ │ +0004f090: 3133 3930 3162 3363 2b31 3539 3630 6132  13901b3c+15960a2
│ │ │ │ +0004f0a0: 6332 2d34 3530 3161 6263 322b 3131 3035  c2-4501abc2+1105
│ │ │ │ +0004f0b0: 6232 6332 2d31 3433 3134 6133 7c0a 7c2d  b2c2-14314a3|.|-
│ │ │ │ +0004f0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0004f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0004f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004f110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0004f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +0004f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0004f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f170: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f1c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f210: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f260: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f290: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f2b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f300: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f350: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f380: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f3a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f3f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f440: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f4e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f510: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f530: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f560: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f580: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f5d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f600: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f620: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f650: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f670: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f6c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f710: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f740: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f760: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f790: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f7b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f7e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f800: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f830: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f850: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f8a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f8f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f920: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f970: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f990: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004f9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004f9e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fa20: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ -0004fa30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fa00: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0004fa10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0004fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fa70: 2020 2020 2020 2020 2020 3020 2020 2020            0     
│ │ │ │ -0004fa80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fa50: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ +0004fa60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0004fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fac0: 2020 2020 2020 2020 2020 3132 3336 6133            1236a3
│ │ │ │ -0004fad0: 2b38 3932 3261 3262 2d33 3538 397c 0a7c  +8922a2b-3589|.|
│ │ │ │ -0004fae0: 3264 2b36 3034 3061 6364 2d38 3032 3262  2d+6040acd-8022b
│ │ │ │ -0004faf0: 6364 2d33 3936 3863 3264 2b33 3136 3461  cd-3968c2d+3164a
│ │ │ │ -0004fb00: 6432 2d32 3935 6264 322b 3736 3530 6364  d2-295bd2+7650cd
│ │ │ │ -0004fb10: 322d 3134 3338 3864 3320 3130 3337 3061  2-14388d3 10370a
│ │ │ │ -0004fb20: 6232 2d37 3039 3262 332d 3937 307c 0a7c  b2-7092b3-970|.|
│ │ │ │ +0004faa0: 2020 2020 2020 2020 2031 3233 3661 332b           1236a3+
│ │ │ │ +0004fab0: 3839 3232 6132 622d 3335 3839 7c0a 7c32  8922a2b-3589|.|2
│ │ │ │ +0004fac0: 642b 3630 3430 6163 642d 3830 3232 6263  d+6040acd-8022bc
│ │ │ │ +0004fad0: 642d 3339 3638 6332 642b 3331 3634 6164  d-3968c2d+3164ad
│ │ │ │ +0004fae0: 322d 3239 3562 6432 2b37 3635 3063 6432  2-295bd2+7650cd2
│ │ │ │ +0004faf0: 2d31 3433 3838 6433 2031 3033 3730 6162  -14388d3 10370ab
│ │ │ │ +0004fb00: 322d 3730 3932 6233 2d39 3730 7c0a 7c20  2-7092b3-970|.| 
│ │ │ │ +0004fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fb50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fb70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fbc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fbf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fc10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fc40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fc60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fc90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fcb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fce0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fd00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fd50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0004fd60: 642d 3135 3935 3761 3263 642b 3135 3239  d-15957a2cd+1529
│ │ │ │ -0004fd70: 3361 6263 642d 3138 3331 6232 6364 2b36  3abcd-1831b2cd+6
│ │ │ │ -0004fd80: 3034 3261 6332 642d 3235 3631 6263 3264  042ac2d-2561bc2d
│ │ │ │ -0004fd90: 2d38 3730 3961 3264 322d 3133 3231 3961  -8709a2d2-13219a
│ │ │ │ -0004fda0: 6264 322b 3432 3039 6232 6432 2b7c 0a7c  bd2+4209b2d2+|.|
│ │ │ │ +0004fd30: 2020 2020 2020 2020 2020 2020 7c0a 7c64              |.|d
│ │ │ │ +0004fd40: 2d31 3539 3537 6132 6364 2b31 3532 3933  -15957a2cd+15293
│ │ │ │ +0004fd50: 6162 6364 2d31 3833 3162 3263 642b 3630  abcd-1831b2cd+60
│ │ │ │ +0004fd60: 3432 6163 3264 2d32 3536 3162 6332 642d  42ac2d-2561bc2d-
│ │ │ │ +0004fd70: 3837 3039 6132 6432 2d31 3332 3139 6162  8709a2d2-13219ab
│ │ │ │ +0004fd80: 6432 2b34 3230 3962 3264 322b 7c0a 7c20  d2+4209b2d2+|.| 
│ │ │ │ +0004fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fdd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fdf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fe20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fe40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fe70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fe90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004fec0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004fee0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ff10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ff30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ff40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ff60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ff60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ff80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0004ffb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0004ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0004ffd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0004ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0004fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050000: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050020: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000500a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000500a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000500b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000500c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000500c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000500d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000500e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000500f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000500f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050110: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050140: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050160: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000501a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000501b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000501b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000501c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000501d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000501e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000501e0: 2020 2020 2030 2020 2020 2020 7c0a 7c64       0      |.|d
│ │ │ │  000501f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050200: 2020 2020 2020 3020 2020 2020 207c 0a7c        0      |.|
│ │ │ │ -00050210: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ +00050200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050230: 2020 2020 2030 2020 2020 2020 7c0a 7c20       0      |.| 
│ │ │ │  00050240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050250: 2020 2020 2020 3020 2020 2020 207c 0a7c        0      |.|
│ │ │ │ +00050250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000502a0: 2020 2020 2020 3130 3761 332b 347c 0a7c        107a3+4|.|
│ │ │ │ -000502b0: 3235 6162 642d 3535 3439 6232 642d 3930  25abd-5549b2d-90
│ │ │ │ -000502c0: 3333 6163 642d 3239 3938 6263 642b 3230  33acd-2998bcd+20
│ │ │ │ -000502d0: 3336 6332 642d 3531 3037 6164 322d 3536  36c2d-5107ad2-56
│ │ │ │ -000502e0: 3739 6264 322b 3633 3235 6364 322b 3131  79bd2+6325cd2+11
│ │ │ │ -000502f0: 3734 3064 3320 3832 3331 6162 327c 0a7c  740d3 8231ab2|.|
│ │ │ │ +00050280: 2020 2020 2031 3037 6133 2b34 7c0a 7c32       107a3+4|.|2
│ │ │ │ +00050290: 3561 6264 2d35 3534 3962 3264 2d39 3033  5abd-5549b2d-903
│ │ │ │ +000502a0: 3361 6364 2d32 3939 3862 6364 2b32 3033  3acd-2998bcd+203
│ │ │ │ +000502b0: 3663 3264 2d35 3130 3761 6432 2d35 3637  6c2d-5107ad2-567
│ │ │ │ +000502c0: 3962 6432 2b36 3332 3563 6432 2b31 3137  9bd2+6325cd2+117
│ │ │ │ +000502d0: 3430 6433 2038 3233 3161 6232 7c0a 7c20  40d3 8231ab2|.| 
│ │ │ │ +000502e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000502f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050320: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050340: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050370: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050390: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000503a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000503b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000503c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000503c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000503d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000503e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000503e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000503f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050410: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050430: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050460: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050480: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000504a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000504b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000504b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000504c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000504d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000504d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000504e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000504f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050520: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00050530: 6132 6264 2d31 3138 3630 6162 3264 2d31  a2bd-11860ab2d-1
│ │ │ │ -00050540: 3430 3534 6233 642b 3135 3239 3961 3263  4054b3d+15299a2c
│ │ │ │ -00050550: 642d 3731 3861 6263 642d 3134 3638 3562  d-718abcd-14685b
│ │ │ │ -00050560: 3263 642d 3130 3538 3161 6332 642b 3532  2cd-10581ac2d+52
│ │ │ │ -00050570: 3238 6263 3264 2d31 3538 3636 617c 0a7c  28bc2d-15866a|.|
│ │ │ │ +00050500: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +00050510: 3262 642d 3131 3836 3061 6232 642d 3134  2bd-11860ab2d-14
│ │ │ │ +00050520: 3035 3462 3364 2b31 3532 3939 6132 6364  054b3d+15299a2cd
│ │ │ │ +00050530: 2d37 3138 6162 6364 2d31 3436 3835 6232  -718abcd-14685b2
│ │ │ │ +00050540: 6364 2d31 3035 3831 6163 3264 2b35 3232  cd-10581ac2d+522
│ │ │ │ +00050550: 3862 6332 642d 3135 3836 3661 7c0a 7c20  8bc2d-15866a|.| 
│ │ │ │ +00050560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000505a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000505a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000505b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000505c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000505c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000505d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000505e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000505f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000505f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050660: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000506a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000506b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000506b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000506c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000506d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000506e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000506e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000506f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050700: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050750: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000507a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000507b0: 642b 3132 3735 3761 3262 642b 3132 3832  d+12757a2bd+1282
│ │ │ │ -000507c0: 3261 6232 642d 3131 3735 3262 3364 2d37  2ab2d-11752b3d-7
│ │ │ │ -000507d0: 3232 3061 3263 642d 3134 3139 6162 6364  220a2cd-1419abcd
│ │ │ │ -000507e0: 2d39 3839 3462 3263 642b 3834 3430 6163  -9894b2cd+8440ac
│ │ │ │ -000507f0: 3264 2b31 3036 3933 6263 3264 2b7c 0a7c  2d+10693bc2d+|.|
│ │ │ │ +00050780: 2020 2020 2020 2020 2020 2020 7c0a 7c64              |.|d
│ │ │ │ +00050790: 2b31 3237 3537 6132 6264 2b31 3238 3232  +12757a2bd+12822
│ │ │ │ +000507a0: 6162 3264 2d31 3137 3532 6233 642d 3732  ab2d-11752b3d-72
│ │ │ │ +000507b0: 3230 6132 6364 2d31 3431 3961 6263 642d  20a2cd-1419abcd-
│ │ │ │ +000507c0: 3938 3934 6232 6364 2b38 3434 3061 6332  9894b2cd+8440ac2
│ │ │ │ +000507d0: 642b 3130 3639 3362 6332 642b 7c0a 7c2d  d+10693bc2d+|.|-
│ │ │ │ +000507e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000507f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00050810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00050840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ +00050820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00050830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050890: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000508a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000508b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000508c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000508c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000508d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000508e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000508e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000508f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050910: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050930: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050960: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050980: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000509a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000509b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000509b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000509c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000509d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000509d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000509e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000509f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050a20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050a70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050b10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050b60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050bb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050c00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050c50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050d20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050d40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050d70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050d90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050de0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050e30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050e80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050ed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050f20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050f70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00050fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00050fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00050fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00050ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00050ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051010: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051060: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000510a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000510b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000510b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000510c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000510d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000510e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000510e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000510f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051130: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000511a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000511b0: 6162 322b 3839 3731 6233 2d35 3030 3661  ab2+8971b3-5006a
│ │ │ │ -000511c0: 3263 2d35 3539 3961 6263 2d31 3431 3635  2c-5599abc-14165
│ │ │ │ -000511d0: 6232 632b 3838 3830 6132 642b 3432 3539  b2c+8880a2d+4259
│ │ │ │ -000511e0: 6162 642d 3330 3032 6232 642d 3133 3839  abd-3002b2d-1389
│ │ │ │ -000511f0: 3261 6364 2d31 3035 3231 6263 647c 0a7c  2acd-10521bcd|.|
│ │ │ │ -00051200: 3261 6263 2d36 3632 3762 3263 2b38 3838  2abc-6627b2c+888
│ │ │ │ -00051210: 3661 6264 2b34 3730 3062 3264 2d31 3561  6abd+4700b2d-15a
│ │ │ │ -00051220: 6364 2b35 3936 3962 6364 2020 2020 2020  cd+5969bcd      
│ │ │ │ +00051180: 2020 2020 2020 2020 2020 2020 7c0a 7c61              |.|a
│ │ │ │ +00051190: 6232 2b38 3937 3162 332d 3530 3036 6132  b2+8971b3-5006a2
│ │ │ │ +000511a0: 632d 3535 3939 6162 632d 3134 3136 3562  c-5599abc-14165b
│ │ │ │ +000511b0: 3263 2b38 3838 3061 3264 2b34 3235 3961  2c+8880a2d+4259a
│ │ │ │ +000511c0: 6264 2d33 3030 3262 3264 2d31 3338 3932  bd-3002b2d-13892
│ │ │ │ +000511d0: 6163 642d 3130 3532 3162 6364 7c0a 7c32  acd-10521bcd|.|2
│ │ │ │ +000511e0: 6162 632d 3636 3237 6232 632b 3838 3836  abc-6627b2c+8886
│ │ │ │ +000511f0: 6162 642b 3437 3030 6232 642d 3135 6163  abd+4700b2d-15ac
│ │ │ │ +00051200: 642b 3539 3639 6263 6420 2020 2020 2020  d+5969bcd       
│ │ │ │ +00051210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051270: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051290: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000512a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000512b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000512c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000512c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000512d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000512e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000512e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000512f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051380: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000513a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000513b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000513b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000513c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000513d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000513d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000513e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000513f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051400: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051420: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051470: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00051480: 3132 3232 3561 6364 322d 3236 3035 6263  12225acd2-2605bc
│ │ │ │ -00051490: 6432 2d39 3263 3264 322b 3135 3936 3861  d2-92c2d2+15968a
│ │ │ │ -000514a0: 6433 2b31 3438 3630 6264 332d 3838 3239  d3+14860bd3-8829
│ │ │ │ -000514b0: 6364 332d 3131 3237 3464 3420 7c20 2020  cd3-11274d4 |   
│ │ │ │ -000514c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051450: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +00051460: 3232 3235 6163 6432 2d32 3630 3562 6364  2225acd2-2605bcd
│ │ │ │ +00051470: 322d 3932 6332 6432 2b31 3539 3638 6164  2-92c2d2+15968ad
│ │ │ │ +00051480: 332b 3134 3836 3062 6433 2d38 3832 3963  3+14860bd3-8829c
│ │ │ │ +00051490: 6433 2d31 3132 3734 6434 207c 2020 2020  d3-11274d4 |    
│ │ │ │ +000514a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000514b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000514c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000514d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000514e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000514f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000514f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051510: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051560: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051590: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000515a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000515b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000515b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000515c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000515d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000515e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000515e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000515f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051600: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051630: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051650: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000516a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000516a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000516b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000516c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000516d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000516d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000516e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000516f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000516f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051720: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051740: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051790: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000517a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000517b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000517c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000517c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000517d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000517e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000517e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000517f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051810: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051830: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051880: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000518a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000518b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000518b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000518c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000518d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000518d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000518e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000518f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051900: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051920: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051970: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00051980: 3337 3661 3262 2b33 3738 3361 6232 2b31  376a2b+3783ab2+1
│ │ │ │ -00051990: 3033 3539 6233 2d35 3537 3061 3263 2d35  0359b3-5570a2c-5
│ │ │ │ -000519a0: 3330 3761 6263 2d37 3436 3462 3263 2b33  307abc-7464b2c+3
│ │ │ │ -000519b0: 3138 3761 3264 2b38 3537 3061 6264 2d38  187a2d+8570abd-8
│ │ │ │ -000519c0: 3235 3162 3264 2b38 3434 3461 637c 0a7c  251b2d+8444ac|.|
│ │ │ │ -000519d0: 2b31 3331 3737 6233 2b35 3836 3461 6263  +13177b3+5864abc
│ │ │ │ -000519e0: 2b31 3339 3930 6232 632b 3530 3236 6162  +13990b2c+5026ab
│ │ │ │ -000519f0: 642d 3131 3532 3162 3264 2d37 3530 3161  d-11521b2d-7501a
│ │ │ │ -00051a00: 6364 2d31 3737 3962 6364 2020 2020 2020  cd-1779bcd      
│ │ │ │ -00051a10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051950: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +00051960: 3736 6132 622b 3337 3833 6162 322b 3130  76a2b+3783ab2+10
│ │ │ │ +00051970: 3335 3962 332d 3535 3730 6132 632d 3533  359b3-5570a2c-53
│ │ │ │ +00051980: 3037 6162 632d 3734 3634 6232 632b 3331  07abc-7464b2c+31
│ │ │ │ +00051990: 3837 6132 642b 3835 3730 6162 642d 3832  87a2d+8570abd-82
│ │ │ │ +000519a0: 3531 6232 642b 3834 3434 6163 7c0a 7c2b  51b2d+8444ac|.|+
│ │ │ │ +000519b0: 3133 3137 3762 332b 3538 3634 6162 632b  13177b3+5864abc+
│ │ │ │ +000519c0: 3133 3939 3062 3263 2b35 3032 3661 6264  13990b2c+5026abd
│ │ │ │ +000519d0: 2d31 3135 3231 6232 642d 3735 3031 6163  -11521b2d-7501ac
│ │ │ │ +000519e0: 642d 3137 3739 6263 6420 2020 2020 2020  d-1779bcd       
│ │ │ │ +000519f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00051a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051a60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051a90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051ab0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051ae0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051b00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051b30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051b50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051bd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051c40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00051c50: 3264 322d 3137 3933 6162 6432 2d31 3232  2d2-1793abd2-122
│ │ │ │ -00051c60: 3435 6232 6432 2b31 3232 3833 6163 6432  45b2d2+12283acd2
│ │ │ │ -00051c70: 2b36 3733 3062 6364 322b 3835 3237 6332  +6730bcd2+8527c2
│ │ │ │ -00051c80: 6432 2b35 3934 3261 6433 2b31 3439 3235  d2+5942ad3+14925
│ │ │ │ -00051c90: 6264 332d 3239 3663 6433 2b38 307c 0a7c  bd3-296cd3+80|.|
│ │ │ │ +00051c20: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +00051c30: 6432 2d31 3739 3361 6264 322d 3132 3234  d2-1793abd2-1224
│ │ │ │ +00051c40: 3562 3264 322b 3132 3238 3361 6364 322b  5b2d2+12283acd2+
│ │ │ │ +00051c50: 3637 3330 6263 6432 2b38 3532 3763 3264  6730bcd2+8527c2d
│ │ │ │ +00051c60: 322b 3539 3432 6164 332b 3134 3932 3562  2+5942ad3+14925b
│ │ │ │ +00051c70: 6433 2d32 3936 6364 332b 3830 7c0a 7c20  d3-296cd3+80|.| 
│ │ │ │ +00051c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051ce0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051d30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051d60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051d80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051db0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051e20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051e50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051e70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051ec0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00051ed0: 3132 3036 3261 3264 322d 3133 3632 3661  12062a2d2-13626a
│ │ │ │ -00051ee0: 6264 322b 3432 3437 6232 6432 2d31 3237  bd2+4247b2d2-127
│ │ │ │ -00051ef0: 3033 6163 6432 2b32 3935 3762 6364 322d  03acd2+2957bcd2-
│ │ │ │ -00051f00: 3533 3735 6332 6432 2d31 3033 3035 6164  5375c2d2-10305ad
│ │ │ │ -00051f10: 332b 3832 3937 6264 332b 3130 367c 0a7c  3+8297bd3+106|.|
│ │ │ │ +00051ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +00051eb0: 3230 3632 6132 6432 2d31 3336 3236 6162  2062a2d2-13626ab
│ │ │ │ +00051ec0: 6432 2b34 3234 3762 3264 322d 3132 3730  d2+4247b2d2-1270
│ │ │ │ +00051ed0: 3361 6364 322b 3239 3537 6263 6432 2d35  3acd2+2957bcd2-5
│ │ │ │ +00051ee0: 3337 3563 3264 322d 3130 3330 3561 6433  375c2d2-10305ad3
│ │ │ │ +00051ef0: 2b38 3239 3762 6433 2b31 3036 7c0a 7c2d  +8297bd3+106|.|-
│ │ │ │ +00051f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00051f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00051f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00051f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -00051f70: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ +00051f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00051f50: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │ +00051f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00051fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00051fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00051fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00051fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00051ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052030: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052080: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000520a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000520a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000520b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000520c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000520d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000520d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000520e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000520f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000520f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052120: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052170: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052190: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000521a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000521b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000521c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000521c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000521d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000521e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000521e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000521f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052210: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052230: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052260: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052280: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000522a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000522b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000522b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000522c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000522d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000522d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000522e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000522f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052300: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052320: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052350: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052370: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000523a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000523a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000523b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000523c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000523c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000523d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000523e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000523f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000523f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052410: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052440: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052490: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000524a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000524b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000524b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000524c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000524d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000524e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000524e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000524f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052530: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052580: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000525a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000525a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000525b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000525c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000525d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000525d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000525e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000525f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000525f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052620: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052640: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052670: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000526a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000526b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000526c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000526c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000526d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000526e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000526e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000526f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052710: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052760: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052780: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000527a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000527b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000527b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000527c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000527d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000527d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000527e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000527f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00052830: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00052800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052810: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052870: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00052880: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00052850: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052860: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000528a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000528b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000528c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000528d0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +000528a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000528b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000528c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000528d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000528e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000528f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052910: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00052920: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +000528f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052900: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00052910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052940: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052960: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052990: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000529a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000529b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000529b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000529c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000529d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000529e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000529e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000529f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052a00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052af0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052b40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052b70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052b90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052be0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052c30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052c80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052cd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052d70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052dc0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052e40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052e90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052f00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052f30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052f50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00052f80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00052f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052fa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00052fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00052fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00052ff0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00053000: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +00052fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00052fe0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00052ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053040: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00053050: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +00053020: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00053030: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +00053040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000530a0: 642b 3530 3731 6263 6420 7c20 2020 2020  d+5071bcd |     
│ │ │ │ +00053070: 2020 2020 2020 2020 2020 2020 7c0a 7c64              |.|d
│ │ │ │ +00053080: 2b35 3037 3162 6364 207c 2020 2020 2020  +5071bcd |      
│ │ │ │ +00053090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000530a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000530b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000530c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000530d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000530e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000530f0: 2020 2020 2020 2020 2020 7c20 2020 2020            |     
│ │ │ │ +000530c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000530d0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ +000530e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000530f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053110: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053130: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053160: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053180: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000531a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000531b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000531b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000531c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000531d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000531d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000531e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000531f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053200: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053220: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053250: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053270: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000532a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000532a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000532b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000532c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000532c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000532d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000532e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000532f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000532f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053310: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053360: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00053370: 3834 6434 207c 2020 2020 2020 2020 2020  84d4 |          
│ │ │ │ +00053340: 2020 2020 2020 2020 2020 2020 7c0a 7c38              |.|8
│ │ │ │ +00053350: 3464 3420 7c20 2020 2020 2020 2020 2020  4d4 |           
│ │ │ │ +00053360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053390: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000533a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000533b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000533b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000533c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000533d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000533e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000533e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000533f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053400: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053430: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000534a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000534a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000534b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000534c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000534d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000534d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000534e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000534f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000534f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053520: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053540: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053570: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00053580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053590: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00053590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000535a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000535b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000535c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000535d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000535e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000535f0: 3633 6364 332b 3633 3431 6434 207c 2020  63cd3+6341d4 |  
│ │ │ │ +000535c0: 2020 2020 2020 2020 2020 2020 7c0a 7c36              |.|6
│ │ │ │ +000535d0: 3363 6433 2b36 3334 3164 3420 7c20 2020  3cd3+6341d4 |   
│ │ │ │ +000535e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000535f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053630: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00053610: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00053620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00053690: 5765 2063 616e 2073 6565 2074 6861 7420  We can see that 
│ │ │ │ -000536a0: 616c 6c20 3620 706f 7465 6e74 6961 6c20  all 6 potential 
│ │ │ │ -000536b0: 6869 6768 6572 2068 6f6d 6f74 6f70 6965  higher homotopie
│ │ │ │ -000536c0: 7320 6172 6520 6e6f 6e74 7269 7669 616c  s are nontrivial
│ │ │ │ -000536d0: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :..+------------
│ │ │ │ +00053660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57  ------------+..W
│ │ │ │ +00053670: 6520 6361 6e20 7365 6520 7468 6174 2061  e can see that a
│ │ │ │ +00053680: 6c6c 2036 2070 6f74 656e 7469 616c 2068  ll 6 potential h
│ │ │ │ +00053690: 6967 6865 7220 686f 6d6f 746f 7069 6573  igher homotopies
│ │ │ │ +000536a0: 2061 7265 206e 6f6e 7472 6976 6961 6c3a   are nontrivial:
│ │ │ │ +000536b0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ +000536c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000536d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000536e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000536f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053720: 2d2b 0a7c 6931 3420 3a20 4c20 3d20 736f  -+.|i14 : L = so
│ │ │ │ -00053730: 7274 2073 656c 6563 7428 6b65 7973 2068  rt select(keys h
│ │ │ │ -00053740: 6f6d 6f74 2c20 6b2d 3e28 686f 6d6f 7423  omot, k->(homot#
│ │ │ │ -00053750: 6b21 3d30 2061 6e64 2073 756d 286b 5f30  k!=0 and sum(k_0
│ │ │ │ -00053760: 293e 3129 2920 2020 2020 2020 2020 2020  )>1))           
│ │ │ │ -00053770: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00053700: 2b0a 7c69 3134 203a 204c 203d 2073 6f72  +.|i14 : L = sor
│ │ │ │ +00053710: 7420 7365 6c65 6374 286b 6579 7320 686f  t select(keys ho
│ │ │ │ +00053720: 6d6f 742c 206b 2d3e 2868 6f6d 6f74 236b  mot, k->(homot#k
│ │ │ │ +00053730: 213d 3020 616e 6420 7375 6d28 6b5f 3029  !=0 and sum(k_0)
│ │ │ │ +00053740: 3e31 2929 2020 2020 2020 2020 2020 2020  >1))            
│ │ │ │ +00053750: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00053760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000537a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000537b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000537c0: 207c 0a7c 6f31 3420 3d20 7b7b 7b30 2c20   |.|o14 = {{{0, 
│ │ │ │ -000537d0: 327d 2c20 307d 2c20 7b7b 302c 2032 7d2c  2}, 0}, {{0, 2},
│ │ │ │ -000537e0: 2031 7d2c 207b 7b31 2c20 317d 2c20 307d   1}, {{1, 1}, 0}
│ │ │ │ -000537f0: 2c20 7b7b 312c 2031 7d2c 2031 7d2c 207b  , {{1, 1}, 1}, {
│ │ │ │ -00053800: 7b32 2c20 307d 2c20 307d 2c20 7b7b 322c  {2, 0}, 0}, {{2,
│ │ │ │ -00053810: 207c 0a7c 2020 2020 2020 2d2d 2d2d 2d2d   |.|      ------
│ │ │ │ +000537a0: 7c0a 7c6f 3134 203d 207b 7b7b 302c 2032  |.|o14 = {{{0, 2
│ │ │ │ +000537b0: 7d2c 2030 7d2c 207b 7b30 2c20 327d 2c20  }, 0}, {{0, 2}, 
│ │ │ │ +000537c0: 317d 2c20 7b7b 312c 2031 7d2c 2030 7d2c  1}, {{1, 1}, 0},
│ │ │ │ +000537d0: 207b 7b31 2c20 317d 2c20 317d 2c20 7b7b   {{1, 1}, 1}, {{
│ │ │ │ +000537e0: 322c 2030 7d2c 2030 7d2c 207b 7b32 2c20  2, 0}, 0}, {{2, 
│ │ │ │ +000537f0: 7c0a 7c20 2020 2020 202d 2d2d 2d2d 2d2d  |.|      -------
│ │ │ │ +00053800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053860: 2d7c 0a7c 2020 2020 2020 307d 2c20 317d  -|.|      0}, 1}
│ │ │ │ -00053870: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00053840: 7c0a 7c20 2020 2020 2030 7d2c 2031 7d7d  |.|      0}, 1}}
│ │ │ │ +00053850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053890: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000538a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000538b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000538b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000538c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000538d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000538e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000538e0: 7c0a 7c6f 3134 203a 204c 6973 7420 2020  |.|o14 : List   
│ │ │ │  000538f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053900: 207c 0a7c 6f31 3420 3a20 4c69 7374 2020   |.|o14 : List  
│ │ │ │ +00053900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053950: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00053930: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00053940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000539a0: 2d2b 0a7c 6931 3520 3a20 234c 2020 2020  -+.|i15 : #L    
│ │ │ │ +00053980: 2b0a 7c69 3135 203a 2023 4c20 2020 2020  +.|i15 : #L     
│ │ │ │ +00053990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000539a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000539b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000539c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000539d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000539d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000539e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000539f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000539f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053a20: 7c0a 7c6f 3135 203d 2036 2020 2020 2020  |.|o15 = 6      
│ │ │ │  00053a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a40: 207c 0a7c 6f31 3520 3d20 3620 2020 2020   |.|o15 = 6     
│ │ │ │ +00053a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053a90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00053a70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00053a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053ae0: 2d2b 0a7c 6931 3620 3a20 6e65 744c 6973  -+.|i16 : netLis
│ │ │ │ -00053af0: 7420 4c20 2020 2020 2020 2020 2020 2020  t L             
│ │ │ │ +00053ac0: 2b0a 7c69 3136 203a 206e 6574 4c69 7374  +.|i16 : netList
│ │ │ │ +00053ad0: 204c 2020 2020 2020 2020 2020 2020 2020   L              
│ │ │ │ +00053ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053b10: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00053b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00053b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053b80: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053b90: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053b60: 7c0a 7c20 2020 2020 202b 2d2d 2d2d 2d2d  |.|      +------
│ │ │ │ +00053b70: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ +00053b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053bd0: 207c 0a7c 6f31 3620 3d20 7c7b 302c 2032   |.|o16 = |{0, 2
│ │ │ │ -00053be0: 7d7c 307c 2020 2020 2020 2020 2020 2020  }|0|            
│ │ │ │ +00053bb0: 7c0a 7c6f 3136 203d 207c 7b30 2c20 327d  |.|o16 = |{0, 2}
│ │ │ │ +00053bc0: 7c30 7c20 2020 2020 2020 2020 2020 2020  |0|             
│ │ │ │ +00053bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c20: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053c30: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053c00: 7c0a 7c20 2020 2020 202b 2d2d 2d2d 2d2d  |.|      +------
│ │ │ │ +00053c10: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ +00053c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053c70: 207c 0a7c 2020 2020 2020 7c7b 302c 2032   |.|      |{0, 2
│ │ │ │ -00053c80: 7d7c 317c 2020 2020 2020 2020 2020 2020  }|1|            
│ │ │ │ +00053c50: 7c0a 7c20 2020 2020 207c 7b30 2c20 327d  |.|      |{0, 2}
│ │ │ │ +00053c60: 7c31 7c20 2020 2020 2020 2020 2020 2020  |1|             
│ │ │ │ +00053c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053cc0: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053cd0: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053ca0: 7c0a 7c20 2020 2020 202b 2d2d 2d2d 2d2d  |.|      +------
│ │ │ │ +00053cb0: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ +00053cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d10: 207c 0a7c 2020 2020 2020 7c7b 312c 2031   |.|      |{1, 1
│ │ │ │ -00053d20: 7d7c 307c 2020 2020 2020 2020 2020 2020  }|0|            
│ │ │ │ +00053cf0: 7c0a 7c20 2020 2020 207c 7b31 2c20 317d  |.|      |{1, 1}
│ │ │ │ +00053d00: 7c30 7c20 2020 2020 2020 2020 2020 2020  |0|             
│ │ │ │ +00053d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d60: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053d70: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053d40: 7c0a 7c20 2020 2020 202b 2d2d 2d2d 2d2d  |.|      +------
│ │ │ │ +00053d50: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ +00053d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053db0: 207c 0a7c 2020 2020 2020 7c7b 312c 2031   |.|      |{1, 1
│ │ │ │ -00053dc0: 7d7c 317c 2020 2020 2020 2020 2020 2020  }|1|            
│ │ │ │ +00053d90: 7c0a 7c20 2020 2020 207c 7b31 2c20 317d  |.|      |{1, 1}
│ │ │ │ +00053da0: 7c31 7c20 2020 2020 2020 2020 2020 2020  |1|             
│ │ │ │ +00053db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e00: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053e10: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053de0: 7c0a 7c20 2020 2020 202b 2d2d 2d2d 2d2d  |.|      +------
│ │ │ │ +00053df0: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ +00053e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e50: 207c 0a7c 2020 2020 2020 7c7b 322c 2030   |.|      |{2, 0
│ │ │ │ -00053e60: 7d7c 307c 2020 2020 2020 2020 2020 2020  }|0|            
│ │ │ │ +00053e30: 7c0a 7c20 2020 2020 207c 7b32 2c20 307d  |.|      |{2, 0}
│ │ │ │ +00053e40: 7c30 7c20 2020 2020 2020 2020 2020 2020  |0|             
│ │ │ │ +00053e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ea0: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053eb0: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053e80: 7c0a 7c20 2020 2020 202b 2d2d 2d2d 2d2d  |.|      +------
│ │ │ │ +00053e90: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ +00053ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053ef0: 207c 0a7c 2020 2020 2020 7c7b 322c 2030   |.|      |{2, 0
│ │ │ │ -00053f00: 7d7c 317c 2020 2020 2020 2020 2020 2020  }|1|            
│ │ │ │ +00053ed0: 7c0a 7c20 2020 2020 207c 7b32 2c20 307d  |.|      |{2, 0}
│ │ │ │ +00053ee0: 7c31 7c20 2020 2020 2020 2020 2020 2020  |1|             
│ │ │ │ +00053ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f40: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d   |.|      +-----
│ │ │ │ -00053f50: 2d2b 2d2b 2020 2020 2020 2020 2020 2020  -+-+            
│ │ │ │ +00053f20: 7c0a 7c20 2020 2020 202b 2d2d 2d2d 2d2d  |.|      +------
│ │ │ │ +00053f30: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ +00053f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00053f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00053f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00053f90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00053f70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00053f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00053fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00053fe0: 2d2b 0a0a 466f 7220 6578 616d 706c 6520  -+..For example 
│ │ │ │ -00053ff0: 7765 2068 6176 653a 0a0a 2b2d 2d2d 2d2d  we have:..+-----
│ │ │ │ +00053fc0: 2b0a 0a46 6f72 2065 7861 6d70 6c65 2077  +..For example w
│ │ │ │ +00053fd0: 6520 6861 7665 3a0a 0a2b 2d2d 2d2d 2d2d  e have:..+------
│ │ │ │ +00053fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00053ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054040: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a  --------+.|i17 :
│ │ │ │ -00054050: 2068 6f6d 6f74 2328 4c5f 3029 2020 2020   homot#(L_0)    
│ │ │ │ +00054020: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 3a20  -------+.|i17 : 
│ │ │ │ +00054030: 686f 6d6f 7423 284c 5f30 2920 2020 2020  homot#(L_0)     
│ │ │ │ +00054040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054070: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00054080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054090: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00054090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000540a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000540b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000540c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000540d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000540e0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ -000540f0: 207b 367d 207c 202d 3133 3739 3561 342b   {6} | -13795a4+
│ │ │ │ -00054100: 3230 3139 6133 622b 3133 3736 3961 3262  2019a3b+13769a2b
│ │ │ │ -00054110: 322b 3735 3836 6162 332b 3836 3439 6234  2+7586ab3+8649b4
│ │ │ │ -00054120: 2b36 3435 3461 3363 2d31 3031 3837 6132  +6454a3c-10187a2
│ │ │ │ -00054130: 6263 2d31 3738 3361 7c0a 7c20 2020 2020  bc-1783a|.|     
│ │ │ │ -00054140: 207b 367d 207c 2031 3131 3532 6134 2d31   {6} | 11152a4-1
│ │ │ │ -00054150: 3333 3661 3362 2b31 3138 3436 6132 6232  336a3b+11846a2b2
│ │ │ │ -00054160: 2b31 3032 3634 6162 332b 3631 3862 342d  +10264ab3+618b4-
│ │ │ │ -00054170: 3131 3035 3161 3363 2b31 3231 3239 6132  11051a3c+12129a2
│ │ │ │ -00054180: 6263 2b35 3932 3761 7c0a 7c20 2020 2020  bc+5927a|.|     
│ │ │ │ -00054190: 207b 367d 207c 202d 3633 3338 6134 2b31   {6} | -6338a4+1
│ │ │ │ -000541a0: 3030 3235 6133 622b 3134 3938 3761 3363  0025a3b+14987a3c
│ │ │ │ -000541b0: 2d39 3935 3961 3262 632d 3131 3639 3161  -9959a2bc-11691a
│ │ │ │ -000541c0: 3263 322b 3132 3333 3661 6263 322d 3737  2c2+12336abc2-77
│ │ │ │ -000541d0: 3836 6133 642d 3131 7c0a 7c20 2020 2020  86a3d-11|.|     
│ │ │ │ -000541e0: 207b 367d 207c 2032 3237 3561 342d 3233   {6} | 2275a4-23
│ │ │ │ -000541f0: 3961 3362 2b31 3435 3934 6132 6232 2d38  9a3b+14594a2b2-8
│ │ │ │ -00054200: 3135 3361 6233 2d31 3139 3435 6234 2d38  153ab3-11945b4-8
│ │ │ │ -00054210: 3431 3661 3363 2b36 3235 3161 3262 632d  416a3c+6251a2bc-
│ │ │ │ -00054220: 3330 3233 6162 3263 7c0a 7c20 2020 2020  3023ab2c|.|     
│ │ │ │ -00054230: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +000540c0: 2020 2020 2020 207c 0a7c 6f31 3720 3d20         |.|o17 = 
│ │ │ │ +000540d0: 7b36 7d20 7c20 2d31 3337 3935 6134 2b32  {6} | -13795a4+2
│ │ │ │ +000540e0: 3031 3961 3362 2b31 3337 3639 6132 6232  019a3b+13769a2b2
│ │ │ │ +000540f0: 2b37 3538 3661 6233 2b38 3634 3962 342b  +7586ab3+8649b4+
│ │ │ │ +00054100: 3634 3534 6133 632d 3130 3138 3761 3262  6454a3c-10187a2b
│ │ │ │ +00054110: 632d 3137 3833 617c 0a7c 2020 2020 2020  c-1783a|.|      
│ │ │ │ +00054120: 7b36 7d20 7c20 3131 3135 3261 342d 3133  {6} | 11152a4-13
│ │ │ │ +00054130: 3336 6133 622b 3131 3834 3661 3262 322b  36a3b+11846a2b2+
│ │ │ │ +00054140: 3130 3236 3461 6233 2b36 3138 6234 2d31  10264ab3+618b4-1
│ │ │ │ +00054150: 3130 3531 6133 632b 3132 3132 3961 3262  1051a3c+12129a2b
│ │ │ │ +00054160: 632b 3539 3237 617c 0a7c 2020 2020 2020  c+5927a|.|      
│ │ │ │ +00054170: 7b36 7d20 7c20 2d36 3333 3861 342b 3130  {6} | -6338a4+10
│ │ │ │ +00054180: 3032 3561 3362 2b31 3439 3837 6133 632d  025a3b+14987a3c-
│ │ │ │ +00054190: 3939 3539 6132 6263 2d31 3136 3931 6132  9959a2bc-11691a2
│ │ │ │ +000541a0: 6332 2b31 3233 3336 6162 6332 2d37 3738  c2+12336abc2-778
│ │ │ │ +000541b0: 3661 3364 2d31 317c 0a7c 2020 2020 2020  6a3d-11|.|      
│ │ │ │ +000541c0: 7b36 7d20 7c20 3232 3735 6134 2d32 3339  {6} | 2275a4-239
│ │ │ │ +000541d0: 6133 622b 3134 3539 3461 3262 322d 3831  a3b+14594a2b2-81
│ │ │ │ +000541e0: 3533 6162 332d 3131 3934 3562 342d 3834  53ab3-11945b4-84
│ │ │ │ +000541f0: 3136 6133 632b 3632 3531 6132 6263 2d33  16a3c+6251a2bc-3
│ │ │ │ +00054200: 3032 3361 6232 637c 0a7c 2020 2020 2020  023ab2c|.|      
│ │ │ │ +00054210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054270: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00054280: 2062 3263 2b39 3231 3962 3363 2b35 3531   b2c+9219b3c+551
│ │ │ │ -00054290: 3361 3263 322b 3130 3535 3861 6263 322b  3a2c2+10558abc2+
│ │ │ │ -000542a0: 3235 3930 6232 6332 2b31 3136 3234 6133  2590b2c2+11624a3
│ │ │ │ -000542b0: 642d 3536 3033 6132 6264 2b31 3430 3538  d-5603a2bd+14058
│ │ │ │ -000542c0: 6162 3264 2d31 3236 7c0a 7c20 2020 2020  ab2d-126|.|     
│ │ │ │ -000542d0: 2062 3263 2b34 3839 6233 632d 3135 3338   b2c+489b3c-1538
│ │ │ │ -000542e0: 3361 3263 322b 3530 3761 6263 322d 3133  3a2c2+507abc2-13
│ │ │ │ -000542f0: 3830 3462 3263 322d 3834 3136 6163 332b  804b2c2-8416ac3+
│ │ │ │ -00054300: 3932 6334 2d31 3130 3537 6133 642d 3531  92c4-11057a3d-51
│ │ │ │ -00054310: 3133 6132 6264 2d32 7c0a 7c20 2020 2020  13a2bd-2|.|     
│ │ │ │ -00054320: 2035 3661 3262 642b 3439 3630 6132 6364   56a2bd+4960a2cd
│ │ │ │ -00054330: 2d35 3538 3961 6263 642d 3831 3633 6163  -5589abcd-8163ac
│ │ │ │ -00054340: 3264 2d31 3839 3562 6332 642b 3934 3634  2d-1895bc2d+9464
│ │ │ │ -00054350: 6132 6432 2d37 3235 3361 6264 322b 3132  a2d2-7253abd2+12
│ │ │ │ -00054360: 3634 3261 6364 322d 7c0a 7c20 2020 2020  642acd2-|.|     
│ │ │ │ -00054370: 202b 3539 3333 6233 632b 3932 6132 6332   +5933b3c+92a2c2
│ │ │ │ -00054380: 2b35 3334 3361 6263 322b 3337 3938 6232  +5343abc2+3798b2
│ │ │ │ -00054390: 6332 2d31 3539 3638 6133 642b 3437 3361  c2-15968a3d+473a
│ │ │ │ -000543a0: 3262 642b 3133 3239 3361 6232 642d 3337  2bd+13293ab2d-37
│ │ │ │ -000543b0: 3631 6233 642d 3737 7c0a 7c20 2020 2020  61b3d-77|.|     
│ │ │ │ -000543c0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +00054250: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ +00054260: 6232 632b 3932 3139 6233 632b 3535 3133  b2c+9219b3c+5513
│ │ │ │ +00054270: 6132 6332 2b31 3035 3538 6162 6332 2b32  a2c2+10558abc2+2
│ │ │ │ +00054280: 3539 3062 3263 322b 3131 3632 3461 3364  590b2c2+11624a3d
│ │ │ │ +00054290: 2d35 3630 3361 3262 642b 3134 3035 3861  -5603a2bd+14058a
│ │ │ │ +000542a0: 6232 642d 3132 367c 0a7c 2020 2020 2020  b2d-126|.|      
│ │ │ │ +000542b0: 6232 632b 3438 3962 3363 2d31 3533 3833  b2c+489b3c-15383
│ │ │ │ +000542c0: 6132 6332 2b35 3037 6162 6332 2d31 3338  a2c2+507abc2-138
│ │ │ │ +000542d0: 3034 6232 6332 2d38 3431 3661 6333 2b39  04b2c2-8416ac3+9
│ │ │ │ +000542e0: 3263 342d 3131 3035 3761 3364 2d35 3131  2c4-11057a3d-511
│ │ │ │ +000542f0: 3361 3262 642d 327c 0a7c 2020 2020 2020  3a2bd-2|.|      
│ │ │ │ +00054300: 3536 6132 6264 2b34 3936 3061 3263 642d  56a2bd+4960a2cd-
│ │ │ │ +00054310: 3535 3839 6162 6364 2d38 3136 3361 6332  5589abcd-8163ac2
│ │ │ │ +00054320: 642d 3138 3935 6263 3264 2b39 3436 3461  d-1895bc2d+9464a
│ │ │ │ +00054330: 3264 322d 3732 3533 6162 6432 2b31 3236  2d2-7253abd2+126
│ │ │ │ +00054340: 3432 6163 6432 2d7c 0a7c 2020 2020 2020  42acd2-|.|      
│ │ │ │ +00054350: 2b35 3933 3362 3363 2b39 3261 3263 322b  +5933b3c+92a2c2+
│ │ │ │ +00054360: 3533 3433 6162 6332 2b33 3739 3862 3263  5343abc2+3798b2c
│ │ │ │ +00054370: 322d 3135 3936 3861 3364 2b34 3733 6132  2-15968a3d+473a2
│ │ │ │ +00054380: 6264 2b31 3332 3933 6162 3264 2d33 3736  bd+13293ab2d-376
│ │ │ │ +00054390: 3162 3364 2d37 377c 0a7c 2020 2020 2020  1b3d-77|.|      
│ │ │ │ +000543a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000543b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000543c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000543d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000543e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000543f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054400: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -00054410: 2031 3562 3364 2b37 3836 3961 3263 642d   15b3d+7869a2cd-
│ │ │ │ -00054420: 3230 3532 6162 6364 2d31 3833 3162 3263  2052abcd-1831b2c
│ │ │ │ -00054430: 642b 3630 3432 6163 3264 2d32 3536 3162  d+6042ac2d-2561b
│ │ │ │ -00054440: 6332 642d 3837 3039 6132 6432 2d31 3332  c2d-8709a2d2-132
│ │ │ │ -00054450: 3139 6162 6432 2b34 7c0a 7c20 2020 2020  19abd2+4|.|     
│ │ │ │ -00054460: 2037 3632 6162 3264 2b31 3430 3935 6233   762ab2d+14095b3
│ │ │ │ -00054470: 642d 3135 3838 6132 6364 2b32 3030 3061  d-1588a2cd+2000a
│ │ │ │ -00054480: 6263 642d 3230 3830 6232 6364 2b39 3137  bcd-2080b2cd+917
│ │ │ │ -00054490: 3561 6332 642d 3634 3962 6332 642b 3838  5ac2d-649bc2d+88
│ │ │ │ -000544a0: 3239 6333 642b 3231 7c0a 7c20 2020 2020  29c3d+21|.|     
│ │ │ │ -000544b0: 2031 3935 3862 6364 3220 2020 2020 2020   1958bcd2       
│ │ │ │ +000543e0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ +000543f0: 3135 6233 642b 3738 3639 6132 6364 2d32  15b3d+7869a2cd-2
│ │ │ │ +00054400: 3035 3261 6263 642d 3138 3331 6232 6364  052abcd-1831b2cd
│ │ │ │ +00054410: 2b36 3034 3261 6332 642d 3235 3631 6263  +6042ac2d-2561bc
│ │ │ │ +00054420: 3264 2d38 3730 3961 3264 322d 3133 3231  2d-8709a2d2-1321
│ │ │ │ +00054430: 3961 6264 322b 347c 0a7c 2020 2020 2020  9abd2+4|.|      
│ │ │ │ +00054440: 3736 3261 6232 642b 3134 3039 3562 3364  762ab2d+14095b3d
│ │ │ │ +00054450: 2d31 3538 3861 3263 642b 3230 3030 6162  -1588a2cd+2000ab
│ │ │ │ +00054460: 6364 2d32 3038 3062 3263 642b 3931 3735  cd-2080b2cd+9175
│ │ │ │ +00054470: 6163 3264 2d36 3439 6263 3264 2b38 3832  ac2d-649bc2d+882
│ │ │ │ +00054480: 3963 3364 2b32 317c 0a7c 2020 2020 2020  9c3d+21|.|      
│ │ │ │ +00054490: 3139 3538 6263 6432 2020 2020 2020 2020  1958bcd2        
│ │ │ │ +000544a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000544b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000544c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000544d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000544e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000544f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054500: 2031 3761 3263 642d 3733 3839 6162 6364   17a2cd-7389abcd
│ │ │ │ -00054510: 2b34 3732 3362 3263 642d 3133 3236 3261  +4723b2cd-13262a
│ │ │ │ -00054520: 6332 642b 3534 3331 6263 3264 2b31 3132  c2d+5431bc2d+112
│ │ │ │ -00054530: 3734 6132 6432 2d32 3137 6162 6432 2b31  74a2d2-217abd2+1
│ │ │ │ -00054540: 3236 3162 3264 322b 7c0a 7c20 2020 2020  261b2d2+|.|     
│ │ │ │ -00054550: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +000544d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000544e0: 3137 6132 6364 2d37 3338 3961 6263 642b  17a2cd-7389abcd+
│ │ │ │ +000544f0: 3437 3233 6232 6364 2d31 3332 3632 6163  4723b2cd-13262ac
│ │ │ │ +00054500: 3264 2b35 3433 3162 6332 642b 3131 3237  2d+5431bc2d+1127
│ │ │ │ +00054510: 3461 3264 322d 3231 3761 6264 322b 3132  4a2d2-217abd2+12
│ │ │ │ +00054520: 3631 6232 6432 2b7c 0a7c 2020 2020 2020  61b2d2+|.|      
│ │ │ │ +00054530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054590: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020  --------|.|     
│ │ │ │ -000545a0: 2032 3039 6232 6432 2b31 3232 3235 6163   209b2d2+12225ac
│ │ │ │ -000545b0: 6432 2d32 3630 3562 6364 322d 3932 6332  d2-2605bcd2-92c2
│ │ │ │ -000545c0: 6432 2b31 3539 3638 6164 332b 3134 3836  d2+15968ad3+1486
│ │ │ │ -000545d0: 3062 6433 2d38 3832 3963 6433 2d31 3132  0bd3-8829cd3-112
│ │ │ │ -000545e0: 3734 6434 207c 2020 7c0a 7c20 2020 2020  74d4 |  |.|     
│ │ │ │ -000545f0: 2036 3461 3264 322b 3836 3335 6162 6432   64a2d2+8635abd2
│ │ │ │ -00054600: 2d37 3136 3162 3264 322b 3939 3761 6364  -7161b2d2+997acd
│ │ │ │ -00054610: 322b 3330 3135 6263 6432 2b31 3132 3734  2+3015bcd2+11274
│ │ │ │ -00054620: 6332 6432 2020 2020 2020 2020 2020 2020  c2d2            
│ │ │ │ -00054630: 2020 2020 207c 2020 7c0a 7c20 2020 2020       |  |.|     
│ │ │ │ +00054570: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ +00054580: 3230 3962 3264 322b 3132 3232 3561 6364  209b2d2+12225acd
│ │ │ │ +00054590: 322d 3236 3035 6263 6432 2d39 3263 3264  2-2605bcd2-92c2d
│ │ │ │ +000545a0: 322b 3135 3936 3861 6433 2b31 3438 3630  2+15968ad3+14860
│ │ │ │ +000545b0: 6264 332d 3838 3239 6364 332d 3131 3237  bd3-8829cd3-1127
│ │ │ │ +000545c0: 3464 3420 7c20 207c 0a7c 2020 2020 2020  4d4 |  |.|      
│ │ │ │ +000545d0: 3634 6132 6432 2b38 3633 3561 6264 322d  64a2d2+8635abd2-
│ │ │ │ +000545e0: 3731 3631 6232 6432 2b39 3937 6163 6432  7161b2d2+997acd2
│ │ │ │ +000545f0: 2b33 3031 3562 6364 322b 3131 3237 3463  +3015bcd2+11274c
│ │ │ │ +00054600: 3264 3220 2020 2020 2020 2020 2020 2020  2d2             
│ │ │ │ +00054610: 2020 2020 7c20 207c 0a7c 2020 2020 2020      |  |.|      
│ │ │ │ +00054620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054680: 2020 2020 207c 2020 7c0a 7c20 2020 2020       |  |.|     
│ │ │ │ -00054690: 2038 3230 3161 6364 322d 3134 3038 3062   8201acd2-14080b
│ │ │ │ -000546a0: 6364 3220 2020 2020 2020 2020 2020 2020  cd2             
│ │ │ │ -000546b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054660: 2020 2020 7c20 207c 0a7c 2020 2020 2020      |  |.|      
│ │ │ │ +00054670: 3832 3031 6163 6432 2d31 3430 3830 6263  8201acd2-14080bc
│ │ │ │ +00054680: 6432 2020 2020 2020 2020 2020 2020 2020  d2              
│ │ │ │ +00054690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000546a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000546b0: 2020 2020 7c20 207c 0a7c 2020 2020 2020      |  |.|      
│ │ │ │  000546c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000546d0: 2020 2020 207c 2020 7c0a 7c20 2020 2020       |  |.|     
│ │ │ │ +000546d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000546e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000546f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054720: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00054730: 2020 2020 2020 2020 2034 2020 2020 2020           4      
│ │ │ │ -00054740: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00054750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054770: 2020 2020 2020 2020 7c0a 7c6f 3137 203a          |.|o17 :
│ │ │ │ -00054780: 204d 6174 7269 7820 5320 203c 2d2d 2053   Matrix S  <-- S
│ │ │ │ +00054700: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00054710: 2020 2020 2020 2020 3420 2020 2020 2031          4      1
│ │ │ │ +00054720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054750: 2020 2020 2020 207c 0a7c 6f31 3720 3a20         |.|o17 : 
│ │ │ │ +00054760: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +00054770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00054790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000547a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000547b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000547c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000547a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000547b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000547c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000547d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000547e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000547f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054810: 2d2d 2d2d 2d2d 2d2d 2b0a 0a42 7574 2061  --------+..But a
│ │ │ │ -00054820: 6c6c 2074 6865 2068 6f6d 6f74 6f70 6965  ll the homotopie
│ │ │ │ -00054830: 7320 6172 6520 6d69 6e69 6d61 6c20 696e  s are minimal in
│ │ │ │ -00054840: 2074 6869 7320 6361 7365 3a0a 0a2b 2d2d   this case:..+--
│ │ │ │ -00054850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054870: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138  ----------+.|i18
│ │ │ │ -00054880: 203a 206b 3120 3d20 535e 312f 2869 6465   : k1 = S^1/(ide
│ │ │ │ -00054890: 616c 2076 6172 7320 5329 3b20 2020 2020  al vars S);     
│ │ │ │ -000548a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000548b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000548c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000548d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a  --------+.|i19 :
│ │ │ │ -000548e0: 2073 656c 6563 7428 6b65 7973 2068 6f6d   select(keys hom
│ │ │ │ -000548f0: 6f74 2c6b 2d3e 286b 312a 2a68 6f6d 6f74  ot,k->(k1**homot
│ │ │ │ -00054900: 236b 2921 3d30 297c 0a7c 2020 2020 2020  #k)!=0)|.|      
│ │ │ │ -00054910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000547f0: 2d2d 2d2d 2d2d 2d2b 0a0a 4275 7420 616c  -------+..But al
│ │ │ │ +00054800: 6c20 7468 6520 686f 6d6f 746f 7069 6573  l the homotopies
│ │ │ │ +00054810: 2061 7265 206d 696e 696d 616c 2069 6e20   are minimal in 
│ │ │ │ +00054820: 7468 6973 2063 6173 653a 0a0a 2b2d 2d2d  this case:..+---
│ │ │ │ +00054830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054850: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820  ---------+.|i18 
│ │ │ │ +00054860: 3a20 6b31 203d 2053 5e31 2f28 6964 6561  : k1 = S^1/(idea
│ │ │ │ +00054870: 6c20 7661 7273 2053 293b 2020 2020 2020  l vars S);      
│ │ │ │ +00054880: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +00054890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000548a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000548b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20  -------+.|i19 : 
│ │ │ │ +000548c0: 7365 6c65 6374 286b 6579 7320 686f 6d6f  select(keys homo
│ │ │ │ +000548d0: 742c 6b2d 3e28 6b31 2a2a 686f 6d6f 7423  t,k->(k1**homot#
│ │ │ │ +000548e0: 6b29 213d 3029 7c0a 7c20 2020 2020 2020  k)!=0)|.|       
│ │ │ │ +000548f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054910: 2020 2020 207c 0a7c 6f31 3920 3d20 7b7d       |.|o19 = {}
│ │ │ │  00054920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054930: 2020 2020 2020 7c0a 7c6f 3139 203d 207b        |.|o19 = {
│ │ │ │ -00054940: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00054930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00054950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054960: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00054970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00054970: 2020 207c 0a7c 6f31 3920 3a20 4c69 7374     |.|o19 : List
│ │ │ │  00054980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00054990: 2020 2020 7c0a 7c6f 3139 203a 204c 6973      |.|o19 : Lis
│ │ │ │ -000549a0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ -000549b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000549c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -000549d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000549e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000549f0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d  --+..See also.==
│ │ │ │ -00054a00: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74  ======..  * *not
│ │ │ │ -00054a10: 6520 6d61 6b65 486f 6d6f 746f 7069 6573  e makeHomotopies
│ │ │ │ -00054a20: 313a 206d 616b 6548 6f6d 6f74 6f70 6965  1: makeHomotopie
│ │ │ │ -00054a30: 7331 2c20 2d2d 2072 6574 7572 6e73 2061  s1, -- returns a
│ │ │ │ -00054a40: 2073 7973 7465 6d20 6f66 2066 6972 7374   system of first
│ │ │ │ -00054a50: 0a20 2020 2068 6f6d 6f74 6f70 6965 730a  .    homotopies.
│ │ │ │ -00054a60: 0a57 6179 7320 746f 2075 7365 206d 616b  .Ways to use mak
│ │ │ │ -00054a70: 6548 6f6d 6f74 6f70 6965 733a 0a3d 3d3d  eHomotopies:.===
│ │ │ │ -00054a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00054a90: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d  ========..  * "m
│ │ │ │ -00054aa0: 616b 6548 6f6d 6f74 6f70 6965 7328 4d61  akeHomotopies(Ma
│ │ │ │ -00054ab0: 7472 6978 2c43 6f6d 706c 6578 2922 0a20  trix,Complex)". 
│ │ │ │ -00054ac0: 202a 2022 6d61 6b65 486f 6d6f 746f 7069   * "makeHomotopi
│ │ │ │ -00054ad0: 6573 284d 6174 7269 782c 436f 6d70 6c65  es(Matrix,Comple
│ │ │ │ -00054ae0: 782c 5a5a 2922 0a0a 466f 7220 7468 6520  x,ZZ)"..For the 
│ │ │ │ -00054af0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00054b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -00054b10: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -00054b20: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20  makeHomotopies: 
│ │ │ │ -00054b30: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20  makeHomotopies, 
│ │ │ │ -00054b40: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -00054b50: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -00054b60: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -00054b70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +00054990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000549a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000549b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000549c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000549d0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d  -+..See also.===
│ │ │ │ +000549e0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ +000549f0: 206d 616b 6548 6f6d 6f74 6f70 6965 7331   makeHomotopies1
│ │ │ │ +00054a00: 3a20 6d61 6b65 486f 6d6f 746f 7069 6573  : makeHomotopies
│ │ │ │ +00054a10: 312c 202d 2d20 7265 7475 726e 7320 6120  1, -- returns a 
│ │ │ │ +00054a20: 7379 7374 656d 206f 6620 6669 7273 740a  system of first.
│ │ │ │ +00054a30: 2020 2020 686f 6d6f 746f 7069 6573 0a0a      homotopies..
│ │ │ │ +00054a40: 5761 7973 2074 6f20 7573 6520 6d61 6b65  Ways to use make
│ │ │ │ +00054a50: 486f 6d6f 746f 7069 6573 3a0a 3d3d 3d3d  Homotopies:.====
│ │ │ │ +00054a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00054a70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6d61  =======..  * "ma
│ │ │ │ +00054a80: 6b65 486f 6d6f 746f 7069 6573 284d 6174  keHomotopies(Mat
│ │ │ │ +00054a90: 7269 782c 436f 6d70 6c65 7829 220a 2020  rix,Complex)".  
│ │ │ │ +00054aa0: 2a20 226d 616b 6548 6f6d 6f74 6f70 6965  * "makeHomotopie
│ │ │ │ +00054ab0: 7328 4d61 7472 6978 2c43 6f6d 706c 6578  s(Matrix,Complex
│ │ │ │ +00054ac0: 2c5a 5a29 220a 0a46 6f72 2074 6865 2070  ,ZZ)"..For the p
│ │ │ │ +00054ad0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +00054ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00054af0: 6520 6f62 6a65 6374 202a 6e6f 7465 206d  e object *note m
│ │ │ │ +00054b00: 616b 6548 6f6d 6f74 6f70 6965 733a 206d  akeHomotopies: m
│ │ │ │ +00054b10: 616b 6548 6f6d 6f74 6f70 6965 732c 2069  akeHomotopies, i
│ │ │ │ +00054b20: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +00054b30: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ +00054b40: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +00054b50: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d  unction,...-----
│ │ │ │ +00054b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00054b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00054b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00054bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -00054bd0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ -00054be0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ -00054bf0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ -00054c00: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -00054c10: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ -00054c20: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -00054c30: 6573 2f0a 436f 6d70 6c65 7465 496e 7465  es/.CompleteInte
│ │ │ │ -00054c40: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -00054c50: 6f6e 732e 6d32 3a33 3737 313a 302e 0a1f  ons.m2:3771:0...
│ │ │ │ -00054c60: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ -00054c70: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -00054c80: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ -00054c90: 653a 206d 616b 6548 6f6d 6f74 6f70 6965  e: makeHomotopie
│ │ │ │ -00054ca0: 7331 2c20 4e65 7874 3a20 6d61 6b65 486f  s1, Next: makeHo
│ │ │ │ -00054cb0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f  motopiesOnHomolo
│ │ │ │ -00054cc0: 6779 2c20 5072 6576 3a20 6d61 6b65 486f  gy, Prev: makeHo
│ │ │ │ -00054cd0: 6d6f 746f 7069 6573 2c20 5570 3a20 546f  motopies, Up: To
│ │ │ │ -00054ce0: 700a 0a6d 616b 6548 6f6d 6f74 6f70 6965  p..makeHomotopie
│ │ │ │ -00054cf0: 7331 202d 2d20 7265 7475 726e 7320 6120  s1 -- returns a 
│ │ │ │ -00054d00: 7379 7374 656d 206f 6620 6669 7273 7420  system of first 
│ │ │ │ -00054d10: 686f 6d6f 746f 7069 6573 0a2a 2a2a 2a2a  homotopies.*****
│ │ │ │ +00054ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ +00054bb0: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ +00054bc0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ +00054bd0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +00054be0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +00054bf0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ +00054c00: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +00054c10: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ +00054c20: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +00054c30: 6e73 2e6d 323a 3337 3731 3a30 2e0a 1f0a  ns.m2:3771:0....
│ │ │ │ +00054c40: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ +00054c50: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +00054c60: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ +00054c70: 3a20 6d61 6b65 486f 6d6f 746f 7069 6573  : makeHomotopies
│ │ │ │ +00054c80: 312c 204e 6578 743a 206d 616b 6548 6f6d  1, Next: makeHom
│ │ │ │ +00054c90: 6f74 6f70 6965 734f 6e48 6f6d 6f6c 6f67  otopiesOnHomolog
│ │ │ │ +00054ca0: 792c 2050 7265 763a 206d 616b 6548 6f6d  y, Prev: makeHom
│ │ │ │ +00054cb0: 6f74 6f70 6965 732c 2055 703a 2054 6f70  otopies, Up: Top
│ │ │ │ +00054cc0: 0a0a 6d61 6b65 486f 6d6f 746f 7069 6573  ..makeHomotopies
│ │ │ │ +00054cd0: 3120 2d2d 2072 6574 7572 6e73 2061 2073  1 -- returns a s
│ │ │ │ +00054ce0: 7973 7465 6d20 6f66 2066 6972 7374 2068  ystem of first h
│ │ │ │ +00054cf0: 6f6d 6f74 6f70 6965 730a 2a2a 2a2a 2a2a  omotopies.******
│ │ │ │ +00054d00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00054d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00054d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054d30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054d40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00054d50: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ -00054d60: 2020 2020 2020 2020 4820 3d20 6d61 6b65          H = make
│ │ │ │ -00054d70: 486f 6d6f 746f 7069 6573 3128 662c 462c  Homotopies1(f,F,
│ │ │ │ -00054d80: 6429 0a20 202a 2049 6e70 7574 733a 0a20  d).  * Inputs:. 
│ │ │ │ -00054d90: 2020 2020 202a 2066 2c20 6120 2a6e 6f74       * f, a *not
│ │ │ │ -00054da0: 6520 6d61 7472 6978 3a20 284d 6163 6175  e matrix: (Macau
│ │ │ │ -00054db0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c  lay2Doc)Matrix,,
│ │ │ │ -00054dc0: 2031 786e 206d 6174 7269 7820 6f66 2065   1xn matrix of e
│ │ │ │ -00054dd0: 6c65 6d65 6e74 7320 6f66 2053 0a20 2020  lements of S.   
│ │ │ │ -00054de0: 2020 202a 2046 2c20 6120 2a6e 6f74 6520     * F, a *note 
│ │ │ │ -00054df0: 636f 6d70 6c65 783a 2028 436f 6d70 6c65  complex: (Comple
│ │ │ │ -00054e00: 7865 7329 436f 6d70 6c65 782c 2c20 6164  xes)Complex,, ad
│ │ │ │ -00054e10: 6d69 7474 696e 6720 686f 6d6f 746f 7069  mitting homotopi
│ │ │ │ -00054e20: 6573 2066 6f72 2074 6865 0a20 2020 2020  es for the.     
│ │ │ │ -00054e30: 2020 2065 6e74 7269 6573 206f 6620 660a     entries of f.
│ │ │ │ -00054e40: 2020 2020 2020 2a20 642c 2061 6e20 2a6e        * d, an *n
│ │ │ │ -00054e50: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -00054e60: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -00054e70: 686f 7720 6661 7220 6261 636b 2074 6f20  how far back to 
│ │ │ │ -00054e80: 636f 6d70 7574 6520 7468 650a 2020 2020  compute the.    
│ │ │ │ -00054e90: 2020 2020 686f 6d6f 746f 7069 6573 2028      homotopies (
│ │ │ │ -00054ea0: 6465 6661 756c 7473 2074 6f20 6c65 6e67  defaults to leng
│ │ │ │ -00054eb0: 7468 206f 6620 4629 0a20 202a 204f 7574  th of F).  * Out
│ │ │ │ -00054ec0: 7075 7473 3a0a 2020 2020 2020 2a20 482c  puts:.      * H,
│ │ │ │ -00054ed0: 2061 202a 6e6f 7465 2068 6173 6820 7461   a *note hash ta
│ │ │ │ -00054ee0: 626c 653a 2028 4d61 6361 756c 6179 3244  ble: (Macaulay2D
│ │ │ │ -00054ef0: 6f63 2948 6173 6854 6162 6c65 2c2c 2067  oc)HashTable,, g
│ │ │ │ -00054f00: 6976 6573 2074 6865 2068 6f6d 6f74 6f70  ives the homotop
│ │ │ │ -00054f10: 790a 2020 2020 2020 2020 6672 6f6d 2046  y.        from F
│ │ │ │ -00054f20: 5f69 2063 6f72 7265 7370 6f6e 6469 6e67  _i corresponding
│ │ │ │ -00054f30: 2074 6f20 665f 6a20 6173 2074 6865 2076   to f_j as the v
│ │ │ │ -00054f40: 616c 7565 2024 4823 5c7b 6a2c 695c 7d24  alue $H#\{j,i\}$
│ │ │ │ -00054f50: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -00054f60: 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 616d 6520  =========..Same 
│ │ │ │ -00054f70: 6173 206d 616b 6548 6f6d 6f74 6f70 6965  as makeHomotopie
│ │ │ │ -00054f80: 732c 2062 7574 206f 6e6c 7920 636f 6d70  s, but only comp
│ │ │ │ -00054f90: 7574 6573 2074 6865 206f 7264 696e 6172  utes the ordinar
│ │ │ │ -00054fa0: 7920 686f 6d6f 746f 7069 6573 2c20 6e6f  y homotopies, no
│ │ │ │ -00054fb0: 7420 7468 650a 6869 6768 6572 206f 6e65  t the.higher one
│ │ │ │ -00054fc0: 732e 2055 7365 6420 696e 2065 7874 6572  s. Used in exter
│ │ │ │ -00054fd0: 696f 7254 6f72 4d6f 6475 6c65 0a0a 5365  iorTorModule..Se
│ │ │ │ -00054fe0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ -00054ff0: 0a20 202a 202a 6e6f 7465 206d 616b 6548  .  * *note makeH
│ │ │ │ -00055000: 6f6d 6f74 6f70 6965 733a 206d 616b 6548  omotopies: makeH
│ │ │ │ -00055010: 6f6d 6f74 6f70 6965 732c 202d 2d20 7265  omotopies, -- re
│ │ │ │ -00055020: 7475 726e 7320 6120 7379 7374 656d 206f  turns a system o
│ │ │ │ -00055030: 6620 6869 6768 6572 0a20 2020 2068 6f6d  f higher.    hom
│ │ │ │ -00055040: 6f74 6f70 6965 730a 2020 2a20 2a6e 6f74  otopies.  * *not
│ │ │ │ -00055050: 6520 6578 7465 7269 6f72 546f 724d 6f64  e exteriorTorMod
│ │ │ │ -00055060: 756c 653a 2065 7874 6572 696f 7254 6f72  ule: exteriorTor
│ │ │ │ -00055070: 4d6f 6475 6c65 2c20 2d2d 2054 6f72 2061  Module, -- Tor a
│ │ │ │ -00055080: 7320 6120 6d6f 6475 6c65 206f 7665 7220  s a module over 
│ │ │ │ -00055090: 616e 0a20 2020 2065 7874 6572 696f 7220  an.    exterior 
│ │ │ │ -000550a0: 616c 6765 6272 6120 6f72 2062 6967 7261  algebra or bigra
│ │ │ │ -000550b0: 6465 6420 616c 6765 6272 610a 0a57 6179  ded algebra..Way
│ │ │ │ -000550c0: 7320 746f 2075 7365 206d 616b 6548 6f6d  s to use makeHom
│ │ │ │ -000550d0: 6f74 6f70 6965 7331 3a0a 3d3d 3d3d 3d3d  otopies1:.======
│ │ │ │ -000550e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000550f0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b  ======..  * "mak
│ │ │ │ -00055100: 6548 6f6d 6f74 6f70 6965 7331 284d 6174  eHomotopies1(Mat
│ │ │ │ -00055110: 7269 782c 436f 6d70 6c65 7829 220a 2020  rix,Complex)".  
│ │ │ │ -00055120: 2a20 226d 616b 6548 6f6d 6f74 6f70 6965  * "makeHomotopie
│ │ │ │ -00055130: 7331 284d 6174 7269 782c 436f 6d70 6c65  s1(Matrix,Comple
│ │ │ │ -00055140: 782c 5a5a 2922 0a0a 466f 7220 7468 6520  x,ZZ)"..For the 
│ │ │ │ -00055150: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -00055160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -00055170: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -00055180: 6d61 6b65 486f 6d6f 746f 7069 6573 313a  makeHomotopies1:
│ │ │ │ -00055190: 206d 616b 6548 6f6d 6f74 6f70 6965 7331   makeHomotopies1
│ │ │ │ -000551a0: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -000551b0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ -000551c0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -000551d0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ +00054d30: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ +00054d40: 2020 2020 2020 2048 203d 206d 616b 6548         H = makeH
│ │ │ │ +00054d50: 6f6d 6f74 6f70 6965 7331 2866 2c46 2c64  omotopies1(f,F,d
│ │ │ │ +00054d60: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +00054d70: 2020 2020 2a20 662c 2061 202a 6e6f 7465      * f, a *note
│ │ │ │ +00054d80: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ +00054d90: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ +00054da0: 3178 6e20 6d61 7472 6978 206f 6620 656c  1xn matrix of el
│ │ │ │ +00054db0: 656d 656e 7473 206f 6620 530a 2020 2020  ements of S.    
│ │ │ │ +00054dc0: 2020 2a20 462c 2061 202a 6e6f 7465 2063    * F, a *note c
│ │ │ │ +00054dd0: 6f6d 706c 6578 3a20 2843 6f6d 706c 6578  omplex: (Complex
│ │ │ │ +00054de0: 6573 2943 6f6d 706c 6578 2c2c 2061 646d  es)Complex,, adm
│ │ │ │ +00054df0: 6974 7469 6e67 2068 6f6d 6f74 6f70 6965  itting homotopie
│ │ │ │ +00054e00: 7320 666f 7220 7468 650a 2020 2020 2020  s for the.      
│ │ │ │ +00054e10: 2020 656e 7472 6965 7320 6f66 2066 0a20    entries of f. 
│ │ │ │ +00054e20: 2020 2020 202a 2064 2c20 616e 202a 6e6f       * d, an *no
│ │ │ │ +00054e30: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +00054e40: 6175 6c61 7932 446f 6329 5a5a 2c2c 2068  aulay2Doc)ZZ,, h
│ │ │ │ +00054e50: 6f77 2066 6172 2062 6163 6b20 746f 2063  ow far back to c
│ │ │ │ +00054e60: 6f6d 7075 7465 2074 6865 0a20 2020 2020  ompute the.     
│ │ │ │ +00054e70: 2020 2068 6f6d 6f74 6f70 6965 7320 2864     homotopies (d
│ │ │ │ +00054e80: 6566 6175 6c74 7320 746f 206c 656e 6774  efaults to lengt
│ │ │ │ +00054e90: 6820 6f66 2046 290a 2020 2a20 4f75 7470  h of F).  * Outp
│ │ │ │ +00054ea0: 7574 733a 0a20 2020 2020 202a 2048 2c20  uts:.      * H, 
│ │ │ │ +00054eb0: 6120 2a6e 6f74 6520 6861 7368 2074 6162  a *note hash tab
│ │ │ │ +00054ec0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +00054ed0: 6329 4861 7368 5461 626c 652c 2c20 6769  c)HashTable,, gi
│ │ │ │ +00054ee0: 7665 7320 7468 6520 686f 6d6f 746f 7079  ves the homotopy
│ │ │ │ +00054ef0: 0a20 2020 2020 2020 2066 726f 6d20 465f  .        from F_
│ │ │ │ +00054f00: 6920 636f 7272 6573 706f 6e64 696e 6720  i corresponding 
│ │ │ │ +00054f10: 746f 2066 5f6a 2061 7320 7468 6520 7661  to f_j as the va
│ │ │ │ +00054f20: 6c75 6520 2448 235c 7b6a 2c69 5c7d 240a  lue $H#\{j,i\}$.
│ │ │ │ +00054f30: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00054f40: 3d3d 3d3d 3d3d 3d3d 0a0a 5361 6d65 2061  ========..Same a
│ │ │ │ +00054f50: 7320 6d61 6b65 486f 6d6f 746f 7069 6573  s makeHomotopies
│ │ │ │ +00054f60: 2c20 6275 7420 6f6e 6c79 2063 6f6d 7075  , but only compu
│ │ │ │ +00054f70: 7465 7320 7468 6520 6f72 6469 6e61 7279  tes the ordinary
│ │ │ │ +00054f80: 2068 6f6d 6f74 6f70 6965 732c 206e 6f74   homotopies, not
│ │ │ │ +00054f90: 2074 6865 0a68 6967 6865 7220 6f6e 6573   the.higher ones
│ │ │ │ +00054fa0: 2e20 5573 6564 2069 6e20 6578 7465 7269  . Used in exteri
│ │ │ │ +00054fb0: 6f72 546f 724d 6f64 756c 650a 0a53 6565  orTorModule..See
│ │ │ │ +00054fc0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ +00054fd0: 2020 2a20 2a6e 6f74 6520 6d61 6b65 486f    * *note makeHo
│ │ │ │ +00054fe0: 6d6f 746f 7069 6573 3a20 6d61 6b65 486f  motopies: makeHo
│ │ │ │ +00054ff0: 6d6f 746f 7069 6573 2c20 2d2d 2072 6574  motopies, -- ret
│ │ │ │ +00055000: 7572 6e73 2061 2073 7973 7465 6d20 6f66  urns a system of
│ │ │ │ +00055010: 2068 6967 6865 720a 2020 2020 686f 6d6f   higher.    homo
│ │ │ │ +00055020: 746f 7069 6573 0a20 202a 202a 6e6f 7465  topies.  * *note
│ │ │ │ +00055030: 2065 7874 6572 696f 7254 6f72 4d6f 6475   exteriorTorModu
│ │ │ │ +00055040: 6c65 3a20 6578 7465 7269 6f72 546f 724d  le: exteriorTorM
│ │ │ │ +00055050: 6f64 756c 652c 202d 2d20 546f 7220 6173  odule, -- Tor as
│ │ │ │ +00055060: 2061 206d 6f64 756c 6520 6f76 6572 2061   a module over a
│ │ │ │ +00055070: 6e0a 2020 2020 6578 7465 7269 6f72 2061  n.    exterior a
│ │ │ │ +00055080: 6c67 6562 7261 206f 7220 6269 6772 6164  lgebra or bigrad
│ │ │ │ +00055090: 6564 2061 6c67 6562 7261 0a0a 5761 7973  ed algebra..Ways
│ │ │ │ +000550a0: 2074 6f20 7573 6520 6d61 6b65 486f 6d6f   to use makeHomo
│ │ │ │ +000550b0: 746f 7069 6573 313a 0a3d 3d3d 3d3d 3d3d  topies1:.=======
│ │ │ │ +000550c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000550d0: 3d3d 3d3d 3d0a 0a20 202a 2022 6d61 6b65  =====..  * "make
│ │ │ │ +000550e0: 486f 6d6f 746f 7069 6573 3128 4d61 7472  Homotopies1(Matr
│ │ │ │ +000550f0: 6978 2c43 6f6d 706c 6578 2922 0a20 202a  ix,Complex)".  *
│ │ │ │ +00055100: 2022 6d61 6b65 486f 6d6f 746f 7069 6573   "makeHomotopies
│ │ │ │ +00055110: 3128 4d61 7472 6978 2c43 6f6d 706c 6578  1(Matrix,Complex
│ │ │ │ +00055120: 2c5a 5a29 220a 0a46 6f72 2074 6865 2070  ,ZZ)"..For the p
│ │ │ │ +00055130: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +00055140: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +00055150: 6520 6f62 6a65 6374 202a 6e6f 7465 206d  e object *note m
│ │ │ │ +00055160: 616b 6548 6f6d 6f74 6f70 6965 7331 3a20  akeHomotopies1: 
│ │ │ │ +00055170: 6d61 6b65 486f 6d6f 746f 7069 6573 312c  makeHomotopies1,
│ │ │ │ +00055180: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +00055190: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ +000551a0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +000551b0: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d  dFunction,...---
│ │ │ │ +000551c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000551d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000551e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000551f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -00055230: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -00055240: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -00055250: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -00055260: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -00055270: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ -00055280: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -00055290: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ -000552a0: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -000552b0: 7469 6f6e 732e 6d32 3a33 3830 313a 302e  tions.m2:3801:0.
│ │ │ │ -000552c0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ -000552d0: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -000552e0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ -000552f0: 6f64 653a 206d 616b 6548 6f6d 6f74 6f70  ode: makeHomotop
│ │ │ │ -00055300: 6965 734f 6e48 6f6d 6f6c 6f67 792c 204e  iesOnHomology, N
│ │ │ │ -00055310: 6578 743a 206d 616b 654d 6f64 756c 652c  ext: makeModule,
│ │ │ │ -00055320: 2050 7265 763a 206d 616b 6548 6f6d 6f74   Prev: makeHomot
│ │ │ │ -00055330: 6f70 6965 7331 2c20 5570 3a20 546f 700a  opies1, Up: Top.
│ │ │ │ -00055340: 0a6d 616b 6548 6f6d 6f74 6f70 6965 734f  .makeHomotopiesO
│ │ │ │ -00055350: 6e48 6f6d 6f6c 6f67 7920 2d2d 2048 6f6d  nHomology -- Hom
│ │ │ │ -00055360: 6f6c 6f67 7920 6f66 2061 2063 6f6d 706c  ology of a compl
│ │ │ │ -00055370: 6578 2061 7320 6578 7465 7269 6f72 206d  ex as exterior m
│ │ │ │ -00055380: 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a 2a2a  odule.**********
│ │ │ │ +00055200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +00055210: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +00055220: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +00055230: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +00055240: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +00055250: 7932 2d31 2e32 362e 3036 2b64 732f 4d32  y2-1.26.06+ds/M2
│ │ │ │ +00055260: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +00055270: 6765 732f 0a43 6f6d 706c 6574 6549 6e74  ges/.CompleteInt
│ │ │ │ +00055280: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00055290: 696f 6e73 2e6d 323a 3338 3031 3a30 2e0a  ions.m2:3801:0..
│ │ │ │ +000552a0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +000552b0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +000552c0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +000552d0: 6465 3a20 6d61 6b65 486f 6d6f 746f 7069  de: makeHomotopi
│ │ │ │ +000552e0: 6573 4f6e 486f 6d6f 6c6f 6779 2c20 4e65  esOnHomology, Ne
│ │ │ │ +000552f0: 7874 3a20 6d61 6b65 4d6f 6475 6c65 2c20  xt: makeModule, 
│ │ │ │ +00055300: 5072 6576 3a20 6d61 6b65 486f 6d6f 746f  Prev: makeHomoto
│ │ │ │ +00055310: 7069 6573 312c 2055 703a 2054 6f70 0a0a  pies1, Up: Top..
│ │ │ │ +00055320: 6d61 6b65 486f 6d6f 746f 7069 6573 4f6e  makeHomotopiesOn
│ │ │ │ +00055330: 486f 6d6f 6c6f 6779 202d 2d20 486f 6d6f  Homology -- Homo
│ │ │ │ +00055340: 6c6f 6779 206f 6620 6120 636f 6d70 6c65  logy of a comple
│ │ │ │ +00055350: 7820 6173 2065 7874 6572 696f 7220 6d6f  x as exterior mo
│ │ │ │ +00055360: 6475 6c65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  dule.***********
│ │ │ │ +00055370: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00055380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00055390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000553a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000553b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000553c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -000553d0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -000553e0: 2848 2c68 2920 3d20 6d61 6b65 486f 6d6f  (H,h) = makeHomo
│ │ │ │ -000553f0: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779  topiesOnHomology
│ │ │ │ -00055400: 2866 662c 2043 290a 2020 2a20 496e 7075  (ff, C).  * Inpu
│ │ │ │ -00055410: 7473 3a0a 2020 2020 2020 2a20 6666 2c20  ts:.      * ff, 
│ │ │ │ -00055420: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ -00055430: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ -00055440: 7472 6978 2c2c 206d 6174 7269 7820 6f66  trix,, matrix of
│ │ │ │ -00055450: 2065 6c65 6d65 6e74 7320 686f 6d6f 746f   elements homoto
│ │ │ │ -00055460: 7069 630a 2020 2020 2020 2020 746f 2030  pic.        to 0
│ │ │ │ -00055470: 206f 6e20 430a 2020 2020 2020 2a20 432c   on C.      * C,
│ │ │ │ -00055480: 2061 202a 6e6f 7465 2063 6f6d 706c 6578   a *note complex
│ │ │ │ -00055490: 3a20 2843 6f6d 706c 6578 6573 2943 6f6d  : (Complexes)Com
│ │ │ │ -000554a0: 706c 6578 2c2c 200a 2020 2a20 4f75 7470  plex,, .  * Outp
│ │ │ │ -000554b0: 7574 733a 0a20 2020 2020 202a 2048 2c20  uts:.      * H, 
│ │ │ │ -000554c0: 6120 2a6e 6f74 6520 6861 7368 2074 6162  a *note hash tab
│ │ │ │ -000554d0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -000554e0: 6329 4861 7368 5461 626c 652c 2c20 486f  c)HashTable,, Ho
│ │ │ │ -000554f0: 6d6f 6c6f 6779 206f 6620 432c 2069 6e64  mology of C, ind
│ │ │ │ -00055500: 6578 6564 0a20 2020 2020 2020 2062 7920  exed.        by 
│ │ │ │ -00055510: 706c 6163 6573 2069 6e20 7468 6520 430a  places in the C.
│ │ │ │ -00055520: 2020 2020 2020 2a20 682c 2061 202a 6e6f        * h, a *no
│ │ │ │ -00055530: 7465 2068 6173 6820 7461 626c 653a 2028  te hash table: (
│ │ │ │ -00055540: 4d61 6361 756c 6179 3244 6f63 2948 6173  Macaulay2Doc)Has
│ │ │ │ -00055550: 6854 6162 6c65 2c2c 2068 6f6d 6f74 6f70  hTable,, homotop
│ │ │ │ -00055560: 6965 7320 666f 720a 2020 2020 2020 2020  ies for.        
│ │ │ │ -00055570: 656c 656d 656e 7473 206f 6620 6620 6f6e  elements of f on
│ │ │ │ -00055580: 2074 6865 2068 6f6d 6f6c 6f67 7920 6f66   the homology of
│ │ │ │ -00055590: 2043 0a0a 4465 7363 7269 7074 696f 6e0a   C..Description.
│ │ │ │ -000555a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -000555b0: 2073 6372 6970 7420 6361 6c6c 7320 6d61   script calls ma
│ │ │ │ -000555c0: 6b65 486f 6d6f 746f 7069 6573 3120 746f  keHomotopies1 to
│ │ │ │ -000555d0: 2070 726f 6475 6365 2068 6f6d 6f74 6f70   produce homotop
│ │ │ │ -000555e0: 6965 7320 666f 7220 7468 6520 6666 5f69  ies for the ff_i
│ │ │ │ -000555f0: 206f 6e20 432c 2061 6e64 0a74 6865 6e20   on C, and.then 
│ │ │ │ -00055600: 636f 6d70 7574 6573 2074 6865 6972 2061  computes their a
│ │ │ │ -00055610: 6374 696f 6e20 6f6e 2074 6865 2048 6f6d  ction on the Hom
│ │ │ │ -00055620: 6f6c 6f67 7920 6f66 2043 2e0a 0a53 6565  ology of C...See
│ │ │ │ -00055630: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -00055640: 2020 2a20 2a6e 6f74 6520 6578 7465 7269    * *note exteri
│ │ │ │ -00055650: 6f72 546f 724d 6f64 756c 653a 2065 7874  orTorModule: ext
│ │ │ │ -00055660: 6572 696f 7254 6f72 4d6f 6475 6c65 2c20  eriorTorModule, 
│ │ │ │ -00055670: 2d2d 2054 6f72 2061 7320 6120 6d6f 6475  -- Tor as a modu
│ │ │ │ -00055680: 6c65 206f 7665 7220 616e 0a20 2020 2065  le over an.    e
│ │ │ │ -00055690: 7874 6572 696f 7220 616c 6765 6272 6120  xterior algebra 
│ │ │ │ -000556a0: 6f72 2062 6967 7261 6465 6420 616c 6765  or bigraded alge
│ │ │ │ -000556b0: 6272 610a 2020 2a20 2a6e 6f74 6520 6578  bra.  * *note ex
│ │ │ │ -000556c0: 7465 7269 6f72 4578 744d 6f64 756c 653a  teriorExtModule:
│ │ │ │ -000556d0: 2065 7874 6572 696f 7245 7874 4d6f 6475   exteriorExtModu
│ │ │ │ -000556e0: 6c65 2c20 2d2d 2045 7874 284d 2c6b 2920  le, -- Ext(M,k) 
│ │ │ │ -000556f0: 6f72 2045 7874 284d 2c4e 2920 6173 2061  or Ext(M,N) as a
│ │ │ │ -00055700: 0a20 2020 206d 6f64 756c 6520 6f76 6572  .    module over
│ │ │ │ -00055710: 2061 6e20 6578 7465 7269 6f72 2061 6c67   an exterior alg
│ │ │ │ -00055720: 6562 7261 0a0a 5761 7973 2074 6f20 7573  ebra..Ways to us
│ │ │ │ -00055730: 6520 6d61 6b65 486f 6d6f 746f 7069 6573  e makeHomotopies
│ │ │ │ -00055740: 4f6e 486f 6d6f 6c6f 6779 3a0a 3d3d 3d3d  OnHomology:.====
│ │ │ │ -00055750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00055760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00055770: 3d0a 0a20 202a 2022 6d61 6b65 486f 6d6f  =..  * "makeHomo
│ │ │ │ -00055780: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779  topiesOnHomology
│ │ │ │ -00055790: 284d 6174 7269 782c 436f 6d70 6c65 7829  (Matrix,Complex)
│ │ │ │ -000557a0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -000557b0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -000557c0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -000557d0: 6a65 6374 202a 6e6f 7465 206d 616b 6548  ject *note makeH
│ │ │ │ -000557e0: 6f6d 6f74 6f70 6965 734f 6e48 6f6d 6f6c  omotopiesOnHomol
│ │ │ │ -000557f0: 6f67 793a 206d 616b 6548 6f6d 6f74 6f70  ogy: makeHomotop
│ │ │ │ -00055800: 6965 734f 6e48 6f6d 6f6c 6f67 792c 2069  iesOnHomology, i
│ │ │ │ -00055810: 7320 6120 2a6e 6f74 650a 6d65 7468 6f64  s a *note.method
│ │ │ │ -00055820: 2066 756e 6374 696f 6e3a 2028 4d61 6361   function: (Maca
│ │ │ │ -00055830: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ -00055840: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d  unction,...-----
│ │ │ │ +000553a0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ +000553b0: 7361 6765 3a20 0a20 2020 2020 2020 2028  sage: .        (
│ │ │ │ +000553c0: 482c 6829 203d 206d 616b 6548 6f6d 6f74  H,h) = makeHomot
│ │ │ │ +000553d0: 6f70 6965 734f 6e48 6f6d 6f6c 6f67 7928  opiesOnHomology(
│ │ │ │ +000553e0: 6666 2c20 4329 0a20 202a 2049 6e70 7574  ff, C).  * Input
│ │ │ │ +000553f0: 733a 0a20 2020 2020 202a 2066 662c 2061  s:.      * ff, a
│ │ │ │ +00055400: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ +00055410: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ +00055420: 7269 782c 2c20 6d61 7472 6978 206f 6620  rix,, matrix of 
│ │ │ │ +00055430: 656c 656d 656e 7473 2068 6f6d 6f74 6f70  elements homotop
│ │ │ │ +00055440: 6963 0a20 2020 2020 2020 2074 6f20 3020  ic.        to 0 
│ │ │ │ +00055450: 6f6e 2043 0a20 2020 2020 202a 2043 2c20  on C.      * C, 
│ │ │ │ +00055460: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a  a *note complex:
│ │ │ │ +00055470: 2028 436f 6d70 6c65 7865 7329 436f 6d70   (Complexes)Comp
│ │ │ │ +00055480: 6c65 782c 2c20 0a20 202a 204f 7574 7075  lex,, .  * Outpu
│ │ │ │ +00055490: 7473 3a0a 2020 2020 2020 2a20 482c 2061  ts:.      * H, a
│ │ │ │ +000554a0: 202a 6e6f 7465 2068 6173 6820 7461 626c   *note hash tabl
│ │ │ │ +000554b0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +000554c0: 2948 6173 6854 6162 6c65 2c2c 2048 6f6d  )HashTable,, Hom
│ │ │ │ +000554d0: 6f6c 6f67 7920 6f66 2043 2c20 696e 6465  ology of C, inde
│ │ │ │ +000554e0: 7865 640a 2020 2020 2020 2020 6279 2070  xed.        by p
│ │ │ │ +000554f0: 6c61 6365 7320 696e 2074 6865 2043 0a20  laces in the C. 
│ │ │ │ +00055500: 2020 2020 202a 2068 2c20 6120 2a6e 6f74       * h, a *not
│ │ │ │ +00055510: 6520 6861 7368 2074 6162 6c65 3a20 284d  e hash table: (M
│ │ │ │ +00055520: 6163 6175 6c61 7932 446f 6329 4861 7368  acaulay2Doc)Hash
│ │ │ │ +00055530: 5461 626c 652c 2c20 686f 6d6f 746f 7069  Table,, homotopi
│ │ │ │ +00055540: 6573 2066 6f72 0a20 2020 2020 2020 2065  es for.        e
│ │ │ │ +00055550: 6c65 6d65 6e74 7320 6f66 2066 206f 6e20  lements of f on 
│ │ │ │ +00055560: 7468 6520 686f 6d6f 6c6f 6779 206f 6620  the homology of 
│ │ │ │ +00055570: 430a 0a44 6573 6372 6970 7469 6f6e 0a3d  C..Description.=
│ │ │ │ +00055580: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ +00055590: 7363 7269 7074 2063 616c 6c73 206d 616b  script calls mak
│ │ │ │ +000555a0: 6548 6f6d 6f74 6f70 6965 7331 2074 6f20  eHomotopies1 to 
│ │ │ │ +000555b0: 7072 6f64 7563 6520 686f 6d6f 746f 7069  produce homotopi
│ │ │ │ +000555c0: 6573 2066 6f72 2074 6865 2066 665f 6920  es for the ff_i 
│ │ │ │ +000555d0: 6f6e 2043 2c20 616e 640a 7468 656e 2063  on C, and.then c
│ │ │ │ +000555e0: 6f6d 7075 7465 7320 7468 6569 7220 6163  omputes their ac
│ │ │ │ +000555f0: 7469 6f6e 206f 6e20 7468 6520 486f 6d6f  tion on the Homo
│ │ │ │ +00055600: 6c6f 6779 206f 6620 432e 0a0a 5365 6520  logy of C...See 
│ │ │ │ +00055610: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +00055620: 202a 202a 6e6f 7465 2065 7874 6572 696f   * *note exterio
│ │ │ │ +00055630: 7254 6f72 4d6f 6475 6c65 3a20 6578 7465  rTorModule: exte
│ │ │ │ +00055640: 7269 6f72 546f 724d 6f64 756c 652c 202d  riorTorModule, -
│ │ │ │ +00055650: 2d20 546f 7220 6173 2061 206d 6f64 756c  - Tor as a modul
│ │ │ │ +00055660: 6520 6f76 6572 2061 6e0a 2020 2020 6578  e over an.    ex
│ │ │ │ +00055670: 7465 7269 6f72 2061 6c67 6562 7261 206f  terior algebra o
│ │ │ │ +00055680: 7220 6269 6772 6164 6564 2061 6c67 6562  r bigraded algeb
│ │ │ │ +00055690: 7261 0a20 202a 202a 6e6f 7465 2065 7874  ra.  * *note ext
│ │ │ │ +000556a0: 6572 696f 7245 7874 4d6f 6475 6c65 3a20  eriorExtModule: 
│ │ │ │ +000556b0: 6578 7465 7269 6f72 4578 744d 6f64 756c  exteriorExtModul
│ │ │ │ +000556c0: 652c 202d 2d20 4578 7428 4d2c 6b29 206f  e, -- Ext(M,k) o
│ │ │ │ +000556d0: 7220 4578 7428 4d2c 4e29 2061 7320 610a  r Ext(M,N) as a.
│ │ │ │ +000556e0: 2020 2020 6d6f 6475 6c65 206f 7665 7220      module over 
│ │ │ │ +000556f0: 616e 2065 7874 6572 696f 7220 616c 6765  an exterior alge
│ │ │ │ +00055700: 6272 610a 0a57 6179 7320 746f 2075 7365  bra..Ways to use
│ │ │ │ +00055710: 206d 616b 6548 6f6d 6f74 6f70 6965 734f   makeHomotopiesO
│ │ │ │ +00055720: 6e48 6f6d 6f6c 6f67 793a 0a3d 3d3d 3d3d  nHomology:.=====
│ │ │ │ +00055730: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00055740: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00055750: 0a0a 2020 2a20 226d 616b 6548 6f6d 6f74  ..  * "makeHomot
│ │ │ │ +00055760: 6f70 6965 734f 6e48 6f6d 6f6c 6f67 7928  opiesOnHomology(
│ │ │ │ +00055770: 4d61 7472 6978 2c43 6f6d 706c 6578 2922  Matrix,Complex)"
│ │ │ │ +00055780: 0a0a 466f 7220 7468 6520 7072 6f67 7261  ..For the progra
│ │ │ │ +00055790: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  mmer.===========
│ │ │ │ +000557a0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a  =======..The obj
│ │ │ │ +000557b0: 6563 7420 2a6e 6f74 6520 6d61 6b65 486f  ect *note makeHo
│ │ │ │ +000557c0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f  motopiesOnHomolo
│ │ │ │ +000557d0: 6779 3a20 6d61 6b65 486f 6d6f 746f 7069  gy: makeHomotopi
│ │ │ │ +000557e0: 6573 4f6e 486f 6d6f 6c6f 6779 2c20 6973  esOnHomology, is
│ │ │ │ +000557f0: 2061 202a 6e6f 7465 0a6d 6574 686f 6420   a *note.method 
│ │ │ │ +00055800: 6675 6e63 7469 6f6e 3a20 284d 6163 6175  function: (Macau
│ │ │ │ +00055810: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +00055820: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ +00055830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00055840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00055860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00055890: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ -000558a0: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ -000558b0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ -000558c0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -000558d0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -000558e0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ -000558f0: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -00055900: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ -00055910: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00055920: 6e73 2e6d 323a 3236 3934 3a30 2e0a 1f0a  ns.m2:2694:0....
│ │ │ │ -00055930: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ -00055940: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -00055950: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ -00055960: 3a20 6d61 6b65 4d6f 6475 6c65 2c20 4e65  : makeModule, Ne
│ │ │ │ -00055970: 7874 3a20 6d61 6b65 542c 2050 7265 763a  xt: makeT, Prev:
│ │ │ │ -00055980: 206d 616b 6548 6f6d 6f74 6f70 6965 734f   makeHomotopiesO
│ │ │ │ -00055990: 6e48 6f6d 6f6c 6f67 792c 2055 703a 2054  nHomology, Up: T
│ │ │ │ -000559a0: 6f70 0a0a 6d61 6b65 4d6f 6475 6c65 202d  op..makeModule -
│ │ │ │ -000559b0: 2d20 6d61 6b65 7320 6120 4d6f 6475 6c65  - makes a Module
│ │ │ │ -000559c0: 206f 7574 206f 6620 6120 636f 6c6c 6563   out of a collec
│ │ │ │ -000559d0: 7469 6f6e 206f 6620 6d6f 6475 6c65 7320  tion of modules 
│ │ │ │ -000559e0: 616e 6420 6d61 7073 0a2a 2a2a 2a2a 2a2a  and maps.*******
│ │ │ │ +00055870: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ +00055880: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ +00055890: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ +000558a0: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ +000558b0: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ +000558c0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ +000558d0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ +000558e0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ +000558f0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00055900: 732e 6d32 3a32 3639 343a 302e 0a1f 0a46  s.m2:2694:0....F
│ │ │ │ +00055910: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ +00055920: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +00055930: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ +00055940: 206d 616b 654d 6f64 756c 652c 204e 6578   makeModule, Nex
│ │ │ │ +00055950: 743a 206d 616b 6554 2c20 5072 6576 3a20  t: makeT, Prev: 
│ │ │ │ +00055960: 6d61 6b65 486f 6d6f 746f 7069 6573 4f6e  makeHomotopiesOn
│ │ │ │ +00055970: 486f 6d6f 6c6f 6779 2c20 5570 3a20 546f  Homology, Up: To
│ │ │ │ +00055980: 700a 0a6d 616b 654d 6f64 756c 6520 2d2d  p..makeModule --
│ │ │ │ +00055990: 206d 616b 6573 2061 204d 6f64 756c 6520   makes a Module 
│ │ │ │ +000559a0: 6f75 7420 6f66 2061 2063 6f6c 6c65 6374  out of a collect
│ │ │ │ +000559b0: 696f 6e20 6f66 206d 6f64 756c 6573 2061  ion of modules a
│ │ │ │ +000559c0: 6e64 206d 6170 730a 2a2a 2a2a 2a2a 2a2a  nd maps.********
│ │ │ │ +000559d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000559e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000559f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00055a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00055a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00055a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ -00055a30: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ -00055a40: 2020 204d 203d 206d 616b 654d 6f64 756c     M = makeModul
│ │ │ │ -00055a50: 6528 482c 452c 7068 6929 0a20 202a 2049  e(H,E,phi).  * I
│ │ │ │ -00055a60: 6e70 7574 733a 0a20 2020 2020 202a 2048  nputs:.      * H
│ │ │ │ -00055a70: 2c20 6120 2a6e 6f74 6520 6861 7368 2074  , a *note hash t
│ │ │ │ -00055a80: 6162 6c65 3a20 284d 6163 6175 6c61 7932  able: (Macaulay2
│ │ │ │ -00055a90: 446f 6329 4861 7368 5461 626c 652c 2c20  Doc)HashTable,, 
│ │ │ │ -00055aa0: 6772 6164 6564 2063 6f6d 706f 6e65 6e74  graded component
│ │ │ │ -00055ab0: 7320 7468 6174 0a20 2020 2020 2020 2061  s that.        a
│ │ │ │ -00055ac0: 7265 206d 6f64 756c 6573 2c20 746f 206d  re modules, to m
│ │ │ │ -00055ad0: 616b 6520 696e 746f 2061 7320 7369 6e67  ake into as sing
│ │ │ │ -00055ae0: 6c65 206d 6f64 756c 650a 2020 2020 2020  le module.      
│ │ │ │ -00055af0: 2a20 452c 2061 202a 6e6f 7465 206d 6174  * E, a *note mat
│ │ │ │ -00055b00: 7269 783a 2028 4d61 6361 756c 6179 3244  rix: (Macaulay2D
│ │ │ │ -00055b10: 6f63 294d 6174 7269 782c 2c20 4d61 7472  oc)Matrix,, Matr
│ │ │ │ -00055b20: 6978 206f 6620 7661 7269 6162 6c65 7320  ix of variables 
│ │ │ │ -00055b30: 7768 6f73 650a 2020 2020 2020 2020 6163  whose.        ac
│ │ │ │ -00055b40: 7469 6f6e 2077 696c 6c20 6465 6669 6e65  tion will define
│ │ │ │ -00055b50: 640a 2020 2020 2020 2a20 7068 692c 2061  d.      * phi, a
│ │ │ │ -00055b60: 202a 6e6f 7465 2068 6173 6820 7461 626c   *note hash tabl
│ │ │ │ -00055b70: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -00055b80: 2948 6173 6854 6162 6c65 2c2c 206d 6170  )HashTable,, map
│ │ │ │ -00055b90: 7320 6265 7477 6565 6e20 7468 650a 2020  s between the.  
│ │ │ │ -00055ba0: 2020 2020 2020 6772 6164 6564 2063 6f6d        graded com
│ │ │ │ -00055bb0: 706f 6e65 6e74 7320 7468 6174 2077 696c  ponents that wil
│ │ │ │ -00055bc0: 6c20 6265 2074 6865 2061 6374 696f 6e20  l be the action 
│ │ │ │ -00055bd0: 6f66 2074 6865 2076 6172 6961 626c 6573  of the variables
│ │ │ │ -00055be0: 2069 6e20 450a 2020 2a20 4f75 7470 7574   in E.  * Output
│ │ │ │ -00055bf0: 733a 0a20 2020 2020 202a 204d 2c20 6120  s:.      * M, a 
│ │ │ │ -00055c00: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ -00055c10: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ -00055c20: 6c65 2c2c 2067 7261 6465 6420 6d6f 6475  le,, graded modu
│ │ │ │ -00055c30: 6c65 7320 7768 6f73 650a 2020 2020 2020  les whose.      
│ │ │ │ -00055c40: 2020 636f 6d70 6f6e 656e 7473 2061 7265    components are
│ │ │ │ -00055c50: 2067 6976 656e 2062 7920 480a 0a44 6573   given by H..Des
│ │ │ │ -00055c60: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ -00055c70: 3d3d 3d3d 0a0a 5468 6520 4861 7368 7461  ====..The Hashta
│ │ │ │ -00055c80: 626c 6520 4820 7368 6f75 6c64 2068 6176  ble H should hav
│ │ │ │ -00055c90: 6520 636f 6e73 6563 7574 6976 6520 696e  e consecutive in
│ │ │ │ -00055ca0: 7465 6765 7220 6b65 7973 2069 5f30 2e2e  teger keys i_0..
│ │ │ │ -00055cb0: 695f 302c 2073 6179 2c20 7769 7468 2076  i_0, say, with v
│ │ │ │ -00055cc0: 616c 7565 730a 4823 6920 7468 6174 2061  alues.H#i that a
│ │ │ │ -00055cd0: 7265 206d 6f64 756c 6573 206f 7665 7220  re modules over 
│ │ │ │ -00055ce0: 6120 7269 6e67 2053 4520 7768 6f73 6520  a ring SE whose 
│ │ │ │ -00055cf0: 7661 7269 6162 6c65 7320 696e 636c 7564  variables includ
│ │ │ │ -00055d00: 6520 7468 6520 656c 656d 656e 7473 206f  e the elements o
│ │ │ │ -00055d10: 6620 452e 0a45 3a20 5c6f 706c 7573 2053  f E..E: \oplus S
│ │ │ │ -00055d20: 455e 7b64 5f69 7d20 5c74 6f20 5345 5e31  E^{d_i} \to SE^1
│ │ │ │ -00055d30: 2069 7320 6120 6d61 7472 6978 206f 6620   is a matrix of 
│ │ │ │ -00055d40: 6320 7661 7269 6162 6c65 7320 6672 6f6d  c variables from
│ │ │ │ -00055d50: 2053 4520 4820 6973 2061 2068 6173 6854   SE H is a hashT
│ │ │ │ -00055d60: 6162 6c65 0a6f 6620 6d20 7061 6972 7320  able.of m pairs 
│ │ │ │ -00055d70: 7b69 2c20 745f 697d 2c20 7768 6572 6520  {i, t_i}, where 
│ │ │ │ -00055d80: 7468 6520 745f 6920 6172 6520 5245 2d6d  the t_i are RE-m
│ │ │ │ -00055d90: 6f64 756c 6573 2c20 616e 6420 7468 6520  odules, and the 
│ │ │ │ -00055da0: 6920 6172 6520 636f 6e73 6563 7574 6976  i are consecutiv
│ │ │ │ -00055db0: 650a 696e 7465 6765 722e 2070 6869 2069  e.integer. phi i
│ │ │ │ -00055dc0: 7320 6120 6861 7368 2d74 6162 6c65 206f  s a hash-table o
│ │ │ │ -00055dd0: 6620 686f 6d6f 6765 6e65 6f75 7320 6d61  f homogeneous ma
│ │ │ │ -00055de0: 7073 2070 6869 237b 6a2c 697d 3a20 4823  ps phi#{j,i}: H#
│ │ │ │ -00055df0: 692a 2a46 5f6a 5c74 6f20 4823 2869 2b31  i**F_j\to H#(i+1
│ │ │ │ -00055e00: 290a 7768 6572 6520 465f 6a20 3d20 736f  ).where F_j = so
│ │ │ │ -00055e10: 7572 6365 2028 455f 7b6a 7d20 3d20 6d61  urce (E_{j} = ma
│ │ │ │ -00055e20: 7472 6978 207b 7b65 5f6a 7d7d 292e 2054  trix {{e_j}}). T
│ │ │ │ -00055e30: 6875 7320 7468 6520 6d61 7073 2070 237b  hus the maps p#{
│ │ │ │ -00055e40: 6a2c 697d 203d 2028 455f 6a20 7c7c 0a2d  j,i} = (E_j ||.-
│ │ │ │ -00055e50: 7068 6923 7b6a 2c69 7d29 3a20 745f 692a  phi#{j,i}): t_i*
│ │ │ │ -00055e60: 2a46 5f6a 205c 746f 2074 5f69 2b2b 745f  *F_j \to t_i++t_
│ │ │ │ -00055e70: 7b28 692b 3129 7d2c 2061 7265 2068 6f6d  {(i+1)}, are hom
│ │ │ │ -00055e80: 6f67 656e 656f 7573 2e20 5468 6520 7363  ogeneous. The sc
│ │ │ │ -00055e90: 7269 7074 2072 6574 7572 6e73 204d 0a3d  ript returns M.=
│ │ │ │ -00055ea0: 205c 6f70 6c75 735f 6920 545f 2061 7320   \oplus_i T_ as 
│ │ │ │ -00055eb0: 616e 2053 452d 6d6f 6475 6c65 2c20 636f  an SE-module, co
│ │ │ │ -00055ec0: 6d70 7574 6564 2061 7320 7468 6520 7175  mputed as the qu
│ │ │ │ -00055ed0: 6f74 6965 6e74 206f 6620 5020 3a3d 205c  otient of P := \
│ │ │ │ -00055ee0: 6f70 6c75 7320 545f 690a 6f62 7461 696e  oplus T_i.obtain
│ │ │ │ -00055ef0: 6564 2062 7920 6661 6374 6f72 696e 6720  ed by factoring 
│ │ │ │ -00055f00: 6f75 7420 7468 6520 7375 6d20 6f66 2074  out the sum of t
│ │ │ │ -00055f10: 6865 2069 6d61 6765 7320 6f66 2074 6865  he images of the
│ │ │ │ -00055f20: 206d 6170 7320 7023 7b6a 2c69 7d0a 0a54   maps p#{j,i}..T
│ │ │ │ -00055f30: 6865 2048 6173 6874 6162 6c65 2070 6869  he Hashtable phi
│ │ │ │ -00055f40: 2068 6173 206b 6579 7320 6f66 2074 6865   has keys of the
│ │ │ │ -00055f50: 2066 6f72 6d20 7b6a 2c69 7d20 7768 6572   form {j,i} wher
│ │ │ │ -00055f60: 6520 6a20 7275 6e73 2066 726f 6d20 3020  e j runs from 0 
│ │ │ │ -00055f70: 746f 2063 2d31 2c20 6920 616e 640a 692b  to c-1, i and.i+
│ │ │ │ -00055f80: 3120 6172 6520 6b65 7973 206f 6620 482c  1 are keys of H,
│ │ │ │ -00055f90: 2061 6e64 2070 6869 237b 6a2c 697d 2069   and phi#{j,i} i
│ │ │ │ -00055fa0: 7320 7468 6520 6d61 7020 6672 6f6d 2028  s the map from (
│ │ │ │ -00055fb0: 736f 7572 6365 2045 5f7b 697d 292a 2a48  source E_{i})**H
│ │ │ │ -00055fc0: 2369 2074 6f20 4823 2869 2b31 290a 7468  #i to H#(i+1).th
│ │ │ │ -00055fd0: 6174 2077 696c 6c20 6265 2069 6465 6e74  at will be ident
│ │ │ │ -00055fe0: 6966 6965 6420 7769 7468 2074 6865 2061  ified with the a
│ │ │ │ -00055ff0: 6374 696f 6e20 6f66 2045 5f7b 6a7d 2e0a  ction of E_{j}..
│ │ │ │ -00056000: 0a54 6865 2073 6372 6970 7420 6973 2075  .The script is u
│ │ │ │ -00056010: 7365 6420 696e 2062 6f74 6820 7468 6520  sed in both the 
│ │ │ │ -00056020: 7369 6e67 6c79 2067 7261 6465 6420 6361  singly graded ca
│ │ │ │ -00056030: 7365 2c20 666f 7220 6578 616d 706c 6520  se, for example 
│ │ │ │ -00056040: 696e 0a65 7874 6572 696f 7254 6f72 4d6f  in.exteriorTorMo
│ │ │ │ -00056050: 6475 6c65 2866 662c 4d29 2061 6e64 2069  dule(ff,M) and i
│ │ │ │ -00056060: 6e20 7468 6520 6269 6772 6164 6564 2063  n the bigraded c
│ │ │ │ -00056070: 6173 652c 2066 6f72 2065 7861 6d70 6c65  ase, for example
│ │ │ │ -00056080: 2069 6e0a 6578 7465 7269 6f72 546f 724d   in.exteriorTorM
│ │ │ │ -00056090: 6f64 756c 6528 6666 2c4d 2c4e 292e 0a0a  odule(ff,M,N)...
│ │ │ │ -000560a0: 496e 2074 6865 2066 6f6c 6c6f 7769 6e67  In the following
│ │ │ │ -000560b0: 2077 6520 7573 6520 6d61 6b65 4d6f 6475   we use makeModu
│ │ │ │ -000560c0: 6c65 2074 6f20 636f 6e73 7472 7563 7420  le to construct 
│ │ │ │ -000560d0: 6279 2068 616e 6420 6120 6672 6565 206d  by hand a free m
│ │ │ │ -000560e0: 6f64 756c 6520 6f66 2072 616e 6b20 310a  odule of rank 1.
│ │ │ │ -000560f0: 6f76 6572 2074 6865 2065 7874 6572 696f  over the exterio
│ │ │ │ -00056100: 7220 616c 6765 6272 6120 6f6e 2078 2c79  r algebra on x,y
│ │ │ │ -00056110: 2c20 7374 6172 7469 6e67 2077 6974 6820  , starting with 
│ │ │ │ -00056120: 7468 6520 636f 6e73 7472 7563 7469 6f6e  the construction
│ │ │ │ -00056130: 206f 6620 6120 6d6f 6475 6c65 0a6f 7665   of a module.ove
│ │ │ │ -00056140: 7220 6120 6269 686f 6d6f 6765 6e65 6f75  r a bihomogeneou
│ │ │ │ -00056150: 7320 7269 6e67 2e0a 0a2b 2d2d 2d2d 2d2d  s ring...+------
│ │ │ │ +00055a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ +00055a10: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ +00055a20: 2020 4d20 3d20 6d61 6b65 4d6f 6475 6c65    M = makeModule
│ │ │ │ +00055a30: 2848 2c45 2c70 6869 290a 2020 2a20 496e  (H,E,phi).  * In
│ │ │ │ +00055a40: 7075 7473 3a0a 2020 2020 2020 2a20 482c  puts:.      * H,
│ │ │ │ +00055a50: 2061 202a 6e6f 7465 2068 6173 6820 7461   a *note hash ta
│ │ │ │ +00055a60: 626c 653a 2028 4d61 6361 756c 6179 3244  ble: (Macaulay2D
│ │ │ │ +00055a70: 6f63 2948 6173 6854 6162 6c65 2c2c 2067  oc)HashTable,, g
│ │ │ │ +00055a80: 7261 6465 6420 636f 6d70 6f6e 656e 7473  raded components
│ │ │ │ +00055a90: 2074 6861 740a 2020 2020 2020 2020 6172   that.        ar
│ │ │ │ +00055aa0: 6520 6d6f 6475 6c65 732c 2074 6f20 6d61  e modules, to ma
│ │ │ │ +00055ab0: 6b65 2069 6e74 6f20 6173 2073 696e 676c  ke into as singl
│ │ │ │ +00055ac0: 6520 6d6f 6475 6c65 0a20 2020 2020 202a  e module.      *
│ │ │ │ +00055ad0: 2045 2c20 6120 2a6e 6f74 6520 6d61 7472   E, a *note matr
│ │ │ │ +00055ae0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ +00055af0: 6329 4d61 7472 6978 2c2c 204d 6174 7269  c)Matrix,, Matri
│ │ │ │ +00055b00: 7820 6f66 2076 6172 6961 626c 6573 2077  x of variables w
│ │ │ │ +00055b10: 686f 7365 0a20 2020 2020 2020 2061 6374  hose.        act
│ │ │ │ +00055b20: 696f 6e20 7769 6c6c 2064 6566 696e 6564  ion will defined
│ │ │ │ +00055b30: 0a20 2020 2020 202a 2070 6869 2c20 6120  .      * phi, a 
│ │ │ │ +00055b40: 2a6e 6f74 6520 6861 7368 2074 6162 6c65  *note hash table
│ │ │ │ +00055b50: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00055b60: 4861 7368 5461 626c 652c 2c20 6d61 7073  HashTable,, maps
│ │ │ │ +00055b70: 2062 6574 7765 656e 2074 6865 0a20 2020   between the.   
│ │ │ │ +00055b80: 2020 2020 2067 7261 6465 6420 636f 6d70       graded comp
│ │ │ │ +00055b90: 6f6e 656e 7473 2074 6861 7420 7769 6c6c  onents that will
│ │ │ │ +00055ba0: 2062 6520 7468 6520 6163 7469 6f6e 206f   be the action o
│ │ │ │ +00055bb0: 6620 7468 6520 7661 7269 6162 6c65 7320  f the variables 
│ │ │ │ +00055bc0: 696e 2045 0a20 202a 204f 7574 7075 7473  in E.  * Outputs
│ │ │ │ +00055bd0: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a  :.      * M, a *
│ │ │ │ +00055be0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ +00055bf0: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ +00055c00: 652c 2c20 6772 6164 6564 206d 6f64 756c  e,, graded modul
│ │ │ │ +00055c10: 6573 2077 686f 7365 0a20 2020 2020 2020  es whose.       
│ │ │ │ +00055c20: 2063 6f6d 706f 6e65 6e74 7320 6172 6520   components are 
│ │ │ │ +00055c30: 6769 7665 6e20 6279 2048 0a0a 4465 7363  given by H..Desc
│ │ │ │ +00055c40: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ +00055c50: 3d3d 3d0a 0a54 6865 2048 6173 6874 6162  ===..The Hashtab
│ │ │ │ +00055c60: 6c65 2048 2073 686f 756c 6420 6861 7665  le H should have
│ │ │ │ +00055c70: 2063 6f6e 7365 6375 7469 7665 2069 6e74   consecutive int
│ │ │ │ +00055c80: 6567 6572 206b 6579 7320 695f 302e 2e69  eger keys i_0..i
│ │ │ │ +00055c90: 5f30 2c20 7361 792c 2077 6974 6820 7661  _0, say, with va
│ │ │ │ +00055ca0: 6c75 6573 0a48 2369 2074 6861 7420 6172  lues.H#i that ar
│ │ │ │ +00055cb0: 6520 6d6f 6475 6c65 7320 6f76 6572 2061  e modules over a
│ │ │ │ +00055cc0: 2072 696e 6720 5345 2077 686f 7365 2076   ring SE whose v
│ │ │ │ +00055cd0: 6172 6961 626c 6573 2069 6e63 6c75 6465  ariables include
│ │ │ │ +00055ce0: 2074 6865 2065 6c65 6d65 6e74 7320 6f66   the elements of
│ │ │ │ +00055cf0: 2045 2e0a 453a 205c 6f70 6c75 7320 5345   E..E: \oplus SE
│ │ │ │ +00055d00: 5e7b 645f 697d 205c 746f 2053 455e 3120  ^{d_i} \to SE^1 
│ │ │ │ +00055d10: 6973 2061 206d 6174 7269 7820 6f66 2063  is a matrix of c
│ │ │ │ +00055d20: 2076 6172 6961 626c 6573 2066 726f 6d20   variables from 
│ │ │ │ +00055d30: 5345 2048 2069 7320 6120 6861 7368 5461  SE H is a hashTa
│ │ │ │ +00055d40: 626c 650a 6f66 206d 2070 6169 7273 207b  ble.of m pairs {
│ │ │ │ +00055d50: 692c 2074 5f69 7d2c 2077 6865 7265 2074  i, t_i}, where t
│ │ │ │ +00055d60: 6865 2074 5f69 2061 7265 2052 452d 6d6f  he t_i are RE-mo
│ │ │ │ +00055d70: 6475 6c65 732c 2061 6e64 2074 6865 2069  dules, and the i
│ │ │ │ +00055d80: 2061 7265 2063 6f6e 7365 6375 7469 7665   are consecutive
│ │ │ │ +00055d90: 0a69 6e74 6567 6572 2e20 7068 6920 6973  .integer. phi is
│ │ │ │ +00055da0: 2061 2068 6173 682d 7461 626c 6520 6f66   a hash-table of
│ │ │ │ +00055db0: 2068 6f6d 6f67 656e 656f 7573 206d 6170   homogeneous map
│ │ │ │ +00055dc0: 7320 7068 6923 7b6a 2c69 7d3a 2048 2369  s phi#{j,i}: H#i
│ │ │ │ +00055dd0: 2a2a 465f 6a5c 746f 2048 2328 692b 3129  **F_j\to H#(i+1)
│ │ │ │ +00055de0: 0a77 6865 7265 2046 5f6a 203d 2073 6f75  .where F_j = sou
│ │ │ │ +00055df0: 7263 6520 2845 5f7b 6a7d 203d 206d 6174  rce (E_{j} = mat
│ │ │ │ +00055e00: 7269 7820 7b7b 655f 6a7d 7d29 2e20 5468  rix {{e_j}}). Th
│ │ │ │ +00055e10: 7573 2074 6865 206d 6170 7320 7023 7b6a  us the maps p#{j
│ │ │ │ +00055e20: 2c69 7d20 3d20 2845 5f6a 207c 7c0a 2d70  ,i} = (E_j ||.-p
│ │ │ │ +00055e30: 6869 237b 6a2c 697d 293a 2074 5f69 2a2a  hi#{j,i}): t_i**
│ │ │ │ +00055e40: 465f 6a20 5c74 6f20 745f 692b 2b74 5f7b  F_j \to t_i++t_{
│ │ │ │ +00055e50: 2869 2b31 297d 2c20 6172 6520 686f 6d6f  (i+1)}, are homo
│ │ │ │ +00055e60: 6765 6e65 6f75 732e 2054 6865 2073 6372  geneous. The scr
│ │ │ │ +00055e70: 6970 7420 7265 7475 726e 7320 4d0a 3d20  ipt returns M.= 
│ │ │ │ +00055e80: 5c6f 706c 7573 5f69 2054 5f20 6173 2061  \oplus_i T_ as a
│ │ │ │ +00055e90: 6e20 5345 2d6d 6f64 756c 652c 2063 6f6d  n SE-module, com
│ │ │ │ +00055ea0: 7075 7465 6420 6173 2074 6865 2071 756f  puted as the quo
│ │ │ │ +00055eb0: 7469 656e 7420 6f66 2050 203a 3d20 5c6f  tient of P := \o
│ │ │ │ +00055ec0: 706c 7573 2054 5f69 0a6f 6274 6169 6e65  plus T_i.obtaine
│ │ │ │ +00055ed0: 6420 6279 2066 6163 746f 7269 6e67 206f  d by factoring o
│ │ │ │ +00055ee0: 7574 2074 6865 2073 756d 206f 6620 7468  ut the sum of th
│ │ │ │ +00055ef0: 6520 696d 6167 6573 206f 6620 7468 6520  e images of the 
│ │ │ │ +00055f00: 6d61 7073 2070 237b 6a2c 697d 0a0a 5468  maps p#{j,i}..Th
│ │ │ │ +00055f10: 6520 4861 7368 7461 626c 6520 7068 6920  e Hashtable phi 
│ │ │ │ +00055f20: 6861 7320 6b65 7973 206f 6620 7468 6520  has keys of the 
│ │ │ │ +00055f30: 666f 726d 207b 6a2c 697d 2077 6865 7265  form {j,i} where
│ │ │ │ +00055f40: 206a 2072 756e 7320 6672 6f6d 2030 2074   j runs from 0 t
│ │ │ │ +00055f50: 6f20 632d 312c 2069 2061 6e64 0a69 2b31  o c-1, i and.i+1
│ │ │ │ +00055f60: 2061 7265 206b 6579 7320 6f66 2048 2c20   are keys of H, 
│ │ │ │ +00055f70: 616e 6420 7068 6923 7b6a 2c69 7d20 6973  and phi#{j,i} is
│ │ │ │ +00055f80: 2074 6865 206d 6170 2066 726f 6d20 2873   the map from (s
│ │ │ │ +00055f90: 6f75 7263 6520 455f 7b69 7d29 2a2a 4823  ource E_{i})**H#
│ │ │ │ +00055fa0: 6920 746f 2048 2328 692b 3129 0a74 6861  i to H#(i+1).tha
│ │ │ │ +00055fb0: 7420 7769 6c6c 2062 6520 6964 656e 7469  t will be identi
│ │ │ │ +00055fc0: 6669 6564 2077 6974 6820 7468 6520 6163  fied with the ac
│ │ │ │ +00055fd0: 7469 6f6e 206f 6620 455f 7b6a 7d2e 0a0a  tion of E_{j}...
│ │ │ │ +00055fe0: 5468 6520 7363 7269 7074 2069 7320 7573  The script is us
│ │ │ │ +00055ff0: 6564 2069 6e20 626f 7468 2074 6865 2073  ed in both the s
│ │ │ │ +00056000: 696e 676c 7920 6772 6164 6564 2063 6173  ingly graded cas
│ │ │ │ +00056010: 652c 2066 6f72 2065 7861 6d70 6c65 2069  e, for example i
│ │ │ │ +00056020: 6e0a 6578 7465 7269 6f72 546f 724d 6f64  n.exteriorTorMod
│ │ │ │ +00056030: 756c 6528 6666 2c4d 2920 616e 6420 696e  ule(ff,M) and in
│ │ │ │ +00056040: 2074 6865 2062 6967 7261 6465 6420 6361   the bigraded ca
│ │ │ │ +00056050: 7365 2c20 666f 7220 6578 616d 706c 6520  se, for example 
│ │ │ │ +00056060: 696e 0a65 7874 6572 696f 7254 6f72 4d6f  in.exteriorTorMo
│ │ │ │ +00056070: 6475 6c65 2866 662c 4d2c 4e29 2e0a 0a49  dule(ff,M,N)...I
│ │ │ │ +00056080: 6e20 7468 6520 666f 6c6c 6f77 696e 6720  n the following 
│ │ │ │ +00056090: 7765 2075 7365 206d 616b 654d 6f64 756c  we use makeModul
│ │ │ │ +000560a0: 6520 746f 2063 6f6e 7374 7275 6374 2062  e to construct b
│ │ │ │ +000560b0: 7920 6861 6e64 2061 2066 7265 6520 6d6f  y hand a free mo
│ │ │ │ +000560c0: 6475 6c65 206f 6620 7261 6e6b 2031 0a6f  dule of rank 1.o
│ │ │ │ +000560d0: 7665 7220 7468 6520 6578 7465 7269 6f72  ver the exterior
│ │ │ │ +000560e0: 2061 6c67 6562 7261 206f 6e20 782c 792c   algebra on x,y,
│ │ │ │ +000560f0: 2073 7461 7274 696e 6720 7769 7468 2074   starting with t
│ │ │ │ +00056100: 6865 2063 6f6e 7374 7275 6374 696f 6e20  he construction 
│ │ │ │ +00056110: 6f66 2061 206d 6f64 756c 650a 6f76 6572  of a module.over
│ │ │ │ +00056120: 2061 2062 6968 6f6d 6f67 656e 656f 7573   a bihomogeneous
│ │ │ │ +00056130: 2072 696e 672e 0a0a 2b2d 2d2d 2d2d 2d2d   ring...+-------
│ │ │ │ +00056140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000561a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053  -------+.|i1 : S
│ │ │ │ -000561b0: 4520 3d20 5a5a 2f31 3031 5b61 2c62 2c63  E = ZZ/101[a,b,c
│ │ │ │ -000561c0: 2c78 2c79 2c44 6567 7265 6573 3d3e 746f  ,x,y,Degrees=>to
│ │ │ │ -000561d0: 4c69 7374 2833 3a7b 312c 307d 297c 746f  List(3:{1,0})|to
│ │ │ │ -000561e0: 4c69 7374 2832 3a7b 312c 317d 292c 2020  List(2:{1,1}),  
│ │ │ │ -000561f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056180: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5345  ------+.|i1 : SE
│ │ │ │ +00056190: 203d 205a 5a2f 3130 315b 612c 622c 632c   = ZZ/101[a,b,c,
│ │ │ │ +000561a0: 782c 792c 4465 6772 6565 733d 3e74 6f4c  x,y,Degrees=>toL
│ │ │ │ +000561b0: 6973 7428 333a 7b31 2c30 7d29 7c74 6f4c  ist(3:{1,0})|toL
│ │ │ │ +000561c0: 6973 7428 323a 7b31 2c31 7d29 2c20 2020  ist(2:{1,1}),   
│ │ │ │ +000561d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000561e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000561f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056220: 2020 2020 2020 7c0a 7c6f 3120 3d20 5345        |.|o1 = SE
│ │ │ │  00056230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056240: 2020 2020 2020 207c 0a7c 6f31 203d 2053         |.|o1 = S
│ │ │ │ -00056250: 4520 2020 2020 2020 2020 2020 2020 2020  E               
│ │ │ │ +00056240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056270: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056290: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000562a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000562b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000562c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000562d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000562e0: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -000562f0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2032  olynomialRing, 2
│ │ │ │ -00056300: 2073 6b65 7720 636f 6d6d 7574 6174 6976   skew commutativ
│ │ │ │ -00056310: 6520 7661 7269 6162 6c65 2873 2920 2020  e variable(s)   
│ │ │ │ -00056320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056330: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +000562c0: 2020 2020 2020 7c0a 7c6f 3120 3a20 506f        |.|o1 : Po
│ │ │ │ +000562d0: 6c79 6e6f 6d69 616c 5269 6e67 2c20 3220  lynomialRing, 2 
│ │ │ │ +000562e0: 736b 6577 2063 6f6d 6d75 7461 7469 7665  skew commutative
│ │ │ │ +000562f0: 2076 6172 6961 626c 6528 7329 2020 2020   variable(s)    
│ │ │ │ +00056300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056310: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00056320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056380: 2d2d 2d2d 2d2d 2d7c 0a7c 536b 6577 436f  -------|.|SkewCo
│ │ │ │ -00056390: 6d6d 7574 6174 6976 653d 3e7b 782c 797d  mmutative=>{x,y}
│ │ │ │ -000563a0: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -000563b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000563c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000563d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00056360: 2d2d 2d2d 2d2d 7c0a 7c53 6b65 7743 6f6d  ------|.|SkewCom
│ │ │ │ +00056370: 6d75 7461 7469 7665 3d3e 7b78 2c79 7d5d  mutative=>{x,y}]
│ │ │ │ +00056380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000563a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000563b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000563c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000563d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000563e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000563f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056420: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052  -------+.|i2 : R
│ │ │ │ -00056430: 4520 3d20 5345 2f69 6465 616c 2261 322c  E = SE/ideal"a2,
│ │ │ │ -00056440: 6232 2c63 3222 2020 2020 2020 2020 2020  b2,c2"          
│ │ │ │ -00056450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056400: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5245  ------+.|i2 : RE
│ │ │ │ +00056410: 203d 2053 452f 6964 6561 6c22 6132 2c62   = SE/ideal"a2,b
│ │ │ │ +00056420: 322c 6332 2220 2020 2020 2020 2020 2020  2,c2"           
│ │ │ │ +00056430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056450: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056470: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000564a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000564a0: 2020 2020 2020 7c0a 7c6f 3220 3d20 5245        |.|o2 = RE
│ │ │ │  000564b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000564c0: 2020 2020 2020 207c 0a7c 6f32 203d 2052         |.|o2 = R
│ │ │ │ -000564d0: 4520 2020 2020 2020 2020 2020 2020 2020  E               
│ │ │ │ +000564c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000564d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000564e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000564f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000564f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056510: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056560: 2020 2020 2020 207c 0a7c 6f32 203a 2051         |.|o2 : Q
│ │ │ │ -00056570: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00056540: 2020 2020 2020 7c0a 7c6f 3220 3a20 5175        |.|o2 : Qu
│ │ │ │ +00056550: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +00056560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000565a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000565b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00056590: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000565a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000565b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000565c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000565d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000565e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000565f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056600: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2054  -------+.|i3 : T
│ │ │ │ -00056610: 203d 2068 6173 6854 6162 6c65 207b 7b30   = hashTable {{0
│ │ │ │ -00056620: 2c52 455e 317d 2c7b 312c 5245 5e7b 323a  ,RE^1},{1,RE^{2:
│ │ │ │ -00056630: 7b20 2d31 2c2d 317d 7d7d 2c20 7b32 2c52  { -1,-1}}}, {2,R
│ │ │ │ -00056640: 455e 7b7b 202d 322c 2d32 7d7d 7d7d 2020  E^{{ -2,-2}}}}  
│ │ │ │ -00056650: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000565e0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 5420  ------+.|i3 : T 
│ │ │ │ +000565f0: 3d20 6861 7368 5461 626c 6520 7b7b 302c  = hashTable {{0,
│ │ │ │ +00056600: 5245 5e31 7d2c 7b31 2c52 455e 7b32 3a7b  RE^1},{1,RE^{2:{
│ │ │ │ +00056610: 202d 312c 2d31 7d7d 7d2c 207b 322c 5245   -1,-1}}}, {2,RE
│ │ │ │ +00056620: 5e7b 7b20 2d32 2c2d 327d 7d7d 7d20 2020  ^{{ -2,-2}}}}   
│ │ │ │ +00056630: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056680: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056690: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +000566a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000566b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566c0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -000566d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000566f0: 2020 2020 2020 207c 0a7c 6f33 203d 2048         |.|o3 = H
│ │ │ │ -00056700: 6173 6854 6162 6c65 7b30 203d 3e20 5245  ashTable{0 => RE
│ │ │ │ -00056710: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ -00056720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056740: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000566c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000566d0: 2020 2020 2020 7c0a 7c6f 3320 3d20 4861        |.|o3 = Ha
│ │ │ │ +000566e0: 7368 5461 626c 657b 3020 3d3e 2052 4520  shTable{0 => RE 
│ │ │ │ +000566f0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00056700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056720: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056730: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00056740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056760: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00056770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056790: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000567a0: 2020 2020 2020 2020 2031 203d 3e20 5245           1 => RE
│ │ │ │ +00056760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056770: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056780: 2020 2020 2020 2020 3120 3d3e 2052 4520          1 => RE 
│ │ │ │ +00056790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000567a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000567b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000567c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000567d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000567e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000567c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000567d0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +000567e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000567f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056800: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00056810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056830: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056840: 2020 2020 2020 2020 2032 203d 3e20 5245           2 => RE
│ │ │ │ +00056800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056810: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056820: 2020 2020 2020 2020 3220 3d3e 2052 4520          2 => RE 
│ │ │ │ +00056830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056860: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056880: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000568a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000568b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000568c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000568d0: 2020 2020 2020 207c 0a7c 6f33 203a 2048         |.|o3 : H
│ │ │ │ -000568e0: 6173 6854 6162 6c65 2020 2020 2020 2020  ashTable        
│ │ │ │ +000568b0: 2020 2020 2020 7c0a 7c6f 3320 3a20 4861        |.|o3 : Ha
│ │ │ │ +000568c0: 7368 5461 626c 6520 2020 2020 2020 2020  shTable         
│ │ │ │ +000568d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000568e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000568f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056920: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00056900: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00056910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056970: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2045  -------+.|i4 : E
│ │ │ │ -00056980: 203d 206d 6174 7269 787b 7b78 2c79 7d7d   = matrix{{x,y}}
│ │ │ │ +00056950: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 4520  ------+.|i4 : E 
│ │ │ │ +00056960: 3d20 6d61 7472 6978 7b7b 782c 797d 7d20  = matrix{{x,y}} 
│ │ │ │ +00056970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000569a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000569a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000569b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000569c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000569c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000569e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000569f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a10: 2020 2020 2020 207c 0a7c 6f34 203d 207c         |.|o4 = |
│ │ │ │ -00056a20: 2078 2079 207c 2020 2020 2020 2020 2020   x y |          
│ │ │ │ +000569f0: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20        |.|o4 = | 
│ │ │ │ +00056a00: 7820 7920 7c20 2020 2020 2020 2020 2020  x y |           
│ │ │ │ +00056a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056a40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056ac0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00056ad0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00056ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056b00: 2020 2020 2020 207c 0a7c 6f34 203a 204d         |.|o4 : M
│ │ │ │ -00056b10: 6174 7269 7820 5245 2020 3c2d 2d20 5245  atrix RE  <-- RE
│ │ │ │ +00056a90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056aa0: 2020 2020 2020 2031 2020 2020 2020 2032         1       2
│ │ │ │ +00056ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056ae0: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d61        |.|o4 : Ma
│ │ │ │ +00056af0: 7472 6978 2052 4520 203c 2d2d 2052 4520  trix RE  <-- RE 
│ │ │ │ +00056b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056b50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00056b30: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00056b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056ba0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2046  -------+.|i5 : F
│ │ │ │ -00056bb0: 3d61 7070 6c79 2832 2c20 6a2d 3e20 736f  =apply(2, j-> so
│ │ │ │ -00056bc0: 7572 6365 2045 5f7b 6a7d 2920 2020 2020  urce E_{j})     
│ │ │ │ -00056bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056b80: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 463d  ------+.|i5 : F=
│ │ │ │ +00056b90: 6170 706c 7928 322c 206a 2d3e 2073 6f75  apply(2, j-> sou
│ │ │ │ +00056ba0: 7263 6520 455f 7b6a 7d29 2020 2020 2020  rce E_{j})      
│ │ │ │ +00056bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056bd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056bf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056c50: 2020 3120 2020 2031 2020 2020 2020 2020    1    1        
│ │ │ │ +00056c20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056c30: 2031 2020 2020 3120 2020 2020 2020 2020   1    1         
│ │ │ │ +00056c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056c90: 2020 2020 2020 207c 0a7c 6f35 203d 207b         |.|o5 = {
│ │ │ │ -00056ca0: 5245 202c 2052 4520 7d20 2020 2020 2020  RE , RE }       
│ │ │ │ +00056c70: 2020 2020 2020 7c0a 7c6f 3520 3d20 7b52        |.|o5 = {R
│ │ │ │ +00056c80: 4520 2c20 5245 207d 2020 2020 2020 2020  E , RE }        
│ │ │ │ +00056c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056cc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ce0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d30: 2020 2020 2020 207c 0a7c 6f35 203a 204c         |.|o5 : L
│ │ │ │ -00056d40: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +00056d10: 2020 2020 2020 7c0a 7c6f 3520 3a20 4c69        |.|o5 : Li
│ │ │ │ +00056d20: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +00056d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056d80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00056d60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00056d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00056d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00056da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00056dd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2070  -------+.|i6 : p
│ │ │ │ -00056de0: 6869 203d 2068 6173 6854 6162 6c65 7b20  hi = hashTable{ 
│ │ │ │ -00056df0: 7b7b 302c 307d 2c20 6d61 7028 5423 312c  {{0,0}, map(T#1,
│ │ │ │ -00056e00: 2046 5f30 2a2a 5423 302c 2054 2331 5f7b   F_0**T#0, T#1_{
│ │ │ │ -00056e10: 307d 297d 2c7b 7b31 2c30 7d2c 206d 6170  0})},{{1,0}, map
│ │ │ │ -00056e20: 2854 2331 2c20 207c 0a7c 2020 2020 2020  (T#1,  |.|      
│ │ │ │ +00056db0: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 7068  ------+.|i6 : ph
│ │ │ │ +00056dc0: 6920 3d20 6861 7368 5461 626c 657b 207b  i = hashTable{ {
│ │ │ │ +00056dd0: 7b30 2c30 7d2c 206d 6170 2854 2331 2c20  {0,0}, map(T#1, 
│ │ │ │ +00056de0: 465f 302a 2a54 2330 2c20 5423 315f 7b30  F_0**T#0, T#1_{0
│ │ │ │ +00056df0: 7d29 7d2c 7b7b 312c 307d 2c20 6d61 7028  })},{{1,0}, map(
│ │ │ │ +00056e00: 5423 312c 2020 7c0a 7c20 2020 2020 2020  T#1,  |.|       
│ │ │ │ +00056e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00056e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056e70: 2020 2020 2020 207c 0a7c 6f36 203d 2048         |.|o6 = H
│ │ │ │ -00056e80: 6173 6854 6162 6c65 7b7b 302c 2030 7d20  ashTable{{0, 0} 
│ │ │ │ -00056e90: 3d3e 207b 312c 2031 7d20 7c20 3120 7c20  => {1, 1} | 1 | 
│ │ │ │ -00056ea0: 2020 7d20 2020 2020 2020 2020 2020 2020    }             
│ │ │ │ +00056e50: 2020 2020 2020 7c0a 7c6f 3620 3d20 4861        |.|o6 = Ha
│ │ │ │ +00056e60: 7368 5461 626c 657b 7b30 2c20 307d 203d  shTable{{0, 0} =
│ │ │ │ +00056e70: 3e20 7b31 2c20 317d 207c 2031 207c 2020  > {1, 1} | 1 |  
│ │ │ │ +00056e80: 207d 2020 2020 2020 2020 2020 2020 2020   }              
│ │ │ │ +00056e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056ea0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056ec0: 2020 7b31 2c20 317d 207c 2030 207c 2020    {1, 1} | 0 |  
│ │ │ │  00056ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ee0: 2020 207b 312c 2031 7d20 7c20 3020 7c20     {1, 1} | 0 | 
│ │ │ │ -00056ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056f20: 2020 2020 2020 2020 207b 302c 2031 7d20           {0, 1} 
│ │ │ │ -00056f30: 3d3e 207b 322c 2032 7d20 7c20 3020 3120  => {2, 2} | 0 1 
│ │ │ │ -00056f40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00056f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056f60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00056f70: 2020 2020 2020 2020 207b 312c 2030 7d20           {1, 0} 
│ │ │ │ -00056f80: 3d3e 207b 312c 2031 7d20 7c20 3020 7c20  => {1, 1} | 0 | 
│ │ │ │ -00056f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056ef0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056f00: 2020 2020 2020 2020 7b30 2c20 317d 203d          {0, 1} =
│ │ │ │ +00056f10: 3e20 7b32 2c20 327d 207c 2030 2031 207c  > {2, 2} | 0 1 |
│ │ │ │ +00056f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056f40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056f50: 2020 2020 2020 2020 7b31 2c20 307d 203d          {1, 0} =
│ │ │ │ +00056f60: 3e20 7b31 2c20 317d 207c 2030 207c 2020  > {1, 1} | 0 |  
│ │ │ │ +00056f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056f90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00056fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056fb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00056fb0: 2020 7b31 2c20 317d 207c 2031 207c 2020    {1, 1} | 1 |  
│ │ │ │  00056fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056fd0: 2020 207b 312c 2031 7d20 7c20 3120 7c20     {1, 1} | 1 | 
│ │ │ │ -00056fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00056ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057000: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057010: 2020 2020 2020 2020 207b 312c 2031 7d20           {1, 1} 
│ │ │ │ -00057020: 3d3e 207b 322c 2032 7d20 7c20 2d31 2030  => {2, 2} | -1 0
│ │ │ │ -00057030: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00056fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00056fe0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00056ff0: 2020 2020 2020 2020 7b31 2c20 317d 203d          {1, 1} =
│ │ │ │ +00057000: 3e20 7b32 2c20 327d 207c 202d 3120 3020  > {2, 2} | -1 0 
│ │ │ │ +00057010: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00057020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057030: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00057040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057050: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570a0: 2020 2020 2020 207c 0a7c 6f36 203a 2048         |.|o6 : H
│ │ │ │ -000570b0: 6173 6854 6162 6c65 2020 2020 2020 2020  ashTable        
│ │ │ │ +00057080: 2020 2020 2020 7c0a 7c6f 3620 3a20 4861        |.|o6 : Ha
│ │ │ │ +00057090: 7368 5461 626c 6520 2020 2020 2020 2020  shTable         
│ │ │ │ +000570a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000570b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000570c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000570f0: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +000570d0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +000570e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000570f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057140: 2d2d 2d2d 2d2d 2d7c 0a7c 465f 312a 2a54  -------|.|F_1**T
│ │ │ │ -00057150: 2330 2c20 5423 315f 7b31 7d29 7d2c 7b7b  #0, T#1_{1})},{{
│ │ │ │ -00057160: 302c 317d 2c20 6d61 7028 5423 322c 2046  0,1}, map(T#2, F
│ │ │ │ -00057170: 5f30 2a2a 5423 312c 2054 2331 5e7b 317d  _0**T#1, T#1^{1}
│ │ │ │ -00057180: 297d 2c20 7b7b 312c 317d 2c20 2d6d 6170  )}, {{1,1}, -map
│ │ │ │ -00057190: 2854 2332 2c20 207c 0a7c 2d2d 2d2d 2d2d  (T#2,  |.|------
│ │ │ │ +00057120: 2d2d 2d2d 2d2d 7c0a 7c46 5f31 2a2a 5423  ------|.|F_1**T#
│ │ │ │ +00057130: 302c 2054 2331 5f7b 317d 297d 2c7b 7b30  0, T#1_{1})},{{0
│ │ │ │ +00057140: 2c31 7d2c 206d 6170 2854 2332 2c20 465f  ,1}, map(T#2, F_
│ │ │ │ +00057150: 302a 2a54 2331 2c20 5423 315e 7b31 7d29  0**T#1, T#1^{1})
│ │ │ │ +00057160: 7d2c 207b 7b31 2c31 7d2c 202d 6d61 7028  }, {{1,1}, -map(
│ │ │ │ +00057170: 5423 322c 2020 7c0a 7c2d 2d2d 2d2d 2d2d  T#2,  |.|-------
│ │ │ │ +00057180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000571a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000571b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000571c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000571d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000571e0: 2d2d 2d2d 2d2d 2d7c 0a7c 465f 312a 2a54  -------|.|F_1**T
│ │ │ │ -000571f0: 2331 2c20 5423 315e 7b30 7d29 7d7d 2020  #1, T#1^{0})}}  
│ │ │ │ +000571c0: 2d2d 2d2d 2d2d 7c0a 7c46 5f31 2a2a 5423  ------|.|F_1**T#
│ │ │ │ +000571d0: 312c 2054 2331 5e7b 307d 297d 7d20 2020  1, T#1^{0})}}   
│ │ │ │ +000571e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000571f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057230: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057210: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00057220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057280: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2061  -------+.|i7 : a
│ │ │ │ -00057290: 7070 6c79 286b 6579 7320 7068 692c 206b  pply(keys phi, k
│ │ │ │ -000572a0: 2d3e 6973 486f 6d6f 6765 6e65 6f75 7320  ->isHomogeneous 
│ │ │ │ -000572b0: 7068 6923 6b29 2020 2020 2020 2020 2020  phi#k)          
│ │ │ │ +00057260: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6170  ------+.|i7 : ap
│ │ │ │ +00057270: 706c 7928 6b65 7973 2070 6869 2c20 6b2d  ply(keys phi, k-
│ │ │ │ +00057280: 3e69 7348 6f6d 6f67 656e 656f 7573 2070  >isHomogeneous p
│ │ │ │ +00057290: 6869 236b 2920 2020 2020 2020 2020 2020  hi#k)           
│ │ │ │ +000572a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000572b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000572c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000572d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000572d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000572e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000572f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057320: 2020 2020 2020 207c 0a7c 6f37 203d 207b         |.|o7 = {
│ │ │ │ -00057330: 7472 7565 2c20 7472 7565 2c20 7472 7565  true, true, true
│ │ │ │ -00057340: 2c20 7472 7565 7d20 2020 2020 2020 2020  , true}         
│ │ │ │ -00057350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057300: 2020 2020 2020 7c0a 7c6f 3720 3d20 7b74        |.|o7 = {t
│ │ │ │ +00057310: 7275 652c 2074 7275 652c 2074 7275 652c  rue, true, true,
│ │ │ │ +00057320: 2074 7275 657d 2020 2020 2020 2020 2020   true}          
│ │ │ │ +00057330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057350: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00057360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057370: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000573a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000573b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000573c0: 2020 2020 2020 207c 0a7c 6f37 203a 204c         |.|o7 : L
│ │ │ │ -000573d0: 6973 7420 2020 2020 2020 2020 2020 2020  ist             
│ │ │ │ +000573a0: 2020 2020 2020 7c0a 7c6f 3720 3a20 4c69        |.|o7 : Li
│ │ │ │ +000573b0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +000573c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000573d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000573e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000573f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057410: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000573f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00057400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057460: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2058  -------+.|i8 : X
│ │ │ │ -00057470: 203d 206d 616b 654d 6f64 756c 6528 542c   = makeModule(T,
│ │ │ │ -00057480: 452c 7068 6929 2020 2020 2020 2020 2020  E,phi)          
│ │ │ │ -00057490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057440: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 5820  ------+.|i8 : X 
│ │ │ │ +00057450: 3d20 6d61 6b65 4d6f 6475 6c65 2854 2c45  = makeModule(T,E
│ │ │ │ +00057460: 2c70 6869 2920 2020 2020 2020 2020 2020  ,phi)           
│ │ │ │ +00057470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057490: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000574a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000574b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000574b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000574c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000574d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000574e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000574f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057500: 2020 2020 2020 207c 0a7c 6f38 203d 2063         |.|o8 = c
│ │ │ │ -00057510: 6f6b 6572 6e65 6c20 7b30 2c20 307d 207c  okernel {0, 0} |
│ │ │ │ -00057520: 202d 7820 3020 2030 2020 2d79 2030 2020   -x 0  0  -y 0  
│ │ │ │ -00057530: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00057540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057550: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057560: 2020 2020 2020 2020 7b31 2c20 317d 207c          {1, 1} |
│ │ │ │ -00057570: 2031 2020 2d78 2030 2020 3020 202d 7920   1  -x 0  0  -y 
│ │ │ │ -00057580: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ -00057590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000575a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000575b0: 2020 2020 2020 2020 7b31 2c20 317d 207c          {1, 1} |
│ │ │ │ -000575c0: 2030 2020 3020 202d 7820 3120 2030 2020   0  0  -x 1  0  
│ │ │ │ -000575d0: 2d79 207c 2020 2020 2020 2020 2020 2020  -y |            
│ │ │ │ -000575e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000575f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057600: 2020 2020 2020 2020 7b32 2c20 327d 207c          {2, 2} |
│ │ │ │ -00057610: 2030 2020 3020 2031 2020 3020 202d 3120   0  0  1  0  -1 
│ │ │ │ -00057620: 3020 207c 2020 2020 2020 2020 2020 2020  0  |            
│ │ │ │ +000574e0: 2020 2020 2020 7c0a 7c6f 3820 3d20 636f        |.|o8 = co
│ │ │ │ +000574f0: 6b65 726e 656c 207b 302c 2030 7d20 7c20  kernel {0, 0} | 
│ │ │ │ +00057500: 2d78 2030 2020 3020 202d 7920 3020 2030  -x 0  0  -y 0  0
│ │ │ │ +00057510: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00057520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057530: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00057540: 2020 2020 2020 207b 312c 2031 7d20 7c20         {1, 1} | 
│ │ │ │ +00057550: 3120 202d 7820 3020 2030 2020 2d79 2030  1  -x 0  0  -y 0
│ │ │ │ +00057560: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00057570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057580: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00057590: 2020 2020 2020 207b 312c 2031 7d20 7c20         {1, 1} | 
│ │ │ │ +000575a0: 3020 2030 2020 2d78 2031 2020 3020 202d  0  0  -x 1  0  -
│ │ │ │ +000575b0: 7920 7c20 2020 2020 2020 2020 2020 2020  y |             
│ │ │ │ +000575c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000575d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000575e0: 2020 2020 2020 207b 322c 2032 7d20 7c20         {2, 2} | 
│ │ │ │ +000575f0: 3020 2030 2020 3120 2030 2020 2d31 2030  0  0  1  0  -1 0
│ │ │ │ +00057600: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00057610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057620: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00057630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057640: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057670: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00057680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057690: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057690: 2020 2020 2020 2034 2020 2020 2020 2020         4        
│ │ │ │  000576a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000576b0: 2020 2020 2020 2020 3420 2020 2020 2020          4       
│ │ │ │ -000576c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000576d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000576e0: 2020 2020 2020 207c 0a7c 6f38 203a 2052         |.|o8 : R
│ │ │ │ -000576f0: 452d 6d6f 6475 6c65 2c20 7175 6f74 6965  E-module, quotie
│ │ │ │ -00057700: 6e74 206f 6620 5245 2020 2020 2020 2020  nt of RE        
│ │ │ │ -00057710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057730: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000576b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000576c0: 2020 2020 2020 7c0a 7c6f 3820 3a20 5245        |.|o8 : RE
│ │ │ │ +000576d0: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +000576e0: 7420 6f66 2052 4520 2020 2020 2020 2020  t of RE         
│ │ │ │ +000576f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057710: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00057720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057780: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2069  -------+.|i9 : i
│ │ │ │ -00057790: 7348 6f6d 6f67 656e 656f 7573 2058 2020  sHomogeneous X  
│ │ │ │ +00057760: 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 6973  ------+.|i9 : is
│ │ │ │ +00057770: 486f 6d6f 6765 6e65 6f75 7320 5820 2020  Homogeneous X   
│ │ │ │ +00057780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000577a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000577b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000577b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000577c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000577d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000577d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000577e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000577f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057820: 2020 2020 2020 207c 0a7c 6f39 203d 2074         |.|o9 = t
│ │ │ │ -00057830: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ +00057800: 2020 2020 2020 7c0a 7c6f 3920 3d20 7472        |.|o9 = tr
│ │ │ │ +00057810: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ +00057820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057870: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057850: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00057860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000578a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000578b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000578c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │ -000578d0: 7120 3d20 6d61 7028 5a5a 2f31 3031 5b78  q = map(ZZ/101[x
│ │ │ │ -000578e0: 2c79 2c20 536b 6577 436f 6d6d 7574 6174  ,y, SkewCommutat
│ │ │ │ -000578f0: 6976 6520 3d3e 2074 7275 652c 2044 6567  ive => true, Deg
│ │ │ │ -00057900: 7265 654d 6170 203d 3e20 642d 3e7b 645f  reeMap => d->{d_
│ │ │ │ -00057910: 317d 5d2c 2020 207c 0a7c 2020 2020 2020  1}],   |.|      
│ │ │ │ +000578a0: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2071  ------+.|i10 : q
│ │ │ │ +000578b0: 203d 206d 6170 285a 5a2f 3130 315b 782c   = map(ZZ/101[x,
│ │ │ │ +000578c0: 792c 2053 6b65 7743 6f6d 6d75 7461 7469  y, SkewCommutati
│ │ │ │ +000578d0: 7665 203d 3e20 7472 7565 2c20 4465 6772  ve => true, Degr
│ │ │ │ +000578e0: 6565 4d61 7020 3d3e 2064 2d3e 7b64 5f31  eeMap => d->{d_1
│ │ │ │ +000578f0: 7d5d 2c20 2020 7c0a 7c20 2020 2020 2020  }],   |.|       
│ │ │ │ +00057900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057960: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057970: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ +00057940: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00057950: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │ +00057960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000579a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000579b0: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ -000579c0: 6d61 7020 282d 2d2d 5b78 2e2e 795d 2c20  map (---[x..y], 
│ │ │ │ -000579d0: 5245 2c20 7b30 2c20 302c 2030 2c20 782c  RE, {0, 0, 0, x,
│ │ │ │ -000579e0: 2079 7d29 2020 2020 2020 2020 2020 2020   y})            
│ │ │ │ -000579f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057a00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057a10: 2020 2020 2031 3031 2020 2020 2020 2020       101        
│ │ │ │ +00057990: 2020 2020 2020 7c0a 7c6f 3130 203d 206d        |.|o10 = m
│ │ │ │ +000579a0: 6170 2028 2d2d 2d5b 782e 2e79 5d2c 2052  ap (---[x..y], R
│ │ │ │ +000579b0: 452c 207b 302c 2030 2c20 302c 2078 2c20  E, {0, 0, 0, x, 
│ │ │ │ +000579c0: 797d 2920 2020 2020 2020 2020 2020 2020  y})             
│ │ │ │ +000579d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000579e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000579f0: 2020 2020 3130 3120 2020 2020 2020 2020      101         
│ │ │ │ +00057a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057a30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00057a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057a50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057aa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057ab0: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ +00057a80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00057a90: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +00057aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057af0: 2020 2020 2020 207c 0a7c 6f31 3020 3a20         |.|o10 : 
│ │ │ │ -00057b00: 5269 6e67 4d61 7020 2d2d 2d5b 782e 2e79  RingMap ---[x..y
│ │ │ │ -00057b10: 5d20 3c2d 2d20 5245 2020 2020 2020 2020  ] <-- RE        
│ │ │ │ -00057b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057b40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057b50: 2020 2020 2020 2020 3130 3120 2020 2020          101     
│ │ │ │ +00057ad0: 2020 2020 2020 7c0a 7c6f 3130 203a 2052        |.|o10 : R
│ │ │ │ +00057ae0: 696e 674d 6170 202d 2d2d 5b78 2e2e 795d  ingMap ---[x..y]
│ │ │ │ +00057af0: 203c 2d2d 2052 4520 2020 2020 2020 2020   <-- RE         
│ │ │ │ +00057b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057b20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00057b30: 2020 2020 2020 2031 3031 2020 2020 2020         101      
│ │ │ │ +00057b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057b90: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +00057b70: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00057b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057be0: 2d2d 2d2d 2d2d 2d7c 0a7c 7269 6e67 2058  -------|.|ring X
│ │ │ │ -00057bf0: 2c20 7b33 3a30 2c78 2c79 7d29 2020 2020  , {3:0,x,y})    
│ │ │ │ +00057bc0: 2d2d 2d2d 2d2d 7c0a 7c72 696e 6720 582c  ------|.|ring X,
│ │ │ │ +00057bd0: 207b 333a 302c 782c 797d 2920 2020 2020   {3:0,x,y})     
│ │ │ │ +00057be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057c30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057c10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00057c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057c80: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ -00057c90: 7072 756e 6520 636f 6b65 7220 7120 7072  prune coker q pr
│ │ │ │ -00057ca0: 6573 656e 7461 7469 6f6e 2058 2020 2020  esentation X    
│ │ │ │ -00057cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057c60: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2070  ------+.|i11 : p
│ │ │ │ +00057c70: 7275 6e65 2063 6f6b 6572 2071 2070 7265  rune coker q pre
│ │ │ │ +00057c80: 7365 6e74 6174 696f 6e20 5820 2020 2020  sentation X     
│ │ │ │ +00057c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057cb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00057cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057cd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057d20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057d30: 2020 5a5a 2020 2020 2020 2031 2020 2020    ZZ       1    
│ │ │ │ +00057d00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00057d10: 205a 5a20 2020 2020 2020 3120 2020 2020   ZZ       1     
│ │ │ │ +00057d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057d70: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ -00057d80: 282d 2d2d 5b78 2e2e 795d 2920 2020 2020  (---[x..y])     
│ │ │ │ +00057d50: 2020 2020 2020 7c0a 7c6f 3131 203d 2028        |.|o11 = (
│ │ │ │ +00057d60: 2d2d 2d5b 782e 2e79 5d29 2020 2020 2020  ---[x..y])      
│ │ │ │ +00057d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057dc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057dd0: 2031 3031 2020 2020 2020 2020 2020 2020   101            
│ │ │ │ +00057da0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00057db0: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │ +00057dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057df0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00057e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00057e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057e70: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +00057e40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00057e50: 5a5a 2020 2020 2020 2020 2020 2020 2020  ZZ              
│ │ │ │ +00057e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057eb0: 2020 2020 2020 207c 0a7c 6f31 3120 3a20         |.|o11 : 
│ │ │ │ -00057ec0: 2d2d 2d5b 782e 2e79 5d2d 6d6f 6475 6c65  ---[x..y]-module
│ │ │ │ -00057ed0: 2c20 6672 6565 2020 2020 2020 2020 2020  , free          
│ │ │ │ -00057ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00057f10: 3130 3120 2020 2020 2020 2020 2020 2020  101             
│ │ │ │ +00057e90: 2020 2020 2020 7c0a 7c6f 3131 203a 202d        |.|o11 : -
│ │ │ │ +00057ea0: 2d2d 5b78 2e2e 795d 2d6d 6f64 756c 652c  --[x..y]-module,
│ │ │ │ +00057eb0: 2066 7265 6520 2020 2020 2020 2020 2020   free           
│ │ │ │ +00057ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057ee0: 2020 2020 2020 7c0a 7c20 2020 2020 2031        |.|      1
│ │ │ │ +00057ef0: 3031 2020 2020 2020 2020 2020 2020 2020  01              
│ │ │ │ +00057f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00057f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00057f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00057f50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00057f30: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00057f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00057f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00057f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00057fa0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -00057fb0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -00057fc0: 202a 6e6f 7465 2065 7874 6572 696f 7248   *note exteriorH
│ │ │ │ -00057fd0: 6f6d 6f6c 6f67 794d 6f64 756c 653a 2065  omologyModule: e
│ │ │ │ -00057fe0: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d  xteriorHomologyM
│ │ │ │ -00057ff0: 6f64 756c 652c 202d 2d20 4d61 6b65 2074  odule, -- Make t
│ │ │ │ -00058000: 6865 2068 6f6d 6f6c 6f67 790a 2020 2020  he homology.    
│ │ │ │ -00058010: 6f66 2061 2063 6f6d 706c 6578 2069 6e74  of a complex int
│ │ │ │ -00058020: 6f20 6120 6d6f 6475 6c65 206f 7665 7220  o a module over 
│ │ │ │ -00058030: 616e 2065 7874 6572 696f 7220 616c 6765  an exterior alge
│ │ │ │ -00058040: 6272 610a 2020 2a20 2a6e 6f74 6520 6578  bra.  * *note ex
│ │ │ │ -00058050: 7465 7269 6f72 546f 724d 6f64 756c 653a  teriorTorModule:
│ │ │ │ -00058060: 2065 7874 6572 696f 7254 6f72 4d6f 6475   exteriorTorModu
│ │ │ │ -00058070: 6c65 2c20 2d2d 2054 6f72 2061 7320 6120  le, -- Tor as a 
│ │ │ │ -00058080: 6d6f 6475 6c65 206f 7665 7220 616e 0a20  module over an. 
│ │ │ │ -00058090: 2020 2065 7874 6572 696f 7220 616c 6765     exterior alge
│ │ │ │ -000580a0: 6272 6120 6f72 2062 6967 7261 6465 6420  bra or bigraded 
│ │ │ │ -000580b0: 616c 6765 6272 610a 2020 2a20 2a6e 6f74  algebra.  * *not
│ │ │ │ -000580c0: 6520 6578 7465 7269 6f72 4578 744d 6f64  e exteriorExtMod
│ │ │ │ -000580d0: 756c 653a 2065 7874 6572 696f 7245 7874  ule: exteriorExt
│ │ │ │ -000580e0: 4d6f 6475 6c65 2c20 2d2d 2045 7874 284d  Module, -- Ext(M
│ │ │ │ -000580f0: 2c6b 2920 6f72 2045 7874 284d 2c4e 2920  ,k) or Ext(M,N) 
│ │ │ │ -00058100: 6173 2061 0a20 2020 206d 6f64 756c 6520  as a.    module 
│ │ │ │ -00058110: 6f76 6572 2061 6e20 6578 7465 7269 6f72  over an exterior
│ │ │ │ -00058120: 2061 6c67 6562 7261 0a0a 5761 7973 2074   algebra..Ways t
│ │ │ │ -00058130: 6f20 7573 6520 6d61 6b65 4d6f 6475 6c65  o use makeModule
│ │ │ │ -00058140: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -00058150: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -00058160: 6d61 6b65 4d6f 6475 6c65 2848 6173 6854  makeModule(HashT
│ │ │ │ -00058170: 6162 6c65 2c4d 6174 7269 782c 4861 7368  able,Matrix,Hash
│ │ │ │ -00058180: 5461 626c 6529 220a 0a46 6f72 2074 6865  Table)"..For the
│ │ │ │ -00058190: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ -000581a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -000581b0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ -000581c0: 206d 616b 654d 6f64 756c 653a 206d 616b   makeModule: mak
│ │ │ │ -000581d0: 654d 6f64 756c 652c 2069 7320 6120 2a6e  eModule, is a *n
│ │ │ │ -000581e0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -000581f0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244  ion:.(Macaulay2D
│ │ │ │ -00058200: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -00058210: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  n,...-----------
│ │ │ │ +00057f80: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ +00057f90: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00057fa0: 2a6e 6f74 6520 6578 7465 7269 6f72 486f  *note exteriorHo
│ │ │ │ +00057fb0: 6d6f 6c6f 6779 4d6f 6475 6c65 3a20 6578  mologyModule: ex
│ │ │ │ +00057fc0: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f  teriorHomologyMo
│ │ │ │ +00057fd0: 6475 6c65 2c20 2d2d 204d 616b 6520 7468  dule, -- Make th
│ │ │ │ +00057fe0: 6520 686f 6d6f 6c6f 6779 0a20 2020 206f  e homology.    o
│ │ │ │ +00057ff0: 6620 6120 636f 6d70 6c65 7820 696e 746f  f a complex into
│ │ │ │ +00058000: 2061 206d 6f64 756c 6520 6f76 6572 2061   a module over a
│ │ │ │ +00058010: 6e20 6578 7465 7269 6f72 2061 6c67 6562  n exterior algeb
│ │ │ │ +00058020: 7261 0a20 202a 202a 6e6f 7465 2065 7874  ra.  * *note ext
│ │ │ │ +00058030: 6572 696f 7254 6f72 4d6f 6475 6c65 3a20  eriorTorModule: 
│ │ │ │ +00058040: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ +00058050: 652c 202d 2d20 546f 7220 6173 2061 206d  e, -- Tor as a m
│ │ │ │ +00058060: 6f64 756c 6520 6f76 6572 2061 6e0a 2020  odule over an.  
│ │ │ │ +00058070: 2020 6578 7465 7269 6f72 2061 6c67 6562    exterior algeb
│ │ │ │ +00058080: 7261 206f 7220 6269 6772 6164 6564 2061  ra or bigraded a
│ │ │ │ +00058090: 6c67 6562 7261 0a20 202a 202a 6e6f 7465  lgebra.  * *note
│ │ │ │ +000580a0: 2065 7874 6572 696f 7245 7874 4d6f 6475   exteriorExtModu
│ │ │ │ +000580b0: 6c65 3a20 6578 7465 7269 6f72 4578 744d  le: exteriorExtM
│ │ │ │ +000580c0: 6f64 756c 652c 202d 2d20 4578 7428 4d2c  odule, -- Ext(M,
│ │ │ │ +000580d0: 6b29 206f 7220 4578 7428 4d2c 4e29 2061  k) or Ext(M,N) a
│ │ │ │ +000580e0: 7320 610a 2020 2020 6d6f 6475 6c65 206f  s a.    module o
│ │ │ │ +000580f0: 7665 7220 616e 2065 7874 6572 696f 7220  ver an exterior 
│ │ │ │ +00058100: 616c 6765 6272 610a 0a57 6179 7320 746f  algebra..Ways to
│ │ │ │ +00058110: 2075 7365 206d 616b 654d 6f64 756c 653a   use makeModule:
│ │ │ │ +00058120: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00058130: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d  ========..  * "m
│ │ │ │ +00058140: 616b 654d 6f64 756c 6528 4861 7368 5461  akeModule(HashTa
│ │ │ │ +00058150: 626c 652c 4d61 7472 6978 2c48 6173 6854  ble,Matrix,HashT
│ │ │ │ +00058160: 6162 6c65 2922 0a0a 466f 7220 7468 6520  able)"..For the 
│ │ │ │ +00058170: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ +00058180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ +00058190: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ +000581a0: 6d61 6b65 4d6f 6475 6c65 3a20 6d61 6b65  makeModule: make
│ │ │ │ +000581b0: 4d6f 6475 6c65 2c20 6973 2061 202a 6e6f  Module, is a *no
│ │ │ │ +000581c0: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +000581d0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ +000581e0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +000581f0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +00058200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00058230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058260: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -00058270: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -00058280: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00058290: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -000582a0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e  /macaulay2-1.26.
│ │ │ │ -000582b0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61  06+ds/M2/Macaula
│ │ │ │ -000582c0: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ -000582d0: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -000582e0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ -000582f0: 3237 3539 3a30 2e0a 1f0a 4669 6c65 3a20  2759:0....File: 
│ │ │ │ -00058300: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00058310: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00058320: 696e 666f 2c20 4e6f 6465 3a20 6d61 6b65  info, Node: make
│ │ │ │ -00058330: 542c 204e 6578 743a 206d 6174 7269 7846  T, Next: matrixF
│ │ │ │ -00058340: 6163 746f 7269 7a61 7469 6f6e 2c20 5072  actorization, Pr
│ │ │ │ -00058350: 6576 3a20 6d61 6b65 4d6f 6475 6c65 2c20  ev: makeModule, 
│ │ │ │ -00058360: 5570 3a20 546f 700a 0a6d 616b 6554 202d  Up: Top..makeT -
│ │ │ │ -00058370: 2d20 6d61 6b65 2074 6865 2043 4920 6f70  - make the CI op
│ │ │ │ -00058380: 6572 6174 6f72 7320 6f6e 2061 2063 6f6d  erators on a com
│ │ │ │ -00058390: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  plex.***********
│ │ │ │ -000583a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000583b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000583c0: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -000583d0: 2020 2020 2020 5420 3d20 6d61 6b65 5428        T = makeT(
│ │ │ │ -000583e0: 6666 2c46 2c69 290a 2020 2020 2020 2020  ff,F,i).        
│ │ │ │ -000583f0: 5420 3d20 6d61 6b65 5428 6666 2c46 2c74  T = makeT(ff,F,t
│ │ │ │ -00058400: 302c 6929 0a20 202a 2049 6e70 7574 733a  0,i).  * Inputs:
│ │ │ │ -00058410: 0a20 2020 2020 202a 2066 662c 2061 202a  .      * ff, a *
│ │ │ │ -00058420: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ -00058430: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ -00058440: 782c 2c20 3178 6320 6d61 7472 6978 2077  x,, 1xc matrix w
│ │ │ │ -00058450: 686f 7365 2065 6e74 7269 6573 2061 7265  hose entries are
│ │ │ │ -00058460: 0a20 2020 2020 2020 2061 2063 6f6d 706c  .        a compl
│ │ │ │ -00058470: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ -00058480: 2069 6e20 530a 2020 2020 2020 2a20 462c   in S.      * F,
│ │ │ │ -00058490: 2061 202a 6e6f 7465 2063 6f6d 706c 6578   a *note complex
│ │ │ │ -000584a0: 3a20 2843 6f6d 706c 6578 6573 2943 6f6d  : (Complexes)Com
│ │ │ │ -000584b0: 706c 6578 2c2c 206f 7665 7220 532f 6964  plex,, over S/id
│ │ │ │ -000584c0: 6561 6c20 6666 0a20 2020 2020 202a 2074  eal ff.      * t
│ │ │ │ -000584d0: 302c 2061 202a 6e6f 7465 206d 6174 7269  0, a *note matri
│ │ │ │ -000584e0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ -000584f0: 294d 6174 7269 782c 2c20 4349 2d6f 7065  )Matrix,, CI-ope
│ │ │ │ -00058500: 7261 746f 7220 6f6e 2046 2066 6f72 2066  rator on F for f
│ │ │ │ -00058510: 665f 3020 746f 0a20 2020 2020 2020 2062  f_0 to.        b
│ │ │ │ -00058520: 6520 7072 6573 6572 7665 640a 2020 2020  e preserved.    
│ │ │ │ -00058530: 2020 2a20 692c 2061 6e20 2a6e 6f74 6520    * i, an *note 
│ │ │ │ -00058540: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -00058550: 6179 3244 6f63 295a 5a2c 2c20 6465 6669  ay2Doc)ZZ,, defi
│ │ │ │ -00058560: 6e65 2043 4920 6f70 6572 6174 6f72 7320  ne CI operators 
│ │ │ │ -00058570: 6672 6f6d 2046 5f69 0a20 2020 2020 2020  from F_i.       
│ │ │ │ -00058580: 205c 746f 2046 5f7b 692d 327d 0a20 202a   \to F_{i-2}.  *
│ │ │ │ -00058590: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -000585a0: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973  * L, a *note lis
│ │ │ │ -000585b0: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -000585c0: 294c 6973 742c 2c20 6f66 2043 4920 6f70  )List,, of CI op
│ │ │ │ -000585d0: 6572 6174 6f72 7320 465f 6920 5c74 6f20  erators F_i \to 
│ │ │ │ -000585e0: 465f 7b69 2d32 7d0a 2020 2020 2020 2020  F_{i-2}.        
│ │ │ │ -000585f0: 636f 7272 6573 706f 6e64 696e 6720 746f  corresponding to
│ │ │ │ -00058600: 2065 6e74 7269 6573 206f 6620 6666 0a0a   entries of ff..
│ │ │ │ -00058610: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00058620: 3d3d 3d3d 3d3d 3d0a 0a73 7562 7374 6974  =======..substit
│ │ │ │ -00058630: 7574 6520 6d61 7472 6963 6573 206f 6620  ute matrices of 
│ │ │ │ -00058640: 7477 6f20 6469 6666 6572 656e 7469 616c  two differential
│ │ │ │ -00058650: 7320 6f66 2046 2069 6e74 6f20 5320 3d20  s of F into S = 
│ │ │ │ -00058660: 7269 6e67 2066 662c 2063 6f6d 706f 7365  ring ff, compose
│ │ │ │ -00058670: 2074 6865 6d2c 0a61 6e64 2064 6976 6964   them,.and divid
│ │ │ │ -00058680: 6520 6279 2065 6e74 7269 6573 206f 6620  e by entries of 
│ │ │ │ -00058690: 6666 2c20 696e 206f 7264 6572 2e20 4966  ff, in order. If
│ │ │ │ -000586a0: 2074 6865 2073 6563 6f6e 6420 4d61 7472   the second Matr
│ │ │ │ -000586b0: 6978 2061 7267 756d 656e 7420 7430 2069  ix argument t0 i
│ │ │ │ -000586c0: 730a 7072 6573 656e 742c 2075 7365 2069  s.present, use i
│ │ │ │ -000586d0: 7420 6173 2074 6865 2066 6972 7374 2043  t as the first C
│ │ │ │ -000586e0: 4920 6f70 6572 6174 6f72 2e0a 0a54 6865  I operator...The
│ │ │ │ -000586f0: 2064 6567 7265 6573 206f 6620 7468 6520   degrees of the 
│ │ │ │ -00058700: 7461 7267 6574 7320 6f66 2074 6865 2054  targets of the T
│ │ │ │ -00058710: 5f6a 2061 7265 2063 6861 6e67 6564 2062  _j are changed b
│ │ │ │ -00058720: 7920 7468 6520 6465 6772 6565 7320 6f66  y the degrees of
│ │ │ │ -00058730: 2074 6865 2066 5f6a 2074 6f0a 6d61 6b65   the f_j to.make
│ │ │ │ -00058740: 2074 6865 2054 5f6a 2068 6f6d 6f67 656e   the T_j homogen
│ │ │ │ -00058750: 656f 7573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  eous...+--------
│ │ │ │ -00058760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -00058790: 3120 3a20 5320 3d20 5a5a 2f31 3031 5b78  1 : S = ZZ/101[x
│ │ │ │ -000587a0: 2c79 2c7a 5d3b 2020 2020 2020 2020 2020  ,y,z];          
│ │ │ │ -000587b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000587c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -000587d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000587e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000587f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ -00058800: 3a20 6666 203d 206d 6174 7269 7822 7833  : ff = matrix"x3
│ │ │ │ -00058810: 2c79 332c 7a33 223b 2020 2020 2020 2020  ,y3,z3";        
│ │ │ │ +00058240: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +00058250: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +00058260: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +00058270: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +00058280: 6d61 6361 756c 6179 322d 312e 3236 2e30  macaulay2-1.26.0
│ │ │ │ +00058290: 362b 6473 2f4d 322f 4d61 6361 756c 6179  6+ds/M2/Macaulay
│ │ │ │ +000582a0: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ +000582b0: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ +000582c0: 5265 736f 6c75 7469 6f6e 732e 6d32 3a32  Resolutions.m2:2
│ │ │ │ +000582d0: 3735 393a 302e 0a1f 0a46 696c 653a 2043  759:0....File: C
│ │ │ │ +000582e0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +000582f0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ +00058300: 6e66 6f2c 204e 6f64 653a 206d 616b 6554  nfo, Node: makeT
│ │ │ │ +00058310: 2c20 4e65 7874 3a20 6d61 7472 6978 4661  , Next: matrixFa
│ │ │ │ +00058320: 6374 6f72 697a 6174 696f 6e2c 2050 7265  ctorization, Pre
│ │ │ │ +00058330: 763a 206d 616b 654d 6f64 756c 652c 2055  v: makeModule, U
│ │ │ │ +00058340: 703a 2054 6f70 0a0a 6d61 6b65 5420 2d2d  p: Top..makeT --
│ │ │ │ +00058350: 206d 616b 6520 7468 6520 4349 206f 7065   make the CI ope
│ │ │ │ +00058360: 7261 746f 7273 206f 6e20 6120 636f 6d70  rators on a comp
│ │ │ │ +00058370: 6c65 780a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  lex.************
│ │ │ │ +00058380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00058390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +000583a0: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +000583b0: 2020 2020 2054 203d 206d 616b 6554 2866       T = makeT(f
│ │ │ │ +000583c0: 662c 462c 6929 0a20 2020 2020 2020 2054  f,F,i).        T
│ │ │ │ +000583d0: 203d 206d 616b 6554 2866 662c 462c 7430   = makeT(ff,F,t0
│ │ │ │ +000583e0: 2c69 290a 2020 2a20 496e 7075 7473 3a0a  ,i).  * Inputs:.
│ │ │ │ +000583f0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e        * ff, a *n
│ │ │ │ +00058400: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ +00058410: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ +00058420: 2c2c 2031 7863 206d 6174 7269 7820 7768  ,, 1xc matrix wh
│ │ │ │ +00058430: 6f73 6520 656e 7472 6965 7320 6172 650a  ose entries are.
│ │ │ │ +00058440: 2020 2020 2020 2020 6120 636f 6d70 6c65          a comple
│ │ │ │ +00058450: 7465 2069 6e74 6572 7365 6374 696f 6e20  te intersection 
│ │ │ │ +00058460: 696e 2053 0a20 2020 2020 202a 2046 2c20  in S.      * F, 
│ │ │ │ +00058470: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a  a *note complex:
│ │ │ │ +00058480: 2028 436f 6d70 6c65 7865 7329 436f 6d70   (Complexes)Comp
│ │ │ │ +00058490: 6c65 782c 2c20 6f76 6572 2053 2f69 6465  lex,, over S/ide
│ │ │ │ +000584a0: 616c 2066 660a 2020 2020 2020 2a20 7430  al ff.      * t0
│ │ │ │ +000584b0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978  , a *note matrix
│ │ │ │ +000584c0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000584d0: 4d61 7472 6978 2c2c 2043 492d 6f70 6572  Matrix,, CI-oper
│ │ │ │ +000584e0: 6174 6f72 206f 6e20 4620 666f 7220 6666  ator on F for ff
│ │ │ │ +000584f0: 5f30 2074 6f0a 2020 2020 2020 2020 6265  _0 to.        be
│ │ │ │ +00058500: 2070 7265 7365 7276 6564 0a20 2020 2020   preserved.     
│ │ │ │ +00058510: 202a 2069 2c20 616e 202a 6e6f 7465 2069   * i, an *note i
│ │ │ │ +00058520: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ +00058530: 7932 446f 6329 5a5a 2c2c 2064 6566 696e  y2Doc)ZZ,, defin
│ │ │ │ +00058540: 6520 4349 206f 7065 7261 746f 7273 2066  e CI operators f
│ │ │ │ +00058550: 726f 6d20 465f 690a 2020 2020 2020 2020  rom F_i.        
│ │ │ │ +00058560: 5c74 6f20 465f 7b69 2d32 7d0a 2020 2a20  \to F_{i-2}.  * 
│ │ │ │ +00058570: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +00058580: 204c 2c20 6120 2a6e 6f74 6520 6c69 7374   L, a *note list
│ │ │ │ +00058590: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +000585a0: 4c69 7374 2c2c 206f 6620 4349 206f 7065  List,, of CI ope
│ │ │ │ +000585b0: 7261 746f 7273 2046 5f69 205c 746f 2046  rators F_i \to F
│ │ │ │ +000585c0: 5f7b 692d 327d 0a20 2020 2020 2020 2063  _{i-2}.        c
│ │ │ │ +000585d0: 6f72 7265 7370 6f6e 6469 6e67 2074 6f20  orresponding to 
│ │ │ │ +000585e0: 656e 7472 6965 7320 6f66 2066 660a 0a44  entries of ff..D
│ │ │ │ +000585f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00058600: 3d3d 3d3d 3d3d 0a0a 7375 6273 7469 7475  ======..substitu
│ │ │ │ +00058610: 7465 206d 6174 7269 6365 7320 6f66 2074  te matrices of t
│ │ │ │ +00058620: 776f 2064 6966 6665 7265 6e74 6961 6c73  wo differentials
│ │ │ │ +00058630: 206f 6620 4620 696e 746f 2053 203d 2072   of F into S = r
│ │ │ │ +00058640: 696e 6720 6666 2c20 636f 6d70 6f73 6520  ing ff, compose 
│ │ │ │ +00058650: 7468 656d 2c0a 616e 6420 6469 7669 6465  them,.and divide
│ │ │ │ +00058660: 2062 7920 656e 7472 6965 7320 6f66 2066   by entries of f
│ │ │ │ +00058670: 662c 2069 6e20 6f72 6465 722e 2049 6620  f, in order. If 
│ │ │ │ +00058680: 7468 6520 7365 636f 6e64 204d 6174 7269  the second Matri
│ │ │ │ +00058690: 7820 6172 6775 6d65 6e74 2074 3020 6973  x argument t0 is
│ │ │ │ +000586a0: 0a70 7265 7365 6e74 2c20 7573 6520 6974  .present, use it
│ │ │ │ +000586b0: 2061 7320 7468 6520 6669 7273 7420 4349   as the first CI
│ │ │ │ +000586c0: 206f 7065 7261 746f 722e 0a0a 5468 6520   operator...The 
│ │ │ │ +000586d0: 6465 6772 6565 7320 6f66 2074 6865 2074  degrees of the t
│ │ │ │ +000586e0: 6172 6765 7473 206f 6620 7468 6520 545f  argets of the T_
│ │ │ │ +000586f0: 6a20 6172 6520 6368 616e 6765 6420 6279  j are changed by
│ │ │ │ +00058700: 2074 6865 2064 6567 7265 6573 206f 6620   the degrees of 
│ │ │ │ +00058710: 7468 6520 665f 6a20 746f 0a6d 616b 6520  the f_j to.make 
│ │ │ │ +00058720: 7468 6520 545f 6a20 686f 6d6f 6765 6e65  the T_j homogene
│ │ │ │ +00058730: 6f75 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ous...+---------
│ │ │ │ +00058740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00058770: 203a 2053 203d 205a 5a2f 3130 315b 782c   : S = ZZ/101[x,
│ │ │ │ +00058780: 792c 7a5d 3b20 2020 2020 2020 2020 2020  y,z];           
│ │ │ │ +00058790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000587a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000587b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000587c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000587d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ +000587e0: 2066 6620 3d20 6d61 7472 6978 2278 332c   ff = matrix"x3,
│ │ │ │ +000587f0: 7933 2c7a 3322 3b20 2020 2020 2020 2020  y3,z3";         
│ │ │ │ +00058800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058810: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  00058820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058830: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00058840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058860: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00058870: 2020 2020 2020 2020 3120 2020 2020 2033          1      3
│ │ │ │ -00058880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000588a0: 0a7c 6f32 203a 204d 6174 7269 7820 5320  .|o2 : Matrix S 
│ │ │ │ -000588b0: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ -000588c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000588d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -000588e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000588f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00058910: 6933 203a 2052 203d 2053 2f69 6465 616c  i3 : R = S/ideal
│ │ │ │ -00058920: 2066 663b 2020 2020 2020 2020 2020 2020   ff;            
│ │ │ │ -00058930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00058950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00058980: 203a 204d 203d 2063 6f6b 6572 206d 6174   : M = coker mat
│ │ │ │ -00058990: 7269 7822 782c 792c 7a3b 792c 7a2c 7822  rix"x,y,z;y,z,x"
│ │ │ │ -000589a0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ -000589b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -000589c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000589d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000589e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -000589f0: 2062 6574 7469 2028 4620 3d20 6672 6565   betti (F = free
│ │ │ │ -00058a00: 5265 736f 6c75 7469 6f6e 284d 2c20 4c65  Resolution(M, Le
│ │ │ │ -00058a10: 6e67 7468 4c69 6d69 7420 3d3e 2033 2929  ngthLimit => 3))
│ │ │ │ -00058a20: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00058a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00058a60: 2020 2020 2020 3020 3120 3220 3320 2020        0 1 2 3   
│ │ │ │ -00058a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058a80: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00058a90: 7c6f 3520 3d20 746f 7461 6c3a 2032 2033  |o5 = total: 2 3
│ │ │ │ -00058aa0: 2035 2036 2020 2020 2020 2020 2020 2020   5 6            
│ │ │ │ -00058ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ac0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00058ad0: 2030 3a20 3220 3320 2e20 2e20 2020 2020   0: 2 3 . .     
│ │ │ │ -00058ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00058b00: 2020 2020 2020 2020 313a 202e 202e 2035          1: . . 5
│ │ │ │ -00058b10: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ +00058830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058840: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00058850: 2020 2020 2020 2031 2020 2020 2020 3320         1      3 
│ │ │ │ +00058860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058870: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00058880: 7c6f 3220 3a20 4d61 7472 6978 2053 2020  |o2 : Matrix S  
│ │ │ │ +00058890: 3c2d 2d20 5320 2020 2020 2020 2020 2020  <-- S           
│ │ │ │ +000588a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000588b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000588c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000588d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000588e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000588f0: 3320 3a20 5220 3d20 532f 6964 6561 6c20  3 : R = S/ideal 
│ │ │ │ +00058900: 6666 3b20 2020 2020 2020 2020 2020 2020  ff;             
│ │ │ │ +00058910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058920: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00058930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058950: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420  ----------+.|i4 
│ │ │ │ +00058960: 3a20 4d20 3d20 636f 6b65 7220 6d61 7472  : M = coker matr
│ │ │ │ +00058970: 6978 2278 2c79 2c7a 3b79 2c7a 2c78 223b  ix"x,y,z;y,z,x";
│ │ │ │ +00058980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058990: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000589a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000589b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000589c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +000589d0: 6265 7474 6920 2846 203d 2066 7265 6552  betti (F = freeR
│ │ │ │ +000589e0: 6573 6f6c 7574 696f 6e28 4d2c 204c 656e  esolution(M, Len
│ │ │ │ +000589f0: 6774 684c 696d 6974 203d 3e20 3329 297c  gthLimit => 3))|
│ │ │ │ +00058a00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00058a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058a30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00058a40: 2020 2020 2030 2031 2032 2033 2020 2020       0 1 2 3    
│ │ │ │ +00058a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058a60: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058a70: 6f35 203d 2074 6f74 616c 3a20 3220 3320  o5 = total: 2 3 
│ │ │ │ +00058a80: 3520 3620 2020 2020 2020 2020 2020 2020  5 6             
│ │ │ │ +00058a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058aa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00058ab0: 303a 2032 2033 202e 202e 2020 2020 2020  0: 2 3 . .      
│ │ │ │ +00058ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058ad0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00058ae0: 2020 2020 2020 2031 3a20 2e20 2e20 3520         1: . . 5 
│ │ │ │ +00058af0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +00058b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058b10: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00058b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00058b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b60: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -00058b70: 3a20 4265 7474 6954 616c 6c79 2020 2020  : BettiTally    
│ │ │ │ -00058b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ba0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -00058bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058bd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -00058be0: 5420 3d20 6d61 6b65 5428 6666 2c46 2c33  T = makeT(ff,F,3
│ │ │ │ -00058bf0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ -00058c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00058c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00058c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058c40: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6e65  ------+.|i7 : ne
│ │ │ │ -00058c50: 744c 6973 7420 5420 2020 2020 2020 2020  tList T         
│ │ │ │ +00058b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058b40: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ +00058b50: 2042 6574 7469 5461 6c6c 7920 2020 2020   BettiTally     
│ │ │ │ +00058b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058b80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00058b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058bb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2054  -------+.|i6 : T
│ │ │ │ +00058bc0: 203d 206d 616b 6554 2866 662c 462c 3329   = makeT(ff,F,3)
│ │ │ │ +00058bd0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00058be0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00058bf0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00058c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058c20: 2d2d 2d2d 2d2b 0a7c 6937 203a 206e 6574  -----+.|i7 : net
│ │ │ │ +00058c30: 4c69 7374 2054 2020 2020 2020 2020 2020  List T          
│ │ │ │ +00058c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00058c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058cb0: 2020 2020 7c0a 7c20 2020 2020 2b2d 2d2d      |.|     +---
│ │ │ │ -00058cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058cd0: 2d2d 2d2d 2d2b 2020 2020 2020 2020 2020  -----+          
│ │ │ │ -00058ce0: 2020 2020 2020 2020 2020 207c 0a7c 6f37             |.|o7
│ │ │ │ -00058cf0: 203d 207c 7b34 7d20 7c20 3020 3020 3020   = |{4} | 0 0 0 
│ │ │ │ -00058d00: 3020 2031 2030 207c 2020 2020 7c20 2020  0  1 0 |    |   
│ │ │ │ -00058d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058d20: 2020 7c0a 7c20 2020 2020 7c7b 347d 207c    |.|     |{4} |
│ │ │ │ -00058d30: 2030 2030 2030 202d 3120 3020 3020 7c20   0 0 0 -1 0 0 | 
│ │ │ │ -00058d40: 2020 207c 2020 2020 2020 2020 2020 2020     |            
│ │ │ │ -00058d50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00058d60: 207c 7b34 7d20 7c20 3020 3020 3020 3020   |{4} | 0 0 0 0 
│ │ │ │ -00058d70: 2030 2031 207c 2020 2020 7c20 2020 2020   0 1 |    |     
│ │ │ │ -00058d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058d90: 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d 2d2d  |.|     +-------
│ │ │ │ -00058da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058db0: 2d2b 2020 2020 2020 2020 2020 2020 2020  -+              
│ │ │ │ -00058dc0: 2020 2020 2020 207c 0a7c 2020 2020 207c         |.|     |
│ │ │ │ -00058dd0: 7b34 7d20 7c20 3020 3120 3020 3020 3020  {4} | 0 1 0 0 0 
│ │ │ │ -00058de0: 3020 7c20 2020 2020 7c20 2020 2020 2020  0 |     |       
│ │ │ │ -00058df0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00058e00: 7c20 2020 2020 7c7b 347d 207c 2031 2030  |     |{4} | 1 0
│ │ │ │ -00058e10: 2030 2030 2030 2030 207c 2020 2020 207c   0 0 0 0 |     |
│ │ │ │ -00058e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058e30: 2020 2020 207c 0a7c 2020 2020 207c 7b34       |.|     |{4
│ │ │ │ -00058e40: 7d20 7c20 3020 3020 3120 3020 3020 3020  } | 0 0 1 0 0 0 
│ │ │ │ -00058e50: 7c20 2020 2020 7c20 2020 2020 2020 2020  |     |         
│ │ │ │ -00058e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00058e70: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d      +-----------
│ │ │ │ -00058e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020  -------------+  
│ │ │ │ -00058e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058ea0: 2020 207c 0a7c 2020 2020 207c 7b34 7d20     |.|     |{4} 
│ │ │ │ -00058eb0: 7c20 3020 202d 3120 3020 2030 202d 3120  | 0  -1 0  0 -1 
│ │ │ │ -00058ec0: 3020 207c 7c20 2020 2020 2020 2020 2020  0  ||           
│ │ │ │ -00058ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00058ee0: 2020 7c7b 347d 207c 202d 3120 3020 2030    |{4} | -1 0  0
│ │ │ │ -00058ef0: 2020 3120 3020 2030 2020 7c7c 2020 2020    1 0  0  ||    
│ │ │ │ -00058f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058f10: 207c 0a7c 2020 2020 207c 7b34 7d20 7c20   |.|     |{4} | 
│ │ │ │ -00058f20: 3020 2030 2020 2d31 2030 2030 2020 2d31  0  0  -1 0 0  -1
│ │ │ │ -00058f30: 207c 7c20 2020 2020 2020 2020 2020 2020   ||             
│ │ │ │ -00058f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00058f50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00058f60: 2d2d 2d2d 2d2d 2d2d 2d2b 2020 2020 2020  ---------+      
│ │ │ │ -00058f70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00058f80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00058f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00058fb0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 6973  ------+.|i8 : is
│ │ │ │ -00058fc0: 486f 6d6f 6765 6e65 6f75 7320 545f 3220  Homogeneous T_2 
│ │ │ │ +00058c90: 2020 207c 0a7c 2020 2020 202b 2d2d 2d2d     |.|     +----
│ │ │ │ +00058ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058cb0: 2d2d 2d2d 2b20 2020 2020 2020 2020 2020  ----+           
│ │ │ │ +00058cc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720            |.|o7 
│ │ │ │ +00058cd0: 3d20 7c7b 347d 207c 2030 2030 2030 2030  = |{4} | 0 0 0 0
│ │ │ │ +00058ce0: 2020 3120 3020 7c20 2020 207c 2020 2020    1 0 |    |    
│ │ │ │ +00058cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058d00: 207c 0a7c 2020 2020 207c 7b34 7d20 7c20   |.|     |{4} | 
│ │ │ │ +00058d10: 3020 3020 3020 2d31 2030 2030 207c 2020  0 0 0 -1 0 0 |  
│ │ │ │ +00058d20: 2020 7c20 2020 2020 2020 2020 2020 2020    |             
│ │ │ │ +00058d30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00058d40: 7c7b 347d 207c 2030 2030 2030 2030 2020  |{4} | 0 0 0 0  
│ │ │ │ +00058d50: 3020 3120 7c20 2020 207c 2020 2020 2020  0 1 |    |      
│ │ │ │ +00058d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00058d70: 0a7c 2020 2020 202b 2d2d 2d2d 2d2d 2d2d  .|     +--------
│ │ │ │ +00058d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058d90: 2b20 2020 2020 2020 2020 2020 2020 2020  +               
│ │ │ │ +00058da0: 2020 2020 2020 7c0a 7c20 2020 2020 7c7b        |.|     |{
│ │ │ │ +00058db0: 347d 207c 2030 2031 2030 2030 2030 2030  4} | 0 1 0 0 0 0
│ │ │ │ +00058dc0: 207c 2020 2020 207c 2020 2020 2020 2020   |     |        
│ │ │ │ +00058dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058de0: 2020 2020 207c 7b34 7d20 7c20 3120 3020       |{4} | 1 0 
│ │ │ │ +00058df0: 3020 3020 3020 3020 7c20 2020 2020 7c20  0 0 0 0 |     | 
│ │ │ │ +00058e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058e10: 2020 2020 7c0a 7c20 2020 2020 7c7b 347d      |.|     |{4}
│ │ │ │ +00058e20: 207c 2030 2030 2031 2030 2030 2030 207c   | 0 0 1 0 0 0 |
│ │ │ │ +00058e30: 2020 2020 207c 2020 2020 2020 2020 2020       |          
│ │ │ │ +00058e40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00058e50: 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     +------------
│ │ │ │ +00058e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020  ------------+   
│ │ │ │ +00058e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058e80: 2020 7c0a 7c20 2020 2020 7c7b 347d 207c    |.|     |{4} |
│ │ │ │ +00058e90: 2030 2020 2d31 2030 2020 3020 2d31 2030   0  -1 0  0 -1 0
│ │ │ │ +00058ea0: 2020 7c7c 2020 2020 2020 2020 2020 2020    ||            
│ │ │ │ +00058eb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00058ec0: 207c 7b34 7d20 7c20 2d31 2030 2020 3020   |{4} | -1 0  0 
│ │ │ │ +00058ed0: 2031 2030 2020 3020 207c 7c20 2020 2020   1 0  0  ||     
│ │ │ │ +00058ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058ef0: 7c0a 7c20 2020 2020 7c7b 347d 207c 2030  |.|     |{4} | 0
│ │ │ │ +00058f00: 2020 3020 202d 3120 3020 3020 202d 3120    0  -1 0 0  -1 
│ │ │ │ +00058f10: 7c7c 2020 2020 2020 2020 2020 2020 2020  ||              
│ │ │ │ +00058f20: 2020 2020 2020 207c 0a7c 2020 2020 202b         |.|     +
│ │ │ │ +00058f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058f40: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020  --------+       
│ │ │ │ +00058f50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00058f60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00058f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00058f90: 2d2d 2d2d 2d2b 0a7c 6938 203a 2069 7348  -----+.|i8 : isH
│ │ │ │ +00058fa0: 6f6d 6f67 656e 656f 7573 2054 5f32 2020  omogeneous T_2  
│ │ │ │ +00058fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00058fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00058fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00058fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00058fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00058ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059000: 2020 207c 0a7c 6f38 203d 2074 7275 6520     |.|o8 = true 
│ │ │ │  00059010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059020: 2020 2020 7c0a 7c6f 3820 3d20 7472 7565      |.|o8 = true
│ │ │ │ -00059030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059050: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00059020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059030: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00059040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059090: 2d2d 2b0a 0a43 6176 6561 740a 3d3d 3d3d  --+..Caveat.====
│ │ │ │ -000590a0: 3d3d 0a0a 5363 7269 7074 2061 7373 756d  ==..Script assum
│ │ │ │ -000590b0: 6573 2074 6861 7420 7269 6e67 2046 203d  es that ring F =
│ │ │ │ -000590c0: 3d20 2872 696e 6720 6666 292f 2869 6465  = (ring ff)/(ide
│ │ │ │ -000590d0: 616c 2066 6629 2e20 4974 206d 6967 6874  al ff). It might
│ │ │ │ -000590e0: 2062 6520 6d6f 7265 2075 7365 6675 6c20   be more useful 
│ │ │ │ -000590f0: 746f 0a72 6574 7572 6e20 7468 6520 6f70  to.return the op
│ │ │ │ -00059100: 6572 6174 6f72 7320 6173 206d 6174 7269  erators as matri
│ │ │ │ -00059110: 6365 7320 6f76 6572 2053 2072 6174 6865  ces over S rathe
│ │ │ │ -00059120: 7220 7468 616e 206f 7665 7220 522c 2073  r than over R, s
│ │ │ │ -00059130: 696e 6365 2074 6869 7320 6973 2077 6861  ince this is wha
│ │ │ │ -00059140: 740a 7765 2764 206e 6565 6420 666f 7220  t.we'd need for 
│ │ │ │ -00059150: 7468 696e 6773 206c 696b 6520 6d61 7472  things like matr
│ │ │ │ -00059160: 6978 4661 6374 6f72 697a 6174 696f 6e20  ixFactorization 
│ │ │ │ -00059170: 2877 6865 7265 2074 6869 7320 7072 6f63  (where this proc
│ │ │ │ -00059180: 6573 7320 6375 7272 656e 746c 790a 646f  ess currently.do
│ │ │ │ -00059190: 6e65 206f 6e20 7468 6520 666c 792c 206e  ne on the fly, n
│ │ │ │ -000591a0: 6f74 2063 616c 6c69 6e67 206d 616b 6554  ot calling makeT
│ │ │ │ -000591b0: 290a 0a57 6179 7320 746f 2075 7365 206d  )..Ways to use m
│ │ │ │ -000591c0: 616b 6554 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  akeT:.==========
│ │ │ │ -000591d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d  ========..  * "m
│ │ │ │ -000591e0: 616b 6554 284d 6174 7269 782c 436f 6d70  akeT(Matrix,Comp
│ │ │ │ -000591f0: 6c65 782c 5a5a 2922 0a0a 466f 7220 7468  lex,ZZ)"..For th
│ │ │ │ -00059200: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00059210: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00059220: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00059230: 6520 6d61 6b65 543a 206d 616b 6554 2c20  e makeT: makeT, 
│ │ │ │ -00059240: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -00059250: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -00059260: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -00059270: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +00059070: 2d2b 0a0a 4361 7665 6174 0a3d 3d3d 3d3d  -+..Caveat.=====
│ │ │ │ +00059080: 3d0a 0a53 6372 6970 7420 6173 7375 6d65  =..Script assume
│ │ │ │ +00059090: 7320 7468 6174 2072 696e 6720 4620 3d3d  s that ring F ==
│ │ │ │ +000590a0: 2028 7269 6e67 2066 6629 2f28 6964 6561   (ring ff)/(idea
│ │ │ │ +000590b0: 6c20 6666 292e 2049 7420 6d69 6768 7420  l ff). It might 
│ │ │ │ +000590c0: 6265 206d 6f72 6520 7573 6566 756c 2074  be more useful t
│ │ │ │ +000590d0: 6f0a 7265 7475 726e 2074 6865 206f 7065  o.return the ope
│ │ │ │ +000590e0: 7261 746f 7273 2061 7320 6d61 7472 6963  rators as matric
│ │ │ │ +000590f0: 6573 206f 7665 7220 5320 7261 7468 6572  es over S rather
│ │ │ │ +00059100: 2074 6861 6e20 6f76 6572 2052 2c20 7369   than over R, si
│ │ │ │ +00059110: 6e63 6520 7468 6973 2069 7320 7768 6174  nce this is what
│ │ │ │ +00059120: 0a77 6527 6420 6e65 6564 2066 6f72 2074  .we'd need for t
│ │ │ │ +00059130: 6869 6e67 7320 6c69 6b65 206d 6174 7269  hings like matri
│ │ │ │ +00059140: 7846 6163 746f 7269 7a61 7469 6f6e 2028  xFactorization (
│ │ │ │ +00059150: 7768 6572 6520 7468 6973 2070 726f 6365  where this proce
│ │ │ │ +00059160: 7373 2063 7572 7265 6e74 6c79 0a64 6f6e  ss currently.don
│ │ │ │ +00059170: 6520 6f6e 2074 6865 2066 6c79 2c20 6e6f  e on the fly, no
│ │ │ │ +00059180: 7420 6361 6c6c 696e 6720 6d61 6b65 5429  t calling makeT)
│ │ │ │ +00059190: 0a0a 5761 7973 2074 6f20 7573 6520 6d61  ..Ways to use ma
│ │ │ │ +000591a0: 6b65 543a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  keT:.===========
│ │ │ │ +000591b0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6d61  =======..  * "ma
│ │ │ │ +000591c0: 6b65 5428 4d61 7472 6978 2c43 6f6d 706c  keT(Matrix,Compl
│ │ │ │ +000591d0: 6578 2c5a 5a29 220a 0a46 6f72 2074 6865  ex,ZZ)"..For the
│ │ │ │ +000591e0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +000591f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00059200: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00059210: 206d 616b 6554 3a20 6d61 6b65 542c 2069   makeT: makeT, i
│ │ │ │ +00059220: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +00059230: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ +00059240: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +00059250: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d  unction,...-----
│ │ │ │ +00059260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000592a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000592b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000592c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -000592d0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ -000592e0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ -000592f0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ -00059300: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -00059310: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ -00059320: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -00059330: 6573 2f0a 436f 6d70 6c65 7465 496e 7465  es/.CompleteInte
│ │ │ │ -00059340: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -00059350: 6f6e 732e 6d32 3a33 3534 383a 302e 0a1f  ons.m2:3548:0...
│ │ │ │ -00059360: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ -00059370: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -00059380: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ -00059390: 653a 206d 6174 7269 7846 6163 746f 7269  e: matrixFactori
│ │ │ │ -000593a0: 7a61 7469 6f6e 2c20 4e65 7874 3a20 6d66  zation, Next: mf
│ │ │ │ -000593b0: 426f 756e 642c 2050 7265 763a 206d 616b  Bound, Prev: mak
│ │ │ │ -000593c0: 6554 2c20 5570 3a20 546f 700a 0a6d 6174  eT, Up: Top..mat
│ │ │ │ -000593d0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -000593e0: 202d 2d20 4d61 7073 2069 6e20 6120 6869   -- Maps in a hi
│ │ │ │ -000593f0: 6768 6572 2063 6f64 696d 656e 7369 6f6e  gher codimension
│ │ │ │ -00059400: 206d 6174 7269 7820 6661 6374 6f72 697a   matrix factoriz
│ │ │ │ -00059410: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a  ation.**********
│ │ │ │ +000592a0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ +000592b0: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ +000592c0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ +000592d0: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +000592e0: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +000592f0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ +00059300: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +00059310: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ +00059320: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +00059330: 6e73 2e6d 323a 3335 3438 3a30 2e0a 1f0a  ns.m2:3548:0....
│ │ │ │ +00059340: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ +00059350: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +00059360: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ +00059370: 3a20 6d61 7472 6978 4661 6374 6f72 697a  : matrixFactoriz
│ │ │ │ +00059380: 6174 696f 6e2c 204e 6578 743a 206d 6642  ation, Next: mfB
│ │ │ │ +00059390: 6f75 6e64 2c20 5072 6576 3a20 6d61 6b65  ound, Prev: make
│ │ │ │ +000593a0: 542c 2055 703a 2054 6f70 0a0a 6d61 7472  T, Up: Top..matr
│ │ │ │ +000593b0: 6978 4661 6374 6f72 697a 6174 696f 6e20  ixFactorization 
│ │ │ │ +000593c0: 2d2d 204d 6170 7320 696e 2061 2068 6967  -- Maps in a hig
│ │ │ │ +000593d0: 6865 7220 636f 6469 6d65 6e73 696f 6e20  her codimension 
│ │ │ │ +000593e0: 6d61 7472 6978 2066 6163 746f 7269 7a61  matrix factoriza
│ │ │ │ +000593f0: 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  tion.***********
│ │ │ │ +00059400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00059410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00059420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00059430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00059440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00059450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -00059460: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -00059470: 2020 2020 4d46 203d 206d 6174 7269 7846      MF = matrixF
│ │ │ │ -00059480: 6163 746f 7269 7a61 7469 6f6e 2866 662c  actorization(ff,
│ │ │ │ -00059490: 4d29 0a20 202a 2049 6e70 7574 733a 0a20  M).  * Inputs:. 
│ │ │ │ -000594a0: 2020 2020 202a 2066 662c 2061 202a 6e6f       * ff, a *no
│ │ │ │ -000594b0: 7465 206d 6174 7269 783a 2028 4d61 6361  te matrix: (Maca
│ │ │ │ -000594c0: 756c 6179 3244 6f63 294d 6174 7269 782c  ulay2Doc)Matrix,
│ │ │ │ -000594d0: 2c20 6120 7375 6666 6963 6965 6e74 6c79  , a sufficiently
│ │ │ │ -000594e0: 2067 656e 6572 616c 0a20 2020 2020 2020   general.       
│ │ │ │ -000594f0: 2072 6567 756c 6172 2073 6571 7565 6e63   regular sequenc
│ │ │ │ -00059500: 6520 696e 2061 2072 696e 6720 530a 2020  e in a ring S.  
│ │ │ │ -00059510: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ -00059520: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ -00059530: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ -00059540: 6120 6d61 7869 6d61 6c20 436f 6865 6e2d  a maximal Cohen-
│ │ │ │ -00059550: 4d61 6361 756c 6179 0a20 2020 2020 2020  Macaulay.       
│ │ │ │ -00059560: 206d 6f64 756c 6520 6f76 6572 2053 2f69   module over S/i
│ │ │ │ -00059570: 6465 616c 2066 660a 2020 2a20 2a6e 6f74  deal ff.  * *not
│ │ │ │ -00059580: 6520 4f70 7469 6f6e 616c 2069 6e70 7574  e Optional input
│ │ │ │ -00059590: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ -000595a0: 2975 7369 6e67 2066 756e 6374 696f 6e73  )using functions
│ │ │ │ -000595b0: 2077 6974 6820 6f70 7469 6f6e 616c 2069   with optional i
│ │ │ │ -000595c0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20  nputs,:.      * 
│ │ │ │ -000595d0: 4175 676d 656e 7461 7469 6f6e 203d 3e20  Augmentation => 
│ │ │ │ -000595e0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -000595f0: 7565 2074 7275 650a 2020 2020 2020 2a20  ue true.      * 
│ │ │ │ -00059600: 4368 6563 6b20 3d3e 202e 2e2e 2c20 6465  Check => ..., de
│ │ │ │ -00059610: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ -00059620: 650a 2020 2020 2020 2a20 4c61 7965 7265  e.      * Layere
│ │ │ │ -00059630: 6420 3d3e 202e 2e2e 2c20 6465 6661 756c  d => ..., defaul
│ │ │ │ -00059640: 7420 7661 6c75 6520 7472 7565 0a20 2020  t value true.   
│ │ │ │ -00059650: 2020 202a 2056 6572 626f 7365 203d 3e20     * Verbose => 
│ │ │ │ -00059660: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -00059670: 7565 2066 616c 7365 0a20 202a 204f 7574  ue false.  * Out
│ │ │ │ -00059680: 7075 7473 3a0a 2020 2020 2020 2a20 4d46  puts:.      * MF
│ │ │ │ -00059690: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -000596a0: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -000596b0: 7374 2c2c 205c 7b64 2c68 2c67 616d 6d61  st,, \{d,h,gamma
│ │ │ │ -000596c0: 5c7d 2c20 7768 6572 6520 643a 415f 3120  \}, where d:A_1 
│ │ │ │ -000596d0: 5c74 6f0a 2020 2020 2020 2020 415f 3020  \to.        A_0 
│ │ │ │ -000596e0: 616e 6420 683a 205c 6f70 6c75 7320 415f  and h: \oplus A_
│ │ │ │ -000596f0: 3028 7029 205c 746f 2041 5f31 2069 7320  0(p) \to A_1 is 
│ │ │ │ -00059700: 7468 6520 6469 7265 6374 2073 756d 206f  the direct sum o
│ │ │ │ -00059710: 6620 7061 7274 6961 6c0a 2020 2020 2020  f partial.      
│ │ │ │ -00059720: 2020 686f 6d6f 746f 7069 6573 2c20 616e    homotopies, an
│ │ │ │ -00059730: 6420 6761 6d6d 613a 2041 5f30 202d 3e4d  d gamma: A_0 ->M
│ │ │ │ -00059740: 2069 7320 7468 6520 6175 676d 656e 7461   is the augmenta
│ │ │ │ -00059750: 7469 6f6e 2028 7265 7475 726e 6564 206f  tion (returned o
│ │ │ │ -00059760: 6e6c 7920 6966 0a20 2020 2020 2020 2041  nly if.        A
│ │ │ │ -00059770: 7567 6d65 6e74 6174 696f 6e20 3d3e 7472  ugmentation =>tr
│ │ │ │ -00059780: 7565 290a 0a44 6573 6372 6970 7469 6f6e  ue)..Description
│ │ │ │ -00059790: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  .===========..Th
│ │ │ │ -000597a0: 6520 696e 7075 7420 6d6f 6475 6c65 204d  e input module M
│ │ │ │ -000597b0: 2073 686f 756c 6420 6265 2061 206d 6178   should be a max
│ │ │ │ -000597c0: 696d 616c 2043 6f68 656e 2d4d 6163 6175  imal Cohen-Macau
│ │ │ │ -000597d0: 6c61 7920 6d6f 6475 6c65 206f 7665 7220  lay module over 
│ │ │ │ -000597e0: 5220 3d20 532f 6964 6561 6c0a 6666 2e20  R = S/ideal.ff. 
│ │ │ │ -000597f0: 2049 6620 4d20 6973 2069 6e20 6661 6374   If M is in fact
│ │ │ │ -00059800: 2061 2022 6869 6768 2073 797a 7967 7922   a "high syzygy"
│ │ │ │ -00059810: 2c20 7468 656e 2074 6865 2066 756e 6374  , then the funct
│ │ │ │ -00059820: 696f 6e0a 6d61 7472 6978 4661 6374 6f72  ion.matrixFactor
│ │ │ │ -00059830: 697a 6174 696f 6e28 6666 2c4d 2c4c 6179  ization(ff,M,Lay
│ │ │ │ -00059840: 6572 6564 3d3e 6661 6c73 6529 2075 7365  ered=>false) use
│ │ │ │ -00059850: 7320 6120 6469 6666 6572 656e 742c 2066  s a different, f
│ │ │ │ -00059860: 6173 7465 7220 616c 676f 7269 7468 6d0a  aster algorithm.
│ │ │ │ -00059870: 7768 6963 6820 6f6e 6c79 2077 6f72 6b73  which only works
│ │ │ │ -00059880: 2069 6e20 7468 6520 6869 6768 2073 797a   in the high syz
│ │ │ │ -00059890: 7967 7920 6361 7365 2e0a 0a49 6e20 616c  ygy case...In al
│ │ │ │ -000598a0: 6c20 6578 616d 706c 6573 2077 6520 6b6e  l examples we kn
│ │ │ │ -000598b0: 6f77 2c20 4d20 6361 6e20 6265 2063 6f6e  ow, M can be con
│ │ │ │ -000598c0: 7369 6465 7265 6420 6120 2268 6967 6820  sidered a "high 
│ │ │ │ -000598d0: 7379 7a79 6779 2220 6173 206c 6f6e 6720  syzygy" as long 
│ │ │ │ -000598e0: 6173 0a45 7874 5e7b 6576 656e 7d5f 5228  as.Ext^{even}_R(
│ │ │ │ -000598f0: 4d2c 6b29 2061 6e64 2045 7874 5e7b 6f64  M,k) and Ext^{od
│ │ │ │ -00059900: 647d 5f52 284d 2c6b 2920 6861 7665 206e  d}_R(M,k) have n
│ │ │ │ -00059910: 6567 6174 6976 6520 7265 6775 6c61 7269  egative regulari
│ │ │ │ -00059920: 7479 206f 7665 7220 7468 6520 7269 6e67  ty over the ring
│ │ │ │ -00059930: 0a6f 6620 4349 206f 7065 7261 746f 7273  .of CI operators
│ │ │ │ -00059940: 2028 7265 6772 6164 6564 2077 6974 6820   (regraded with 
│ │ │ │ -00059950: 7661 7269 6162 6c65 7320 6f66 2064 6567  variables of deg
│ │ │ │ -00059960: 7265 6520 312e 2048 6f77 6576 6572 2c20  ree 1. However, 
│ │ │ │ -00059970: 7468 6520 6265 7374 2072 6573 756c 740a  the best result.
│ │ │ │ -00059980: 7765 2063 616e 2070 726f 7665 2069 7320  we can prove is 
│ │ │ │ -00059990: 7468 6174 2069 7420 7375 6666 6963 6573  that it suffices
│ │ │ │ -000599a0: 2074 6f20 6861 7665 2072 6567 756c 6172   to have regular
│ │ │ │ -000599b0: 6974 7920 3c20 2d28 322a 6469 6d20 522b  ity < -(2*dim R+
│ │ │ │ -000599c0: 3129 2e0a 0a57 6865 6e20 7468 6520 6f70  1)...When the op
│ │ │ │ -000599d0: 7469 6f6e 616c 2069 6e70 7574 2043 6865  tional input Che
│ │ │ │ -000599e0: 636b 3d3d 7472 7565 2028 7468 6520 6465  ck==true (the de
│ │ │ │ -000599f0: 6661 756c 7420 6973 2043 6865 636b 3d3d  fault is Check==
│ │ │ │ -00059a00: 6661 6c73 6529 2c20 7468 650a 7072 6f70  false), the.prop
│ │ │ │ -00059a10: 6572 7469 6573 2069 6e20 7468 6520 6465  erties in the de
│ │ │ │ -00059a20: 6669 6e69 7469 6f6e 206f 6620 4d61 7472  finition of Matr
│ │ │ │ -00059a30: 6978 2046 6163 746f 7269 7a61 7469 6f6e  ix Factorization
│ │ │ │ -00059a40: 2061 7265 2076 6572 6966 6965 640a 0a54   are verified..T
│ │ │ │ -00059a50: 6865 206f 7574 7075 7420 6973 2061 206c  he output is a l
│ │ │ │ -00059a60: 6973 7420 6f66 206d 6170 7320 5c7b 642c  ist of maps \{d,
│ │ │ │ -00059a70: 685c 7d20 6f72 205c 7b64 2c68 2c67 616d  h\} or \{d,h,gam
│ │ │ │ -00059a80: 6d61 5c7d 2c20 7768 6572 6520 6761 6d6d  ma\}, where gamm
│ │ │ │ -00059a90: 6120 6973 2061 6e0a 6175 676d 656e 7461  a is an.augmenta
│ │ │ │ -00059aa0: 7469 6f6e 2c20 7468 6174 2069 732c 2061  tion, that is, a
│ │ │ │ -00059ab0: 206d 6170 2066 726f 6d20 7461 7267 6574   map from target
│ │ │ │ -00059ac0: 2064 2074 6f20 4d2e 0a0a 5468 6520 6d61   d to M...The ma
│ │ │ │ -00059ad0: 7020 6420 6973 2061 2073 7065 6369 616c  p d is a special
│ │ │ │ -00059ae0: 206c 6966 7469 6e67 2074 6f20 5320 6f66   lifting to S of
│ │ │ │ -00059af0: 2061 2070 7265 7365 6e74 6174 696f 6e20   a presentation 
│ │ │ │ -00059b00: 6f66 204d 206f 7665 7220 522e 2054 6f20  of M over R. To 
│ │ │ │ -00059b10: 6578 706c 6169 6e0a 7468 6520 636f 6e74  explain.the cont
│ │ │ │ -00059b20: 656e 7473 2c20 7765 2069 6e74 726f 6475  ents, we introdu
│ │ │ │ -00059b30: 6365 2073 6f6d 6520 6e6f 7461 7469 6f6e  ce some notation
│ │ │ │ -00059b40: 2028 6672 6f6d 2045 6973 656e 6275 6420   (from Eisenbud 
│ │ │ │ -00059b50: 616e 6420 5065 6576 612c 2022 4d69 6e69  and Peeva, "Mini
│ │ │ │ -00059b60: 6d61 6c0a 6672 6565 2072 6573 6f6c 7574  mal.free resolut
│ │ │ │ -00059b70: 696f 6e73 206f 7665 7220 636f 6d70 6c65  ions over comple
│ │ │ │ -00059b80: 7465 2069 6e74 6572 7365 6374 696f 6e73  te intersections
│ │ │ │ -00059b90: 2220 4c65 6374 7572 6520 4e6f 7465 7320  " Lecture Notes 
│ │ │ │ -00059ba0: 696e 204d 6174 6865 6d61 7469 6373 2c0a  in Mathematics,.
│ │ │ │ -00059bb0: 3231 3532 2e20 5370 7269 6e67 6572 2c20  2152. Springer, 
│ │ │ │ -00059bc0: 4368 616d 2c20 3230 3136 2e20 782b 3130  Cham, 2016. x+10
│ │ │ │ -00059bd0: 3720 7070 2e20 4953 424e 3a20 3937 382d  7 pp. ISBN: 978-
│ │ │ │ -00059be0: 332d 3331 392d 3236 3433 362d 333b 0a39  3-319-26436-3;.9
│ │ │ │ -00059bf0: 3738 2d33 2d33 3139 2d32 3634 3337 2d30  78-3-319-26437-0
│ │ │ │ -00059c00: 292e 0a0a 5228 6929 203d 2053 2f28 6666  )...R(i) = S/(ff
│ │ │ │ -00059c10: 5f30 2c2e 2e2c 6666 5f7b 692d 317d 292e  _0,..,ff_{i-1}).
│ │ │ │ -00059c20: 2048 6572 6520 303c 3d20 6920 3c3d 2063   Here 0<= i <= c
│ │ │ │ -00059c30: 2c20 616e 6420 5220 3d20 5228 6329 2061  , and R = R(c) a
│ │ │ │ -00059c40: 6e64 2053 203d 2052 2830 292e 0a0a 4228  nd S = R(0)...B(
│ │ │ │ -00059c50: 6929 203d 2074 6865 206d 6174 7269 7820  i) = the matrix 
│ │ │ │ -00059c60: 286f 7665 7220 5329 2072 6570 7265 7365  (over S) represe
│ │ │ │ -00059c70: 6e74 696e 6720 645f 693a 2042 5f31 2869  nting d_i: B_1(i
│ │ │ │ -00059c80: 2920 5c74 6f20 425f 3028 6929 0a0a 6428  ) \to B_0(i)..d(
│ │ │ │ -00059c90: 6929 3a20 415f 3128 6929 205c 746f 2041  i): A_1(i) \to A
│ │ │ │ -00059ca0: 5f30 2869 2920 7468 6520 7265 7374 7269  _0(i) the restri
│ │ │ │ -00059cb0: 6374 696f 6e20 6f66 2064 203d 2064 2863  ction of d = d(c
│ │ │ │ -00059cc0: 292e 2077 6865 7265 2041 2869 2920 3d0a  ). where A(i) =.
│ │ │ │ -00059cd0: 5c6f 706c 7573 5f7b 693d 317d 5e70 2042  \oplus_{i=1}^p B
│ │ │ │ -00059ce0: 2869 290a 0a0a 0a54 6865 206d 6170 2068  (i)....The map h
│ │ │ │ -00059cf0: 2069 7320 6120 6469 7265 6374 2073 756d   is a direct sum
│ │ │ │ -00059d00: 206f 6620 6d61 7073 2074 6172 6765 7420   of maps target 
│ │ │ │ -00059d10: 6428 7029 205c 746f 2073 6f75 7263 6520  d(p) \to source 
│ │ │ │ -00059d20: 6428 7029 2074 6861 7420 6172 650a 686f  d(p) that are.ho
│ │ │ │ -00059d30: 6d6f 746f 7069 6573 2066 6f72 2066 665f  motopies for ff_
│ │ │ │ -00059d40: 7020 6f6e 2074 6865 2072 6573 7472 6963  p on the restric
│ │ │ │ -00059d50: 7469 6f6e 2064 2870 293a 206f 7665 7220  tion d(p): over 
│ │ │ │ -00059d60: 7468 6520 7269 6e67 2052 2328 702d 3129  the ring R#(p-1)
│ │ │ │ -00059d70: 203d 0a53 2f28 6666 2331 2e2e 6666 2328   =.S/(ff#1..ff#(
│ │ │ │ -00059d80: 702d 3129 2c20 736f 2064 2870 2920 2a20  p-1), so d(p) * 
│ │ │ │ -00059d90: 6823 7020 3d20 6666 2370 206d 6f64 2028  h#p = ff#p mod (
│ │ │ │ -00059da0: 6666 2331 2e2e 6666 2328 702d 3129 2e0a  ff#1..ff#(p-1)..
│ │ │ │ -00059db0: 0a49 6e20 6164 6469 7469 6f6e 2c20 6823  .In addition, h#
│ │ │ │ -00059dc0: 7020 2a20 6428 7029 2069 6e64 7563 6573  p * d(p) induces
│ │ │ │ -00059dd0: 2066 6623 7020 6f6e 2042 3123 7020 6d6f   ff#p on B1#p mo
│ │ │ │ -00059de0: 6420 2866 6623 312e 2e66 6623 2870 2d31  d (ff#1..ff#(p-1
│ │ │ │ -00059df0: 292e 0a0a 4865 7265 2069 7320 6120 7369  )...Here is a si
│ │ │ │ -00059e00: 6d70 6c65 2065 7861 6d70 6c65 3a0a 0a2b  mple example:..+
│ │ │ │ +00059430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ +00059440: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +00059450: 2020 204d 4620 3d20 6d61 7472 6978 4661     MF = matrixFa
│ │ │ │ +00059460: 6374 6f72 697a 6174 696f 6e28 6666 2c4d  ctorization(ff,M
│ │ │ │ +00059470: 290a 2020 2a20 496e 7075 7473 3a0a 2020  ).  * Inputs:.  
│ │ │ │ +00059480: 2020 2020 2a20 6666 2c20 6120 2a6e 6f74      * ff, a *not
│ │ │ │ +00059490: 6520 6d61 7472 6978 3a20 284d 6163 6175  e matrix: (Macau
│ │ │ │ +000594a0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c  lay2Doc)Matrix,,
│ │ │ │ +000594b0: 2061 2073 7566 6669 6369 656e 746c 7920   a sufficiently 
│ │ │ │ +000594c0: 6765 6e65 7261 6c0a 2020 2020 2020 2020  general.        
│ │ │ │ +000594d0: 7265 6775 6c61 7220 7365 7175 656e 6365  regular sequence
│ │ │ │ +000594e0: 2069 6e20 6120 7269 6e67 2053 0a20 2020   in a ring S.   
│ │ │ │ +000594f0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ +00059500: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ +00059510: 7932 446f 6329 4d6f 6475 6c65 2c2c 2061  y2Doc)Module,, a
│ │ │ │ +00059520: 206d 6178 696d 616c 2043 6f68 656e 2d4d   maximal Cohen-M
│ │ │ │ +00059530: 6163 6175 6c61 790a 2020 2020 2020 2020  acaulay.        
│ │ │ │ +00059540: 6d6f 6475 6c65 206f 7665 7220 532f 6964  module over S/id
│ │ │ │ +00059550: 6561 6c20 6666 0a20 202a 202a 6e6f 7465  eal ff.  * *note
│ │ │ │ +00059560: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ +00059570: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00059580: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ +00059590: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ +000595a0: 7075 7473 2c3a 0a20 2020 2020 202a 2041  puts,:.      * A
│ │ │ │ +000595b0: 7567 6d65 6e74 6174 696f 6e20 3d3e 202e  ugmentation => .
│ │ │ │ +000595c0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +000595d0: 6520 7472 7565 0a20 2020 2020 202a 2043  e true.      * C
│ │ │ │ +000595e0: 6865 636b 203d 3e20 2e2e 2e2c 2064 6566  heck => ..., def
│ │ │ │ +000595f0: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ +00059600: 0a20 2020 2020 202a 204c 6179 6572 6564  .      * Layered
│ │ │ │ +00059610: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +00059620: 2076 616c 7565 2074 7275 650a 2020 2020   value true.    
│ │ │ │ +00059630: 2020 2a20 5665 7262 6f73 6520 3d3e 202e    * Verbose => .
│ │ │ │ +00059640: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +00059650: 6520 6661 6c73 650a 2020 2a20 4f75 7470  e false.  * Outp
│ │ │ │ +00059660: 7574 733a 0a20 2020 2020 202a 204d 462c  uts:.      * MF,
│ │ │ │ +00059670: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ +00059680: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ +00059690: 742c 2c20 5c7b 642c 682c 6761 6d6d 615c  t,, \{d,h,gamma\
│ │ │ │ +000596a0: 7d2c 2077 6865 7265 2064 3a41 5f31 205c  }, where d:A_1 \
│ │ │ │ +000596b0: 746f 0a20 2020 2020 2020 2041 5f30 2061  to.        A_0 a
│ │ │ │ +000596c0: 6e64 2068 3a20 5c6f 706c 7573 2041 5f30  nd h: \oplus A_0
│ │ │ │ +000596d0: 2870 2920 5c74 6f20 415f 3120 6973 2074  (p) \to A_1 is t
│ │ │ │ +000596e0: 6865 2064 6972 6563 7420 7375 6d20 6f66  he direct sum of
│ │ │ │ +000596f0: 2070 6172 7469 616c 0a20 2020 2020 2020   partial.       
│ │ │ │ +00059700: 2068 6f6d 6f74 6f70 6965 732c 2061 6e64   homotopies, and
│ │ │ │ +00059710: 2067 616d 6d61 3a20 415f 3020 2d3e 4d20   gamma: A_0 ->M 
│ │ │ │ +00059720: 6973 2074 6865 2061 7567 6d65 6e74 6174  is the augmentat
│ │ │ │ +00059730: 696f 6e20 2872 6574 7572 6e65 6420 6f6e  ion (returned on
│ │ │ │ +00059740: 6c79 2069 660a 2020 2020 2020 2020 4175  ly if.        Au
│ │ │ │ +00059750: 676d 656e 7461 7469 6f6e 203d 3e74 7275  gmentation =>tru
│ │ │ │ +00059760: 6529 0a0a 4465 7363 7269 7074 696f 6e0a  e)..Description.
│ │ │ │ +00059770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +00059780: 2069 6e70 7574 206d 6f64 756c 6520 4d20   input module M 
│ │ │ │ +00059790: 7368 6f75 6c64 2062 6520 6120 6d61 7869  should be a maxi
│ │ │ │ +000597a0: 6d61 6c20 436f 6865 6e2d 4d61 6361 756c  mal Cohen-Macaul
│ │ │ │ +000597b0: 6179 206d 6f64 756c 6520 6f76 6572 2052  ay module over R
│ │ │ │ +000597c0: 203d 2053 2f69 6465 616c 0a66 662e 2020   = S/ideal.ff.  
│ │ │ │ +000597d0: 4966 204d 2069 7320 696e 2066 6163 7420  If M is in fact 
│ │ │ │ +000597e0: 6120 2268 6967 6820 7379 7a79 6779 222c  a "high syzygy",
│ │ │ │ +000597f0: 2074 6865 6e20 7468 6520 6675 6e63 7469   then the functi
│ │ │ │ +00059800: 6f6e 0a6d 6174 7269 7846 6163 746f 7269  on.matrixFactori
│ │ │ │ +00059810: 7a61 7469 6f6e 2866 662c 4d2c 4c61 7965  zation(ff,M,Laye
│ │ │ │ +00059820: 7265 643d 3e66 616c 7365 2920 7573 6573  red=>false) uses
│ │ │ │ +00059830: 2061 2064 6966 6665 7265 6e74 2c20 6661   a different, fa
│ │ │ │ +00059840: 7374 6572 2061 6c67 6f72 6974 686d 0a77  ster algorithm.w
│ │ │ │ +00059850: 6869 6368 206f 6e6c 7920 776f 726b 7320  hich only works 
│ │ │ │ +00059860: 696e 2074 6865 2068 6967 6820 7379 7a79  in the high syzy
│ │ │ │ +00059870: 6779 2063 6173 652e 0a0a 496e 2061 6c6c  gy case...In all
│ │ │ │ +00059880: 2065 7861 6d70 6c65 7320 7765 206b 6e6f   examples we kno
│ │ │ │ +00059890: 772c 204d 2063 616e 2062 6520 636f 6e73  w, M can be cons
│ │ │ │ +000598a0: 6964 6572 6564 2061 2022 6869 6768 2073  idered a "high s
│ │ │ │ +000598b0: 797a 7967 7922 2061 7320 6c6f 6e67 2061  yzygy" as long a
│ │ │ │ +000598c0: 730a 4578 745e 7b65 7665 6e7d 5f52 284d  s.Ext^{even}_R(M
│ │ │ │ +000598d0: 2c6b 2920 616e 6420 4578 745e 7b6f 6464  ,k) and Ext^{odd
│ │ │ │ +000598e0: 7d5f 5228 4d2c 6b29 2068 6176 6520 6e65  }_R(M,k) have ne
│ │ │ │ +000598f0: 6761 7469 7665 2072 6567 756c 6172 6974  gative regularit
│ │ │ │ +00059900: 7920 6f76 6572 2074 6865 2072 696e 670a  y over the ring.
│ │ │ │ +00059910: 6f66 2043 4920 6f70 6572 6174 6f72 7320  of CI operators 
│ │ │ │ +00059920: 2872 6567 7261 6465 6420 7769 7468 2076  (regraded with v
│ │ │ │ +00059930: 6172 6961 626c 6573 206f 6620 6465 6772  ariables of degr
│ │ │ │ +00059940: 6565 2031 2e20 486f 7765 7665 722c 2074  ee 1. However, t
│ │ │ │ +00059950: 6865 2062 6573 7420 7265 7375 6c74 0a77  he best result.w
│ │ │ │ +00059960: 6520 6361 6e20 7072 6f76 6520 6973 2074  e can prove is t
│ │ │ │ +00059970: 6861 7420 6974 2073 7566 6669 6365 7320  hat it suffices 
│ │ │ │ +00059980: 746f 2068 6176 6520 7265 6775 6c61 7269  to have regulari
│ │ │ │ +00059990: 7479 203c 202d 2832 2a64 696d 2052 2b31  ty < -(2*dim R+1
│ │ │ │ +000599a0: 292e 0a0a 5768 656e 2074 6865 206f 7074  )...When the opt
│ │ │ │ +000599b0: 696f 6e61 6c20 696e 7075 7420 4368 6563  ional input Chec
│ │ │ │ +000599c0: 6b3d 3d74 7275 6520 2874 6865 2064 6566  k==true (the def
│ │ │ │ +000599d0: 6175 6c74 2069 7320 4368 6563 6b3d 3d66  ault is Check==f
│ │ │ │ +000599e0: 616c 7365 292c 2074 6865 0a70 726f 7065  alse), the.prope
│ │ │ │ +000599f0: 7274 6965 7320 696e 2074 6865 2064 6566  rties in the def
│ │ │ │ +00059a00: 696e 6974 696f 6e20 6f66 204d 6174 7269  inition of Matri
│ │ │ │ +00059a10: 7820 4661 6374 6f72 697a 6174 696f 6e20  x Factorization 
│ │ │ │ +00059a20: 6172 6520 7665 7269 6669 6564 0a0a 5468  are verified..Th
│ │ │ │ +00059a30: 6520 6f75 7470 7574 2069 7320 6120 6c69  e output is a li
│ │ │ │ +00059a40: 7374 206f 6620 6d61 7073 205c 7b64 2c68  st of maps \{d,h
│ │ │ │ +00059a50: 5c7d 206f 7220 5c7b 642c 682c 6761 6d6d  \} or \{d,h,gamm
│ │ │ │ +00059a60: 615c 7d2c 2077 6865 7265 2067 616d 6d61  a\}, where gamma
│ │ │ │ +00059a70: 2069 7320 616e 0a61 7567 6d65 6e74 6174   is an.augmentat
│ │ │ │ +00059a80: 696f 6e2c 2074 6861 7420 6973 2c20 6120  ion, that is, a 
│ │ │ │ +00059a90: 6d61 7020 6672 6f6d 2074 6172 6765 7420  map from target 
│ │ │ │ +00059aa0: 6420 746f 204d 2e0a 0a54 6865 206d 6170  d to M...The map
│ │ │ │ +00059ab0: 2064 2069 7320 6120 7370 6563 6961 6c20   d is a special 
│ │ │ │ +00059ac0: 6c69 6674 696e 6720 746f 2053 206f 6620  lifting to S of 
│ │ │ │ +00059ad0: 6120 7072 6573 656e 7461 7469 6f6e 206f  a presentation o
│ │ │ │ +00059ae0: 6620 4d20 6f76 6572 2052 2e20 546f 2065  f M over R. To e
│ │ │ │ +00059af0: 7870 6c61 696e 0a74 6865 2063 6f6e 7465  xplain.the conte
│ │ │ │ +00059b00: 6e74 732c 2077 6520 696e 7472 6f64 7563  nts, we introduc
│ │ │ │ +00059b10: 6520 736f 6d65 206e 6f74 6174 696f 6e20  e some notation 
│ │ │ │ +00059b20: 2866 726f 6d20 4569 7365 6e62 7564 2061  (from Eisenbud a
│ │ │ │ +00059b30: 6e64 2050 6565 7661 2c20 224d 696e 696d  nd Peeva, "Minim
│ │ │ │ +00059b40: 616c 0a66 7265 6520 7265 736f 6c75 7469  al.free resoluti
│ │ │ │ +00059b50: 6f6e 7320 6f76 6572 2063 6f6d 706c 6574  ons over complet
│ │ │ │ +00059b60: 6520 696e 7465 7273 6563 7469 6f6e 7322  e intersections"
│ │ │ │ +00059b70: 204c 6563 7475 7265 204e 6f74 6573 2069   Lecture Notes i
│ │ │ │ +00059b80: 6e20 4d61 7468 656d 6174 6963 732c 0a32  n Mathematics,.2
│ │ │ │ +00059b90: 3135 322e 2053 7072 696e 6765 722c 2043  152. Springer, C
│ │ │ │ +00059ba0: 6861 6d2c 2032 3031 362e 2078 2b31 3037  ham, 2016. x+107
│ │ │ │ +00059bb0: 2070 702e 2049 5342 4e3a 2039 3738 2d33   pp. ISBN: 978-3
│ │ │ │ +00059bc0: 2d33 3139 2d32 3634 3336 2d33 3b0a 3937  -319-26436-3;.97
│ │ │ │ +00059bd0: 382d 332d 3331 392d 3236 3433 372d 3029  8-3-319-26437-0)
│ │ │ │ +00059be0: 2e0a 0a52 2869 2920 3d20 532f 2866 665f  ...R(i) = S/(ff_
│ │ │ │ +00059bf0: 302c 2e2e 2c66 665f 7b69 2d31 7d29 2e20  0,..,ff_{i-1}). 
│ │ │ │ +00059c00: 4865 7265 2030 3c3d 2069 203c 3d20 632c  Here 0<= i <= c,
│ │ │ │ +00059c10: 2061 6e64 2052 203d 2052 2863 2920 616e   and R = R(c) an
│ │ │ │ +00059c20: 6420 5320 3d20 5228 3029 2e0a 0a42 2869  d S = R(0)...B(i
│ │ │ │ +00059c30: 2920 3d20 7468 6520 6d61 7472 6978 2028  ) = the matrix (
│ │ │ │ +00059c40: 6f76 6572 2053 2920 7265 7072 6573 656e  over S) represen
│ │ │ │ +00059c50: 7469 6e67 2064 5f69 3a20 425f 3128 6929  ting d_i: B_1(i)
│ │ │ │ +00059c60: 205c 746f 2042 5f30 2869 290a 0a64 2869   \to B_0(i)..d(i
│ │ │ │ +00059c70: 293a 2041 5f31 2869 2920 5c74 6f20 415f  ): A_1(i) \to A_
│ │ │ │ +00059c80: 3028 6929 2074 6865 2072 6573 7472 6963  0(i) the restric
│ │ │ │ +00059c90: 7469 6f6e 206f 6620 6420 3d20 6428 6329  tion of d = d(c)
│ │ │ │ +00059ca0: 2e20 7768 6572 6520 4128 6929 203d 0a5c  . where A(i) =.\
│ │ │ │ +00059cb0: 6f70 6c75 735f 7b69 3d31 7d5e 7020 4228  oplus_{i=1}^p B(
│ │ │ │ +00059cc0: 6929 0a0a 0a0a 5468 6520 6d61 7020 6820  i)....The map h 
│ │ │ │ +00059cd0: 6973 2061 2064 6972 6563 7420 7375 6d20  is a direct sum 
│ │ │ │ +00059ce0: 6f66 206d 6170 7320 7461 7267 6574 2064  of maps target d
│ │ │ │ +00059cf0: 2870 2920 5c74 6f20 736f 7572 6365 2064  (p) \to source d
│ │ │ │ +00059d00: 2870 2920 7468 6174 2061 7265 0a68 6f6d  (p) that are.hom
│ │ │ │ +00059d10: 6f74 6f70 6965 7320 666f 7220 6666 5f70  otopies for ff_p
│ │ │ │ +00059d20: 206f 6e20 7468 6520 7265 7374 7269 6374   on the restrict
│ │ │ │ +00059d30: 696f 6e20 6428 7029 3a20 6f76 6572 2074  ion d(p): over t
│ │ │ │ +00059d40: 6865 2072 696e 6720 5223 2870 2d31 2920  he ring R#(p-1) 
│ │ │ │ +00059d50: 3d0a 532f 2866 6623 312e 2e66 6623 2870  =.S/(ff#1..ff#(p
│ │ │ │ +00059d60: 2d31 292c 2073 6f20 6428 7029 202a 2068  -1), so d(p) * h
│ │ │ │ +00059d70: 2370 203d 2066 6623 7020 6d6f 6420 2866  #p = ff#p mod (f
│ │ │ │ +00059d80: 6623 312e 2e66 6623 2870 2d31 292e 0a0a  f#1..ff#(p-1)...
│ │ │ │ +00059d90: 496e 2061 6464 6974 696f 6e2c 2068 2370  In addition, h#p
│ │ │ │ +00059da0: 202a 2064 2870 2920 696e 6475 6365 7320   * d(p) induces 
│ │ │ │ +00059db0: 6666 2370 206f 6e20 4231 2370 206d 6f64  ff#p on B1#p mod
│ │ │ │ +00059dc0: 2028 6666 2331 2e2e 6666 2328 702d 3129   (ff#1..ff#(p-1)
│ │ │ │ +00059dd0: 2e0a 0a48 6572 6520 6973 2061 2073 696d  ...Here is a sim
│ │ │ │ +00059de0: 706c 6520 6578 616d 706c 653a 0a0a 2b2d  ple example:..+-
│ │ │ │ +00059df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00059e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059e40: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e  --+.|i1 : setRan
│ │ │ │ -00059e50: 646f 6d53 6565 6420 3020 2020 2020 2020  domSeed 0       
│ │ │ │ -00059e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059e70: 2020 2020 2020 207c 0a7c 202d 2d20 7365         |.| -- se
│ │ │ │ -00059e80: 7474 696e 6720 7261 6e64 6f6d 2073 6565  tting random see
│ │ │ │ -00059e90: 6420 746f 2030 2020 2020 2020 2020 2020  d to 0          
│ │ │ │ -00059ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00059e20: 2d2b 0a7c 6931 203a 2073 6574 5261 6e64  -+.|i1 : setRand
│ │ │ │ +00059e30: 6f6d 5365 6564 2030 2020 2020 2020 2020  omSeed 0        
│ │ │ │ +00059e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059e50: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574        |.| -- set
│ │ │ │ +00059e60: 7469 6e67 2072 616e 646f 6d20 7365 6564  ting random seed
│ │ │ │ +00059e70: 2074 6f20 3020 2020 2020 2020 2020 2020   to 0           
│ │ │ │ +00059e80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00059e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00059eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059ec0: 7c0a 7c6f 3120 3d20 3020 2020 2020 2020  |.|o1 = 0       
│ │ │ │  00059ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059ee0: 207c 0a7c 6f31 203d 2030 2020 2020 2020   |.|o1 = 0      
│ │ │ │ -00059ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059f10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00059f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00059f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00059f50: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020   : kk = ZZ/101  
│ │ │ │ -00059f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059ef0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00059f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00059f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00059f30: 3a20 6b6b 203d 205a 5a2f 3130 3120 2020  : kk = ZZ/101   
│ │ │ │ +00059f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00059f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00059f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059f80: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00059f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059f90: 2020 2020 7c0a 7c6f 3220 3d20 6b6b 2020      |.|o2 = kk  
│ │ │ │  00059fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059fb0: 2020 2020 207c 0a7c 6f32 203d 206b 6b20       |.|o2 = kk 
│ │ │ │ -00059fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059fc0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00059fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00059fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00059ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005a020: 0a7c 6f32 203a 2051 756f 7469 656e 7452  .|o2 : QuotientR
│ │ │ │ -0005a030: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0005a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a050: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -0005a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a080: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0005a090: 2053 203d 206b 6b5b 612c 622c 752c 765d   S = kk[a,b,u,v]
│ │ │ │ +00059fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00059ff0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005a000: 7c6f 3220 3a20 5175 6f74 6965 6e74 5269  |o2 : QuotientRi
│ │ │ │ +0005a010: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0005a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a030: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a060: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0005a070: 5320 3d20 6b6b 5b61 2c62 2c75 2c76 5d20  S = kk[a,b,u,v] 
│ │ │ │ +0005a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a090: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0005a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005a0c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0005a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a0d0: 2020 7c0a 7c6f 3320 3d20 5320 2020 2020    |.|o3 = S     
│ │ │ │  0005a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a0f0: 2020 207c 0a7c 6f33 203d 2053 2020 2020     |.|o3 = S    
│ │ │ │ -0005a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a100: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0005a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a120: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0005a130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a150: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0005a160: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52  o3 : PolynomialR
│ │ │ │ -0005a170: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │ -0005a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0005a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a1c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066  -------+.|i4 : f
│ │ │ │ -0005a1d0: 6620 3d20 6d61 7472 6978 2261 752c 6276  f = matrix"au,bv
│ │ │ │ -0005a1e0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -0005a1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005a120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a130: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0005a140: 3320 3a20 506f 6c79 6e6f 6d69 616c 5269  3 : PolynomialRi
│ │ │ │ +0005a150: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ +0005a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a170: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0005a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a1a0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6666  ------+.|i4 : ff
│ │ │ │ +0005a1b0: 203d 206d 6174 7269 7822 6175 2c62 7622   = matrix"au,bv"
│ │ │ │ +0005a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a1d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a230: 207c 0a7c 6f34 203d 207c 2061 7520 6276   |.|o4 = | au bv
│ │ │ │ -0005a240: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0005a210: 7c0a 7c6f 3420 3d20 7c20 6175 2062 7620  |.|o4 = | au bv 
│ │ │ │ +0005a220: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a240: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0005a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a260: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0005a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a290: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005a2a0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0005a2b0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0005a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a2d0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053  |.|o4 : Matrix S
│ │ │ │ -0005a2e0: 2020 3c2d 2d20 5320 2020 2020 2020 2020    <-- S         
│ │ │ │ -0005a2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a300: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -0005a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ -0005a340: 3a20 5220 3d20 532f 6964 6561 6c20 6666  : R = S/ideal ff
│ │ │ │ -0005a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005a370: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0005a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a270: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0005a280: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ +0005a290: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0005a2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a2b0: 0a7c 6f34 203a 204d 6174 7269 7820 5320  .|o4 : Matrix S 
│ │ │ │ +0005a2c0: 203c 2d2d 2053 2020 2020 2020 2020 2020   <-- S          
│ │ │ │ +0005a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a2e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a310: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ +0005a320: 2052 203d 2053 2f69 6465 616c 2066 6620   R = S/ideal ff 
│ │ │ │ +0005a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005a350: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a380: 2020 207c 0a7c 6f35 203d 2052 2020 2020     |.|o5 = R    
│ │ │ │  0005a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a3a0: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020      |.|o5 = R   
│ │ │ │ -0005a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a3b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a400: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005a410: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269  |o5 : QuotientRi
│ │ │ │ -0005a420: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -0005a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0005a450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ -0005a480: 4d30 203d 2052 5e31 2f69 6465 616c 2261  M0 = R^1/ideal"a
│ │ │ │ -0005a490: 2c62 2220 2020 2020 2020 2020 2020 2020  ,b"             
│ │ │ │ -0005a4a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a3e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005a3f0: 6f35 203a 2051 756f 7469 656e 7452 696e  o5 : QuotientRin
│ │ │ │ +0005a400: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0005a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a420: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a450: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d  -------+.|i6 : M
│ │ │ │ +0005a460: 3020 3d20 525e 312f 6964 6561 6c22 612c  0 = R^1/ideal"a,
│ │ │ │ +0005a470: 6222 2020 2020 2020 2020 2020 2020 2020  b"              
│ │ │ │ +0005a480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005a490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a4e0: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e    |.|o6 = cokern
│ │ │ │ -0005a4f0: 656c 207c 2061 2062 207c 2020 2020 2020  el | a b |      
│ │ │ │ +0005a4c0: 207c 0a7c 6f36 203d 2063 6f6b 6572 6e65   |.|o6 = cokerne
│ │ │ │ +0005a4d0: 6c20 7c20 6120 6220 7c20 2020 2020 2020  l | a b |       
│ │ │ │ +0005a4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a4f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0005a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a510: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005a520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a520: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0005a530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005a540: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │  0005a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a560: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │ -0005a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a580: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c   |.|o6 : R-modul
│ │ │ │ -0005a590: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ -0005a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a5b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0005a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ -0005a5f0: 203a 204d 203d 2068 6967 6853 797a 7967   : M = highSyzyg
│ │ │ │ -0005a600: 7920 4d30 2020 2020 2020 2020 2020 2020  y M0            
│ │ │ │ +0005a560: 7c0a 7c6f 3620 3a20 522d 6d6f 6475 6c65  |.|o6 : R-module
│ │ │ │ +0005a570: 2c20 7175 6f74 6965 6e74 206f 6620 5220  , quotient of R 
│ │ │ │ +0005a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a590: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0005a5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720  ----------+.|i7 
│ │ │ │ +0005a5d0: 3a20 4d20 3d20 6869 6768 5379 7a79 6779  : M = highSyzygy
│ │ │ │ +0005a5e0: 204d 3020 2020 2020 2020 2020 2020 2020   M0             
│ │ │ │ +0005a5f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a600: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005a610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a620: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0005a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a650: 2020 2020 207c 0a7c 6f37 203d 2063 6f6b       |.|o7 = cok
│ │ │ │ -0005a660: 6572 6e65 6c20 7b32 7d20 7c20 6220 2d61  ernel {2} | b -a
│ │ │ │ -0005a670: 2030 2030 207c 2020 2020 2020 2020 2020   0 0 |          
│ │ │ │ -0005a680: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0005a690: 2020 2020 2020 2020 2020 207b 327d 207c             {2} |
│ │ │ │ -0005a6a0: 2030 2030 2020 6120 6220 7c20 2020 2020   0 0  a b |     
│ │ │ │ -0005a6b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005a6c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0005a6d0: 7b32 7d20 7c20 3020 7620 2030 2075 207c  {2} | 0 v  0 u |
│ │ │ │ +0005a620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a630: 2020 2020 7c0a 7c6f 3720 3d20 636f 6b65      |.|o7 = coke
│ │ │ │ +0005a640: 726e 656c 207b 327d 207c 2062 202d 6120  rnel {2} | b -a 
│ │ │ │ +0005a650: 3020 3020 7c20 2020 2020 2020 2020 2020  0 0 |           
│ │ │ │ +0005a660: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005a670: 2020 2020 2020 2020 2020 7b32 7d20 7c20            {2} | 
│ │ │ │ +0005a680: 3020 3020 2061 2062 207c 2020 2020 2020  0 0  a b |      
│ │ │ │ +0005a690: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005a6a0: 7c20 2020 2020 2020 2020 2020 2020 207b  |              {
│ │ │ │ +0005a6b0: 327d 207c 2030 2076 2020 3020 7520 7c20  2} | 0 v  0 u | 
│ │ │ │ +0005a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a6d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a6f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0005a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a700: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a720: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a740: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -0005a750: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005a760: 7c6f 3720 3a20 522d 6d6f 6475 6c65 2c20  |o7 : R-module, 
│ │ │ │ -0005a770: 7175 6f74 6965 6e74 206f 6620 5220 2020  quotient of R   
│ │ │ │ -0005a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a790: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0005a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ -0005a7d0: 4d46 203d 206d 6174 7269 7846 6163 746f  MF = matrixFacto
│ │ │ │ -0005a7e0: 7269 7a61 7469 6f6e 2866 662c 4d29 3b20  rization(ff,M); 
│ │ │ │ -0005a7f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0005a720: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +0005a730: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005a740: 6f37 203a 2052 2d6d 6f64 756c 652c 2071  o7 : R-module, q
│ │ │ │ +0005a750: 756f 7469 656e 7420 6f66 2052 2020 2020  uotient of R    
│ │ │ │ +0005a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a770: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a7a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 204d  -------+.|i8 : M
│ │ │ │ +0005a7b0: 4620 3d20 6d61 7472 6978 4661 6374 6f72  F = matrixFactor
│ │ │ │ +0005a7c0: 697a 6174 696f 6e28 6666 2c4d 293b 2020  ization(ff,M);  
│ │ │ │ +0005a7d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0005a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005a800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a830: 2d2d 2b0a 7c69 3920 3a20 6e65 744c 6973  --+.|i9 : netLis
│ │ │ │ -0005a840: 7420 4252 616e 6b73 204d 4620 2020 2020  t BRanks MF     
│ │ │ │ +0005a810: 2d2b 0a7c 6939 203a 206e 6574 4c69 7374  -+.|i9 : netList
│ │ │ │ +0005a820: 2042 5261 6e6b 7320 4d46 2020 2020 2020   BRanks MF      
│ │ │ │ +0005a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a840: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0005a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a860: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a890: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0005a8a0: 2020 2020 2b2d 2b2d 2b20 2020 2020 2020      +-+-+       
│ │ │ │ -0005a8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a870: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005a880: 2020 202b 2d2b 2d2b 2020 2020 2020 2020     +-+-+        
│ │ │ │ +0005a890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a8b0: 7c0a 7c6f 3920 3d20 7c32 7c32 7c20 2020  |.|o9 = |2|2|   
│ │ │ │  0005a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a8d0: 207c 0a7c 6f39 203d 207c 327c 327c 2020   |.|o9 = |2|2|  
│ │ │ │ -0005a8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a900: 2020 2020 2020 7c0a 7c20 2020 2020 2b2d        |.|     +-
│ │ │ │ -0005a910: 2b2d 2b20 2020 2020 2020 2020 2020 2020  +-+             
│ │ │ │ -0005a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a930: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005a940: 2020 207c 317c 327c 2020 2020 2020 2020     |1|2|        
│ │ │ │ -0005a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a8e0: 2020 2020 207c 0a7c 2020 2020 202b 2d2b       |.|     +-+
│ │ │ │ +0005a8f0: 2d2b 2020 2020 2020 2020 2020 2020 2020  -+              
│ │ │ │ +0005a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a910: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0005a920: 2020 7c31 7c32 7c20 2020 2020 2020 2020    |1|2|         
│ │ │ │ +0005a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005a950: 0a7c 2020 2020 202b 2d2b 2d2b 2020 2020  .|     +-+-+    
│ │ │ │  0005a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a970: 7c0a 7c20 2020 2020 2b2d 2b2d 2b20 2020  |.|     +-+-+   
│ │ │ │ -0005a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005a9a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -0005a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ -0005a9e0: 203a 206e 6574 4c69 7374 2062 4d61 7073   : netList bMaps
│ │ │ │ -0005a9f0: 204d 4620 2020 2020 2020 2020 2020 2020   MF             
│ │ │ │ -0005aa00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005aa10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0005aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa40: 2020 2020 7c0a 7c20 2020 2020 202b 2d2d      |.|      +--
│ │ │ │ -0005aa50: 2d2d 2d2d 2d2d 2d2d 2d2b 2020 2020 2020  ---------+      
│ │ │ │ -0005aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aa70: 2020 2020 2020 2020 207c 0a7c 6f31 3020           |.|o10 
│ │ │ │ -0005aa80: 3d20 7c7b 327d 207c 2061 2062 207c 7c20  = |{2} | a b || 
│ │ │ │ -0005aa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aaa0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005aab0: 7c20 2020 2020 207c 7b32 7d20 7c20 3020  |      |{2} | 0 
│ │ │ │ -0005aac0: 7520 7c7c 2020 2020 2020 2020 2020 2020  u ||            
│ │ │ │ -0005aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aae0: 2020 207c 0a7c 2020 2020 2020 2b2d 2d2d     |.|      +---
│ │ │ │ -0005aaf0: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020  --------+       
│ │ │ │ -0005ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ab10: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0005ab20: 207c 7b32 7d20 7c20 6220 6120 7c7c 2020   |{2} | b a ||  
│ │ │ │ -0005ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ab40: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0005ab50: 2020 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d        +---------
│ │ │ │ -0005ab60: 2d2d 2b20 2020 2020 2020 2020 2020 2020  --+             
│ │ │ │ -0005ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ab80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ -0005ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005abb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │ -0005abc0: 6265 7474 6920 6672 6565 5265 736f 6c75  betti freeResolu
│ │ │ │ -0005abd0: 7469 6f6e 284d 2c20 4c65 6e67 7468 4c69  tion(M, LengthLi
│ │ │ │ -0005abe0: 6d69 7420 3d3e 2037 2920 2020 7c0a 7c20  mit => 7)   |.| 
│ │ │ │ +0005a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005a980: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005a990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005a9b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020  ---------+.|i10 
│ │ │ │ +0005a9c0: 3a20 6e65 744c 6973 7420 624d 6170 7320  : netList bMaps 
│ │ │ │ +0005a9d0: 4d46 2020 2020 2020 2020 2020 2020 2020  MF              
│ │ │ │ +0005a9e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005a9f0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aa20: 2020 207c 0a7c 2020 2020 2020 2b2d 2d2d     |.|      +---
│ │ │ │ +0005aa30: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020  --------+       
│ │ │ │ +0005aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aa50: 2020 2020 2020 2020 7c0a 7c6f 3130 203d          |.|o10 =
│ │ │ │ +0005aa60: 207c 7b32 7d20 7c20 6120 6220 7c7c 2020   |{2} | a b ||  
│ │ │ │ +0005aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aa80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005aa90: 2020 2020 2020 7c7b 327d 207c 2030 2075        |{2} | 0 u
│ │ │ │ +0005aaa0: 207c 7c20 2020 2020 2020 2020 2020 2020   ||             
│ │ │ │ +0005aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aac0: 2020 7c0a 7c20 2020 2020 202b 2d2d 2d2d    |.|      +----
│ │ │ │ +0005aad0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020  -------+        
│ │ │ │ +0005aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aaf0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0005ab00: 7c7b 327d 207c 2062 2061 207c 7c20 2020  |{2} | b a ||   
│ │ │ │ +0005ab10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0005ab30: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d       +----------
│ │ │ │ +0005ab40: 2d2b 2020 2020 2020 2020 2020 2020 2020  -+              
│ │ │ │ +0005ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ab60: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0005ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ab90: 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a 2062  ------+.|i11 : b
│ │ │ │ +0005aba0: 6574 7469 2066 7265 6552 6573 6f6c 7574  etti freeResolut
│ │ │ │ +0005abb0: 696f 6e28 4d2c 204c 656e 6774 684c 696d  ion(M, LengthLim
│ │ │ │ +0005abc0: 6974 203d 3e20 3729 2020 207c 0a7c 2020  it => 7)   |.|  
│ │ │ │ +0005abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ac20: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0005ac30: 2030 2031 2032 2033 2034 2035 2036 2020   0 1 2 3 4 5 6  
│ │ │ │ -0005ac40: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ -0005ac50: 2020 2020 2020 7c0a 7c6f 3131 203d 2074        |.|o11 = t
│ │ │ │ -0005ac60: 6f74 616c 3a20 3320 3420 3520 3620 3720  otal: 3 4 5 6 7 
│ │ │ │ -0005ac70: 3820 3920 3130 2020 2020 2020 2020 2020  8 9 10          
│ │ │ │ -0005ac80: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005ac90: 2020 2020 2020 2020 323a 2033 2034 2035          2: 3 4 5
│ │ │ │ -0005aca0: 2036 2037 2038 2039 2031 3020 2020 2020   6 7 8 9 10     
│ │ │ │ +0005ac00: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0005ac10: 3020 3120 3220 3320 3420 3520 3620 2037  0 1 2 3 4 5 6  7
│ │ │ │ +0005ac20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ac30: 2020 2020 207c 0a7c 6f31 3120 3d20 746f       |.|o11 = to
│ │ │ │ +0005ac40: 7461 6c3a 2033 2034 2035 2036 2037 2038  tal: 3 4 5 6 7 8
│ │ │ │ +0005ac50: 2039 2031 3020 2020 2020 2020 2020 2020   9 10           
│ │ │ │ +0005ac60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0005ac70: 2020 2020 2020 2032 3a20 3320 3420 3520         2: 3 4 5 
│ │ │ │ +0005ac80: 3620 3720 3820 3920 3130 2020 2020 2020  6 7 8 9 10      
│ │ │ │ +0005ac90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005aca0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0005acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005acc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0005acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005acf0: 2020 2020 207c 0a7c 6f31 3120 3a20 4265       |.|o11 : Be
│ │ │ │ -0005ad00: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ -0005ad10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ad20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ -0005ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0005ad60: 0a7c 6931 3220 3a20 696e 6669 6e69 7465  .|i12 : infinite
│ │ │ │ -0005ad70: 4265 7474 694e 756d 6265 7273 2028 4d46  BettiNumbers (MF
│ │ │ │ -0005ad80: 2c37 2920 2020 2020 2020 2020 2020 2020  ,7)             
│ │ │ │ -0005ad90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0005ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005adc0: 2020 2020 2020 2020 207c 0a7c 6f31 3220           |.|o12 
│ │ │ │ -0005add0: 3d20 7b33 2c20 342c 2035 2c20 362c 2037  = {3, 4, 5, 6, 7
│ │ │ │ -0005ade0: 2c20 382c 2039 2c20 3130 7d20 2020 2020  , 8, 9, 10}     
│ │ │ │ -0005adf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005ae00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0005ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005acd0: 2020 2020 7c0a 7c6f 3131 203a 2042 6574      |.|o11 : Bet
│ │ │ │ +0005ace0: 7469 5461 6c6c 7920 2020 2020 2020 2020  tiTally         
│ │ │ │ +0005acf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ad00: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ad20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005ad40: 7c69 3132 203a 2069 6e66 696e 6974 6542  |i12 : infiniteB
│ │ │ │ +0005ad50: 6574 7469 4e75 6d62 6572 7320 284d 462c  ettiNumbers (MF,
│ │ │ │ +0005ad60: 3729 2020 2020 2020 2020 2020 2020 2020  7)              
│ │ │ │ +0005ad70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0005ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ada0: 2020 2020 2020 2020 7c0a 7c6f 3132 203d          |.|o12 =
│ │ │ │ +0005adb0: 207b 332c 2034 2c20 352c 2036 2c20 372c   {3, 4, 5, 6, 7,
│ │ │ │ +0005adc0: 2038 2c20 392c 2031 307d 2020 2020 2020   8, 9, 10}      
│ │ │ │ +0005add0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ae10: 2020 7c0a 7c6f 3132 203a 204c 6973 7420    |.|o12 : List 
│ │ │ │  0005ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae30: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374     |.|o12 : List
│ │ │ │ -0005ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ae60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0005ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0005aea0: 6931 3320 3a20 6265 7474 6920 6672 6565  i13 : betti free
│ │ │ │ -0005aeb0: 5265 736f 6c75 7469 6f6e 2070 7573 6846  Resolution pushF
│ │ │ │ -0005aec0: 6f72 7761 7264 286d 6170 2852 2c53 292c  orward(map(R,S),
│ │ │ │ -0005aed0: 4d29 7c0a 7c20 2020 2020 2020 2020 2020  M)|.|           
│ │ │ │ -0005aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005af00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0005af10: 2020 2020 2020 2030 2031 2032 2020 2020         0 1 2    
│ │ │ │ -0005af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005af30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005af40: 3133 203d 2074 6f74 616c 3a20 3320 3520  13 = total: 3 5 
│ │ │ │ -0005af50: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0005af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005af70: 207c 0a7c 2020 2020 2020 2020 2020 323a   |.|          2:
│ │ │ │ -0005af80: 2033 2034 202e 2020 2020 2020 2020 2020   3 4 .          
│ │ │ │ -0005af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005afa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0005afb0: 2020 2033 3a20 2e20 3120 3220 2020 2020     3: . 1 2     
│ │ │ │ +0005ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ae40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0005ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0005ae80: 3133 203a 2062 6574 7469 2066 7265 6552  13 : betti freeR
│ │ │ │ +0005ae90: 6573 6f6c 7574 696f 6e20 7075 7368 466f  esolution pushFo
│ │ │ │ +0005aea0: 7277 6172 6428 6d61 7028 522c 5329 2c4d  rward(map(R,S),M
│ │ │ │ +0005aeb0: 297c 0a7c 2020 2020 2020 2020 2020 2020  )|.|            
│ │ │ │ +0005aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005aee0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0005aef0: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ +0005af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af10: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0005af20: 3320 3d20 746f 7461 6c3a 2033 2035 2032  3 = total: 3 5 2
│ │ │ │ +0005af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af50: 7c0a 7c20 2020 2020 2020 2020 2032 3a20  |.|          2: 
│ │ │ │ +0005af60: 3320 3420 2e20 2020 2020 2020 2020 2020  3 4 .           
│ │ │ │ +0005af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005af80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0005af90: 2020 333a 202e 2031 2032 2020 2020 2020    3: . 1 2      
│ │ │ │ +0005afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005afb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0005afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005afd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0005afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b010: 7c0a 7c6f 3133 203a 2042 6574 7469 5461  |.|o13 : BettiTa
│ │ │ │ -0005b020: 6c6c 7920 2020 2020 2020 2020 2020 2020  lly             
│ │ │ │ -0005b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b040: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -0005b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134  ----------+.|i14
│ │ │ │ -0005b080: 203a 2066 696e 6974 6542 6574 7469 4e75   : finiteBettiNu
│ │ │ │ -0005b090: 6d62 6572 7320 4d46 2020 2020 2020 2020  mbers MF        
│ │ │ │ -0005b0a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005b0b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0005b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b0e0: 2020 2020 7c0a 7c6f 3134 203d 207b 332c      |.|o14 = {3,
│ │ │ │ -0005b0f0: 2035 2c20 327d 2020 2020 2020 2020 2020   5, 2}          
│ │ │ │ +0005afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005afe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0005aff0: 0a7c 6f31 3320 3a20 4265 7474 6954 616c  .|o13 : BettiTal
│ │ │ │ +0005b000: 6c79 2020 2020 2020 2020 2020 2020 2020  ly              
│ │ │ │ +0005b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b020: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005b030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b050: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420  ---------+.|i14 
│ │ │ │ +0005b060: 3a20 6669 6e69 7465 4265 7474 694e 756d  : finiteBettiNum
│ │ │ │ +0005b070: 6265 7273 204d 4620 2020 2020 2020 2020  bers MF         
│ │ │ │ +0005b080: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005b090: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0005b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b0c0: 2020 207c 0a7c 6f31 3420 3d20 7b33 2c20     |.|o14 = {3, 
│ │ │ │ +0005b0d0: 352c 2032 7d20 2020 2020 2020 2020 2020  5, 2}           
│ │ │ │ +0005b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b110: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b140: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005b150: 7c6f 3134 203a 204c 6973 7420 2020 2020  |o14 : List     
│ │ │ │ -0005b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005b180: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0005b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b1b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061  --------+..See a
│ │ │ │ -0005b1c0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -0005b1d0: 2a20 2a6e 6f74 6520 6669 6e69 7465 4265  * *note finiteBe
│ │ │ │ -0005b1e0: 7474 694e 756d 6265 7273 3a20 6669 6e69  ttiNumbers: fini
│ │ │ │ -0005b1f0: 7465 4265 7474 694e 756d 6265 7273 2c20  teBettiNumbers, 
│ │ │ │ -0005b200: 2d2d 2062 6574 7469 206e 756d 6265 7273  -- betti numbers
│ │ │ │ -0005b210: 206f 6620 6669 6e69 7465 0a20 2020 2072   of finite.    r
│ │ │ │ -0005b220: 6573 6f6c 7574 696f 6e20 636f 6d70 7574  esolution comput
│ │ │ │ -0005b230: 6564 2066 726f 6d20 6120 6d61 7472 6978  ed from a matrix
│ │ │ │ -0005b240: 2066 6163 746f 7269 7a61 7469 6f6e 0a20   factorization. 
│ │ │ │ -0005b250: 202a 202a 6e6f 7465 2069 6e66 696e 6974   * *note infinit
│ │ │ │ -0005b260: 6542 6574 7469 4e75 6d62 6572 733a 2069  eBettiNumbers: i
│ │ │ │ -0005b270: 6e66 696e 6974 6542 6574 7469 4e75 6d62  nfiniteBettiNumb
│ │ │ │ -0005b280: 6572 732c 202d 2d20 6265 7474 6920 6e75  ers, -- betti nu
│ │ │ │ -0005b290: 6d62 6572 7320 6f66 0a20 2020 2066 696e  mbers of.    fin
│ │ │ │ -0005b2a0: 6974 6520 7265 736f 6c75 7469 6f6e 2063  ite resolution c
│ │ │ │ -0005b2b0: 6f6d 7075 7465 6420 6672 6f6d 2061 206d  omputed from a m
│ │ │ │ -0005b2c0: 6174 7269 7820 6661 6374 6f72 697a 6174  atrix factorizat
│ │ │ │ -0005b2d0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 6869  ion.  * *note hi
│ │ │ │ -0005b2e0: 6768 5379 7a79 6779 3a20 6869 6768 5379  ghSyzygy: highSy
│ │ │ │ -0005b2f0: 7a79 6779 2c20 2d2d 2052 6574 7572 6e73  zygy, -- Returns
│ │ │ │ -0005b300: 2061 2073 797a 7967 7920 6d6f 6475 6c65   a syzygy module
│ │ │ │ -0005b310: 206f 6e65 2062 6579 6f6e 6420 7468 650a   one beyond the.
│ │ │ │ -0005b320: 2020 2020 7265 6775 6c61 7269 7479 206f      regularity o
│ │ │ │ -0005b330: 6620 4578 7428 4d2c 6b29 0a20 202a 202a  f Ext(M,k).  * *
│ │ │ │ -0005b340: 6e6f 7465 2062 4d61 7073 3a20 624d 6170  note bMaps: bMap
│ │ │ │ -0005b350: 732c 202d 2d20 6c69 7374 2074 6865 206d  s, -- list the m
│ │ │ │ -0005b360: 6170 7320 2064 5f70 3a42 5f31 2870 292d  aps  d_p:B_1(p)-
│ │ │ │ -0005b370: 2d3e 425f 3028 7029 2069 6e20 610a 2020  ->B_0(p) in a.  
│ │ │ │ -0005b380: 2020 6d61 7472 6978 4661 6374 6f72 697a    matrixFactoriz
│ │ │ │ -0005b390: 6174 696f 6e0a 2020 2a20 2a6e 6f74 6520  ation.  * *note 
│ │ │ │ -0005b3a0: 4252 616e 6b73 3a20 4252 616e 6b73 2c20  BRanks: BRanks, 
│ │ │ │ -0005b3b0: 2d2d 2072 616e 6b73 206f 6620 7468 6520  -- ranks of the 
│ │ │ │ -0005b3c0: 6d6f 6475 6c65 7320 425f 6928 6429 2069  modules B_i(d) i
│ │ │ │ -0005b3d0: 6e20 610a 2020 2020 6d61 7472 6978 4661  n a.    matrixFa
│ │ │ │ -0005b3e0: 6374 6f72 697a 6174 696f 6e0a 0a57 6179  ctorization..Way
│ │ │ │ -0005b3f0: 7320 746f 2075 7365 206d 6174 7269 7846  s to use matrixF
│ │ │ │ -0005b400: 6163 746f 7269 7a61 7469 6f6e 3a0a 3d3d  actorization:.==
│ │ │ │ -0005b410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0005b420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ -0005b430: 2020 2a20 226d 6174 7269 7846 6163 746f    * "matrixFacto
│ │ │ │ -0005b440: 7269 7a61 7469 6f6e 284d 6174 7269 782c  rization(Matrix,
│ │ │ │ -0005b450: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ -0005b460: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -0005b470: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0005b480: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -0005b490: 6520 6d61 7472 6978 4661 6374 6f72 697a  e matrixFactoriz
│ │ │ │ -0005b4a0: 6174 696f 6e3a 206d 6174 7269 7846 6163  ation: matrixFac
│ │ │ │ -0005b4b0: 746f 7269 7a61 7469 6f6e 2c20 6973 2061  torization, is a
│ │ │ │ -0005b4c0: 202a 6e6f 7465 206d 6574 686f 640a 6675   *note method.fu
│ │ │ │ -0005b4d0: 6e63 7469 6f6e 2077 6974 6820 6f70 7469  nction with opti
│ │ │ │ -0005b4e0: 6f6e 733a 2028 4d61 6361 756c 6179 3244  ons: (Macaulay2D
│ │ │ │ -0005b4f0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -0005b500: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a  nWithOptions,...
│ │ │ │ +0005b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b120: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005b130: 6f31 3420 3a20 4c69 7374 2020 2020 2020  o14 : List      
│ │ │ │ +0005b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005b160: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005b170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b190: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ +0005b1a0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +0005b1b0: 202a 6e6f 7465 2066 696e 6974 6542 6574   *note finiteBet
│ │ │ │ +0005b1c0: 7469 4e75 6d62 6572 733a 2066 696e 6974  tiNumbers: finit
│ │ │ │ +0005b1d0: 6542 6574 7469 4e75 6d62 6572 732c 202d  eBettiNumbers, -
│ │ │ │ +0005b1e0: 2d20 6265 7474 6920 6e75 6d62 6572 7320  - betti numbers 
│ │ │ │ +0005b1f0: 6f66 2066 696e 6974 650a 2020 2020 7265  of finite.    re
│ │ │ │ +0005b200: 736f 6c75 7469 6f6e 2063 6f6d 7075 7465  solution compute
│ │ │ │ +0005b210: 6420 6672 6f6d 2061 206d 6174 7269 7820  d from a matrix 
│ │ │ │ +0005b220: 6661 6374 6f72 697a 6174 696f 6e0a 2020  factorization.  
│ │ │ │ +0005b230: 2a20 2a6e 6f74 6520 696e 6669 6e69 7465  * *note infinite
│ │ │ │ +0005b240: 4265 7474 694e 756d 6265 7273 3a20 696e  BettiNumbers: in
│ │ │ │ +0005b250: 6669 6e69 7465 4265 7474 694e 756d 6265  finiteBettiNumbe
│ │ │ │ +0005b260: 7273 2c20 2d2d 2062 6574 7469 206e 756d  rs, -- betti num
│ │ │ │ +0005b270: 6265 7273 206f 660a 2020 2020 6669 6e69  bers of.    fini
│ │ │ │ +0005b280: 7465 2072 6573 6f6c 7574 696f 6e20 636f  te resolution co
│ │ │ │ +0005b290: 6d70 7574 6564 2066 726f 6d20 6120 6d61  mputed from a ma
│ │ │ │ +0005b2a0: 7472 6978 2066 6163 746f 7269 7a61 7469  trix factorizati
│ │ │ │ +0005b2b0: 6f6e 0a20 202a 202a 6e6f 7465 2068 6967  on.  * *note hig
│ │ │ │ +0005b2c0: 6853 797a 7967 793a 2068 6967 6853 797a  hSyzygy: highSyz
│ │ │ │ +0005b2d0: 7967 792c 202d 2d20 5265 7475 726e 7320  ygy, -- Returns 
│ │ │ │ +0005b2e0: 6120 7379 7a79 6779 206d 6f64 756c 6520  a syzygy module 
│ │ │ │ +0005b2f0: 6f6e 6520 6265 796f 6e64 2074 6865 0a20  one beyond the. 
│ │ │ │ +0005b300: 2020 2072 6567 756c 6172 6974 7920 6f66     regularity of
│ │ │ │ +0005b310: 2045 7874 284d 2c6b 290a 2020 2a20 2a6e   Ext(M,k).  * *n
│ │ │ │ +0005b320: 6f74 6520 624d 6170 733a 2062 4d61 7073  ote bMaps: bMaps
│ │ │ │ +0005b330: 2c20 2d2d 206c 6973 7420 7468 6520 6d61  , -- list the ma
│ │ │ │ +0005b340: 7073 2020 645f 703a 425f 3128 7029 2d2d  ps  d_p:B_1(p)--
│ │ │ │ +0005b350: 3e42 5f30 2870 2920 696e 2061 0a20 2020  >B_0(p) in a.   
│ │ │ │ +0005b360: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0005b370: 7469 6f6e 0a20 202a 202a 6e6f 7465 2042  tion.  * *note B
│ │ │ │ +0005b380: 5261 6e6b 733a 2042 5261 6e6b 732c 202d  Ranks: BRanks, -
│ │ │ │ +0005b390: 2d20 7261 6e6b 7320 6f66 2074 6865 206d  - ranks of the m
│ │ │ │ +0005b3a0: 6f64 756c 6573 2042 5f69 2864 2920 696e  odules B_i(d) in
│ │ │ │ +0005b3b0: 2061 0a20 2020 206d 6174 7269 7846 6163   a.    matrixFac
│ │ │ │ +0005b3c0: 746f 7269 7a61 7469 6f6e 0a0a 5761 7973  torization..Ways
│ │ │ │ +0005b3d0: 2074 6f20 7573 6520 6d61 7472 6978 4661   to use matrixFa
│ │ │ │ +0005b3e0: 6374 6f72 697a 6174 696f 6e3a 0a3d 3d3d  ctorization:.===
│ │ │ │ +0005b3f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005b400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ +0005b410: 202a 2022 6d61 7472 6978 4661 6374 6f72   * "matrixFactor
│ │ │ │ +0005b420: 697a 6174 696f 6e28 4d61 7472 6978 2c4d  ization(Matrix,M
│ │ │ │ +0005b430: 6f64 756c 6529 220a 0a46 6f72 2074 6865  odule)"..For the
│ │ │ │ +0005b440: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +0005b450: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0005b460: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +0005b470: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +0005b480: 7469 6f6e 3a20 6d61 7472 6978 4661 6374  tion: matrixFact
│ │ │ │ +0005b490: 6f72 697a 6174 696f 6e2c 2069 7320 6120  orization, is a 
│ │ │ │ +0005b4a0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ +0005b4b0: 6374 696f 6e20 7769 7468 206f 7074 696f  ction with optio
│ │ │ │ +0005b4c0: 6e73 3a20 284d 6163 6175 6c61 7932 446f  ns: (Macaulay2Do
│ │ │ │ +0005b4d0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +0005b4e0: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d  WithOptions,...-
│ │ │ │ +0005b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005b500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -0005b560: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -0005b570: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ -0005b580: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ -0005b590: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -0005b5a0: 756c 6179 322d 312e 3236 2e30 362b 6473  ulay2-1.26.06+ds
│ │ │ │ -0005b5b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -0005b5c0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465  ckages/.Complete
│ │ │ │ -0005b5d0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ -0005b5e0: 6c75 7469 6f6e 732e 6d32 3a34 3033 333a  lutions.m2:4033:
│ │ │ │ -0005b5f0: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c  0....File: Compl
│ │ │ │ -0005b600: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -0005b610: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ -0005b620: 204e 6f64 653a 206d 6642 6f75 6e64 2c20   Node: mfBound, 
│ │ │ │ -0005b630: 4e65 7874 3a20 6d6f 6475 6c65 4173 4578  Next: moduleAsEx
│ │ │ │ -0005b640: 742c 2050 7265 763a 206d 6174 7269 7846  t, Prev: matrixF
│ │ │ │ -0005b650: 6163 746f 7269 7a61 7469 6f6e 2c20 5570  actorization, Up
│ │ │ │ -0005b660: 3a20 546f 700a 0a6d 6642 6f75 6e64 202d  : Top..mfBound -
│ │ │ │ -0005b670: 2d20 6465 7465 726d 696e 6573 2068 6f77  - determines how
│ │ │ │ -0005b680: 2068 6967 6820 6120 7379 7a79 6779 2074   high a syzygy t
│ │ │ │ -0005b690: 6f20 7461 6b65 2066 6f72 2022 6d61 7472  o take for "matr
│ │ │ │ -0005b6a0: 6978 4661 6374 6f72 697a 6174 696f 6e22  ixFactorization"
│ │ │ │ -0005b6b0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +0005b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0005b540: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +0005b550: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +0005b560: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0005b570: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0005b580: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ +0005b590: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +0005b5a0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ +0005b5b0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0005b5c0: 7574 696f 6e73 2e6d 323a 3430 3333 3a30  utions.m2:4033:0
│ │ │ │ +0005b5d0: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ +0005b5e0: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +0005b5f0: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ +0005b600: 4e6f 6465 3a20 6d66 426f 756e 642c 204e  Node: mfBound, N
│ │ │ │ +0005b610: 6578 743a 206d 6f64 756c 6541 7345 7874  ext: moduleAsExt
│ │ │ │ +0005b620: 2c20 5072 6576 3a20 6d61 7472 6978 4661  , Prev: matrixFa
│ │ │ │ +0005b630: 6374 6f72 697a 6174 696f 6e2c 2055 703a  ctorization, Up:
│ │ │ │ +0005b640: 2054 6f70 0a0a 6d66 426f 756e 6420 2d2d   Top..mfBound --
│ │ │ │ +0005b650: 2064 6574 6572 6d69 6e65 7320 686f 7720   determines how 
│ │ │ │ +0005b660: 6869 6768 2061 2073 797a 7967 7920 746f  high a syzygy to
│ │ │ │ +0005b670: 2074 616b 6520 666f 7220 226d 6174 7269   take for "matri
│ │ │ │ +0005b680: 7846 6163 746f 7269 7a61 7469 6f6e 220a  xFactorization".
│ │ │ │ +0005b690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0005b6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0005b6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005b6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005b6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005b6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005b6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ -0005b700: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ -0005b710: 7020 3d20 6d66 426f 756e 6420 4d0a 2020  p = mfBound M.  
│ │ │ │ -0005b720: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -0005b730: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64  * M, a *note mod
│ │ │ │ -0005b740: 756c 653a 2028 4d61 6361 756c 6179 3244  ule: (Macaulay2D
│ │ │ │ -0005b750: 6f63 294d 6f64 756c 652c 2c20 6f76 6572  oc)Module,, over
│ │ │ │ -0005b760: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ -0005b770: 7273 6563 7469 6f6e 0a20 202a 204f 7574  rsection.  * Out
│ │ │ │ -0005b780: 7075 7473 3a0a 2020 2020 2020 2a20 702c  puts:.      * p,
│ │ │ │ -0005b790: 2061 6e20 2a6e 6f74 6520 696e 7465 6765   an *note intege
│ │ │ │ -0005b7a0: 723a 2028 4d61 6361 756c 6179 3244 6f63  r: (Macaulay2Doc
│ │ │ │ -0005b7b0: 295a 5a2c 2c20 0a0a 4465 7363 7269 7074  )ZZ,, ..Descript
│ │ │ │ -0005b7c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -0005b7d0: 0a49 6620 7020 3d20 6d66 426f 756e 6420  .If p = mfBound 
│ │ │ │ -0005b7e0: 4d2c 2074 6865 6e20 7468 6520 702d 7468  M, then the p-th
│ │ │ │ -0005b7f0: 2073 797a 7967 7920 6f66 204d 2c20 7768   syzygy of M, wh
│ │ │ │ -0005b800: 6963 6820 6973 2063 6f6d 7075 7465 6420  ich is computed 
│ │ │ │ -0005b810: 6279 0a68 6967 6853 797a 7967 7928 4d29  by.highSyzygy(M)
│ │ │ │ -0005b820: 2c20 7368 6f75 6c64 2028 7468 6973 2069  , should (this i
│ │ │ │ -0005b830: 7320 6120 636f 6e6a 6563 7475 7265 2920  s a conjecture) 
│ │ │ │ -0005b840: 6265 2061 2022 6869 6768 2053 797a 7967  be a "high Syzyg
│ │ │ │ -0005b850: 7922 2069 6e20 7468 6520 7365 6e73 650a  y" in the sense.
│ │ │ │ -0005b860: 7265 7175 6972 6564 2066 6f72 206d 6174  required for mat
│ │ │ │ -0005b870: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0005b880: 2e20 496e 2065 7861 6d70 6c65 732c 2074  . In examples, t
│ │ │ │ -0005b890: 6865 2065 7374 696d 6174 6520 7365 656d  he estimate seem
│ │ │ │ -0005b8a0: 7320 7368 6172 7020 2865 7863 6570 740a  s sharp (except.
│ │ │ │ -0005b8b0: 7768 656e 204d 2069 7320 616c 7265 6164  when M is alread
│ │ │ │ -0005b8c0: 7920 6120 6869 6768 2073 797a 7967 7929  y a high syzygy)
│ │ │ │ -0005b8d0: 2e0a 0a54 6865 2061 6374 7561 6c20 666f  ...The actual fo
│ │ │ │ -0005b8e0: 726d 756c 6120 7573 6564 2069 733a 0a0a  rmula used is:..
│ │ │ │ -0005b8f0: 6d66 426f 756e 6420 4d20 3d20 6d61 7828  mfBound M = max(
│ │ │ │ -0005b900: 322a 725f 7b65 7665 6e7d 2c20 312b 322a  2*r_{even}, 1+2*
│ │ │ │ -0005b910: 725f 7b6f 6464 7d29 0a0a 7768 6572 6520  r_{odd})..where 
│ │ │ │ -0005b920: 725f 7b65 7665 6e7d 203d 2072 6567 756c  r_{even} = regul
│ │ │ │ -0005b930: 6172 6974 7920 6576 656e 4578 744d 6f64  arity evenExtMod
│ │ │ │ -0005b940: 756c 6520 4d20 616e 6420 725f 7b6f 6464  ule M and r_{odd
│ │ │ │ -0005b950: 7d20 3d20 7265 6775 6c61 7269 7479 0a6f  } = regularity.o
│ │ │ │ -0005b960: 6464 4578 744d 6f64 756c 6520 4d2e 2048  ddExtModule M. H
│ │ │ │ -0005b970: 6572 6520 6576 656e 4578 744d 6f64 756c  ere evenExtModul
│ │ │ │ -0005b980: 6520 4d20 6973 2074 6865 2065 7665 6e20  e M is the even 
│ │ │ │ -0005b990: 6465 6772 6565 2070 6172 7420 6f66 2045  degree part of E
│ │ │ │ -0005b9a0: 7874 284d 2c20 2872 6573 6964 7565 0a63  xt(M, (residue.c
│ │ │ │ -0005b9b0: 6c61 7373 2066 6965 6c64 2929 2e0a 0a53  lass field))...S
│ │ │ │ -0005b9c0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -0005b9d0: 0a0a 2020 2a20 2a6e 6f74 6520 6869 6768  ..  * *note high
│ │ │ │ -0005b9e0: 5379 7a79 6779 3a20 6869 6768 5379 7a79  Syzygy: highSyzy
│ │ │ │ -0005b9f0: 6779 2c20 2d2d 2052 6574 7572 6e73 2061  gy, -- Returns a
│ │ │ │ -0005ba00: 2073 797a 7967 7920 6d6f 6475 6c65 206f   syzygy module o
│ │ │ │ -0005ba10: 6e65 2062 6579 6f6e 6420 7468 650a 2020  ne beyond the.  
│ │ │ │ -0005ba20: 2020 7265 6775 6c61 7269 7479 206f 6620    regularity of 
│ │ │ │ -0005ba30: 4578 7428 4d2c 6b29 0a20 202a 202a 6e6f  Ext(M,k).  * *no
│ │ │ │ -0005ba40: 7465 2065 7665 6e45 7874 4d6f 6475 6c65  te evenExtModule
│ │ │ │ -0005ba50: 3a20 6576 656e 4578 744d 6f64 756c 652c  : evenExtModule,
│ │ │ │ -0005ba60: 202d 2d20 6576 656e 2070 6172 7420 6f66   -- even part of
│ │ │ │ -0005ba70: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ -0005ba80: 2061 0a20 2020 2063 6f6d 706c 6574 6520   a.    complete 
│ │ │ │ -0005ba90: 696e 7465 7273 6563 7469 6f6e 2061 7320  intersection as 
│ │ │ │ -0005baa0: 6d6f 6475 6c65 206f 7665 7220 4349 206f  module over CI o
│ │ │ │ -0005bab0: 7065 7261 746f 7220 7269 6e67 0a20 202a  perator ring.  *
│ │ │ │ -0005bac0: 202a 6e6f 7465 206f 6464 4578 744d 6f64   *note oddExtMod
│ │ │ │ -0005bad0: 756c 653a 206f 6464 4578 744d 6f64 756c  ule: oddExtModul
│ │ │ │ -0005bae0: 652c 202d 2d20 6f64 6420 7061 7274 206f  e, -- odd part o
│ │ │ │ -0005baf0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ -0005bb00: 7220 6120 636f 6d70 6c65 7465 0a20 2020  r a complete.   
│ │ │ │ -0005bb10: 2069 6e74 6572 7365 6374 696f 6e20 6173   intersection as
│ │ │ │ -0005bb20: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ -0005bb30: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ -0005bb40: 2a20 2a6e 6f74 6520 6d61 7472 6978 4661  * *note matrixFa
│ │ │ │ -0005bb50: 6374 6f72 697a 6174 696f 6e3a 206d 6174  ctorization: mat
│ │ │ │ -0005bb60: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -0005bb70: 2c20 2d2d 204d 6170 7320 696e 2061 2068  , -- Maps in a h
│ │ │ │ -0005bb80: 6967 6865 720a 2020 2020 636f 6469 6d65  igher.    codime
│ │ │ │ -0005bb90: 6e73 696f 6e20 6d61 7472 6978 2066 6163  nsion matrix fac
│ │ │ │ -0005bba0: 746f 7269 7a61 7469 6f6e 0a0a 5761 7973  torization..Ways
│ │ │ │ -0005bbb0: 2074 6f20 7573 6520 6d66 426f 756e 643a   to use mfBound:
│ │ │ │ -0005bbc0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0005bbd0: 3d3d 3d3d 3d0a 0a20 202a 2022 6d66 426f  =====..  * "mfBo
│ │ │ │ -0005bbe0: 756e 6428 4d6f 6475 6c65 2922 0a0a 466f  und(Module)"..Fo
│ │ │ │ -0005bbf0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -0005bc00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0005bc10: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -0005bc20: 2a6e 6f74 6520 6d66 426f 756e 643a 206d  *note mfBound: m
│ │ │ │ -0005bc30: 6642 6f75 6e64 2c20 6973 2061 202a 6e6f  fBound, is a *no
│ │ │ │ -0005bc40: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ -0005bc50: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ -0005bc60: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ -0005bc70: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +0005b6d0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ +0005b6e0: 7361 6765 3a20 0a20 2020 2020 2020 2070  sage: .        p
│ │ │ │ +0005b6f0: 203d 206d 6642 6f75 6e64 204d 0a20 202a   = mfBound M.  *
│ │ │ │ +0005b700: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +0005b710: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ +0005b720: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ +0005b730: 6329 4d6f 6475 6c65 2c2c 206f 7665 7220  c)Module,, over 
│ │ │ │ +0005b740: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ +0005b750: 7365 6374 696f 6e0a 2020 2a20 4f75 7470  section.  * Outp
│ │ │ │ +0005b760: 7574 733a 0a20 2020 2020 202a 2070 2c20  uts:.      * p, 
│ │ │ │ +0005b770: 616e 202a 6e6f 7465 2069 6e74 6567 6572  an *note integer
│ │ │ │ +0005b780: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0005b790: 5a5a 2c2c 200a 0a44 6573 6372 6970 7469  ZZ,, ..Descripti
│ │ │ │ +0005b7a0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ +0005b7b0: 4966 2070 203d 206d 6642 6f75 6e64 204d  If p = mfBound M
│ │ │ │ +0005b7c0: 2c20 7468 656e 2074 6865 2070 2d74 6820  , then the p-th 
│ │ │ │ +0005b7d0: 7379 7a79 6779 206f 6620 4d2c 2077 6869  syzygy of M, whi
│ │ │ │ +0005b7e0: 6368 2069 7320 636f 6d70 7574 6564 2062  ch is computed b
│ │ │ │ +0005b7f0: 790a 6869 6768 5379 7a79 6779 284d 292c  y.highSyzygy(M),
│ │ │ │ +0005b800: 2073 686f 756c 6420 2874 6869 7320 6973   should (this is
│ │ │ │ +0005b810: 2061 2063 6f6e 6a65 6374 7572 6529 2062   a conjecture) b
│ │ │ │ +0005b820: 6520 6120 2268 6967 6820 5379 7a79 6779  e a "high Syzygy
│ │ │ │ +0005b830: 2220 696e 2074 6865 2073 656e 7365 0a72  " in the sense.r
│ │ │ │ +0005b840: 6571 7569 7265 6420 666f 7220 6d61 7472  equired for matr
│ │ │ │ +0005b850: 6978 4661 6374 6f72 697a 6174 696f 6e2e  ixFactorization.
│ │ │ │ +0005b860: 2049 6e20 6578 616d 706c 6573 2c20 7468   In examples, th
│ │ │ │ +0005b870: 6520 6573 7469 6d61 7465 2073 6565 6d73  e estimate seems
│ │ │ │ +0005b880: 2073 6861 7270 2028 6578 6365 7074 0a77   sharp (except.w
│ │ │ │ +0005b890: 6865 6e20 4d20 6973 2061 6c72 6561 6479  hen M is already
│ │ │ │ +0005b8a0: 2061 2068 6967 6820 7379 7a79 6779 292e   a high syzygy).
│ │ │ │ +0005b8b0: 0a0a 5468 6520 6163 7475 616c 2066 6f72  ..The actual for
│ │ │ │ +0005b8c0: 6d75 6c61 2075 7365 6420 6973 3a0a 0a6d  mula used is:..m
│ │ │ │ +0005b8d0: 6642 6f75 6e64 204d 203d 206d 6178 2832  fBound M = max(2
│ │ │ │ +0005b8e0: 2a72 5f7b 6576 656e 7d2c 2031 2b32 2a72  *r_{even}, 1+2*r
│ │ │ │ +0005b8f0: 5f7b 6f64 647d 290a 0a77 6865 7265 2072  _{odd})..where r
│ │ │ │ +0005b900: 5f7b 6576 656e 7d20 3d20 7265 6775 6c61  _{even} = regula
│ │ │ │ +0005b910: 7269 7479 2065 7665 6e45 7874 4d6f 6475  rity evenExtModu
│ │ │ │ +0005b920: 6c65 204d 2061 6e64 2072 5f7b 6f64 647d  le M and r_{odd}
│ │ │ │ +0005b930: 203d 2072 6567 756c 6172 6974 790a 6f64   = regularity.od
│ │ │ │ +0005b940: 6445 7874 4d6f 6475 6c65 204d 2e20 4865  dExtModule M. He
│ │ │ │ +0005b950: 7265 2065 7665 6e45 7874 4d6f 6475 6c65  re evenExtModule
│ │ │ │ +0005b960: 204d 2069 7320 7468 6520 6576 656e 2064   M is the even d
│ │ │ │ +0005b970: 6567 7265 6520 7061 7274 206f 6620 4578  egree part of Ex
│ │ │ │ +0005b980: 7428 4d2c 2028 7265 7369 6475 650a 636c  t(M, (residue.cl
│ │ │ │ +0005b990: 6173 7320 6669 656c 6429 292e 0a0a 5365  ass field))...Se
│ │ │ │ +0005b9a0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +0005b9b0: 0a20 202a 202a 6e6f 7465 2068 6967 6853  .  * *note highS
│ │ │ │ +0005b9c0: 797a 7967 793a 2068 6967 6853 797a 7967  yzygy: highSyzyg
│ │ │ │ +0005b9d0: 792c 202d 2d20 5265 7475 726e 7320 6120  y, -- Returns a 
│ │ │ │ +0005b9e0: 7379 7a79 6779 206d 6f64 756c 6520 6f6e  syzygy module on
│ │ │ │ +0005b9f0: 6520 6265 796f 6e64 2074 6865 0a20 2020  e beyond the.   
│ │ │ │ +0005ba00: 2072 6567 756c 6172 6974 7920 6f66 2045   regularity of E
│ │ │ │ +0005ba10: 7874 284d 2c6b 290a 2020 2a20 2a6e 6f74  xt(M,k).  * *not
│ │ │ │ +0005ba20: 6520 6576 656e 4578 744d 6f64 756c 653a  e evenExtModule:
│ │ │ │ +0005ba30: 2065 7665 6e45 7874 4d6f 6475 6c65 2c20   evenExtModule, 
│ │ │ │ +0005ba40: 2d2d 2065 7665 6e20 7061 7274 206f 6620  -- even part of 
│ │ │ │ +0005ba50: 4578 745e 2a28 4d2c 6b29 206f 7665 7220  Ext^*(M,k) over 
│ │ │ │ +0005ba60: 610a 2020 2020 636f 6d70 6c65 7465 2069  a.    complete i
│ │ │ │ +0005ba70: 6e74 6572 7365 6374 696f 6e20 6173 206d  ntersection as m
│ │ │ │ +0005ba80: 6f64 756c 6520 6f76 6572 2043 4920 6f70  odule over CI op
│ │ │ │ +0005ba90: 6572 6174 6f72 2072 696e 670a 2020 2a20  erator ring.  * 
│ │ │ │ +0005baa0: 2a6e 6f74 6520 6f64 6445 7874 4d6f 6475  *note oddExtModu
│ │ │ │ +0005bab0: 6c65 3a20 6f64 6445 7874 4d6f 6475 6c65  le: oddExtModule
│ │ │ │ +0005bac0: 2c20 2d2d 206f 6464 2070 6172 7420 6f66  , -- odd part of
│ │ │ │ +0005bad0: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ +0005bae0: 2061 2063 6f6d 706c 6574 650a 2020 2020   a complete.    
│ │ │ │ +0005baf0: 696e 7465 7273 6563 7469 6f6e 2061 7320  intersection as 
│ │ │ │ +0005bb00: 6d6f 6475 6c65 206f 7665 7220 4349 206f  module over CI o
│ │ │ │ +0005bb10: 7065 7261 746f 7220 7269 6e67 0a20 202a  perator ring.  *
│ │ │ │ +0005bb20: 202a 6e6f 7465 206d 6174 7269 7846 6163   *note matrixFac
│ │ │ │ +0005bb30: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472  torization: matr
│ │ │ │ +0005bb40: 6978 4661 6374 6f72 697a 6174 696f 6e2c  ixFactorization,
│ │ │ │ +0005bb50: 202d 2d20 4d61 7073 2069 6e20 6120 6869   -- Maps in a hi
│ │ │ │ +0005bb60: 6768 6572 0a20 2020 2063 6f64 696d 656e  gher.    codimen
│ │ │ │ +0005bb70: 7369 6f6e 206d 6174 7269 7820 6661 6374  sion matrix fact
│ │ │ │ +0005bb80: 6f72 697a 6174 696f 6e0a 0a57 6179 7320  orization..Ways 
│ │ │ │ +0005bb90: 746f 2075 7365 206d 6642 6f75 6e64 3a0a  to use mfBound:.
│ │ │ │ +0005bba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005bbb0: 3d3d 3d3d 0a0a 2020 2a20 226d 6642 6f75  ====..  * "mfBou
│ │ │ │ +0005bbc0: 6e64 284d 6f64 756c 6529 220a 0a46 6f72  nd(Module)"..For
│ │ │ │ +0005bbd0: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +0005bbe0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005bbf0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +0005bc00: 6e6f 7465 206d 6642 6f75 6e64 3a20 6d66  note mfBound: mf
│ │ │ │ +0005bc10: 426f 756e 642c 2069 7320 6120 2a6e 6f74  Bound, is a *not
│ │ │ │ +0005bc20: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ +0005bc30: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ +0005bc40: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ +0005bc50: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0005bc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005bcc0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ -0005bcd0: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ -0005bce0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ -0005bcf0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ -0005bd00: 6d61 6361 756c 6179 322d 312e 3236 2e30  macaulay2-1.26.0
│ │ │ │ -0005bd10: 362b 6473 2f4d 322f 4d61 6361 756c 6179  6+ds/M2/Macaulay
│ │ │ │ -0005bd20: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ -0005bd30: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -0005bd40: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33  Resolutions.m2:3
│ │ │ │ -0005bd50: 3334 393a 302e 0a1f 0a46 696c 653a 2043  349:0....File: C
│ │ │ │ -0005bd60: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ -0005bd70: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ -0005bd80: 6e66 6f2c 204e 6f64 653a 206d 6f64 756c  nfo, Node: modul
│ │ │ │ -0005bd90: 6541 7345 7874 2c20 4e65 7874 3a20 6e65  eAsExt, Next: ne
│ │ │ │ -0005bda0: 7745 7874 2c20 5072 6576 3a20 6d66 426f  wExt, Prev: mfBo
│ │ │ │ -0005bdb0: 756e 642c 2055 703a 2054 6f70 0a0a 6d6f  und, Up: Top..mo
│ │ │ │ -0005bdc0: 6475 6c65 4173 4578 7420 2d2d 2046 696e  duleAsExt -- Fin
│ │ │ │ -0005bdd0: 6420 6120 6d6f 6475 6c65 2077 6974 6820  d a module with 
│ │ │ │ -0005bde0: 6769 7665 6e20 6173 796d 7074 6f74 6963  given asymptotic
│ │ │ │ -0005bdf0: 2072 6573 6f6c 7574 696f 6e0a 2a2a 2a2a   resolution.****
│ │ │ │ +0005bca0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ +0005bcb0: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ +0005bcc0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ +0005bcd0: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ +0005bce0: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ +0005bcf0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ +0005bd00: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ +0005bd10: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +0005bd20: 6573 6f6c 7574 696f 6e73 2e6d 323a 3333  esolutions.m2:33
│ │ │ │ +0005bd30: 3439 3a30 2e0a 1f0a 4669 6c65 3a20 436f  49:0....File: Co
│ │ │ │ +0005bd40: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ +0005bd50: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ +0005bd60: 666f 2c20 4e6f 6465 3a20 6d6f 6475 6c65  fo, Node: module
│ │ │ │ +0005bd70: 4173 4578 742c 204e 6578 743a 206e 6577  AsExt, Next: new
│ │ │ │ +0005bd80: 4578 742c 2050 7265 763a 206d 6642 6f75  Ext, Prev: mfBou
│ │ │ │ +0005bd90: 6e64 2c20 5570 3a20 546f 700a 0a6d 6f64  nd, Up: Top..mod
│ │ │ │ +0005bda0: 756c 6541 7345 7874 202d 2d20 4669 6e64  uleAsExt -- Find
│ │ │ │ +0005bdb0: 2061 206d 6f64 756c 6520 7769 7468 2067   a module with g
│ │ │ │ +0005bdc0: 6976 656e 2061 7379 6d70 746f 7469 6320  iven asymptotic 
│ │ │ │ +0005bdd0: 7265 736f 6c75 7469 6f6e 0a2a 2a2a 2a2a  resolution.*****
│ │ │ │ +0005bde0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0005bdf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005be00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005be10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005be20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005be30: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -0005be40: 7361 6765 3a20 0a20 2020 2020 2020 204d  sage: .        M
│ │ │ │ -0005be50: 203d 206d 6f64 756c 6541 7345 7874 284d   = moduleAsExt(M
│ │ │ │ -0005be60: 4d2c 5229 0a20 202a 2049 6e70 7574 733a  M,R).  * Inputs:
│ │ │ │ -0005be70: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
│ │ │ │ -0005be80: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ -0005be90: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ -0005bea0: 2c2c 206d 6f64 756c 6520 6f76 6572 2070  ,, module over p
│ │ │ │ -0005beb0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20  olynomial ring. 
│ │ │ │ -0005bec0: 2020 2020 2020 2077 6974 6820 6320 7661         with c va
│ │ │ │ -0005bed0: 7269 6162 6c65 730a 2020 2020 2020 2a20  riables.      * 
│ │ │ │ -0005bee0: 522c 2061 202a 6e6f 7465 2072 696e 673a  R, a *note ring:
│ │ │ │ -0005bef0: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ -0005bf00: 696e 672c 2c20 2867 7261 6465 6429 2063  ing,, (graded) c
│ │ │ │ -0005bf10: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ -0005bf20: 7469 6f6e 0a20 2020 2020 2020 2072 696e  tion.        rin
│ │ │ │ -0005bf30: 6720 6f66 2063 6f64 696d 656e 7369 6f6e  g of codimension
│ │ │ │ -0005bf40: 2063 2c20 656d 6265 6464 696e 6720 6469   c, embedding di
│ │ │ │ -0005bf50: 6d65 6e73 696f 6e20 6e0a 2020 2a20 4f75  mension n.  * Ou
│ │ │ │ -0005bf60: 7470 7574 733a 0a20 2020 2020 202a 204e  tputs:.      * N
│ │ │ │ -0005bf70: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ -0005bf80: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -0005bf90: 4d6f 6475 6c65 2c2c 206d 6f64 756c 6520  Module,, module 
│ │ │ │ -0005bfa0: 6f76 6572 2052 2073 7563 6820 7468 6174  over R such that
│ │ │ │ -0005bfb0: 0a20 2020 2020 2020 2045 7874 5f52 284e  .        Ext_R(N
│ │ │ │ -0005bfc0: 2c6b 2920 3d20 4d5c 6f74 696d 6573 205c  ,k) = M\otimes \
│ │ │ │ -0005bfd0: 7765 6467 6528 6b5e 6e29 2069 6e20 6c61  wedge(k^n) in la
│ │ │ │ -0005bfe0: 7267 6520 686f 6d6f 6c6f 6769 6361 6c20  rge homological 
│ │ │ │ -0005bff0: 6465 6772 6565 2e0a 0a44 6573 6372 6970  degree...Descrip
│ │ │ │ -0005c000: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -0005c010: 0a0a 5468 6520 726f 7574 696e 6520 6060  ..The routine ``
│ │ │ │ -0005c020: 6d6f 6475 6c65 4173 4578 7427 2720 6973  moduleAsExt'' is
│ │ │ │ -0005c030: 2061 2070 6172 7469 616c 2069 6e76 6572   a partial inver
│ │ │ │ -0005c040: 7365 2074 6f20 7468 6520 726f 7574 696e  se to the routin
│ │ │ │ -0005c050: 6520 4578 744d 6f64 756c 652c 0a63 6f6d  e ExtModule,.com
│ │ │ │ -0005c060: 7075 7465 6420 666f 6c6c 6f77 696e 6720  puted following 
│ │ │ │ -0005c070: 6964 6561 7320 6f66 2041 7672 616d 6f76  ideas of Avramov
│ │ │ │ -0005c080: 2061 6e64 204a 6f72 6765 6e73 656e 3a20   and Jorgensen: 
│ │ │ │ -0005c090: 6769 7665 6e20 6120 6d6f 6475 6c65 2045  given a module E
│ │ │ │ -0005c0a0: 206f 7665 7220 610a 706f 6c79 6e6f 6d69   over a.polynomi
│ │ │ │ -0005c0b0: 616c 2072 696e 6720 6b5b 785f 312e 2e78  al ring k[x_1..x
│ │ │ │ -0005c0c0: 5f63 5d2c 2069 7420 7072 6f76 6964 6573  _c], it provides
│ │ │ │ -0005c0d0: 2061 206d 6f64 756c 6520 4e20 6f76 6572   a module N over
│ │ │ │ -0005c0e0: 2061 2073 7065 6369 6669 6564 2070 6f6c   a specified pol
│ │ │ │ -0005c0f0: 796e 6f6d 6961 6c0a 7269 6e67 2069 6e20  ynomial.ring in 
│ │ │ │ -0005c100: 6e20 7661 7269 6162 6c65 7320 7375 6368  n variables such
│ │ │ │ -0005c110: 2074 6861 7420 4578 7428 4e2c 6b29 2061   that Ext(N,k) a
│ │ │ │ -0005c120: 6772 6565 7320 7769 7468 2024 4527 3d45  grees with $E'=E
│ │ │ │ -0005c130: 5c6f 7469 6d65 7320 5c77 6564 6765 286b  \otimes \wedge(k
│ │ │ │ -0005c140: 5e6e 2924 0a61 6674 6572 2074 7275 6e63  ^n)$.after trunc
│ │ │ │ -0005c150: 6174 696f 6e2e 2048 6572 6520 7468 6520  ation. Here the 
│ │ │ │ -0005c160: 6772 6164 696e 6720 6f6e 2045 2069 7320  grading on E is 
│ │ │ │ -0005c170: 7461 6b65 6e20 746f 2062 6520 6576 656e  taken to be even
│ │ │ │ -0005c180: 2c20 7768 696c 650a 245c 7765 6467 6528  , while.$\wedge(
│ │ │ │ -0005c190: 6b5e 6e29 2420 6861 7320 6765 6e65 7261  k^n)$ has genera
│ │ │ │ -0005c1a0: 746f 7273 2069 6e20 6465 6772 6565 2031  tors in degree 1
│ │ │ │ -0005c1b0: 2e20 5468 6520 726f 7574 696e 6520 6866  . The routine hf
│ │ │ │ -0005c1c0: 4d6f 6475 6c65 4173 4578 7420 636f 6d70  ModuleAsExt comp
│ │ │ │ -0005c1d0: 7574 6573 0a74 6865 2072 6573 756c 7469  utes.the resulti
│ │ │ │ -0005c1e0: 6e67 2068 696c 6265 7274 2066 756e 6374  ng hilbert funct
│ │ │ │ -0005c1f0: 696f 6e20 666f 7220 4527 2e20 5468 6973  ion for E'. This
│ │ │ │ -0005c200: 2075 7365 7320 6964 6561 7320 6f66 2041   uses ideas of A
│ │ │ │ -0005c210: 7672 616d 6f76 2061 6e64 0a4a 6f72 6765  vramov and.Jorge
│ │ │ │ -0005c220: 6e73 656e 2e20 4e6f 7465 2074 6861 7420  nsen. Note that 
│ │ │ │ -0005c230: 7468 6520 6d6f 6475 6c65 2045 7874 284e  the module Ext(N
│ │ │ │ -0005c240: 2c6b 2920 2874 7275 6e63 6174 6564 2920  ,k) (truncated) 
│ │ │ │ -0005c250: 7769 6c6c 2061 7574 6f6d 6174 6963 616c  will automatical
│ │ │ │ -0005c260: 6c79 2062 6520 6672 6565 0a6f 7665 7220  ly be free.over 
│ │ │ │ -0005c270: 7468 6520 6578 7465 7269 6f72 2061 6c67  the exterior alg
│ │ │ │ -0005c280: 6562 7261 2024 5c77 6564 6765 286b 5e6e  ebra $\wedge(k^n
│ │ │ │ -0005c290: 2924 2067 656e 6572 6174 6564 2062 7920  )$ generated by 
│ │ │ │ -0005c2a0: 4578 745e 3128 6b2c 6b29 3b20 6e6f 7420  Ext^1(k,k); not 
│ │ │ │ -0005c2b0: 6120 7479 7069 6361 6c0a 4578 7420 6d6f  a typical.Ext mo
│ │ │ │ -0005c2c0: 6475 6c65 2e0a 0a4d 6f72 6520 7072 6563  dule...More prec
│ │ │ │ -0005c2d0: 6973 656c 793a 0a0a 5375 7070 6f73 6520  isely:..Suppose 
│ │ │ │ -0005c2e0: 7468 6174 2024 5220 3d20 6b5b 615f 312c  that $R = k[a_1,
│ │ │ │ -0005c2f0: 5c64 6f74 732c 2061 5f6e 5d2f 2866 5f31  \dots, a_n]/(f_1
│ │ │ │ -0005c300: 2c5c 646f 7473 2c66 5f63 2924 206c 6574  ,\dots,f_c)$ let
│ │ │ │ -0005c310: 2024 4b4b 203d 0a6b 5b78 5f31 2c5c 646f   $KK =.k[x_1,\do
│ │ │ │ -0005c320: 7473 2c78 5f63 5d24 2c20 616e 6420 6c65  ts,x_c]$, and le
│ │ │ │ -0005c330: 7420 245c 4c61 6d62 6461 203d 205c 7765  t $\Lambda = \we
│ │ │ │ -0005c340: 6467 6520 6b5e 6e24 2e20 2445 203d 204b  dge k^n$. $E = K
│ │ │ │ -0005c350: 4b5c 6f74 696d 6573 5c4c 616d 6264 6124  K\otimes\Lambda$
│ │ │ │ -0005c360: 2c20 736f 0a74 6861 7420 7468 6520 6d69  , so.that the mi
│ │ │ │ -0005c370: 6e69 6d61 6c20 2452 242d 6672 6565 2072  nimal $R$-free r
│ │ │ │ -0005c380: 6573 6f6c 7574 696f 6e20 6f66 2024 6b24  esolution of $k$
│ │ │ │ -0005c390: 2068 6173 2075 6e64 6572 6c79 696e 6720   has underlying 
│ │ │ │ -0005c3a0: 6d6f 6475 6c65 2024 525c 6f74 696d 6573  module $R\otimes
│ │ │ │ -0005c3b0: 0a45 5e2a 242c 2077 6865 7265 2024 455e  .E^*$, where $E^
│ │ │ │ -0005c3c0: 2a24 2069 7320 7468 6520 6772 6164 6564  *$ is the graded
│ │ │ │ -0005c3d0: 2076 6563 746f 7220 7370 6163 6520 6475   vector space du
│ │ │ │ -0005c3e0: 616c 206f 6620 2445 242e 0a0a 4c65 7420  al of $E$...Let 
│ │ │ │ -0005c3f0: 4d4d 2062 6520 7468 6520 7265 7375 6c74  MM be the result
│ │ │ │ -0005c400: 206f 6620 7472 756e 6361 7469 6e67 204d   of truncating M
│ │ │ │ -0005c410: 2061 7420 6974 7320 7265 6775 6c61 7269   at its regulari
│ │ │ │ -0005c420: 7479 2061 6e64 2073 6869 6674 696e 6720  ty and shifting 
│ │ │ │ -0005c430: 6974 2073 6f20 7468 6174 0a69 7420 6973  it so that.it is
│ │ │ │ -0005c440: 2067 656e 6572 6174 6564 2069 6e20 6465   generated in de
│ │ │ │ -0005c450: 6772 6565 2030 2e20 4c65 7420 2446 2420  gree 0. Let $F$ 
│ │ │ │ -0005c460: 6265 2061 2024 4b4b 242d 6672 6565 2072  be a $KK$-free r
│ │ │ │ -0005c470: 6573 6f6c 7574 696f 6e20 6f66 2024 4d4d  esolution of $MM
│ │ │ │ -0005c480: 242c 2061 6e64 0a77 7269 7465 2024 465f  $, and.write $F_
│ │ │ │ -0005c490: 6920 3d20 4b4b 5c6f 7469 6d65 7320 565f  i = KK\otimes V_
│ │ │ │ -0005c4a0: 692e 2420 5369 6e63 6520 6c69 6e65 6172  i.$ Since linear
│ │ │ │ -0005c4b0: 2066 6f72 6d73 206f 7665 7220 244b 4b24   forms over $KK$
│ │ │ │ -0005c4c0: 2063 6f72 7265 7370 6f6e 6420 746f 2043   correspond to C
│ │ │ │ -0005c4d0: 490a 6f70 6572 6174 6f72 7320 6f66 2064  I.operators of d
│ │ │ │ -0005c4e0: 6567 7265 6520 2d32 206f 6e20 7468 6520  egree -2 on the 
│ │ │ │ -0005c4f0: 7265 736f 6c75 7469 6f6e 2047 206f 6620  resolution G of 
│ │ │ │ -0005c500: 6b20 6f76 6572 2052 2c20 7765 206d 6179  k over R, we may
│ │ │ │ -0005c510: 2066 6f72 6d20 6120 6d61 7020 2424 0a64   form a map $$.d
│ │ │ │ -0005c520: 5f31 2b64 5f32 3a20 5c73 756d 5f7b 693d  _1+d_2: \sum_{i=
│ │ │ │ -0005c530: 307d 5e6d 2047 5f7b 692b 317d 5c6f 7469  0}^m G_{i+1}\oti
│ │ │ │ -0005c540: 6d65 7320 565f 7b6d 2d69 7d5e 2a20 5c74  mes V_{m-i}^* \t
│ │ │ │ -0005c550: 6f20 5c73 756d 5f7b 693d 307d 5e6d 2047  o \sum_{i=0}^m G
│ │ │ │ -0005c560: 5f69 5c6f 7469 6d65 730a 565f 7b6d 2d69  _i\otimes.V_{m-i
│ │ │ │ -0005c570: 7d5e 2a20 2424 2077 6865 7265 2024 645f  }^* $$ where $d_
│ │ │ │ -0005c580: 3124 2069 7320 7468 6520 6469 7265 6374  1$ is the direct
│ │ │ │ -0005c590: 2073 756d 206f 6620 7468 6520 6469 6666   sum of the diff
│ │ │ │ -0005c5a0: 6572 656e 7469 616c 7320 2428 475f 7b69  erentials $(G_{i
│ │ │ │ -0005c5b0: 2b31 7d5c 746f 0a47 5f69 295c 6f74 696d  +1}\to.G_i)\otim
│ │ │ │ -0005c5c0: 6573 2056 5f69 5e2a 2420 616e 6420 2464  es V_i^*$ and $d
│ │ │ │ -0005c5d0: 5f32 2420 6973 2074 6865 2064 6972 6563  _2$ is the direc
│ │ │ │ -0005c5e0: 7420 7375 6d20 6f66 2074 6865 206d 6170  t sum of the map
│ │ │ │ -0005c5f0: 7320 245c 7068 695f 6924 2064 6566 696e  s $\phi_i$ defin
│ │ │ │ -0005c600: 6564 0a66 726f 6d20 7468 6520 6469 6666  ed.from the diff
│ │ │ │ -0005c610: 6572 656e 7469 616c 7320 6f66 2024 4624  erentials of $F$
│ │ │ │ -0005c620: 2062 7920 7375 6273 7469 7475 7469 6e67   by substituting
│ │ │ │ -0005c630: 2043 4920 6f70 6572 6174 6f72 7320 666f   CI operators fo
│ │ │ │ -0005c640: 7220 6c69 6e65 6172 2066 6f72 6d73 2c0a  r linear forms,.
│ │ │ │ -0005c650: 245c 7068 695f 693a 2047 5f7b 692b 317d  $\phi_i: G_{i+1}
│ │ │ │ -0005c660: 5c6f 7469 6d65 7320 565f 6920 5c74 6f20  \otimes V_i \to 
│ │ │ │ -0005c670: 475f 7b69 2d31 7d5c 6f74 696d 6573 2056  G_{i-1}\otimes V
│ │ │ │ -0005c680: 5f7b 692d 317d 242e 2054 6865 2073 6372  _{i-1}$. The scr
│ │ │ │ -0005c690: 6970 7420 7265 7475 726e 7320 7468 650a  ipt returns the.
│ │ │ │ -0005c6a0: 6d6f 6475 6c65 204e 2074 6861 7420 6973  module N that is
│ │ │ │ -0005c6b0: 2074 6865 2063 6f6b 6572 6e65 6c20 6f66   the cokernel of
│ │ │ │ -0005c6c0: 2024 645f 312b 645f 3224 2e0a 0a54 6865   $d_1+d_2$...The
│ │ │ │ -0005c6d0: 206d 6f64 756c 6520 2445 7874 5f52 284e   module $Ext_R(N
│ │ │ │ -0005c6e0: 2c6b 2924 2061 6772 6565 732c 2061 6674  ,k)$ agrees, aft
│ │ │ │ -0005c6f0: 6572 2061 2066 6577 2073 7465 7073 2c20  er a few steps, 
│ │ │ │ -0005c700: 7769 7468 2074 6865 206d 6f64 756c 6520  with the module 
│ │ │ │ -0005c710: 6465 7269 7665 6420 6672 6f6d 0a24 4d4d  derived from.$MM
│ │ │ │ -0005c720: 2420 6279 2074 656e 736f 7269 6e67 2069  $ by tensoring i
│ │ │ │ -0005c730: 7420 7769 7468 2024 5c4c 616d 6264 6124  t with $\Lambda$
│ │ │ │ -0005c740: 2c20 7468 6174 2069 732c 2077 6974 6820  , that is, with 
│ │ │ │ -0005c750: 7468 6520 6d6f 6475 6c65 c39f 2024 2420  the module.. $$ 
│ │ │ │ -0005c760: 4d4d 2720 3d20 5c73 756d 5f6a 0a28 4d4d  MM' = \sum_j.(MM
│ │ │ │ -0005c770: 2728 6a29 5c6f 7469 6d65 7320 5c4c 616d  '(j)\otimes \Lam
│ │ │ │ -0005c780: 6264 615f 6a29 2024 2420 736f 2074 6861  bda_j) $$ so tha
│ │ │ │ -0005c790: 7420 244d 4d27 5f70 203d 2028 4d4d 5f70  t $MM'_p = (MM_p
│ │ │ │ -0005c7a0: 5c6f 7469 6d65 7320 4c61 6d62 6461 5f30  \otimes Lambda_0
│ │ │ │ -0005c7b0: 2920 5c6f 706c 7573 0a28 4d4d 5f7b 702d  ) \oplus.(MM_{p-
│ │ │ │ -0005c7c0: 317d 5c6f 7469 6d65 7320 4c61 6d62 6461  1}\otimes Lambda
│ │ │ │ -0005c7d0: 5f31 2920 5c6f 706c 7573 5c63 646f 7473  _1) \oplus\cdots
│ │ │ │ -0005c7e0: 242e 0a0a 5468 6520 6675 6e63 7469 6f6e  $...The function
│ │ │ │ -0005c7f0: 2068 664d 6f64 756c 6541 7345 7874 2063   hfModuleAsExt c
│ │ │ │ -0005c800: 6f6d 7075 7465 7320 7468 6520 4869 6c62  omputes the Hilb
│ │ │ │ -0005c810: 6572 7420 6675 6e63 7469 6f6e 206f 6620  ert function of 
│ │ │ │ -0005c820: 4d4d 2720 6e75 6d65 7269 6361 6c6c 790a  MM' numerically.
│ │ │ │ -0005c830: 6672 6f6d 2074 6861 7420 6f66 204d 4d2e  from that of MM.
│ │ │ │ -0005c840: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..+-------------
│ │ │ │ -0005c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c870: 2d2d 2d2b 0a7c 6931 203a 206b 6b20 3d20  ---+.|i1 : kk = 
│ │ │ │ -0005c880: 5a5a 2f31 3031 3b20 2020 2020 2020 2020  ZZ/101;         
│ │ │ │ -0005c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c8a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -0005c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c8d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -0005c8e0: 2053 203d 206b 6b5b 612c 622c 635d 3b20   S = kk[a,b,c]; 
│ │ │ │ -0005c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ -0005c910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0005c940: 0a7c 6933 203a 2066 6620 3d20 6d61 7472  .|i3 : ff = matr
│ │ │ │ -0005c950: 6978 7b7b 615e 342c 2062 5e34 2c63 5e34  ix{{a^4, b^4,c^4
│ │ │ │ -0005c960: 7d7d 3b20 2020 2020 2020 2020 2020 2020  }};             
│ │ │ │ -0005c970: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0005c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c9a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005c9b0: 2020 2020 2031 2020 2020 2020 3320 2020       1      3   
│ │ │ │ -0005c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005c9d0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ -0005c9e0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ -0005c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ca00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0005ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0005ca40: 7c69 3420 3a20 5220 3d20 532f 6964 6561  |i4 : R = S/idea
│ │ │ │ -0005ca50: 6c20 6666 3b20 2020 2020 2020 2020 2020  l ff;           
│ │ │ │ -0005ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ca70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0005ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005caa0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4f70 7320  ----+.|i5 : Ops 
│ │ │ │ -0005cab0: 3d20 6b6b 5b78 5f31 2c78 5f32 2c78 5f33  = kk[x_1,x_2,x_3
│ │ │ │ -0005cac0: 5d3b 2020 2020 2020 2020 2020 2020 2020  ];              
│ │ │ │ -0005cad0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0005cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ -0005cb10: 3a20 4d4d 203d 204f 7073 5e31 2f28 785f  : MM = Ops^1/(x_
│ │ │ │ -0005cb20: 312a 6964 6561 6c28 785f 325e 322c 785f  1*ideal(x_2^2,x_
│ │ │ │ -0005cb30: 3329 293b 2020 2020 2020 2020 207c 0a2b  3));         |.+
│ │ │ │ -0005cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cb70: 2b0a 7c69 3720 3a20 4e20 3d20 6d6f 6475  +.|i7 : N = modu
│ │ │ │ -0005cb80: 6c65 4173 4578 7428 4d4d 2c52 293b 2020  leAsExt(MM,R);  
│ │ │ │ -0005cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cba0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0005cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cbd0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 6265  ------+.|i8 : be
│ │ │ │ -0005cbe0: 7474 6920 6672 6565 5265 736f 6c75 7469  tti freeResoluti
│ │ │ │ -0005cbf0: 6f6e 2820 4e2c 204c 656e 6774 684c 696d  on( N, LengthLim
│ │ │ │ -0005cc00: 6974 203d 3e20 3130 297c 0a7c 2020 2020  it => 10)|.|    
│ │ │ │ -0005cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0005cc40: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ -0005cc50: 2020 3220 2033 2020 3420 2035 2020 3620    2  3  4  5  6 
│ │ │ │ -0005cc60: 2037 2020 3820 2039 2031 3020 2020 207c   7  8  9 10    |
│ │ │ │ -0005cc70: 0a7c 6f38 203d 2074 6f74 616c 3a20 3336  .|o8 = total: 36
│ │ │ │ -0005cc80: 2032 3720 3239 2033 3120 3333 2033 3520   27 29 31 33 35 
│ │ │ │ -0005cc90: 3337 2033 3920 3431 2034 3320 3435 2020  37 39 41 43 45  
│ │ │ │ -0005cca0: 2020 7c0a 7c20 2020 2020 2020 202d 363a    |.|        -6:
│ │ │ │ -0005ccb0: 2031 3820 2036 2020 2e20 202e 2020 2e20   18  6  .  .  . 
│ │ │ │ -0005ccc0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0005ccd0: 2e20 2020 207c 0a7c 2020 2020 2020 2020  .    |.|        
│ │ │ │ -0005cce0: 2d35 3a20 202e 2020 2e20 202e 2020 2e20  -5:  .  .  .  . 
│ │ │ │ -0005ccf0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0005cd00: 2e20 202e 2020 2020 7c0a 7c20 2020 2020  .  .    |.|     
│ │ │ │ -0005cd10: 2020 202d 343a 2031 3820 3231 2032 3120     -4: 18 21 21 
│ │ │ │ -0005cd20: 2037 2020 2e20 202e 2020 2e20 202e 2020   7  .  .  .  .  
│ │ │ │ -0005cd30: 2e20 202e 2020 2e20 2020 207c 0a7c 2020  .  .  .    |.|  
│ │ │ │ -0005cd40: 2020 2020 2020 2d33 3a20 202e 2020 2e20        -3:  .  . 
│ │ │ │ -0005cd50: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0005cd60: 2e20 202e 2020 2e20 202e 2020 2020 7c0a  .  .  .  .    |.
│ │ │ │ -0005cd70: 7c20 2020 2020 2020 202d 323a 2020 2e20  |        -2:  . 
│ │ │ │ -0005cd80: 202e 2020 3820 3234 2032 3420 2038 2020   .  8 24 24  8  
│ │ │ │ -0005cd90: 2e20 202e 2020 2e20 202e 2020 2e20 2020  .  .  .  .  .   
│ │ │ │ -0005cda0: 207c 0a7c 2020 2020 2020 2020 2d31 3a20   |.|        -1: 
│ │ │ │ -0005cdb0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ -0005cdc0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0005cdd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0005cde0: 303a 2020 2e20 202e 2020 2e20 202e 2020  0:  .  .  .  .  
│ │ │ │ -0005cdf0: 3920 3237 2032 3720 2039 2020 2e20 202e  9 27 27  9  .  .
│ │ │ │ -0005ce00: 2020 2e20 2020 207c 0a7c 2020 2020 2020    .    |.|      
│ │ │ │ -0005ce10: 2020 2031 3a20 202e 2020 2e20 202e 2020     1:  .  .  .  
│ │ │ │ -0005ce20: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0005ce30: 2020 2e20 202e 2020 2020 7c0a 7c20 2020    .  .    |.|   
│ │ │ │ -0005ce40: 2020 2020 2020 323a 2020 2e20 202e 2020        2:  .  .  
│ │ │ │ -0005ce50: 2e20 202e 2020 2e20 202e 2031 3020 3330  .  .  .  . 10 30
│ │ │ │ -0005ce60: 2033 3020 3130 2020 2e20 2020 207c 0a7c   30 10  .    |.|
│ │ │ │ -0005ce70: 2020 2020 2020 2020 2033 3a20 202e 2020           3:  .  
│ │ │ │ -0005ce80: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0005ce90: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ -0005cea0: 7c0a 7c20 2020 2020 2020 2020 343a 2020  |.|         4:  
│ │ │ │ -0005ceb0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ -0005cec0: 2020 2e20 202e 2031 3120 3333 2033 3320    .  . 11 33 33 
│ │ │ │ -0005ced0: 2020 207c 0a7c 2020 2020 2020 2020 2035     |.|         5
│ │ │ │ -0005cee0: 3a20 202e 2020 2e20 202e 2020 2e20 202e  :  .  .  .  .  .
│ │ │ │ -0005cef0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005cf00: 202e 2020 2020 7c0a 7c20 2020 2020 2020   .    |.|       
│ │ │ │ -0005cf10: 2020 363a 2020 2e20 202e 2020 2e20 202e    6:  .  .  .  .
│ │ │ │ -0005cf20: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ -0005cf30: 202e 2031 3220 2020 207c 0a7c 2020 2020   . 12    |.|    
│ │ │ │ -0005cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0005cf70: 3820 3a20 4265 7474 6954 616c 6c79 2020  8 : BettiTally  
│ │ │ │ -0005cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005cf90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0005cfa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0005cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005cfd0: 2d2d 2b0a 7c69 3920 3a20 6866 4d6f 6475  --+.|i9 : hfModu
│ │ │ │ -0005cfe0: 6c65 4173 4578 7428 3132 2c4d 4d2c 3329  leAsExt(12,MM,3)
│ │ │ │ +0005be10: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +0005be20: 6167 653a 200a 2020 2020 2020 2020 4d20  age: .        M 
│ │ │ │ +0005be30: 3d20 6d6f 6475 6c65 4173 4578 7428 4d4d  = moduleAsExt(MM
│ │ │ │ +0005be40: 2c52 290a 2020 2a20 496e 7075 7473 3a0a  ,R).  * Inputs:.
│ │ │ │ +0005be50: 2020 2020 2020 2a20 4d2c 2061 202a 6e6f        * M, a *no
│ │ │ │ +0005be60: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ +0005be70: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ +0005be80: 2c20 6d6f 6475 6c65 206f 7665 7220 706f  , module over po
│ │ │ │ +0005be90: 6c79 6e6f 6d69 616c 2072 696e 670a 2020  lynomial ring.  
│ │ │ │ +0005bea0: 2020 2020 2020 7769 7468 2063 2076 6172        with c var
│ │ │ │ +0005beb0: 6961 626c 6573 0a20 2020 2020 202a 2052  iables.      * R
│ │ │ │ +0005bec0: 2c20 6120 2a6e 6f74 6520 7269 6e67 3a20  , a *note ring: 
│ │ │ │ +0005bed0: 284d 6163 6175 6c61 7932 446f 6329 5269  (Macaulay2Doc)Ri
│ │ │ │ +0005bee0: 6e67 2c2c 2028 6772 6164 6564 2920 636f  ng,, (graded) co
│ │ │ │ +0005bef0: 6d70 6c65 7465 2069 6e74 6572 7365 6374  mplete intersect
│ │ │ │ +0005bf00: 696f 6e0a 2020 2020 2020 2020 7269 6e67  ion.        ring
│ │ │ │ +0005bf10: 206f 6620 636f 6469 6d65 6e73 696f 6e20   of codimension 
│ │ │ │ +0005bf20: 632c 2065 6d62 6564 6469 6e67 2064 696d  c, embedding dim
│ │ │ │ +0005bf30: 656e 7369 6f6e 206e 0a20 202a 204f 7574  ension n.  * Out
│ │ │ │ +0005bf40: 7075 7473 3a0a 2020 2020 2020 2a20 4e2c  puts:.      * N,
│ │ │ │ +0005bf50: 2061 202a 6e6f 7465 206d 6f64 756c 653a   a *note module:
│ │ │ │ +0005bf60: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +0005bf70: 6f64 756c 652c 2c20 6d6f 6475 6c65 206f  odule,, module o
│ │ │ │ +0005bf80: 7665 7220 5220 7375 6368 2074 6861 740a  ver R such that.
│ │ │ │ +0005bf90: 2020 2020 2020 2020 4578 745f 5228 4e2c          Ext_R(N,
│ │ │ │ +0005bfa0: 6b29 203d 204d 5c6f 7469 6d65 7320 5c77  k) = M\otimes \w
│ │ │ │ +0005bfb0: 6564 6765 286b 5e6e 2920 696e 206c 6172  edge(k^n) in lar
│ │ │ │ +0005bfc0: 6765 2068 6f6d 6f6c 6f67 6963 616c 2064  ge homological d
│ │ │ │ +0005bfd0: 6567 7265 652e 0a0a 4465 7363 7269 7074  egree...Descript
│ │ │ │ +0005bfe0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +0005bff0: 0a54 6865 2072 6f75 7469 6e65 2060 606d  .The routine ``m
│ │ │ │ +0005c000: 6f64 756c 6541 7345 7874 2727 2069 7320  oduleAsExt'' is 
│ │ │ │ +0005c010: 6120 7061 7274 6961 6c20 696e 7665 7273  a partial invers
│ │ │ │ +0005c020: 6520 746f 2074 6865 2072 6f75 7469 6e65  e to the routine
│ │ │ │ +0005c030: 2045 7874 4d6f 6475 6c65 2c0a 636f 6d70   ExtModule,.comp
│ │ │ │ +0005c040: 7574 6564 2066 6f6c 6c6f 7769 6e67 2069  uted following i
│ │ │ │ +0005c050: 6465 6173 206f 6620 4176 7261 6d6f 7620  deas of Avramov 
│ │ │ │ +0005c060: 616e 6420 4a6f 7267 656e 7365 6e3a 2067  and Jorgensen: g
│ │ │ │ +0005c070: 6976 656e 2061 206d 6f64 756c 6520 4520  iven a module E 
│ │ │ │ +0005c080: 6f76 6572 2061 0a70 6f6c 796e 6f6d 6961  over a.polynomia
│ │ │ │ +0005c090: 6c20 7269 6e67 206b 5b78 5f31 2e2e 785f  l ring k[x_1..x_
│ │ │ │ +0005c0a0: 635d 2c20 6974 2070 726f 7669 6465 7320  c], it provides 
│ │ │ │ +0005c0b0: 6120 6d6f 6475 6c65 204e 206f 7665 7220  a module N over 
│ │ │ │ +0005c0c0: 6120 7370 6563 6966 6965 6420 706f 6c79  a specified poly
│ │ │ │ +0005c0d0: 6e6f 6d69 616c 0a72 696e 6720 696e 206e  nomial.ring in n
│ │ │ │ +0005c0e0: 2076 6172 6961 626c 6573 2073 7563 6820   variables such 
│ │ │ │ +0005c0f0: 7468 6174 2045 7874 284e 2c6b 2920 6167  that Ext(N,k) ag
│ │ │ │ +0005c100: 7265 6573 2077 6974 6820 2445 273d 455c  rees with $E'=E\
│ │ │ │ +0005c110: 6f74 696d 6573 205c 7765 6467 6528 6b5e  otimes \wedge(k^
│ │ │ │ +0005c120: 6e29 240a 6166 7465 7220 7472 756e 6361  n)$.after trunca
│ │ │ │ +0005c130: 7469 6f6e 2e20 4865 7265 2074 6865 2067  tion. Here the g
│ │ │ │ +0005c140: 7261 6469 6e67 206f 6e20 4520 6973 2074  rading on E is t
│ │ │ │ +0005c150: 616b 656e 2074 6f20 6265 2065 7665 6e2c  aken to be even,
│ │ │ │ +0005c160: 2077 6869 6c65 0a24 5c77 6564 6765 286b   while.$\wedge(k
│ │ │ │ +0005c170: 5e6e 2924 2068 6173 2067 656e 6572 6174  ^n)$ has generat
│ │ │ │ +0005c180: 6f72 7320 696e 2064 6567 7265 6520 312e  ors in degree 1.
│ │ │ │ +0005c190: 2054 6865 2072 6f75 7469 6e65 2068 664d   The routine hfM
│ │ │ │ +0005c1a0: 6f64 756c 6541 7345 7874 2063 6f6d 7075  oduleAsExt compu
│ │ │ │ +0005c1b0: 7465 730a 7468 6520 7265 7375 6c74 696e  tes.the resultin
│ │ │ │ +0005c1c0: 6720 6869 6c62 6572 7420 6675 6e63 7469  g hilbert functi
│ │ │ │ +0005c1d0: 6f6e 2066 6f72 2045 272e 2054 6869 7320  on for E'. This 
│ │ │ │ +0005c1e0: 7573 6573 2069 6465 6173 206f 6620 4176  uses ideas of Av
│ │ │ │ +0005c1f0: 7261 6d6f 7620 616e 640a 4a6f 7267 656e  ramov and.Jorgen
│ │ │ │ +0005c200: 7365 6e2e 204e 6f74 6520 7468 6174 2074  sen. Note that t
│ │ │ │ +0005c210: 6865 206d 6f64 756c 6520 4578 7428 4e2c  he module Ext(N,
│ │ │ │ +0005c220: 6b29 2028 7472 756e 6361 7465 6429 2077  k) (truncated) w
│ │ │ │ +0005c230: 696c 6c20 6175 746f 6d61 7469 6361 6c6c  ill automaticall
│ │ │ │ +0005c240: 7920 6265 2066 7265 650a 6f76 6572 2074  y be free.over t
│ │ │ │ +0005c250: 6865 2065 7874 6572 696f 7220 616c 6765  he exterior alge
│ │ │ │ +0005c260: 6272 6120 245c 7765 6467 6528 6b5e 6e29  bra $\wedge(k^n)
│ │ │ │ +0005c270: 2420 6765 6e65 7261 7465 6420 6279 2045  $ generated by E
│ │ │ │ +0005c280: 7874 5e31 286b 2c6b 293b 206e 6f74 2061  xt^1(k,k); not a
│ │ │ │ +0005c290: 2074 7970 6963 616c 0a45 7874 206d 6f64   typical.Ext mod
│ │ │ │ +0005c2a0: 756c 652e 0a0a 4d6f 7265 2070 7265 6369  ule...More preci
│ │ │ │ +0005c2b0: 7365 6c79 3a0a 0a53 7570 706f 7365 2074  sely:..Suppose t
│ │ │ │ +0005c2c0: 6861 7420 2452 203d 206b 5b61 5f31 2c5c  hat $R = k[a_1,\
│ │ │ │ +0005c2d0: 646f 7473 2c20 615f 6e5d 2f28 665f 312c  dots, a_n]/(f_1,
│ │ │ │ +0005c2e0: 5c64 6f74 732c 665f 6329 2420 6c65 7420  \dots,f_c)$ let 
│ │ │ │ +0005c2f0: 244b 4b20 3d0a 6b5b 785f 312c 5c64 6f74  $KK =.k[x_1,\dot
│ │ │ │ +0005c300: 732c 785f 635d 242c 2061 6e64 206c 6574  s,x_c]$, and let
│ │ │ │ +0005c310: 2024 5c4c 616d 6264 6120 3d20 5c77 6564   $\Lambda = \wed
│ │ │ │ +0005c320: 6765 206b 5e6e 242e 2024 4520 3d20 4b4b  ge k^n$. $E = KK
│ │ │ │ +0005c330: 5c6f 7469 6d65 735c 4c61 6d62 6461 242c  \otimes\Lambda$,
│ │ │ │ +0005c340: 2073 6f0a 7468 6174 2074 6865 206d 696e   so.that the min
│ │ │ │ +0005c350: 696d 616c 2024 5224 2d66 7265 6520 7265  imal $R$-free re
│ │ │ │ +0005c360: 736f 6c75 7469 6f6e 206f 6620 246b 2420  solution of $k$ 
│ │ │ │ +0005c370: 6861 7320 756e 6465 726c 7969 6e67 206d  has underlying m
│ │ │ │ +0005c380: 6f64 756c 6520 2452 5c6f 7469 6d65 730a  odule $R\otimes.
│ │ │ │ +0005c390: 455e 2a24 2c20 7768 6572 6520 2445 5e2a  E^*$, where $E^*
│ │ │ │ +0005c3a0: 2420 6973 2074 6865 2067 7261 6465 6420  $ is the graded 
│ │ │ │ +0005c3b0: 7665 6374 6f72 2073 7061 6365 2064 7561  vector space dua
│ │ │ │ +0005c3c0: 6c20 6f66 2024 4524 2e0a 0a4c 6574 204d  l of $E$...Let M
│ │ │ │ +0005c3d0: 4d20 6265 2074 6865 2072 6573 756c 7420  M be the result 
│ │ │ │ +0005c3e0: 6f66 2074 7275 6e63 6174 696e 6720 4d20  of truncating M 
│ │ │ │ +0005c3f0: 6174 2069 7473 2072 6567 756c 6172 6974  at its regularit
│ │ │ │ +0005c400: 7920 616e 6420 7368 6966 7469 6e67 2069  y and shifting i
│ │ │ │ +0005c410: 7420 736f 2074 6861 740a 6974 2069 7320  t so that.it is 
│ │ │ │ +0005c420: 6765 6e65 7261 7465 6420 696e 2064 6567  generated in deg
│ │ │ │ +0005c430: 7265 6520 302e 204c 6574 2024 4624 2062  ree 0. Let $F$ b
│ │ │ │ +0005c440: 6520 6120 244b 4b24 2d66 7265 6520 7265  e a $KK$-free re
│ │ │ │ +0005c450: 736f 6c75 7469 6f6e 206f 6620 244d 4d24  solution of $MM$
│ │ │ │ +0005c460: 2c20 616e 640a 7772 6974 6520 2446 5f69  , and.write $F_i
│ │ │ │ +0005c470: 203d 204b 4b5c 6f74 696d 6573 2056 5f69   = KK\otimes V_i
│ │ │ │ +0005c480: 2e24 2053 696e 6365 206c 696e 6561 7220  .$ Since linear 
│ │ │ │ +0005c490: 666f 726d 7320 6f76 6572 2024 4b4b 2420  forms over $KK$ 
│ │ │ │ +0005c4a0: 636f 7272 6573 706f 6e64 2074 6f20 4349  correspond to CI
│ │ │ │ +0005c4b0: 0a6f 7065 7261 746f 7273 206f 6620 6465  .operators of de
│ │ │ │ +0005c4c0: 6772 6565 202d 3220 6f6e 2074 6865 2072  gree -2 on the r
│ │ │ │ +0005c4d0: 6573 6f6c 7574 696f 6e20 4720 6f66 206b  esolution G of k
│ │ │ │ +0005c4e0: 206f 7665 7220 522c 2077 6520 6d61 7920   over R, we may 
│ │ │ │ +0005c4f0: 666f 726d 2061 206d 6170 2024 240a 645f  form a map $$.d_
│ │ │ │ +0005c500: 312b 645f 323a 205c 7375 6d5f 7b69 3d30  1+d_2: \sum_{i=0
│ │ │ │ +0005c510: 7d5e 6d20 475f 7b69 2b31 7d5c 6f74 696d  }^m G_{i+1}\otim
│ │ │ │ +0005c520: 6573 2056 5f7b 6d2d 697d 5e2a 205c 746f  es V_{m-i}^* \to
│ │ │ │ +0005c530: 205c 7375 6d5f 7b69 3d30 7d5e 6d20 475f   \sum_{i=0}^m G_
│ │ │ │ +0005c540: 695c 6f74 696d 6573 0a56 5f7b 6d2d 697d  i\otimes.V_{m-i}
│ │ │ │ +0005c550: 5e2a 2024 2420 7768 6572 6520 2464 5f31  ^* $$ where $d_1
│ │ │ │ +0005c560: 2420 6973 2074 6865 2064 6972 6563 7420  $ is the direct 
│ │ │ │ +0005c570: 7375 6d20 6f66 2074 6865 2064 6966 6665  sum of the diffe
│ │ │ │ +0005c580: 7265 6e74 6961 6c73 2024 2847 5f7b 692b  rentials $(G_{i+
│ │ │ │ +0005c590: 317d 5c74 6f0a 475f 6929 5c6f 7469 6d65  1}\to.G_i)\otime
│ │ │ │ +0005c5a0: 7320 565f 695e 2a24 2061 6e64 2024 645f  s V_i^*$ and $d_
│ │ │ │ +0005c5b0: 3224 2069 7320 7468 6520 6469 7265 6374  2$ is the direct
│ │ │ │ +0005c5c0: 2073 756d 206f 6620 7468 6520 6d61 7073   sum of the maps
│ │ │ │ +0005c5d0: 2024 5c70 6869 5f69 2420 6465 6669 6e65   $\phi_i$ define
│ │ │ │ +0005c5e0: 640a 6672 6f6d 2074 6865 2064 6966 6665  d.from the diffe
│ │ │ │ +0005c5f0: 7265 6e74 6961 6c73 206f 6620 2446 2420  rentials of $F$ 
│ │ │ │ +0005c600: 6279 2073 7562 7374 6974 7574 696e 6720  by substituting 
│ │ │ │ +0005c610: 4349 206f 7065 7261 746f 7273 2066 6f72  CI operators for
│ │ │ │ +0005c620: 206c 696e 6561 7220 666f 726d 732c 0a24   linear forms,.$
│ │ │ │ +0005c630: 5c70 6869 5f69 3a20 475f 7b69 2b31 7d5c  \phi_i: G_{i+1}\
│ │ │ │ +0005c640: 6f74 696d 6573 2056 5f69 205c 746f 2047  otimes V_i \to G
│ │ │ │ +0005c650: 5f7b 692d 317d 5c6f 7469 6d65 7320 565f  _{i-1}\otimes V_
│ │ │ │ +0005c660: 7b69 2d31 7d24 2e20 5468 6520 7363 7269  {i-1}$. The scri
│ │ │ │ +0005c670: 7074 2072 6574 7572 6e73 2074 6865 0a6d  pt returns the.m
│ │ │ │ +0005c680: 6f64 756c 6520 4e20 7468 6174 2069 7320  odule N that is 
│ │ │ │ +0005c690: 7468 6520 636f 6b65 726e 656c 206f 6620  the cokernel of 
│ │ │ │ +0005c6a0: 2464 5f31 2b64 5f32 242e 0a0a 5468 6520  $d_1+d_2$...The 
│ │ │ │ +0005c6b0: 6d6f 6475 6c65 2024 4578 745f 5228 4e2c  module $Ext_R(N,
│ │ │ │ +0005c6c0: 6b29 2420 6167 7265 6573 2c20 6166 7465  k)$ agrees, afte
│ │ │ │ +0005c6d0: 7220 6120 6665 7720 7374 6570 732c 2077  r a few steps, w
│ │ │ │ +0005c6e0: 6974 6820 7468 6520 6d6f 6475 6c65 2064  ith the module d
│ │ │ │ +0005c6f0: 6572 6976 6564 2066 726f 6d0a 244d 4d24  erived from.$MM$
│ │ │ │ +0005c700: 2062 7920 7465 6e73 6f72 696e 6720 6974   by tensoring it
│ │ │ │ +0005c710: 2077 6974 6820 245c 4c61 6d62 6461 242c   with $\Lambda$,
│ │ │ │ +0005c720: 2074 6861 7420 6973 2c20 7769 7468 2074   that is, with t
│ │ │ │ +0005c730: 6865 206d 6f64 756c 65c3 9f20 2424 204d  he module.. $$ M
│ │ │ │ +0005c740: 4d27 203d 205c 7375 6d5f 6a0a 284d 4d27  M' = \sum_j.(MM'
│ │ │ │ +0005c750: 286a 295c 6f74 696d 6573 205c 4c61 6d62  (j)\otimes \Lamb
│ │ │ │ +0005c760: 6461 5f6a 2920 2424 2073 6f20 7468 6174  da_j) $$ so that
│ │ │ │ +0005c770: 2024 4d4d 275f 7020 3d20 284d 4d5f 705c   $MM'_p = (MM_p\
│ │ │ │ +0005c780: 6f74 696d 6573 204c 616d 6264 615f 3029  otimes Lambda_0)
│ │ │ │ +0005c790: 205c 6f70 6c75 730a 284d 4d5f 7b70 2d31   \oplus.(MM_{p-1
│ │ │ │ +0005c7a0: 7d5c 6f74 696d 6573 204c 616d 6264 615f  }\otimes Lambda_
│ │ │ │ +0005c7b0: 3129 205c 6f70 6c75 735c 6364 6f74 7324  1) \oplus\cdots$
│ │ │ │ +0005c7c0: 2e0a 0a54 6865 2066 756e 6374 696f 6e20  ...The function 
│ │ │ │ +0005c7d0: 6866 4d6f 6475 6c65 4173 4578 7420 636f  hfModuleAsExt co
│ │ │ │ +0005c7e0: 6d70 7574 6573 2074 6865 2048 696c 6265  mputes the Hilbe
│ │ │ │ +0005c7f0: 7274 2066 756e 6374 696f 6e20 6f66 204d  rt function of M
│ │ │ │ +0005c800: 4d27 206e 756d 6572 6963 616c 6c79 0a66  M' numerically.f
│ │ │ │ +0005c810: 726f 6d20 7468 6174 206f 6620 4d4d 2e0a  rom that of MM..
│ │ │ │ +0005c820: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0005c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c850: 2d2d 2b0a 7c69 3120 3a20 6b6b 203d 205a  --+.|i1 : kk = Z
│ │ │ │ +0005c860: 5a2f 3130 313b 2020 2020 2020 2020 2020  Z/101;          
│ │ │ │ +0005c870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c880: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0005c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c8b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ +0005c8c0: 5320 3d20 6b6b 5b61 2c62 2c63 5d3b 2020  S = kk[a,b,c];  
│ │ │ │ +0005c8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c8e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0005c8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005c910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0005c920: 7c69 3320 3a20 6666 203d 206d 6174 7269  |i3 : ff = matri
│ │ │ │ +0005c930: 787b 7b61 5e34 2c20 625e 342c 635e 347d  x{{a^4, b^4,c^4}
│ │ │ │ +0005c940: 7d3b 2020 2020 2020 2020 2020 2020 2020  };              
│ │ │ │ +0005c950: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0005c960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c980: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005c990: 2020 2020 3120 2020 2020 2033 2020 2020      1      3    
│ │ │ │ +0005c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c9b0: 2020 2020 2020 207c 0a7c 6f33 203a 204d         |.|o3 : M
│ │ │ │ +0005c9c0: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +0005c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005c9e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0005c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0005ca20: 6934 203a 2052 203d 2053 2f69 6465 616c  i4 : R = S/ideal
│ │ │ │ +0005ca30: 2066 663b 2020 2020 2020 2020 2020 2020   ff;            
│ │ │ │ +0005ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ca50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0005ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ca80: 2d2d 2d2b 0a7c 6935 203a 204f 7073 203d  ---+.|i5 : Ops =
│ │ │ │ +0005ca90: 206b 6b5b 785f 312c 785f 322c 785f 335d   kk[x_1,x_2,x_3]
│ │ │ │ +0005caa0: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +0005cab0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0005cac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cae0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ +0005caf0: 204d 4d20 3d20 4f70 735e 312f 2878 5f31   MM = Ops^1/(x_1
│ │ │ │ +0005cb00: 2a69 6465 616c 2878 5f32 5e32 2c78 5f33  *ideal(x_2^2,x_3
│ │ │ │ +0005cb10: 2929 3b20 2020 2020 2020 2020 7c0a 2b2d  ));         |.+-
│ │ │ │ +0005cb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0005cb50: 0a7c 6937 203a 204e 203d 206d 6f64 756c  .|i7 : N = modul
│ │ │ │ +0005cb60: 6541 7345 7874 284d 4d2c 5229 3b20 2020  eAsExt(MM,R);   
│ │ │ │ +0005cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cb80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0005cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cbb0: 2d2d 2d2d 2d2b 0a7c 6938 203a 2062 6574  -----+.|i8 : bet
│ │ │ │ +0005cbc0: 7469 2066 7265 6552 6573 6f6c 7574 696f  ti freeResolutio
│ │ │ │ +0005cbd0: 6e28 204e 2c20 4c65 6e67 7468 4c69 6d69  n( N, LengthLimi
│ │ │ │ +0005cbe0: 7420 3d3e 2031 3029 7c0a 7c20 2020 2020  t => 10)|.|     
│ │ │ │ +0005cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cc10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0005cc20: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ +0005cc30: 2032 2020 3320 2034 2020 3520 2036 2020   2  3  4  5  6  
│ │ │ │ +0005cc40: 3720 2038 2020 3920 3130 2020 2020 7c0a  7  8  9 10    |.
│ │ │ │ +0005cc50: 7c6f 3820 3d20 746f 7461 6c3a 2033 3620  |o8 = total: 36 
│ │ │ │ +0005cc60: 3237 2032 3920 3331 2033 3320 3335 2033  27 29 31 33 35 3
│ │ │ │ +0005cc70: 3720 3339 2034 3120 3433 2034 3520 2020  7 39 41 43 45   
│ │ │ │ +0005cc80: 207c 0a7c 2020 2020 2020 2020 2d36 3a20   |.|        -6: 
│ │ │ │ +0005cc90: 3138 2020 3620 202e 2020 2e20 202e 2020  18  6  .  .  .  
│ │ │ │ +0005cca0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0005ccb0: 2020 2020 7c0a 7c20 2020 2020 2020 202d      |.|        -
│ │ │ │ +0005ccc0: 353a 2020 2e20 202e 2020 2e20 202e 2020  5:  .  .  .  .  
│ │ │ │ +0005ccd0: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0005cce0: 2020 2e20 2020 207c 0a7c 2020 2020 2020    .    |.|      
│ │ │ │ +0005ccf0: 2020 2d34 3a20 3138 2032 3120 3231 2020    -4: 18 21 21  
│ │ │ │ +0005cd00: 3720 202e 2020 2e20 202e 2020 2e20 202e  7  .  .  .  .  .
│ │ │ │ +0005cd10: 2020 2e20 202e 2020 2020 7c0a 7c20 2020    .  .    |.|   
│ │ │ │ +0005cd20: 2020 2020 202d 333a 2020 2e20 202e 2020       -3:  .  .  
│ │ │ │ +0005cd30: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0005cd40: 2020 2e20 202e 2020 2e20 2020 207c 0a7c    .  .  .    |.|
│ │ │ │ +0005cd50: 2020 2020 2020 2020 2d32 3a20 202e 2020          -2:  .  
│ │ │ │ +0005cd60: 2e20 2038 2032 3420 3234 2020 3820 202e  .  8 24 24  8  .
│ │ │ │ +0005cd70: 2020 2e20 202e 2020 2e20 202e 2020 2020    .  .  .  .    
│ │ │ │ +0005cd80: 7c0a 7c20 2020 2020 2020 202d 313a 2020  |.|        -1:  
│ │ │ │ +0005cd90: 2e20 202e 2020 2e20 202e 2020 2e20 202e  .  .  .  .  .  .
│ │ │ │ +0005cda0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005cdb0: 2020 207c 0a7c 2020 2020 2020 2020 2030     |.|         0
│ │ │ │ +0005cdc0: 3a20 202e 2020 2e20 202e 2020 2e20 2039  :  .  .  .  .  9
│ │ │ │ +0005cdd0: 2032 3720 3237 2020 3920 202e 2020 2e20   27 27  9  .  . 
│ │ │ │ +0005cde0: 202e 2020 2020 7c0a 7c20 2020 2020 2020   .    |.|       
│ │ │ │ +0005cdf0: 2020 313a 2020 2e20 202e 2020 2e20 202e    1:  .  .  .  .
│ │ │ │ +0005ce00: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005ce10: 202e 2020 2e20 2020 207c 0a7c 2020 2020   .  .    |.|    
│ │ │ │ +0005ce20: 2020 2020 2032 3a20 202e 2020 2e20 202e       2:  .  .  .
│ │ │ │ +0005ce30: 2020 2e20 202e 2020 2e20 3130 2033 3020    .  .  . 10 30 
│ │ │ │ +0005ce40: 3330 2031 3020 202e 2020 2020 7c0a 7c20  30 10  .    |.| 
│ │ │ │ +0005ce50: 2020 2020 2020 2020 333a 2020 2e20 202e          3:  .  .
│ │ │ │ +0005ce60: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005ce70: 202e 2020 2e20 202e 2020 2e20 2020 207c   .  .  .  .    |
│ │ │ │ +0005ce80: 0a7c 2020 2020 2020 2020 2034 3a20 202e  .|         4:  .
│ │ │ │ +0005ce90: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005cea0: 202e 2020 2e20 3131 2033 3320 3333 2020   .  . 11 33 33  
│ │ │ │ +0005ceb0: 2020 7c0a 7c20 2020 2020 2020 2020 353a    |.|         5:
│ │ │ │ +0005cec0: 2020 2e20 202e 2020 2e20 202e 2020 2e20    .  .  .  .  . 
│ │ │ │ +0005ced0: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005cee0: 2e20 2020 207c 0a7c 2020 2020 2020 2020  .    |.|        
│ │ │ │ +0005cef0: 2036 3a20 202e 2020 2e20 202e 2020 2e20   6:  .  .  .  . 
│ │ │ │ +0005cf00: 202e 2020 2e20 202e 2020 2e20 202e 2020   .  .  .  .  .  
│ │ │ │ +0005cf10: 2e20 3132 2020 2020 7c0a 7c20 2020 2020  . 12    |.|     
│ │ │ │ +0005cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cf40: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +0005cf50: 203a 2042 6574 7469 5461 6c6c 7920 2020   : BettiTally   
│ │ │ │ +0005cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cf70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0005cf80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0005cf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005cfb0: 2d2b 0a7c 6939 203a 2068 664d 6f64 756c  -+.|i9 : hfModul
│ │ │ │ +0005cfc0: 6541 7345 7874 2831 322c 4d4d 2c33 2920  eAsExt(12,MM,3) 
│ │ │ │ +0005cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005cfe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0005cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0005d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d030: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20          |.|o9 = 
│ │ │ │ -0005d040: 2832 332c 2032 352c 2032 372c 2032 392c  (23, 25, 27, 29,
│ │ │ │ -0005d050: 2033 312c 2033 332c 2033 352c 2033 372c   31, 33, 35, 37,
│ │ │ │ -0005d060: 2033 392c 2034 3129 2020 207c 0a7c 2020   39, 41)   |.|  
│ │ │ │ -0005d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d090: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0005d0a0: 7c6f 3920 3a20 5365 7175 656e 6365 2020  |o9 : Sequence  
│ │ │ │ -0005d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005d0d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ -0005d0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d100: 2d2d 2d2d 2b0a 0a43 6176 6561 740a 3d3d  ----+..Caveat.==
│ │ │ │ -0005d110: 3d3d 3d3d 0a0a 5468 6520 656c 656d 656e  ====..The elemen
│ │ │ │ -0005d120: 7473 2066 5f31 2e2e 665f 6320 6d75 7374  ts f_1..f_c must
│ │ │ │ -0005d130: 2062 6520 686f 6d6f 6765 6e65 6f75 7320   be homogeneous 
│ │ │ │ -0005d140: 6f66 2074 6865 2073 616d 6520 6465 6772  of the same degr
│ │ │ │ -0005d150: 6565 2e20 5468 6520 7363 7269 7074 2063  ee. The script c
│ │ │ │ -0005d160: 6f75 6c64 0a62 6520 7265 7772 6974 7465  ould.be rewritte
│ │ │ │ -0005d170: 6e20 746f 2061 6363 6f6d 6d6f 6461 7465  n to accommodate
│ │ │ │ -0005d180: 2064 6966 6665 7265 6e74 2064 6567 7265   different degre
│ │ │ │ -0005d190: 6573 2c20 6275 7420 6f6e 6c79 2062 7920  es, but only by 
│ │ │ │ -0005d1a0: 676f 696e 6720 746f 2074 6865 206c 6f63  going to the loc
│ │ │ │ -0005d1b0: 616c 0a63 6174 6567 6f72 790a 0a53 6565  al.category..See
│ │ │ │ -0005d1c0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -0005d1d0: 2020 2a20 2a6e 6f74 6520 4578 744d 6f64    * *note ExtMod
│ │ │ │ -0005d1e0: 756c 653a 2045 7874 4d6f 6475 6c65 2c20  ule: ExtModule, 
│ │ │ │ -0005d1f0: 2d2d 2045 7874 5e2a 284d 2c6b 2920 6f76  -- Ext^*(M,k) ov
│ │ │ │ -0005d200: 6572 2061 2063 6f6d 706c 6574 6520 696e  er a complete in
│ │ │ │ -0005d210: 7465 7273 6563 7469 6f6e 2061 730a 2020  tersection as.  
│ │ │ │ -0005d220: 2020 6d6f 6475 6c65 206f 7665 7220 4349    module over CI
│ │ │ │ -0005d230: 206f 7065 7261 746f 7220 7269 6e67 0a20   operator ring. 
│ │ │ │ -0005d240: 202a 202a 6e6f 7465 2065 7665 6e45 7874   * *note evenExt
│ │ │ │ -0005d250: 4d6f 6475 6c65 3a20 6576 656e 4578 744d  Module: evenExtM
│ │ │ │ -0005d260: 6f64 756c 652c 202d 2d20 6576 656e 2070  odule, -- even p
│ │ │ │ -0005d270: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ -0005d280: 2920 6f76 6572 2061 0a20 2020 2063 6f6d  ) over a.    com
│ │ │ │ -0005d290: 706c 6574 6520 696e 7465 7273 6563 7469  plete intersecti
│ │ │ │ -0005d2a0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ -0005d2b0: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ -0005d2c0: 6e67 0a20 202a 202a 6e6f 7465 206f 6464  ng.  * *note odd
│ │ │ │ -0005d2d0: 4578 744d 6f64 756c 653a 206f 6464 4578  ExtModule: oddEx
│ │ │ │ -0005d2e0: 744d 6f64 756c 652c 202d 2d20 6f64 6420  tModule, -- odd 
│ │ │ │ -0005d2f0: 7061 7274 206f 6620 4578 745e 2a28 4d2c  part of Ext^*(M,
│ │ │ │ -0005d300: 6b29 206f 7665 7220 6120 636f 6d70 6c65  k) over a comple
│ │ │ │ -0005d310: 7465 0a20 2020 2069 6e74 6572 7365 6374  te.    intersect
│ │ │ │ -0005d320: 696f 6e20 6173 206d 6f64 756c 6520 6f76  ion as module ov
│ │ │ │ -0005d330: 6572 2043 4920 6f70 6572 6174 6f72 2072  er CI operator r
│ │ │ │ -0005d340: 696e 670a 2020 2a20 2a6e 6f74 6520 4578  ing.  * *note Ex
│ │ │ │ -0005d350: 744d 6f64 756c 6544 6174 613a 2045 7874  tModuleData: Ext
│ │ │ │ -0005d360: 4d6f 6475 6c65 4461 7461 2c20 2d2d 2045  ModuleData, -- E
│ │ │ │ -0005d370: 7665 6e20 616e 6420 6f64 6420 4578 7420  ven and odd Ext 
│ │ │ │ -0005d380: 6d6f 6475 6c65 7320 616e 6420 7468 6569  modules and thei
│ │ │ │ -0005d390: 720a 2020 2020 7265 6775 6c61 7269 7479  r.    regularity
│ │ │ │ -0005d3a0: 0a20 202a 202a 6e6f 7465 2068 664d 6f64  .  * *note hfMod
│ │ │ │ -0005d3b0: 756c 6541 7345 7874 3a20 6866 4d6f 6475  uleAsExt: hfModu
│ │ │ │ -0005d3c0: 6c65 4173 4578 742c 202d 2d20 7072 6564  leAsExt, -- pred
│ │ │ │ -0005d3d0: 6963 7420 6265 7474 6920 6e75 6d62 6572  ict betti number
│ │ │ │ -0005d3e0: 7320 6f66 0a20 2020 206d 6f64 756c 6541  s of.    moduleA
│ │ │ │ -0005d3f0: 7345 7874 284d 2c52 290a 0a57 6179 7320  sExt(M,R)..Ways 
│ │ │ │ -0005d400: 746f 2075 7365 206d 6f64 756c 6541 7345  to use moduleAsE
│ │ │ │ -0005d410: 7874 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  xt:.============
│ │ │ │ -0005d420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -0005d430: 2a20 226d 6f64 756c 6541 7345 7874 284d  * "moduleAsExt(M
│ │ │ │ -0005d440: 6f64 756c 652c 5269 6e67 2922 0a0a 466f  odule,Ring)"..Fo
│ │ │ │ -0005d450: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -0005d460: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0005d470: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -0005d480: 2a6e 6f74 6520 6d6f 6475 6c65 4173 4578  *note moduleAsEx
│ │ │ │ -0005d490: 743a 206d 6f64 756c 6541 7345 7874 2c20  t: moduleAsExt, 
│ │ │ │ -0005d4a0: 6973 2061 202a 6e6f 7465 206d 6574 686f  is a *note metho
│ │ │ │ -0005d4b0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163  d function:.(Mac
│ │ │ │ -0005d4c0: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ -0005d4d0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d  Function,...----
│ │ │ │ +0005d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d010: 2020 2020 2020 207c 0a7c 6f39 203d 2028         |.|o9 = (
│ │ │ │ +0005d020: 3233 2c20 3235 2c20 3237 2c20 3239 2c20  23, 25, 27, 29, 
│ │ │ │ +0005d030: 3331 2c20 3333 2c20 3335 2c20 3337 2c20  31, 33, 35, 37, 
│ │ │ │ +0005d040: 3339 2c20 3431 2920 2020 7c0a 7c20 2020  39, 41)   |.|   
│ │ │ │ +0005d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d070: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0005d080: 6f39 203a 2053 6571 7565 6e63 6520 2020  o9 : Sequence   
│ │ │ │ +0005d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005d0b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0005d0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005d0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005d0e0: 2d2d 2d2b 0a0a 4361 7665 6174 0a3d 3d3d  ---+..Caveat.===
│ │ │ │ +0005d0f0: 3d3d 3d0a 0a54 6865 2065 6c65 6d65 6e74  ===..The element
│ │ │ │ +0005d100: 7320 665f 312e 2e66 5f63 206d 7573 7420  s f_1..f_c must 
│ │ │ │ +0005d110: 6265 2068 6f6d 6f67 656e 656f 7573 206f  be homogeneous o
│ │ │ │ +0005d120: 6620 7468 6520 7361 6d65 2064 6567 7265  f the same degre
│ │ │ │ +0005d130: 652e 2054 6865 2073 6372 6970 7420 636f  e. The script co
│ │ │ │ +0005d140: 756c 640a 6265 2072 6577 7269 7474 656e  uld.be rewritten
│ │ │ │ +0005d150: 2074 6f20 6163 636f 6d6d 6f64 6174 6520   to accommodate 
│ │ │ │ +0005d160: 6469 6666 6572 656e 7420 6465 6772 6565  different degree
│ │ │ │ +0005d170: 732c 2062 7574 206f 6e6c 7920 6279 2067  s, but only by g
│ │ │ │ +0005d180: 6f69 6e67 2074 6f20 7468 6520 6c6f 6361  oing to the loca
│ │ │ │ +0005d190: 6c0a 6361 7465 676f 7279 0a0a 5365 6520  l.category..See 
│ │ │ │ +0005d1a0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +0005d1b0: 202a 202a 6e6f 7465 2045 7874 4d6f 6475   * *note ExtModu
│ │ │ │ +0005d1c0: 6c65 3a20 4578 744d 6f64 756c 652c 202d  le: ExtModule, -
│ │ │ │ +0005d1d0: 2d20 4578 745e 2a28 4d2c 6b29 206f 7665  - Ext^*(M,k) ove
│ │ │ │ +0005d1e0: 7220 6120 636f 6d70 6c65 7465 2069 6e74  r a complete int
│ │ │ │ +0005d1f0: 6572 7365 6374 696f 6e20 6173 0a20 2020  ersection as.   
│ │ │ │ +0005d200: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ +0005d210: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ +0005d220: 2a20 2a6e 6f74 6520 6576 656e 4578 744d  * *note evenExtM
│ │ │ │ +0005d230: 6f64 756c 653a 2065 7665 6e45 7874 4d6f  odule: evenExtMo
│ │ │ │ +0005d240: 6475 6c65 2c20 2d2d 2065 7665 6e20 7061  dule, -- even pa
│ │ │ │ +0005d250: 7274 206f 6620 4578 745e 2a28 4d2c 6b29  rt of Ext^*(M,k)
│ │ │ │ +0005d260: 206f 7665 7220 610a 2020 2020 636f 6d70   over a.    comp
│ │ │ │ +0005d270: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ +0005d280: 6e20 6173 206d 6f64 756c 6520 6f76 6572  n as module over
│ │ │ │ +0005d290: 2043 4920 6f70 6572 6174 6f72 2072 696e   CI operator rin
│ │ │ │ +0005d2a0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445  g.  * *note oddE
│ │ │ │ +0005d2b0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874  xtModule: oddExt
│ │ │ │ +0005d2c0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070  Module, -- odd p
│ │ │ │ +0005d2d0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b  art of Ext^*(M,k
│ │ │ │ +0005d2e0: 2920 6f76 6572 2061 2063 6f6d 706c 6574  ) over a complet
│ │ │ │ +0005d2f0: 650a 2020 2020 696e 7465 7273 6563 7469  e.    intersecti
│ │ │ │ +0005d300: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665  on as module ove
│ │ │ │ +0005d310: 7220 4349 206f 7065 7261 746f 7220 7269  r CI operator ri
│ │ │ │ +0005d320: 6e67 0a20 202a 202a 6e6f 7465 2045 7874  ng.  * *note Ext
│ │ │ │ +0005d330: 4d6f 6475 6c65 4461 7461 3a20 4578 744d  ModuleData: ExtM
│ │ │ │ +0005d340: 6f64 756c 6544 6174 612c 202d 2d20 4576  oduleData, -- Ev
│ │ │ │ +0005d350: 656e 2061 6e64 206f 6464 2045 7874 206d  en and odd Ext m
│ │ │ │ +0005d360: 6f64 756c 6573 2061 6e64 2074 6865 6972  odules and their
│ │ │ │ +0005d370: 0a20 2020 2072 6567 756c 6172 6974 790a  .    regularity.
│ │ │ │ +0005d380: 2020 2a20 2a6e 6f74 6520 6866 4d6f 6475    * *note hfModu
│ │ │ │ +0005d390: 6c65 4173 4578 743a 2068 664d 6f64 756c  leAsExt: hfModul
│ │ │ │ +0005d3a0: 6541 7345 7874 2c20 2d2d 2070 7265 6469  eAsExt, -- predi
│ │ │ │ +0005d3b0: 6374 2062 6574 7469 206e 756d 6265 7273  ct betti numbers
│ │ │ │ +0005d3c0: 206f 660a 2020 2020 6d6f 6475 6c65 4173   of.    moduleAs
│ │ │ │ +0005d3d0: 4578 7428 4d2c 5229 0a0a 5761 7973 2074  Ext(M,R)..Ways t
│ │ │ │ +0005d3e0: 6f20 7573 6520 6d6f 6475 6c65 4173 4578  o use moduleAsEx
│ │ │ │ +0005d3f0: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  t:.=============
│ │ │ │ +0005d400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +0005d410: 2022 6d6f 6475 6c65 4173 4578 7428 4d6f   "moduleAsExt(Mo
│ │ │ │ +0005d420: 6475 6c65 2c52 696e 6729 220a 0a46 6f72  dule,Ring)"..For
│ │ │ │ +0005d430: 2074 6865 2070 726f 6772 616d 6d65 720a   the programmer.
│ │ │ │ +0005d440: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0005d450: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a  ==..The object *
│ │ │ │ +0005d460: 6e6f 7465 206d 6f64 756c 6541 7345 7874  note moduleAsExt
│ │ │ │ +0005d470: 3a20 6d6f 6475 6c65 4173 4578 742c 2069  : moduleAsExt, i
│ │ │ │ +0005d480: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64  s a *note method
│ │ │ │ +0005d490: 2066 756e 6374 696f 6e3a 0a28 4d61 6361   function:.(Maca
│ │ │ │ +0005d4a0: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ +0005d4b0: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d  unction,...-----
│ │ │ │ +0005d4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005d4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ -0005d530: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ -0005d540: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ -0005d550: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ -0005d560: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ -0005d570: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ -0005d580: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ -0005d590: 6573 2f0a 436f 6d70 6c65 7465 496e 7465  es/.CompleteInte
│ │ │ │ -0005d5a0: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -0005d5b0: 6f6e 732e 6d32 3a33 3039 363a 302e 0a1f  ons.m2:3096:0...
│ │ │ │ -0005d5c0: 0a46 696c 653a 2043 6f6d 706c 6574 6549  .File: CompleteI
│ │ │ │ -0005d5d0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ -0005d5e0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64  utions.info, Nod
│ │ │ │ -0005d5f0: 653a 206e 6577 4578 742c 204e 6578 743a  e: newExt, Next:
│ │ │ │ -0005d600: 206f 6464 4578 744d 6f64 756c 652c 2050   oddExtModule, P
│ │ │ │ -0005d610: 7265 763a 206d 6f64 756c 6541 7345 7874  rev: moduleAsExt
│ │ │ │ -0005d620: 2c20 5570 3a20 546f 700a 0a6e 6577 4578  , Up: Top..newEx
│ │ │ │ -0005d630: 7420 2d2d 2047 6c6f 6261 6c20 4578 7420  t -- Global Ext 
│ │ │ │ -0005d640: 666f 7220 6d6f 6475 6c65 7320 6f76 6572  for modules over
│ │ │ │ -0005d650: 2061 2063 6f6d 706c 6574 6520 496e 7465   a complete Inte
│ │ │ │ -0005d660: 7273 6563 7469 6f6e 0a2a 2a2a 2a2a 2a2a  rsection.*******
│ │ │ │ +0005d500: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ +0005d510: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ +0005d520: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ +0005d530: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ +0005d540: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ +0005d550: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ +0005d560: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ +0005d570: 732f 0a43 6f6d 706c 6574 6549 6e74 6572  s/.CompleteInter
│ │ │ │ +0005d580: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +0005d590: 6e73 2e6d 323a 3330 3936 3a30 2e0a 1f0a  ns.m2:3096:0....
│ │ │ │ +0005d5a0: 4669 6c65 3a20 436f 6d70 6c65 7465 496e  File: CompleteIn
│ │ │ │ +0005d5b0: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ +0005d5c0: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465  tions.info, Node
│ │ │ │ +0005d5d0: 3a20 6e65 7745 7874 2c20 4e65 7874 3a20  : newExt, Next: 
│ │ │ │ +0005d5e0: 6f64 6445 7874 4d6f 6475 6c65 2c20 5072  oddExtModule, Pr
│ │ │ │ +0005d5f0: 6576 3a20 6d6f 6475 6c65 4173 4578 742c  ev: moduleAsExt,
│ │ │ │ +0005d600: 2055 703a 2054 6f70 0a0a 6e65 7745 7874   Up: Top..newExt
│ │ │ │ +0005d610: 202d 2d20 476c 6f62 616c 2045 7874 2066   -- Global Ext f
│ │ │ │ +0005d620: 6f72 206d 6f64 756c 6573 206f 7665 7220  or modules over 
│ │ │ │ +0005d630: 6120 636f 6d70 6c65 7465 2049 6e74 6572  a complete Inter
│ │ │ │ +0005d640: 7365 6374 696f 6e0a 2a2a 2a2a 2a2a 2a2a  section.********
│ │ │ │ +0005d650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0005d660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0005d670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005d680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005d690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0005d6a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -0005d6b0: 653a 200a 2020 2020 2020 2020 4520 3d20  e: .        E = 
│ │ │ │ -0005d6c0: 6e65 7745 7874 284d 2c4e 290a 2020 2a20  newExt(M,N).  * 
│ │ │ │ -0005d6d0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ -0005d6e0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ -0005d6f0: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ -0005d700: 294d 6f64 756c 652c 2c20 6f76 6572 2061  )Module,, over a
│ │ │ │ -0005d710: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -0005d720: 6563 7469 6f6e 0a20 2020 2020 2020 2052  ection.        R
│ │ │ │ -0005d730: 6261 720a 2020 2020 2020 2a20 4e2c 2061  bar.      * N, a
│ │ │ │ -0005d740: 202a 6e6f 7465 206d 6f64 756c 653a 2028   *note module: (
│ │ │ │ -0005d750: 4d61 6361 756c 6179 3244 6f63 294d 6f64  Macaulay2Doc)Mod
│ │ │ │ -0005d760: 756c 652c 2c20 6f76 6572 2052 6261 720a  ule,, over Rbar.
│ │ │ │ -0005d770: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ -0005d780: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
│ │ │ │ -0005d790: 756c 6179 3244 6f63 2975 7369 6e67 2066  ulay2Doc)using f
│ │ │ │ -0005d7a0: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ -0005d7b0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ -0005d7c0: 2020 2020 2020 2a20 4368 6563 6b20 3d3e        * Check =>
│ │ │ │ -0005d7d0: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ -0005d7e0: 6c75 6520 6661 6c73 650a 2020 2020 2020  lue false.      
│ │ │ │ -0005d7f0: 2a20 4772 6164 696e 6720 3d3e 202e 2e2e  * Grading => ...
│ │ │ │ -0005d800: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -0005d810: 320a 2020 2020 2020 2a20 4c69 6674 203d  2.      * Lift =
│ │ │ │ -0005d820: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -0005d830: 616c 7565 2066 616c 7365 0a20 2020 2020  alue false.     
│ │ │ │ -0005d840: 202a 2056 6172 6961 626c 6573 203d 3e20   * Variables => 
│ │ │ │ -0005d850: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ -0005d860: 7565 2073 0a20 202a 204f 7574 7075 7473  ue s.  * Outputs
│ │ │ │ -0005d870: 3a0a 2020 2020 2020 2a20 452c 2061 202a  :.      * E, a *
│ │ │ │ -0005d880: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0005d890: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0005d8a0: 652c 2c20 6f76 6572 2061 2072 696e 6720  e,, over a ring 
│ │ │ │ -0005d8b0: 5320 6d61 6465 2066 726f 6d20 7269 6e67  S made from ring
│ │ │ │ -0005d8c0: 0a20 2020 2020 2020 2070 7265 7365 6e74  .        present
│ │ │ │ -0005d8d0: 6174 696f 6e20 5262 6172 2077 6974 6820  ation Rbar with 
│ │ │ │ -0005d8e0: 636f 6469 6d20 5262 6172 206e 6577 2076  codim Rbar new v
│ │ │ │ -0005d8f0: 6172 6961 626c 6573 0a0a 4465 7363 7269  ariables..Descri
│ │ │ │ -0005d900: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -0005d910: 3d0a 0a4c 6574 2052 6261 7220 3d20 522f  =..Let Rbar = R/
│ │ │ │ -0005d920: 2866 312e 2e66 6329 2c20 6120 636f 6d70  (f1..fc), a comp
│ │ │ │ -0005d930: 6c65 7465 2069 6e74 6572 7365 6374 696f  lete intersectio
│ │ │ │ -0005d940: 6e20 6f66 2063 6f64 696d 656e 7369 6f6e  n of codimension
│ │ │ │ -0005d950: 2063 2c20 616e 6420 6c65 7420 4d2c 4e20   c, and let M,N 
│ │ │ │ -0005d960: 6265 0a52 6261 722d 6d6f 6475 6c65 732e  be.Rbar-modules.
│ │ │ │ -0005d970: 2057 6520 6173 7375 6d65 2074 6861 7420   We assume that 
│ │ │ │ -0005d980: 7468 6520 7075 7368 466f 7277 6172 6420  the pushForward 
│ │ │ │ -0005d990: 6f66 204d 2074 6f20 5220 6861 7320 6669  of M to R has fi
│ │ │ │ -0005d9a0: 6e69 7465 2066 7265 650a 7265 736f 6c75  nite free.resolu
│ │ │ │ -0005d9b0: 7469 6f6e 2e20 5468 6520 7363 7269 7074  tion. The script
│ │ │ │ -0005d9c0: 2074 6865 6e20 636f 6d70 7574 6573 2074   then computes t
│ │ │ │ -0005d9d0: 6865 2074 6f74 616c 2045 7874 284d 2c4e  he total Ext(M,N
│ │ │ │ -0005d9e0: 2920 6173 2061 206d 6f64 756c 6520 6f76  ) as a module ov
│ │ │ │ -0005d9f0: 6572 2053 203d 0a6b 6b28 735f 312e 2e73  er S =.kk(s_1..s
│ │ │ │ -0005da00: 5f63 2c67 656e 7320 5229 2c20 7573 696e  _c,gens R), usin
│ │ │ │ -0005da10: 6720 4569 7365 6e62 7564 5368 616d 6173  g EisenbudShamas
│ │ │ │ -0005da20: 6854 6f74 616c 2e0a 0a49 6620 4368 6563  hTotal...If Chec
│ │ │ │ -0005da30: 6b20 3d3e 2074 7275 652c 2074 6865 6e20  k => true, then 
│ │ │ │ -0005da40: 7468 6520 7265 7375 6c74 2069 7320 636f  the result is co
│ │ │ │ -0005da50: 6d70 6172 6564 2077 6974 6820 7468 6520  mpared with the 
│ │ │ │ -0005da60: 6275 696c 742d 696e 2067 6c6f 6261 6c20  built-in global 
│ │ │ │ -0005da70: 4578 740a 7772 6974 7465 6e20 6279 2041  Ext.written by A
│ │ │ │ -0005da80: 7672 616d 6f76 2061 6e64 2047 7261 7973  vramov and Grays
│ │ │ │ -0005da90: 6f6e 2028 6275 7420 6e6f 7465 2074 6865  on (but note the
│ │ │ │ -0005daa0: 2064 6966 6665 7265 6e63 652c 2065 7870   difference, exp
│ │ │ │ -0005dab0: 6c61 696e 6564 2062 656c 6f77 292e 0a0a  lained below)...
│ │ │ │ -0005dac0: 4966 204c 6966 7420 3d3e 2066 616c 7365  If Lift => false
│ │ │ │ -0005dad0: 2074 6865 2072 6573 756c 7420 6973 2072   the result is r
│ │ │ │ -0005dae0: 6574 7572 6e65 6420 6f76 6572 2061 6e64  eturned over and
│ │ │ │ -0005daf0: 2065 7874 656e 7369 6f6e 206f 6620 5262   extension of Rb
│ │ │ │ -0005db00: 6172 3b20 6966 204c 6966 7420 3d3e 0a74  ar; if Lift =>.t
│ │ │ │ -0005db10: 7275 6520 7468 6520 7265 7375 6c74 2069  rue the result i
│ │ │ │ -0005db20: 7320 7265 7475 726e 6564 206f 7665 7220  s returned over 
│ │ │ │ -0005db30: 616e 6420 6578 7465 6e73 696f 6e20 6f66  and extension of
│ │ │ │ -0005db40: 2052 2e0a 0a49 6620 4772 6164 696e 6720   R...If Grading 
│ │ │ │ -0005db50: 3d3e 2032 2c20 7468 6520 6465 6661 756c  => 2, the defaul
│ │ │ │ -0005db60: 742c 2074 6865 6e20 7468 6520 7265 7375  t, then the resu
│ │ │ │ -0005db70: 6c74 2069 7320 6269 6772 6164 6564 2028  lt is bigraded (
│ │ │ │ -0005db80: 7468 6973 2069 7320 6e65 6365 7373 6172  this is necessar
│ │ │ │ -0005db90: 790a 7768 656e 2043 6865 636b 3d3e 7472  y.when Check=>tr
│ │ │ │ -0005dba0: 7565 0a0a 5468 6520 6465 6661 756c 7420  ue..The default 
│ │ │ │ -0005dbb0: 5661 7269 6162 6c65 7320 3d3e 2073 796d  Variables => sym
│ │ │ │ -0005dbc0: 626f 6c20 2273 2220 6769 7665 7320 7468  bol "s" gives th
│ │ │ │ -0005dbd0: 6520 6e65 7720 7661 7269 6162 6c65 7320  e new variables 
│ │ │ │ -0005dbe0: 7468 6520 6e61 6d65 2073 5f69 2c0a 693d  the name s_i,.i=
│ │ │ │ -0005dbf0: 302e 2e63 2d31 2e20 286e 6f74 6520 7468  0..c-1. (note th
│ │ │ │ -0005dc00: 6174 2074 6865 2062 7569 6c74 696e 2045  at the builtin E
│ │ │ │ -0005dc10: 7874 2075 7365 7320 585f 312e 2e58 5f63  xt uses X_1..X_c
│ │ │ │ -0005dc20: 2e0a 0a4f 6e20 536f 6d65 2065 7861 6d70  ...On Some examp
│ │ │ │ -0005dc30: 6c65 7320 6e65 7745 7874 2069 7320 6661  les newExt is fa
│ │ │ │ -0005dc40: 7374 6572 2074 6861 6e20 4578 743b 206f  ster than Ext; o
│ │ │ │ -0005dc50: 6e20 6f74 6865 7273 2069 7427 7320 736c  n others it's sl
│ │ │ │ -0005dc60: 6f77 6572 2e0a 0a41 2073 696d 706c 6520  ower...A simple 
│ │ │ │ -0005dc70: 6578 616d 706c 653a 2069 6620 5220 3d20  example: if R = 
│ │ │ │ -0005dc80: 6b5b 785f 312e 2e78 5f6e 5d20 616e 6420  k[x_1..x_n] and 
│ │ │ │ -0005dc90: 4920 6973 2063 6f6e 7461 696e 6564 2069  I is contained i
│ │ │ │ -0005dca0: 6e20 7468 6520 6375 6265 206f 6620 7468  n the cube of th
│ │ │ │ -0005dcb0: 650a 6d61 7869 6d61 6c20 6964 6561 6c2c  e.maximal ideal,
│ │ │ │ -0005dcc0: 2074 6865 6e20 4578 7428 6b2c 6b29 2069   then Ext(k,k) i
│ │ │ │ -0005dcd0: 7320 6120 6672 6565 2053 2f28 785f 312e  s a free S/(x_1.
│ │ │ │ -0005dce0: 2e78 5f6e 2920 3d20 6b5b 735f 302e 2e73  .x_n) = k[s_0..s
│ │ │ │ -0005dcf0: 5f28 632d 3129 5d2d 206d 6f64 756c 650a  _(c-1)]- module.
│ │ │ │ -0005dd00: 7769 7468 2062 696e 6f6d 6961 6c28 6e2c  with binomial(n,
│ │ │ │ -0005dd10: 6929 2067 656e 6572 6174 6f72 7320 696e  i) generators in
│ │ │ │ -0005dd20: 2064 6567 7265 6520 690a 0a2b 2d2d 2d2d   degree i..+----
│ │ │ │ +0005d680: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +0005d690: 3a20 0a20 2020 2020 2020 2045 203d 206e  : .        E = n
│ │ │ │ +0005d6a0: 6577 4578 7428 4d2c 4e29 0a20 202a 2049  ewExt(M,N).  * I
│ │ │ │ +0005d6b0: 6e70 7574 733a 0a20 2020 2020 202a 204d  nputs:.      * M
│ │ │ │ +0005d6c0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65  , a *note module
│ │ │ │ +0005d6d0: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +0005d6e0: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120  Module,, over a 
│ │ │ │ +0005d6f0: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +0005d700: 6374 696f 6e0a 2020 2020 2020 2020 5262  ction.        Rb
│ │ │ │ +0005d710: 6172 0a20 2020 2020 202a 204e 2c20 6120  ar.      * N, a 
│ │ │ │ +0005d720: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d  *note module: (M
│ │ │ │ +0005d730: 6163 6175 6c61 7932 446f 6329 4d6f 6475  acaulay2Doc)Modu
│ │ │ │ +0005d740: 6c65 2c2c 206f 7665 7220 5262 6172 0a20  le,, over Rbar. 
│ │ │ │ +0005d750: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ +0005d760: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ +0005d770: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ +0005d780: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ +0005d790: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ +0005d7a0: 2020 2020 202a 2043 6865 636b 203d 3e20       * Check => 
│ │ │ │ +0005d7b0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +0005d7c0: 7565 2066 616c 7365 0a20 2020 2020 202a  ue false.      *
│ │ │ │ +0005d7d0: 2047 7261 6469 6e67 203d 3e20 2e2e 2e2c   Grading => ...,
│ │ │ │ +0005d7e0: 2064 6566 6175 6c74 2076 616c 7565 2032   default value 2
│ │ │ │ +0005d7f0: 0a20 2020 2020 202a 204c 6966 7420 3d3e  .      * Lift =>
│ │ │ │ +0005d800: 202e 2e2e 2c20 6465 6661 756c 7420 7661   ..., default va
│ │ │ │ +0005d810: 6c75 6520 6661 6c73 650a 2020 2020 2020  lue false.      
│ │ │ │ +0005d820: 2a20 5661 7269 6162 6c65 7320 3d3e 202e  * Variables => .
│ │ │ │ +0005d830: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +0005d840: 6520 730a 2020 2a20 4f75 7470 7574 733a  e s.  * Outputs:
│ │ │ │ +0005d850: 0a20 2020 2020 202a 2045 2c20 6120 2a6e  .      * E, a *n
│ │ │ │ +0005d860: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +0005d870: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +0005d880: 2c2c 206f 7665 7220 6120 7269 6e67 2053  ,, over a ring S
│ │ │ │ +0005d890: 206d 6164 6520 6672 6f6d 2072 696e 670a   made from ring.
│ │ │ │ +0005d8a0: 2020 2020 2020 2020 7072 6573 656e 7461          presenta
│ │ │ │ +0005d8b0: 7469 6f6e 2052 6261 7220 7769 7468 2063  tion Rbar with c
│ │ │ │ +0005d8c0: 6f64 696d 2052 6261 7220 6e65 7720 7661  odim Rbar new va
│ │ │ │ +0005d8d0: 7269 6162 6c65 730a 0a44 6573 6372 6970  riables..Descrip
│ │ │ │ +0005d8e0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +0005d8f0: 0a0a 4c65 7420 5262 6172 203d 2052 2f28  ..Let Rbar = R/(
│ │ │ │ +0005d900: 6631 2e2e 6663 292c 2061 2063 6f6d 706c  f1..fc), a compl
│ │ │ │ +0005d910: 6574 6520 696e 7465 7273 6563 7469 6f6e  ete intersection
│ │ │ │ +0005d920: 206f 6620 636f 6469 6d65 6e73 696f 6e20   of codimension 
│ │ │ │ +0005d930: 632c 2061 6e64 206c 6574 204d 2c4e 2062  c, and let M,N b
│ │ │ │ +0005d940: 650a 5262 6172 2d6d 6f64 756c 6573 2e20  e.Rbar-modules. 
│ │ │ │ +0005d950: 5765 2061 7373 756d 6520 7468 6174 2074  We assume that t
│ │ │ │ +0005d960: 6865 2070 7573 6846 6f72 7761 7264 206f  he pushForward o
│ │ │ │ +0005d970: 6620 4d20 746f 2052 2068 6173 2066 696e  f M to R has fin
│ │ │ │ +0005d980: 6974 6520 6672 6565 0a72 6573 6f6c 7574  ite free.resolut
│ │ │ │ +0005d990: 696f 6e2e 2054 6865 2073 6372 6970 7420  ion. The script 
│ │ │ │ +0005d9a0: 7468 656e 2063 6f6d 7075 7465 7320 7468  then computes th
│ │ │ │ +0005d9b0: 6520 746f 7461 6c20 4578 7428 4d2c 4e29  e total Ext(M,N)
│ │ │ │ +0005d9c0: 2061 7320 6120 6d6f 6475 6c65 206f 7665   as a module ove
│ │ │ │ +0005d9d0: 7220 5320 3d0a 6b6b 2873 5f31 2e2e 735f  r S =.kk(s_1..s_
│ │ │ │ +0005d9e0: 632c 6765 6e73 2052 292c 2075 7369 6e67  c,gens R), using
│ │ │ │ +0005d9f0: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ +0005da00: 546f 7461 6c2e 0a0a 4966 2043 6865 636b  Total...If Check
│ │ │ │ +0005da10: 203d 3e20 7472 7565 2c20 7468 656e 2074   => true, then t
│ │ │ │ +0005da20: 6865 2072 6573 756c 7420 6973 2063 6f6d  he result is com
│ │ │ │ +0005da30: 7061 7265 6420 7769 7468 2074 6865 2062  pared with the b
│ │ │ │ +0005da40: 7569 6c74 2d69 6e20 676c 6f62 616c 2045  uilt-in global E
│ │ │ │ +0005da50: 7874 0a77 7269 7474 656e 2062 7920 4176  xt.written by Av
│ │ │ │ +0005da60: 7261 6d6f 7620 616e 6420 4772 6179 736f  ramov and Grayso
│ │ │ │ +0005da70: 6e20 2862 7574 206e 6f74 6520 7468 6520  n (but note the 
│ │ │ │ +0005da80: 6469 6666 6572 656e 6365 2c20 6578 706c  difference, expl
│ │ │ │ +0005da90: 6169 6e65 6420 6265 6c6f 7729 2e0a 0a49  ained below)...I
│ │ │ │ +0005daa0: 6620 4c69 6674 203d 3e20 6661 6c73 6520  f Lift => false 
│ │ │ │ +0005dab0: 7468 6520 7265 7375 6c74 2069 7320 7265  the result is re
│ │ │ │ +0005dac0: 7475 726e 6564 206f 7665 7220 616e 6420  turned over and 
│ │ │ │ +0005dad0: 6578 7465 6e73 696f 6e20 6f66 2052 6261  extension of Rba
│ │ │ │ +0005dae0: 723b 2069 6620 4c69 6674 203d 3e0a 7472  r; if Lift =>.tr
│ │ │ │ +0005daf0: 7565 2074 6865 2072 6573 756c 7420 6973  ue the result is
│ │ │ │ +0005db00: 2072 6574 7572 6e65 6420 6f76 6572 2061   returned over a
│ │ │ │ +0005db10: 6e64 2065 7874 656e 7369 6f6e 206f 6620  nd extension of 
│ │ │ │ +0005db20: 522e 0a0a 4966 2047 7261 6469 6e67 203d  R...If Grading =
│ │ │ │ +0005db30: 3e20 322c 2074 6865 2064 6566 6175 6c74  > 2, the default
│ │ │ │ +0005db40: 2c20 7468 656e 2074 6865 2072 6573 756c  , then the resul
│ │ │ │ +0005db50: 7420 6973 2062 6967 7261 6465 6420 2874  t is bigraded (t
│ │ │ │ +0005db60: 6869 7320 6973 206e 6563 6573 7361 7279  his is necessary
│ │ │ │ +0005db70: 0a77 6865 6e20 4368 6563 6b3d 3e74 7275  .when Check=>tru
│ │ │ │ +0005db80: 650a 0a54 6865 2064 6566 6175 6c74 2056  e..The default V
│ │ │ │ +0005db90: 6172 6961 626c 6573 203d 3e20 7379 6d62  ariables => symb
│ │ │ │ +0005dba0: 6f6c 2022 7322 2067 6976 6573 2074 6865  ol "s" gives the
│ │ │ │ +0005dbb0: 206e 6577 2076 6172 6961 626c 6573 2074   new variables t
│ │ │ │ +0005dbc0: 6865 206e 616d 6520 735f 692c 0a69 3d30  he name s_i,.i=0
│ │ │ │ +0005dbd0: 2e2e 632d 312e 2028 6e6f 7465 2074 6861  ..c-1. (note tha
│ │ │ │ +0005dbe0: 7420 7468 6520 6275 696c 7469 6e20 4578  t the builtin Ex
│ │ │ │ +0005dbf0: 7420 7573 6573 2058 5f31 2e2e 585f 632e  t uses X_1..X_c.
│ │ │ │ +0005dc00: 0a0a 4f6e 2053 6f6d 6520 6578 616d 706c  ..On Some exampl
│ │ │ │ +0005dc10: 6573 206e 6577 4578 7420 6973 2066 6173  es newExt is fas
│ │ │ │ +0005dc20: 7465 7220 7468 616e 2045 7874 3b20 6f6e  ter than Ext; on
│ │ │ │ +0005dc30: 206f 7468 6572 7320 6974 2773 2073 6c6f   others it's slo
│ │ │ │ +0005dc40: 7765 722e 0a0a 4120 7369 6d70 6c65 2065  wer...A simple e
│ │ │ │ +0005dc50: 7861 6d70 6c65 3a20 6966 2052 203d 206b  xample: if R = k
│ │ │ │ +0005dc60: 5b78 5f31 2e2e 785f 6e5d 2061 6e64 2049  [x_1..x_n] and I
│ │ │ │ +0005dc70: 2069 7320 636f 6e74 6169 6e65 6420 696e   is contained in
│ │ │ │ +0005dc80: 2074 6865 2063 7562 6520 6f66 2074 6865   the cube of the
│ │ │ │ +0005dc90: 0a6d 6178 696d 616c 2069 6465 616c 2c20  .maximal ideal, 
│ │ │ │ +0005dca0: 7468 656e 2045 7874 286b 2c6b 2920 6973  then Ext(k,k) is
│ │ │ │ +0005dcb0: 2061 2066 7265 6520 532f 2878 5f31 2e2e   a free S/(x_1..
│ │ │ │ +0005dcc0: 785f 6e29 203d 206b 5b73 5f30 2e2e 735f  x_n) = k[s_0..s_
│ │ │ │ +0005dcd0: 2863 2d31 295d 2d20 6d6f 6475 6c65 0a77  (c-1)]- module.w
│ │ │ │ +0005dce0: 6974 6820 6269 6e6f 6d69 616c 286e 2c69  ith binomial(n,i
│ │ │ │ +0005dcf0: 2920 6765 6e65 7261 746f 7273 2069 6e20  ) generators in 
│ │ │ │ +0005dd00: 6465 6772 6565 2069 0a0a 2b2d 2d2d 2d2d  degree i..+-----
│ │ │ │ +0005dd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005dd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dd70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a  ---------+.|i1 :
│ │ │ │ -0005dd80: 206e 203d 2033 3b63 3d32 3b20 2020 2020   n = 3;c=2;     
│ │ │ │ +0005dd50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +0005dd60: 6e20 3d20 333b 633d 323b 2020 2020 2020  n = 3;c=2;      
│ │ │ │ +0005dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ddb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ddc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005dda0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ddc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ddf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005de10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a  ---------+.|i3 :
│ │ │ │ -0005de20: 2052 203d 205a 5a2f 3130 315b 785f 302e   R = ZZ/101[x_0.
│ │ │ │ -0005de30: 2e78 5f28 6e2d 3129 5d20 2020 2020 2020  .x_(n-1)]       
│ │ │ │ -0005de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ddf0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ +0005de00: 5220 3d20 5a5a 2f31 3031 5b78 5f30 2e2e  R = ZZ/101[x_0..
│ │ │ │ +0005de10: 785f 286e 2d31 295d 2020 2020 2020 2020  x_(n-1)]        
│ │ │ │ +0005de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005de40: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005deb0: 2020 2020 2020 2020 207c 0a7c 6f33 203d           |.|o3 =
│ │ │ │ -0005dec0: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0005de90: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20          |.|o3 = 
│ │ │ │ +0005dea0: 5220 2020 2020 2020 2020 2020 2020 2020  R               
│ │ │ │ +0005deb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005dec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ded0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005dee0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005def0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df50: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -0005df60: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ +0005df30: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +0005df40: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +0005df50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005dfa0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005df80: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005dff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a  ---------+.|i4 :
│ │ │ │ -0005e000: 2052 6261 7220 3d20 522f 2869 6465 616c   Rbar = R/(ideal
│ │ │ │ -0005e010: 2061 7070 6c79 2863 2c20 692d 3e20 525f   apply(c, i-> R_
│ │ │ │ -0005e020: 695e 3329 2920 2020 2020 2020 2020 2020  i^3))           
│ │ │ │ +0005dfd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ +0005dfe0: 5262 6172 203d 2052 2f28 6964 6561 6c20  Rbar = R/(ideal 
│ │ │ │ +0005dff0: 6170 706c 7928 632c 2069 2d3e 2052 5f69  apply(c, i-> R_i
│ │ │ │ +0005e000: 5e33 2929 2020 2020 2020 2020 2020 2020  ^3))            
│ │ │ │ +0005e010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e020: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005e030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e040: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e090: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ -0005e0a0: 2052 6261 7220 2020 2020 2020 2020 2020   Rbar           
│ │ │ │ +0005e070: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20          |.|o4 = 
│ │ │ │ +0005e080: 5262 6172 2020 2020 2020 2020 2020 2020  Rbar            
│ │ │ │ +0005e090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e0c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e0e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e130: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ -0005e140: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ +0005e110: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +0005e120: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +0005e130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e180: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005e160: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e1d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -0005e1e0: 204d 6261 7220 3d20 4e62 6172 203d 2063   Mbar = Nbar = c
│ │ │ │ -0005e1f0: 6f6b 6572 2076 6172 7320 5262 6172 2020  oker vars Rbar  
│ │ │ │ -0005e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e1b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20  --------+.|i5 : 
│ │ │ │ +0005e1c0: 4d62 6172 203d 204e 6261 7220 3d20 636f  Mbar = Nbar = co
│ │ │ │ +0005e1d0: 6b65 7220 7661 7273 2052 6261 7220 2020  ker vars Rbar   
│ │ │ │ +0005e1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e200: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e220: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e270: 2020 2020 2020 2020 207c 0a7c 6f35 203d           |.|o5 =
│ │ │ │ -0005e280: 2063 6f6b 6572 6e65 6c20 7c20 785f 3020   cokernel | x_0 
│ │ │ │ -0005e290: 785f 3120 785f 3220 7c20 2020 2020 2020  x_1 x_2 |       
│ │ │ │ -0005e2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e250: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20          |.|o5 = 
│ │ │ │ +0005e260: 636f 6b65 726e 656c 207c 2078 5f30 2078  cokernel | x_0 x
│ │ │ │ +0005e270: 5f31 2078 5f32 207c 2020 2020 2020 2020  _1 x_2 |        
│ │ │ │ +0005e280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e2a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005e2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e2c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e2f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e310: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e310: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │  0005e320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e330: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -0005e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e360: 2020 2020 2020 2020 207c 0a7c 6f35 203a           |.|o5 :
│ │ │ │ -0005e370: 2052 6261 722d 6d6f 6475 6c65 2c20 7175   Rbar-module, qu
│ │ │ │ -0005e380: 6f74 6965 6e74 206f 6620 5262 6172 2020  otient of Rbar  
│ │ │ │ -0005e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e3b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e340: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20          |.|o5 : 
│ │ │ │ +0005e350: 5262 6172 2d6d 6f64 756c 652c 2071 756f  Rbar-module, quo
│ │ │ │ +0005e360: 7469 656e 7420 6f66 2052 6261 7220 2020  tient of Rbar   
│ │ │ │ +0005e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e390: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e400: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a  ---------+.|i6 :
│ │ │ │ -0005e410: 2045 203d 206e 6577 4578 7428 4d62 6172   E = newExt(Mbar
│ │ │ │ -0005e420: 2c4e 6261 7229 2020 2020 2020 2020 2020  ,Nbar)          
│ │ │ │ -0005e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e3e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20  --------+.|i6 : 
│ │ │ │ +0005e3f0: 4520 3d20 6e65 7745 7874 284d 6261 722c  E = newExt(Mbar,
│ │ │ │ +0005e400: 4e62 6172 2920 2020 2020 2020 2020 2020  Nbar)           
│ │ │ │ +0005e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e430: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e450: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005e450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e4a0: 2020 2020 2020 2020 207c 0a7c 6f36 203d           |.|o6 =
│ │ │ │ -0005e4b0: 2063 6f6b 6572 6e65 6c20 7b30 2c20 307d   cokernel {0, 0}
│ │ │ │ -0005e4c0: 2020 207c 2078 5f32 2078 5f31 2078 5f30     | x_2 x_1 x_0
│ │ │ │ -0005e4d0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e4e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e4f0: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e500: 2020 2020 2020 2020 2020 7b2d 322c 202d            {-2, -
│ │ │ │ -0005e510: 327d 207c 2030 2020 2030 2020 2030 2020  2} | 0   0   0  
│ │ │ │ -0005e520: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e530: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e540: 2030 2020 2078 5f32 207c 0a7c 2020 2020   0   x_2 |.|    
│ │ │ │ -0005e550: 2020 2020 2020 2020 2020 7b2d 322c 202d            {-2, -
│ │ │ │ -0005e560: 327d 207c 2030 2020 2030 2020 2030 2020  2} | 0   0   0  
│ │ │ │ -0005e570: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e580: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e590: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e5a0: 2020 2020 2020 2020 2020 7b2d 322c 202d            {-2, -
│ │ │ │ -0005e5b0: 327d 207c 2030 2020 2030 2020 2030 2020  2} | 0   0   0  
│ │ │ │ -0005e5c0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e5d0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e5e0: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e5f0: 2020 2020 2020 2020 2020 7b2d 312c 202d            {-1, -
│ │ │ │ -0005e600: 317d 207c 2030 2020 2030 2020 2030 2020  1} | 0   0   0  
│ │ │ │ -0005e610: 2078 5f32 2078 5f31 2078 5f30 2030 2020   x_2 x_1 x_0 0  
│ │ │ │ -0005e620: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e630: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e640: 2020 2020 2020 2020 2020 7b2d 312c 202d            {-1, -
│ │ │ │ -0005e650: 317d 207c 2030 2020 2030 2020 2030 2020  1} | 0   0   0  
│ │ │ │ -0005e660: 2030 2020 2030 2020 2030 2020 2078 5f32   0   0   0   x_2
│ │ │ │ -0005e670: 2078 5f31 2078 5f30 2030 2020 2030 2020   x_1 x_0 0   0  
│ │ │ │ -0005e680: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ -0005e690: 2020 2020 2020 2020 2020 7b2d 312c 202d            {-1, -
│ │ │ │ -0005e6a0: 317d 207c 2030 2020 2030 2020 2030 2020  1} | 0   0   0  
│ │ │ │ -0005e6b0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e6c0: 2030 2020 2030 2020 2078 5f32 2078 5f31   0   0   x_2 x_1
│ │ │ │ -0005e6d0: 2078 5f30 2030 2020 207c 0a7c 2020 2020   x_0 0   |.|    
│ │ │ │ -0005e6e0: 2020 2020 2020 2020 2020 7b2d 332c 202d            {-3, -
│ │ │ │ -0005e6f0: 337d 207c 2030 2020 2030 2020 2030 2020  3} | 0   0   0  
│ │ │ │ -0005e700: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e710: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -0005e720: 2030 2020 2030 2020 207c 0a7c 2020 2020   0   0   |.|    
│ │ │ │ +0005e480: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20          |.|o6 = 
│ │ │ │ +0005e490: 636f 6b65 726e 656c 207b 302c 2030 7d20  cokernel {0, 0} 
│ │ │ │ +0005e4a0: 2020 7c20 785f 3220 785f 3120 785f 3020    | x_2 x_1 x_0 
│ │ │ │ +0005e4b0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e4c0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e4d0: 3020 2020 3020 2020 7c0a 7c20 2020 2020  0   0   |.|     
│ │ │ │ +0005e4e0: 2020 2020 2020 2020 207b 2d32 2c20 2d32           {-2, -2
│ │ │ │ +0005e4f0: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005e500: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e510: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e520: 3020 2020 785f 3220 7c0a 7c20 2020 2020  0   x_2 |.|     
│ │ │ │ +0005e530: 2020 2020 2020 2020 207b 2d32 2c20 2d32           {-2, -2
│ │ │ │ +0005e540: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005e550: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e560: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e570: 3020 2020 3020 2020 7c0a 7c20 2020 2020  0   0   |.|     
│ │ │ │ +0005e580: 2020 2020 2020 2020 207b 2d32 2c20 2d32           {-2, -2
│ │ │ │ +0005e590: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005e5a0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e5b0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e5c0: 3020 2020 3020 2020 7c0a 7c20 2020 2020  0   0   |.|     
│ │ │ │ +0005e5d0: 2020 2020 2020 2020 207b 2d31 2c20 2d31           {-1, -1
│ │ │ │ +0005e5e0: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005e5f0: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
│ │ │ │ +0005e600: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e610: 3020 2020 3020 2020 7c0a 7c20 2020 2020  0   0   |.|     
│ │ │ │ +0005e620: 2020 2020 2020 2020 207b 2d31 2c20 2d31           {-1, -1
│ │ │ │ +0005e630: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005e640: 3020 2020 3020 2020 3020 2020 785f 3220  0   0   0   x_2 
│ │ │ │ +0005e650: 785f 3120 785f 3020 3020 2020 3020 2020  x_1 x_0 0   0   
│ │ │ │ +0005e660: 3020 2020 3020 2020 7c0a 7c20 2020 2020  0   0   |.|     
│ │ │ │ +0005e670: 2020 2020 2020 2020 207b 2d31 2c20 2d31           {-1, -1
│ │ │ │ +0005e680: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005e690: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e6a0: 3020 2020 3020 2020 785f 3220 785f 3120  0   0   x_2 x_1 
│ │ │ │ +0005e6b0: 785f 3020 3020 2020 7c0a 7c20 2020 2020  x_0 0   |.|     
│ │ │ │ +0005e6c0: 2020 2020 2020 2020 207b 2d33 2c20 2d33           {-3, -3
│ │ │ │ +0005e6d0: 7d20 7c20 3020 2020 3020 2020 3020 2020  } | 0   0   0   
│ │ │ │ +0005e6e0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e6f0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +0005e700: 3020 2020 3020 2020 7c0a 7c20 2020 2020  0   0   |.|     
│ │ │ │ +0005e710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005e740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e770: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e780: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │ -0005e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e7a0: 2020 2020 2020 2020 202f 205a 5a20 2020           / ZZ   
│ │ │ │ -0005e7b0: 2020 2020 2020 2020 2020 2020 205c 2020               \  
│ │ │ │ -0005e7c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e7d0: 202d 2d2d 5b73 202e 2e73 202c 2078 202e   ---[s ..s , x .
│ │ │ │ -0005e7e0: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │ -0005e7f0: 2020 2020 2020 2020 207c 2d2d 2d5b 7320           |---[s 
│ │ │ │ -0005e800: 2e2e 7320 2c20 7820 2e2e 7820 5d7c 2020  ..s , x ..x ]|  
│ │ │ │ -0005e810: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e820: 2031 3031 2020 3020 2020 3120 2020 3020   101  0   1   0 
│ │ │ │ -0005e830: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0005e840: 2020 2020 2020 2020 207c 3130 3120 2030           |101  0
│ │ │ │ -0005e850: 2020 2031 2020 2030 2020 2032 207c 3820     1   0   2 |8 
│ │ │ │ -0005e860: 2020 2020 2020 2020 207c 0a7c 6f36 203a           |.|o6 :
│ │ │ │ -0005e870: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ -0005e880: 2d2d 2d2d 2d6d 6f64 756c 652c 2071 756f  -----module, quo
│ │ │ │ -0005e890: 7469 656e 7420 6f66 207c 2d2d 2d2d 2d2d  tient of |------
│ │ │ │ -0005e8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 2020  -------------|  
│ │ │ │ -0005e8b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e8c0: 2020 2020 2020 2020 2033 2020 2033 2020           3   3  
│ │ │ │ -0005e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e8e0: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ -0005e8f0: 2020 3320 2020 3320 2020 2020 207c 2020    3   3      |  
│ │ │ │ -0005e900: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e910: 2020 2020 2020 2028 7820 2c20 7820 2920         (x , x ) 
│ │ │ │ -0005e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e930: 2020 2020 2020 2020 207c 2020 2020 2020           |      
│ │ │ │ -0005e940: 2878 202c 2078 2029 2020 2020 207c 2020  (x , x )     |  
│ │ │ │ -0005e950: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005e960: 2020 2020 2020 2020 2030 2020 2031 2020           0   1  
│ │ │ │ -0005e970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005e980: 2020 2020 2020 2020 205c 2020 2020 2020           \      
│ │ │ │ -0005e990: 2020 3020 2020 3120 2020 2020 202f 2020    0   1      /  
│ │ │ │ -0005e9a0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +0005e750: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005e760: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +0005e770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e780: 2020 2020 2020 2020 2f20 5a5a 2020 2020          / ZZ    
│ │ │ │ +0005e790: 2020 2020 2020 2020 2020 2020 5c20 2020              \   
│ │ │ │ +0005e7a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005e7b0: 2d2d 2d5b 7320 2e2e 7320 2c20 7820 2e2e  ---[s ..s , x ..
│ │ │ │ +0005e7c0: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │ +0005e7d0: 2020 2020 2020 2020 7c2d 2d2d 5b73 202e          |---[s .
│ │ │ │ +0005e7e0: 2e73 202c 2078 202e 2e78 205d 7c20 2020  .s , x ..x ]|   
│ │ │ │ +0005e7f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005e800: 3130 3120 2030 2020 2031 2020 2030 2020  101  0   1   0  
│ │ │ │ +0005e810: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0005e820: 2020 2020 2020 2020 7c31 3031 2020 3020          |101  0 
│ │ │ │ +0005e830: 2020 3120 2020 3020 2020 3220 7c38 2020    1   0   2 |8  
│ │ │ │ +0005e840: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20          |.|o6 : 
│ │ │ │ +0005e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005e860: 2d2d 2d2d 6d6f 6475 6c65 2c20 7175 6f74  ----module, quot
│ │ │ │ +0005e870: 6965 6e74 206f 6620 7c2d 2d2d 2d2d 2d2d  ient of |-------
│ │ │ │ +0005e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c20 2020  ------------|   
│ │ │ │ +0005e890: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005e8a0: 2020 2020 2020 2020 3320 2020 3320 2020          3   3   
│ │ │ │ +0005e8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e8c0: 2020 2020 2020 2020 7c20 2020 2020 2020          |       
│ │ │ │ +0005e8d0: 2033 2020 2033 2020 2020 2020 7c20 2020   3   3      |   
│ │ │ │ +0005e8e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005e8f0: 2020 2020 2020 2878 202c 2078 2029 2020        (x , x )  
│ │ │ │ +0005e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e910: 2020 2020 2020 2020 7c20 2020 2020 2028          |      (
│ │ │ │ +0005e920: 7820 2c20 7820 2920 2020 2020 7c20 2020  x , x )     |   
│ │ │ │ +0005e930: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005e940: 2020 2020 2020 2020 3020 2020 3120 2020          0   1   
│ │ │ │ +0005e950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005e960: 2020 2020 2020 2020 5c20 2020 2020 2020          \       
│ │ │ │ +0005e970: 2030 2020 2031 2020 2020 2020 2f20 2020   0   1      /   
│ │ │ │ +0005e980: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d          |.|-----
│ │ │ │ +0005e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005e9f0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020  ---------|.|0   
│ │ │ │ -0005ea00: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ea10: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ea20: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ea30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea40: 2020 2020 2020 2020 207c 0a7c 785f 3120           |.|x_1 
│ │ │ │ -0005ea50: 785f 3020 3020 2020 3020 2020 3020 2020  x_0 0   0   0   
│ │ │ │ -0005ea60: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ea70: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ea90: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005eaa0: 3020 2020 785f 3220 785f 3120 785f 3020  0   x_2 x_1 x_0 
│ │ │ │ -0005eab0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eac0: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eae0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005eaf0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eb00: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
│ │ │ │ -0005eb10: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005eb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb30: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005eb40: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eb50: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eb60: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005eb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eb80: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005eb90: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005eba0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ebb0: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ebd0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005ebe0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ebf0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ec00: 3020 2020 3020 2020 7c20 2020 2020 2020  0   0   |       
│ │ │ │ -0005ec10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ec20: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ -0005ec30: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ -0005ec40: 3020 2020 3020 2020 3020 2020 785f 3220  0   0   0   x_2 
│ │ │ │ -0005ec50: 785f 3120 785f 3020 7c20 2020 2020 2020  x_1 x_0 |       
│ │ │ │ -0005ec60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ec70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005e9d0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c30 2020 2030  --------|.|0   0
│ │ │ │ +0005e9e0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005e9f0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005ea00: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +0005ea10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ea20: 2020 2020 2020 2020 7c0a 7c78 5f31 2078          |.|x_1 x
│ │ │ │ +0005ea30: 5f30 2030 2020 2030 2020 2030 2020 2030  _0 0   0   0   0
│ │ │ │ +0005ea40: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005ea50: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +0005ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ea70: 2020 2020 2020 2020 7c0a 7c30 2020 2030          |.|0   0
│ │ │ │ +0005ea80: 2020 2078 5f32 2078 5f31 2078 5f30 2030     x_2 x_1 x_0 0
│ │ │ │ +0005ea90: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005eaa0: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +0005eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eac0: 2020 2020 2020 2020 7c0a 7c30 2020 2030          |.|0   0
│ │ │ │ +0005ead0: 2020 2030 2020 2030 2020 2030 2020 2078     0   0   0   x
│ │ │ │ +0005eae0: 5f32 2078 5f31 2078 5f30 2030 2020 2030  _2 x_1 x_0 0   0
│ │ │ │ +0005eaf0: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +0005eb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eb10: 2020 2020 2020 2020 7c0a 7c30 2020 2030          |.|0   0
│ │ │ │ +0005eb20: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005eb30: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005eb40: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +0005eb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005eb60: 2020 2020 2020 2020 7c0a 7c30 2020 2030          |.|0   0
│ │ │ │ +0005eb70: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005eb80: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005eb90: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +0005eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ebb0: 2020 2020 2020 2020 7c0a 7c30 2020 2030          |.|0   0
│ │ │ │ +0005ebc0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005ebd0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005ebe0: 2020 2030 2020 207c 2020 2020 2020 2020     0   |        
│ │ │ │ +0005ebf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ec00: 2020 2020 2020 2020 7c0a 7c30 2020 2030          |.|0   0
│ │ │ │ +0005ec10: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ +0005ec20: 2020 2030 2020 2030 2020 2078 5f32 2078     0   0   x_2 x
│ │ │ │ +0005ec30: 5f31 2078 5f30 207c 2020 2020 2020 2020  _1 x_0 |        
│ │ │ │ +0005ec40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ec50: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ecc0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a  ---------+.|i7 :
│ │ │ │ -0005ecd0: 2074 616c 6c79 2064 6567 7265 6573 2045   tally degrees E
│ │ │ │ +0005eca0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20  --------+.|i7 : 
│ │ │ │ +0005ecb0: 7461 6c6c 7920 6465 6772 6565 7320 4520  tally degrees E 
│ │ │ │ +0005ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ecf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005ed10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ed30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ed60: 2020 2020 2020 2020 207c 0a7c 6f37 203d           |.|o7 =
│ │ │ │ -0005ed70: 2054 616c 6c79 7b7b 2d31 2c20 2d31 7d20   Tally{{-1, -1} 
│ │ │ │ -0005ed80: 3d3e 2033 7d20 2020 2020 2020 2020 2020  => 3}           
│ │ │ │ -0005ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005edb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005edc0: 2020 2020 2020 207b 2d32 2c20 2d32 7d20         {-2, -2} 
│ │ │ │ -0005edd0: 3d3e 2033 2020 2020 2020 2020 2020 2020  => 3            
│ │ │ │ -0005ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005ee10: 2020 2020 2020 207b 2d33 2c20 2d33 7d20         {-3, -3} 
│ │ │ │ -0005ee20: 3d3e 2031 2020 2020 2020 2020 2020 2020  => 1            
│ │ │ │ -0005ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ee50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005ee60: 2020 2020 2020 207b 302c 2030 7d20 3d3e         {0, 0} =>
│ │ │ │ -0005ee70: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │ -0005ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ed40: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20          |.|o7 = 
│ │ │ │ +0005ed50: 5461 6c6c 797b 7b2d 312c 202d 317d 203d  Tally{{-1, -1} =
│ │ │ │ +0005ed60: 3e20 337d 2020 2020 2020 2020 2020 2020  > 3}            
│ │ │ │ +0005ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ed90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005eda0: 2020 2020 2020 7b2d 322c 202d 327d 203d        {-2, -2} =
│ │ │ │ +0005edb0: 3e20 3320 2020 2020 2020 2020 2020 2020  > 3             
│ │ │ │ +0005edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ede0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005edf0: 2020 2020 2020 7b2d 332c 202d 337d 203d        {-3, -3} =
│ │ │ │ +0005ee00: 3e20 3120 2020 2020 2020 2020 2020 2020  > 1             
│ │ │ │ +0005ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ee30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005ee40: 2020 2020 2020 7b30 2c20 307d 203d 3e20        {0, 0} => 
│ │ │ │ +0005ee50: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0005ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ee80: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eea0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005eef0: 2020 2020 2020 2020 207c 0a7c 6f37 203a           |.|o7 :
│ │ │ │ -0005ef00: 2054 616c 6c79 2020 2020 2020 2020 2020   Tally          
│ │ │ │ +0005eed0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20          |.|o7 : 
│ │ │ │ +0005eee0: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +0005eef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ef00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ef10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ef30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ef40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005ef20: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ef60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ef70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ef80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ef90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a  ---------+.|i8 :
│ │ │ │ -0005efa0: 2061 6e6e 6968 696c 6174 6f72 2045 2020   annihilator E  
│ │ │ │ +0005ef70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20  --------+.|i8 : 
│ │ │ │ +0005ef80: 616e 6e69 6869 6c61 746f 7220 4520 2020  annihilator E   
│ │ │ │ +0005ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005efc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005efe0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f030: 2020 2020 2020 2020 207c 0a7c 6f38 203d           |.|o8 =
│ │ │ │ -0005f040: 2069 6465 616c 2028 7820 2c20 7820 2c20   ideal (x , x , 
│ │ │ │ -0005f050: 7820 2920 2020 2020 2020 2020 2020 2020  x )             
│ │ │ │ -0005f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f080: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f090: 2020 2020 2020 2020 2032 2020 2031 2020           2   1  
│ │ │ │ -0005f0a0: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ -0005f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f010: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20          |.|o8 = 
│ │ │ │ +0005f020: 6964 6561 6c20 2878 202c 2078 202c 2078  ideal (x , x , x
│ │ │ │ +0005f030: 2029 2020 2020 2020 2020 2020 2020 2020   )              
│ │ │ │ +0005f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f060: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005f070: 2020 2020 2020 2020 3220 2020 3120 2020          2   1   
│ │ │ │ +0005f080: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +0005f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f0b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f0d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f120: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f130: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ +0005f100: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005f110: 2020 2020 2020 2020 2020 5a5a 2020 2020            ZZ    
│ │ │ │ +0005f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f170: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f180: 2020 2020 2020 2020 2020 2d2d 2d5b 7320            ---[s 
│ │ │ │ -0005f190: 2e2e 7320 2c20 7820 2e2e 7820 5d20 2020  ..s , x ..x ]   
│ │ │ │ -0005f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f1c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0005f1d0: 2020 2020 2020 2020 2020 3130 3120 2030            101  0
│ │ │ │ -0005f1e0: 2020 2031 2020 2030 2020 2032 2020 2020     1   0   2    
│ │ │ │ -0005f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f210: 2020 2020 2020 2020 207c 0a7c 6f38 203a           |.|o8 :
│ │ │ │ -0005f220: 2049 6465 616c 206f 6620 2d2d 2d2d 2d2d   Ideal of ------
│ │ │ │ -0005f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d20 2020  -------------   
│ │ │ │ -0005f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f150: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005f160: 2020 2020 2020 2020 202d 2d2d 5b73 202e           ---[s .
│ │ │ │ +0005f170: 2e73 202c 2078 202e 2e78 205d 2020 2020  .s , x ..x ]    
│ │ │ │ +0005f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f1a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005f1b0: 2020 2020 2020 2020 2031 3031 2020 3020           101  0 
│ │ │ │ +0005f1c0: 2020 3120 2020 3020 2020 3220 2020 2020    1   0   2     
│ │ │ │ +0005f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f1f0: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20          |.|o8 : 
│ │ │ │ +0005f200: 4964 6561 6c20 6f66 202d 2d2d 2d2d 2d2d  Ideal of -------
│ │ │ │ +0005f210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020  ------------    
│ │ │ │ +0005f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f240: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f260: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f260: 2033 2020 2033 2020 2020 2020 2020 2020   3   3          
│ │ │ │  0005f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f280: 2020 3320 2020 3320 2020 2020 2020 2020    3   3         
│ │ │ │ -0005f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0005f2a0: 2020 2020 2020 2020 2020 2020 2020 2028                 (
│ │ │ │ +0005f2b0: 7820 2c20 7820 2920 2020 2020 2020 2020  x , x )         
│ │ │ │  0005f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f2d0: 2878 202c 2078 2029 2020 2020 2020 2020  (x , x )        
│ │ │ │ -0005f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f2e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0005f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f300: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0005f300: 2030 2020 2031 2020 2020 2020 2020 2020   0   1          
│ │ │ │  0005f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f320: 2020 3020 2020 3120 2020 2020 2020 2020    0   1         
│ │ │ │ -0005f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f350: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0005f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f330: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +0005f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f3a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 416e 2065  ---------+..An e
│ │ │ │ -0005f3b0: 7861 6d70 6c65 2077 6865 7265 2074 6865  xample where the
│ │ │ │ -0005f3c0: 2062 7569 6c74 2d69 6e20 676c 6f62 616c   built-in global
│ │ │ │ -0005f3d0: 2045 7874 2069 7320 6861 7264 2074 6f20   Ext is hard to 
│ │ │ │ -0005f3e0: 636f 6d70 6172 6520 6469 7265 6374 6c79  compare directly
│ │ │ │ -0005f3f0: 2077 6974 6820 6f75 720a 6d65 7468 6f64   with our.method
│ │ │ │ -0005f400: 206f 6620 636f 6d70 7574 6174 696f 6e3a   of computation:
│ │ │ │ -0005f410: 2049 202a 6775 6573 732a 2074 6861 7420   I *guess* that 
│ │ │ │ -0005f420: 7468 6520 7369 676e 2063 686f 6963 6573  the sign choices
│ │ │ │ -0005f430: 2069 6e20 7468 6520 6275 696c 742d 696e   in the built-in
│ │ │ │ -0005f440: 2061 6d6f 756e 740a 6573 7365 6e74 6961   amount.essentia
│ │ │ │ -0005f450: 6c6c 7920 746f 2061 2063 6861 6e67 6520  lly to a change 
│ │ │ │ -0005f460: 6f66 2076 6172 6961 626c 6520 696e 2074  of variable in t
│ │ │ │ -0005f470: 6865 206e 6577 2076 6172 6961 626c 6573  he new variables
│ │ │ │ -0005f480: 2c20 616e 6420 7370 6f69 6c20 616e 2065  , and spoil an e
│ │ │ │ -0005f490: 6173 790a 636f 6d70 6172 6973 6f6e 2e20  asy.comparison. 
│ │ │ │ -0005f4a0: 4275 7420 666f 7220 6578 616d 706c 6520  But for example 
│ │ │ │ -0005f4b0: 7468 6520 6269 2d67 7261 6465 6420 4265  the bi-graded Be
│ │ │ │ -0005f4c0: 7474 6920 6e75 6d62 6572 7320 6172 6520  tti numbers are 
│ │ │ │ -0005f4d0: 6571 7561 6c2e 2074 6869 7320 7365 656d  equal. this seem
│ │ │ │ -0005f4e0: 730a 746f 2073 7461 7274 2077 6974 6820  s.to start with 
│ │ │ │ -0005f4f0: 633d 332e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  c=3...+---------
│ │ │ │ +0005f380: 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6e20 6578  --------+..An ex
│ │ │ │ +0005f390: 616d 706c 6520 7768 6572 6520 7468 6520  ample where the 
│ │ │ │ +0005f3a0: 6275 696c 742d 696e 2067 6c6f 6261 6c20  built-in global 
│ │ │ │ +0005f3b0: 4578 7420 6973 2068 6172 6420 746f 2063  Ext is hard to c
│ │ │ │ +0005f3c0: 6f6d 7061 7265 2064 6972 6563 746c 7920  ompare directly 
│ │ │ │ +0005f3d0: 7769 7468 206f 7572 0a6d 6574 686f 6420  with our.method 
│ │ │ │ +0005f3e0: 6f66 2063 6f6d 7075 7461 7469 6f6e 3a20  of computation: 
│ │ │ │ +0005f3f0: 4920 2a67 7565 7373 2a20 7468 6174 2074  I *guess* that t
│ │ │ │ +0005f400: 6865 2073 6967 6e20 6368 6f69 6365 7320  he sign choices 
│ │ │ │ +0005f410: 696e 2074 6865 2062 7569 6c74 2d69 6e20  in the built-in 
│ │ │ │ +0005f420: 616d 6f75 6e74 0a65 7373 656e 7469 616c  amount.essential
│ │ │ │ +0005f430: 6c79 2074 6f20 6120 6368 616e 6765 206f  ly to a change o
│ │ │ │ +0005f440: 6620 7661 7269 6162 6c65 2069 6e20 7468  f variable in th
│ │ │ │ +0005f450: 6520 6e65 7720 7661 7269 6162 6c65 732c  e new variables,
│ │ │ │ +0005f460: 2061 6e64 2073 706f 696c 2061 6e20 6561   and spoil an ea
│ │ │ │ +0005f470: 7379 0a63 6f6d 7061 7269 736f 6e2e 2042  sy.comparison. B
│ │ │ │ +0005f480: 7574 2066 6f72 2065 7861 6d70 6c65 2074  ut for example t
│ │ │ │ +0005f490: 6865 2062 692d 6772 6164 6564 2042 6574  he bi-graded Bet
│ │ │ │ +0005f4a0: 7469 206e 756d 6265 7273 2061 7265 2065  ti numbers are e
│ │ │ │ +0005f4b0: 7175 616c 2e20 7468 6973 2073 6565 6d73  qual. this seems
│ │ │ │ +0005f4c0: 0a74 6f20 7374 6172 7420 7769 7468 2063  .to start with c
│ │ │ │ +0005f4d0: 3d33 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  =3...+----------
│ │ │ │ +0005f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f540: 2d2d 2d2d 2b0a 7c69 3920 3a20 7365 7452  ----+.|i9 : setR
│ │ │ │ -0005f550: 616e 646f 6d53 6565 6420 3020 2020 2020  andomSeed 0     
│ │ │ │ +0005f520: 2d2d 2d2b 0a7c 6939 203a 2073 6574 5261  ---+.|i9 : setRa
│ │ │ │ +0005f530: 6e64 6f6d 5365 6564 2030 2020 2020 2020  ndomSeed 0      
│ │ │ │ +0005f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f590: 2020 2020 7c0a 7c20 2d2d 2073 6574 7469      |.| -- setti
│ │ │ │ -0005f5a0: 6e67 2072 616e 646f 6d20 7365 6564 2074  ng random seed t
│ │ │ │ -0005f5b0: 6f20 3020 2020 2020 2020 2020 2020 2020  o 0             
│ │ │ │ -0005f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f570: 2020 207c 0a7c 202d 2d20 7365 7474 696e     |.| -- settin
│ │ │ │ +0005f580: 6720 7261 6e64 6f6d 2073 6565 6420 746f  g random seed to
│ │ │ │ +0005f590: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +0005f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f5c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f610: 2020 207c 0a7c 6f39 203d 2030 2020 2020     |.|o9 = 0    
│ │ │ │  0005f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f630: 2020 2020 7c0a 7c6f 3920 3d20 3020 2020      |.|o9 = 0   
│ │ │ │ +0005f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f680: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005f660: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f6d0: 2d2d 2d2d 2b0a 7c69 3130 203a 206e 203d  ----+.|i10 : n =
│ │ │ │ -0005f6e0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0005f6b0: 2d2d 2d2b 0a7c 6931 3020 3a20 6e20 3d20  ---+.|i10 : n = 
│ │ │ │ +0005f6c0: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0005f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f700: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f720: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f750: 2020 207c 0a7c 6f31 3020 3d20 3320 2020     |.|o10 = 3   
│ │ │ │  0005f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f770: 2020 2020 7c0a 7c6f 3130 203d 2033 2020      |.|o10 = 3  
│ │ │ │ +0005f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f7c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005f7a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f810: 2d2d 2d2d 2b0a 7c69 3131 203a 2063 203d  ----+.|i11 : c =
│ │ │ │ -0005f820: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +0005f7f0: 2d2d 2d2b 0a7c 6931 3120 3a20 6320 3d20  ---+.|i11 : c = 
│ │ │ │ +0005f800: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +0005f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f840: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005f850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f860: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005f860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f890: 2020 207c 0a7c 6f31 3120 3d20 3320 2020     |.|o11 = 3   
│ │ │ │  0005f8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8b0: 2020 2020 7c0a 7c6f 3131 203d 2033 2020      |.|o11 = 3  
│ │ │ │ +0005f8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f900: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005f8e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005f8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005f950: 2d2d 2d2d 2b0a 7c69 3132 203a 206b 6b20  ----+.|i12 : kk 
│ │ │ │ -0005f960: 3d20 5a5a 2f31 3031 2020 2020 2020 2020  = ZZ/101        
│ │ │ │ +0005f930: 2d2d 2d2b 0a7c 6931 3220 3a20 6b6b 203d  ---+.|i12 : kk =
│ │ │ │ +0005f940: 205a 5a2f 3130 3120 2020 2020 2020 2020   ZZ/101         
│ │ │ │ +0005f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f980: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005f990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f9a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005f9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005f9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005f9d0: 2020 207c 0a7c 6f31 3220 3d20 6b6b 2020     |.|o12 = kk  
│ │ │ │  0005f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005f9f0: 2020 2020 7c0a 7c6f 3132 203d 206b 6b20      |.|o12 = kk 
│ │ │ │ +0005f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fa20: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa40: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fa90: 2020 2020 7c0a 7c6f 3132 203a 2051 756f      |.|o12 : Quo
│ │ │ │ -0005faa0: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +0005fa70: 2020 207c 0a7c 6f31 3220 3a20 5175 6f74     |.|o12 : Quot
│ │ │ │ +0005fa80: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +0005fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fae0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005fac0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005fad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fb30: 2d2d 2d2d 2b0a 7c69 3133 203a 2052 203d  ----+.|i13 : R =
│ │ │ │ -0005fb40: 206b 6b5b 785f 302e 2e78 5f28 6e2d 3129   kk[x_0..x_(n-1)
│ │ │ │ -0005fb50: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │ -0005fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fb10: 2d2d 2d2b 0a7c 6931 3320 3a20 5220 3d20  ---+.|i13 : R = 
│ │ │ │ +0005fb20: 6b6b 5b78 5f30 2e2e 785f 286e 2d31 295d  kk[x_0..x_(n-1)]
│ │ │ │ +0005fb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fb60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fb80: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fbb0: 2020 207c 0a7c 6f31 3320 3d20 5220 2020     |.|o13 = R   
│ │ │ │  0005fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fbd0: 2020 2020 7c0a 7c6f 3133 203d 2052 2020      |.|o13 = R  
│ │ │ │ +0005fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fc00: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc20: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fc70: 2020 2020 7c0a 7c6f 3133 203a 2050 6f6c      |.|o13 : Pol
│ │ │ │ -0005fc80: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +0005fc50: 2020 207c 0a7c 6f31 3320 3a20 506f 6c79     |.|o13 : Poly
│ │ │ │ +0005fc60: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +0005fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fcc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005fca0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005fcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005fd10: 2d2d 2d2d 2b0a 7c69 3134 203a 2049 203d  ----+.|i14 : I =
│ │ │ │ -0005fd20: 2069 6465 616c 2061 7070 6c79 2863 2c20   ideal apply(c, 
│ │ │ │ -0005fd30: 692d 3e52 5f69 5e32 2920 2020 2020 2020  i->R_i^2)       
│ │ │ │ -0005fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fcf0: 2d2d 2d2b 0a7c 6931 3420 3a20 4920 3d20  ---+.|i14 : I = 
│ │ │ │ +0005fd00: 6964 6561 6c20 6170 706c 7928 632c 2069  ideal apply(c, i
│ │ │ │ +0005fd10: 2d3e 525f 695e 3229 2020 2020 2020 2020  ->R_i^2)        
│ │ │ │ +0005fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fd40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fdb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0005fdc0: 2020 2020 2032 2020 2032 2020 2032 2020       2   2   2  
│ │ │ │ +0005fd90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0005fda0: 2020 2020 3220 2020 3220 2020 3220 2020      2   2   2   
│ │ │ │ +0005fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe00: 2020 2020 7c0a 7c6f 3134 203d 2069 6465      |.|o14 = ide
│ │ │ │ -0005fe10: 616c 2028 7820 2c20 7820 2c20 7820 2920  al (x , x , x ) 
│ │ │ │ +0005fde0: 2020 207c 0a7c 6f31 3420 3d20 6964 6561     |.|o14 = idea
│ │ │ │ +0005fdf0: 6c20 2878 202c 2078 202c 2078 2029 2020  l (x , x , x )  
│ │ │ │ +0005fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0005fe60: 2020 2020 2030 2020 2031 2020 2032 2020       0   1   2  
│ │ │ │ +0005fe30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +0005fe40: 2020 2020 3020 2020 3120 2020 3220 2020      0   1   2   
│ │ │ │ +0005fe50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fe60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fe70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005fe80: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005fef0: 2020 2020 7c0a 7c6f 3134 203a 2049 6465      |.|o14 : Ide
│ │ │ │ -0005ff00: 616c 206f 6620 5220 2020 2020 2020 2020  al of R         
│ │ │ │ +0005fed0: 2020 207c 0a7c 6f31 3420 3a20 4964 6561     |.|o14 : Idea
│ │ │ │ +0005fee0: 6c20 6f66 2052 2020 2020 2020 2020 2020  l of R          
│ │ │ │ +0005fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ff40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0005ff20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0005ff30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0005ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0005ff60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ff70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ff80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0005ff90: 2d2d 2d2d 2b0a 7c69 3135 203a 2066 6620  ----+.|i15 : ff 
│ │ │ │ -0005ffa0: 3d20 6765 6e73 2049 2020 2020 2020 2020  = gens I        
│ │ │ │ +0005ff70: 2d2d 2d2b 0a7c 6931 3520 3a20 6666 203d  ---+.|i15 : ff =
│ │ │ │ +0005ff80: 2067 656e 7320 4920 2020 2020 2020 2020   gens I         
│ │ │ │ +0005ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0005ffc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0005ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0005ffe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0005ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0005fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060030: 2020 2020 7c0a 7c6f 3135 203d 207c 2078      |.|o15 = | x
│ │ │ │ -00060040: 5f30 5e32 2078 5f31 5e32 2078 5f32 5e32  _0^2 x_1^2 x_2^2
│ │ │ │ -00060050: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00060060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060010: 2020 207c 0a7c 6f31 3520 3d20 7c20 785f     |.|o15 = | x_
│ │ │ │ +00060020: 305e 3220 785f 315e 3220 785f 325e 3220  0^2 x_1^2 x_2^2 
│ │ │ │ +00060030: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00060040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060060: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00060070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060080: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000600a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000600b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000600c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000600d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000600e0: 2020 2020 2031 2020 2020 2020 3320 2020       1      3   
│ │ │ │ +000600b0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000600c0: 2020 2020 3120 2020 2020 2033 2020 2020      1      3    
│ │ │ │ +000600d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000600e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000600f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060120: 2020 2020 7c0a 7c6f 3135 203a 204d 6174      |.|o15 : Mat
│ │ │ │ -00060130: 7269 7820 5220 203c 2d2d 2052 2020 2020  rix R  <-- R    
│ │ │ │ +00060100: 2020 207c 0a7c 6f31 3520 3a20 4d61 7472     |.|o15 : Matr
│ │ │ │ +00060110: 6978 2052 2020 3c2d 2d20 5220 2020 2020  ix R  <-- R     
│ │ │ │ +00060120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060170: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00060150: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00060160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00060170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000601a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000601b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000601c0: 2d2d 2d2d 2b0a 7c69 3136 203a 2052 6261  ----+.|i16 : Rba
│ │ │ │ -000601d0: 7220 3d20 522f 4920 2020 2020 2020 2020  r = R/I         
│ │ │ │ +000601a0: 2d2d 2d2b 0a7c 6931 3620 3a20 5262 6172  ---+.|i16 : Rbar
│ │ │ │ +000601b0: 203d 2052 2f49 2020 2020 2020 2020 2020   = R/I          
│ │ │ │ +000601c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000601d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000601e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000601f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000601f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00060200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060210: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060240: 2020 207c 0a7c 6f31 3620 3d20 5262 6172     |.|o16 = Rbar
│ │ │ │  00060250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060260: 2020 2020 7c0a 7c6f 3136 203d 2052 6261      |.|o16 = Rba
│ │ │ │ -00060270: 7220 2020 2020 2020 2020 2020 2020 2020  r               
│ │ │ │ +00060260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000602a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000602b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000602b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000602c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000602d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000602e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000602f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060300: 2020 2020 7c0a 7c6f 3136 203a 2051 756f      |.|o16 : Quo
│ │ │ │ -00060310: 7469 656e 7452 696e 6720 2020 2020 2020  tientRing       
│ │ │ │ +000602e0: 2020 207c 0a7c 6f31 3620 3a20 5175 6f74     |.|o16 : Quot
│ │ │ │ +000602f0: 6965 6e74 5269 6e67 2020 2020 2020 2020  ientRing        
│ │ │ │ +00060300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060350: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00060330: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00060340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00060350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000603a0: 2d2d 2d2d 2b0a 7c69 3137 203a 2062 6172  ----+.|i17 : bar
│ │ │ │ -000603b0: 203d 206d 6170 2852 6261 722c 2052 2920   = map(Rbar, R) 
│ │ │ │ +00060380: 2d2d 2d2b 0a7c 6931 3720 3a20 6261 7220  ---+.|i17 : bar 
│ │ │ │ +00060390: 3d20 6d61 7028 5262 6172 2c20 5229 2020  = map(Rbar, R)  
│ │ │ │ +000603a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000603b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000603c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000603d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000603d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000603e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000603f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000603f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060440: 2020 2020 7c0a 7c6f 3137 203d 206d 6170      |.|o17 = map
│ │ │ │ -00060450: 2028 5262 6172 2c20 522c 207b 7820 2c20   (Rbar, R, {x , 
│ │ │ │ -00060460: 7820 2c20 7820 7d29 2020 2020 2020 2020  x , x })        
│ │ │ │ -00060470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060490: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -000604a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000604b0: 2031 2020 2032 2020 2020 2020 2020 2020   1   2          
│ │ │ │ -000604c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060420: 2020 207c 0a7c 6f31 3720 3d20 6d61 7020     |.|o17 = map 
│ │ │ │ +00060430: 2852 6261 722c 2052 2c20 7b78 202c 2078  (Rbar, R, {x , x
│ │ │ │ +00060440: 202c 2078 207d 2920 2020 2020 2020 2020   , x })         
│ │ │ │ +00060450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060470: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00060480: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00060490: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │ +000604a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000604b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000604c0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000604d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000604e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000604e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000604f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060530: 2020 2020 7c0a 7c6f 3137 203a 2052 696e      |.|o17 : Rin
│ │ │ │ -00060540: 674d 6170 2052 6261 7220 3c2d 2d20 5220  gMap Rbar <-- R 
│ │ │ │ +00060510: 2020 207c 0a7c 6f31 3720 3a20 5269 6e67     |.|o17 : Ring
│ │ │ │ +00060520: 4d61 7020 5262 6172 203c 2d2d 2052 2020  Map Rbar <-- R  
│ │ │ │ +00060530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00060560: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00060570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00060580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000605a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000605b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000605c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000605d0: 2d2d 2d2d 2b0a 7c69 3138 203a 204b 203d  ----+.|i18 : K =
│ │ │ │ -000605e0: 2063 6f6b 6572 2076 6172 7320 5262 6172   coker vars Rbar
│ │ │ │ +000605b0: 2d2d 2d2b 0a7c 6931 3820 3a20 4b20 3d20  ---+.|i18 : K = 
│ │ │ │ +000605c0: 636f 6b65 7220 7661 7273 2052 6261 7220  coker vars Rbar 
│ │ │ │ +000605d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000605e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000605f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060600: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00060610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060620: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060670: 2020 2020 7c0a 7c6f 3138 203d 2063 6f6b      |.|o18 = cok
│ │ │ │ -00060680: 6572 6e65 6c20 7c20 785f 3020 785f 3120  ernel | x_0 x_1 
│ │ │ │ -00060690: 785f 3220 7c20 2020 2020 2020 2020 2020  x_2 |           
│ │ │ │ -000606a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060650: 2020 207c 0a7c 6f31 3820 3d20 636f 6b65     |.|o18 = coke
│ │ │ │ +00060660: 726e 656c 207c 2078 5f30 2078 5f31 2078  rnel | x_0 x_1 x
│ │ │ │ +00060670: 5f32 207c 2020 2020 2020 2020 2020 2020  _2 |            
│ │ │ │ +00060680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000606a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000606b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000606c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +000606c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000606d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000606e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000606f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000606f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00060700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060710: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060710: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │  00060720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060730: 2020 2020 2020 2020 2020 3120 2020 2020            1     
│ │ │ │ -00060740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060760: 2020 2020 7c0a 7c6f 3138 203a 2052 6261      |.|o18 : Rba
│ │ │ │ -00060770: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965  r-module, quotie
│ │ │ │ -00060780: 6e74 206f 6620 5262 6172 2020 2020 2020  nt of Rbar      
│ │ │ │ -00060790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000607a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000607b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00060730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060740: 2020 207c 0a7c 6f31 3820 3a20 5262 6172     |.|o18 : Rbar
│ │ │ │ +00060750: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +00060760: 7420 6f66 2052 6261 7220 2020 2020 2020  t of Rbar       
│ │ │ │ +00060770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060790: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +000607a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000607b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000607c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000607d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000607e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000607f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060800: 2d2d 2d2d 2b0a 7c69 3139 203a 204d 6261  ----+.|i19 : Mba
│ │ │ │ -00060810: 7220 3d20 7072 756e 6520 636f 6b65 7220  r = prune coker 
│ │ │ │ -00060820: 7261 6e64 6f6d 2852 6261 725e 322c 2052  random(Rbar^2, R
│ │ │ │ -00060830: 6261 725e 7b2d 322c 2d32 7d29 2020 2020  bar^{-2,-2})    
│ │ │ │ +000607e0: 2d2d 2d2b 0a7c 6931 3920 3a20 4d62 6172  ---+.|i19 : Mbar
│ │ │ │ +000607f0: 203d 2070 7275 6e65 2063 6f6b 6572 2072   = prune coker r
│ │ │ │ +00060800: 616e 646f 6d28 5262 6172 5e32 2c20 5262  andom(Rbar^2, Rb
│ │ │ │ +00060810: 6172 5e7b 2d32 2c2d 327d 2920 2020 2020  ar^{-2,-2})     
│ │ │ │ +00060820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060830: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00060840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060850: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000608a0: 2020 2020 7c0a 7c6f 3139 203d 2063 6f6b      |.|o19 = cok
│ │ │ │ -000608b0: 6572 6e65 6c20 7c20 785f 3078 5f31 2b31  ernel | x_0x_1+1
│ │ │ │ -000608c0: 3578 5f30 785f 322b 3338 785f 3178 5f32  5x_0x_2+38x_1x_2
│ │ │ │ -000608d0: 2034 3578 5f30 785f 322b 3239 785f 3178   45x_0x_2+29x_1x
│ │ │ │ -000608e0: 5f32 2020 2020 2020 2020 7c20 2020 2020  _2        |     
│ │ │ │ -000608f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00060900: 2020 2020 2020 7c20 3335 785f 3078 5f32        | 35x_0x_2
│ │ │ │ -00060910: 2d33 3078 5f31 785f 3220 2020 2020 2020  -30x_1x_2       
│ │ │ │ -00060920: 2078 5f30 785f 312d 3130 785f 3078 5f32   x_0x_1-10x_0x_2
│ │ │ │ -00060930: 2d32 3278 5f31 785f 3220 7c20 2020 2020  -22x_1x_2 |     
│ │ │ │ -00060940: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060880: 2020 207c 0a7c 6f31 3920 3d20 636f 6b65     |.|o19 = coke
│ │ │ │ +00060890: 726e 656c 207c 2078 5f30 785f 312b 3135  rnel | x_0x_1+15
│ │ │ │ +000608a0: 785f 3078 5f32 2b33 3878 5f31 785f 3220  x_0x_2+38x_1x_2 
│ │ │ │ +000608b0: 3435 785f 3078 5f32 2b32 3978 5f31 785f  45x_0x_2+29x_1x_
│ │ │ │ +000608c0: 3220 2020 2020 2020 207c 2020 2020 2020  2        |      
│ │ │ │ +000608d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000608e0: 2020 2020 207c 2033 3578 5f30 785f 322d       | 35x_0x_2-
│ │ │ │ +000608f0: 3330 785f 3178 5f32 2020 2020 2020 2020  30x_1x_2        
│ │ │ │ +00060900: 785f 3078 5f31 2d31 3078 5f30 785f 322d  x_0x_1-10x_0x_2-
│ │ │ │ +00060910: 3232 785f 3178 5f32 207c 2020 2020 2020  22x_1x_2 |      
│ │ │ │ +00060920: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00060930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060970: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00060980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060990: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060990: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  000609a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609b0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -000609c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000609e0: 2020 2020 7c0a 7c6f 3139 203a 2052 6261      |.|o19 : Rba
│ │ │ │ -000609f0: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965  r-module, quotie
│ │ │ │ -00060a00: 6e74 206f 6620 5262 6172 2020 2020 2020  nt of Rbar      
│ │ │ │ -00060a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000609b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000609c0: 2020 207c 0a7c 6f31 3920 3a20 5262 6172     |.|o19 : Rbar
│ │ │ │ +000609d0: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ +000609e0: 7420 6f66 2052 6261 7220 2020 2020 2020  t of Rbar       
│ │ │ │ +000609f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060a10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00060a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00060a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060a80: 2d2d 2d2d 2b0a 7c69 3230 203a 2045 5320  ----+.|i20 : ES 
│ │ │ │ -00060a90: 3d20 6e65 7745 7874 284d 6261 722c 4b2c  = newExt(Mbar,K,
│ │ │ │ -00060aa0: 4c69 6674 203d 3e20 7472 7565 2920 2020  Lift => true)   
│ │ │ │ -00060ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060a60: 2d2d 2d2b 0a7c 6932 3020 3a20 4553 203d  ---+.|i20 : ES =
│ │ │ │ +00060a70: 206e 6577 4578 7428 4d62 6172 2c4b 2c4c   newExt(Mbar,K,L
│ │ │ │ +00060a80: 6966 7420 3d3e 2074 7275 6529 2020 2020  ift => true)    
│ │ │ │ +00060a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060ab0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00060ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060b20: 2020 2020 7c0a 7c6f 3230 203d 2063 6f6b      |.|o20 = cok
│ │ │ │ -00060b30: 6572 6e65 6c20 7b30 2c20 307d 2020 207c  ernel {0, 0}   |
│ │ │ │ -00060b40: 2078 5f32 2078 5f31 2078 5f30 2030 2020   x_2 x_1 x_0 0  
│ │ │ │ -00060b50: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060b60: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060b70: 2073 5f32 7c0a 7c20 2020 2020 2020 2020   s_2|.|         
│ │ │ │ -00060b80: 2020 2020 2020 7b30 2c20 307d 2020 207c        {0, 0}   |
│ │ │ │ -00060b90: 2030 2020 2030 2020 2030 2020 2078 5f32   0   0   0   x_2
│ │ │ │ -00060ba0: 2078 5f31 2078 5f30 2030 2020 2030 2020   x_1 x_0 0   0  
│ │ │ │ -00060bb0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060bc0: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060bd0: 2020 2020 2020 7b2d 322c 202d 337d 207c        {-2, -3} |
│ │ │ │ -00060be0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060bf0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c00: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c10: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060c20: 2020 2020 2020 7b2d 322c 202d 337d 207c        {-2, -3} |
│ │ │ │ -00060c30: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c40: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c50: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c60: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060c70: 2020 2020 2020 7b2d 322c 202d 337d 207c        {-2, -3} |
│ │ │ │ -00060c80: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060c90: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060ca0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060cb0: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060cc0: 2020 2020 2020 7b2d 322c 202d 337d 207c        {-2, -3} |
│ │ │ │ -00060cd0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060ce0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060cf0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060d00: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060d10: 2020 2020 2020 7b2d 312c 202d 327d 207c        {-1, -2} |
│ │ │ │ -00060d20: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060d30: 2030 2020 2030 2020 2078 5f32 2078 5f31   0   0   x_2 x_1
│ │ │ │ -00060d40: 2078 5f30 2030 2020 2030 2020 2030 2020   x_0 0   0   0  
│ │ │ │ -00060d50: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ -00060d60: 2020 2020 2020 7b2d 312c 202d 327d 207c        {-1, -2} |
│ │ │ │ -00060d70: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060d80: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00060d90: 2030 2020 2078 5f32 2078 5f31 2078 5f30   0   x_2 x_1 x_0
│ │ │ │ -00060da0: 2030 2020 7c0a 7c20 2020 2020 2020 2020   0  |.|         
│ │ │ │ +00060b00: 2020 207c 0a7c 6f32 3020 3d20 636f 6b65     |.|o20 = coke
│ │ │ │ +00060b10: 726e 656c 207b 302c 2030 7d20 2020 7c20  rnel {0, 0}   | 
│ │ │ │ +00060b20: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
│ │ │ │ +00060b30: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060b40: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060b50: 735f 327c 0a7c 2020 2020 2020 2020 2020  s_2|.|          
│ │ │ │ +00060b60: 2020 2020 207b 302c 2030 7d20 2020 7c20       {0, 0}   | 
│ │ │ │ +00060b70: 3020 2020 3020 2020 3020 2020 785f 3220  0   0   0   x_2 
│ │ │ │ +00060b80: 785f 3120 785f 3020 3020 2020 3020 2020  x_1 x_0 0   0   
│ │ │ │ +00060b90: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060ba0: 3020 207c 0a7c 2020 2020 2020 2020 2020  0  |.|          
│ │ │ │ +00060bb0: 2020 2020 207b 2d32 2c20 2d33 7d20 7c20       {-2, -3} | 
│ │ │ │ +00060bc0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060bd0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060be0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060bf0: 3020 207c 0a7c 2020 2020 2020 2020 2020  0  |.|          
│ │ │ │ +00060c00: 2020 2020 207b 2d32 2c20 2d33 7d20 7c20       {-2, -3} | 
│ │ │ │ +00060c10: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060c20: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060c30: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060c40: 3020 207c 0a7c 2020 2020 2020 2020 2020  0  |.|          
│ │ │ │ +00060c50: 2020 2020 207b 2d32 2c20 2d33 7d20 7c20       {-2, -3} | 
│ │ │ │ +00060c60: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060c70: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060c80: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060c90: 3020 207c 0a7c 2020 2020 2020 2020 2020  0  |.|          
│ │ │ │ +00060ca0: 2020 2020 207b 2d32 2c20 2d33 7d20 7c20       {-2, -3} | 
│ │ │ │ +00060cb0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060cc0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060cd0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060ce0: 3020 207c 0a7c 2020 2020 2020 2020 2020  0  |.|          
│ │ │ │ +00060cf0: 2020 2020 207b 2d31 2c20 2d32 7d20 7c20       {-1, -2} | 
│ │ │ │ +00060d00: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060d10: 3020 2020 3020 2020 785f 3220 785f 3120  0   0   x_2 x_1 
│ │ │ │ +00060d20: 785f 3020 3020 2020 3020 2020 3020 2020  x_0 0   0   0   
│ │ │ │ +00060d30: 3020 207c 0a7c 2020 2020 2020 2020 2020  0  |.|          
│ │ │ │ +00060d40: 2020 2020 207b 2d31 2c20 2d32 7d20 7c20       {-1, -2} | 
│ │ │ │ +00060d50: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060d60: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00060d70: 3020 2020 785f 3220 785f 3120 785f 3020  0   x_2 x_1 x_0 
│ │ │ │ +00060d80: 3020 207c 0a7c 2020 2020 2020 2020 2020  0  |.|          
│ │ │ │ +00060d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060dd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00060de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060df0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00060df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00060e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060e30: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │ -00060e40: 2020 2020 7c0a 7c6f 3230 203a 206b 6b5b      |.|o20 : kk[
│ │ │ │ -00060e50: 7320 2e2e 7320 2c20 7820 2e2e 7820 5d2d  s ..s , x ..x ]-
│ │ │ │ -00060e60: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ -00060e70: 206f 6620 286b 6b5b 7320 2e2e 7320 2c20   of (kk[s ..s , 
│ │ │ │ -00060e80: 7820 2e2e 7820 5d29 2020 2020 2020 2020  x ..x ])        
│ │ │ │ -00060e90: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -00060ea0: 2030 2020 2032 2020 2030 2020 2032 2020   0   2   0   2  
│ │ │ │ -00060eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00060ec0: 2020 2020 2020 2020 2030 2020 2032 2020           0   2  
│ │ │ │ -00060ed0: 2030 2020 2032 2020 2020 2020 2020 2020   0   2          
│ │ │ │ -00060ee0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00060e10: 2020 2020 2020 2038 2020 2020 2020 2020         8        
│ │ │ │ +00060e20: 2020 207c 0a7c 6f32 3020 3a20 6b6b 5b73     |.|o20 : kk[s
│ │ │ │ +00060e30: 202e 2e73 202c 2078 202e 2e78 205d 2d6d   ..s , x ..x ]-m
│ │ │ │ +00060e40: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ +00060e50: 6f66 2028 6b6b 5b73 202e 2e73 202c 2078  of (kk[s ..s , x
│ │ │ │ +00060e60: 202e 2e78 205d 2920 2020 2020 2020 2020   ..x ])         
│ │ │ │ +00060e70: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00060e80: 3020 2020 3220 2020 3020 2020 3220 2020  0   2   0   2   
│ │ │ │ +00060e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00060ea0: 2020 2020 2020 2020 3020 2020 3220 2020          0   2   
│ │ │ │ +00060eb0: 3020 2020 3220 2020 2020 2020 2020 2020  0   2           
│ │ │ │ +00060ec0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +00060ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00060ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00060f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00060f30: 2d2d 2d2d 7c0a 7c73 5f31 2073 5f30 2030  ----|.|s_1 s_0 0
│ │ │ │ -00060f40: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060f50: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060f60: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060f70: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00060f80: 2020 2020 7c0a 7c30 2020 2030 2020 2073      |.|0   0   s
│ │ │ │ -00060f90: 5f32 2073 5f31 2073 5f30 2030 2020 2030  _2 s_1 s_0 0   0
│ │ │ │ -00060fa0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060fb0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00060fc0: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00060fd0: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00060fe0: 2020 2030 2020 2030 2020 2078 5f32 2078     0   0   x_2 x
│ │ │ │ -00060ff0: 5f31 2078 5f30 2030 2020 2030 2020 2030  _1 x_0 0   0   0
│ │ │ │ -00061000: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061010: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00061020: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00061030: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061040: 2020 2030 2020 2078 5f32 2078 5f31 2078     0   x_2 x_1 x
│ │ │ │ -00061050: 5f30 2030 2020 2030 2020 2030 2020 2030  _0 0   0   0   0
│ │ │ │ -00061060: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00061070: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00061080: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061090: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -000610a0: 2020 2078 5f32 2078 5f31 2078 5f30 2030     x_2 x_1 x_0 0
│ │ │ │ -000610b0: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -000610c0: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -000610d0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -000610e0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -000610f0: 2020 2030 2020 2030 2020 2030 2020 2078     0   0   0   x
│ │ │ │ -00061100: 5f32 2078 5f31 2078 5f30 2020 2020 2020  _2 x_1 x_0      
│ │ │ │ -00061110: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00061120: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061130: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061140: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061150: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -00061160: 2020 2020 7c0a 7c30 2020 2030 2020 2030      |.|0   0   0
│ │ │ │ -00061170: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061180: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00061190: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -000611a0: 2020 2030 2020 2030 2020 2020 2020 2020     0   0        
│ │ │ │ -000611b0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00060f10: 2d2d 2d7c 0a7c 735f 3120 735f 3020 3020  ---|.|s_1 s_0 0 
│ │ │ │ +00060f20: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060f30: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060f40: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060f50: 2020 3020 2020 3020 2020 2020 2020 2020    0   0         
│ │ │ │ +00060f60: 2020 207c 0a7c 3020 2020 3020 2020 735f     |.|0   0   s_
│ │ │ │ +00060f70: 3220 735f 3120 735f 3020 3020 2020 3020  2 s_1 s_0 0   0 
│ │ │ │ +00060f80: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060f90: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060fa0: 2020 3020 2020 3020 2020 2020 2020 2020    0   0         
│ │ │ │ +00060fb0: 2020 207c 0a7c 3020 2020 3020 2020 3020     |.|0   0   0 
│ │ │ │ +00060fc0: 2020 3020 2020 3020 2020 785f 3220 785f    0   0   x_2 x_
│ │ │ │ +00060fd0: 3120 785f 3020 3020 2020 3020 2020 3020  1 x_0 0   0   0 
│ │ │ │ +00060fe0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00060ff0: 2020 3020 2020 3020 2020 2020 2020 2020    0   0         
│ │ │ │ +00061000: 2020 207c 0a7c 3020 2020 3020 2020 3020     |.|0   0   0 
│ │ │ │ +00061010: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061020: 2020 3020 2020 785f 3220 785f 3120 785f    0   x_2 x_1 x_
│ │ │ │ +00061030: 3020 3020 2020 3020 2020 3020 2020 3020  0 0   0   0   0 
│ │ │ │ +00061040: 2020 3020 2020 3020 2020 2020 2020 2020    0   0         
│ │ │ │ +00061050: 2020 207c 0a7c 3020 2020 3020 2020 3020     |.|0   0   0 
│ │ │ │ +00061060: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061070: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061080: 2020 785f 3220 785f 3120 785f 3020 3020    x_2 x_1 x_0 0 
│ │ │ │ +00061090: 2020 3020 2020 3020 2020 2020 2020 2020    0   0         
│ │ │ │ +000610a0: 2020 207c 0a7c 3020 2020 3020 2020 3020     |.|0   0   0 
│ │ │ │ +000610b0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000610c0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +000610d0: 2020 3020 2020 3020 2020 3020 2020 785f    0   0   0   x_
│ │ │ │ +000610e0: 3220 785f 3120 785f 3020 2020 2020 2020  2 x_1 x_0       
│ │ │ │ +000610f0: 2020 207c 0a7c 3020 2020 3020 2020 3020     |.|0   0   0 
│ │ │ │ +00061100: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061110: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061120: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061130: 2020 3020 2020 3020 2020 2020 2020 2020    0   0         
│ │ │ │ +00061140: 2020 207c 0a7c 3020 2020 3020 2020 3020     |.|0   0   0 
│ │ │ │ +00061150: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061160: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061170: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00061180: 2020 3020 2020 3020 2020 2020 2020 2020    0   0         
│ │ │ │ +00061190: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +000611a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000611b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000611c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000611d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000611e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000611f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061200: 2d2d 2d2d 7c0a 7c30 2020 2020 2020 2020  ----|.|0        
│ │ │ │ -00061210: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061220: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061230: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -00061240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061250: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -00061260: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061270: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061280: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -00061290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000612a0: 2020 2020 7c0a 7c73 5f30 2d31 3173 5f31      |.|s_0-11s_1
│ │ │ │ -000612b0: 2d34 3073 5f32 202d 735f 3120 2020 2020  -40s_2 -s_1     
│ │ │ │ -000612c0: 2020 2020 2020 2039 735f 312d 3233 735f         9s_1-23s_
│ │ │ │ -000612d0: 3220 2020 2020 3137 735f 3120 2020 2020  2     17s_1     
│ │ │ │ -000612e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000612f0: 2020 2020 7c0a 7c34 3573 5f31 2d33 3573      |.|45s_1-35s
│ │ │ │ -00061300: 5f32 2020 2020 2033 3873 5f31 2020 2020  _2     38s_1    
│ │ │ │ -00061310: 2020 2020 2020 2073 5f30 2d37 735f 312b         s_0-7s_1+
│ │ │ │ -00061320: 3135 735f 3220 735f 3120 2020 2020 2020  15s_2 s_1       
│ │ │ │ -00061330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061340: 2020 2020 7c0a 7c2d 3130 735f 312d 3236      |.|-10s_1-26
│ │ │ │ -00061350: 735f 3220 2020 2073 5f30 2b34 3973 5f31  s_2    s_0+49s_1
│ │ │ │ -00061360: 2d34 3073 5f32 2033 3473 5f31 2b34 735f  -40s_2 34s_1+4s_
│ │ │ │ -00061370: 3220 2020 2020 3973 5f31 2d32 3373 5f32  2     9s_1-23s_2
│ │ │ │ -00061380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061390: 2020 2020 7c0a 7c35 3073 5f31 2d73 5f32      |.|50s_1-s_2
│ │ │ │ -000613a0: 2020 2020 2020 2034 3573 5f31 2d33 3573         45s_1-35s
│ │ │ │ -000613b0: 5f32 2020 2020 2031 3073 5f31 2b32 3673  _2     10s_1+26s
│ │ │ │ -000613c0: 5f32 2020 2020 735f 302d 3438 735f 312b  _2    s_0-48s_1+
│ │ │ │ -000613d0: 3135 735f 3220 2020 2020 2020 2020 2020  15s_2           
│ │ │ │ -000613e0: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -000613f0: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061400: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061410: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -00061420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061430: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -00061440: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061450: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -00061460: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -00061470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061480: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +000611e0: 2d2d 2d7c 0a7c 3020 2020 2020 2020 2020  ---|.|0         
│ │ │ │ +000611f0: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00061200: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00061210: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00061220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061230: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │ +00061240: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00061250: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00061260: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00061270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061280: 2020 207c 0a7c 735f 302d 3131 735f 312d     |.|s_0-11s_1-
│ │ │ │ +00061290: 3430 735f 3220 2d73 5f31 2020 2020 2020  40s_2 -s_1      
│ │ │ │ +000612a0: 2020 2020 2020 3973 5f31 2d32 3373 5f32        9s_1-23s_2
│ │ │ │ +000612b0: 2020 2020 2031 3773 5f31 2020 2020 2020       17s_1      
│ │ │ │ +000612c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000612d0: 2020 207c 0a7c 3435 735f 312d 3335 735f     |.|45s_1-35s_
│ │ │ │ +000612e0: 3220 2020 2020 3338 735f 3120 2020 2020  2     38s_1     
│ │ │ │ +000612f0: 2020 2020 2020 735f 302d 3773 5f31 2b31        s_0-7s_1+1
│ │ │ │ +00061300: 3573 5f32 2073 5f31 2020 2020 2020 2020  5s_2 s_1        
│ │ │ │ +00061310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061320: 2020 207c 0a7c 2d31 3073 5f31 2d32 3673     |.|-10s_1-26s
│ │ │ │ +00061330: 5f32 2020 2020 735f 302b 3439 735f 312d  _2    s_0+49s_1-
│ │ │ │ +00061340: 3430 735f 3220 3334 735f 312b 3473 5f32  40s_2 34s_1+4s_2
│ │ │ │ +00061350: 2020 2020 2039 735f 312d 3233 735f 3220       9s_1-23s_2 
│ │ │ │ +00061360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061370: 2020 207c 0a7c 3530 735f 312d 735f 3220     |.|50s_1-s_2 
│ │ │ │ +00061380: 2020 2020 2020 3435 735f 312d 3335 735f        45s_1-35s_
│ │ │ │ +00061390: 3220 2020 2020 3130 735f 312b 3236 735f  2     10s_1+26s_
│ │ │ │ +000613a0: 3220 2020 2073 5f30 2d34 3873 5f31 2b31  2    s_0-48s_1+1
│ │ │ │ +000613b0: 3573 5f32 2020 2020 2020 2020 2020 2020  5s_2            
│ │ │ │ +000613c0: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │ +000613d0: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +000613e0: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +000613f0: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00061400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061410: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │ +00061420: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00061430: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ +00061440: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +00061450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061460: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +00061470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000614a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000614b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000614c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000614d0: 2d2d 2d2d 7c0a 7c30 2020 2020 2020 2020  ----|.|0        
│ │ │ │ +000614b0: 2d2d 2d7c 0a7c 3020 2020 2020 2020 2020  ---|.|0         
│ │ │ │ +000614c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000614d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000614e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000614f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061500: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  00061510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061520: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +00061520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061550: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  00061560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061570: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +00061570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000615a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000615a0: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  000615b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000615c0: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +000615c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000615d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000615e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000615f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000615f0: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  00061600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061610: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +00061610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061640: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  00061650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061660: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +00061660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000616a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000616b0: 2020 2020 7c0a 7c73 5f30 5e32 2b34 3273      |.|s_0^2+42s
│ │ │ │ -000616c0: 5f30 735f 312d 3330 735f 315e 322d 3235  _0s_1-30s_1^2-25
│ │ │ │ -000616d0: 735f 3073 5f32 2d33 3573 5f31 735f 322b  s_0s_2-35s_1s_2+
│ │ │ │ -000616e0: 3973 5f32 5e32 2020 2020 2020 2020 2020  9s_2^2          
│ │ │ │ +00061690: 2020 207c 0a7c 735f 305e 322b 3432 735f     |.|s_0^2+42s_
│ │ │ │ +000616a0: 3073 5f31 2d33 3073 5f31 5e32 2d32 3573  0s_1-30s_1^2-25s
│ │ │ │ +000616b0: 5f30 735f 322d 3335 735f 3173 5f32 2b39  _0s_2-35s_1s_2+9
│ │ │ │ +000616c0: 735f 325e 3220 2020 2020 2020 2020 2020  s_2^2           
│ │ │ │ +000616d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000616e0: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  000616f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061700: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ +00061700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061750: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +00061730: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │ +00061740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000617a0: 2d2d 2d2d 7c0a 7c30 2020 2020 2020 2020  ----|.|0        
│ │ │ │ -000617b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061780: 2d2d 2d7c 0a7c 3020 2020 2020 2020 2020  ---|.|0         
│ │ │ │ +00061790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000617a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000617b0: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │  000617c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000617d0: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +000617d0: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  000617e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000617f0: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -00061800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000617f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061800: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │  00061810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061820: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +00061820: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  00061830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061840: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -00061850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061850: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │  00061860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061870: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +00061870: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  00061880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061890: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -000618a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000618a0: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │  000618b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000618c0: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +000618c0: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  000618d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000618e0: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -000618f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000618e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000618f0: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │  00061900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061910: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +00061910: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  00061920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061930: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -00061940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061940: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │  00061950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061960: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ +00061960: 2020 207c 0a7c 3020 2020 2020 2020 2020     |.|0         
│ │ │ │  00061970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061980: 2020 2020 7c0a 7c30 2020 2020 2020 2020      |.|0        
│ │ │ │ -00061990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061990: 2020 2020 2020 7c20 2020 2020 2020 2020        |         
│ │ │ │  000619a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000619b0: 2020 2020 2020 207c 2020 2020 2020 2020         |        
│ │ │ │ -000619c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000619d0: 2020 2020 7c0a 7c73 5f30 5e32 2b34 3273      |.|s_0^2+42s
│ │ │ │ -000619e0: 5f30 735f 312d 3330 735f 315e 322d 3235  _0s_1-30s_1^2-25
│ │ │ │ -000619f0: 735f 3073 5f32 2d33 3573 5f31 735f 322b  s_0s_2-35s_1s_2+
│ │ │ │ -00061a00: 3973 5f32 5e32 207c 2020 2020 2020 2020  9s_2^2 |        
│ │ │ │ -00061a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061a20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000619b0: 2020 207c 0a7c 735f 305e 322b 3432 735f     |.|s_0^2+42s_
│ │ │ │ +000619c0: 3073 5f31 2d33 3073 5f31 5e32 2d32 3573  0s_1-30s_1^2-25s
│ │ │ │ +000619d0: 5f30 735f 322d 3335 735f 3173 5f32 2b39  _0s_2-35s_1s_2+9
│ │ │ │ +000619e0: 735f 325e 3220 7c20 2020 2020 2020 2020  s_2^2 |         
│ │ │ │ +000619f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061a00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00061a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061a70: 2d2d 2d2d 2b0a 7c69 3231 203a 2053 203d  ----+.|i21 : S =
│ │ │ │ -00061a80: 2072 696e 6720 4553 2020 2020 2020 2020   ring ES        
│ │ │ │ +00061a50: 2d2d 2d2b 0a7c 6932 3120 3a20 5320 3d20  ---+.|i21 : S = 
│ │ │ │ +00061a60: 7269 6e67 2045 5320 2020 2020 2020 2020  ring ES         
│ │ │ │ +00061a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061aa0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00061ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00061ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061af0: 2020 207c 0a7c 6f32 3120 3d20 5320 2020     |.|o21 = S   
│ │ │ │  00061b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b10: 2020 2020 7c0a 7c6f 3231 203d 2053 2020      |.|o21 = S  
│ │ │ │ +00061b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061b40: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00061b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00061b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061bb0: 2020 2020 7c0a 7c6f 3231 203a 2050 6f6c      |.|o21 : Pol
│ │ │ │ -00061bc0: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +00061b90: 2020 207c 0a7c 6f32 3120 3a20 506f 6c79     |.|o21 : Poly
│ │ │ │ +00061ba0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +00061bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061c00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00061be0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00061bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061c50: 2d2d 2d2d 2b0a 0a63 6f6d 7061 7265 2077  ----+..compare w
│ │ │ │ -00061c60: 6974 6820 7468 6520 6275 696c 742d 696e  ith the built-in
│ │ │ │ -00061c70: 2045 7874 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d   Ext..+---------
│ │ │ │ -00061c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061cb0: 2b0a 7c69 3232 203a 2045 4520 3d20 4578  +.|i22 : EE = Ex
│ │ │ │ -00061cc0: 7428 4d62 6172 2c4b 293b 2020 2020 2020  t(Mbar,K);      
│ │ │ │ -00061cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061ce0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00061c30: 2d2d 2d2b 0a0a 636f 6d70 6172 6520 7769  ---+..compare wi
│ │ │ │ +00061c40: 7468 2074 6865 2062 7569 6c74 2d69 6e20  th the built-in 
│ │ │ │ +00061c50: 4578 740a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  Ext..+----------
│ │ │ │ +00061c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00061c90: 0a7c 6932 3220 3a20 4545 203d 2045 7874  .|i22 : EE = Ext
│ │ │ │ +00061ca0: 284d 6261 722c 4b29 3b20 2020 2020 2020  (Mbar,K);       
│ │ │ │ +00061cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061cc0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00061cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061d20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a  --------+.|i23 :
│ │ │ │ -00061d30: 2053 2720 3d20 7269 6e67 2045 4520 2d2d   S' = ring EE --
│ │ │ │ -00061d40: 206e 6f74 6520 7468 6174 2053 2720 6973   note that S' is
│ │ │ │ -00061d50: 2074 6865 2070 6f6c 796e 6f6d 6961 6c20   the polynomial 
│ │ │ │ -00061d60: 7269 6e67 7c0a 7c20 2020 2020 2020 2020  ring|.|         
│ │ │ │ -00061d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061d00: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20  -------+.|i23 : 
│ │ │ │ +00061d10: 5327 203d 2072 696e 6720 4545 202d 2d20  S' = ring EE -- 
│ │ │ │ +00061d20: 6e6f 7465 2074 6861 7420 5327 2069 7320  note that S' is 
│ │ │ │ +00061d30: 7468 6520 706f 6c79 6e6f 6d69 616c 2072  the polynomial r
│ │ │ │ +00061d40: 696e 677c 0a7c 2020 2020 2020 2020 2020  ing|.|          
│ │ │ │ +00061d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00061d80: 0a7c 6f32 3320 3d20 5327 2020 2020 2020  .|o23 = S'      
│ │ │ │  00061d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061da0: 7c0a 7c6f 3233 203d 2053 2720 2020 2020  |.|o23 = S'     
│ │ │ │ -00061db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061db0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00061dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00061dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e10: 2020 2020 2020 2020 7c0a 7c6f 3233 203a          |.|o23 :
│ │ │ │ -00061e20: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ -00061e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061e50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ -00061e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061e90: 2b0a 0a54 6865 2074 776f 2076 6572 7369  +..The two versi
│ │ │ │ -00061ea0: 6f6e 7320 6f66 2045 7874 2061 7070 6561  ons of Ext appea
│ │ │ │ -00061eb0: 7220 746f 2062 6520 7468 6520 7361 6d65  r to be the same
│ │ │ │ -00061ec0: 2075 7020 746f 2063 6861 6e67 6520 6f66   up to change of
│ │ │ │ -00061ed0: 2076 6172 6961 626c 6573 3a0a 0a2b 2d2d   variables:..+--
│ │ │ │ +00061df0: 2020 2020 2020 207c 0a7c 6f32 3320 3a20         |.|o23 : 
│ │ │ │ +00061e00: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +00061e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061e30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00061e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00061e70: 0a0a 5468 6520 7477 6f20 7665 7273 696f  ..The two versio
│ │ │ │ +00061e80: 6e73 206f 6620 4578 7420 6170 7065 6172  ns of Ext appear
│ │ │ │ +00061e90: 2074 6f20 6265 2074 6865 2073 616d 6520   to be the same 
│ │ │ │ +00061ea0: 7570 2074 6f20 6368 616e 6765 206f 6620  up to change of 
│ │ │ │ +00061eb0: 7661 7269 6162 6c65 733a 0a0a 2b2d 2d2d  variables:..+---
│ │ │ │ +00061ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00061ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00061ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00061f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00061f20: 6932 3420 3a20 4120 3d20 6672 6565 5265  i24 : A = freeRe
│ │ │ │ -00061f30: 736f 6c75 7469 6f6e 2045 5320 2020 2020  solution ES     
│ │ │ │ -00061f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00061f60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00061ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00061f00: 3234 203a 2041 203d 2066 7265 6552 6573  24 : A = freeRes
│ │ │ │ +00061f10: 6f6c 7574 696f 6e20 4553 2020 2020 2020  olution ES      
│ │ │ │ +00061f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061f30: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00061f40: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00061f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00061f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00061f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00061fa0: 207c 0a7c 2020 2020 2020 2038 2020 2020   |.|       8    
│ │ │ │ -00061fb0: 2020 3336 2020 2020 2020 3636 2020 2020    36      66    
│ │ │ │ -00061fc0: 2020 3634 2020 2020 2020 3336 2020 2020    64      36    
│ │ │ │ -00061fd0: 2020 3132 2020 2020 2020 3220 2020 2020    12      2     
│ │ │ │ -00061fe0: 2020 207c 0a7c 6f32 3420 3d20 5320 203c     |.|o24 = S  <
│ │ │ │ -00061ff0: 2d2d 2053 2020 203c 2d2d 2053 2020 203c  -- S   <-- S   <
│ │ │ │ -00062000: 2d2d 2053 2020 203c 2d2d 2053 2020 203c  -- S   <-- S   <
│ │ │ │ -00062010: 2d2d 2053 2020 203c 2d2d 2053 2020 2020  -- S   <-- S    
│ │ │ │ -00062020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00061f80: 7c0a 7c20 2020 2020 2020 3820 2020 2020  |.|       8     
│ │ │ │ +00061f90: 2033 3620 2020 2020 2036 3620 2020 2020   36      66     
│ │ │ │ +00061fa0: 2036 3420 2020 2020 2033 3620 2020 2020   64      36     
│ │ │ │ +00061fb0: 2031 3220 2020 2020 2032 2020 2020 2020   12      2      
│ │ │ │ +00061fc0: 2020 7c0a 7c6f 3234 203d 2053 2020 3c2d    |.|o24 = S  <-
│ │ │ │ +00061fd0: 2d20 5320 2020 3c2d 2d20 5320 2020 3c2d  - S   <-- S   <-
│ │ │ │ +00061fe0: 2d20 5320 2020 3c2d 2d20 5320 2020 3c2d  - S   <-- S   <-
│ │ │ │ +00061ff0: 2d20 5320 2020 3c2d 2d20 5320 2020 2020  - S   <-- S     
│ │ │ │ +00062000: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00062010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062060: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00062070: 3020 2020 2020 2031 2020 2020 2020 2032  0      1       2
│ │ │ │ -00062080: 2020 2020 2020 2033 2020 2020 2020 2034         3       4
│ │ │ │ -00062090: 2020 2020 2020 2035 2020 2020 2020 2036         5       6
│ │ │ │ -000620a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00062040: 2020 2020 2020 7c0a 7c20 2020 2020 2030        |.|      0
│ │ │ │ +00062050: 2020 2020 2020 3120 2020 2020 2020 3220        1       2 
│ │ │ │ +00062060: 2020 2020 2020 3320 2020 2020 2020 3420        3       4 
│ │ │ │ +00062070: 2020 2020 2020 3520 2020 2020 2020 3620        5       6 
│ │ │ │ +00062080: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00062090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000620a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000620b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000620c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000620d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000620e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -000620f0: 3420 3a20 436f 6d70 6c65 7820 2020 2020  4 : Complex     
│ │ │ │ -00062100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062120: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +000620c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3234            |.|o24
│ │ │ │ +000620d0: 203a 2043 6f6d 706c 6578 2020 2020 2020   : Complex      
│ │ │ │ +000620e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000620f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062100: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +00062110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00062120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00062170: 0a7c 6932 3520 3a20 4220 3d20 6672 6565  .|i25 : B = free
│ │ │ │ -00062180: 5265 736f 6c75 7469 6f6e 2045 4520 2020  Resolution EE   
│ │ │ │ -00062190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00062150: 7c69 3235 203a 2042 203d 2066 7265 6552  |i25 : B = freeR
│ │ │ │ +00062160: 6573 6f6c 7574 696f 6e20 4545 2020 2020  esolution EE    
│ │ │ │ +00062170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062190: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000621a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +000621b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000621c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000621f0: 2020 207c 0a7c 2020 2020 2020 2020 3820     |.|        8 
│ │ │ │ -00062200: 2020 2020 2020 3336 2020 2020 2020 2036        36       6
│ │ │ │ -00062210: 3620 2020 2020 2020 3634 2020 2020 2020  6       64      
│ │ │ │ -00062220: 2033 3620 2020 2020 2020 3132 2020 2020   36       12    
│ │ │ │ -00062230: 2020 2032 207c 0a7c 6f32 3520 3d20 5327     2 |.|o25 = S'
│ │ │ │ -00062240: 2020 3c2d 2d20 5327 2020 203c 2d2d 2053    <-- S'   <-- S
│ │ │ │ -00062250: 2720 2020 3c2d 2d20 5327 2020 203c 2d2d  '   <-- S'   <--
│ │ │ │ -00062260: 2053 2720 2020 3c2d 2d20 5327 2020 203c   S'   <-- S'   <
│ │ │ │ -00062270: 2d2d 2053 2720 207c 0a7c 2020 2020 2020  -- S'  |.|      
│ │ │ │ +000621d0: 2020 7c0a 7c20 2020 2020 2020 2038 2020    |.|        8  
│ │ │ │ +000621e0: 2020 2020 2033 3620 2020 2020 2020 3636       36       66
│ │ │ │ +000621f0: 2020 2020 2020 2036 3420 2020 2020 2020         64       
│ │ │ │ +00062200: 3336 2020 2020 2020 2031 3220 2020 2020  36       12     
│ │ │ │ +00062210: 2020 3220 7c0a 7c6f 3235 203d 2053 2720    2 |.|o25 = S' 
│ │ │ │ +00062220: 203c 2d2d 2053 2720 2020 3c2d 2d20 5327   <-- S'   <-- S'
│ │ │ │ +00062230: 2020 203c 2d2d 2053 2720 2020 3c2d 2d20     <-- S'   <-- 
│ │ │ │ +00062240: 5327 2020 203c 2d2d 2053 2720 2020 3c2d  S'   <-- S'   <-
│ │ │ │ +00062250: 2d20 5327 2020 7c0a 7c20 2020 2020 2020  - S'  |.|       
│ │ │ │ +00062260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000622a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000622b0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000622c0: 2020 3020 2020 2020 2020 3120 2020 2020    0       1     
│ │ │ │ -000622d0: 2020 2032 2020 2020 2020 2020 3320 2020     2        3   
│ │ │ │ -000622e0: 2020 2020 2034 2020 2020 2020 2020 3520       4        5 
│ │ │ │ -000622f0: 2020 2020 2020 2036 2020 207c 0a7c 2020         6   |.|  
│ │ │ │ +00062290: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +000622a0: 2030 2020 2020 2020 2031 2020 2020 2020   0       1      
│ │ │ │ +000622b0: 2020 3220 2020 2020 2020 2033 2020 2020    2        3    
│ │ │ │ +000622c0: 2020 2020 3420 2020 2020 2020 2035 2020      4        5  
│ │ │ │ +000622d0: 2020 2020 2020 3620 2020 7c0a 7c20 2020        6   |.|   
│ │ │ │ +000622e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000622f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062330: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00062340: 6f32 3520 3a20 436f 6d70 6c65 7820 2020  o25 : Complex   
│ │ │ │ -00062350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00062380: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00062310: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +00062320: 3235 203a 2043 6f6d 706c 6578 2020 2020  25 : Complex    
│ │ │ │ +00062330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062350: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00062360: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00062370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00062380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000623a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000623b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000623c0: 2d2b 0a7c 6932 3620 3a20 616c 6c28 6c65  -+.|i26 : all(le
│ │ │ │ -000623d0: 6e67 7468 2041 2b31 2c20 692d 3e20 736f  ngth A+1, i-> so
│ │ │ │ -000623e0: 7274 2064 6567 7265 6573 2041 5f69 203d  rt degrees A_i =
│ │ │ │ -000623f0: 3d20 736f 7274 2064 6567 7265 6573 2042  = sort degrees B
│ │ │ │ -00062400: 5f69 297c 0a7c 2020 2020 2020 2020 2020  _i)|.|          
│ │ │ │ +000623a0: 2b0a 7c69 3236 203a 2061 6c6c 286c 656e  +.|i26 : all(len
│ │ │ │ +000623b0: 6774 6820 412b 312c 2069 2d3e 2073 6f72  gth A+1, i-> sor
│ │ │ │ +000623c0: 7420 6465 6772 6565 7320 415f 6920 3d3d  t degrees A_i ==
│ │ │ │ +000623d0: 2073 6f72 7420 6465 6772 6565 7320 425f   sort degrees B_
│ │ │ │ +000623e0: 6929 7c0a 7c20 2020 2020 2020 2020 2020  i)|.|           
│ │ │ │ +000623f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062440: 2020 2020 207c 0a7c 6f32 3620 3d20 7472       |.|o26 = tr
│ │ │ │ -00062450: 7565 2020 2020 2020 2020 2020 2020 2020  ue              
│ │ │ │ -00062460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062480: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00062420: 2020 2020 7c0a 7c6f 3236 203d 2074 7275      |.|o26 = tru
│ │ │ │ +00062430: 6520 2020 2020 2020 2020 2020 2020 2020  e               
│ │ │ │ +00062440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062460: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00062470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00062480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000624a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000624b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000624c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 6275 7420  ---------+..but 
│ │ │ │ -000624d0: 7468 6579 2068 6176 6520 6170 7061 7265  they have appare
│ │ │ │ -000624e0: 6e74 6c79 2064 6966 6665 7265 6e74 2061  ntly different a
│ │ │ │ -000624f0: 6e6e 6968 696c 6174 6f72 730a 0a2b 2d2d  nnihilators..+--
│ │ │ │ +000624a0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a62 7574 2074  --------+..but t
│ │ │ │ +000624b0: 6865 7920 6861 7665 2061 7070 6172 656e  hey have apparen
│ │ │ │ +000624c0: 746c 7920 6469 6666 6572 656e 7420 616e  tly different an
│ │ │ │ +000624d0: 6e69 6869 6c61 746f 7273 0a0a 2b2d 2d2d  nihilators..+---
│ │ │ │ +000624e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000624f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062540: 2d2d 2d2b 0a7c 6932 3720 3a20 616e 6e20  ---+.|i27 : ann 
│ │ │ │ -00062550: 4545 2020 2020 2020 2020 2020 2020 2020  EE              
│ │ │ │ -00062560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062520: 2d2d 2b0a 7c69 3237 203a 2061 6e6e 2045  --+.|i27 : ann E
│ │ │ │ +00062530: 4520 2020 2020 2020 2020 2020 2020 2020  E               
│ │ │ │ +00062540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062560: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00062570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062580: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00062580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000625a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000625b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000625c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000625d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000625b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000625c0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000625d0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  000625e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000625f0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00062600: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00062610: 2020 2020 2020 2020 2032 207c 0a7c 6f32           2 |.|o2
│ │ │ │ -00062620: 3720 3d20 6964 6561 6c20 2878 202c 2078  7 = ideal (x , x
│ │ │ │ -00062630: 202c 2078 202c 2058 2020 2b20 3431 5820   , x , X  + 41X 
│ │ │ │ -00062640: 5820 202d 2033 3758 2020 2d20 3134 5820  X  - 37X  - 14X 
│ │ │ │ -00062650: 5820 202d 2032 3958 2058 2020 2b20 3435  X  - 29X X  + 45
│ │ │ │ -00062660: 5820 297c 0a7c 2020 2020 2020 2020 2020  X )|.|          
│ │ │ │ -00062670: 2020 2020 3220 2020 3120 2020 3020 2020      2   1   0   
│ │ │ │ -00062680: 3120 2020 2020 2031 2032 2020 2020 2020  1      1 2      
│ │ │ │ -00062690: 3220 2020 2020 2031 2033 2020 2020 2020  2      1 3      
│ │ │ │ -000626a0: 3220 3320 2020 2020 2033 207c 0a7c 2020  2 3      3 |.|  
│ │ │ │ +000625f0: 2020 2020 2020 2020 3220 7c0a 7c6f 3237          2 |.|o27
│ │ │ │ +00062600: 203d 2069 6465 616c 2028 7820 2c20 7820   = ideal (x , x 
│ │ │ │ +00062610: 2c20 7820 2c20 5820 202b 2034 3158 2058  , x , X  + 41X X
│ │ │ │ +00062620: 2020 2d20 3337 5820 202d 2031 3458 2058    - 37X  - 14X X
│ │ │ │ +00062630: 2020 2d20 3239 5820 5820 202b 2034 3558    - 29X X  + 45X
│ │ │ │ +00062640: 2029 7c0a 7c20 2020 2020 2020 2020 2020   )|.|           
│ │ │ │ +00062650: 2020 2032 2020 2031 2020 2030 2020 2031     2   1   0   1
│ │ │ │ +00062660: 2020 2020 2020 3120 3220 2020 2020 2032        1 2      2
│ │ │ │ +00062670: 2020 2020 2020 3120 3320 2020 2020 2032        1 3      2
│ │ │ │ +00062680: 2033 2020 2020 2020 3320 7c0a 7c20 2020   3      3 |.|   
│ │ │ │ +00062690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000626a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000626b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000626c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000626f0: 2020 207c 0a7c 6f32 3720 3a20 4964 6561     |.|o27 : Idea
│ │ │ │ -00062700: 6c20 6f66 2053 2720 2020 2020 2020 2020  l of S'         
│ │ │ │ -00062710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062730: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000626d0: 2020 7c0a 7c6f 3237 203a 2049 6465 616c    |.|o27 : Ideal
│ │ │ │ +000626e0: 206f 6620 5327 2020 2020 2020 2020 2020   of S'          
│ │ │ │ +000626f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062710: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00062720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00062730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062780: 2d2d 2d2b 0a7c 6932 3820 3a20 616e 6e20  ---+.|i28 : ann 
│ │ │ │ -00062790: 4553 2020 2020 2020 2020 2020 2020 2020  ES              
│ │ │ │ -000627a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062760: 2d2d 2b0a 7c69 3238 203a 2061 6e6e 2045  --+.|i28 : ann E
│ │ │ │ +00062770: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ +00062780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000627a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000627b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000627c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000627c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000627d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000627e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000627f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062810: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000627f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +00062800: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00062810: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00062820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062830: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00062840: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00062850: 2020 2020 2020 2020 3220 207c 0a7c 6f32          2  |.|o2
│ │ │ │ -00062860: 3820 3d20 6964 6561 6c20 2878 202c 2078  8 = ideal (x , x
│ │ │ │ -00062870: 202c 2078 202c 2073 2020 2b20 3432 7320   , x , s  + 42s 
│ │ │ │ -00062880: 7320 202d 2033 3073 2020 2d20 3235 7320  s  - 30s  - 25s 
│ │ │ │ -00062890: 7320 202d 2033 3573 2073 2020 2b20 3973  s  - 35s s  + 9s
│ │ │ │ -000628a0: 2029 207c 0a7c 2020 2020 2020 2020 2020   ) |.|          
│ │ │ │ -000628b0: 2020 2020 3220 2020 3120 2020 3020 2020      2   1   0   
│ │ │ │ -000628c0: 3020 2020 2020 2030 2031 2020 2020 2020  0      0 1      
│ │ │ │ -000628d0: 3120 2020 2020 2030 2032 2020 2020 2020  1      0 2      
│ │ │ │ -000628e0: 3120 3220 2020 2020 3220 207c 0a7c 2020  1 2     2  |.|  
│ │ │ │ +00062830: 2020 2020 2020 2032 2020 7c0a 7c6f 3238         2  |.|o28
│ │ │ │ +00062840: 203d 2069 6465 616c 2028 7820 2c20 7820   = ideal (x , x 
│ │ │ │ +00062850: 2c20 7820 2c20 7320 202b 2034 3273 2073  , x , s  + 42s s
│ │ │ │ +00062860: 2020 2d20 3330 7320 202d 2032 3573 2073    - 30s  - 25s s
│ │ │ │ +00062870: 2020 2d20 3335 7320 7320 202b 2039 7320    - 35s s  + 9s 
│ │ │ │ +00062880: 2920 7c0a 7c20 2020 2020 2020 2020 2020  ) |.|           
│ │ │ │ +00062890: 2020 2032 2020 2031 2020 2030 2020 2030     2   1   0   0
│ │ │ │ +000628a0: 2020 2020 2020 3020 3120 2020 2020 2031        0 1      1
│ │ │ │ +000628b0: 2020 2020 2020 3020 3220 2020 2020 2031        0 2      1
│ │ │ │ +000628c0: 2032 2020 2020 2032 2020 7c0a 7c20 2020   2     2  |.|   
│ │ │ │ +000628d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000628e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000628f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062930: 2020 207c 0a7c 6f32 3820 3a20 4964 6561     |.|o28 : Idea
│ │ │ │ -00062940: 6c20 6f66 2053 2020 2020 2020 2020 2020  l of S          
│ │ │ │ -00062950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062970: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00062910: 2020 7c0a 7c6f 3238 203a 2049 6465 616c    |.|o28 : Ideal
│ │ │ │ +00062920: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ +00062930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062950: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00062960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00062970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000629a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000629b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000629c0: 2d2d 2d2b 0a0a 616e 6420 696e 2066 6163  ---+..and in fac
│ │ │ │ -000629d0: 7420 7468 6579 2061 7265 206e 6f74 2069  t they are not i
│ │ │ │ -000629e0: 736f 6d6f 7270 6869 633a 0a0a 2b2d 2d2d  somorphic:..+---
│ │ │ │ +000629a0: 2d2d 2b0a 0a61 6e64 2069 6e20 6661 6374  --+..and in fact
│ │ │ │ +000629b0: 2074 6865 7920 6172 6520 6e6f 7420 6973   they are not is
│ │ │ │ +000629c0: 6f6d 6f72 7068 6963 3a0a 0a2b 2d2d 2d2d  omorphic:..+----
│ │ │ │ +000629d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000629e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000629f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3239  ----------+.|i29
│ │ │ │ -00062a40: 203a 2045 4574 6f45 5320 3d20 6d61 7028   : EEtoES = map(
│ │ │ │ -00062a50: 7269 6e67 2045 532c 7269 6e67 2045 452c  ring ES,ring EE,
│ │ │ │ -00062a60: 2067 656e 7320 7269 6e67 2045 5329 2020   gens ring ES)  
│ │ │ │ +00062a10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3920  ---------+.|i29 
│ │ │ │ +00062a20: 3a20 4545 746f 4553 203d 206d 6170 2872  : EEtoES = map(r
│ │ │ │ +00062a30: 696e 6720 4553 2c72 696e 6720 4545 2c20  ing ES,ring EE, 
│ │ │ │ +00062a40: 6765 6e73 2072 696e 6720 4553 2920 2020  gens ring ES)   
│ │ │ │ +00062a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062a60: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00062a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062a80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00062a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ad0: 2020 2020 2020 2020 2020 7c0a 7c6f 3239            |.|o29
│ │ │ │ -00062ae0: 203d 206d 6170 2028 532c 2053 272c 207b   = map (S, S', {
│ │ │ │ -00062af0: 7320 2c20 7320 2c20 7320 2c20 7820 2c20  s , s , s , x , 
│ │ │ │ -00062b00: 7820 2c20 7820 7d29 2020 2020 2020 2020  x , x })        
│ │ │ │ +00062ab0: 2020 2020 2020 2020 207c 0a7c 6f32 3920           |.|o29 
│ │ │ │ +00062ac0: 3d20 6d61 7020 2853 2c20 5327 2c20 7b73  = map (S, S', {s
│ │ │ │ +00062ad0: 202c 2073 202c 2073 202c 2078 202c 2078   , s , s , x , x
│ │ │ │ +00062ae0: 202c 2078 207d 2920 2020 2020 2020 2020   , x })         
│ │ │ │ +00062af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062b00: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00062b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062b20: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00062b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062b40: 2030 2020 2031 2020 2032 2020 2030 2020   0   1   2   0  
│ │ │ │ -00062b50: 2031 2020 2032 2020 2020 2020 2020 2020   1   2          
│ │ │ │ +00062b20: 3020 2020 3120 2020 3220 2020 3020 2020  0   1   2   0   
│ │ │ │ +00062b30: 3120 2020 3220 2020 2020 2020 2020 2020  1   2           
│ │ │ │ +00062b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062b50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00062b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062b70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00062b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062bc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3239            |.|o29
│ │ │ │ -00062bd0: 203a 2052 696e 674d 6170 2053 203c 2d2d   : RingMap S <--
│ │ │ │ -00062be0: 2053 2720 2020 2020 2020 2020 2020 2020   S'             
│ │ │ │ -00062bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062c10: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00062ba0: 2020 2020 2020 2020 207c 0a7c 6f32 3920           |.|o29 
│ │ │ │ +00062bb0: 3a20 5269 6e67 4d61 7020 5320 3c2d 2d20  : RingMap S <-- 
│ │ │ │ +00062bc0: 5327 2020 2020 2020 2020 2020 2020 2020  S'              
│ │ │ │ +00062bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062bf0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00062c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00062c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00062c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00062c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3330  ----------+.|i30
│ │ │ │ -00062c70: 203a 2045 4527 203d 2063 6f6b 6572 2045   : EE' = coker E
│ │ │ │ -00062c80: 4574 6f45 5320 7072 6573 656e 7461 7469  EtoES presentati
│ │ │ │ -00062c90: 6f6e 2045 4520 2020 2020 2020 2020 2020  on EE           
│ │ │ │ +00062c40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3020  ---------+.|i30 
│ │ │ │ +00062c50: 3a20 4545 2720 3d20 636f 6b65 7220 4545  : EE' = coker EE
│ │ │ │ +00062c60: 746f 4553 2070 7265 7365 6e74 6174 696f  toES presentatio
│ │ │ │ +00062c70: 6e20 4545 2020 2020 2020 2020 2020 2020  n EE            
│ │ │ │ +00062c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062c90: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00062ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062cb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00062cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062d00: 2020 2020 2020 2020 2020 7c0a 7c6f 3330            |.|o30
│ │ │ │ -00062d10: 203d 2063 6f6b 6572 6e65 6c20 7b30 2c20   = cokernel {0, 
│ │ │ │ -00062d20: 307d 2020 207c 2078 5f32 2078 5f31 2078  0}   | x_2 x_1 x
│ │ │ │ -00062d30: 5f30 2030 2020 2030 2020 2030 2020 2030  _0 0   0   0   0
│ │ │ │ -00062d40: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062d50: 2020 2030 2020 2073 5f32 7c0a 7c20 2020     0   s_2|.|   
│ │ │ │ -00062d60: 2020 2020 2020 2020 2020 2020 7b30 2c20              {0, 
│ │ │ │ -00062d70: 307d 2020 207c 2030 2020 2030 2020 2030  0}   | 0   0   0
│ │ │ │ -00062d80: 2020 2078 5f32 2078 5f31 2078 5f30 2030     x_2 x_1 x_0 0
│ │ │ │ -00062d90: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062da0: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062db0: 2020 2020 2020 2020 2020 2020 7b2d 312c              {-1,
│ │ │ │ -00062dc0: 202d 327d 207c 2030 2020 2030 2020 2030   -2} | 0   0   0
│ │ │ │ -00062dd0: 2020 2030 2020 2030 2020 2030 2020 2078     0   0   0   x
│ │ │ │ -00062de0: 5f32 2078 5f31 2078 5f30 2030 2020 2030  _2 x_1 x_0 0   0
│ │ │ │ -00062df0: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062e00: 2020 2020 2020 2020 2020 2020 7b2d 312c              {-1,
│ │ │ │ -00062e10: 202d 327d 207c 2030 2020 2030 2020 2030   -2} | 0   0   0
│ │ │ │ -00062e20: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062e30: 2020 2030 2020 2030 2020 2078 5f32 2078     0   0   x_2 x
│ │ │ │ -00062e40: 5f31 2078 5f30 2030 2020 7c0a 7c20 2020  _1 x_0 0  |.|   
│ │ │ │ -00062e50: 2020 2020 2020 2020 2020 2020 7b2d 322c              {-2,
│ │ │ │ -00062e60: 202d 337d 207c 2030 2020 2030 2020 2030   -3} | 0   0   0
│ │ │ │ -00062e70: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062e80: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062e90: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062ea0: 2020 2020 2020 2020 2020 2020 7b2d 322c              {-2,
│ │ │ │ -00062eb0: 202d 337d 207c 2030 2020 2030 2020 2030   -3} | 0   0   0
│ │ │ │ -00062ec0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062ed0: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062ee0: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062ef0: 2020 2020 2020 2020 2020 2020 7b2d 322c              {-2,
│ │ │ │ -00062f00: 202d 337d 207c 2030 2020 2030 2020 2030   -3} | 0   0   0
│ │ │ │ -00062f10: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062f20: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062f30: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ -00062f40: 2020 2020 2020 2020 2020 2020 7b2d 322c              {-2,
│ │ │ │ -00062f50: 202d 337d 207c 2030 2020 2030 2020 2030   -3} | 0   0   0
│ │ │ │ -00062f60: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062f70: 2020 2030 2020 2030 2020 2030 2020 2030     0   0   0   0
│ │ │ │ -00062f80: 2020 2030 2020 2030 2020 7c0a 7c20 2020     0   0  |.|   
│ │ │ │ +00062ce0: 2020 2020 2020 2020 207c 0a7c 6f33 3020           |.|o30 
│ │ │ │ +00062cf0: 3d20 636f 6b65 726e 656c 207b 302c 2030  = cokernel {0, 0
│ │ │ │ +00062d00: 7d20 2020 7c20 785f 3220 785f 3120 785f  }   | x_2 x_1 x_
│ │ │ │ +00062d10: 3020 3020 2020 3020 2020 3020 2020 3020  0 0   0   0   0 
│ │ │ │ +00062d20: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062d30: 2020 3020 2020 735f 327c 0a7c 2020 2020    0   s_2|.|    
│ │ │ │ +00062d40: 2020 2020 2020 2020 2020 207b 302c 2030             {0, 0
│ │ │ │ +00062d50: 7d20 2020 7c20 3020 2020 3020 2020 3020  }   | 0   0   0 
│ │ │ │ +00062d60: 2020 785f 3220 785f 3120 785f 3020 3020    x_2 x_1 x_0 0 
│ │ │ │ +00062d70: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062d80: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00062d90: 2020 2020 2020 2020 2020 207b 2d31 2c20             {-1, 
│ │ │ │ +00062da0: 2d32 7d20 7c20 3020 2020 3020 2020 3020  -2} | 0   0   0 
│ │ │ │ +00062db0: 2020 3020 2020 3020 2020 3020 2020 785f    0   0   0   x_
│ │ │ │ +00062dc0: 3220 785f 3120 785f 3020 3020 2020 3020  2 x_1 x_0 0   0 
│ │ │ │ +00062dd0: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00062de0: 2020 2020 2020 2020 2020 207b 2d31 2c20             {-1, 
│ │ │ │ +00062df0: 2d32 7d20 7c20 3020 2020 3020 2020 3020  -2} | 0   0   0 
│ │ │ │ +00062e00: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062e10: 2020 3020 2020 3020 2020 785f 3220 785f    0   0   x_2 x_
│ │ │ │ +00062e20: 3120 785f 3020 3020 207c 0a7c 2020 2020  1 x_0 0  |.|    
│ │ │ │ +00062e30: 2020 2020 2020 2020 2020 207b 2d32 2c20             {-2, 
│ │ │ │ +00062e40: 2d33 7d20 7c20 3020 2020 3020 2020 3020  -3} | 0   0   0 
│ │ │ │ +00062e50: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062e60: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062e70: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00062e80: 2020 2020 2020 2020 2020 207b 2d32 2c20             {-2, 
│ │ │ │ +00062e90: 2d33 7d20 7c20 3020 2020 3020 2020 3020  -3} | 0   0   0 
│ │ │ │ +00062ea0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062eb0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062ec0: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00062ed0: 2020 2020 2020 2020 2020 207b 2d32 2c20             {-2, 
│ │ │ │ +00062ee0: 2d33 7d20 7c20 3020 2020 3020 2020 3020  -3} | 0   0   0 
│ │ │ │ +00062ef0: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062f00: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062f10: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00062f20: 2020 2020 2020 2020 2020 207b 2d32 2c20             {-2, 
│ │ │ │ +00062f30: 2d33 7d20 7c20 3020 2020 3020 2020 3020  -3} | 0   0   0 
│ │ │ │ +00062f40: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062f50: 2020 3020 2020 3020 2020 3020 2020 3020    0   0   0   0 
│ │ │ │ +00062f60: 2020 3020 2020 3020 207c 0a7c 2020 2020    0   0  |.|    
│ │ │ │ +00062f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00062fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00062fb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00062fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062fd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00062fd0: 2020 2020 2020 2020 2038 2020 2020 2020           8      
│ │ │ │  00062fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00062ff0: 2020 2020 2020 2020 2020 3820 2020 2020            8     
│ │ │ │ -00063000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063020: 2020 2020 2020 2020 2020 7c0a 7c6f 3330            |.|o30
│ │ │ │ -00063030: 203a 2053 2d6d 6f64 756c 652c 2071 756f   : S-module, quo
│ │ │ │ -00063040: 7469 656e 7420 6f66 2053 2020 2020 2020  tient of S      
│ │ │ │ -00063050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063070: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +00062ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063000: 2020 2020 2020 2020 207c 0a7c 6f33 3020           |.|o30 
│ │ │ │ +00063010: 3a20 532d 6d6f 6475 6c65 2c20 7175 6f74  : S-module, quot
│ │ │ │ +00063020: 6965 6e74 206f 6620 5320 2020 2020 2020  ient of S       
│ │ │ │ +00063030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063050: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +00063060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00063070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000630a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000630b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000630c0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c73 5f31  ----------|.|s_1
│ │ │ │ -000630d0: 2073 5f30 2030 2020 2030 2020 2030 2020   s_0 0   0   0  
│ │ │ │ -000630e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000630f0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063100: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063110: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063120: 2030 2020 2073 5f32 2073 5f31 2073 5f30   0   s_2 s_1 s_0
│ │ │ │ -00063130: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063140: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063150: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063160: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063170: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063180: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063190: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631a0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631b0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -000631c0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631d0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631e0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000631f0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063200: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063210: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063220: 2078 5f32 2078 5f31 2078 5f30 2030 2020   x_2 x_1 x_0 0  
│ │ │ │ -00063230: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063240: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063250: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063260: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063270: 2030 2020 2030 2020 2030 2020 2078 5f32   0   0   0   x_2
│ │ │ │ -00063280: 2078 5f31 2078 5f30 2030 2020 2030 2020   x_1 x_0 0   0  
│ │ │ │ -00063290: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000632a0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -000632b0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000632c0: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -000632d0: 2030 2020 2030 2020 2078 5f32 2078 5f31   0   0   x_2 x_1
│ │ │ │ -000632e0: 2078 5f30 2030 2020 2030 2020 2030 2020   x_0 0   0   0  
│ │ │ │ -000632f0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063300: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063310: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063320: 2030 2020 2030 2020 2030 2020 2030 2020   0   0   0   0  
│ │ │ │ -00063330: 2030 2020 2078 5f32 2078 5f31 2078 5f30   0   x_2 x_1 x_0
│ │ │ │ -00063340: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +000630a0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 735f 3120  ---------|.|s_1 
│ │ │ │ +000630b0: 735f 3020 3020 2020 3020 2020 3020 2020  s_0 0   0   0   
│ │ │ │ +000630c0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000630d0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000630e0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000630f0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +00063100: 3020 2020 735f 3220 735f 3120 735f 3020  0   s_2 s_1 s_0 
│ │ │ │ +00063110: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063120: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063130: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063140: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +00063150: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063160: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063170: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063180: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063190: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +000631a0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000631b0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000631c0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000631d0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000631e0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +000631f0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063200: 785f 3220 785f 3120 785f 3020 3020 2020  x_2 x_1 x_0 0   
│ │ │ │ +00063210: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063220: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063230: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +00063240: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063250: 3020 2020 3020 2020 3020 2020 785f 3220  0   0   0   x_2 
│ │ │ │ +00063260: 785f 3120 785f 3020 3020 2020 3020 2020  x_1 x_0 0   0   
│ │ │ │ +00063270: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063280: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +00063290: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000632a0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000632b0: 3020 2020 3020 2020 785f 3220 785f 3120  0   0   x_2 x_1 
│ │ │ │ +000632c0: 785f 3020 3020 2020 3020 2020 3020 2020  x_0 0   0   0   
│ │ │ │ +000632d0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +000632e0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +000632f0: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063300: 3020 2020 3020 2020 3020 2020 3020 2020  0   0   0   0   
│ │ │ │ +00063310: 3020 2020 785f 3220 785f 3120 785f 3020  0   x_2 x_1 x_0 
│ │ │ │ +00063320: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +00063330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00063340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063390: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c30 2020  ----------|.|0  
│ │ │ │ -000633a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000633b0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000633c0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000633d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000633e0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -000633f0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063400: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063410: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063430: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063440: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063450: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063460: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -00063470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063480: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063490: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000634a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000634b0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ -000634c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000634d0: 2020 2020 2020 2020 2020 7c0a 7c73 5f30            |.|s_0
│ │ │ │ -000634e0: 2d31 3873 5f31 2d33 3273 5f32 202d 3237  -18s_1-32s_2 -27
│ │ │ │ -000634f0: 735f 312b 3235 735f 3220 2020 2034 3273  s_1+25s_2    42s
│ │ │ │ -00063500: 5f31 2020 2020 2020 2020 2020 202d 3232  _1           -22
│ │ │ │ -00063510: 735f 3120 2020 2020 2020 2020 2020 2020  s_1             
│ │ │ │ -00063520: 2020 2020 2020 2020 2020 7c0a 7c32 3373            |.|23s
│ │ │ │ -00063530: 5f31 2d34 3173 5f32 2020 2020 2073 5f30  _1-41s_2     s_0
│ │ │ │ -00063540: 2d34 3273 5f31 2b31 3873 5f32 202d 3435  -42s_1+18s_2 -45
│ │ │ │ -00063550: 735f 3120 2020 2020 2020 2020 202d 3432  s_1          -42
│ │ │ │ -00063560: 735f 3120 2020 2020 2020 2020 2020 2020  s_1             
│ │ │ │ -00063570: 2020 2020 2020 2020 2020 7c0a 7c2d 3432            |.|-42
│ │ │ │ -00063580: 735f 3220 2020 2020 2020 2020 2032 3273  s_2          22s
│ │ │ │ -00063590: 5f32 2020 2020 2020 2020 2020 2073 5f30  _2           s_0
│ │ │ │ -000635a0: 2d31 3873 5f31 2d33 3273 5f32 202d 3237  -18s_1-32s_2 -27
│ │ │ │ -000635b0: 735f 312b 3235 735f 3220 2020 2020 2020  s_1+25s_2       
│ │ │ │ -000635c0: 2020 2020 2020 2020 2020 7c0a 7c34 3573            |.|45s
│ │ │ │ -000635d0: 5f32 2020 2020 2020 2020 2020 2034 3273  _2           42s
│ │ │ │ -000635e0: 5f32 2020 2020 2020 2020 2020 2032 3373  _2           23s
│ │ │ │ -000635f0: 5f31 2d34 3173 5f32 2020 2020 2073 5f30  _1-41s_2     s_0
│ │ │ │ -00063600: 2d34 3273 5f31 2b31 3873 5f32 2020 2020  -42s_1+18s_2    
│ │ │ │ -00063610: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +00063370: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020  ---------|.|0   
│ │ │ │ +00063380: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00063390: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +000633a0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +000633b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000633c0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +000633d0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +000633e0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +000633f0: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00063400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063410: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +00063420: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00063430: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00063440: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00063450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063460: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │ +00063470: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00063480: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +00063490: 2020 2020 2020 2020 2020 2020 3020 2020              0   
│ │ │ │ +000634a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000634b0: 2020 2020 2020 2020 207c 0a7c 735f 302d           |.|s_0-
│ │ │ │ +000634c0: 3138 735f 312d 3332 735f 3220 2d32 3773  18s_1-32s_2 -27s
│ │ │ │ +000634d0: 5f31 2b32 3573 5f32 2020 2020 3432 735f  _1+25s_2    42s_
│ │ │ │ +000634e0: 3120 2020 2020 2020 2020 2020 2d32 3273  1           -22s
│ │ │ │ +000634f0: 5f31 2020 2020 2020 2020 2020 2020 2020  _1              
│ │ │ │ +00063500: 2020 2020 2020 2020 207c 0a7c 3233 735f           |.|23s_
│ │ │ │ +00063510: 312d 3431 735f 3220 2020 2020 735f 302d  1-41s_2     s_0-
│ │ │ │ +00063520: 3432 735f 312b 3138 735f 3220 2d34 3573  42s_1+18s_2 -45s
│ │ │ │ +00063530: 5f31 2020 2020 2020 2020 2020 2d34 3273  _1          -42s
│ │ │ │ +00063540: 5f31 2020 2020 2020 2020 2020 2020 2020  _1              
│ │ │ │ +00063550: 2020 2020 2020 2020 207c 0a7c 2d34 3273           |.|-42s
│ │ │ │ +00063560: 5f32 2020 2020 2020 2020 2020 3232 735f  _2          22s_
│ │ │ │ +00063570: 3220 2020 2020 2020 2020 2020 735f 302d  2           s_0-
│ │ │ │ +00063580: 3138 735f 312d 3332 735f 3220 2d32 3773  18s_1-32s_2 -27s
│ │ │ │ +00063590: 5f31 2b32 3573 5f32 2020 2020 2020 2020  _1+25s_2        
│ │ │ │ +000635a0: 2020 2020 2020 2020 207c 0a7c 3435 735f           |.|45s_
│ │ │ │ +000635b0: 3220 2020 2020 2020 2020 2020 3432 735f  2           42s_
│ │ │ │ +000635c0: 3220 2020 2020 2020 2020 2020 3233 735f  2           23s_
│ │ │ │ +000635d0: 312d 3431 735f 3220 2020 2020 735f 302d  1-41s_2     s_0-
│ │ │ │ +000635e0: 3432 735f 312b 3138 735f 3220 2020 2020  42s_1+18s_2     
│ │ │ │ +000635f0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +00063600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00063610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063660: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c30 2020  ----------|.|0  
│ │ │ │ +00063640: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020  ---------|.|0   
│ │ │ │ +00063650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063690: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  000636a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636b0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +000636b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000636c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000636d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000636f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063700: 2020 2020 2020 2020 2020 7c0a 7c73 5f30            |.|s_0
│ │ │ │ -00063710: 5e32 2b34 3173 5f30 735f 312d 3337 735f  ^2+41s_0s_1-37s_
│ │ │ │ -00063720: 315e 322d 3134 735f 3073 5f32 2d32 3973  1^2-14s_0s_2-29s
│ │ │ │ -00063730: 5f31 735f 322b 3435 735f 325e 3220 2020  _1s_2+45s_2^2   
│ │ │ │ +000636e0: 2020 2020 2020 2020 207c 0a7c 735f 305e           |.|s_0^
│ │ │ │ +000636f0: 322b 3431 735f 3073 5f31 2d33 3773 5f31  2+41s_0s_1-37s_1
│ │ │ │ +00063700: 5e32 2d31 3473 5f30 735f 322d 3239 735f  ^2-14s_0s_2-29s_
│ │ │ │ +00063710: 3173 5f32 2b34 3573 5f32 5e32 2020 2020  1s_2+45s_2^2    
│ │ │ │ +00063720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063730: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063750: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063780: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637a0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +000637a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000637b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000637c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000637d0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  000637e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000637f0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +000637f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063820: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063840: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063870: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063890: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ +00063890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000638a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000638b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000638c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000638d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000638e0: 2020 2020 2020 2020 2020 7c0a 7c2d 2d2d            |.|---
│ │ │ │ +000638c0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d           |.|----
│ │ │ │ +000638d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000638e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000638f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063930: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c30 2020  ----------|.|0  
│ │ │ │ -00063940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063910: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3020 2020  ---------|.|0   
│ │ │ │ +00063920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063940: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │  00063950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063960: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +00063960: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063980: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063990: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │  000639a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000639b0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +000639b0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  000639c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000639d0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -000639e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000639d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000639e0: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │  000639f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a00: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -00063a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a20: 2020 2020 2020 2020 2020 7c0a 7c73 5f30            |.|s_0
│ │ │ │ -00063a30: 5e32 2b34 3173 5f30 735f 312d 3337 735f  ^2+41s_0s_1-37s_
│ │ │ │ -00063a40: 315e 322d 3134 735f 3073 5f32 2d32 3973  1^2-14s_0s_2-29s
│ │ │ │ -00063a50: 5f31 735f 322b 3435 735f 325e 3220 7c20  _1s_2+45s_2^2 | 
│ │ │ │ +00063a00: 2020 2020 2020 2020 207c 0a7c 735f 305e           |.|s_0^
│ │ │ │ +00063a10: 322b 3431 735f 3073 5f31 2d33 3773 5f31  2+41s_0s_1-37s_1
│ │ │ │ +00063a20: 5e32 2d31 3473 5f30 735f 322d 3239 735f  ^2-14s_0s_2-29s_
│ │ │ │ +00063a30: 3173 5f32 2b34 3573 5f32 5e32 207c 2020  1s_2+45s_2^2 |  
│ │ │ │ +00063a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063a50: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063a70: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063a80: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │  00063a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063aa0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +00063aa0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063ac0: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063ad0: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │  00063ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063af0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +00063af0: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b10: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063b20: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │  00063b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b40: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +00063b40: 2020 2020 2020 2020 207c 0a7c 3020 2020           |.|0   
│ │ │ │  00063b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b60: 2020 2020 2020 2020 2020 7c0a 7c30 2020            |.|0  
│ │ │ │ -00063b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063b70: 2020 2020 2020 2020 2020 2020 207c 2020               |  
│ │ │ │  00063b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063b90: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ -00063ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063bb0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00063b90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00063ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00063bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3331  ----------+.|i31
│ │ │ │ -00063c10: 203a 2048 203d 2048 6f6d 2845 4527 2c45   : H = Hom(EE',E
│ │ │ │ -00063c20: 5329 3b20 2020 2020 2020 2020 2020 2020  S);             
│ │ │ │ -00063c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063c50: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00063be0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3120  ---------+.|i31 
│ │ │ │ +00063bf0: 3a20 4820 3d20 486f 6d28 4545 272c 4553  : H = Hom(EE',ES
│ │ │ │ +00063c00: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +00063c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063c30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00063c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00063c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3332  ----------+.|i32
│ │ │ │ -00063cb0: 203a 2051 203d 2070 6f73 6974 696f 6e73   : Q = positions
│ │ │ │ -00063cc0: 2864 6567 7265 6573 2074 6172 6765 7420  (degrees target 
│ │ │ │ -00063cd0: 7072 6573 656e 7461 7469 6f6e 2048 2c20  presentation H, 
│ │ │ │ -00063ce0: 692d 3e20 6920 3d3d 207b 302c 307d 2920  i-> i == {0,0}) 
│ │ │ │ -00063cf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00063c80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3220  ---------+.|i32 
│ │ │ │ +00063c90: 3a20 5120 3d20 706f 7369 7469 6f6e 7328  : Q = positions(
│ │ │ │ +00063ca0: 6465 6772 6565 7320 7461 7267 6574 2070  degrees target p
│ │ │ │ +00063cb0: 7265 7365 6e74 6174 696f 6e20 482c 2069  resentation H, i
│ │ │ │ +00063cc0: 2d3e 2069 203d 3d20 7b30 2c30 7d29 2020  -> i == {0,0})  
│ │ │ │ +00063cd0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00063ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d40: 2020 2020 2020 2020 2020 7c0a 7c6f 3332            |.|o32
│ │ │ │ -00063d50: 203d 207b 382c 2039 2c20 3130 2c20 3131   = {8, 9, 10, 11
│ │ │ │ -00063d60: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ -00063d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063d20: 2020 2020 2020 2020 207c 0a7c 6f33 3220           |.|o32 
│ │ │ │ +00063d30: 3d20 7b38 2c20 392c 2031 302c 2031 317d  = {8, 9, 10, 11}
│ │ │ │ +00063d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063d70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00063d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063d90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00063d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063de0: 2020 2020 2020 2020 2020 7c0a 7c6f 3332            |.|o32
│ │ │ │ -00063df0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00063dc0: 2020 2020 2020 2020 207c 0a7c 6f33 3220           |.|o32 
│ │ │ │ +00063dd0: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +00063de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063e30: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00063e10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00063e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00063e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00063e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00063e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3333  ----------+.|i33
│ │ │ │ -00063e90: 203a 2066 203d 2073 756d 2851 2c20 702d   : f = sum(Q, p-
│ │ │ │ -00063ea0: 3e20 7261 6e64 6f6d 2028 535e 312c 2053  > random (S^1, S
│ │ │ │ -00063eb0: 5e31 292a 2a68 6f6d 6f6d 6f72 7068 6973  ^1)**homomorphis
│ │ │ │ -00063ec0: 6d20 485f 7b70 7d29 2020 2020 2020 2020  m H_{p})        
│ │ │ │ -00063ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00063e60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3320  ---------+.|i33 
│ │ │ │ +00063e70: 3a20 6620 3d20 7375 6d28 512c 2070 2d3e  : f = sum(Q, p->
│ │ │ │ +00063e80: 2072 616e 646f 6d20 2853 5e31 2c20 535e   random (S^1, S^
│ │ │ │ +00063e90: 3129 2a2a 686f 6d6f 6d6f 7270 6869 736d  1)**homomorphism
│ │ │ │ +00063ea0: 2048 5f7b 707d 2920 2020 2020 2020 2020   H_{p})         
│ │ │ │ +00063eb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00063ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00063ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063f20: 2020 2020 2020 2020 2020 7c0a 7c6f 3333            |.|o33
│ │ │ │ -00063f30: 203d 207b 302c 2030 7d20 2020 7c20 2d33   = {0, 0}   | -3
│ │ │ │ -00063f40: 3820 3339 2030 2030 2030 2030 2030 2030  8 39 0 0 0 0 0 0
│ │ │ │ -00063f50: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00063f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063f70: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00063f80: 2020 207b 302c 2030 7d20 2020 7c20 2d31     {0, 0}   | -1
│ │ │ │ -00063f90: 3620 3231 2030 2030 2030 2030 2030 2030  6 21 0 0 0 0 0 0
│ │ │ │ -00063fa0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00063fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00063fc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00063fd0: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020     {-2, -3} | 0 
│ │ │ │ -00063fe0: 2020 3020 2030 2030 2030 2030 2030 2030    0  0 0 0 0 0 0
│ │ │ │ -00063ff0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00064000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064010: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00064020: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020     {-2, -3} | 0 
│ │ │ │ -00064030: 2020 3020 2030 2030 2030 2030 2030 2030    0  0 0 0 0 0 0
│ │ │ │ -00064040: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00064050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064060: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00064070: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020     {-2, -3} | 0 
│ │ │ │ -00064080: 2020 3020 2030 2030 2030 2030 2030 2030    0  0 0 0 0 0 0
│ │ │ │ -00064090: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -000640a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000640b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -000640c0: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020     {-2, -3} | 0 
│ │ │ │ -000640d0: 2020 3020 2030 2030 2030 2030 2030 2030    0  0 0 0 0 0 0
│ │ │ │ -000640e0: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -000640f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064100: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00064110: 2020 207b 2d31 2c20 2d32 7d20 7c20 3020     {-1, -2} | 0 
│ │ │ │ -00064120: 2020 3020 2030 2030 2030 2030 2030 2030    0  0 0 0 0 0 0
│ │ │ │ -00064130: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ -00064140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064150: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00064160: 2020 207b 2d31 2c20 2d32 7d20 7c20 3020     {-1, -2} | 0 
│ │ │ │ -00064170: 2020 3020 2030 2030 2030 2030 2030 2030    0  0 0 0 0 0 0
│ │ │ │ -00064180: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00063f00: 2020 2020 2020 2020 207c 0a7c 6f33 3320           |.|o33 
│ │ │ │ +00063f10: 3d20 7b30 2c20 307d 2020 207c 202d 3338  = {0, 0}   | -38
│ │ │ │ +00063f20: 2033 3920 3020 3020 3020 3020 3020 3020   39 0 0 0 0 0 0 
│ │ │ │ +00063f30: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00063f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063f50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00063f60: 2020 7b30 2c20 307d 2020 207c 202d 3136    {0, 0}   | -16
│ │ │ │ +00063f70: 2032 3120 3020 3020 3020 3020 3020 3020   21 0 0 0 0 0 0 
│ │ │ │ +00063f80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00063f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063fa0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00063fb0: 2020 7b2d 322c 202d 337d 207c 2030 2020    {-2, -3} | 0  
│ │ │ │ +00063fc0: 2030 2020 3020 3020 3020 3020 3020 3020   0  0 0 0 0 0 0 
│ │ │ │ +00063fd0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00063fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00063ff0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00064000: 2020 7b2d 322c 202d 337d 207c 2030 2020    {-2, -3} | 0  
│ │ │ │ +00064010: 2030 2020 3020 3020 3020 3020 3020 3020   0  0 0 0 0 0 0 
│ │ │ │ +00064020: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064040: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00064050: 2020 7b2d 322c 202d 337d 207c 2030 2020    {-2, -3} | 0  
│ │ │ │ +00064060: 2030 2020 3020 3020 3020 3020 3020 3020   0  0 0 0 0 0 0 
│ │ │ │ +00064070: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064090: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000640a0: 2020 7b2d 322c 202d 337d 207c 2030 2020    {-2, -3} | 0  
│ │ │ │ +000640b0: 2030 2020 3020 3020 3020 3020 3020 3020   0  0 0 0 0 0 0 
│ │ │ │ +000640c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000640d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000640e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +000640f0: 2020 7b2d 312c 202d 327d 207c 2030 2020    {-1, -2} | 0  
│ │ │ │ +00064100: 2030 2020 3020 3020 3020 3020 3020 3020   0  0 0 0 0 0 0 
│ │ │ │ +00064110: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064130: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00064140: 2020 7b2d 312c 202d 327d 207c 2030 2020    {-1, -2} | 0  
│ │ │ │ +00064150: 2030 2020 3020 3020 3020 3020 3020 3020   0  0 0 0 0 0 0 
│ │ │ │ +00064160: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064180: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00064190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000641a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000641a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000641b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000641c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000641d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000641e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000641f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3333            |.|o33
│ │ │ │ -00064200: 203a 204d 6174 7269 7820 4553 203c 2d2d   : Matrix ES <--
│ │ │ │ -00064210: 2045 4527 2020 2020 2020 2020 2020 2020   EE'            
│ │ │ │ -00064220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064240: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +000641d0: 2020 2020 2020 2020 207c 0a7c 6f33 3320           |.|o33 
│ │ │ │ +000641e0: 3a20 4d61 7472 6978 2045 5320 3c2d 2d20  : Matrix ES <-- 
│ │ │ │ +000641f0: 4545 2720 2020 2020 2020 2020 2020 2020  EE'             
│ │ │ │ +00064200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064220: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +00064230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00064240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064290: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6620  ----------+..If 
│ │ │ │ -000642a0: 4545 2061 6e64 2045 5320 7765 7265 2069  EE and ES were i
│ │ │ │ -000642b0: 736f 6d6f 7270 6869 632c 2077 6520 776f  somorphic, we wo
│ │ │ │ -000642c0: 756c 6420 6578 7065 6374 2063 6f6b 6572  uld expect coker
│ │ │ │ -000642d0: 2066 2074 6f20 6265 2030 2c20 616e 6420   f to be 0, and 
│ │ │ │ -000642e0: 6974 2773 206e 6f74 2e0a 7072 756e 6520  it's not..prune 
│ │ │ │ -000642f0: 636f 6b65 7220 660a 0a53 6565 2061 6c73  coker f..See als
│ │ │ │ -00064300: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ -00064310: 2a6e 6f74 6520 4578 743a 2028 4d61 6361  *note Ext: (Maca
│ │ │ │ -00064320: 756c 6179 3244 6f63 2945 7874 2c20 2d2d  ulay2Doc)Ext, --
│ │ │ │ -00064330: 2063 6f6d 7075 7465 2061 6e20 4578 7420   compute an Ext 
│ │ │ │ -00064340: 6d6f 6475 6c65 0a20 202a 202a 6e6f 7465  module.  * *note
│ │ │ │ -00064350: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ -00064360: 546f 7461 6c3a 2045 6973 656e 6275 6453  Total: EisenbudS
│ │ │ │ -00064370: 6861 6d61 7368 546f 7461 6c2c 202d 2d20  hamashTotal, -- 
│ │ │ │ -00064380: 5072 6563 7572 736f 7220 636f 6d70 6c65  Precursor comple
│ │ │ │ -00064390: 7820 6f66 0a20 2020 2074 6f74 616c 2045  x of.    total E
│ │ │ │ -000643a0: 7874 0a0a 5761 7973 2074 6f20 7573 6520  xt..Ways to use 
│ │ │ │ -000643b0: 6e65 7745 7874 3a0a 3d3d 3d3d 3d3d 3d3d  newExt:.========
│ │ │ │ -000643c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -000643d0: 2022 6e65 7745 7874 284d 6f64 756c 652c   "newExt(Module,
│ │ │ │ -000643e0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468  Module)"..For th
│ │ │ │ -000643f0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -00064400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -00064410: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -00064420: 6520 6e65 7745 7874 3a20 6e65 7745 7874  e newExt: newExt
│ │ │ │ -00064430: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -00064440: 686f 6420 6675 6e63 7469 6f6e 2077 6974  hod function wit
│ │ │ │ -00064450: 6820 6f70 7469 6f6e 733a 0a28 4d61 6361  h options:.(Maca
│ │ │ │ -00064460: 756c 6179 3244 6f63 294d 6574 686f 6446  ulay2Doc)MethodF
│ │ │ │ -00064470: 756e 6374 696f 6e57 6974 684f 7074 696f  unctionWithOptio
│ │ │ │ -00064480: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  ns,...----------
│ │ │ │ +00064270: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4966 2045  ---------+..If E
│ │ │ │ +00064280: 4520 616e 6420 4553 2077 6572 6520 6973  E and ES were is
│ │ │ │ +00064290: 6f6d 6f72 7068 6963 2c20 7765 2077 6f75  omorphic, we wou
│ │ │ │ +000642a0: 6c64 2065 7870 6563 7420 636f 6b65 7220  ld expect coker 
│ │ │ │ +000642b0: 6620 746f 2062 6520 302c 2061 6e64 2069  f to be 0, and i
│ │ │ │ +000642c0: 7427 7320 6e6f 742e 0a70 7275 6e65 2063  t's not..prune c
│ │ │ │ +000642d0: 6f6b 6572 2066 0a0a 5365 6520 616c 736f  oker f..See also
│ │ │ │ +000642e0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +000642f0: 6e6f 7465 2045 7874 3a20 284d 6163 6175  note Ext: (Macau
│ │ │ │ +00064300: 6c61 7932 446f 6329 4578 742c 202d 2d20  lay2Doc)Ext, -- 
│ │ │ │ +00064310: 636f 6d70 7574 6520 616e 2045 7874 206d  compute an Ext m
│ │ │ │ +00064320: 6f64 756c 650a 2020 2a20 2a6e 6f74 6520  odule.  * *note 
│ │ │ │ +00064330: 4569 7365 6e62 7564 5368 616d 6173 6854  EisenbudShamashT
│ │ │ │ +00064340: 6f74 616c 3a20 4569 7365 6e62 7564 5368  otal: EisenbudSh
│ │ │ │ +00064350: 616d 6173 6854 6f74 616c 2c20 2d2d 2050  amashTotal, -- P
│ │ │ │ +00064360: 7265 6375 7273 6f72 2063 6f6d 706c 6578  recursor complex
│ │ │ │ +00064370: 206f 660a 2020 2020 746f 7461 6c20 4578   of.    total Ex
│ │ │ │ +00064380: 740a 0a57 6179 7320 746f 2075 7365 206e  t..Ways to use n
│ │ │ │ +00064390: 6577 4578 743a 0a3d 3d3d 3d3d 3d3d 3d3d  ewExt:.=========
│ │ │ │ +000643a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +000643b0: 226e 6577 4578 7428 4d6f 6475 6c65 2c4d  "newExt(Module,M
│ │ │ │ +000643c0: 6f64 756c 6529 220a 0a46 6f72 2074 6865  odule)"..For the
│ │ │ │ +000643d0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +000643e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +000643f0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00064400: 206e 6577 4578 743a 206e 6577 4578 742c   newExt: newExt,
│ │ │ │ +00064410: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +00064420: 6f64 2066 756e 6374 696f 6e20 7769 7468  od function with
│ │ │ │ +00064430: 206f 7074 696f 6e73 3a0a 284d 6163 6175   options:.(Macau
│ │ │ │ +00064440: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ +00064450: 6e63 7469 6f6e 5769 7468 4f70 7469 6f6e  nctionWithOption
│ │ │ │ +00064460: 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  s,...-----------
│ │ │ │ +00064470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00064480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000644a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000644b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000644c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000644d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -000644e0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -000644f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -00064500: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -00064510: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ -00064520: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ -00064530: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ -00064540: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -00064550: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
│ │ │ │ -00064560: 3a32 3536 343a 302e 0a1f 0a46 696c 653a  :2564:0....File:
│ │ │ │ -00064570: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ -00064580: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ -00064590: 2e69 6e66 6f2c 204e 6f64 653a 206f 6464  .info, Node: odd
│ │ │ │ -000645a0: 4578 744d 6f64 756c 652c 204e 6578 743a  ExtModule, Next:
│ │ │ │ -000645b0: 204f 7074 696d 6973 6d2c 2050 7265 763a   Optimism, Prev:
│ │ │ │ -000645c0: 206e 6577 4578 742c 2055 703a 2054 6f70   newExt, Up: Top
│ │ │ │ -000645d0: 0a0a 6f64 6445 7874 4d6f 6475 6c65 202d  ..oddExtModule -
│ │ │ │ -000645e0: 2d20 6f64 6420 7061 7274 206f 6620 4578  - odd part of Ex
│ │ │ │ -000645f0: 745e 2a28 4d2c 6b29 206f 7665 7220 6120  t^*(M,k) over a 
│ │ │ │ -00064600: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ -00064610: 6374 696f 6e20 6173 206d 6f64 756c 6520  ction as module 
│ │ │ │ -00064620: 6f76 6572 2043 4920 6f70 6572 6174 6f72  over CI operator
│ │ │ │ -00064630: 2072 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a   ring.**********
│ │ │ │ +000644b0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ +000644c0: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ +000644d0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ +000644e0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ +000644f0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e  /macaulay2-1.26.
│ │ │ │ +00064500: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61  06+ds/M2/Macaula
│ │ │ │ +00064510: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ +00064520: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ +00064530: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ +00064540: 3235 3634 3a30 2e0a 1f0a 4669 6c65 3a20  2564:0....File: 
│ │ │ │ +00064550: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +00064560: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +00064570: 696e 666f 2c20 4e6f 6465 3a20 6f64 6445  info, Node: oddE
│ │ │ │ +00064580: 7874 4d6f 6475 6c65 2c20 4e65 7874 3a20  xtModule, Next: 
│ │ │ │ +00064590: 4f70 7469 6d69 736d 2c20 5072 6576 3a20  Optimism, Prev: 
│ │ │ │ +000645a0: 6e65 7745 7874 2c20 5570 3a20 546f 700a  newExt, Up: Top.
│ │ │ │ +000645b0: 0a6f 6464 4578 744d 6f64 756c 6520 2d2d  .oddExtModule --
│ │ │ │ +000645c0: 206f 6464 2070 6172 7420 6f66 2045 7874   odd part of Ext
│ │ │ │ +000645d0: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063  ^*(M,k) over a c
│ │ │ │ +000645e0: 6f6d 706c 6574 6520 696e 7465 7273 6563  omplete intersec
│ │ │ │ +000645f0: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f  tion as module o
│ │ │ │ +00064600: 7665 7220 4349 206f 7065 7261 746f 7220  ver CI operator 
│ │ │ │ +00064610: 7269 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ring.***********
│ │ │ │ +00064620: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00064630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00064660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00064670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00064680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00064690: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -000646a0: 7361 6765 3a20 0a20 2020 2020 2020 2045  sage: .        E
│ │ │ │ -000646b0: 203d 206f 6464 4578 744d 6f64 756c 6520   = oddExtModule 
│ │ │ │ -000646c0: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020  M.  * Inputs:.  
│ │ │ │ -000646d0: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ -000646e0: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ -000646f0: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ -00064700: 6f76 6572 2061 2063 6f6d 706c 6574 6520  over a complete 
│ │ │ │ -00064710: 696e 7465 7273 6563 7469 6f6e 0a20 2020  intersection.   
│ │ │ │ -00064720: 2020 2020 2072 696e 670a 2020 2a20 2a6e       ring.  * *n
│ │ │ │ -00064730: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70  ote Optional inp
│ │ │ │ -00064740: 7574 733a 2028 4d61 6361 756c 6179 3244  uts: (Macaulay2D
│ │ │ │ -00064750: 6f63 2975 7369 6e67 2066 756e 6374 696f  oc)using functio
│ │ │ │ -00064760: 6e73 2077 6974 6820 6f70 7469 6f6e 616c  ns with optional
│ │ │ │ -00064770: 2069 6e70 7574 732c 3a0a 2020 2020 2020   inputs,:.      
│ │ │ │ -00064780: 2a20 4f75 7452 696e 6720 3d3e 202e 2e2e  * OutRing => ...
│ │ │ │ -00064790: 2c20 6465 6661 756c 7420 7661 6c75 6520  , default value 
│ │ │ │ -000647a0: 300a 2020 2a20 4f75 7470 7574 733a 0a20  0.  * Outputs:. 
│ │ │ │ -000647b0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74       * E, a *not
│ │ │ │ -000647c0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ -000647d0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ -000647e0: 206f 7665 7220 6120 706f 6c79 6e6f 6d69   over a polynomi
│ │ │ │ -000647f0: 616c 2072 696e 6720 7769 7468 0a20 2020  al ring with.   
│ │ │ │ -00064800: 2020 2020 2067 656e 7320 696e 2064 6567       gens in deg
│ │ │ │ -00064810: 7265 6520 310a 0a44 6573 6372 6970 7469  ree 1..Descripti
│ │ │ │ -00064820: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  on.===========..
│ │ │ │ -00064830: 4578 7472 6163 7473 2074 6865 206f 6464  Extracts the odd
│ │ │ │ -00064840: 2064 6567 7265 6520 7061 7274 2066 726f   degree part fro
│ │ │ │ -00064850: 6d20 4578 744d 6f64 756c 6520 4d2e 2049  m ExtModule M. I
│ │ │ │ -00064860: 6620 7468 6520 6f70 7469 6f6e 616c 2061  f the optional a
│ │ │ │ -00064870: 7267 756d 656e 7420 4f75 7452 696e 670a  rgument OutRing.
│ │ │ │ -00064880: 3d3e 2054 2069 7320 6769 7665 6e2c 2061  => T is given, a
│ │ │ │ -00064890: 6e64 2063 6c61 7373 2054 203d 3d3d 2050  nd class T === P
│ │ │ │ -000648a0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2074  olynomialRing, t
│ │ │ │ -000648b0: 6865 6e20 7468 6520 6f75 7470 7574 2077  hen the output w
│ │ │ │ -000648c0: 696c 6c20 6265 2061 206d 6f64 756c 650a  ill be a module.
│ │ │ │ -000648d0: 6f76 6572 2054 2e0a 0a2b 2d2d 2d2d 2d2d  over T...+------
│ │ │ │ +00064670: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +00064680: 6167 653a 200a 2020 2020 2020 2020 4520  age: .        E 
│ │ │ │ +00064690: 3d20 6f64 6445 7874 4d6f 6475 6c65 204d  = oddExtModule M
│ │ │ │ +000646a0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ +000646b0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ +000646c0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ +000646d0: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f  y2Doc)Module,, o
│ │ │ │ +000646e0: 7665 7220 6120 636f 6d70 6c65 7465 2069  ver a complete i
│ │ │ │ +000646f0: 6e74 6572 7365 6374 696f 6e0a 2020 2020  ntersection.    
│ │ │ │ +00064700: 2020 2020 7269 6e67 0a20 202a 202a 6e6f      ring.  * *no
│ │ │ │ +00064710: 7465 204f 7074 696f 6e61 6c20 696e 7075  te Optional inpu
│ │ │ │ +00064720: 7473 3a20 284d 6163 6175 6c61 7932 446f  ts: (Macaulay2Do
│ │ │ │ +00064730: 6329 7573 696e 6720 6675 6e63 7469 6f6e  c)using function
│ │ │ │ +00064740: 7320 7769 7468 206f 7074 696f 6e61 6c20  s with optional 
│ │ │ │ +00064750: 696e 7075 7473 2c3a 0a20 2020 2020 202a  inputs,:.      *
│ │ │ │ +00064760: 204f 7574 5269 6e67 203d 3e20 2e2e 2e2c   OutRing => ...,
│ │ │ │ +00064770: 2064 6566 6175 6c74 2076 616c 7565 2030   default value 0
│ │ │ │ +00064780: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ +00064790: 2020 2020 2a20 452c 2061 202a 6e6f 7465      * E, a *note
│ │ │ │ +000647a0: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +000647b0: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +000647c0: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961  over a polynomia
│ │ │ │ +000647d0: 6c20 7269 6e67 2077 6974 680a 2020 2020  l ring with.    
│ │ │ │ +000647e0: 2020 2020 6765 6e73 2069 6e20 6465 6772      gens in degr
│ │ │ │ +000647f0: 6565 2031 0a0a 4465 7363 7269 7074 696f  ee 1..Descriptio
│ │ │ │ +00064800: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a45  n.===========..E
│ │ │ │ +00064810: 7874 7261 6374 7320 7468 6520 6f64 6420  xtracts the odd 
│ │ │ │ +00064820: 6465 6772 6565 2070 6172 7420 6672 6f6d  degree part from
│ │ │ │ +00064830: 2045 7874 4d6f 6475 6c65 204d 2e20 4966   ExtModule M. If
│ │ │ │ +00064840: 2074 6865 206f 7074 696f 6e61 6c20 6172   the optional ar
│ │ │ │ +00064850: 6775 6d65 6e74 204f 7574 5269 6e67 0a3d  gument OutRing.=
│ │ │ │ +00064860: 3e20 5420 6973 2067 6976 656e 2c20 616e  > T is given, an
│ │ │ │ +00064870: 6420 636c 6173 7320 5420 3d3d 3d20 506f  d class T === Po
│ │ │ │ +00064880: 6c79 6e6f 6d69 616c 5269 6e67 2c20 7468  lynomialRing, th
│ │ │ │ +00064890: 656e 2074 6865 206f 7574 7075 7420 7769  en the output wi
│ │ │ │ +000648a0: 6c6c 2062 6520 6120 6d6f 6475 6c65 0a6f  ll be a module.o
│ │ │ │ +000648b0: 7665 7220 542e 0a0a 2b2d 2d2d 2d2d 2d2d  ver T...+-------
│ │ │ │ +000648c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000648d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000648e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000648f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064910: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a  ---+.|i1 : kk= Z
│ │ │ │ -00064920: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │ -00064930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00064960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064980: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -00064990: 203d 206b 6b20 2020 2020 2020 2020 2020   = kk           
│ │ │ │ -000649a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000648f0: 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20 5a5a  --+.|i1 : kk= ZZ
│ │ │ │ +00064900: 2f31 3031 2020 2020 2020 2020 2020 2020  /101            
│ │ │ │ +00064910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064920: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00064930: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064960: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +00064970: 3d20 6b6b 2020 2020 2020 2020 2020 2020  = kk            
│ │ │ │ +00064980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000649a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000649b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000649c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000649c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000649d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000649e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000649f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064a00: 2020 207c 0a7c 6f31 203a 2051 756f 7469     |.|o1 : Quoti
│ │ │ │ -00064a10: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ -00064a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064a40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00064a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00064a80: 203a 2053 203d 206b 6b5b 782c 792c 7a5d   : S = kk[x,y,z]
│ │ │ │ -00064a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000649e0: 2020 7c0a 7c6f 3120 3a20 5175 6f74 6965    |.|o1 : Quotie
│ │ │ │ +000649f0: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +00064a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064a10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00064a20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00064a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00064a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00064a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220  ----------+.|i2 
│ │ │ │ +00064a60: 3a20 5320 3d20 6b6b 5b78 2c79 2c7a 5d20  : S = kk[x,y,z] 
│ │ │ │ +00064a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064a90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00064aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00064ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064ad0: 2020 7c0a 7c6f 3220 3d20 5320 2020 2020    |.|o2 = S     
│ │ │ │  00064ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064af0: 2020 207c 0a7c 6f32 203d 2053 2020 2020     |.|o2 = S    
│ │ │ │ -00064b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064b30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00064b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064b60: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -00064b70: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e   : PolynomialRin
│ │ │ │ -00064b80: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ -00064b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ba0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00064af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064b00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00064b10: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064b40: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00064b50: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67  : PolynomialRing
│ │ │ │ +00064b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064b80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00064b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00064ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064be0: 2d2d 2d2b 0a7c 6933 203a 2049 3220 3d20  ---+.|i3 : I2 = 
│ │ │ │ -00064bf0: 6964 6561 6c22 7833 2c79 7a22 2020 2020  ideal"x3,yz"    
│ │ │ │ -00064c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00064c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00064c60: 2020 2020 2020 2020 2020 2033 2020 2020             3    
│ │ │ │ -00064c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064c90: 2020 2020 2020 207c 0a7c 6f33 203d 2069         |.|o3 = i
│ │ │ │ -00064ca0: 6465 616c 2028 7820 2c20 792a 7a29 2020  deal (x , y*z)  
│ │ │ │ -00064cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064bc0: 2d2d 2b0a 7c69 3320 3a20 4932 203d 2069  --+.|i3 : I2 = i
│ │ │ │ +00064bd0: 6465 616c 2278 332c 797a 2220 2020 2020  deal"x3,yz"     
│ │ │ │ +00064be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064bf0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00064c00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00064c40: 2020 2020 2020 2020 2020 3320 2020 2020            3     
│ │ │ │ +00064c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064c70: 2020 2020 2020 7c0a 7c6f 3320 3d20 6964        |.|o3 = id
│ │ │ │ +00064c80: 6561 6c20 2878 202c 2079 2a7a 2920 2020  eal (x , y*z)   
│ │ │ │ +00064c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064cb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00064cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064cd0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00064ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064d10: 0a7c 6f33 203a 2049 6465 616c 206f 6620  .|o3 : Ideal of 
│ │ │ │ -00064d20: 5320 2020 2020 2020 2020 2020 2020 2020  S               
│ │ │ │ -00064d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064d40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +00064cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064ce0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00064cf0: 7c6f 3320 3a20 4964 6561 6c20 6f66 2053  |o3 : Ideal of S
│ │ │ │ +00064d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064d20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00064d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00064d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00064d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052  -------+.|i4 : R
│ │ │ │ -00064d90: 3220 3d20 532f 4932 2020 2020 2020 2020  2 = S/I2        
│ │ │ │ -00064da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064d60: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 5232  ------+.|i4 : R2
│ │ │ │ +00064d70: 203d 2053 2f49 3220 2020 2020 2020 2020   = S/I2         
│ │ │ │ +00064d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064da0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00064db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064dc0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00064dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064e00: 0a7c 6f34 203d 2052 3220 2020 2020 2020  .|o4 = R2       
│ │ │ │ -00064e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064dd0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00064de0: 7c6f 3420 3d20 5232 2020 2020 2020 2020  |o4 = R2        
│ │ │ │ +00064df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064e10: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00064e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064e30: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00064e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064e70: 2020 2020 2020 207c 0a7c 6f34 203a 2051         |.|o4 : Q
│ │ │ │ -00064e80: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ -00064e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064eb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00064ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00064ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00064ef0: 0a7c 6935 203a 204d 3220 3d20 5232 5e31  .|i5 : M2 = R2^1
│ │ │ │ -00064f00: 2f69 6465 616c 2278 322c 792c 7a22 2020  /ideal"x2,y,z"  
│ │ │ │ +00064e50: 2020 2020 2020 7c0a 7c6f 3420 3a20 5175        |.|o4 : Qu
│ │ │ │ +00064e60: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +00064e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064e90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00064ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00064eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00064ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00064ed0: 7c69 3520 3a20 4d32 203d 2052 325e 312f  |i5 : M2 = R2^1/
│ │ │ │ +00064ee0: 6964 6561 6c22 7832 2c79 2c7a 2220 2020  ideal"x2,y,z"   
│ │ │ │ +00064ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064f00: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00064f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064f20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00064f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00064f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064f60: 2020 2020 2020 207c 0a7c 6f35 203d 2063         |.|o5 = c
│ │ │ │ -00064f70: 6f6b 6572 6e65 6c20 7c20 7832 2079 207a  okernel | x2 y z
│ │ │ │ -00064f80: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +00064f40: 2020 2020 2020 7c0a 7c6f 3520 3d20 636f        |.|o5 = co
│ │ │ │ +00064f50: 6b65 726e 656c 207c 2078 3220 7920 7a20  kernel | x2 y z 
│ │ │ │ +00064f60: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064f80: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00064f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064fa0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00064fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00064fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00064fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00064ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065000: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00065010: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -00065020: 203a 2052 322d 6d6f 6475 6c65 2c20 7175   : R2-module, qu
│ │ │ │ -00065030: 6f74 6965 6e74 206f 6620 5232 2020 2020  otient of R2    
│ │ │ │ -00065040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065050: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00064fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064fb0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00064fc0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00064fd0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00064fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00064ff0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ +00065000: 3a20 5232 2d6d 6f64 756c 652c 2071 756f  : R2-module, quo
│ │ │ │ +00065010: 7469 656e 7420 6f66 2052 3220 2020 2020  tient of R2     
│ │ │ │ +00065020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065030: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00065040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065090: 2d2d 2d2b 0a7c 6936 203a 2062 6574 7469  ---+.|i6 : betti
│ │ │ │ -000650a0: 2066 7265 6552 6573 6f6c 7574 696f 6e20   freeResolution 
│ │ │ │ -000650b0: 284d 322c 204c 656e 6774 684c 696d 6974  (M2, LengthLimit
│ │ │ │ -000650c0: 203d 3e31 3029 2020 2020 2020 2020 207c   =>10)         |
│ │ │ │ -000650d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000650e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000650f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065100: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00065110: 2020 2020 2020 2020 2020 3020 3120 3220            0 1 2 
│ │ │ │ -00065120: 3320 3420 2035 2020 3620 2037 2020 3820  3 4  5  6  7  8 
│ │ │ │ -00065130: 2039 2031 3020 2020 2020 2020 2020 2020   9 10           
│ │ │ │ -00065140: 2020 2020 2020 207c 0a7c 6f36 203d 2074         |.|o6 = t
│ │ │ │ -00065150: 6f74 616c 3a20 3120 3320 3520 3720 3920  otal: 1 3 5 7 9 
│ │ │ │ -00065160: 3131 2031 3320 3135 2031 3720 3139 2032  11 13 15 17 19 2
│ │ │ │ -00065170: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ -00065180: 2020 207c 0a7c 2020 2020 2020 2020 2030     |.|         0
│ │ │ │ -00065190: 3a20 3120 3220 3220 3220 3220 2032 2020  : 1 2 2 2 2  2  
│ │ │ │ -000651a0: 3220 2032 2020 3220 2032 2020 3220 2020  2  2  2  2  2   
│ │ │ │ -000651b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000651c0: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ -000651d0: 3120 3320 3420 3420 2034 2020 3420 2034  1 3 4 4  4  4  4
│ │ │ │ -000651e0: 2020 3420 2034 2020 3420 2020 2020 2020    4  4  4       
│ │ │ │ -000651f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00065200: 2020 2020 2020 2032 3a20 2e20 2e20 2e20         2: . . . 
│ │ │ │ -00065210: 3120 3320 2034 2020 3420 2034 2020 3420  1 3  4  4  4  4 
│ │ │ │ -00065220: 2034 2020 3420 2020 2020 2020 2020 2020   4  4           
│ │ │ │ -00065230: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00065240: 2020 2033 3a20 2e20 2e20 2e20 2e20 2e20     3: . . . . . 
│ │ │ │ -00065250: 2031 2020 3320 2034 2020 3420 2034 2020   1  3  4  4  4  
│ │ │ │ -00065260: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ -00065270: 2020 207c 0a7c 2020 2020 2020 2020 2034     |.|         4
│ │ │ │ -00065280: 3a20 2e20 2e20 2e20 2e20 2e20 202e 2020  : . . . . .  .  
│ │ │ │ -00065290: 2e20 2031 2020 3320 2034 2020 3420 2020  .  1  3  4  4   
│ │ │ │ -000652a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000652b0: 0a7c 2020 2020 2020 2020 2035 3a20 2e20  .|         5: . 
│ │ │ │ -000652c0: 2e20 2e20 2e20 2e20 202e 2020 2e20 202e  . . . .  .  .  .
│ │ │ │ -000652d0: 2020 2e20 2031 2020 3320 2020 2020 2020    .  1  3       
│ │ │ │ -000652e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00065070: 2d2d 2b0a 7c69 3620 3a20 6265 7474 6920  --+.|i6 : betti 
│ │ │ │ +00065080: 6672 6565 5265 736f 6c75 7469 6f6e 2028  freeResolution (
│ │ │ │ +00065090: 4d32 2c20 4c65 6e67 7468 4c69 6d69 7420  M2, LengthLimit 
│ │ │ │ +000650a0: 3d3e 3130 2920 2020 2020 2020 2020 7c0a  =>10)         |.
│ │ │ │ +000650b0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +000650c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000650d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000650e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000650f0: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ +00065100: 2034 2020 3520 2036 2020 3720 2038 2020   4  5  6  7  8  
│ │ │ │ +00065110: 3920 3130 2020 2020 2020 2020 2020 2020  9 10            
│ │ │ │ +00065120: 2020 2020 2020 7c0a 7c6f 3620 3d20 746f        |.|o6 = to
│ │ │ │ +00065130: 7461 6c3a 2031 2033 2035 2037 2039 2031  tal: 1 3 5 7 9 1
│ │ │ │ +00065140: 3120 3133 2031 3520 3137 2031 3920 3231  1 13 15 17 19 21
│ │ │ │ +00065150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065160: 2020 7c0a 7c20 2020 2020 2020 2020 303a    |.|         0:
│ │ │ │ +00065170: 2031 2032 2032 2032 2032 2020 3220 2032   1 2 2 2 2  2  2
│ │ │ │ +00065180: 2020 3220 2032 2020 3220 2032 2020 2020    2  2  2  2    
│ │ │ │ +00065190: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000651a0: 7c20 2020 2020 2020 2020 313a 202e 2031  |         1: . 1
│ │ │ │ +000651b0: 2033 2034 2034 2020 3420 2034 2020 3420   3 4 4  4  4  4 
│ │ │ │ +000651c0: 2034 2020 3420 2034 2020 2020 2020 2020   4  4  4        
│ │ │ │ +000651d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000651e0: 2020 2020 2020 323a 202e 202e 202e 2031        2: . . . 1
│ │ │ │ +000651f0: 2033 2020 3420 2034 2020 3420 2034 2020   3  4  4  4  4  
│ │ │ │ +00065200: 3420 2034 2020 2020 2020 2020 2020 2020  4  4            
│ │ │ │ +00065210: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00065220: 2020 333a 202e 202e 202e 202e 202e 2020    3: . . . . .  
│ │ │ │ +00065230: 3120 2033 2020 3420 2034 2020 3420 2034  1  3  4  4  4  4
│ │ │ │ +00065240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065250: 2020 7c0a 7c20 2020 2020 2020 2020 343a    |.|         4:
│ │ │ │ +00065260: 202e 202e 202e 202e 202e 2020 2e20 202e   . . . . .  .  .
│ │ │ │ +00065270: 2020 3120 2033 2020 3420 2034 2020 2020    1  3  4  4    
│ │ │ │ +00065280: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00065290: 7c20 2020 2020 2020 2020 353a 202e 202e  |         5: . .
│ │ │ │ +000652a0: 202e 202e 202e 2020 2e20 202e 2020 2e20   . . .  .  .  . 
│ │ │ │ +000652b0: 202e 2020 3120 2033 2020 2020 2020 2020   .  1  3        
│ │ │ │ +000652c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +000652d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000652e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000652f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065320: 2020 2020 2020 207c 0a7c 6f36 203a 2042         |.|o6 : B
│ │ │ │ -00065330: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ -00065340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065360: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00065370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -000653a0: 0a7c 6937 203a 2045 203d 2045 7874 4d6f  .|i7 : E = ExtMo
│ │ │ │ -000653b0: 6475 6c65 204d 3220 2020 2020 2020 2020  dule M2         
│ │ │ │ +00065300: 2020 2020 2020 7c0a 7c6f 3620 3a20 4265        |.|o6 : Be
│ │ │ │ +00065310: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +00065320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065340: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +00065350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00065380: 7c69 3720 3a20 4520 3d20 4578 744d 6f64  |i7 : E = ExtMod
│ │ │ │ +00065390: 756c 6520 4d32 2020 2020 2020 2020 2020  ule M2          
│ │ │ │ +000653a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000653b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000653c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000653d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000653d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000653e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000653f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00065420: 2020 2020 2020 2020 2020 2038 2020 2020             8    
│ │ │ │ -00065430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065450: 2020 207c 0a7c 6f37 203d 2028 6b6b 5b58     |.|o7 = (kk[X
│ │ │ │ -00065460: 202e 2e58 205d 2920 2020 2020 2020 2020   ..X ])         
│ │ │ │ -00065470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065490: 0a7c 2020 2020 2020 2020 2020 3020 2020  .|          0   
│ │ │ │ -000654a0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000653f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00065400: 2020 2020 2020 2020 2020 3820 2020 2020            8     
│ │ │ │ +00065410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065430: 2020 7c0a 7c6f 3720 3d20 286b 6b5b 5820    |.|o7 = (kk[X 
│ │ │ │ +00065440: 2e2e 5820 5d29 2020 2020 2020 2020 2020  ..X ])          
│ │ │ │ +00065450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065460: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00065470: 7c20 2020 2020 2020 2020 2030 2020 2031  |          0   1
│ │ │ │ +00065480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000654a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  000654b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000654c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000654c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000654d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000654e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000654f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065500: 2020 2020 2020 207c 0a7c 6f37 203a 206b         |.|o7 : k
│ │ │ │ -00065510: 6b5b 5820 2e2e 5820 5d2d 6d6f 6475 6c65  k[X ..X ]-module
│ │ │ │ -00065520: 2c20 6672 6565 2c20 6465 6772 6565 7320  , free, degrees 
│ │ │ │ -00065530: 7b30 2e2e 312c 2032 3a31 2c20 333a 322c  {0..1, 2:1, 3:2,
│ │ │ │ -00065540: 2033 7d7c 0a7c 2020 2020 2020 2020 2030   3}|.|         0
│ │ │ │ -00065550: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ -00065560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00065590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000655a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000655b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ -000655c0: 203a 2061 7070 6c79 2874 6f4c 6973 7428   : apply(toList(
│ │ │ │ -000655d0: 302e 2e31 3029 2c20 692d 3e68 696c 6265  0..10), i->hilbe
│ │ │ │ -000655e0: 7274 4675 6e63 7469 6f6e 2869 2c20 4529  rtFunction(i, E)
│ │ │ │ -000655f0: 2920 2020 2020 207c 0a7c 2020 2020 2020  )      |.|      
│ │ │ │ +000654e0: 2020 2020 2020 7c0a 7c6f 3720 3a20 6b6b        |.|o7 : kk
│ │ │ │ +000654f0: 5b58 202e 2e58 205d 2d6d 6f64 756c 652c  [X ..X ]-module,
│ │ │ │ +00065500: 2066 7265 652c 2064 6567 7265 6573 207b   free, degrees {
│ │ │ │ +00065510: 302e 2e31 2c20 323a 312c 2033 3a32 2c20  0..1, 2:1, 3:2, 
│ │ │ │ +00065520: 337d 7c0a 7c20 2020 2020 2020 2020 3020  3}|.|         0 
│ │ │ │ +00065530: 2020 3120 2020 2020 2020 2020 2020 2020    1             
│ │ │ │ +00065540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065550: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00065560: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00065570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065590: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820  ----------+.|i8 
│ │ │ │ +000655a0: 3a20 6170 706c 7928 746f 4c69 7374 2830  : apply(toList(0
│ │ │ │ +000655b0: 2e2e 3130 292c 2069 2d3e 6869 6c62 6572  ..10), i->hilber
│ │ │ │ +000655c0: 7446 756e 6374 696f 6e28 692c 2045 2929  tFunction(i, E))
│ │ │ │ +000655d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000655e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000655f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065630: 2020 207c 0a7c 6f38 203d 207b 312c 2033     |.|o8 = {1, 3
│ │ │ │ -00065640: 2c20 352c 2037 2c20 392c 2031 312c 2031  , 5, 7, 9, 11, 1
│ │ │ │ -00065650: 332c 2031 352c 2031 372c 2031 392c 2032  3, 15, 17, 19, 2
│ │ │ │ -00065660: 317d 2020 2020 2020 2020 2020 2020 207c  1}             |
│ │ │ │ -00065670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00065680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656a0: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ -000656b0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ -000656c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000656e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00065610: 2020 7c0a 7c6f 3820 3d20 7b31 2c20 332c    |.|o8 = {1, 3,
│ │ │ │ +00065620: 2035 2c20 372c 2039 2c20 3131 2c20 3133   5, 7, 9, 11, 13
│ │ │ │ +00065630: 2c20 3135 2c20 3137 2c20 3139 2c20 3231  , 15, 17, 19, 21
│ │ │ │ +00065640: 7d20 2020 2020 2020 2020 2020 2020 7c0a  }             |.
│ │ │ │ +00065650: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00065660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065680: 2020 2020 2020 2020 2020 7c0a 7c6f 3820            |.|o8 
│ │ │ │ +00065690: 3a20 4c69 7374 2020 2020 2020 2020 2020  : List          
│ │ │ │ +000656a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000656b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000656c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000656d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000656e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000656f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065720: 2d2d 2d2b 0a7c 6939 203a 2045 6f64 6420  ---+.|i9 : Eodd 
│ │ │ │ -00065730: 3d20 6f64 6445 7874 4d6f 6475 6c65 204d  = oddExtModule M
│ │ │ │ -00065740: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -00065750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00065770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000657a0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ -000657b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000657c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000657d0: 2020 2020 2020 207c 0a7c 6f39 203d 2028         |.|o9 = (
│ │ │ │ -000657e0: 6b6b 5b58 202e 2e58 205d 2920 2020 2020  kk[X ..X ])     
│ │ │ │ -000657f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065810: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00065820: 3020 2020 3120 2020 2020 2020 2020 2020  0   1           
│ │ │ │ -00065830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065850: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00065860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065880: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ -00065890: 203a 206b 6b5b 5820 2e2e 5820 5d2d 6d6f   : kk[X ..X ]-mo
│ │ │ │ -000658a0: 6475 6c65 2c20 6672 6565 2c20 6465 6772  dule, free, degr
│ │ │ │ -000658b0: 6565 7320 7b33 3a30 2c20 317d 2020 2020  ees {3:0, 1}    
│ │ │ │ -000658c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000658d0: 2020 2030 2020 2031 2020 2020 2020 2020     0   1        
│ │ │ │ -000658e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000658f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065900: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -00065910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00065940: 0a7c 6931 3020 3a20 6170 706c 7928 746f  .|i10 : apply(to
│ │ │ │ -00065950: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68  List(0..5), i->h
│ │ │ │ -00065960: 696c 6265 7274 4675 6e63 7469 6f6e 2869  ilbertFunction(i
│ │ │ │ -00065970: 2c20 456f 6464 2929 2020 207c 0a7c 2020  , Eodd))   |.|  
│ │ │ │ +00065700: 2d2d 2b0a 7c69 3920 3a20 456f 6464 203d  --+.|i9 : Eodd =
│ │ │ │ +00065710: 206f 6464 4578 744d 6f64 756c 6520 4d32   oddExtModule M2
│ │ │ │ +00065720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00065740: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00065750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065770: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00065780: 2020 2020 2020 2020 2020 2020 2020 3420                4 
│ │ │ │ +00065790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000657a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000657b0: 2020 2020 2020 7c0a 7c6f 3920 3d20 286b        |.|o9 = (k
│ │ │ │ +000657c0: 6b5b 5820 2e2e 5820 5d29 2020 2020 2020  k[X ..X ])      
│ │ │ │ +000657d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000657e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000657f0: 2020 7c0a 7c20 2020 2020 2020 2020 2030    |.|          0
│ │ │ │ +00065800: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +00065810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065820: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00065830: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00065840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065860: 2020 2020 2020 2020 2020 7c0a 7c6f 3920            |.|o9 
│ │ │ │ +00065870: 3a20 6b6b 5b58 202e 2e58 205d 2d6d 6f64  : kk[X ..X ]-mod
│ │ │ │ +00065880: 756c 652c 2066 7265 652c 2064 6567 7265  ule, free, degre
│ │ │ │ +00065890: 6573 207b 333a 302c 2031 7d20 2020 2020  es {3:0, 1}     
│ │ │ │ +000658a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000658b0: 2020 3020 2020 3120 2020 2020 2020 2020    0   1         
│ │ │ │ +000658c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000658d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000658e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000658f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00065920: 7c69 3130 203a 2061 7070 6c79 2874 6f4c  |i10 : apply(toL
│ │ │ │ +00065930: 6973 7428 302e 2e35 292c 2069 2d3e 6869  ist(0..5), i->hi
│ │ │ │ +00065940: 6c62 6572 7446 756e 6374 696f 6e28 692c  lbertFunction(i,
│ │ │ │ +00065950: 2045 6f64 6429 2920 2020 7c0a 7c20 2020   Eodd))   |.|   
│ │ │ │ +00065960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00065980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000659a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000659b0: 2020 2020 2020 207c 0a7c 6f31 3020 3d20         |.|o10 = 
│ │ │ │ -000659c0: 7b33 2c20 372c 2031 312c 2031 352c 2031  {3, 7, 11, 15, 1
│ │ │ │ -000659d0: 392c 2032 337d 2020 2020 2020 2020 2020  9, 23}          
│ │ │ │ +00065990: 2020 2020 2020 7c0a 7c6f 3130 203d 207b        |.|o10 = {
│ │ │ │ +000659a0: 332c 2037 2c20 3131 2c20 3135 2c20 3139  3, 7, 11, 15, 19
│ │ │ │ +000659b0: 2c20 3233 7d20 2020 2020 2020 2020 2020  , 23}           
│ │ │ │ +000659c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000659d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  000659e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000659f0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00065a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00065a30: 0a7c 6f31 3020 3a20 4c69 7374 2020 2020  .|o10 : List    
│ │ │ │ -00065a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00065a60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +000659f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065a00: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00065a10: 7c6f 3130 203a 204c 6973 7420 2020 2020  |o10 : List     
│ │ │ │ +00065a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00065a40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00065a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065aa0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c  -------+..See al
│ │ │ │ -00065ab0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ -00065ac0: 202a 6e6f 7465 2045 7874 4d6f 6475 6c65   *note ExtModule
│ │ │ │ -00065ad0: 3a20 4578 744d 6f64 756c 652c 202d 2d20  : ExtModule, -- 
│ │ │ │ -00065ae0: 4578 745e 2a28 4d2c 6b29 206f 7665 7220  Ext^*(M,k) over 
│ │ │ │ -00065af0: 6120 636f 6d70 6c65 7465 2069 6e74 6572  a complete inter
│ │ │ │ -00065b00: 7365 6374 696f 6e20 6173 0a20 2020 206d  section as.    m
│ │ │ │ -00065b10: 6f64 756c 6520 6f76 6572 2043 4920 6f70  odule over CI op
│ │ │ │ -00065b20: 6572 6174 6f72 2072 696e 670a 2020 2a20  erator ring.  * 
│ │ │ │ -00065b30: 2a6e 6f74 6520 6576 656e 4578 744d 6f64  *note evenExtMod
│ │ │ │ -00065b40: 756c 653a 2065 7665 6e45 7874 4d6f 6475  ule: evenExtModu
│ │ │ │ -00065b50: 6c65 2c20 2d2d 2065 7665 6e20 7061 7274  le, -- even part
│ │ │ │ -00065b60: 206f 6620 4578 745e 2a28 4d2c 6b29 206f   of Ext^*(M,k) o
│ │ │ │ -00065b70: 7665 7220 610a 2020 2020 636f 6d70 6c65  ver a.    comple
│ │ │ │ -00065b80: 7465 2069 6e74 6572 7365 6374 696f 6e20  te intersection 
│ │ │ │ -00065b90: 6173 206d 6f64 756c 6520 6f76 6572 2043  as module over C
│ │ │ │ -00065ba0: 4920 6f70 6572 6174 6f72 2072 696e 670a  I operator ring.
│ │ │ │ -00065bb0: 2020 2a20 2a6e 6f74 6520 4f75 7452 696e    * *note OutRin
│ │ │ │ -00065bc0: 673a 204f 7574 5269 6e67 2c20 2d2d 204f  g: OutRing, -- O
│ │ │ │ -00065bd0: 7074 696f 6e20 616c 6c6f 7769 6e67 2073  ption allowing s
│ │ │ │ -00065be0: 7065 6369 6669 6361 7469 6f6e 206f 6620  pecification of 
│ │ │ │ -00065bf0: 7468 6520 7269 6e67 206f 7665 720a 2020  the ring over.  
│ │ │ │ -00065c00: 2020 7768 6963 6820 7468 6520 6f75 7470    which the outp
│ │ │ │ -00065c10: 7574 2069 7320 6465 6669 6e65 640a 0a57  ut is defined..W
│ │ │ │ -00065c20: 6179 7320 746f 2075 7365 206f 6464 4578  ays to use oddEx
│ │ │ │ -00065c30: 744d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d  tModule:.=======
│ │ │ │ -00065c40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00065c50: 3d3d 0a0a 2020 2a20 226f 6464 4578 744d  ==..  * "oddExtM
│ │ │ │ -00065c60: 6f64 756c 6528 4d6f 6475 6c65 2922 0a0a  odule(Module)"..
│ │ │ │ -00065c70: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -00065c80: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -00065c90: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -00065ca0: 7420 2a6e 6f74 6520 6f64 6445 7874 4d6f  t *note oddExtMo
│ │ │ │ -00065cb0: 6475 6c65 3a20 6f64 6445 7874 4d6f 6475  dule: oddExtModu
│ │ │ │ -00065cc0: 6c65 2c20 6973 2061 202a 6e6f 7465 206d  le, is a *note m
│ │ │ │ -00065cd0: 6574 686f 6420 6675 6e63 7469 6f6e 2077  ethod function w
│ │ │ │ -00065ce0: 6974 680a 6f70 7469 6f6e 733a 2028 4d61  ith.options: (Ma
│ │ │ │ -00065cf0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00065d00: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
│ │ │ │ -00065d10: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  ions,...--------
│ │ │ │ +00065a80: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73  ------+..See als
│ │ │ │ +00065a90: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00065aa0: 2a6e 6f74 6520 4578 744d 6f64 756c 653a  *note ExtModule:
│ │ │ │ +00065ab0: 2045 7874 4d6f 6475 6c65 2c20 2d2d 2045   ExtModule, -- E
│ │ │ │ +00065ac0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ +00065ad0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ +00065ae0: 6563 7469 6f6e 2061 730a 2020 2020 6d6f  ection as.    mo
│ │ │ │ +00065af0: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ +00065b00: 7261 746f 7220 7269 6e67 0a20 202a 202a  rator ring.  * *
│ │ │ │ +00065b10: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475  note evenExtModu
│ │ │ │ +00065b20: 6c65 3a20 6576 656e 4578 744d 6f64 756c  le: evenExtModul
│ │ │ │ +00065b30: 652c 202d 2d20 6576 656e 2070 6172 7420  e, -- even part 
│ │ │ │ +00065b40: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76  of Ext^*(M,k) ov
│ │ │ │ +00065b50: 6572 2061 0a20 2020 2063 6f6d 706c 6574  er a.    complet
│ │ │ │ +00065b60: 6520 696e 7465 7273 6563 7469 6f6e 2061  e intersection a
│ │ │ │ +00065b70: 7320 6d6f 6475 6c65 206f 7665 7220 4349  s module over CI
│ │ │ │ +00065b80: 206f 7065 7261 746f 7220 7269 6e67 0a20   operator ring. 
│ │ │ │ +00065b90: 202a 202a 6e6f 7465 204f 7574 5269 6e67   * *note OutRing
│ │ │ │ +00065ba0: 3a20 4f75 7452 696e 672c 202d 2d20 4f70  : OutRing, -- Op
│ │ │ │ +00065bb0: 7469 6f6e 2061 6c6c 6f77 696e 6720 7370  tion allowing sp
│ │ │ │ +00065bc0: 6563 6966 6963 6174 696f 6e20 6f66 2074  ecification of t
│ │ │ │ +00065bd0: 6865 2072 696e 6720 6f76 6572 0a20 2020  he ring over.   
│ │ │ │ +00065be0: 2077 6869 6368 2074 6865 206f 7574 7075   which the outpu
│ │ │ │ +00065bf0: 7420 6973 2064 6566 696e 6564 0a0a 5761  t is defined..Wa
│ │ │ │ +00065c00: 7973 2074 6f20 7573 6520 6f64 6445 7874  ys to use oddExt
│ │ │ │ +00065c10: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d  Module:.========
│ │ │ │ +00065c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00065c30: 3d0a 0a20 202a 2022 6f64 6445 7874 4d6f  =..  * "oddExtMo
│ │ │ │ +00065c40: 6475 6c65 284d 6f64 756c 6529 220a 0a46  dule(Module)"..F
│ │ │ │ +00065c50: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +00065c60: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +00065c70: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +00065c80: 202a 6e6f 7465 206f 6464 4578 744d 6f64   *note oddExtMod
│ │ │ │ +00065c90: 756c 653a 206f 6464 4578 744d 6f64 756c  ule: oddExtModul
│ │ │ │ +00065ca0: 652c 2069 7320 6120 2a6e 6f74 6520 6d65  e, is a *note me
│ │ │ │ +00065cb0: 7468 6f64 2066 756e 6374 696f 6e20 7769  thod function wi
│ │ │ │ +00065cc0: 7468 0a6f 7074 696f 6e73 3a20 284d 6163  th.options: (Mac
│ │ │ │ +00065cd0: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +00065ce0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469  FunctionWithOpti
│ │ │ │ +00065cf0: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ons,...---------
│ │ │ │ +00065d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00065d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00065d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00065d60: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00065d70: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -00065d80: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -00065d90: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -00065da0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00065db0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ -00065dc0: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -00065dd0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00065de0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00065df0: 6d32 3a33 3637 393a 302e 0a1f 0a46 696c  m2:3679:0....Fil
│ │ │ │ -00065e00: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -00065e10: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00065e20: 6e73 2e69 6e66 6f2c 204e 6f64 653a 204f  ns.info, Node: O
│ │ │ │ -00065e30: 7074 696d 6973 6d2c 204e 6578 743a 204f  ptimism, Next: O
│ │ │ │ -00065e40: 7574 5269 6e67 2c20 5072 6576 3a20 6f64  utRing, Prev: od
│ │ │ │ -00065e50: 6445 7874 4d6f 6475 6c65 2c20 5570 3a20  dExtModule, Up: 
│ │ │ │ -00065e60: 546f 700a 0a4f 7074 696d 6973 6d20 2d2d  Top..Optimism --
│ │ │ │ -00065e70: 204f 7074 696f 6e20 746f 2068 6967 6853   Option to highS
│ │ │ │ -00065e80: 797a 7967 790a 2a2a 2a2a 2a2a 2a2a 2a2a  yzygy.**********
│ │ │ │ -00065e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00065ea0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -00065eb0: 653a 200a 2020 2020 2020 2020 6869 6768  e: .        high
│ │ │ │ -00065ec0: 5379 7a79 6779 284d 2c20 4f70 7469 6d69  Syzygy(M, Optimi
│ │ │ │ -00065ed0: 736d 203d 3e20 3129 0a20 202a 2049 6e70  sm => 1).  * Inp
│ │ │ │ -00065ee0: 7574 733a 0a20 2020 2020 202a 204f 7074  uts:.      * Opt
│ │ │ │ -00065ef0: 696d 6973 6d2c 2061 6e20 2a6e 6f74 6520  imism, an *note 
│ │ │ │ -00065f00: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ -00065f10: 6179 3244 6f63 295a 5a2c 2c20 0a0a 4465  ay2Doc)ZZ,, ..De
│ │ │ │ -00065f20: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -00065f30: 3d3d 3d3d 3d0a 0a49 6620 6869 6768 5379  =====..If highSy
│ │ │ │ -00065f40: 7a79 6779 284d 2920 6368 6f6f 7365 7320  zygy(M) chooses 
│ │ │ │ -00065f50: 7468 6520 702d 7468 2073 797a 7967 792c  the p-th syzygy,
│ │ │ │ -00065f60: 2074 6865 6e20 6869 6768 5379 7a79 6779   then highSyzygy
│ │ │ │ -00065f70: 284d 2c4f 7074 696d 6973 6d3d 3e72 290a  (M,Optimism=>r).
│ │ │ │ -00065f80: 6368 6f6f 7365 7320 7468 6520 2870 2d72  chooses the (p-r
│ │ │ │ -00065f90: 292d 7468 2073 797a 7967 792e 2028 506f  )-th syzygy. (Po
│ │ │ │ -00065fa0: 7369 7469 7665 204f 7074 696d 6973 6d20  sitive Optimism 
│ │ │ │ -00065fb0: 6368 6f6f 7365 7320 6120 6c6f 7765 7220  chooses a lower 
│ │ │ │ -00065fc0: 2268 6967 6822 2073 797a 7967 792c 0a6e  "high" syzygy,.n
│ │ │ │ -00065fd0: 6567 6174 6976 6520 4f70 7469 6d69 736d  egative Optimism
│ │ │ │ -00065fe0: 2061 2068 6967 6865 7220 2268 6967 6822   a higher "high"
│ │ │ │ -00065ff0: 2073 797a 7967 792e 0a0a 4361 7665 6174   syzygy...Caveat
│ │ │ │ -00066000: 0a3d 3d3d 3d3d 3d0a 0a41 7265 2074 6865  .======..Are the
│ │ │ │ -00066010: 7265 2063 6173 6573 2077 6865 6e20 706f  re cases when po
│ │ │ │ -00066020: 7369 7469 7665 204f 7074 696d 6973 6d20  sitive Optimism 
│ │ │ │ -00066030: 6973 206a 7573 7469 6669 6564 3f0a 0a53  is justified?..S
│ │ │ │ -00066040: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d  ee also.========
│ │ │ │ -00066050: 0a0a 2020 2a20 2a6e 6f74 6520 6d66 426f  ..  * *note mfBo
│ │ │ │ -00066060: 756e 643a 206d 6642 6f75 6e64 2c20 2d2d  und: mfBound, --
│ │ │ │ -00066070: 2064 6574 6572 6d69 6e65 7320 686f 7720   determines how 
│ │ │ │ -00066080: 6869 6768 2061 2073 797a 7967 7920 746f  high a syzygy to
│ │ │ │ -00066090: 2074 616b 6520 666f 720a 2020 2020 226d   take for.    "m
│ │ │ │ -000660a0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -000660b0: 6f6e 220a 2020 2a20 2a6e 6f74 6520 6869  on".  * *note hi
│ │ │ │ -000660c0: 6768 5379 7a79 6779 3a20 6869 6768 5379  ghSyzygy: highSy
│ │ │ │ -000660d0: 7a79 6779 2c20 2d2d 2052 6574 7572 6e73  zygy, -- Returns
│ │ │ │ -000660e0: 2061 2073 797a 7967 7920 6d6f 6475 6c65   a syzygy module
│ │ │ │ -000660f0: 206f 6e65 2062 6579 6f6e 6420 7468 650a   one beyond the.
│ │ │ │ -00066100: 2020 2020 7265 6775 6c61 7269 7479 206f      regularity o
│ │ │ │ -00066110: 6620 4578 7428 4d2c 6b29 0a0a 4675 6e63  f Ext(M,k)..Func
│ │ │ │ -00066120: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ -00066130: 6e61 6c20 6172 6775 6d65 6e74 206e 616d  nal argument nam
│ │ │ │ -00066140: 6564 204f 7074 696d 6973 6d3a 0a3d 3d3d  ed Optimism:.===
│ │ │ │ -00066150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00066160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00066170: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20  =============.. 
│ │ │ │ -00066180: 202a 2022 6869 6768 5379 7a79 6779 282e   * "highSyzygy(.
│ │ │ │ -00066190: 2e2e 2c4f 7074 696d 6973 6d3d 3e2e 2e2e  ..,Optimism=>...
│ │ │ │ -000661a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -000661b0: 6869 6768 5379 7a79 6779 3a20 6869 6768  highSyzygy: high
│ │ │ │ -000661c0: 5379 7a79 6779 2c20 2d2d 0a20 2020 2052  Syzygy, --.    R
│ │ │ │ -000661d0: 6574 7572 6e73 2061 2073 797a 7967 7920  eturns a syzygy 
│ │ │ │ -000661e0: 6d6f 6475 6c65 206f 6e65 2062 6579 6f6e  module one beyon
│ │ │ │ -000661f0: 6420 7468 6520 7265 6775 6c61 7269 7479  d the regularity
│ │ │ │ -00066200: 206f 6620 4578 7428 4d2c 6b29 0a20 202a   of Ext(M,k).  *
│ │ │ │ -00066210: 2022 7477 6f4d 6f6e 6f6d 6961 6c73 282e   "twoMonomials(.
│ │ │ │ -00066220: 2e2e 2c4f 7074 696d 6973 6d3d 3e2e 2e2e  ..,Optimism=>...
│ │ │ │ -00066230: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -00066240: 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 7477  twoMonomials: tw
│ │ │ │ -00066250: 6f4d 6f6e 6f6d 6961 6c73 2c0a 2020 2020  oMonomials,.    
│ │ │ │ -00066260: 2d2d 2074 616c 6c79 2074 6865 2073 6571  -- tally the seq
│ │ │ │ -00066270: 7565 6e63 6573 206f 6620 4252 616e 6b73  uences of BRanks
│ │ │ │ -00066280: 2066 6f72 2063 6572 7461 696e 2065 7861   for certain exa
│ │ │ │ -00066290: 6d70 6c65 730a 0a46 6f72 2074 6865 2070  mples..For the p
│ │ │ │ -000662a0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -000662b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -000662c0: 6520 6f62 6a65 6374 202a 6e6f 7465 204f  e object *note O
│ │ │ │ -000662d0: 7074 696d 6973 6d3a 204f 7074 696d 6973  ptimism: Optimis
│ │ │ │ -000662e0: 6d2c 2069 7320 6120 2a6e 6f74 6520 7379  m, is a *note sy
│ │ │ │ -000662f0: 6d62 6f6c 3a20 284d 6163 6175 6c61 7932  mbol: (Macaulay2
│ │ │ │ -00066300: 446f 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d  Doc)Symbol,...--
│ │ │ │ +00065d40: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +00065d50: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +00065d60: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00065d70: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00065d80: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00065d90: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ +00065da0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ +00065db0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +00065dc0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ +00065dd0: 323a 3336 3739 3a30 2e0a 1f0a 4669 6c65  2:3679:0....File
│ │ │ │ +00065de0: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ +00065df0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00065e00: 732e 696e 666f 2c20 4e6f 6465 3a20 4f70  s.info, Node: Op
│ │ │ │ +00065e10: 7469 6d69 736d 2c20 4e65 7874 3a20 4f75  timism, Next: Ou
│ │ │ │ +00065e20: 7452 696e 672c 2050 7265 763a 206f 6464  tRing, Prev: odd
│ │ │ │ +00065e30: 4578 744d 6f64 756c 652c 2055 703a 2054  ExtModule, Up: T
│ │ │ │ +00065e40: 6f70 0a0a 4f70 7469 6d69 736d 202d 2d20  op..Optimism -- 
│ │ │ │ +00065e50: 4f70 7469 6f6e 2074 6f20 6869 6768 5379  Option to highSy
│ │ │ │ +00065e60: 7a79 6779 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  zygy.***********
│ │ │ │ +00065e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00065e80: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +00065e90: 3a20 0a20 2020 2020 2020 2068 6967 6853  : .        highS
│ │ │ │ +00065ea0: 797a 7967 7928 4d2c 204f 7074 696d 6973  yzygy(M, Optimis
│ │ │ │ +00065eb0: 6d20 3d3e 2031 290a 2020 2a20 496e 7075  m => 1).  * Inpu
│ │ │ │ +00065ec0: 7473 3a0a 2020 2020 2020 2a20 4f70 7469  ts:.      * Opti
│ │ │ │ +00065ed0: 6d69 736d 2c20 616e 202a 6e6f 7465 2069  mism, an *note i
│ │ │ │ +00065ee0: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ +00065ef0: 7932 446f 6329 5a5a 2c2c 200a 0a44 6573  y2Doc)ZZ,, ..Des
│ │ │ │ +00065f00: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +00065f10: 3d3d 3d3d 0a0a 4966 2068 6967 6853 797a  ====..If highSyz
│ │ │ │ +00065f20: 7967 7928 4d29 2063 686f 6f73 6573 2074  ygy(M) chooses t
│ │ │ │ +00065f30: 6865 2070 2d74 6820 7379 7a79 6779 2c20  he p-th syzygy, 
│ │ │ │ +00065f40: 7468 656e 2068 6967 6853 797a 7967 7928  then highSyzygy(
│ │ │ │ +00065f50: 4d2c 4f70 7469 6d69 736d 3d3e 7229 0a63  M,Optimism=>r).c
│ │ │ │ +00065f60: 686f 6f73 6573 2074 6865 2028 702d 7229  hooses the (p-r)
│ │ │ │ +00065f70: 2d74 6820 7379 7a79 6779 2e20 2850 6f73  -th syzygy. (Pos
│ │ │ │ +00065f80: 6974 6976 6520 4f70 7469 6d69 736d 2063  itive Optimism c
│ │ │ │ +00065f90: 686f 6f73 6573 2061 206c 6f77 6572 2022  hooses a lower "
│ │ │ │ +00065fa0: 6869 6768 2220 7379 7a79 6779 2c0a 6e65  high" syzygy,.ne
│ │ │ │ +00065fb0: 6761 7469 7665 204f 7074 696d 6973 6d20  gative Optimism 
│ │ │ │ +00065fc0: 6120 6869 6768 6572 2022 6869 6768 2220  a higher "high" 
│ │ │ │ +00065fd0: 7379 7a79 6779 2e0a 0a43 6176 6561 740a  syzygy...Caveat.
│ │ │ │ +00065fe0: 3d3d 3d3d 3d3d 0a0a 4172 6520 7468 6572  ======..Are ther
│ │ │ │ +00065ff0: 6520 6361 7365 7320 7768 656e 2070 6f73  e cases when pos
│ │ │ │ +00066000: 6974 6976 6520 4f70 7469 6d69 736d 2069  itive Optimism i
│ │ │ │ +00066010: 7320 6a75 7374 6966 6965 643f 0a0a 5365  s justified?..Se
│ │ │ │ +00066020: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00066030: 0a20 202a 202a 6e6f 7465 206d 6642 6f75  .  * *note mfBou
│ │ │ │ +00066040: 6e64 3a20 6d66 426f 756e 642c 202d 2d20  nd: mfBound, -- 
│ │ │ │ +00066050: 6465 7465 726d 696e 6573 2068 6f77 2068  determines how h
│ │ │ │ +00066060: 6967 6820 6120 7379 7a79 6779 2074 6f20  igh a syzygy to 
│ │ │ │ +00066070: 7461 6b65 2066 6f72 0a20 2020 2022 6d61  take for.    "ma
│ │ │ │ +00066080: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +00066090: 6e22 0a20 202a 202a 6e6f 7465 2068 6967  n".  * *note hig
│ │ │ │ +000660a0: 6853 797a 7967 793a 2068 6967 6853 797a  hSyzygy: highSyz
│ │ │ │ +000660b0: 7967 792c 202d 2d20 5265 7475 726e 7320  ygy, -- Returns 
│ │ │ │ +000660c0: 6120 7379 7a79 6779 206d 6f64 756c 6520  a syzygy module 
│ │ │ │ +000660d0: 6f6e 6520 6265 796f 6e64 2074 6865 0a20  one beyond the. 
│ │ │ │ +000660e0: 2020 2072 6567 756c 6172 6974 7920 6f66     regularity of
│ │ │ │ +000660f0: 2045 7874 284d 2c6b 290a 0a46 756e 6374   Ext(M,k)..Funct
│ │ │ │ +00066100: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +00066110: 616c 2061 7267 756d 656e 7420 6e61 6d65  al argument name
│ │ │ │ +00066120: 6420 4f70 7469 6d69 736d 3a0a 3d3d 3d3d  d Optimism:.====
│ │ │ │ +00066130: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00066140: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00066150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ +00066160: 2a20 2268 6967 6853 797a 7967 7928 2e2e  * "highSyzygy(..
│ │ │ │ +00066170: 2e2c 4f70 7469 6d69 736d 3d3e 2e2e 2e29  .,Optimism=>...)
│ │ │ │ +00066180: 2220 2d2d 2073 6565 202a 6e6f 7465 2068  " -- see *note h
│ │ │ │ +00066190: 6967 6853 797a 7967 793a 2068 6967 6853  ighSyzygy: highS
│ │ │ │ +000661a0: 797a 7967 792c 202d 2d0a 2020 2020 5265  yzygy, --.    Re
│ │ │ │ +000661b0: 7475 726e 7320 6120 7379 7a79 6779 206d  turns a syzygy m
│ │ │ │ +000661c0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64  odule one beyond
│ │ │ │ +000661d0: 2074 6865 2072 6567 756c 6172 6974 7920   the regularity 
│ │ │ │ +000661e0: 6f66 2045 7874 284d 2c6b 290a 2020 2a20  of Ext(M,k).  * 
│ │ │ │ +000661f0: 2274 776f 4d6f 6e6f 6d69 616c 7328 2e2e  "twoMonomials(..
│ │ │ │ +00066200: 2e2c 4f70 7469 6d69 736d 3d3e 2e2e 2e29  .,Optimism=>...)
│ │ │ │ +00066210: 2220 2d2d 2073 6565 202a 6e6f 7465 2074  " -- see *note t
│ │ │ │ +00066220: 776f 4d6f 6e6f 6d69 616c 733a 2074 776f  woMonomials: two
│ │ │ │ +00066230: 4d6f 6e6f 6d69 616c 732c 0a20 2020 202d  Monomials,.    -
│ │ │ │ +00066240: 2d20 7461 6c6c 7920 7468 6520 7365 7175  - tally the sequ
│ │ │ │ +00066250: 656e 6365 7320 6f66 2042 5261 6e6b 7320  ences of BRanks 
│ │ │ │ +00066260: 666f 7220 6365 7274 6169 6e20 6578 616d  for certain exam
│ │ │ │ +00066270: 706c 6573 0a0a 466f 7220 7468 6520 7072  ples..For the pr
│ │ │ │ +00066280: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +00066290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +000662a0: 206f 626a 6563 7420 2a6e 6f74 6520 4f70   object *note Op
│ │ │ │ +000662b0: 7469 6d69 736d 3a20 4f70 7469 6d69 736d  timism: Optimism
│ │ │ │ +000662c0: 2c20 6973 2061 202a 6e6f 7465 2073 796d  , is a *note sym
│ │ │ │ +000662d0: 626f 6c3a 2028 4d61 6361 756c 6179 3244  bol: (Macaulay2D
│ │ │ │ +000662e0: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d  oc)Symbol,...---
│ │ │ │ +000662f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00066300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -00066360: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -00066370: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -00066380: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -00066390: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -000663a0: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ -000663b0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -000663c0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ -000663d0: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -000663e0: 7469 6f6e 732e 6d32 3a33 3136 353a 302e  tions.m2:3165:0.
│ │ │ │ -000663f0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ -00066400: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -00066410: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ -00066420: 6f64 653a 204f 7574 5269 6e67 2c20 4e65  ode: OutRing, Ne
│ │ │ │ -00066430: 7874 3a20 7073 694d 6170 732c 2050 7265  xt: psiMaps, Pre
│ │ │ │ -00066440: 763a 204f 7074 696d 6973 6d2c 2055 703a  v: Optimism, Up:
│ │ │ │ -00066450: 2054 6f70 0a0a 4f75 7452 696e 6720 2d2d   Top..OutRing --
│ │ │ │ -00066460: 204f 7074 696f 6e20 616c 6c6f 7769 6e67   Option allowing
│ │ │ │ -00066470: 2073 7065 6369 6669 6361 7469 6f6e 206f   specification o
│ │ │ │ -00066480: 6620 7468 6520 7269 6e67 206f 7665 7220  f the ring over 
│ │ │ │ -00066490: 7768 6963 6820 7468 6520 6f75 7470 7574  which the output
│ │ │ │ -000664a0: 2069 7320 6465 6669 6e65 640a 2a2a 2a2a   is defined.****
│ │ │ │ +00066330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +00066340: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +00066350: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +00066360: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +00066370: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +00066380: 7932 2d31 2e32 362e 3036 2b64 732f 4d32  y2-1.26.06+ds/M2
│ │ │ │ +00066390: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +000663a0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74  ges/.CompleteInt
│ │ │ │ +000663b0: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +000663c0: 696f 6e73 2e6d 323a 3331 3635 3a30 2e0a  ions.m2:3165:0..
│ │ │ │ +000663d0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +000663e0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +000663f0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +00066400: 6465 3a20 4f75 7452 696e 672c 204e 6578  de: OutRing, Nex
│ │ │ │ +00066410: 743a 2070 7369 4d61 7073 2c20 5072 6576  t: psiMaps, Prev
│ │ │ │ +00066420: 3a20 4f70 7469 6d69 736d 2c20 5570 3a20  : Optimism, Up: 
│ │ │ │ +00066430: 546f 700a 0a4f 7574 5269 6e67 202d 2d20  Top..OutRing -- 
│ │ │ │ +00066440: 4f70 7469 6f6e 2061 6c6c 6f77 696e 6720  Option allowing 
│ │ │ │ +00066450: 7370 6563 6966 6963 6174 696f 6e20 6f66  specification of
│ │ │ │ +00066460: 2074 6865 2072 696e 6720 6f76 6572 2077   the ring over w
│ │ │ │ +00066470: 6869 6368 2074 6865 206f 7574 7075 7420  hich the output 
│ │ │ │ +00066480: 6973 2064 6566 696e 6564 0a2a 2a2a 2a2a  is defined.*****
│ │ │ │ +00066490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000664a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000664b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000664c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000664d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000664e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000664f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00066500: 2a0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  *..See also.====
│ │ │ │ -00066510: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -00066520: 6576 656e 4578 744d 6f64 756c 653a 2065  evenExtModule: e
│ │ │ │ -00066530: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d  venExtModule, --
│ │ │ │ -00066540: 2065 7665 6e20 7061 7274 206f 6620 4578   even part of Ex
│ │ │ │ -00066550: 745e 2a28 4d2c 6b29 206f 7665 7220 610a  t^*(M,k) over a.
│ │ │ │ -00066560: 2020 2020 636f 6d70 6c65 7465 2069 6e74      complete int
│ │ │ │ -00066570: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ -00066580: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ -00066590: 6174 6f72 2072 696e 670a 2020 2a20 2a6e  ator ring.  * *n
│ │ │ │ -000665a0: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65  ote oddExtModule
│ │ │ │ -000665b0: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20  : oddExtModule, 
│ │ │ │ -000665c0: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ -000665d0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -000665e0: 2063 6f6d 706c 6574 650a 2020 2020 696e   complete.    in
│ │ │ │ -000665f0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f  tersection as mo
│ │ │ │ -00066600: 6475 6c65 206f 7665 7220 4349 206f 7065  dule over CI ope
│ │ │ │ -00066610: 7261 746f 7220 7269 6e67 0a0a 4675 6e63  rator ring..Func
│ │ │ │ -00066620: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ -00066630: 6e61 6c20 6172 6775 6d65 6e74 206e 616d  nal argument nam
│ │ │ │ -00066640: 6564 204f 7574 5269 6e67 3a0a 3d3d 3d3d  ed OutRing:.====
│ │ │ │ -00066650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00066660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00066670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00066680: 2022 6576 656e 4578 744d 6f64 756c 6528   "evenExtModule(
│ │ │ │ -00066690: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e  ...,OutRing=>...
│ │ │ │ -000666a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -000666b0: 6576 656e 4578 744d 6f64 756c 653a 0a20  evenExtModule:. 
│ │ │ │ -000666c0: 2020 2065 7665 6e45 7874 4d6f 6475 6c65     evenExtModule
│ │ │ │ -000666d0: 2c20 2d2d 2065 7665 6e20 7061 7274 206f  , -- even part o
│ │ │ │ -000666e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665  f Ext^*(M,k) ove
│ │ │ │ -000666f0: 7220 6120 636f 6d70 6c65 7465 2069 6e74  r a complete int
│ │ │ │ -00066700: 6572 7365 6374 696f 6e20 6173 0a20 2020  ersection as.   
│ │ │ │ -00066710: 206d 6f64 756c 6520 6f76 6572 2043 4920   module over CI 
│ │ │ │ -00066720: 6f70 6572 6174 6f72 2072 696e 670a 2020  operator ring.  
│ │ │ │ -00066730: 2a20 226f 6464 4578 744d 6f64 756c 6528  * "oddExtModule(
│ │ │ │ -00066740: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e  ...,OutRing=>...
│ │ │ │ -00066750: 2922 202d 2d20 7365 6520 2a6e 6f74 6520  )" -- see *note 
│ │ │ │ -00066760: 6f64 6445 7874 4d6f 6475 6c65 3a20 6f64  oddExtModule: od
│ │ │ │ -00066770: 6445 7874 4d6f 6475 6c65 2c0a 2020 2020  dExtModule,.    
│ │ │ │ -00066780: 2d2d 206f 6464 2070 6172 7420 6f66 2045  -- odd part of E
│ │ │ │ -00066790: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061  xt^*(M,k) over a
│ │ │ │ -000667a0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -000667b0: 6563 7469 6f6e 2061 7320 6d6f 6475 6c65  ection as module
│ │ │ │ -000667c0: 206f 7665 7220 4349 0a20 2020 206f 7065   over CI.    ope
│ │ │ │ -000667d0: 7261 746f 7220 7269 6e67 0a0a 466f 7220  rator ring..For 
│ │ │ │ -000667e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -000667f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00066800: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -00066810: 6f74 6520 4f75 7452 696e 673a 204f 7574  ote OutRing: Out
│ │ │ │ -00066820: 5269 6e67 2c20 6973 2061 202a 6e6f 7465  Ring, is a *note
│ │ │ │ -00066830: 2073 796d 626f 6c3a 2028 4d61 6361 756c   symbol: (Macaul
│ │ │ │ -00066840: 6179 3244 6f63 2953 796d 626f 6c2c 2e0a  ay2Doc)Symbol,..
│ │ │ │ -00066850: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +000664e0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ +000664f0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065  ===..  * *note e
│ │ │ │ +00066500: 7665 6e45 7874 4d6f 6475 6c65 3a20 6576  venExtModule: ev
│ │ │ │ +00066510: 656e 4578 744d 6f64 756c 652c 202d 2d20  enExtModule, -- 
│ │ │ │ +00066520: 6576 656e 2070 6172 7420 6f66 2045 7874  even part of Ext
│ │ │ │ +00066530: 5e2a 284d 2c6b 2920 6f76 6572 2061 0a20  ^*(M,k) over a. 
│ │ │ │ +00066540: 2020 2063 6f6d 706c 6574 6520 696e 7465     complete inte
│ │ │ │ +00066550: 7273 6563 7469 6f6e 2061 7320 6d6f 6475  rsection as modu
│ │ │ │ +00066560: 6c65 206f 7665 7220 4349 206f 7065 7261  le over CI opera
│ │ │ │ +00066570: 746f 7220 7269 6e67 0a20 202a 202a 6e6f  tor ring.  * *no
│ │ │ │ +00066580: 7465 206f 6464 4578 744d 6f64 756c 653a  te oddExtModule:
│ │ │ │ +00066590: 206f 6464 4578 744d 6f64 756c 652c 202d   oddExtModule, -
│ │ │ │ +000665a0: 2d20 6f64 6420 7061 7274 206f 6620 4578  - odd part of Ex
│ │ │ │ +000665b0: 745e 2a28 4d2c 6b29 206f 7665 7220 6120  t^*(M,k) over a 
│ │ │ │ +000665c0: 636f 6d70 6c65 7465 0a20 2020 2069 6e74  complete.    int
│ │ │ │ +000665d0: 6572 7365 6374 696f 6e20 6173 206d 6f64  ersection as mod
│ │ │ │ +000665e0: 756c 6520 6f76 6572 2043 4920 6f70 6572  ule over CI oper
│ │ │ │ +000665f0: 6174 6f72 2072 696e 670a 0a46 756e 6374  ator ring..Funct
│ │ │ │ +00066600: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +00066610: 616c 2061 7267 756d 656e 7420 6e61 6d65  al argument name
│ │ │ │ +00066620: 6420 4f75 7452 696e 673a 0a3d 3d3d 3d3d  d OutRing:.=====
│ │ │ │ +00066630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00066640: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00066650: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00066660: 2265 7665 6e45 7874 4d6f 6475 6c65 282e  "evenExtModule(.
│ │ │ │ +00066670: 2e2e 2c4f 7574 5269 6e67 3d3e 2e2e 2e29  ..,OutRing=>...)
│ │ │ │ +00066680: 2220 2d2d 2073 6565 202a 6e6f 7465 2065  " -- see *note e
│ │ │ │ +00066690: 7665 6e45 7874 4d6f 6475 6c65 3a0a 2020  venExtModule:.  
│ │ │ │ +000666a0: 2020 6576 656e 4578 744d 6f64 756c 652c    evenExtModule,
│ │ │ │ +000666b0: 202d 2d20 6576 656e 2070 6172 7420 6f66   -- even part of
│ │ │ │ +000666c0: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572   Ext^*(M,k) over
│ │ │ │ +000666d0: 2061 2063 6f6d 706c 6574 6520 696e 7465   a complete inte
│ │ │ │ +000666e0: 7273 6563 7469 6f6e 2061 730a 2020 2020  rsection as.    
│ │ │ │ +000666f0: 6d6f 6475 6c65 206f 7665 7220 4349 206f  module over CI o
│ │ │ │ +00066700: 7065 7261 746f 7220 7269 6e67 0a20 202a  perator ring.  *
│ │ │ │ +00066710: 2022 6f64 6445 7874 4d6f 6475 6c65 282e   "oddExtModule(.
│ │ │ │ +00066720: 2e2e 2c4f 7574 5269 6e67 3d3e 2e2e 2e29  ..,OutRing=>...)
│ │ │ │ +00066730: 2220 2d2d 2073 6565 202a 6e6f 7465 206f  " -- see *note o
│ │ │ │ +00066740: 6464 4578 744d 6f64 756c 653a 206f 6464  ddExtModule: odd
│ │ │ │ +00066750: 4578 744d 6f64 756c 652c 0a20 2020 202d  ExtModule,.    -
│ │ │ │ +00066760: 2d20 6f64 6420 7061 7274 206f 6620 4578  - odd part of Ex
│ │ │ │ +00066770: 745e 2a28 4d2c 6b29 206f 7665 7220 6120  t^*(M,k) over a 
│ │ │ │ +00066780: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +00066790: 6374 696f 6e20 6173 206d 6f64 756c 6520  ction as module 
│ │ │ │ +000667a0: 6f76 6572 2043 490a 2020 2020 6f70 6572  over CI.    oper
│ │ │ │ +000667b0: 6174 6f72 2072 696e 670a 0a46 6f72 2074  ator ring..For t
│ │ │ │ +000667c0: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +000667d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +000667e0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +000667f0: 7465 204f 7574 5269 6e67 3a20 4f75 7452  te OutRing: OutR
│ │ │ │ +00066800: 696e 672c 2069 7320 6120 2a6e 6f74 6520  ing, is a *note 
│ │ │ │ +00066810: 7379 6d62 6f6c 3a20 284d 6163 6175 6c61  symbol: (Macaula
│ │ │ │ +00066820: 7932 446f 6329 5379 6d62 6f6c 2c2e 0a0a  y2Doc)Symbol,...
│ │ │ │ +00066830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00066840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00066850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000668a0: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -000668b0: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ -000668c0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ -000668d0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -000668e0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64  aulay2-1.26.06+d
│ │ │ │ -000668f0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -00066900: 6163 6b61 6765 732f 0a43 6f6d 706c 6574  ackages/.Complet
│ │ │ │ -00066910: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -00066920: 6f6c 7574 696f 6e73 2e6d 323a 3336 3035  olutions.m2:3605
│ │ │ │ -00066930: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70  :0....File: Comp
│ │ │ │ -00066940: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -00066950: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ -00066960: 2c20 4e6f 6465 3a20 7073 694d 6170 732c  , Node: psiMaps,
│ │ │ │ -00066970: 204e 6578 743a 2072 6567 756c 6172 6974   Next: regularit
│ │ │ │ -00066980: 7953 6571 7565 6e63 652c 2050 7265 763a  ySequence, Prev:
│ │ │ │ -00066990: 204f 7574 5269 6e67 2c20 5570 3a20 546f   OutRing, Up: To
│ │ │ │ -000669a0: 700a 0a70 7369 4d61 7073 202d 2d20 6c69  p..psiMaps -- li
│ │ │ │ -000669b0: 7374 2074 6865 206d 6170 7320 2070 7369  st the maps  psi
│ │ │ │ -000669c0: 2870 293a 2042 5f31 2870 2920 2d2d 3e20  (p): B_1(p) --> 
│ │ │ │ -000669d0: 415f 3028 702d 3129 2069 6e20 6120 6d61  A_0(p-1) in a ma
│ │ │ │ -000669e0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ -000669f0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n.**************
│ │ │ │ +00066870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +00066880: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ +00066890: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ +000668a0: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ +000668b0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +000668c0: 756c 6179 322d 312e 3236 2e30 362b 6473  ulay2-1.26.06+ds
│ │ │ │ +000668d0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +000668e0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465  ckages/.Complete
│ │ │ │ +000668f0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +00066900: 6c75 7469 6f6e 732e 6d32 3a33 3630 353a  lutions.m2:3605:
│ │ │ │ +00066910: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c  0....File: Compl
│ │ │ │ +00066920: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +00066930: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ +00066940: 204e 6f64 653a 2070 7369 4d61 7073 2c20   Node: psiMaps, 
│ │ │ │ +00066950: 4e65 7874 3a20 7265 6775 6c61 7269 7479  Next: regularity
│ │ │ │ +00066960: 5365 7175 656e 6365 2c20 5072 6576 3a20  Sequence, Prev: 
│ │ │ │ +00066970: 4f75 7452 696e 672c 2055 703a 2054 6f70  OutRing, Up: Top
│ │ │ │ +00066980: 0a0a 7073 694d 6170 7320 2d2d 206c 6973  ..psiMaps -- lis
│ │ │ │ +00066990: 7420 7468 6520 6d61 7073 2020 7073 6928  t the maps  psi(
│ │ │ │ +000669a0: 7029 3a20 425f 3128 7029 202d 2d3e 2041  p): B_1(p) --> A
│ │ │ │ +000669b0: 5f30 2870 2d31 2920 696e 2061 206d 6174  _0(p-1) in a mat
│ │ │ │ +000669c0: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ +000669d0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  .***************
│ │ │ │ +000669e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000669f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00066a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00066a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00066a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00066a30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00066a40: 0a0a 2020 2a20 5573 6167 653a 200a 2020  ..  * Usage: .  
│ │ │ │ -00066a50: 2020 2020 2020 7073 6d61 7073 203d 2070        psmaps = p
│ │ │ │ -00066a60: 7369 4d61 7073 206d 660a 2020 2a20 496e  siMaps mf.  * In
│ │ │ │ -00066a70: 7075 7473 3a0a 2020 2020 2020 2a20 6d66  puts:.      * mf
│ │ │ │ -00066a80: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20  , a *note list: 
│ │ │ │ -00066a90: 284d 6163 6175 6c61 7932 446f 6329 4c69  (Macaulay2Doc)Li
│ │ │ │ -00066aa0: 7374 2c2c 206f 7574 7075 7420 6f66 2061  st,, output of a
│ │ │ │ -00066ab0: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ -00066ac0: 7469 6f6e 0a20 2020 2020 2020 2063 6f6d  tion.        com
│ │ │ │ -00066ad0: 7075 7461 7469 6f6e 0a20 202a 204f 7574  putation.  * Out
│ │ │ │ -00066ae0: 7075 7473 3a0a 2020 2020 2020 2a20 7073  puts:.      * ps
│ │ │ │ -00066af0: 6d61 7073 2c20 6120 2a6e 6f74 6520 6c69  maps, a *note li
│ │ │ │ -00066b00: 7374 3a20 284d 6163 6175 6c61 7932 446f  st: (Macaulay2Do
│ │ │ │ -00066b10: 6329 4c69 7374 2c2c 206c 6973 7420 6d61  c)List,, list ma
│ │ │ │ -00066b20: 7472 6963 6573 2024 645f 703a 0a20 2020  trices $d_p:.   
│ │ │ │ -00066b30: 2020 2020 2042 5f31 2870 295c 746f 2041       B_1(p)\to A
│ │ │ │ -00066b40: 5f30 2870 2d31 2924 0a0a 4465 7363 7269  _0(p-1)$..Descri
│ │ │ │ -00066b50: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ -00066b60: 3d0a 0a53 6565 2074 6865 2064 6f63 756d  =..See the docum
│ │ │ │ -00066b70: 656e 7461 7469 6f6e 2066 6f72 206d 6174  entation for mat
│ │ │ │ -00066b80: 7269 7846 6163 746f 7269 7a61 7469 6f6e  rixFactorization
│ │ │ │ -00066b90: 2066 6f72 2061 6e20 6578 616d 706c 652e   for an example.
│ │ │ │ -00066ba0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -00066bb0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d  ===..  * *note m
│ │ │ │ -00066bc0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -00066bd0: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72  on: matrixFactor
│ │ │ │ -00066be0: 697a 6174 696f 6e2c 202d 2d20 4d61 7073  ization, -- Maps
│ │ │ │ -00066bf0: 2069 6e20 6120 6869 6768 6572 0a20 2020   in a higher.   
│ │ │ │ -00066c00: 2063 6f64 696d 656e 7369 6f6e 206d 6174   codimension mat
│ │ │ │ -00066c10: 7269 7820 6661 6374 6f72 697a 6174 696f  rix factorizatio
│ │ │ │ -00066c20: 6e0a 2020 2a20 2a6e 6f74 6520 4252 616e  n.  * *note BRan
│ │ │ │ -00066c30: 6b73 3a20 4252 616e 6b73 2c20 2d2d 2072  ks: BRanks, -- r
│ │ │ │ -00066c40: 616e 6b73 206f 6620 7468 6520 6d6f 6475  anks of the modu
│ │ │ │ -00066c50: 6c65 7320 425f 6928 6429 2069 6e20 610a  les B_i(d) in a.
│ │ │ │ -00066c60: 2020 2020 6d61 7472 6978 4661 6374 6f72      matrixFactor
│ │ │ │ -00066c70: 697a 6174 696f 6e0a 2020 2a20 2a6e 6f74  ization.  * *not
│ │ │ │ -00066c80: 6520 624d 6170 733a 2062 4d61 7073 2c20  e bMaps: bMaps, 
│ │ │ │ -00066c90: 2d2d 206c 6973 7420 7468 6520 6d61 7073  -- list the maps
│ │ │ │ -00066ca0: 2020 645f 703a 425f 3128 7029 2d2d 3e42    d_p:B_1(p)-->B
│ │ │ │ -00066cb0: 5f30 2870 2920 696e 2061 0a20 2020 206d  _0(p) in a.    m
│ │ │ │ -00066cc0: 6174 7269 7846 6163 746f 7269 7a61 7469  atrixFactorizati
│ │ │ │ -00066cd0: 6f6e 0a20 202a 202a 6e6f 7465 2064 4d61  on.  * *note dMa
│ │ │ │ -00066ce0: 7073 3a20 644d 6170 732c 202d 2d20 6c69  ps: dMaps, -- li
│ │ │ │ -00066cf0: 7374 2074 6865 206d 6170 7320 2064 2870  st the maps  d(p
│ │ │ │ -00066d00: 293a 415f 3128 7029 2d2d 3e20 415f 3028  ):A_1(p)--> A_0(
│ │ │ │ -00066d10: 7029 2069 6e20 610a 2020 2020 6d61 7472  p) in a.    matr
│ │ │ │ -00066d20: 6978 4661 6374 6f72 697a 6174 696f 6e0a  ixFactorization.
│ │ │ │ -00066d30: 2020 2a20 2a6e 6f74 6520 684d 6170 733a    * *note hMaps:
│ │ │ │ -00066d40: 2068 4d61 7073 2c20 2d2d 206c 6973 7420   hMaps, -- list 
│ │ │ │ -00066d50: 7468 6520 6d61 7073 2020 6828 7029 3a20  the maps  h(p): 
│ │ │ │ -00066d60: 415f 3028 7029 2d2d 3e20 415f 3128 7029  A_0(p)--> A_1(p)
│ │ │ │ -00066d70: 2069 6e20 610a 2020 2020 6d61 7472 6978   in a.    matrix
│ │ │ │ -00066d80: 4661 6374 6f72 697a 6174 696f 6e0a 0a57  Factorization..W
│ │ │ │ -00066d90: 6179 7320 746f 2075 7365 2070 7369 4d61  ays to use psiMa
│ │ │ │ -00066da0: 7073 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ps:.============
│ │ │ │ -00066db0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2270  ========..  * "p
│ │ │ │ -00066dc0: 7369 4d61 7073 284c 6973 7429 220a 0a46  siMaps(List)"..F
│ │ │ │ -00066dd0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ -00066de0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ -00066df0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ -00066e00: 202a 6e6f 7465 2070 7369 4d61 7073 3a20   *note psiMaps: 
│ │ │ │ -00066e10: 7073 694d 6170 732c 2069 7320 6120 2a6e  psiMaps, is a *n
│ │ │ │ -00066e20: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -00066e30: 696f 6e3a 0a28 4d61 6361 756c 6179 3244  ion:.(Macaulay2D
│ │ │ │ -00066e40: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ -00066e50: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  n,...-----------
│ │ │ │ +00066a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +00066a20: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ +00066a30: 2020 2020 2070 736d 6170 7320 3d20 7073       psmaps = ps
│ │ │ │ +00066a40: 694d 6170 7320 6d66 0a20 202a 2049 6e70  iMaps mf.  * Inp
│ │ │ │ +00066a50: 7574 733a 0a20 2020 2020 202a 206d 662c  uts:.      * mf,
│ │ │ │ +00066a60: 2061 202a 6e6f 7465 206c 6973 743a 2028   a *note list: (
│ │ │ │ +00066a70: 4d61 6361 756c 6179 3244 6f63 294c 6973  Macaulay2Doc)Lis
│ │ │ │ +00066a80: 742c 2c20 6f75 7470 7574 206f 6620 6120  t,, output of a 
│ │ │ │ +00066a90: 6d61 7472 6978 4661 6374 6f72 697a 6174  matrixFactorizat
│ │ │ │ +00066aa0: 696f 6e0a 2020 2020 2020 2020 636f 6d70  ion.        comp
│ │ │ │ +00066ab0: 7574 6174 696f 6e0a 2020 2a20 4f75 7470  utation.  * Outp
│ │ │ │ +00066ac0: 7574 733a 0a20 2020 2020 202a 2070 736d  uts:.      * psm
│ │ │ │ +00066ad0: 6170 732c 2061 202a 6e6f 7465 206c 6973  aps, a *note lis
│ │ │ │ +00066ae0: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ +00066af0: 294c 6973 742c 2c20 6c69 7374 206d 6174  )List,, list mat
│ │ │ │ +00066b00: 7269 6365 7320 2464 5f70 3a0a 2020 2020  rices $d_p:.    
│ │ │ │ +00066b10: 2020 2020 425f 3128 7029 5c74 6f20 415f      B_1(p)\to A_
│ │ │ │ +00066b20: 3028 702d 3129 240a 0a44 6573 6372 6970  0(p-1)$..Descrip
│ │ │ │ +00066b30: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ +00066b40: 0a0a 5365 6520 7468 6520 646f 6375 6d65  ..See the docume
│ │ │ │ +00066b50: 6e74 6174 696f 6e20 666f 7220 6d61 7472  ntation for matr
│ │ │ │ +00066b60: 6978 4661 6374 6f72 697a 6174 696f 6e20  ixFactorization 
│ │ │ │ +00066b70: 666f 7220 616e 2065 7861 6d70 6c65 2e0a  for an example..
│ │ │ │ +00066b80: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ +00066b90: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6d61  ==..  * *note ma
│ │ │ │ +00066ba0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +00066bb0: 6e3a 206d 6174 7269 7846 6163 746f 7269  n: matrixFactori
│ │ │ │ +00066bc0: 7a61 7469 6f6e 2c20 2d2d 204d 6170 7320  zation, -- Maps 
│ │ │ │ +00066bd0: 696e 2061 2068 6967 6865 720a 2020 2020  in a higher.    
│ │ │ │ +00066be0: 636f 6469 6d65 6e73 696f 6e20 6d61 7472  codimension matr
│ │ │ │ +00066bf0: 6978 2066 6163 746f 7269 7a61 7469 6f6e  ix factorization
│ │ │ │ +00066c00: 0a20 202a 202a 6e6f 7465 2042 5261 6e6b  .  * *note BRank
│ │ │ │ +00066c10: 733a 2042 5261 6e6b 732c 202d 2d20 7261  s: BRanks, -- ra
│ │ │ │ +00066c20: 6e6b 7320 6f66 2074 6865 206d 6f64 756c  nks of the modul
│ │ │ │ +00066c30: 6573 2042 5f69 2864 2920 696e 2061 0a20  es B_i(d) in a. 
│ │ │ │ +00066c40: 2020 206d 6174 7269 7846 6163 746f 7269     matrixFactori
│ │ │ │ +00066c50: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465  zation.  * *note
│ │ │ │ +00066c60: 2062 4d61 7073 3a20 624d 6170 732c 202d   bMaps: bMaps, -
│ │ │ │ +00066c70: 2d20 6c69 7374 2074 6865 206d 6170 7320  - list the maps 
│ │ │ │ +00066c80: 2064 5f70 3a42 5f31 2870 292d 2d3e 425f   d_p:B_1(p)-->B_
│ │ │ │ +00066c90: 3028 7029 2069 6e20 610a 2020 2020 6d61  0(p) in a.    ma
│ │ │ │ +00066ca0: 7472 6978 4661 6374 6f72 697a 6174 696f  trixFactorizatio
│ │ │ │ +00066cb0: 6e0a 2020 2a20 2a6e 6f74 6520 644d 6170  n.  * *note dMap
│ │ │ │ +00066cc0: 733a 2064 4d61 7073 2c20 2d2d 206c 6973  s: dMaps, -- lis
│ │ │ │ +00066cd0: 7420 7468 6520 6d61 7073 2020 6428 7029  t the maps  d(p)
│ │ │ │ +00066ce0: 3a41 5f31 2870 292d 2d3e 2041 5f30 2870  :A_1(p)--> A_0(p
│ │ │ │ +00066cf0: 2920 696e 2061 0a20 2020 206d 6174 7269  ) in a.    matri
│ │ │ │ +00066d00: 7846 6163 746f 7269 7a61 7469 6f6e 0a20  xFactorization. 
│ │ │ │ +00066d10: 202a 202a 6e6f 7465 2068 4d61 7073 3a20   * *note hMaps: 
│ │ │ │ +00066d20: 684d 6170 732c 202d 2d20 6c69 7374 2074  hMaps, -- list t
│ │ │ │ +00066d30: 6865 206d 6170 7320 2068 2870 293a 2041  he maps  h(p): A
│ │ │ │ +00066d40: 5f30 2870 292d 2d3e 2041 5f31 2870 2920  _0(p)--> A_1(p) 
│ │ │ │ +00066d50: 696e 2061 0a20 2020 206d 6174 7269 7846  in a.    matrixF
│ │ │ │ +00066d60: 6163 746f 7269 7a61 7469 6f6e 0a0a 5761  actorization..Wa
│ │ │ │ +00066d70: 7973 2074 6f20 7573 6520 7073 694d 6170  ys to use psiMap
│ │ │ │ +00066d80: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  s:.=============
│ │ │ │ +00066d90: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7073  =======..  * "ps
│ │ │ │ +00066da0: 694d 6170 7328 4c69 7374 2922 0a0a 466f  iMaps(List)"..Fo
│ │ │ │ +00066db0: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ +00066dc0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +00066dd0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ +00066de0: 2a6e 6f74 6520 7073 694d 6170 733a 2070  *note psiMaps: p
│ │ │ │ +00066df0: 7369 4d61 7073 2c20 6973 2061 202a 6e6f  siMaps, is a *no
│ │ │ │ +00066e00: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +00066e10: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f  on:.(Macaulay2Do
│ │ │ │ +00066e20: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e  c)MethodFunction
│ │ │ │ +00066e30: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ,...------------
│ │ │ │ +00066e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00066e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00066e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00066ea0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ -00066eb0: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ -00066ec0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ -00066ed0: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ -00066ee0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e  /macaulay2-1.26.
│ │ │ │ -00066ef0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61  06+ds/M2/Macaula
│ │ │ │ -00066f00: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ -00066f10: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ -00066f20: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ -00066f30: 3434 3832 3a30 2e0a 1f0a 4669 6c65 3a20  4482:0....File: 
│ │ │ │ -00066f40: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00066f50: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00066f60: 696e 666f 2c20 4e6f 6465 3a20 7265 6775  info, Node: regu
│ │ │ │ -00066f70: 6c61 7269 7479 5365 7175 656e 6365 2c20  laritySequence, 
│ │ │ │ -00066f80: 4e65 7874 3a20 5332 2c20 5072 6576 3a20  Next: S2, Prev: 
│ │ │ │ -00066f90: 7073 694d 6170 732c 2055 703a 2054 6f70  psiMaps, Up: Top
│ │ │ │ -00066fa0: 0a0a 7265 6775 6c61 7269 7479 5365 7175  ..regularitySequ
│ │ │ │ -00066fb0: 656e 6365 202d 2d20 7265 6775 6c61 7269  ence -- regulari
│ │ │ │ -00066fc0: 7479 206f 6620 4578 7420 6d6f 6475 6c65  ty of Ext module
│ │ │ │ -00066fd0: 7320 666f 7220 6120 7365 7175 656e 6365  s for a sequence
│ │ │ │ -00066fe0: 206f 6620 4d43 4d20 6170 7072 6f78 696d   of MCM approxim
│ │ │ │ -00066ff0: 6174 696f 6e73 0a2a 2a2a 2a2a 2a2a 2a2a  ations.*********
│ │ │ │ +00066e80: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520  ---..The source 
│ │ │ │ +00066e90: 6f66 2074 6869 7320 646f 6375 6d65 6e74  of this document
│ │ │ │ +00066ea0: 2069 7320 696e 0a2f 6275 696c 642f 7265   is in./build/re
│ │ │ │ +00066eb0: 7072 6f64 7563 6962 6c65 2d70 6174 682f  producible-path/
│ │ │ │ +00066ec0: 6d61 6361 756c 6179 322d 312e 3236 2e30  macaulay2-1.26.0
│ │ │ │ +00066ed0: 362b 6473 2f4d 322f 4d61 6361 756c 6179  6+ds/M2/Macaulay
│ │ │ │ +00066ee0: 322f 7061 636b 6167 6573 2f0a 436f 6d70  2/packages/.Comp
│ │ │ │ +00066ef0: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ +00066f00: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34  Resolutions.m2:4
│ │ │ │ +00066f10: 3438 323a 302e 0a1f 0a46 696c 653a 2043  482:0....File: C
│ │ │ │ +00066f20: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +00066f30: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69  ionResolutions.i
│ │ │ │ +00066f40: 6e66 6f2c 204e 6f64 653a 2072 6567 756c  nfo, Node: regul
│ │ │ │ +00066f50: 6172 6974 7953 6571 7565 6e63 652c 204e  aritySequence, N
│ │ │ │ +00066f60: 6578 743a 2053 322c 2050 7265 763a 2070  ext: S2, Prev: p
│ │ │ │ +00066f70: 7369 4d61 7073 2c20 5570 3a20 546f 700a  siMaps, Up: Top.
│ │ │ │ +00066f80: 0a72 6567 756c 6172 6974 7953 6571 7565  .regularitySeque
│ │ │ │ +00066f90: 6e63 6520 2d2d 2072 6567 756c 6172 6974  nce -- regularit
│ │ │ │ +00066fa0: 7920 6f66 2045 7874 206d 6f64 756c 6573  y of Ext modules
│ │ │ │ +00066fb0: 2066 6f72 2061 2073 6571 7565 6e63 6520   for a sequence 
│ │ │ │ +00066fc0: 6f66 204d 434d 2061 7070 726f 7869 6d61  of MCM approxima
│ │ │ │ +00066fd0: 7469 6f6e 730a 2a2a 2a2a 2a2a 2a2a 2a2a  tions.**********
│ │ │ │ +00066fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00066ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -00067050: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -00067060: 204c 203d 2072 6567 756c 6172 6974 7953   L = regularityS
│ │ │ │ -00067070: 6571 7565 6e63 6520 2852 2c4d 290a 2020  equence (R,M).  
│ │ │ │ -00067080: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ -00067090: 2a20 522c 2061 202a 6e6f 7465 206c 6973  * R, a *note lis
│ │ │ │ -000670a0: 743a 2028 4d61 6361 756c 6179 3244 6f63  t: (Macaulay2Doc
│ │ │ │ -000670b0: 294c 6973 742c 2c20 6c69 7374 206f 6620  )List,, list of 
│ │ │ │ -000670c0: 7269 6e67 7320 525f 6920 3d0a 2020 2020  rings R_i =.    
│ │ │ │ -000670d0: 2020 2020 532f 2866 5f30 2e2e 665f 7b28      S/(f_0..f_{(
│ │ │ │ -000670e0: 692d 3129 7d29 2c20 636f 6d70 6c65 7465  i-1)}), complete
│ │ │ │ -000670f0: 2069 6e74 6572 7365 6374 696f 6e73 0a20   intersections. 
│ │ │ │ -00067100: 2020 2020 202a 204d 2c20 6120 2a6e 6f74       * M, a *not
│ │ │ │ -00067110: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ -00067120: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ -00067130: 206d 6f64 756c 6520 6f76 6572 2052 5f63   module over R_c
│ │ │ │ -00067140: 2077 6865 7265 2063 203d 0a20 2020 2020   where c =.     
│ │ │ │ -00067150: 2020 206c 656e 6774 6820 5220 2d20 312e     length R - 1.
│ │ │ │ -00067160: 0a20 202a 204f 7574 7075 7473 3a0a 2020  .  * Outputs:.  
│ │ │ │ -00067170: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465      * L, a *note
│ │ │ │ -00067180: 206c 6973 743a 2028 4d61 6361 756c 6179   list: (Macaulay
│ │ │ │ -00067190: 3244 6f63 294c 6973 742c 2c20 4c69 7374  2Doc)List,, List
│ │ │ │ -000671a0: 206f 6620 7061 6972 7320 7b72 6567 756c   of pairs {regul
│ │ │ │ -000671b0: 6172 6974 790a 2020 2020 2020 2020 6576  arity.        ev
│ │ │ │ -000671c0: 656e 4578 744d 6f64 756c 6520 4d5f 692c  enExtModule M_i,
│ │ │ │ -000671d0: 2072 6567 756c 6172 6974 7920 6f64 6445   regularity oddE
│ │ │ │ -000671e0: 7874 4d6f 6475 6c65 204d 5f69 290a 0a44  xtModule M_i)..D
│ │ │ │ -000671f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ -00067200: 3d3d 3d3d 3d3d 0a0a 436f 6d70 7574 6573  ======..Computes
│ │ │ │ -00067210: 2074 6865 206e 6f6e 2d66 7265 6520 7061   the non-free pa
│ │ │ │ -00067220: 7274 7320 4d5f 6920 6f66 2074 6865 204d  rts M_i of the M
│ │ │ │ -00067230: 434d 2061 7070 726f 7869 6d61 7469 6f6e  CM approximation
│ │ │ │ -00067240: 2074 6f20 4d20 6f76 6572 2052 5f69 2c0a   to M over R_i,.
│ │ │ │ -00067250: 7374 6f70 7069 6e67 2077 6865 6e20 4d5f  stopping when M_
│ │ │ │ -00067260: 6920 6265 636f 6d65 7320 6672 6565 2c20  i becomes free, 
│ │ │ │ -00067270: 616e 6420 7265 7475 726e 7320 7468 6520  and returns the 
│ │ │ │ -00067280: 6c69 7374 2077 686f 7365 2065 6c65 6d65  list whose eleme
│ │ │ │ -00067290: 6e74 7320 6172 6520 7468 650a 7061 6972  nts are the.pair
│ │ │ │ -000672a0: 7320 6f66 2072 6567 756c 6172 6974 6965  s of regularitie
│ │ │ │ -000672b0: 732c 2073 7461 7274 696e 6720 7769 7468  s, starting with
│ │ │ │ -000672c0: 204d 5f7b 2863 2d31 297d 204e 6f74 6520   M_{(c-1)} Note 
│ │ │ │ -000672d0: 7468 6174 2074 6865 2066 6972 7374 2070  that the first p
│ │ │ │ -000672e0: 6169 7220 6973 2066 6f72 0a74 6865 0a0a  air is for.the..
│ │ │ │ -000672f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -00067300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00067330: 0a7c 6931 203a 2063 203d 2033 3b64 3d32  .|i1 : c = 3;d=2
│ │ │ │ -00067340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067020: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00067030: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00067040: 4c20 3d20 7265 6775 6c61 7269 7479 5365  L = regularitySe
│ │ │ │ +00067050: 7175 656e 6365 2028 522c 4d29 0a20 202a  quence (R,M).  *
│ │ │ │ +00067060: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ +00067070: 2052 2c20 6120 2a6e 6f74 6520 6c69 7374   R, a *note list
│ │ │ │ +00067080: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ +00067090: 4c69 7374 2c2c 206c 6973 7420 6f66 2072  List,, list of r
│ │ │ │ +000670a0: 696e 6773 2052 5f69 203d 0a20 2020 2020  ings R_i =.     
│ │ │ │ +000670b0: 2020 2053 2f28 665f 302e 2e66 5f7b 2869     S/(f_0..f_{(i
│ │ │ │ +000670c0: 2d31 297d 292c 2063 6f6d 706c 6574 6520  -1)}), complete 
│ │ │ │ +000670d0: 696e 7465 7273 6563 7469 6f6e 730a 2020  intersections.  
│ │ │ │ +000670e0: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465      * M, a *note
│ │ │ │ +000670f0: 206d 6f64 756c 653a 2028 4d61 6361 756c   module: (Macaul
│ │ │ │ +00067100: 6179 3244 6f63 294d 6f64 756c 652c 2c20  ay2Doc)Module,, 
│ │ │ │ +00067110: 6d6f 6475 6c65 206f 7665 7220 525f 6320  module over R_c 
│ │ │ │ +00067120: 7768 6572 6520 6320 3d0a 2020 2020 2020  where c =.      
│ │ │ │ +00067130: 2020 6c65 6e67 7468 2052 202d 2031 2e0a    length R - 1..
│ │ │ │ +00067140: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +00067150: 2020 202a 204c 2c20 6120 2a6e 6f74 6520     * L, a *note 
│ │ │ │ +00067160: 6c69 7374 3a20 284d 6163 6175 6c61 7932  list: (Macaulay2
│ │ │ │ +00067170: 446f 6329 4c69 7374 2c2c 204c 6973 7420  Doc)List,, List 
│ │ │ │ +00067180: 6f66 2070 6169 7273 207b 7265 6775 6c61  of pairs {regula
│ │ │ │ +00067190: 7269 7479 0a20 2020 2020 2020 2065 7665  rity.        eve
│ │ │ │ +000671a0: 6e45 7874 4d6f 6475 6c65 204d 5f69 2c20  nExtModule M_i, 
│ │ │ │ +000671b0: 7265 6775 6c61 7269 7479 206f 6464 4578  regularity oddEx
│ │ │ │ +000671c0: 744d 6f64 756c 6520 4d5f 6929 0a0a 4465  tModule M_i)..De
│ │ │ │ +000671d0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +000671e0: 3d3d 3d3d 3d0a 0a43 6f6d 7075 7465 7320  =====..Computes 
│ │ │ │ +000671f0: 7468 6520 6e6f 6e2d 6672 6565 2070 6172  the non-free par
│ │ │ │ +00067200: 7473 204d 5f69 206f 6620 7468 6520 4d43  ts M_i of the MC
│ │ │ │ +00067210: 4d20 6170 7072 6f78 696d 6174 696f 6e20  M approximation 
│ │ │ │ +00067220: 746f 204d 206f 7665 7220 525f 692c 0a73  to M over R_i,.s
│ │ │ │ +00067230: 746f 7070 696e 6720 7768 656e 204d 5f69  topping when M_i
│ │ │ │ +00067240: 2062 6563 6f6d 6573 2066 7265 652c 2061   becomes free, a
│ │ │ │ +00067250: 6e64 2072 6574 7572 6e73 2074 6865 206c  nd returns the l
│ │ │ │ +00067260: 6973 7420 7768 6f73 6520 656c 656d 656e  ist whose elemen
│ │ │ │ +00067270: 7473 2061 7265 2074 6865 0a70 6169 7273  ts are the.pairs
│ │ │ │ +00067280: 206f 6620 7265 6775 6c61 7269 7469 6573   of regularities
│ │ │ │ +00067290: 2c20 7374 6172 7469 6e67 2077 6974 6820  , starting with 
│ │ │ │ +000672a0: 4d5f 7b28 632d 3129 7d20 4e6f 7465 2074  M_{(c-1)} Note t
│ │ │ │ +000672b0: 6861 7420 7468 6520 6669 7273 7420 7061  hat the first pa
│ │ │ │ +000672c0: 6972 2069 7320 666f 720a 7468 650a 0a2b  ir is for.the..+
│ │ │ │ +000672d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000672e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000672f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00067300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00067310: 7c69 3120 3a20 6320 3d20 333b 643d 3220  |i1 : c = 3;d=2 
│ │ │ │ +00067320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00067350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00067360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067370: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00067370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067390: 7c0a 7c6f 3220 3d20 3220 2020 2020 2020  |.|o2 = 2       
│ │ │ │  000673a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673b0: 207c 0a7c 6f32 203d 2032 2020 2020 2020   |.|o2 = 2      
│ │ │ │ +000673b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000673c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000673f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000673d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000673e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000673f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067430: 2d2d 2d2b 0a7c 6933 203a 2052 203d 2073  ---+.|i3 : R = s
│ │ │ │ -00067440: 6574 7570 5269 6e67 7328 632c 6429 3b20  etupRings(c,d); 
│ │ │ │ -00067450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067470: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00067410: 2d2d 2b0a 7c69 3320 3a20 5220 3d20 7365  --+.|i3 : R = se
│ │ │ │ +00067420: 7475 7052 696e 6773 2863 2c64 293b 2020  tupRings(c,d);  
│ │ │ │ +00067430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067450: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +00067460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00067470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000674a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000674b0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 6320  -----+.|i4 : Rc 
│ │ │ │ -000674c0: 3d20 525f 6320 2020 2020 2020 2020 2020  = R_c           
│ │ │ │ -000674d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067490: 2d2d 2d2d 2b0a 7c69 3420 3a20 5263 203d  ----+.|i4 : Rc =
│ │ │ │ +000674a0: 2052 5f63 2020 2020 2020 2020 2020 2020   R_c            
│ │ │ │ +000674b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000674c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000674d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000674e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000674f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000674f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067510: 2020 2020 2020 7c0a 7c6f 3420 3d20 5263        |.|o4 = Rc
│ │ │ │  00067520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067530: 2020 2020 2020 207c 0a7c 6f34 203d 2052         |.|o4 = R
│ │ │ │ -00067540: 6320 2020 2020 2020 2020 2020 2020 2020  c               
│ │ │ │ -00067550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067550: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00067560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067570: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00067570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000675a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000675b0: 2020 2020 2020 2020 207c 0a7c 6f34 203a           |.|o4 :
│ │ │ │ -000675c0: 2051 756f 7469 656e 7452 696e 6720 2020   QuotientRing   
│ │ │ │ -000675d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000675e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000675f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +00067590: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ +000675a0: 5175 6f74 6965 6e74 5269 6e67 2020 2020  QuotientRing    
│ │ │ │ +000675b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000675c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000675d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000675e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000675f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -00067640: 203a 204d 203d 2063 6f6b 6572 206d 6174   : M = coker mat
│ │ │ │ -00067650: 7269 787b 7b52 635f 302c 5263 5f31 2c52  rix{{Rc_0,Rc_1,R
│ │ │ │ -00067660: 635f 327d 2c7b 5263 5f31 2c52 635f 322c  c_2},{Rc_1,Rc_2,
│ │ │ │ -00067670: 5263 5f30 7d7d 2020 2020 2020 7c0a 7c20  Rc_0}}      |.| 
│ │ │ │ +00067610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +00067620: 3a20 4d20 3d20 636f 6b65 7220 6d61 7472  : M = coker matr
│ │ │ │ +00067630: 6978 7b7b 5263 5f30 2c52 635f 312c 5263  ix{{Rc_0,Rc_1,Rc
│ │ │ │ +00067640: 5f32 7d2c 7b52 635f 312c 5263 5f32 2c52  _2},{Rc_1,Rc_2,R
│ │ │ │ +00067650: 635f 307d 7d20 2020 2020 207c 0a7c 2020  c_0}}      |.|  
│ │ │ │ +00067660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00067680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000676a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000676b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000676c0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20  o5 = cokernel | 
│ │ │ │ -000676d0: 785f 3020 785f 3120 785f 3220 7c20 2020  x_0 x_1 x_2 |   
│ │ │ │ -000676e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000676f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00067700: 7c20 2020 2020 2020 2020 2020 2020 207c  |              |
│ │ │ │ -00067710: 2078 5f31 2078 5f32 2078 5f30 207c 2020   x_1 x_2 x_0 |  
│ │ │ │ -00067720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00067740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00067750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067690: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000676a0: 3520 3d20 636f 6b65 726e 656c 207c 2078  5 = cokernel | x
│ │ │ │ +000676b0: 5f30 2078 5f31 2078 5f32 207c 2020 2020  _0 x_1 x_2 |    
│ │ │ │ +000676c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000676d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000676e0: 2020 2020 2020 2020 2020 2020 2020 7c20                | 
│ │ │ │ +000676f0: 785f 3120 785f 3220 785f 3020 7c20 2020  x_1 x_2 x_0 |   
│ │ │ │ +00067700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067710: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00067720: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00067730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00067760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00067770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067780: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00067780: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  00067790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000677a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -000677b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000677c0: 207c 0a7c 6f35 203a 2052 632d 6d6f 6475   |.|o5 : Rc-modu
│ │ │ │ -000677d0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620  le, quotient of 
│ │ │ │ -000677e0: 5263 2020 2020 2020 2020 2020 2020 2020  Rc              
│ │ │ │ -000677f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067800: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +000677a0: 7c0a 7c6f 3520 3a20 5263 2d6d 6f64 756c  |.|o5 : Rc-modul
│ │ │ │ +000677b0: 652c 2071 756f 7469 656e 7420 6f66 2052  e, quotient of R
│ │ │ │ +000677c0: 6320 2020 2020 2020 2020 2020 2020 2020  c               
│ │ │ │ +000677d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000677e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +000677f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00067800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067840: 2d2d 2d2b 0a7c 6936 203a 2072 6567 756c  ---+.|i6 : regul
│ │ │ │ -00067850: 6172 6974 7953 6571 7565 6e63 6528 522c  aritySequence(R,
│ │ │ │ -00067860: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ -00067870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067880: 2020 2020 7c0a 7c72 6567 2065 7665 6e20      |.|reg even 
│ │ │ │ -00067890: 6578 742c 2073 6f63 2064 6567 7320 6576  ext, soc degs ev
│ │ │ │ -000678a0: 656e 2065 7874 2c20 7265 6720 6f64 6420  en ext, reg odd 
│ │ │ │ -000678b0: 6578 742c 2073 6f63 2064 6567 7320 6f64  ext, soc degs od
│ │ │ │ -000678c0: 6420 6578 747c 0a7c 2020 2020 2020 2020  d ext|.|        
│ │ │ │ +00067820: 2d2d 2b0a 7c69 3620 3a20 7265 6775 6c61  --+.|i6 : regula
│ │ │ │ +00067830: 7269 7479 5365 7175 656e 6365 2852 2c4d  ritySequence(R,M
│ │ │ │ +00067840: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00067850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067860: 2020 207c 0a7c 7265 6720 6576 656e 2065     |.|reg even e
│ │ │ │ +00067870: 7874 2c20 736f 6320 6465 6773 2065 7665  xt, soc degs eve
│ │ │ │ +00067880: 6e20 6578 742c 2072 6567 206f 6464 2065  n ext, reg odd e
│ │ │ │ +00067890: 7874 2c20 736f 6320 6465 6773 206f 6464  xt, soc degs odd
│ │ │ │ +000678a0: 2065 7874 7c0a 7c20 2020 2020 2020 2020   ext|.|         
│ │ │ │ +000678b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000678c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000678d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000678e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000678f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067900: 2020 2020 2020 7c0a 7c7b 332c 207b 312c        |.|{3, {1,
│ │ │ │ -00067910: 2031 2c20 317d 2c20 322c 207b 312c 2031   1, 1}, 2, {1, 1
│ │ │ │ -00067920: 7d7d 2020 2020 2020 2020 2020 2020 2020  }}              
│ │ │ │ -00067930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067940: 2020 2020 2020 207c 0a7c 7b32 2c20 7b30         |.|{2, {0
│ │ │ │ -00067950: 2c20 302c 2030 2c20 317d 2c20 322c 207b  , 0, 0, 1}, 2, {
│ │ │ │ -00067960: 302c 2030 2c20 307d 7d20 2020 2020 2020  0, 0, 0}}       
│ │ │ │ -00067970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00067980: 2020 2020 2020 2020 7c0a 7c7b 302c 207b          |.|{0, {
│ │ │ │ -00067990: 7d2c 2030 2c20 7b7d 7d20 2020 2020 2020  }, 0, {}}       
│ │ │ │ -000679a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000679b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000679c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +000678e0: 2020 2020 207c 0a7c 7b33 2c20 7b31 2c20       |.|{3, {1, 
│ │ │ │ +000678f0: 312c 2031 7d2c 2032 2c20 7b31 2c20 317d  1, 1}, 2, {1, 1}
│ │ │ │ +00067900: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +00067910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067920: 2020 2020 2020 7c0a 7c7b 322c 207b 302c        |.|{2, {0,
│ │ │ │ +00067930: 2030 2c20 302c 2031 7d2c 2032 2c20 7b30   0, 0, 1}, 2, {0
│ │ │ │ +00067940: 2c20 302c 2030 7d7d 2020 2020 2020 2020  , 0, 0}}        
│ │ │ │ +00067950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067960: 2020 2020 2020 207c 0a7c 7b30 2c20 7b7d         |.|{0, {}
│ │ │ │ +00067970: 2c20 302c 207b 7d7d 2020 2020 2020 2020  , 0, {}}        
│ │ │ │ +00067980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00067990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000679a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ +000679b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000679c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000679d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000679e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000679f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565  ----------+..See
│ │ │ │ -00067a10: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -00067a20: 2020 2a20 2a6e 6f74 6520 6170 7072 6f78    * *note approx
│ │ │ │ -00067a30: 696d 6174 696f 6e3a 2028 4d43 4d41 7070  imation: (MCMApp
│ │ │ │ -00067a40: 726f 7869 6d61 7469 6f6e 7329 6170 7072  roximations)appr
│ │ │ │ -00067a50: 6f78 696d 6174 696f 6e2c 202d 2d20 7265  oximation, -- re
│ │ │ │ -00067a60: 7475 726e 7320 7061 6972 206f 660a 2020  turns pair of.  
│ │ │ │ -00067a70: 2020 636f 6d70 6f6e 656e 7473 206f 6620    components of 
│ │ │ │ -00067a80: 7468 6520 6d61 7020 6672 6f6d 2074 6865  the map from the
│ │ │ │ -00067a90: 204d 434d 2061 7070 726f 7869 6d61 7469   MCM approximati
│ │ │ │ -00067aa0: 6f6e 0a20 202a 202a 6e6f 7465 2061 7573  on.  * *note aus
│ │ │ │ -00067ab0: 6c61 6e64 6572 496e 7661 7269 616e 743a  landerInvariant:
│ │ │ │ -00067ac0: 2028 4d43 4d41 7070 726f 7869 6d61 7469   (MCMApproximati
│ │ │ │ -00067ad0: 6f6e 7329 6175 736c 616e 6465 7249 6e76  ons)auslanderInv
│ │ │ │ -00067ae0: 6172 6961 6e74 2c20 2d2d 0a20 2020 206d  ariant, --.    m
│ │ │ │ -00067af0: 6561 7375 7265 7320 6661 696c 7572 6520  easures failure 
│ │ │ │ -00067b00: 6f66 2073 7572 6a65 6374 6976 6974 7920  of surjectivity 
│ │ │ │ -00067b10: 6f66 2074 6865 2065 7373 656e 7469 616c  of the essential
│ │ │ │ -00067b20: 204d 434d 2061 7070 726f 7869 6d61 7469   MCM approximati
│ │ │ │ -00067b30: 6f6e 0a0a 5761 7973 2074 6f20 7573 6520  on..Ways to use 
│ │ │ │ -00067b40: 7265 6775 6c61 7269 7479 5365 7175 656e  regularitySequen
│ │ │ │ -00067b50: 6365 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ce:.============
│ │ │ │ -00067b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00067b70: 3d3d 3d0a 0a20 202a 2022 7265 6775 6c61  ===..  * "regula
│ │ │ │ -00067b80: 7269 7479 5365 7175 656e 6365 284c 6973  ritySequence(Lis
│ │ │ │ -00067b90: 742c 4d6f 6475 6c65 2922 0a0a 466f 7220  t,Module)"..For 
│ │ │ │ -00067ba0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -00067bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00067bc0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -00067bd0: 6f74 6520 7265 6775 6c61 7269 7479 5365  ote regularitySe
│ │ │ │ -00067be0: 7175 656e 6365 3a20 7265 6775 6c61 7269  quence: regulari
│ │ │ │ -00067bf0: 7479 5365 7175 656e 6365 2c20 6973 2061  tySequence, is a
│ │ │ │ -00067c00: 202a 6e6f 7465 206d 6574 686f 640a 6675   *note method.fu
│ │ │ │ -00067c10: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61  nction: (Macaula
│ │ │ │ -00067c20: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ -00067c30: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  tion,...--------
│ │ │ │ +000679e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ +000679f0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +00067a00: 202a 202a 6e6f 7465 2061 7070 726f 7869   * *note approxi
│ │ │ │ +00067a10: 6d61 7469 6f6e 3a20 284d 434d 4170 7072  mation: (MCMAppr
│ │ │ │ +00067a20: 6f78 696d 6174 696f 6e73 2961 7070 726f  oximations)appro
│ │ │ │ +00067a30: 7869 6d61 7469 6f6e 2c20 2d2d 2072 6574  ximation, -- ret
│ │ │ │ +00067a40: 7572 6e73 2070 6169 7220 6f66 0a20 2020  urns pair of.   
│ │ │ │ +00067a50: 2063 6f6d 706f 6e65 6e74 7320 6f66 2074   components of t
│ │ │ │ +00067a60: 6865 206d 6170 2066 726f 6d20 7468 6520  he map from the 
│ │ │ │ +00067a70: 4d43 4d20 6170 7072 6f78 696d 6174 696f  MCM approximatio
│ │ │ │ +00067a80: 6e0a 2020 2a20 2a6e 6f74 6520 6175 736c  n.  * *note ausl
│ │ │ │ +00067a90: 616e 6465 7249 6e76 6172 6961 6e74 3a20  anderInvariant: 
│ │ │ │ +00067aa0: 284d 434d 4170 7072 6f78 696d 6174 696f  (MCMApproximatio
│ │ │ │ +00067ab0: 6e73 2961 7573 6c61 6e64 6572 496e 7661  ns)auslanderInva
│ │ │ │ +00067ac0: 7269 616e 742c 202d 2d0a 2020 2020 6d65  riant, --.    me
│ │ │ │ +00067ad0: 6173 7572 6573 2066 6169 6c75 7265 206f  asures failure o
│ │ │ │ +00067ae0: 6620 7375 726a 6563 7469 7669 7479 206f  f surjectivity o
│ │ │ │ +00067af0: 6620 7468 6520 6573 7365 6e74 6961 6c20  f the essential 
│ │ │ │ +00067b00: 4d43 4d20 6170 7072 6f78 696d 6174 696f  MCM approximatio
│ │ │ │ +00067b10: 6e0a 0a57 6179 7320 746f 2075 7365 2072  n..Ways to use r
│ │ │ │ +00067b20: 6567 756c 6172 6974 7953 6571 7565 6e63  egularitySequenc
│ │ │ │ +00067b30: 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  e:.=============
│ │ │ │ +00067b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00067b50: 3d3d 0a0a 2020 2a20 2272 6567 756c 6172  ==..  * "regular
│ │ │ │ +00067b60: 6974 7953 6571 7565 6e63 6528 4c69 7374  itySequence(List
│ │ │ │ +00067b70: 2c4d 6f64 756c 6529 220a 0a46 6f72 2074  ,Module)"..For t
│ │ │ │ +00067b80: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +00067b90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00067ba0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +00067bb0: 7465 2072 6567 756c 6172 6974 7953 6571  te regularitySeq
│ │ │ │ +00067bc0: 7565 6e63 653a 2072 6567 756c 6172 6974  uence: regularit
│ │ │ │ +00067bd0: 7953 6571 7565 6e63 652c 2069 7320 6120  ySequence, is a 
│ │ │ │ +00067be0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ +00067bf0: 6374 696f 6e3a 2028 4d61 6361 756c 6179  ction: (Macaulay
│ │ │ │ +00067c00: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +00067c10: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │ +00067c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00067c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00067c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00067c80: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00067c90: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -00067ca0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -00067cb0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -00067cc0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00067cd0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ -00067ce0: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ -00067cf0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ -00067d00: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ -00067d10: 6d32 3a32 3630 303a 302e 0a1f 0a46 696c  m2:2600:0....Fil
│ │ │ │ -00067d20: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ -00067d30: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ -00067d40: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2053  ns.info, Node: S
│ │ │ │ -00067d50: 322c 204e 6578 743a 2053 6861 6d61 7368  2, Next: Shamash
│ │ │ │ -00067d60: 2c20 5072 6576 3a20 7265 6775 6c61 7269  , Prev: regulari
│ │ │ │ -00067d70: 7479 5365 7175 656e 6365 2c20 5570 3a20  tySequence, Up: 
│ │ │ │ -00067d80: 546f 700a 0a53 3220 2d2d 2055 6e69 7665  Top..S2 -- Unive
│ │ │ │ -00067d90: 7273 616c 206d 6170 2074 6f20 6120 6d6f  rsal map to a mo
│ │ │ │ -00067da0: 6475 6c65 2073 6174 6973 6679 696e 6720  dule satisfying 
│ │ │ │ -00067db0: 5365 7272 6527 7320 636f 6e64 6974 696f  Serre's conditio
│ │ │ │ -00067dc0: 6e20 5332 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a  n S2.***********
│ │ │ │ +00067c60: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +00067c70: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +00067c80: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00067c90: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00067ca0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00067cb0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ +00067cc0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43  lay2/packages/.C
│ │ │ │ +00067cd0: 6f6d 706c 6574 6549 6e74 6572 7365 6374  ompleteIntersect
│ │ │ │ +00067ce0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d  ionResolutions.m
│ │ │ │ +00067cf0: 323a 3236 3030 3a30 2e0a 1f0a 4669 6c65  2:2600:0....File
│ │ │ │ +00067d00: 3a20 436f 6d70 6c65 7465 496e 7465 7273  : CompleteInters
│ │ │ │ +00067d10: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ +00067d20: 732e 696e 666f 2c20 4e6f 6465 3a20 5332  s.info, Node: S2
│ │ │ │ +00067d30: 2c20 4e65 7874 3a20 5368 616d 6173 682c  , Next: Shamash,
│ │ │ │ +00067d40: 2050 7265 763a 2072 6567 756c 6172 6974   Prev: regularit
│ │ │ │ +00067d50: 7953 6571 7565 6e63 652c 2055 703a 2054  ySequence, Up: T
│ │ │ │ +00067d60: 6f70 0a0a 5332 202d 2d20 556e 6976 6572  op..S2 -- Univer
│ │ │ │ +00067d70: 7361 6c20 6d61 7020 746f 2061 206d 6f64  sal map to a mod
│ │ │ │ +00067d80: 756c 6520 7361 7469 7366 7969 6e67 2053  ule satisfying S
│ │ │ │ +00067d90: 6572 7265 2773 2063 6f6e 6469 7469 6f6e  erre's condition
│ │ │ │ +00067da0: 2053 320a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a   S2.************
│ │ │ │ +00067db0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00067dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00067dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00067e00: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -00067e10: 200a 2020 2020 2020 2020 6620 3d20 5332   .        f = S2
│ │ │ │ -00067e20: 2862 2c4d 290a 2020 2a20 496e 7075 7473  (b,M).  * Inputs
│ │ │ │ -00067e30: 3a0a 2020 2020 2020 2a20 622c 2061 6e20  :.      * b, an 
│ │ │ │ -00067e40: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ -00067e50: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ -00067e60: 2c20 6465 6772 6565 2062 6f75 6e64 2074  , degree bound t
│ │ │ │ -00067e70: 6f20 7768 6963 6820 746f 2063 6172 7279  o which to carry
│ │ │ │ -00067e80: 0a20 2020 2020 2020 2074 6865 2063 6f6d  .        the com
│ │ │ │ -00067e90: 7075 7461 7469 6f6e 0a20 2020 2020 202a  putation.      *
│ │ │ │ -00067ea0: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -00067eb0: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -00067ec0: 6329 4d6f 6475 6c65 2c2c 200a 2020 2a20  c)Module,, .  * 
│ │ │ │ -00067ed0: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -00067ee0: 2066 2c20 6120 2a6e 6f74 6520 6d61 7472   f, a *note matr
│ │ │ │ -00067ef0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -00067f00: 6329 4d61 7472 6978 2c2c 2064 6566 696e  c)Matrix,, defin
│ │ │ │ -00067f10: 696e 6720 6120 6d61 7020 4d2d 2d3e 4d27  ing a map M-->M'
│ │ │ │ -00067f20: 2074 6861 740a 2020 2020 2020 2020 6167   that.        ag
│ │ │ │ -00067f30: 7265 6573 2077 6974 6820 7468 6520 5332  rees with the S2
│ │ │ │ -00067f40: 2d69 6669 6361 7469 6f6e 206f 6620 4d20  -ification of M 
│ │ │ │ -00067f50: 696e 2064 6567 7265 6573 2024 5c67 6571  in degrees $\geq
│ │ │ │ -00067f60: 2062 240a 0a44 6573 6372 6970 7469 6f6e   b$..Description
│ │ │ │ -00067f70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966  .===========..If
│ │ │ │ -00067f80: 204d 2069 7320 6120 6772 6164 6564 206d   M is a graded m
│ │ │ │ -00067f90: 6f64 756c 6520 6f76 6572 2061 2072 696e  odule over a rin
│ │ │ │ -00067fa0: 6720 532c 2074 6865 6e20 7468 6520 5332  g S, then the S2
│ │ │ │ -00067fb0: 2d69 6669 6361 7469 6f6e 206f 6620 4d20  -ification of M 
│ │ │ │ -00067fc0: 6973 205c 7375 6d5f 7b64 0a5c 696e 205a  is \sum_{d.\in Z
│ │ │ │ -00067fd0: 5a7d 2048 5e30 2828 7368 6561 6620 4d29  Z} H^0((sheaf M)
│ │ │ │ -00067fe0: 2864 2929 2c20 7768 6963 6820 6d61 7920  (d)), which may 
│ │ │ │ -00067ff0: 6265 2063 6f6d 7075 7465 6420 6173 206c  be computed as l
│ │ │ │ -00068000: 696d 5f7b 642d 3e5c 696e 6674 797d 2048  im_{d->\infty} H
│ │ │ │ -00068010: 6f6d 2849 5f64 2c4d 292c 0a77 6865 7265  om(I_d,M),.where
│ │ │ │ -00068020: 2049 5f64 2069 7320 616e 7920 7365 7175   I_d is any sequ
│ │ │ │ -00068030: 656e 6365 206f 6620 6964 6561 6c73 2063  ence of ideals c
│ │ │ │ -00068040: 6f6e 7461 696e 6564 2069 6e20 6869 6768  ontained in high
│ │ │ │ -00068050: 6572 2061 6e64 2068 6967 6865 7220 706f  er and higher po
│ │ │ │ -00068060: 7765 7273 206f 660a 535f 2b2e 2054 6865  wers of.S_+. The
│ │ │ │ -00068070: 7265 2069 7320 6120 6e61 7475 7261 6c20  re is a natural 
│ │ │ │ -00068080: 7265 7374 7269 6374 696f 6e20 6d61 7020  restriction map 
│ │ │ │ -00068090: 663a 204d 203d 2048 6f6d 2853 2c4d 2920  f: M = Hom(S,M) 
│ │ │ │ -000680a0: 5c74 6f20 486f 6d28 495f 642c 4d29 2e20  \to Hom(I_d,M). 
│ │ │ │ -000680b0: 5765 0a63 6f6d 7075 7465 2061 6c6c 2074  We.compute all t
│ │ │ │ -000680c0: 6869 7320 7573 696e 6720 7468 6520 6964  his using the id
│ │ │ │ -000680d0: 6561 6c73 2049 5f64 2067 656e 6572 6174  eals I_d generat
│ │ │ │ -000680e0: 6564 2062 7920 7468 6520 642d 7468 2070  ed by the d-th p
│ │ │ │ -000680f0: 6f77 6572 7320 6f66 2074 6865 0a76 6172  owers of the.var
│ │ │ │ -00068100: 6961 626c 6573 2069 6e20 532e 0a0a 5369  iables in S...Si
│ │ │ │ -00068110: 6e63 6520 7468 6520 7265 7375 6c74 206d  nce the result m
│ │ │ │ -00068120: 6179 206e 6f74 2062 6520 6669 6e69 7465  ay not be finite
│ │ │ │ -00068130: 6c79 2067 656e 6572 6174 6564 2028 7468  ly generated (th
│ │ │ │ -00068140: 6973 2068 6170 7065 6e73 2069 6620 616e  is happens if an
│ │ │ │ -00068150: 6420 6f6e 6c79 2069 6620 4d0a 6861 7320  d only if M.has 
│ │ │ │ -00068160: 616e 2061 7373 6f63 6961 7465 6420 7072  an associated pr
│ │ │ │ -00068170: 696d 6520 6f66 2064 696d 656e 7369 6f6e  ime of dimension
│ │ │ │ -00068180: 2031 292c 2077 6520 636f 6d70 7574 6520   1), we compute 
│ │ │ │ -00068190: 6f6e 6c79 2075 7020 746f 2061 2073 7065  only up to a spe
│ │ │ │ -000681a0: 6369 6669 6564 0a64 6567 7265 6520 626f  cified.degree bo
│ │ │ │ -000681b0: 756e 6420 622e 2046 6f72 2074 6865 2072  und b. For the r
│ │ │ │ -000681c0: 6573 756c 7420 746f 2062 6520 636f 7272  esult to be corr
│ │ │ │ -000681d0: 6563 7420 646f 776e 2074 6f20 6465 6772  ect down to degr
│ │ │ │ -000681e0: 6565 2062 2c20 6974 2069 7320 7375 6666  ee b, it is suff
│ │ │ │ -000681f0: 6963 6965 6e74 0a74 6f20 636f 6d70 7574  icient.to comput
│ │ │ │ -00068200: 6520 486f 6d28 492c 4d29 2077 6865 7265  e Hom(I,M) where
│ │ │ │ -00068210: 2049 205c 7375 6273 6574 2028 535f 2b29   I \subset (S_+)
│ │ │ │ -00068220: 5e7b 722d 627d 2e0a 0a2b 2d2d 2d2d 2d2d  ^{r-b}...+------
│ │ │ │ +00067de0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +00067df0: 0a20 2020 2020 2020 2066 203d 2053 3228  .        f = S2(
│ │ │ │ +00067e00: 622c 4d29 0a20 202a 2049 6e70 7574 733a  b,M).  * Inputs:
│ │ │ │ +00067e10: 0a20 2020 2020 202a 2062 2c20 616e 202a  .      * b, an *
│ │ │ │ +00067e20: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ +00067e30: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ +00067e40: 2064 6567 7265 6520 626f 756e 6420 746f   degree bound to
│ │ │ │ +00067e50: 2077 6869 6368 2074 6f20 6361 7272 790a   which to carry.
│ │ │ │ +00067e60: 2020 2020 2020 2020 7468 6520 636f 6d70          the comp
│ │ │ │ +00067e70: 7574 6174 696f 6e0a 2020 2020 2020 2a20  utation.      * 
│ │ │ │ +00067e80: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ +00067e90: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +00067ea0: 294d 6f64 756c 652c 2c20 0a20 202a 204f  )Module,, .  * O
│ │ │ │ +00067eb0: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +00067ec0: 662c 2061 202a 6e6f 7465 206d 6174 7269  f, a *note matri
│ │ │ │ +00067ed0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +00067ee0: 294d 6174 7269 782c 2c20 6465 6669 6e69  )Matrix,, defini
│ │ │ │ +00067ef0: 6e67 2061 206d 6170 204d 2d2d 3e4d 2720  ng a map M-->M' 
│ │ │ │ +00067f00: 7468 6174 0a20 2020 2020 2020 2061 6772  that.        agr
│ │ │ │ +00067f10: 6565 7320 7769 7468 2074 6865 2053 322d  ees with the S2-
│ │ │ │ +00067f20: 6966 6963 6174 696f 6e20 6f66 204d 2069  ification of M i
│ │ │ │ +00067f30: 6e20 6465 6772 6565 7320 245c 6765 7120  n degrees $\geq 
│ │ │ │ +00067f40: 6224 0a0a 4465 7363 7269 7074 696f 6e0a  b$..Description.
│ │ │ │ +00067f50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620  ===========..If 
│ │ │ │ +00067f60: 4d20 6973 2061 2067 7261 6465 6420 6d6f  M is a graded mo
│ │ │ │ +00067f70: 6475 6c65 206f 7665 7220 6120 7269 6e67  dule over a ring
│ │ │ │ +00067f80: 2053 2c20 7468 656e 2074 6865 2053 322d   S, then the S2-
│ │ │ │ +00067f90: 6966 6963 6174 696f 6e20 6f66 204d 2069  ification of M i
│ │ │ │ +00067fa0: 7320 5c73 756d 5f7b 640a 5c69 6e20 5a5a  s \sum_{d.\in ZZ
│ │ │ │ +00067fb0: 7d20 485e 3028 2873 6865 6166 204d 2928  } H^0((sheaf M)(
│ │ │ │ +00067fc0: 6429 292c 2077 6869 6368 206d 6179 2062  d)), which may b
│ │ │ │ +00067fd0: 6520 636f 6d70 7574 6564 2061 7320 6c69  e computed as li
│ │ │ │ +00067fe0: 6d5f 7b64 2d3e 5c69 6e66 7479 7d20 486f  m_{d->\infty} Ho
│ │ │ │ +00067ff0: 6d28 495f 642c 4d29 2c0a 7768 6572 6520  m(I_d,M),.where 
│ │ │ │ +00068000: 495f 6420 6973 2061 6e79 2073 6571 7565  I_d is any seque
│ │ │ │ +00068010: 6e63 6520 6f66 2069 6465 616c 7320 636f  nce of ideals co
│ │ │ │ +00068020: 6e74 6169 6e65 6420 696e 2068 6967 6865  ntained in highe
│ │ │ │ +00068030: 7220 616e 6420 6869 6768 6572 2070 6f77  r and higher pow
│ │ │ │ +00068040: 6572 7320 6f66 0a53 5f2b 2e20 5468 6572  ers of.S_+. Ther
│ │ │ │ +00068050: 6520 6973 2061 206e 6174 7572 616c 2072  e is a natural r
│ │ │ │ +00068060: 6573 7472 6963 7469 6f6e 206d 6170 2066  estriction map f
│ │ │ │ +00068070: 3a20 4d20 3d20 486f 6d28 532c 4d29 205c  : M = Hom(S,M) \
│ │ │ │ +00068080: 746f 2048 6f6d 2849 5f64 2c4d 292e 2057  to Hom(I_d,M). W
│ │ │ │ +00068090: 650a 636f 6d70 7574 6520 616c 6c20 7468  e.compute all th
│ │ │ │ +000680a0: 6973 2075 7369 6e67 2074 6865 2069 6465  is using the ide
│ │ │ │ +000680b0: 616c 7320 495f 6420 6765 6e65 7261 7465  als I_d generate
│ │ │ │ +000680c0: 6420 6279 2074 6865 2064 2d74 6820 706f  d by the d-th po
│ │ │ │ +000680d0: 7765 7273 206f 6620 7468 650a 7661 7269  wers of the.vari
│ │ │ │ +000680e0: 6162 6c65 7320 696e 2053 2e0a 0a53 696e  ables in S...Sin
│ │ │ │ +000680f0: 6365 2074 6865 2072 6573 756c 7420 6d61  ce the result ma
│ │ │ │ +00068100: 7920 6e6f 7420 6265 2066 696e 6974 656c  y not be finitel
│ │ │ │ +00068110: 7920 6765 6e65 7261 7465 6420 2874 6869  y generated (thi
│ │ │ │ +00068120: 7320 6861 7070 656e 7320 6966 2061 6e64  s happens if and
│ │ │ │ +00068130: 206f 6e6c 7920 6966 204d 0a68 6173 2061   only if M.has a
│ │ │ │ +00068140: 6e20 6173 736f 6369 6174 6564 2070 7269  n associated pri
│ │ │ │ +00068150: 6d65 206f 6620 6469 6d65 6e73 696f 6e20  me of dimension 
│ │ │ │ +00068160: 3129 2c20 7765 2063 6f6d 7075 7465 206f  1), we compute o
│ │ │ │ +00068170: 6e6c 7920 7570 2074 6f20 6120 7370 6563  nly up to a spec
│ │ │ │ +00068180: 6966 6965 640a 6465 6772 6565 2062 6f75  ified.degree bou
│ │ │ │ +00068190: 6e64 2062 2e20 466f 7220 7468 6520 7265  nd b. For the re
│ │ │ │ +000681a0: 7375 6c74 2074 6f20 6265 2063 6f72 7265  sult to be corre
│ │ │ │ +000681b0: 6374 2064 6f77 6e20 746f 2064 6567 7265  ct down to degre
│ │ │ │ +000681c0: 6520 622c 2069 7420 6973 2073 7566 6669  e b, it is suffi
│ │ │ │ +000681d0: 6369 656e 740a 746f 2063 6f6d 7075 7465  cient.to compute
│ │ │ │ +000681e0: 2048 6f6d 2849 2c4d 2920 7768 6572 6520   Hom(I,M) where 
│ │ │ │ +000681f0: 4920 5c73 7562 7365 7420 2853 5f2b 295e  I \subset (S_+)^
│ │ │ │ +00068200: 7b72 2d62 7d2e 0a0a 2b2d 2d2d 2d2d 2d2d  {r-b}...+-------
│ │ │ │ +00068210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00068220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068270: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b  -------+.|i1 : k
│ │ │ │ -00068280: 6b3d 5a5a 2f31 3031 2020 2020 2020 2020  k=ZZ/101        
│ │ │ │ +00068250: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b  ------+.|i1 : kk
│ │ │ │ +00068260: 3d5a 5a2f 3130 3120 2020 2020 2020 2020  =ZZ/101         
│ │ │ │ +00068270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000682a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000682a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000682b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000682c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000682c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000682d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000682e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000682f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000682f0: 2020 2020 2020 7c0a 7c6f 3120 3d20 6b6b        |.|o1 = kk
│ │ │ │  00068300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068310: 2020 2020 2020 207c 0a7c 6f31 203d 206b         |.|o1 = k
│ │ │ │ -00068320: 6b20 2020 2020 2020 2020 2020 2020 2020  k               
│ │ │ │ +00068310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068340: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068360: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000683a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000683b0: 2020 2020 2020 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ -000683c0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +00068390: 2020 2020 2020 7c0a 7c6f 3120 3a20 5175        |.|o1 : Qu
│ │ │ │ +000683a0: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +000683b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000683c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000683d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000683e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000683f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068400: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000683e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000683f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00068400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068450: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053  -------+.|i2 : S
│ │ │ │ -00068460: 203d 206b 6b5b 612c 622c 632c 645d 2020   = kk[a,b,c,d]  
│ │ │ │ +00068430: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320  ------+.|i2 : S 
│ │ │ │ +00068440: 3d20 6b6b 5b61 2c62 2c63 2c64 5d20 2020  = kk[a,b,c,d]   
│ │ │ │ +00068450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068480: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000684a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000684a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000684b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000684c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000684d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000684d0: 2020 2020 2020 7c0a 7c6f 3220 3d20 5320        |.|o2 = S 
│ │ │ │  000684e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000684f0: 2020 2020 2020 207c 0a7c 6f32 203d 2053         |.|o2 = S
│ │ │ │ +000684f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068520: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068540: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068590: 2020 2020 2020 207c 0a7c 6f32 203a 2050         |.|o2 : P
│ │ │ │ -000685a0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ +00068570: 2020 2020 2020 7c0a 7c6f 3220 3a20 506f        |.|o2 : Po
│ │ │ │ +00068580: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020  lynomialRing    
│ │ │ │ +00068590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000685a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000685b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000685c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000685d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000685e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000685c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000685d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000685e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000685f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068630: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204d  -------+.|i3 : M
│ │ │ │ -00068640: 203d 2074 7275 6e63 6174 6528 332c 535e   = truncate(3,S^
│ │ │ │ -00068650: 3129 2020 2020 2020 2020 2020 2020 2020  1)              
│ │ │ │ -00068660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068610: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 4d20  ------+.|i3 : M 
│ │ │ │ +00068620: 3d20 7472 756e 6361 7465 2833 2c53 5e31  = truncate(3,S^1
│ │ │ │ +00068630: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00068640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068660: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068680: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000686a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000686b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000686c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000686d0: 2020 2020 2020 207c 0a7c 6f33 203d 2069         |.|o3 = i
│ │ │ │ -000686e0: 6d61 6765 207c 2064 3320 6364 3220 6264  mage | d3 cd2 bd
│ │ │ │ -000686f0: 3220 6164 3220 6332 6420 6263 6420 6163  2 ad2 c2d bcd ac
│ │ │ │ -00068700: 6420 6232 6420 6162 6420 6132 6420 6333  d b2d abd a2d c3
│ │ │ │ -00068710: 2062 6332 2061 6332 2062 3263 2061 6263   bc2 ac2 b2c abc
│ │ │ │ -00068720: 2061 3263 2062 337c 0a7c 2020 2020 2020   a2c b3|.|      
│ │ │ │ +000686b0: 2020 2020 2020 7c0a 7c6f 3320 3d20 696d        |.|o3 = im
│ │ │ │ +000686c0: 6167 6520 7c20 6433 2063 6432 2062 6432  age | d3 cd2 bd2
│ │ │ │ +000686d0: 2061 6432 2063 3264 2062 6364 2061 6364   ad2 c2d bcd acd
│ │ │ │ +000686e0: 2062 3264 2061 6264 2061 3264 2063 3320   b2d abd a2d c3 
│ │ │ │ +000686f0: 6263 3220 6163 3220 6232 6320 6162 6320  bc2 ac2 b2c abc 
│ │ │ │ +00068700: 6132 6320 6233 7c0a 7c20 2020 2020 2020  a2c b3|.|       
│ │ │ │ +00068710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068750: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068770: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068770: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │  00068780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068790: 2020 2020 2020 2031 2020 2020 2020 2020         1        
│ │ │ │ -000687a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000687b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000687c0: 2020 2020 2020 207c 0a7c 6f33 203a 2053         |.|o3 : S
│ │ │ │ -000687d0: 2d6d 6f64 756c 652c 2073 7562 6d6f 6475  -module, submodu
│ │ │ │ -000687e0: 6c65 206f 6620 5320 2020 2020 2020 2020  le of S         
│ │ │ │ -000687f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068810: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │ +00068790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000687a0: 2020 2020 2020 7c0a 7c6f 3320 3a20 532d        |.|o3 : S-
│ │ │ │ +000687b0: 6d6f 6475 6c65 2c20 7375 626d 6f64 756c  module, submodul
│ │ │ │ +000687c0: 6520 6f66 2053 2020 2020 2020 2020 2020  e of S          
│ │ │ │ +000687d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000687e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000687f0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00068800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00068810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068860: 2d2d 2d2d 2d2d 2d7c 0a7c 6162 3220 6132  -------|.|ab2 a2
│ │ │ │ -00068870: 6220 6133 207c 2020 2020 2020 2020 2020  b a3 |          
│ │ │ │ +00068840: 2d2d 2d2d 2d2d 7c0a 7c61 6232 2061 3262  ------|.|ab2 a2b
│ │ │ │ +00068850: 2061 3320 7c20 2020 2020 2020 2020 2020   a3 |           
│ │ │ │ +00068860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000688a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000688b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00068890: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000688a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000688b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000688c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000688d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000688e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068900: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2062  -------+.|i4 : b
│ │ │ │ -00068910: 6574 7469 206d 6174 7269 7820 5332 2830  etti matrix S2(0
│ │ │ │ -00068920: 2c4d 2920 2020 2020 2020 2020 2020 2020  ,M)             
│ │ │ │ -00068930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000688e0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6265  ------+.|i4 : be
│ │ │ │ +000688f0: 7474 6920 6d61 7472 6978 2053 3228 302c  tti matrix S2(0,
│ │ │ │ +00068900: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ +00068910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068930: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068950: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000689a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000689b0: 2020 2020 2020 3020 2031 2020 2020 2020        0  1      
│ │ │ │ +00068980: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068990: 2020 2020 2030 2020 3120 2020 2020 2020       0  1       
│ │ │ │ +000689a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000689b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000689c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000689d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000689e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000689f0: 2020 2020 2020 207c 0a7c 6f34 203d 2074         |.|o4 = t
│ │ │ │ -00068a00: 6f74 616c 3a20 3120 3230 2020 2020 2020  otal: 1 20      
│ │ │ │ +000689d0: 2020 2020 2020 7c0a 7c6f 3420 3d20 746f        |.|o4 = to
│ │ │ │ +000689e0: 7461 6c3a 2031 2032 3020 2020 2020 2020  tal: 1 20       
│ │ │ │ +000689f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a40: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068a50: 2020 2030 3a20 3120 202e 2020 2020 2020     0: 1  .      
│ │ │ │ +00068a20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068a30: 2020 303a 2031 2020 2e20 2020 2020 2020    0: 1  .       
│ │ │ │ +00068a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068a90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068aa0: 2020 2031 3a20 2e20 202e 2020 2020 2020     1: .  .      
│ │ │ │ +00068a70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068a80: 2020 313a 202e 2020 2e20 2020 2020 2020    1: .  .       
│ │ │ │ +00068a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ae0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068af0: 2020 2032 3a20 2e20 3230 2020 2020 2020     2: . 20      
│ │ │ │ +00068ac0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068ad0: 2020 323a 202e 2032 3020 2020 2020 2020    2: . 20       
│ │ │ │ +00068ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068b10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b30: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068b80: 2020 2020 2020 207c 0a7c 6f34 203a 2042         |.|o4 : B
│ │ │ │ -00068b90: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00068b60: 2020 2020 2020 7c0a 7c6f 3420 3a20 4265        |.|o4 : Be
│ │ │ │ +00068b70: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +00068b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00068bb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00068bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00068bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2062  -------+.|i5 : b
│ │ │ │ -00068c30: 6574 7469 206d 6174 7269 7820 5332 2831  etti matrix S2(1
│ │ │ │ -00068c40: 2c4d 2920 2020 2020 2020 2020 2020 2020  ,M)             
│ │ │ │ -00068c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068c00: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6265  ------+.|i5 : be
│ │ │ │ +00068c10: 7474 6920 6d61 7472 6978 2053 3228 312c  tti matrix S2(1,
│ │ │ │ +00068c20: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ +00068c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068c50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068c70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068cd0: 2020 2020 2020 3020 2031 2020 2020 2020        0  1      
│ │ │ │ +00068ca0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068cb0: 2020 2020 2030 2020 3120 2020 2020 2020       0  1       
│ │ │ │ +00068cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d10: 2020 2020 2020 207c 0a7c 6f35 203d 2074         |.|o5 = t
│ │ │ │ -00068d20: 6f74 616c 3a20 3120 3230 2020 2020 2020  otal: 1 20      
│ │ │ │ +00068cf0: 2020 2020 2020 7c0a 7c6f 3520 3d20 746f        |.|o5 = to
│ │ │ │ +00068d00: 7461 6c3a 2031 2032 3020 2020 2020 2020  tal: 1 20       
│ │ │ │ +00068d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068d70: 2020 2030 3a20 3120 202e 2020 2020 2020     0: 1  .      
│ │ │ │ +00068d40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068d50: 2020 303a 2031 2020 2e20 2020 2020 2020    0: 1  .       
│ │ │ │ +00068d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068db0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068dc0: 2020 2031 3a20 2e20 202e 2020 2020 2020     1: .  .      
│ │ │ │ +00068d90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068da0: 2020 313a 202e 2020 2e20 2020 2020 2020    1: .  .       
│ │ │ │ +00068db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00068e10: 2020 2032 3a20 2e20 3230 2020 2020 2020     2: . 20      
│ │ │ │ +00068de0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00068df0: 2020 323a 202e 2032 3020 2020 2020 2020    2: . 20       
│ │ │ │ +00068e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068e30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00068e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e50: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00068e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ea0: 2020 2020 2020 207c 0a7c 6f35 203a 2042         |.|o5 : B
│ │ │ │ -00068eb0: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ +00068e80: 2020 2020 2020 7c0a 7c6f 3520 3a20 4265        |.|o5 : Be
│ │ │ │ +00068e90: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +00068ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068ef0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +00068ed0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00068ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00068ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00068f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00068f40: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d  -------+.|i6 : M
│ │ │ │ -00068f50: 203d 2053 5e31 2f69 6e74 6572 7365 6374   = S^1/intersect
│ │ │ │ -00068f60: 2869 6465 616c 2261 2c62 2c63 222c 2069  (ideal"a,b,c", i
│ │ │ │ -00068f70: 6465 616c 2262 2c63 2c64 222c 6964 6561  deal"b,c,d",idea
│ │ │ │ -00068f80: 6c22 632c 642c 6122 2c69 6465 616c 2264  l"c,d,a",ideal"d
│ │ │ │ -00068f90: 2c61 2c62 2229 207c 0a7c 2020 2020 2020  ,a,b") |.|      
│ │ │ │ +00068f20: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20  ------+.|i6 : M 
│ │ │ │ +00068f30: 3d20 535e 312f 696e 7465 7273 6563 7428  = S^1/intersect(
│ │ │ │ +00068f40: 6964 6561 6c22 612c 622c 6322 2c20 6964  ideal"a,b,c", id
│ │ │ │ +00068f50: 6561 6c22 622c 632c 6422 2c69 6465 616c  eal"b,c,d",ideal
│ │ │ │ +00068f60: 2263 2c64 2c61 222c 6964 6561 6c22 642c  "c,d,a",ideal"d,
│ │ │ │ +00068f70: 612c 6222 2920 7c0a 7c20 2020 2020 2020  a,b") |.|       
│ │ │ │ +00068f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00068fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00068fe0: 2020 2020 2020 207c 0a7c 6f36 203d 2063         |.|o6 = c
│ │ │ │ -00068ff0: 6f6b 6572 6e65 6c20 7c20 6364 2062 6420  okernel | cd bd 
│ │ │ │ -00069000: 6164 2062 6320 6163 2061 6220 7c20 2020  ad bc ac ab |   
│ │ │ │ -00069010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00068fc0: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f        |.|o6 = co
│ │ │ │ +00068fd0: 6b65 726e 656c 207c 2063 6420 6264 2061  kernel | cd bd a
│ │ │ │ +00068fe0: 6420 6263 2061 6320 6162 207c 2020 2020  d bc ac ab |    
│ │ │ │ +00068ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069010: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00069020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069030: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069060: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00069070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069080: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069080: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │  00069090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000690a0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -000690b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000690c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000690d0: 2020 2020 2020 207c 0a7c 6f36 203a 2053         |.|o6 : S
│ │ │ │ -000690e0: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -000690f0: 7420 6f66 2053 2020 2020 2020 2020 2020  t of S          
│ │ │ │ -00069100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000690a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000690b0: 2020 2020 2020 7c0a 7c6f 3620 3a20 532d        |.|o6 : S-
│ │ │ │ +000690c0: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +000690d0: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ +000690e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000690f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069100: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069170: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2070  -------+.|i7 : p
│ │ │ │ -00069180: 7275 6e65 2073 6f75 7263 6520 5332 2830  rune source S2(0
│ │ │ │ -00069190: 2c4d 2920 2020 2020 2020 2020 2020 2020  ,M)             
│ │ │ │ -000691a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069150: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 7072  ------+.|i7 : pr
│ │ │ │ +00069160: 756e 6520 736f 7572 6365 2053 3228 302c  une source S2(0,
│ │ │ │ +00069170: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ +00069180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000691a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000691b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000691c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000691c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000691e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000691f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069210: 2020 2020 2020 207c 0a7c 6f37 203d 2063         |.|o7 = c
│ │ │ │ -00069220: 6f6b 6572 6e65 6c20 7c20 6364 2062 6420  okernel | cd bd 
│ │ │ │ -00069230: 6164 2062 6320 6163 2061 6220 7c20 2020  ad bc ac ab |   
│ │ │ │ -00069240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000691f0: 2020 2020 2020 7c0a 7c6f 3720 3d20 636f        |.|o7 = co
│ │ │ │ +00069200: 6b65 726e 656c 207c 2063 6420 6264 2061  kernel | cd bd a
│ │ │ │ +00069210: 6420 6263 2061 6320 6162 207c 2020 2020  d bc ac ab |    
│ │ │ │ +00069220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069240: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00069250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069260: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069290: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000692a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000692b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000692b0: 2020 2020 2031 2020 2020 2020 2020 2020       1          
│ │ │ │  000692c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000692d0: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -000692e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000692f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069300: 2020 2020 2020 207c 0a7c 6f37 203a 2053         |.|o7 : S
│ │ │ │ -00069310: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00069320: 7420 6f66 2053 2020 2020 2020 2020 2020  t of S          
│ │ │ │ -00069330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069350: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000692d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000692e0: 2020 2020 2020 7c0a 7c6f 3720 3a20 532d        |.|o7 : S-
│ │ │ │ +000692f0: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +00069300: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ +00069310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069330: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000693a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2070  -------+.|i8 : p
│ │ │ │ -000693b0: 7275 6e65 2074 6172 6765 7420 5332 2830  rune target S2(0
│ │ │ │ -000693c0: 2c4d 2920 2020 2020 2020 2020 2020 2020  ,M)             
│ │ │ │ -000693d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069380: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 7072  ------+.|i8 : pr
│ │ │ │ +00069390: 756e 6520 7461 7267 6574 2053 3228 302c  une target S2(0,
│ │ │ │ +000693a0: 4d29 2020 2020 2020 2020 2020 2020 2020  M)              
│ │ │ │ +000693b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000693c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000693d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000693e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000693f0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000693f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069440: 2020 2020 2020 207c 0a7c 6f38 203d 2063         |.|o8 = c
│ │ │ │ -00069450: 6f6b 6572 6e65 6c20 7b2d 317d 207c 2064  okernel {-1} | d
│ │ │ │ -00069460: 2063 2062 2030 2030 2030 2030 2030 2030   c b 0 0 0 0 0 0
│ │ │ │ -00069470: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -00069480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000694a0: 2020 2020 2020 2020 7b2d 317d 207c 2030          {-1} | 0
│ │ │ │ -000694b0: 2030 2030 2064 2063 2061 2030 2030 2030   0 0 d c a 0 0 0
│ │ │ │ -000694c0: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -000694d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000694e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000694f0: 2020 2020 2020 2020 7b2d 317d 207c 2030          {-1} | 0
│ │ │ │ -00069500: 2030 2030 2030 2030 2030 2064 2062 2061   0 0 0 0 0 d b a
│ │ │ │ -00069510: 2030 2030 2030 207c 2020 2020 2020 2020   0 0 0 |        
│ │ │ │ -00069520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069530: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00069540: 2020 2020 2020 2020 7b2d 317d 207c 2030          {-1} | 0
│ │ │ │ -00069550: 2030 2030 2030 2030 2030 2030 2030 2030   0 0 0 0 0 0 0 0
│ │ │ │ -00069560: 2063 2062 2061 207c 2020 2020 2020 2020   c b a |        
│ │ │ │ +00069420: 2020 2020 2020 7c0a 7c6f 3820 3d20 636f        |.|o8 = co
│ │ │ │ +00069430: 6b65 726e 656c 207b 2d31 7d20 7c20 6420  kernel {-1} | d 
│ │ │ │ +00069440: 6320 6220 3020 3020 3020 3020 3020 3020  c b 0 0 0 0 0 0 
│ │ │ │ +00069450: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ +00069460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069470: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00069480: 2020 2020 2020 207b 2d31 7d20 7c20 3020         {-1} | 0 
│ │ │ │ +00069490: 3020 3020 6420 6320 6120 3020 3020 3020  0 0 d c a 0 0 0 
│ │ │ │ +000694a0: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ +000694b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000694c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000694d0: 2020 2020 2020 207b 2d31 7d20 7c20 3020         {-1} | 0 
│ │ │ │ +000694e0: 3020 3020 3020 3020 3020 6420 6220 6120  0 0 0 0 0 d b a 
│ │ │ │ +000694f0: 3020 3020 3020 7c20 2020 2020 2020 2020  0 0 0 |         
│ │ │ │ +00069500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069510: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00069520: 2020 2020 2020 207b 2d31 7d20 7c20 3020         {-1} | 0 
│ │ │ │ +00069530: 3020 3020 3020 3020 3020 3020 3020 3020  0 0 0 0 0 0 0 0 
│ │ │ │ +00069540: 6320 6220 6120 7c20 2020 2020 2020 2020  c b a |         
│ │ │ │ +00069550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069560: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  00069570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069580: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00069580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00069590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000695a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000695b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000695b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  000695c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000695d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000695d0: 2020 2020 2034 2020 2020 2020 2020 2020       4          
│ │ │ │  000695e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000695f0: 2020 2020 2020 3420 2020 2020 2020 2020        4         
│ │ │ │ -00069600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069620: 2020 2020 2020 207c 0a7c 6f38 203a 2053         |.|o8 : S
│ │ │ │ -00069630: 2d6d 6f64 756c 652c 2071 756f 7469 656e  -module, quotien
│ │ │ │ -00069640: 7420 6f66 2053 2020 2020 2020 2020 2020  t of S          
│ │ │ │ -00069650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069670: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000695f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069600: 2020 2020 2020 7c0a 7c6f 3820 3a20 532d        |.|o8 : S-
│ │ │ │ +00069610: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74  module, quotient
│ │ │ │ +00069620: 206f 6620 5320 2020 2020 2020 2020 2020   of S           
│ │ │ │ +00069630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069650: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00069690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000696a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000696c0: 2d2d 2d2d 2d2d 2d2b 0a0a 4174 206f 6e65  -------+..At one
│ │ │ │ -000696d0: 2074 696d 6520 4445 2068 6f70 6564 2074   time DE hoped t
│ │ │ │ -000696e0: 6861 742c 2069 6620 4d20 7765 7265 2061  hat, if M were a
│ │ │ │ -000696f0: 206d 6f64 756c 6520 6f76 6572 2074 6865   module over the
│ │ │ │ -00069700: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -00069710: 6563 7469 6f6e 2052 0a77 6974 6820 7265  ection R.with re
│ │ │ │ -00069720: 7369 6475 6520 6669 656c 6420 6b2c 2074  sidue field k, t
│ │ │ │ -00069730: 6865 6e20 7468 6520 6e61 7475 7261 6c20  hen the natural 
│ │ │ │ -00069740: 6d61 7020 6672 6f6d 2022 636f 6d70 6c65  map from "comple
│ │ │ │ -00069750: 7465 2220 4578 7420 6d6f 6475 6c65 2022  te" Ext module "
│ │ │ │ -00069760: 2877 6964 6568 6174 0a45 7874 295f 5228  (widehat.Ext)_R(
│ │ │ │ -00069770: 4d2c 6b29 2220 746f 2074 6865 2053 322d  M,k)" to the S2-
│ │ │ │ -00069780: 6966 6963 6174 696f 6e20 6f66 2045 7874  ification of Ext
│ │ │ │ -00069790: 5f52 284d 2c6b 2920 776f 756c 6420 6265  _R(M,k) would be
│ │ │ │ -000697a0: 2073 7572 6a65 6374 6976 653b 0a65 7175   surjective;.equ
│ │ │ │ -000697b0: 6976 616c 656e 746c 792c 2069 6620 4e20  ivalently, if N 
│ │ │ │ -000697c0: 7765 7265 2061 2073 7566 6669 6369 656e  were a sufficien
│ │ │ │ -000697d0: 746c 7920 6e65 6761 7469 7665 2073 797a  tly negative syz
│ │ │ │ -000697e0: 7967 7920 6f66 204d 2c20 7468 656e 2074  ygy of M, then t
│ │ │ │ -000697f0: 6865 2066 6972 7374 0a6c 6f63 616c 2063  he first.local c
│ │ │ │ -00069800: 6f68 6f6d 6f6c 6f67 7920 6d6f 6475 6c65  ohomology module
│ │ │ │ -00069810: 206f 6620 4578 745f 5228 4d2c 6b29 2077   of Ext_R(M,k) w
│ │ │ │ -00069820: 6f75 6c64 2062 6520 7a65 726f 2e20 5468  ould be zero. Th
│ │ │ │ -00069830: 6973 2069 7320 6661 6c73 652c 2061 7320  is is false, as 
│ │ │ │ -00069840: 7368 6f77 6e20 6279 0a74 6865 2066 6f6c  shown by.the fol
│ │ │ │ -00069850: 6c6f 7769 6e67 2065 7861 6d70 6c65 3a0a  lowing example:.
│ │ │ │ -00069860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00069870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069890: 2d2d 2d2b 0a7c 6939 203a 2053 203d 205a  ---+.|i9 : S = Z
│ │ │ │ -000698a0: 5a2f 3130 315b 785f 302e 2e78 5f32 5d3b  Z/101[x_0..x_2];
│ │ │ │ -000698b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000698c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000698d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000698e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000698f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069900: 3020 3a20 6666 203d 2061 7070 6c79 2833  0 : ff = apply(3
│ │ │ │ -00069910: 2c20 692d 3e78 5f69 5e32 293b 2020 2020  , i->x_i^2);    
│ │ │ │ -00069920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069930: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00069940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069960: 2d2d 2d2b 0a7c 6931 3120 3a20 5220 3d20  ---+.|i11 : R = 
│ │ │ │ -00069970: 532f 6964 6561 6c20 6666 3b20 2020 2020  S/ideal ff;     
│ │ │ │ -00069980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069990: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -000699a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000699b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000699c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -000699d0: 3220 3a20 4d20 3d20 636f 6b65 726e 656c  2 : M = cokernel
│ │ │ │ -000699e0: 206d 6174 7269 7820 7b7b 785f 302c 2078   matrix {{x_0, x
│ │ │ │ -000699f0: 5f31 2a78 5f32 7d7d 3b20 2020 2020 207c  _1*x_2}};      |
│ │ │ │ -00069a00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00069a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069a30: 2d2d 2d2b 0a7c 6931 3320 3a20 6220 3d20  ---+.|i13 : b = 
│ │ │ │ -00069a40: 353b 2020 2020 2020 2020 2020 2020 2020  5;              
│ │ │ │ -00069a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069a60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00069a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069aa0: 3420 3a20 4d62 203d 2070 7275 6e65 2073  4 : Mb = prune s
│ │ │ │ -00069ab0: 797a 7967 794d 6f64 756c 6528 2d62 2c4d  yzygyModule(-b,M
│ │ │ │ -00069ac0: 293b 2020 2020 2020 2020 2020 2020 207c  );             |
│ │ │ │ -00069ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00069ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069b00: 2d2d 2d2b 0a7c 6931 3520 3a20 4520 3d20  ---+.|i15 : E = 
│ │ │ │ -00069b10: 7072 756e 6520 6576 656e 4578 744d 6f64  prune evenExtMod
│ │ │ │ -00069b20: 756c 6520 4d62 3b20 2020 2020 2020 2020  ule Mb;         
│ │ │ │ -00069b30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00069b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069b70: 3620 3a20 5332 6d61 7020 3d20 5332 2830  6 : S2map = S2(0
│ │ │ │ -00069b80: 2c45 293b 2020 2020 2020 2020 2020 2020  ,E);            
│ │ │ │ -00069b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00069bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069bd0: 2020 207c 0a7c 6f31 3620 3a20 4d61 7472     |.|o16 : Matr
│ │ │ │ -00069be0: 6978 2020 2020 2020 2020 2020 2020 2020  ix              
│ │ │ │ -00069bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069c00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00069c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069c40: 3720 3a20 5345 203d 2070 7275 6e65 2074  7 : SE = prune t
│ │ │ │ -00069c50: 6172 6765 7420 5332 6d61 703b 2020 2020  arget S2map;    
│ │ │ │ -00069c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069c70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00069c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069ca0: 2d2d 2d2b 0a7c 6931 3820 3a20 6578 7472  ---+.|i18 : extr
│ │ │ │ -00069cb0: 6120 3d20 7072 756e 6520 636f 6b65 7220  a = prune coker 
│ │ │ │ -00069cc0: 5332 6d61 703b 2020 2020 2020 2020 2020  S2map;          
│ │ │ │ -00069cd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00069ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00069d10: 3920 3a20 4b45 203d 2070 7275 6e65 206b  9 : KE = prune k
│ │ │ │ -00069d20: 6572 2053 326d 6170 3b20 2020 2020 2020  er S2map;       
│ │ │ │ -00069d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00069d40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -00069d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069d70: 2d2d 2d2b 0a7c 6932 3020 3a20 6265 7474  ---+.|i20 : bett
│ │ │ │ -00069d80: 6920 6672 6565 5265 736f 6c75 7469 6f6e  i freeResolution
│ │ │ │ -00069d90: 284d 622c 204c 656e 6774 684c 696d 6974  (Mb, LengthLimit
│ │ │ │ -00069da0: 203d 3e20 3130 297c 0a7c 2020 2020 2020   => 10)|.|      
│ │ │ │ -00069db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00069de0: 2020 2020 2020 2020 2020 2020 3020 2031              0  1
│ │ │ │ -00069df0: 2032 2033 2034 2035 2036 2037 2038 2020   2 3 4 5 6 7 8  
│ │ │ │ -00069e00: 3920 3130 2020 2020 2020 2020 2020 207c  9 10           |
│ │ │ │ -00069e10: 0a7c 6f32 3020 3d20 746f 7461 6c3a 2032  .|o20 = total: 2
│ │ │ │ -00069e20: 3020 3134 2039 2035 2032 2031 2032 2034  0 14 9 5 2 1 2 4
│ │ │ │ -00069e30: 2037 2031 3120 3136 2020 2020 2020 2020   7 11 16        
│ │ │ │ -00069e40: 2020 207c 0a7c 2020 2020 2020 2020 202d     |.|         -
│ │ │ │ -00069e50: 363a 2032 3020 3134 2039 2035 2032 202e  6: 20 14 9 5 2 .
│ │ │ │ -00069e60: 202e 202e 202e 2020 2e20 202e 2020 2020   . . .  .  .    
│ │ │ │ -00069e70: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00069e80: 2020 202d 353a 2020 2e20 202e 202e 202e     -5:  .  . . .
│ │ │ │ -00069e90: 202e 2031 2031 2031 2031 2020 3120 2031   . 1 1 1 1  1  1
│ │ │ │ -00069ea0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00069eb0: 2020 2020 2020 202d 343a 2020 2e20 202e         -4:  .  .
│ │ │ │ -00069ec0: 202e 202e 202e 202e 2031 2033 2036 2031   . . . . 1 3 6 1
│ │ │ │ -00069ed0: 3020 3135 2020 2020 2020 2020 2020 207c  0 15           |
│ │ │ │ -00069ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00069ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f10: 2020 207c 0a7c 6f32 3020 3a20 4265 7474     |.|o20 : Bett
│ │ │ │ -00069f20: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ -00069f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069f40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -00069f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00069f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -00069f80: 3120 3a20 6170 706c 7920 2835 2c20 692d  1 : apply (5, i-
│ │ │ │ -00069f90: 3e20 6869 6c62 6572 7446 756e 6374 696f  > hilbertFunctio
│ │ │ │ -00069fa0: 6e28 692c 204b 4529 2920 2020 2020 207c  n(i, KE))      |
│ │ │ │ -00069fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00069fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00069fe0: 2020 207c 0a7c 6f32 3120 3d20 7b32 302c     |.|o21 = {20,
│ │ │ │ -00069ff0: 2039 2c20 322c 2030 2c20 307d 2020 2020   9, 2, 0, 0}    
│ │ │ │ +000696a0: 2d2d 2d2d 2d2d 2b0a 0a41 7420 6f6e 6520  ------+..At one 
│ │ │ │ +000696b0: 7469 6d65 2044 4520 686f 7065 6420 7468  time DE hoped th
│ │ │ │ +000696c0: 6174 2c20 6966 204d 2077 6572 6520 6120  at, if M were a 
│ │ │ │ +000696d0: 6d6f 6475 6c65 206f 7665 7220 7468 6520  module over the 
│ │ │ │ +000696e0: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +000696f0: 6374 696f 6e20 520a 7769 7468 2072 6573  ction R.with res
│ │ │ │ +00069700: 6964 7565 2066 6965 6c64 206b 2c20 7468  idue field k, th
│ │ │ │ +00069710: 656e 2074 6865 206e 6174 7572 616c 206d  en the natural m
│ │ │ │ +00069720: 6170 2066 726f 6d20 2263 6f6d 706c 6574  ap from "complet
│ │ │ │ +00069730: 6522 2045 7874 206d 6f64 756c 6520 2228  e" Ext module "(
│ │ │ │ +00069740: 7769 6465 6861 740a 4578 7429 5f52 284d  widehat.Ext)_R(M
│ │ │ │ +00069750: 2c6b 2922 2074 6f20 7468 6520 5332 2d69  ,k)" to the S2-i
│ │ │ │ +00069760: 6669 6361 7469 6f6e 206f 6620 4578 745f  fication of Ext_
│ │ │ │ +00069770: 5228 4d2c 6b29 2077 6f75 6c64 2062 6520  R(M,k) would be 
│ │ │ │ +00069780: 7375 726a 6563 7469 7665 3b0a 6571 7569  surjective;.equi
│ │ │ │ +00069790: 7661 6c65 6e74 6c79 2c20 6966 204e 2077  valently, if N w
│ │ │ │ +000697a0: 6572 6520 6120 7375 6666 6963 6965 6e74  ere a sufficient
│ │ │ │ +000697b0: 6c79 206e 6567 6174 6976 6520 7379 7a79  ly negative syzy
│ │ │ │ +000697c0: 6779 206f 6620 4d2c 2074 6865 6e20 7468  gy of M, then th
│ │ │ │ +000697d0: 6520 6669 7273 740a 6c6f 6361 6c20 636f  e first.local co
│ │ │ │ +000697e0: 686f 6d6f 6c6f 6779 206d 6f64 756c 6520  homology module 
│ │ │ │ +000697f0: 6f66 2045 7874 5f52 284d 2c6b 2920 776f  of Ext_R(M,k) wo
│ │ │ │ +00069800: 756c 6420 6265 207a 6572 6f2e 2054 6869  uld be zero. Thi
│ │ │ │ +00069810: 7320 6973 2066 616c 7365 2c20 6173 2073  s is false, as s
│ │ │ │ +00069820: 686f 776e 2062 790a 7468 6520 666f 6c6c  hown by.the foll
│ │ │ │ +00069830: 6f77 696e 6720 6578 616d 706c 653a 0a0a  owing example:..
│ │ │ │ +00069840: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00069850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069870: 2d2d 2b0a 7c69 3920 3a20 5320 3d20 5a5a  --+.|i9 : S = ZZ
│ │ │ │ +00069880: 2f31 3031 5b78 5f30 2e2e 785f 325d 3b20  /101[x_0..x_2]; 
│ │ │ │ +00069890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000698a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000698b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000698c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000698d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130  ----------+.|i10
│ │ │ │ +000698e0: 203a 2066 6620 3d20 6170 706c 7928 332c   : ff = apply(3,
│ │ │ │ +000698f0: 2069 2d3e 785f 695e 3229 3b20 2020 2020   i->x_i^2);     
│ │ │ │ +00069900: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00069910: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00069920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069940: 2d2d 2b0a 7c69 3131 203a 2052 203d 2053  --+.|i11 : R = S
│ │ │ │ +00069950: 2f69 6465 616c 2066 663b 2020 2020 2020  /ideal ff;      
│ │ │ │ +00069960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069970: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000699a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ +000699b0: 203a 204d 203d 2063 6f6b 6572 6e65 6c20   : M = cokernel 
│ │ │ │ +000699c0: 6d61 7472 6978 207b 7b78 5f30 2c20 785f  matrix {{x_0, x_
│ │ │ │ +000699d0: 312a 785f 327d 7d3b 2020 2020 2020 7c0a  1*x_2}};      |.
│ │ │ │ +000699e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +000699f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069a10: 2d2d 2b0a 7c69 3133 203a 2062 203d 2035  --+.|i13 : b = 5
│ │ │ │ +00069a20: 3b20 2020 2020 2020 2020 2020 2020 2020  ;               
│ │ │ │ +00069a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069a40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134  ----------+.|i14
│ │ │ │ +00069a80: 203a 204d 6220 3d20 7072 756e 6520 7379   : Mb = prune sy
│ │ │ │ +00069a90: 7a79 6779 4d6f 6475 6c65 282d 622c 4d29  zygyModule(-b,M)
│ │ │ │ +00069aa0: 3b20 2020 2020 2020 2020 2020 2020 7c0a  ;             |.
│ │ │ │ +00069ab0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00069ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069ae0: 2d2d 2b0a 7c69 3135 203a 2045 203d 2070  --+.|i15 : E = p
│ │ │ │ +00069af0: 7275 6e65 2065 7665 6e45 7874 4d6f 6475  rune evenExtModu
│ │ │ │ +00069b00: 6c65 204d 623b 2020 2020 2020 2020 2020  le Mb;          
│ │ │ │ +00069b10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136  ----------+.|i16
│ │ │ │ +00069b50: 203a 2053 326d 6170 203d 2053 3228 302c   : S2map = S2(0,
│ │ │ │ +00069b60: 4529 3b20 2020 2020 2020 2020 2020 2020  E);             
│ │ │ │ +00069b70: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00069b80: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00069b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069bb0: 2020 7c0a 7c6f 3136 203a 204d 6174 7269    |.|o16 : Matri
│ │ │ │ +00069bc0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +00069bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069be0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137  ----------+.|i17
│ │ │ │ +00069c20: 203a 2053 4520 3d20 7072 756e 6520 7461   : SE = prune ta
│ │ │ │ +00069c30: 7267 6574 2053 326d 6170 3b20 2020 2020  rget S2map;     
│ │ │ │ +00069c40: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00069c50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00069c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069c80: 2d2d 2b0a 7c69 3138 203a 2065 7874 7261  --+.|i18 : extra
│ │ │ │ +00069c90: 203d 2070 7275 6e65 2063 6f6b 6572 2053   = prune coker S
│ │ │ │ +00069ca0: 326d 6170 3b20 2020 2020 2020 2020 2020  2map;           
│ │ │ │ +00069cb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139  ----------+.|i19
│ │ │ │ +00069cf0: 203a 204b 4520 3d20 7072 756e 6520 6b65   : KE = prune ke
│ │ │ │ +00069d00: 7220 5332 6d61 703b 2020 2020 2020 2020  r S2map;        
│ │ │ │ +00069d10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00069d20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00069d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069d50: 2d2d 2b0a 7c69 3230 203a 2062 6574 7469  --+.|i20 : betti
│ │ │ │ +00069d60: 2066 7265 6552 6573 6f6c 7574 696f 6e28   freeResolution(
│ │ │ │ +00069d70: 4d62 2c20 4c65 6e67 7468 4c69 6d69 7420  Mb, LengthLimit 
│ │ │ │ +00069d80: 3d3e 2031 3029 7c0a 7c20 2020 2020 2020  => 10)|.|       
│ │ │ │ +00069d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069db0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069dc0: 2020 2020 2020 2020 2020 2030 2020 3120             0  1 
│ │ │ │ +00069dd0: 3220 3320 3420 3520 3620 3720 3820 2039  2 3 4 5 6 7 8  9
│ │ │ │ +00069de0: 2031 3020 2020 2020 2020 2020 2020 7c0a   10           |.
│ │ │ │ +00069df0: 7c6f 3230 203d 2074 6f74 616c 3a20 3230  |o20 = total: 20
│ │ │ │ +00069e00: 2031 3420 3920 3520 3220 3120 3220 3420   14 9 5 2 1 2 4 
│ │ │ │ +00069e10: 3720 3131 2031 3620 2020 2020 2020 2020  7 11 16         
│ │ │ │ +00069e20: 2020 7c0a 7c20 2020 2020 2020 2020 2d36    |.|         -6
│ │ │ │ +00069e30: 3a20 3230 2031 3420 3920 3520 3220 2e20  : 20 14 9 5 2 . 
│ │ │ │ +00069e40: 2e20 2e20 2e20 202e 2020 2e20 2020 2020  . . .  .  .     
│ │ │ │ +00069e50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00069e60: 2020 2d35 3a20 202e 2020 2e20 2e20 2e20    -5:  .  . . . 
│ │ │ │ +00069e70: 2e20 3120 3120 3120 3120 2031 2020 3120  . 1 1 1 1  1  1 
│ │ │ │ +00069e80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +00069e90: 2020 2020 2020 2d34 3a20 202e 2020 2e20        -4:  .  . 
│ │ │ │ +00069ea0: 2e20 2e20 2e20 2e20 3120 3320 3620 3130  . . . . 1 3 6 10
│ │ │ │ +00069eb0: 2031 3520 2020 2020 2020 2020 2020 7c0a   15           |.
│ │ │ │ +00069ec0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00069ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069ef0: 2020 7c0a 7c6f 3230 203a 2042 6574 7469    |.|o20 : Betti
│ │ │ │ +00069f00: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +00069f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069f20: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00069f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00069f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231  ----------+.|i21
│ │ │ │ +00069f60: 203a 2061 7070 6c79 2028 352c 2069 2d3e   : apply (5, i->
│ │ │ │ +00069f70: 2068 696c 6265 7274 4675 6e63 7469 6f6e   hilbertFunction
│ │ │ │ +00069f80: 2869 2c20 4b45 2929 2020 2020 2020 7c0a  (i, KE))      |.
│ │ │ │ +00069f90: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00069fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069fc0: 2020 7c0a 7c6f 3231 203d 207b 3230 2c20    |.|o21 = {20, 
│ │ │ │ +00069fd0: 392c 2032 2c20 302c 2030 7d20 2020 2020  9, 2, 0, 0}     
│ │ │ │ +00069fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00069ff0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0006a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a010: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0006a020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a040: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006a050: 3120 3a20 4c69 7374 2020 2020 2020 2020  1 : List        
│ │ │ │ -0006a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006a080: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0006a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a0b0: 2d2d 2d2b 0a7c 6932 3220 3a20 6170 706c  ---+.|i22 : appl
│ │ │ │ -0006a0c0: 7920 2835 2c20 692d 3e20 6869 6c62 6572  y (5, i-> hilber
│ │ │ │ -0006a0d0: 7446 756e 6374 696f 6e28 692c 2045 2929  tFunction(i, E))
│ │ │ │ -0006a0e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0006a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a110: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006a120: 3220 3d20 7b32 302c 2039 2c20 322c 2032  2 = {20, 9, 2, 2
│ │ │ │ -0006a130: 2c20 377d 2020 2020 2020 2020 2020 2020  , 7}            
│ │ │ │ -0006a140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006a150: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0006a160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a020: 2020 2020 2020 2020 2020 7c0a 7c6f 3231            |.|o21
│ │ │ │ +0006a030: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +0006a040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006a060: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0006a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a090: 2d2d 2b0a 7c69 3232 203a 2061 7070 6c79  --+.|i22 : apply
│ │ │ │ +0006a0a0: 2028 352c 2069 2d3e 2068 696c 6265 7274   (5, i-> hilbert
│ │ │ │ +0006a0b0: 4675 6e63 7469 6f6e 2869 2c20 4529 2920  Function(i, E)) 
│ │ │ │ +0006a0c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006a0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a0f0: 2020 2020 2020 2020 2020 7c0a 7c6f 3232            |.|o22
│ │ │ │ +0006a100: 203d 207b 3230 2c20 392c 2032 2c20 322c   = {20, 9, 2, 2,
│ │ │ │ +0006a110: 2037 7d20 2020 2020 2020 2020 2020 2020   7}             
│ │ │ │ +0006a120: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006a130: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a160: 2020 7c0a 7c6f 3232 203a 204c 6973 7420    |.|o22 : List 
│ │ │ │  0006a170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a180: 2020 207c 0a7c 6f32 3220 3a20 4c69 7374     |.|o22 : List
│ │ │ │ -0006a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a1b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0006a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ -0006a1f0: 3320 3a20 6170 706c 7920 2835 2c20 692d  3 : apply (5, i-
│ │ │ │ -0006a200: 3e20 6869 6c62 6572 7446 756e 6374 696f  > hilbertFunctio
│ │ │ │ -0006a210: 6e28 692c 2053 4529 2920 2020 2020 207c  n(i, SE))      |
│ │ │ │ -0006a220: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0006a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a250: 2020 207c 0a7c 6f32 3320 3d20 7b31 2c20     |.|o23 = {1, 
│ │ │ │ -0006a260: 312c 2031 2c20 322c 2037 7d20 2020 2020  1, 1, 2, 7}     
│ │ │ │ +0006a180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a190: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0006a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233  ----------+.|i23
│ │ │ │ +0006a1d0: 203a 2061 7070 6c79 2028 352c 2069 2d3e   : apply (5, i->
│ │ │ │ +0006a1e0: 2068 696c 6265 7274 4675 6e63 7469 6f6e   hilbertFunction
│ │ │ │ +0006a1f0: 2869 2c20 5345 2929 2020 2020 2020 7c0a  (i, SE))      |.
│ │ │ │ +0006a200: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a230: 2020 7c0a 7c6f 3233 203d 207b 312c 2031    |.|o23 = {1, 1
│ │ │ │ +0006a240: 2c20 312c 2032 2c20 377d 2020 2020 2020  , 1, 2, 7}      
│ │ │ │ +0006a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a260: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0006a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0006a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a2b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006a2c0: 3320 3a20 4c69 7374 2020 2020 2020 2020  3 : List        
│ │ │ │ -0006a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a2e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006a2f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ -0006a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a320: 2d2d 2d2b 0a7c 6932 3420 3a20 6170 706c  ---+.|i24 : appl
│ │ │ │ -0006a330: 7920 2835 2c20 692d 3e20 6869 6c62 6572  y (5, i-> hilber
│ │ │ │ -0006a340: 7446 756e 6374 696f 6e28 692c 2065 7874  tFunction(i, ext
│ │ │ │ -0006a350: 7261 2929 2020 207c 0a7c 2020 2020 2020  ra))   |.|      
│ │ │ │ -0006a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a380: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006a390: 3420 3d20 7b31 2c20 312c 2031 2c20 302c  4 = {1, 1, 1, 0,
│ │ │ │ -0006a3a0: 2030 7d20 2020 2020 2020 2020 2020 2020   0}             
│ │ │ │ -0006a3b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006a3c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0006a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a290: 2020 2020 2020 2020 2020 7c0a 7c6f 3233            |.|o23
│ │ │ │ +0006a2a0: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +0006a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a2c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006a2d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +0006a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a300: 2d2d 2b0a 7c69 3234 203a 2061 7070 6c79  --+.|i24 : apply
│ │ │ │ +0006a310: 2028 352c 2069 2d3e 2068 696c 6265 7274   (5, i-> hilbert
│ │ │ │ +0006a320: 4675 6e63 7469 6f6e 2869 2c20 6578 7472  Function(i, extr
│ │ │ │ +0006a330: 6129 2920 2020 7c0a 7c20 2020 2020 2020  a))   |.|       
│ │ │ │ +0006a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a360: 2020 2020 2020 2020 2020 7c0a 7c6f 3234            |.|o24
│ │ │ │ +0006a370: 203d 207b 312c 2031 2c20 312c 2030 2c20   = {1, 1, 1, 0, 
│ │ │ │ +0006a380: 307d 2020 2020 2020 2020 2020 2020 2020  0}              
│ │ │ │ +0006a390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006a3a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a3d0: 2020 7c0a 7c6f 3234 203a 204c 6973 7420    |.|o24 : List 
│ │ │ │  0006a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a3f0: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374     |.|o24 : List
│ │ │ │ -0006a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006a420: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0006a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361  -----------+..Ca
│ │ │ │ -0006a460: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 6578  veat.======..Tex
│ │ │ │ -0006a470: 7420 5332 2d69 6669 6361 7469 6f6e 2069  t S2-ification i
│ │ │ │ -0006a480: 7320 7265 6c61 7465 6420 746f 2063 6f6d  s related to com
│ │ │ │ -0006a490: 7075 7469 6e67 2063 6f68 6f6d 6f6c 6f67  puting cohomolog
│ │ │ │ -0006a4a0: 7920 616e 6420 746f 2063 6f6d 7075 7469  y and to computi
│ │ │ │ -0006a4b0: 6e67 2069 6e74 6567 7261 6c0a 636c 6f73  ng integral.clos
│ │ │ │ -0006a4c0: 7572 653b 2074 6865 7265 2061 7265 2073  ure; there are s
│ │ │ │ -0006a4d0: 6372 6970 7473 2069 6e20 7468 6f73 6520  cripts in those 
│ │ │ │ -0006a4e0: 7061 636b 6167 6573 2074 6861 7420 7072  packages that pr
│ │ │ │ -0006a4f0: 6f64 7563 6520 616e 2053 322d 6966 6963  oduce an S2-ific
│ │ │ │ -0006a500: 6174 696f 6e2c 2062 7574 0a6f 6e65 2074  ation, but.one t
│ │ │ │ -0006a510: 616b 6573 2061 2072 696e 6720 6173 2061  akes a ring as a
│ │ │ │ -0006a520: 7267 756d 656e 7420 616e 6420 7468 6520  rgument and the 
│ │ │ │ -0006a530: 6f74 6865 7220 646f 6573 6e27 7420 7072  other doesn't pr
│ │ │ │ -0006a540: 6f64 7563 6520 7468 6520 636f 6d70 6172  oduce the compar
│ │ │ │ -0006a550: 6973 6f6e 206d 6170 2e0a 0a53 6565 2061  ison map...See a
│ │ │ │ -0006a560: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020  lso.========..  
│ │ │ │ -0006a570: 2a20 2a6e 6f74 6520 496e 7465 6772 616c  * *note Integral
│ │ │ │ -0006a580: 436c 6f73 7572 653a 2028 496e 7465 6772  Closure: (Integr
│ │ │ │ -0006a590: 616c 436c 6f73 7572 6529 546f 702c 202d  alClosure)Top, -
│ │ │ │ -0006a5a0: 2d20 696e 7465 6772 616c 2063 6c6f 7375  - integral closu
│ │ │ │ -0006a5b0: 7265 0a20 202a 202a 6e6f 7465 206d 616b  re.  * *note mak
│ │ │ │ -0006a5c0: 6553 323a 2028 496e 7465 6772 616c 436c  eS2: (IntegralCl
│ │ │ │ -0006a5d0: 6f73 7572 6529 6d61 6b65 5332 2c20 2d2d  osure)makeS2, --
│ │ │ │ -0006a5e0: 2063 6f6d 7075 7465 2074 6865 2053 3269   compute the S2i
│ │ │ │ -0006a5f0: 6669 6361 7469 6f6e 206f 6620 610a 2020  fication of a.  
│ │ │ │ -0006a600: 2020 7265 6475 6365 6420 7269 6e67 0a20    reduced ring. 
│ │ │ │ -0006a610: 202a 202a 6e6f 7465 2042 4747 3a20 2842   * *note BGG: (B
│ │ │ │ -0006a620: 4747 2954 6f70 2c20 2d2d 2042 6572 6e73  GG)Top, -- Berns
│ │ │ │ -0006a630: 7465 696e 2d47 656c 6661 6e64 2d47 656c  tein-Gelfand-Gel
│ │ │ │ -0006a640: 6661 6e64 2063 6f72 7265 7370 6f6e 6465  fand corresponde
│ │ │ │ -0006a650: 6e63 650a 2020 2a20 2a6e 6f74 6520 636f  nce.  * *note co
│ │ │ │ -0006a660: 686f 6d6f 6c6f 6779 3a20 284d 6163 6175  homology: (Macau
│ │ │ │ -0006a670: 6c61 7932 446f 6329 636f 686f 6d6f 6c6f  lay2Doc)cohomolo
│ │ │ │ -0006a680: 6779 2c20 2d2d 2067 656e 6572 616c 2063  gy, -- general c
│ │ │ │ -0006a690: 6f68 6f6d 6f6c 6f67 7920 6675 6e63 746f  ohomology functo
│ │ │ │ -0006a6a0: 720a 2020 2a20 4848 5e5a 5a20 5375 6d4f  r.  * HH^ZZ SumO
│ │ │ │ -0006a6b0: 6654 7769 7374 7320 286d 6973 7369 6e67  fTwists (missing
│ │ │ │ -0006a6c0: 2064 6f63 756d 656e 7461 7469 6f6e 290a   documentation).
│ │ │ │ -0006a6d0: 0a57 6179 7320 746f 2075 7365 2053 323a  .Ways to use S2:
│ │ │ │ -0006a6e0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0006a6f0: 0a0a 2020 2a20 2253 3228 5a5a 2c4d 6f64  ..  * "S2(ZZ,Mod
│ │ │ │ -0006a700: 756c 6529 220a 0a46 6f72 2074 6865 2070  ule)"..For the p
│ │ │ │ -0006a710: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ -0006a720: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ -0006a730: 6520 6f62 6a65 6374 202a 6e6f 7465 2053  e object *note S
│ │ │ │ -0006a740: 323a 2053 322c 2069 7320 6120 2a6e 6f74  2: S2, is a *not
│ │ │ │ -0006a750: 6520 6d65 7468 6f64 2066 756e 6374 696f  e method functio
│ │ │ │ -0006a760: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63  n:.(Macaulay2Doc
│ │ │ │ -0006a770: 294d 6574 686f 6446 756e 6374 696f 6e2c  )MethodFunction,
│ │ │ │ -0006a780: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...-------------
│ │ │ │ +0006a3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006a400: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0006a410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 6176  ----------+..Cav
│ │ │ │ +0006a440: 6561 740a 3d3d 3d3d 3d3d 0a0a 5465 7874  eat.======..Text
│ │ │ │ +0006a450: 2053 322d 6966 6963 6174 696f 6e20 6973   S2-ification is
│ │ │ │ +0006a460: 2072 656c 6174 6564 2074 6f20 636f 6d70   related to comp
│ │ │ │ +0006a470: 7574 696e 6720 636f 686f 6d6f 6c6f 6779  uting cohomology
│ │ │ │ +0006a480: 2061 6e64 2074 6f20 636f 6d70 7574 696e   and to computin
│ │ │ │ +0006a490: 6720 696e 7465 6772 616c 0a63 6c6f 7375  g integral.closu
│ │ │ │ +0006a4a0: 7265 3b20 7468 6572 6520 6172 6520 7363  re; there are sc
│ │ │ │ +0006a4b0: 7269 7074 7320 696e 2074 686f 7365 2070  ripts in those p
│ │ │ │ +0006a4c0: 6163 6b61 6765 7320 7468 6174 2070 726f  ackages that pro
│ │ │ │ +0006a4d0: 6475 6365 2061 6e20 5332 2d69 6669 6361  duce an S2-ifica
│ │ │ │ +0006a4e0: 7469 6f6e 2c20 6275 740a 6f6e 6520 7461  tion, but.one ta
│ │ │ │ +0006a4f0: 6b65 7320 6120 7269 6e67 2061 7320 6172  kes a ring as ar
│ │ │ │ +0006a500: 6775 6d65 6e74 2061 6e64 2074 6865 206f  gument and the o
│ │ │ │ +0006a510: 7468 6572 2064 6f65 736e 2774 2070 726f  ther doesn't pro
│ │ │ │ +0006a520: 6475 6365 2074 6865 2063 6f6d 7061 7269  duce the compari
│ │ │ │ +0006a530: 736f 6e20 6d61 702e 0a0a 5365 6520 616c  son map...See al
│ │ │ │ +0006a540: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  so.========..  *
│ │ │ │ +0006a550: 202a 6e6f 7465 2049 6e74 6567 7261 6c43   *note IntegralC
│ │ │ │ +0006a560: 6c6f 7375 7265 3a20 2849 6e74 6567 7261  losure: (Integra
│ │ │ │ +0006a570: 6c43 6c6f 7375 7265 2954 6f70 2c20 2d2d  lClosure)Top, --
│ │ │ │ +0006a580: 2069 6e74 6567 7261 6c20 636c 6f73 7572   integral closur
│ │ │ │ +0006a590: 650a 2020 2a20 2a6e 6f74 6520 6d61 6b65  e.  * *note make
│ │ │ │ +0006a5a0: 5332 3a20 2849 6e74 6567 7261 6c43 6c6f  S2: (IntegralClo
│ │ │ │ +0006a5b0: 7375 7265 296d 616b 6553 322c 202d 2d20  sure)makeS2, -- 
│ │ │ │ +0006a5c0: 636f 6d70 7574 6520 7468 6520 5332 6966  compute the S2if
│ │ │ │ +0006a5d0: 6963 6174 696f 6e20 6f66 2061 0a20 2020  ication of a.   
│ │ │ │ +0006a5e0: 2072 6564 7563 6564 2072 696e 670a 2020   reduced ring.  
│ │ │ │ +0006a5f0: 2a20 2a6e 6f74 6520 4247 473a 2028 4247  * *note BGG: (BG
│ │ │ │ +0006a600: 4729 546f 702c 202d 2d20 4265 726e 7374  G)Top, -- Bernst
│ │ │ │ +0006a610: 6569 6e2d 4765 6c66 616e 642d 4765 6c66  ein-Gelfand-Gelf
│ │ │ │ +0006a620: 616e 6420 636f 7272 6573 706f 6e64 656e  and corresponden
│ │ │ │ +0006a630: 6365 0a20 202a 202a 6e6f 7465 2063 6f68  ce.  * *note coh
│ │ │ │ +0006a640: 6f6d 6f6c 6f67 793a 2028 4d61 6361 756c  omology: (Macaul
│ │ │ │ +0006a650: 6179 3244 6f63 2963 6f68 6f6d 6f6c 6f67  ay2Doc)cohomolog
│ │ │ │ +0006a660: 792c 202d 2d20 6765 6e65 7261 6c20 636f  y, -- general co
│ │ │ │ +0006a670: 686f 6d6f 6c6f 6779 2066 756e 6374 6f72  homology functor
│ │ │ │ +0006a680: 0a20 202a 2048 485e 5a5a 2053 756d 4f66  .  * HH^ZZ SumOf
│ │ │ │ +0006a690: 5477 6973 7473 2028 6d69 7373 696e 6720  Twists (missing 
│ │ │ │ +0006a6a0: 646f 6375 6d65 6e74 6174 696f 6e29 0a0a  documentation)..
│ │ │ │ +0006a6b0: 5761 7973 2074 6f20 7573 6520 5332 3a0a  Ways to use S2:.
│ │ │ │ +0006a6c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +0006a6d0: 0a20 202a 2022 5332 285a 5a2c 4d6f 6475  .  * "S2(ZZ,Modu
│ │ │ │ +0006a6e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072  le)"..For the pr
│ │ │ │ +0006a6f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ +0006a700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ +0006a710: 206f 626a 6563 7420 2a6e 6f74 6520 5332   object *note S2
│ │ │ │ +0006a720: 3a20 5332 2c20 6973 2061 202a 6e6f 7465  : S2, is a *note
│ │ │ │ +0006a730: 206d 6574 686f 6420 6675 6e63 7469 6f6e   method function
│ │ │ │ +0006a740: 3a0a 284d 6163 6175 6c61 7932 446f 6329  :.(Macaulay2Doc)
│ │ │ │ +0006a750: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e  MethodFunction,.
│ │ │ │ +0006a760: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ..--------------
│ │ │ │ +0006a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006a7d0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f  --..The source o
│ │ │ │ -0006a7e0: 6620 7468 6973 2064 6f63 756d 656e 7420  f this document 
│ │ │ │ -0006a7f0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570  is in./build/rep
│ │ │ │ -0006a800: 726f 6475 6369 626c 652d 7061 7468 2f6d  roducible-path/m
│ │ │ │ -0006a810: 6163 6175 6c61 7932 2d31 2e32 362e 3036  acaulay2-1.26.06
│ │ │ │ -0006a820: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932  +ds/M2/Macaulay2
│ │ │ │ -0006a830: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c  /packages/.Compl
│ │ │ │ -0006a840: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -0006a850: 6573 6f6c 7574 696f 6e73 2e6d 323a 3338  esolutions.m2:38
│ │ │ │ -0006a860: 3833 3a30 2e0a 1f0a 4669 6c65 3a20 436f  83:0....File: Co
│ │ │ │ -0006a870: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -0006a880: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e  onResolutions.in
│ │ │ │ -0006a890: 666f 2c20 4e6f 6465 3a20 5368 616d 6173  fo, Node: Shamas
│ │ │ │ -0006a8a0: 682c 204e 6578 743a 2073 706c 6974 7469  h, Next: splitti
│ │ │ │ -0006a8b0: 6e67 732c 2050 7265 763a 2053 322c 2055  ngs, Prev: S2, U
│ │ │ │ -0006a8c0: 703a 2054 6f70 0a0a 5368 616d 6173 6820  p: Top..Shamash 
│ │ │ │ -0006a8d0: 2d2d 2043 6f6d 7075 7465 7320 7468 6520  -- Computes the 
│ │ │ │ -0006a8e0: 5368 616d 6173 6820 436f 6d70 6c65 780a  Shamash Complex.
│ │ │ │ -0006a8f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006a900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006a910: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ -0006a920: 6765 3a20 0a20 2020 2020 2020 2046 4620  ge: .        FF 
│ │ │ │ -0006a930: 3d20 5368 616d 6173 6828 6666 2c46 2c6c  = Shamash(ff,F,l
│ │ │ │ -0006a940: 656e 290a 2020 2020 2020 2020 4646 203d  en).        FF =
│ │ │ │ -0006a950: 2053 6861 6d61 7368 2852 6261 722c 462c   Shamash(Rbar,F,
│ │ │ │ -0006a960: 6c65 6e29 0a20 202a 2049 6e70 7574 733a  len).  * Inputs:
│ │ │ │ -0006a970: 0a20 2020 2020 202a 2066 662c 2061 202a  .      * ff, a *
│ │ │ │ -0006a980: 6e6f 7465 206d 6174 7269 783a 2028 4d61  note matrix: (Ma
│ │ │ │ -0006a990: 6361 756c 6179 3244 6f63 294d 6174 7269  caulay2Doc)Matri
│ │ │ │ -0006a9a0: 782c 2c20 3120 7820 3120 4d61 7472 6978  x,, 1 x 1 Matrix
│ │ │ │ -0006a9b0: 206f 7665 7220 7269 6e67 2046 2e0a 2020   over ring F..  
│ │ │ │ -0006a9c0: 2020 2020 2a20 5262 6172 2c20 6120 2a6e      * Rbar, a *n
│ │ │ │ -0006a9d0: 6f74 6520 7269 6e67 3a20 284d 6163 6175  ote ring: (Macau
│ │ │ │ -0006a9e0: 6c61 7932 446f 6329 5269 6e67 2c2c 2072  lay2Doc)Ring,, r
│ │ │ │ -0006a9f0: 696e 6720 4620 6d6f 6420 6964 6561 6c20  ing F mod ideal 
│ │ │ │ -0006aa00: 6666 0a20 2020 2020 202a 2046 2c20 6120  ff.      * F, a 
│ │ │ │ -0006aa10: 2a6e 6f74 6520 636f 6d70 6c65 783a 2028  *note complex: (
│ │ │ │ -0006aa20: 436f 6d70 6c65 7865 7329 436f 6d70 6c65  Complexes)Comple
│ │ │ │ -0006aa30: 782c 2c20 6465 6669 6e65 6420 6f76 6572  x,, defined over
│ │ │ │ -0006aa40: 2072 696e 6720 6666 0a20 2020 2020 202a   ring ff.      *
│ │ │ │ -0006aa50: 206c 656e 2c20 616e 202a 6e6f 7465 2069   len, an *note i
│ │ │ │ -0006aa60: 6e74 6567 6572 3a20 284d 6163 6175 6c61  nteger: (Macaula
│ │ │ │ -0006aa70: 7932 446f 6329 5a5a 2c2c 200a 2020 2a20  y2Doc)ZZ,, .  * 
│ │ │ │ -0006aa80: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ -0006aa90: 2046 462c 2061 202a 6e6f 7465 2063 6f6d   FF, a *note com
│ │ │ │ -0006aaa0: 706c 6578 3a20 2843 6f6d 706c 6578 6573  plex: (Complexes
│ │ │ │ -0006aab0: 2943 6f6d 706c 6578 2c2c 2063 6861 696e  )Complex,, chain
│ │ │ │ -0006aac0: 2063 6f6d 706c 6578 206f 7665 7220 2872   complex over (r
│ │ │ │ -0006aad0: 696e 670a 2020 2020 2020 2020 4629 2f28  ing.        F)/(
│ │ │ │ -0006aae0: 6964 6561 6c20 6666 290a 0a44 6573 6372  ideal ff)..Descr
│ │ │ │ -0006aaf0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -0006ab00: 3d3d 0a0a 4c65 7420 5220 3d20 7269 6e67  ==..Let R = ring
│ │ │ │ -0006ab10: 2046 203d 2072 696e 6720 6666 2c20 616e   F = ring ff, an
│ │ │ │ -0006ab20: 6420 5262 6172 203d 2052 2f28 6964 6561  d Rbar = R/(idea
│ │ │ │ -0006ab30: 6c20 6629 2c20 7768 6572 6520 6666 203d  l f), where ff =
│ │ │ │ -0006ab40: 206d 6174 7269 787b 7b66 7d7d 2069 7320   matrix{{f}} is 
│ │ │ │ -0006ab50: 610a 3178 3120 6d61 7472 6978 2077 686f  a.1x1 matrix who
│ │ │ │ -0006ab60: 7365 2065 6e74 7279 2069 7320 6120 6e6f  se entry is a no
│ │ │ │ -0006ab70: 6e7a 6572 6f64 6976 6973 6f72 2069 6e20  nzerodivisor in 
│ │ │ │ -0006ab80: 522e 2054 6865 2063 6f6d 706c 6578 2046  R. The complex F
│ │ │ │ -0006ab90: 2073 686f 756c 6420 6164 6d69 7420 610a   should admit a.
│ │ │ │ -0006aba0: 7379 7374 656d 206f 6620 6869 6768 6572  system of higher
│ │ │ │ -0006abb0: 2068 6f6d 6f74 6f70 6965 7320 666f 7220   homotopies for 
│ │ │ │ -0006abc0: 7468 6520 656e 7472 7920 6f66 2066 662c  the entry of ff,
│ │ │ │ -0006abd0: 2072 6574 7572 6e65 6420 6279 2074 6865   returned by the
│ │ │ │ -0006abe0: 2063 616c 6c0a 6d61 6b65 486f 6d6f 746f   call.makeHomoto
│ │ │ │ -0006abf0: 7069 6573 2866 662c 4629 2e0a 0a54 6865  pies(ff,F)...The
│ │ │ │ -0006ac00: 2063 6f6d 706c 6578 2046 4620 6861 7320   complex FF has 
│ │ │ │ -0006ac10: 7465 726d 730a 0a46 465f 7b32 2a69 7d20  terms..FF_{2*i} 
│ │ │ │ -0006ac20: 3d20 5262 6172 2a2a 2846 5f30 202b 2b20  = Rbar**(F_0 ++ 
│ │ │ │ -0006ac30: 465f 3220 2b2b 202e 2e20 2b2b 2046 5f69  F_2 ++ .. ++ F_i
│ │ │ │ -0006ac40: 290a 0a46 465f 7b32 2a69 2b31 7d20 3d20  )..FF_{2*i+1} = 
│ │ │ │ -0006ac50: 5262 6172 2a2a 2846 5f31 202b 2b20 465f  Rbar**(F_1 ++ F_
│ │ │ │ -0006ac60: 3320 2b2b 2e2e 2b2b 465f 7b32 2a69 2b31  3 ++..++F_{2*i+1
│ │ │ │ -0006ac70: 7d29 0a0a 616e 6420 6d61 7073 206d 6164  })..and maps mad
│ │ │ │ -0006ac80: 6520 6672 6f6d 2074 6865 2068 6967 6865  e from the highe
│ │ │ │ -0006ac90: 7220 686f 6d6f 746f 7069 6573 2e0a 0a46  r homotopies...F
│ │ │ │ -0006aca0: 6f72 2074 6865 2063 6173 6520 6f66 2061  or the case of a
│ │ │ │ -0006acb0: 2063 6f6d 706c 6574 6520 696e 7465 7273   complete inters
│ │ │ │ -0006acc0: 6563 7469 6f6e 206f 6620 6869 6768 6572  ection of higher
│ │ │ │ -0006acd0: 2063 6f64 696d 656e 7369 6f6e 2c20 6f72   codimension, or
│ │ │ │ -0006ace0: 2074 6f20 7365 6520 7468 650a 636f 6d70   to see the.comp
│ │ │ │ -0006acf0: 6f6e 656e 7473 206f 6620 7468 6520 7265  onents of the re
│ │ │ │ -0006ad00: 736f 6c75 7469 6f6e 2061 7320 7375 6d6d  solution as summ
│ │ │ │ -0006ad10: 616e 6473 206f 6620 4646 5f6a 2c20 7573  ands of FF_j, us
│ │ │ │ -0006ad20: 6520 7468 6520 726f 7574 696e 650a 4569  e the routine.Ei
│ │ │ │ -0006ad30: 7365 6e62 7564 5368 616d 6173 6820 696e  senbudShamash in
│ │ │ │ -0006ad40: 7374 6561 642e 0a0a 2b2d 2d2d 2d2d 2d2d  stead...+-------
│ │ │ │ -0006ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -0006ad80: 6931 203a 2053 203d 205a 5a2f 3130 315b  i1 : S = ZZ/101[
│ │ │ │ -0006ad90: 782c 792c 7a5d 2020 2020 2020 2020 2020  x,y,z]          
│ │ │ │ +0006a7b0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66  -..The source of
│ │ │ │ +0006a7c0: 2074 6869 7320 646f 6375 6d65 6e74 2069   this document i
│ │ │ │ +0006a7d0: 7320 696e 0a2f 6275 696c 642f 7265 7072  s in./build/repr
│ │ │ │ +0006a7e0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61  oducible-path/ma
│ │ │ │ +0006a7f0: 6361 756c 6179 322d 312e 3236 2e30 362b  caulay2-1.26.06+
│ │ │ │ +0006a800: 6473 2f4d 322f 4d61 6361 756c 6179 322f  ds/M2/Macaulay2/
│ │ │ │ +0006a810: 7061 636b 6167 6573 2f0a 436f 6d70 6c65  packages/.Comple
│ │ │ │ +0006a820: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +0006a830: 736f 6c75 7469 6f6e 732e 6d32 3a33 3838  solutions.m2:388
│ │ │ │ +0006a840: 333a 302e 0a1f 0a46 696c 653a 2043 6f6d  3:0....File: Com
│ │ │ │ +0006a850: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ +0006a860: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66  nResolutions.inf
│ │ │ │ +0006a870: 6f2c 204e 6f64 653a 2053 6861 6d61 7368  o, Node: Shamash
│ │ │ │ +0006a880: 2c20 4e65 7874 3a20 7370 6c69 7474 696e  , Next: splittin
│ │ │ │ +0006a890: 6773 2c20 5072 6576 3a20 5332 2c20 5570  gs, Prev: S2, Up
│ │ │ │ +0006a8a0: 3a20 546f 700a 0a53 6861 6d61 7368 202d  : Top..Shamash -
│ │ │ │ +0006a8b0: 2d20 436f 6d70 7574 6573 2074 6865 2053  - Computes the S
│ │ │ │ +0006a8c0: 6861 6d61 7368 2043 6f6d 706c 6578 0a2a  hamash Complex.*
│ │ │ │ +0006a8d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0006a8e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0006a8f0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ +0006a900: 653a 200a 2020 2020 2020 2020 4646 203d  e: .        FF =
│ │ │ │ +0006a910: 2053 6861 6d61 7368 2866 662c 462c 6c65   Shamash(ff,F,le
│ │ │ │ +0006a920: 6e29 0a20 2020 2020 2020 2046 4620 3d20  n).        FF = 
│ │ │ │ +0006a930: 5368 616d 6173 6828 5262 6172 2c46 2c6c  Shamash(Rbar,F,l
│ │ │ │ +0006a940: 656e 290a 2020 2a20 496e 7075 7473 3a0a  en).  * Inputs:.
│ │ │ │ +0006a950: 2020 2020 2020 2a20 6666 2c20 6120 2a6e        * ff, a *n
│ │ │ │ +0006a960: 6f74 6520 6d61 7472 6978 3a20 284d 6163  ote matrix: (Mac
│ │ │ │ +0006a970: 6175 6c61 7932 446f 6329 4d61 7472 6978  aulay2Doc)Matrix
│ │ │ │ +0006a980: 2c2c 2031 2078 2031 204d 6174 7269 7820  ,, 1 x 1 Matrix 
│ │ │ │ +0006a990: 6f76 6572 2072 696e 6720 462e 0a20 2020  over ring F..   
│ │ │ │ +0006a9a0: 2020 202a 2052 6261 722c 2061 202a 6e6f     * Rbar, a *no
│ │ │ │ +0006a9b0: 7465 2072 696e 673a 2028 4d61 6361 756c  te ring: (Macaul
│ │ │ │ +0006a9c0: 6179 3244 6f63 2952 696e 672c 2c20 7269  ay2Doc)Ring,, ri
│ │ │ │ +0006a9d0: 6e67 2046 206d 6f64 2069 6465 616c 2066  ng F mod ideal f
│ │ │ │ +0006a9e0: 660a 2020 2020 2020 2a20 462c 2061 202a  f.      * F, a *
│ │ │ │ +0006a9f0: 6e6f 7465 2063 6f6d 706c 6578 3a20 2843  note complex: (C
│ │ │ │ +0006aa00: 6f6d 706c 6578 6573 2943 6f6d 706c 6578  omplexes)Complex
│ │ │ │ +0006aa10: 2c2c 2064 6566 696e 6564 206f 7665 7220  ,, defined over 
│ │ │ │ +0006aa20: 7269 6e67 2066 660a 2020 2020 2020 2a20  ring ff.      * 
│ │ │ │ +0006aa30: 6c65 6e2c 2061 6e20 2a6e 6f74 6520 696e  len, an *note in
│ │ │ │ +0006aa40: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
│ │ │ │ +0006aa50: 3244 6f63 295a 5a2c 2c20 0a20 202a 204f  2Doc)ZZ,, .  * O
│ │ │ │ +0006aa60: 7574 7075 7473 3a0a 2020 2020 2020 2a20  utputs:.      * 
│ │ │ │ +0006aa70: 4646 2c20 6120 2a6e 6f74 6520 636f 6d70  FF, a *note comp
│ │ │ │ +0006aa80: 6c65 783a 2028 436f 6d70 6c65 7865 7329  lex: (Complexes)
│ │ │ │ +0006aa90: 436f 6d70 6c65 782c 2c20 6368 6169 6e20  Complex,, chain 
│ │ │ │ +0006aaa0: 636f 6d70 6c65 7820 6f76 6572 2028 7269  complex over (ri
│ │ │ │ +0006aab0: 6e67 0a20 2020 2020 2020 2046 292f 2869  ng.        F)/(i
│ │ │ │ +0006aac0: 6465 616c 2066 6629 0a0a 4465 7363 7269  deal ff)..Descri
│ │ │ │ +0006aad0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d  ption.==========
│ │ │ │ +0006aae0: 3d0a 0a4c 6574 2052 203d 2072 696e 6720  =..Let R = ring 
│ │ │ │ +0006aaf0: 4620 3d20 7269 6e67 2066 662c 2061 6e64  F = ring ff, and
│ │ │ │ +0006ab00: 2052 6261 7220 3d20 522f 2869 6465 616c   Rbar = R/(ideal
│ │ │ │ +0006ab10: 2066 292c 2077 6865 7265 2066 6620 3d20   f), where ff = 
│ │ │ │ +0006ab20: 6d61 7472 6978 7b7b 667d 7d20 6973 2061  matrix{{f}} is a
│ │ │ │ +0006ab30: 0a31 7831 206d 6174 7269 7820 7768 6f73  .1x1 matrix whos
│ │ │ │ +0006ab40: 6520 656e 7472 7920 6973 2061 206e 6f6e  e entry is a non
│ │ │ │ +0006ab50: 7a65 726f 6469 7669 736f 7220 696e 2052  zerodivisor in R
│ │ │ │ +0006ab60: 2e20 5468 6520 636f 6d70 6c65 7820 4620  . The complex F 
│ │ │ │ +0006ab70: 7368 6f75 6c64 2061 646d 6974 2061 0a73  should admit a.s
│ │ │ │ +0006ab80: 7973 7465 6d20 6f66 2068 6967 6865 7220  ystem of higher 
│ │ │ │ +0006ab90: 686f 6d6f 746f 7069 6573 2066 6f72 2074  homotopies for t
│ │ │ │ +0006aba0: 6865 2065 6e74 7279 206f 6620 6666 2c20  he entry of ff, 
│ │ │ │ +0006abb0: 7265 7475 726e 6564 2062 7920 7468 6520  returned by the 
│ │ │ │ +0006abc0: 6361 6c6c 0a6d 616b 6548 6f6d 6f74 6f70  call.makeHomotop
│ │ │ │ +0006abd0: 6965 7328 6666 2c46 292e 0a0a 5468 6520  ies(ff,F)...The 
│ │ │ │ +0006abe0: 636f 6d70 6c65 7820 4646 2068 6173 2074  complex FF has t
│ │ │ │ +0006abf0: 6572 6d73 0a0a 4646 5f7b 322a 697d 203d  erms..FF_{2*i} =
│ │ │ │ +0006ac00: 2052 6261 722a 2a28 465f 3020 2b2b 2046   Rbar**(F_0 ++ F
│ │ │ │ +0006ac10: 5f32 202b 2b20 2e2e 202b 2b20 465f 6929  _2 ++ .. ++ F_i)
│ │ │ │ +0006ac20: 0a0a 4646 5f7b 322a 692b 317d 203d 2052  ..FF_{2*i+1} = R
│ │ │ │ +0006ac30: 6261 722a 2a28 465f 3120 2b2b 2046 5f33  bar**(F_1 ++ F_3
│ │ │ │ +0006ac40: 202b 2b2e 2e2b 2b46 5f7b 322a 692b 317d   ++..++F_{2*i+1}
│ │ │ │ +0006ac50: 290a 0a61 6e64 206d 6170 7320 6d61 6465  )..and maps made
│ │ │ │ +0006ac60: 2066 726f 6d20 7468 6520 6869 6768 6572   from the higher
│ │ │ │ +0006ac70: 2068 6f6d 6f74 6f70 6965 732e 0a0a 466f   homotopies...Fo
│ │ │ │ +0006ac80: 7220 7468 6520 6361 7365 206f 6620 6120  r the case of a 
│ │ │ │ +0006ac90: 636f 6d70 6c65 7465 2069 6e74 6572 7365  complete interse
│ │ │ │ +0006aca0: 6374 696f 6e20 6f66 2068 6967 6865 7220  ction of higher 
│ │ │ │ +0006acb0: 636f 6469 6d65 6e73 696f 6e2c 206f 7220  codimension, or 
│ │ │ │ +0006acc0: 746f 2073 6565 2074 6865 0a63 6f6d 706f  to see the.compo
│ │ │ │ +0006acd0: 6e65 6e74 7320 6f66 2074 6865 2072 6573  nents of the res
│ │ │ │ +0006ace0: 6f6c 7574 696f 6e20 6173 2073 756d 6d61  olution as summa
│ │ │ │ +0006acf0: 6e64 7320 6f66 2046 465f 6a2c 2075 7365  nds of FF_j, use
│ │ │ │ +0006ad00: 2074 6865 2072 6f75 7469 6e65 0a45 6973   the routine.Eis
│ │ │ │ +0006ad10: 656e 6275 6453 6861 6d61 7368 2069 6e73  enbudShamash ins
│ │ │ │ +0006ad20: 7465 6164 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  tead...+--------
│ │ │ │ +0006ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0006ad60: 3120 3a20 5320 3d20 5a5a 2f31 3031 5b78  1 : S = ZZ/101[x
│ │ │ │ +0006ad70: 2c79 2c7a 5d20 2020 2020 2020 2020 2020  ,y,z]           
│ │ │ │ +0006ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ad90: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  0006ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006adb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ -0006adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ade0: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ -0006adf0: 203d 2053 2020 2020 2020 2020 2020 2020   = S            
│ │ │ │ -0006ae00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006adc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120            |.|o1 
│ │ │ │ +0006add0: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │ +0006ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ae00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0006ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae20: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0006ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae50: 2020 2020 2020 2020 207c 0a7c 6f31 203a           |.|o1 :
│ │ │ │ -0006ae60: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720   PolynomialRing 
│ │ │ │ -0006ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ae90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0006aea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006aec0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052  -------+.|i2 : R
│ │ │ │ -0006aed0: 203d 2053 2f69 6465 616c 2278 332c 7933   = S/ideal"x3,y3
│ │ │ │ -0006aee0: 2220 2020 2020 2020 2020 2020 2020 2020  "               
│ │ │ │ -0006aef0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006af00: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0006af10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ae30: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20          |.|o1 : 
│ │ │ │ +0006ae40: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020  PolynomialRing  
│ │ │ │ +0006ae50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ae60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006ae70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0006ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006aea0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5220  ------+.|i2 : R 
│ │ │ │ +0006aeb0: 3d20 532f 6964 6561 6c22 7833 2c79 3322  = S/ideal"x3,y3"
│ │ │ │ +0006aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006aed0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006aef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006af00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006af10: 2020 2020 7c0a 7c6f 3220 3d20 5220 2020      |.|o2 = R   
│ │ │ │  0006af20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006af30: 2020 2020 207c 0a7c 6f32 203d 2052 2020       |.|o2 = R  
│ │ │ │ -0006af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006af40: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  0006af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006af60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006afa0: 2020 207c 0a7c 6f32 203a 2051 756f 7469     |.|o2 : Quoti
│ │ │ │ -0006afb0: 656e 7452 696e 6720 2020 2020 2020 2020  entRing         
│ │ │ │ -0006afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006afd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006af80: 2020 7c0a 7c6f 3220 3a20 5175 6f74 6965    |.|o2 : Quotie
│ │ │ │ +0006af90: 6e74 5269 6e67 2020 2020 2020 2020 2020  ntRing          
│ │ │ │ +0006afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006afb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0006afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b010: 2d2b 0a7c 6933 203a 204d 203d 2052 5e31  -+.|i3 : M = R^1
│ │ │ │ -0006b020: 2f69 6465 616c 2878 2c79 2c7a 2920 2020  /ideal(x,y,z)   
│ │ │ │ +0006aff0: 2b0a 7c69 3320 3a20 4d20 3d20 525e 312f  +.|i3 : M = R^1/
│ │ │ │ +0006b000: 6964 6561 6c28 782c 792c 7a29 2020 2020  ideal(x,y,z)    
│ │ │ │ +0006b010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b020: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b040: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006b080: 0a7c 6f33 203d 2063 6f6b 6572 6e65 6c20  .|o3 = cokernel 
│ │ │ │ -0006b090: 7c20 7820 7920 7a20 7c20 2020 2020 2020  | x y z |       
│ │ │ │ +0006b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b050: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006b060: 7c6f 3320 3d20 636f 6b65 726e 656c 207c  |o3 = cokernel |
│ │ │ │ +0006b070: 2078 2079 207a 207c 2020 2020 2020 2020   x y z |        
│ │ │ │ +0006b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0006b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b0b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0006b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0006b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b0e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b0e0: 2020 2020 2020 2020 2020 2031 2020 2020             1    
│ │ │ │  0006b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b100: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ -0006b110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b120: 2020 2020 7c0a 7c6f 3320 3a20 522d 6d6f      |.|o3 : R-mo
│ │ │ │ -0006b130: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ -0006b140: 6620 5220 2020 2020 2020 2020 2020 2020  f R             
│ │ │ │ -0006b150: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0006b100: 2020 207c 0a7c 6f33 203a 2052 2d6d 6f64     |.|o3 : R-mod
│ │ │ │ +0006b110: 756c 652c 2071 756f 7469 656e 7420 6f66  ule, quotient of
│ │ │ │ +0006b120: 2052 2020 2020 2020 2020 2020 2020 2020   R              
│ │ │ │ +0006b130: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006b140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006b150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b190: 2d2d 2b0a 7c69 3420 3a20 4620 3d20 6672  --+.|i4 : F = fr
│ │ │ │ -0006b1a0: 6565 5265 736f 6c75 7469 6f6e 284d 2c20  eeResolution(M, 
│ │ │ │ -0006b1b0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2034  LengthLimit => 4
│ │ │ │ -0006b1c0: 2920 2020 2020 2020 207c 0a7c 2020 2020  )        |.|    
│ │ │ │ -0006b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b200: 7c0a 7c20 2020 2020 2031 2020 2020 2020  |.|      1      
│ │ │ │ -0006b210: 3320 2020 2020 2035 2020 2020 2020 3720  3      5      7 
│ │ │ │ -0006b220: 2020 2020 2039 2020 2020 2020 2020 2020       9          
│ │ │ │ -0006b230: 2020 2020 2020 207c 0a7c 6f34 203d 2052         |.|o4 = R
│ │ │ │ -0006b240: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020    <-- R  <-- R  
│ │ │ │ -0006b250: 3c2d 2d20 5220 203c 2d2d 2052 2020 2020  <-- R  <-- R    
│ │ │ │ -0006b260: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006b270: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -0006b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b2a0: 2020 2020 207c 0a7c 2020 2020 2030 2020       |.|     0  
│ │ │ │ -0006b2b0: 2020 2020 3120 2020 2020 2032 2020 2020      1      2    
│ │ │ │ -0006b2c0: 2020 3320 2020 2020 2034 2020 2020 2020    3      4      
│ │ │ │ -0006b2d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006b170: 2d2b 0a7c 6934 203a 2046 203d 2066 7265  -+.|i4 : F = fre
│ │ │ │ +0006b180: 6552 6573 6f6c 7574 696f 6e28 4d2c 204c  eResolution(M, L
│ │ │ │ +0006b190: 656e 6774 684c 696d 6974 203d 3e20 3429  engthLimit => 4)
│ │ │ │ +0006b1a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b1d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006b1e0: 0a7c 2020 2020 2020 3120 2020 2020 2033  .|      1      3
│ │ │ │ +0006b1f0: 2020 2020 2020 3520 2020 2020 2037 2020        5      7  
│ │ │ │ +0006b200: 2020 2020 3920 2020 2020 2020 2020 2020      9           
│ │ │ │ +0006b210: 2020 2020 2020 7c0a 7c6f 3420 3d20 5220        |.|o4 = R 
│ │ │ │ +0006b220: 203c 2d2d 2052 2020 3c2d 2d20 5220 203c   <-- R  <-- R  <
│ │ │ │ +0006b230: 2d2d 2052 2020 3c2d 2d20 5220 2020 2020  -- R  <-- R     
│ │ │ │ +0006b240: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b280: 2020 2020 7c0a 7c20 2020 2020 3020 2020      |.|     0   
│ │ │ │ +0006b290: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ +0006b2a0: 2033 2020 2020 2020 3420 2020 2020 2020   3      4       
│ │ │ │ +0006b2b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b310: 2020 207c 0a7c 6f34 203a 2043 6f6d 706c     |.|o4 : Compl
│ │ │ │ -0006b320: 6578 2020 2020 2020 2020 2020 2020 2020  ex              
│ │ │ │ -0006b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b340: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006b2f0: 2020 7c0a 7c6f 3420 3a20 436f 6d70 6c65    |.|o4 : Comple
│ │ │ │ +0006b300: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +0006b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b320: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ +0006b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b380: 2d2b 0a7c 6935 203a 2066 6620 3d20 6d61  -+.|i5 : ff = ma
│ │ │ │ -0006b390: 7472 6978 7b7b 7a5e 337d 7d20 2020 2020  trix{{z^3}}     
│ │ │ │ +0006b360: 2b0a 7c69 3520 3a20 6666 203d 206d 6174  +.|i5 : ff = mat
│ │ │ │ +0006b370: 7269 787b 7b7a 5e33 7d7d 2020 2020 2020  rix{{z^3}}      
│ │ │ │ +0006b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b390: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b3b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b3e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006b3f0: 0a7c 6f35 203d 207c 207a 3320 7c20 2020  .|o5 = | z3 |   
│ │ │ │ -0006b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006b3d0: 7c6f 3520 3d20 7c20 7a33 207c 2020 2020  |o5 = | z3 |    
│ │ │ │ +0006b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b400: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0006b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0006b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b450: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0006b460: 2020 2020 2020 2020 2020 2020 2031 2020               1  
│ │ │ │ -0006b470: 2020 2020 3120 2020 2020 2020 2020 2020      1           
│ │ │ │ -0006b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b490: 2020 2020 7c0a 7c6f 3520 3a20 4d61 7472      |.|o5 : Matr
│ │ │ │ -0006b4a0: 6978 2052 2020 3c2d 2d20 5220 2020 2020  ix R  <-- R     
│ │ │ │ -0006b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b4c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0006b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b430: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006b440: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +0006b450: 2020 2031 2020 2020 2020 2020 2020 2020     1            
│ │ │ │ +0006b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b470: 2020 207c 0a7c 6f35 203a 204d 6174 7269     |.|o5 : Matri
│ │ │ │ +0006b480: 7820 5220 203c 2d2d 2052 2020 2020 2020  x R  <-- R      
│ │ │ │ +0006b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b4a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006b4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006b4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b500: 2d2d 2b0a 7c69 3620 3a20 5231 203d 2052  --+.|i6 : R1 = R
│ │ │ │ -0006b510: 2f69 6465 616c 2066 6620 2020 2020 2020  /ideal ff       
│ │ │ │ +0006b4e0: 2d2b 0a7c 6936 203a 2052 3120 3d20 522f  -+.|i6 : R1 = R/
│ │ │ │ +0006b4f0: 6964 6561 6c20 6666 2020 2020 2020 2020  ideal ff        
│ │ │ │ +0006b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b510: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0006b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b530: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006b550: 0a7c 6f36 203d 2052 3120 2020 2020 2020  .|o6 = R1       
│ │ │ │  0006b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b570: 7c0a 7c6f 3620 3d20 5231 2020 2020 2020  |.|o6 = R1      
│ │ │ │ -0006b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b580: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │  0006b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0006b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006b5e0: 7c6f 3620 3a20 5175 6f74 6965 6e74 5269  |o6 : QuotientRi
│ │ │ │ -0006b5f0: 6e67 2020 2020 2020 2020 2020 2020 2020  ng              
│ │ │ │ -0006b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b610: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ -0006b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ -0006b650: 3720 3a20 6265 7474 6920 4620 2020 2020  7 : betti F     
│ │ │ │ -0006b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b5b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b5c0: 6f36 203a 2051 756f 7469 656e 7452 696e  o6 : QuotientRin
│ │ │ │ +0006b5d0: 6720 2020 2020 2020 2020 2020 2020 2020  g               
│ │ │ │ +0006b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b5f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006b610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +0006b630: 203a 2062 6574 7469 2046 2020 2020 2020   : betti F      
│ │ │ │ +0006b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b660: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  0006b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b680: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0006b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b6b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0006b6c0: 2020 2020 2020 2020 2030 2031 2032 2033           0 1 2 3
│ │ │ │ -0006b6d0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ -0006b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b6f0: 207c 0a7c 6f37 203d 2074 6f74 616c 3a20   |.|o7 = total: 
│ │ │ │ -0006b700: 3120 3320 3520 3720 3920 2020 2020 2020  1 3 5 7 9       
│ │ │ │ -0006b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b720: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006b730: 2020 2020 303a 2031 2033 2033 2031 202e      0: 1 3 3 1 .
│ │ │ │ -0006b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006b760: 0a7c 2020 2020 2020 2020 2031 3a20 2e20  .|         1: . 
│ │ │ │ -0006b770: 2e20 3220 3620 3620 2020 2020 2020 2020  . 2 6 6         
│ │ │ │ -0006b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b790: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0006b7a0: 2020 323a 202e 202e 202e 202e 2033 2020    2: . . . . 3  
│ │ │ │ +0006b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b690: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006b6a0: 2020 2020 2020 2020 3020 3120 3220 3320          0 1 2 3 
│ │ │ │ +0006b6b0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0006b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b6d0: 7c0a 7c6f 3720 3d20 746f 7461 6c3a 2031  |.|o7 = total: 1
│ │ │ │ +0006b6e0: 2033 2035 2037 2039 2020 2020 2020 2020   3 5 7 9        
│ │ │ │ +0006b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b700: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006b710: 2020 2030 3a20 3120 3320 3320 3120 2e20     0: 1 3 3 1 . 
│ │ │ │ +0006b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b730: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006b740: 7c20 2020 2020 2020 2020 313a 202e 202e  |         1: . .
│ │ │ │ +0006b750: 2032 2036 2036 2020 2020 2020 2020 2020   2 6 6          
│ │ │ │ +0006b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006b780: 2032 3a20 2e20 2e20 2e20 2e20 3320 2020   2: . . . . 3   
│ │ │ │ +0006b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0006b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b7c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b800: 2020 2020 7c0a 7c6f 3720 3a20 4265 7474      |.|o7 : Bett
│ │ │ │ -0006b810: 6954 616c 6c79 2020 2020 2020 2020 2020  iTally          
│ │ │ │ -0006b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b830: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ +0006b7e0: 2020 207c 0a7c 6f37 203a 2042 6574 7469     |.|o7 : Betti
│ │ │ │ +0006b7f0: 5461 6c6c 7920 2020 2020 2020 2020 2020  Tally           
│ │ │ │ +0006b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b810: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │ +0006b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006b840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006b870: 2d2d 2b0a 7c69 3820 3a20 4646 203d 2053  --+.|i8 : FF = S
│ │ │ │ -0006b880: 6861 6d61 7368 2866 662c 462c 3429 2020  hamash(ff,F,4)  
│ │ │ │ +0006b850: 2d2b 0a7c 6938 203a 2046 4620 3d20 5368  -+.|i8 : FF = Sh
│ │ │ │ +0006b860: 616d 6173 6828 6666 2c46 2c34 2920 2020  amash(ff,F,4)   
│ │ │ │ +0006b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b880: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0006b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b8a0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b8e0: 7c0a 7c20 2020 2020 2f20 525c 3120 2020  |.|     / R\1   
│ │ │ │ -0006b8f0: 2020 2f20 525c 3320 2020 2020 2f20 525c    / R\3     / R\
│ │ │ │ -0006b900: 3620 2020 2020 2f20 525c 3130 2020 2020  6     / R\10    
│ │ │ │ -0006b910: 202f 2052 5c31 357c 0a7c 6f38 203d 207c   / R\15|.|o8 = |
│ │ │ │ -0006b920: 2d2d 7c20 203c 2d2d 207c 2d2d 7c20 203c  --|  <-- |--|  <
│ │ │ │ -0006b930: 2d2d 207c 2d2d 7c20 203c 2d2d 207c 2d2d  -- |--|  <-- |--
│ │ │ │ -0006b940: 7c20 2020 3c2d 2d20 7c2d 2d7c 2020 7c0a  |   <-- |--|  |.
│ │ │ │ -0006b950: 7c20 2020 2020 7c20 337c 2020 2020 2020  |     | 3|      
│ │ │ │ -0006b960: 7c20 337c 2020 2020 2020 7c20 337c 2020  | 3|      | 3|  
│ │ │ │ -0006b970: 2020 2020 7c20 337c 2020 2020 2020 207c      | 3|       |
│ │ │ │ -0006b980: 2033 7c20 207c 0a7c 2020 2020 205c 7a20   3|  |.|     \z 
│ │ │ │ -0006b990: 2f20 2020 2020 205c 7a20 2f20 2020 2020  /      \z /     
│ │ │ │ -0006b9a0: 205c 7a20 2f20 2020 2020 205c 7a20 2f20   \z /      \z / 
│ │ │ │ -0006b9b0: 2020 2020 2020 5c7a 202f 2020 7c0a 7c20        \z /  |.| 
│ │ │ │ +0006b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b8b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006b8c0: 0a7c 2020 2020 202f 2052 5c31 2020 2020  .|     / R\1    
│ │ │ │ +0006b8d0: 202f 2052 5c33 2020 2020 202f 2052 5c36   / R\3     / R\6
│ │ │ │ +0006b8e0: 2020 2020 202f 2052 5c31 3020 2020 2020       / R\10     
│ │ │ │ +0006b8f0: 2f20 525c 3135 7c0a 7c6f 3820 3d20 7c2d  / R\15|.|o8 = |-
│ │ │ │ +0006b900: 2d7c 2020 3c2d 2d20 7c2d 2d7c 2020 3c2d  -|  <-- |--|  <-
│ │ │ │ +0006b910: 2d20 7c2d 2d7c 2020 3c2d 2d20 7c2d 2d7c  - |--|  <-- |--|
│ │ │ │ +0006b920: 2020 203c 2d2d 207c 2d2d 7c20 207c 0a7c     <-- |--|  |.|
│ │ │ │ +0006b930: 2020 2020 207c 2033 7c20 2020 2020 207c       | 3|      |
│ │ │ │ +0006b940: 2033 7c20 2020 2020 207c 2033 7c20 2020   3|      | 3|   
│ │ │ │ +0006b950: 2020 207c 2033 7c20 2020 2020 2020 7c20     | 3|       | 
│ │ │ │ +0006b960: 337c 2020 7c0a 7c20 2020 2020 5c7a 202f  3|  |.|     \z /
│ │ │ │ +0006b970: 2020 2020 2020 5c7a 202f 2020 2020 2020        \z /      
│ │ │ │ +0006b980: 5c7a 202f 2020 2020 2020 5c7a 202f 2020  \z /      \z /  
│ │ │ │ +0006b990: 2020 2020 205c 7a20 2f20 207c 0a7c 2020       \z /  |.|  
│ │ │ │ +0006b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006b9f0: 2020 207c 0a7c 2020 2020 2030 2020 2020     |.|     0    
│ │ │ │ -0006ba00: 2020 2020 2031 2020 2020 2020 2020 2032       1         2
│ │ │ │ -0006ba10: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ -0006ba20: 2020 2020 3420 2020 2020 7c0a 7c20 2020      4     |.|   
│ │ │ │ +0006b9d0: 2020 7c0a 7c20 2020 2020 3020 2020 2020    |.|     0     
│ │ │ │ +0006b9e0: 2020 2020 3120 2020 2020 2020 2020 3220      1         2 
│ │ │ │ +0006b9f0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +0006ba00: 2020 2034 2020 2020 207c 0a7c 2020 2020     4     |.|    
│ │ │ │ +0006ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ba40: 7c0a 7c6f 3820 3a20 436f 6d70 6c65 7820  |.|o8 : Complex 
│ │ │ │  0006ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba60: 207c 0a7c 6f38 203a 2043 6f6d 706c 6578   |.|o8 : Complex
│ │ │ │ -0006ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ba90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d          |.+-----
│ │ │ │ -0006baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0006bad0: 0a7c 6939 203a 2047 4720 3d20 5368 616d  .|i9 : GG = Sham
│ │ │ │ -0006bae0: 6173 6828 5231 2c46 2c34 2920 2020 2020  ash(R1,F,4)     
│ │ │ │ +0006ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ba70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +0006ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0006bab0: 7c69 3920 3a20 4747 203d 2053 6861 6d61  |i9 : GG = Shama
│ │ │ │ +0006bac0: 7368 2852 312c 462c 3429 2020 2020 2020  sh(R1,F,4)      
│ │ │ │ +0006bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0006baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bb00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0006bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bb30: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0006bb40: 2020 2020 2020 2031 2020 2020 2020 2033         1       3
│ │ │ │ -0006bb50: 2020 2020 2020 2036 2020 2020 2020 2031         6       1
│ │ │ │ -0006bb60: 3020 2020 2020 2020 3135 2020 2020 2020  0       15      
│ │ │ │ -0006bb70: 2020 2020 7c0a 7c6f 3920 3d20 5231 2020      |.|o9 = R1  
│ │ │ │ -0006bb80: 3c2d 2d20 5231 2020 3c2d 2d20 5231 2020  <-- R1  <-- R1  
│ │ │ │ -0006bb90: 3c2d 2d20 5231 2020 203c 2d2d 2052 3120  <-- R1   <-- R1 
│ │ │ │ -0006bba0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bb10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0006bb20: 2020 2020 2020 3120 2020 2020 2020 3320        1       3 
│ │ │ │ +0006bb30: 2020 2020 2020 3620 2020 2020 2020 3130        6       10
│ │ │ │ +0006bb40: 2020 2020 2020 2031 3520 2020 2020 2020         15       
│ │ │ │ +0006bb50: 2020 207c 0a7c 6f39 203d 2052 3120 203c     |.|o9 = R1  <
│ │ │ │ +0006bb60: 2d2d 2052 3120 203c 2d2d 2052 3120 203c  -- R1  <-- R1  <
│ │ │ │ +0006bb70: 2d2d 2052 3120 2020 3c2d 2d20 5231 2020  -- R1   <-- R1  
│ │ │ │ +0006bb80: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bbe0: 2020 7c0a 7c20 2020 2020 3020 2020 2020    |.|     0     
│ │ │ │ -0006bbf0: 2020 3120 2020 2020 2020 3220 2020 2020    1       2     
│ │ │ │ -0006bc00: 2020 3320 2020 2020 2020 2034 2020 2020    3        4    
│ │ │ │ -0006bc10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bbc0: 207c 0a7c 2020 2020 2030 2020 2020 2020   |.|     0      
│ │ │ │ +0006bbd0: 2031 2020 2020 2020 2032 2020 2020 2020   1       2      
│ │ │ │ +0006bbe0: 2033 2020 2020 2020 2020 3420 2020 2020   3        4     
│ │ │ │ +0006bbf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bc20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006bc30: 0a7c 6f39 203a 2043 6f6d 706c 6578 2020  .|o9 : Complex  
│ │ │ │  0006bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc50: 7c0a 7c6f 3920 3a20 436f 6d70 6c65 7820  |.|o9 : Complex 
│ │ │ │ -0006bc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bc80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0006bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0006bcc0: 7c69 3130 203a 2062 6574 7469 2046 4620  |i10 : betti FF 
│ │ │ │ -0006bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bc60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0006bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0006bca0: 6931 3020 3a20 6265 7474 6920 4646 2020  i10 : betti FF  
│ │ │ │ +0006bcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bcd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bcf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0006bd30: 2020 2020 2020 2020 2020 2020 3020 3120              0 1 
│ │ │ │ -0006bd40: 3220 2033 2020 3420 2020 2020 2020 2020  2  3  4         
│ │ │ │ -0006bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd60: 2020 207c 0a7c 6f31 3020 3d20 746f 7461     |.|o10 = tota
│ │ │ │ -0006bd70: 6c3a 2031 2033 2036 2031 3020 3135 2020  l: 1 3 6 10 15  
│ │ │ │ -0006bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bd90: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -0006bda0: 2020 2020 2020 2030 3a20 3120 3320 3320         0: 1 3 3 
│ │ │ │ -0006bdb0: 2031 2020 2e20 2020 2020 2020 2020 2020   1  .           
│ │ │ │ -0006bdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bdd0: 207c 0a7c 2020 2020 2020 2020 2020 313a   |.|          1:
│ │ │ │ -0006bde0: 202e 202e 2033 2020 3920 2039 2020 2020   . . 3  9  9    
│ │ │ │ -0006bdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006be00: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006be10: 2020 2020 2032 3a20 2e20 2e20 2e20 202e       2: . . .  .
│ │ │ │ -0006be20: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ -0006be30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006be40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0006be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006be70: 2020 2020 2020 7c0a 7c6f 3130 203a 2042        |.|o10 : B
│ │ │ │ -0006be80: 6574 7469 5461 6c6c 7920 2020 2020 2020  ettiTally       
│ │ │ │ -0006be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bea0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0006bcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bd00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006bd10: 2020 2020 2020 2020 2020 2030 2031 2032             0 1 2
│ │ │ │ +0006bd20: 2020 3320 2034 2020 2020 2020 2020 2020    3  4          
│ │ │ │ +0006bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bd40: 2020 7c0a 7c6f 3130 203d 2074 6f74 616c    |.|o10 = total
│ │ │ │ +0006bd50: 3a20 3120 3320 3620 3130 2031 3520 2020  : 1 3 6 10 15   
│ │ │ │ +0006bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bd70: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006bd80: 2020 2020 2020 303a 2031 2033 2033 2020        0: 1 3 3  
│ │ │ │ +0006bd90: 3120 202e 2020 2020 2020 2020 2020 2020  1  .            
│ │ │ │ +0006bda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bdb0: 7c0a 7c20 2020 2020 2020 2020 2031 3a20  |.|          1: 
│ │ │ │ +0006bdc0: 2e20 2e20 3320 2039 2020 3920 2020 2020  . . 3  9  9     
│ │ │ │ +0006bdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bde0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006bdf0: 2020 2020 323a 202e 202e 202e 2020 2e20      2: . . .  . 
│ │ │ │ +0006be00: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ +0006be10: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006be20: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006be50: 2020 2020 207c 0a7c 6f31 3020 3a20 4265       |.|o10 : Be
│ │ │ │ +0006be60: 7474 6954 616c 6c79 2020 2020 2020 2020  ttiTally        
│ │ │ │ +0006be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006be80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006be90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006bea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006bee0: 2d2d 2d2d 2b0a 7c69 3131 203a 2062 6574  ----+.|i11 : bet
│ │ │ │ -0006bef0: 7469 2047 4720 2020 2020 2020 2020 2020  ti GG           
│ │ │ │ +0006bec0: 2d2d 2d2b 0a7c 6931 3120 3a20 6265 7474  ---+.|i11 : bett
│ │ │ │ +0006bed0: 6920 4747 2020 2020 2020 2020 2020 2020  i GG            
│ │ │ │ +0006bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bef0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006bf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf50: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ -0006bf60: 2020 3020 3120 3220 2033 2020 3420 2020    0 1 2  3  4   
│ │ │ │ -0006bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bf80: 2020 2020 2020 2020 207c 0a7c 6f31 3120           |.|o11 
│ │ │ │ -0006bf90: 3d20 746f 7461 6c3a 2031 2033 2036 2031  = total: 1 3 6 1
│ │ │ │ -0006bfa0: 3020 3135 2020 2020 2020 2020 2020 2020  0 15            
│ │ │ │ -0006bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bfc0: 7c0a 7c20 2020 2020 2020 2020 2030 3a20  |.|          0: 
│ │ │ │ -0006bfd0: 3120 3320 3320 2031 2020 2e20 2020 2020  1 3 3  1  .     
│ │ │ │ -0006bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006bff0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -0006c000: 2020 2020 313a 202e 202e 2033 2020 3920      1: . . 3  9 
│ │ │ │ -0006c010: 2039 2020 2020 2020 2020 2020 2020 2020   9              
│ │ │ │ -0006c020: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006c030: 7c20 2020 2020 2020 2020 2032 3a20 2e20  |          2: . 
│ │ │ │ -0006c040: 2e20 2e20 202e 2020 3620 2020 2020 2020  . .  .  6       
│ │ │ │ +0006bf30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0006bf40: 2030 2031 2032 2020 3320 2034 2020 2020   0 1 2  3  4    
│ │ │ │ +0006bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bf60: 2020 2020 2020 2020 7c0a 7c6f 3131 203d          |.|o11 =
│ │ │ │ +0006bf70: 2074 6f74 616c 3a20 3120 3320 3620 3130   total: 1 3 6 10
│ │ │ │ +0006bf80: 2031 3520 2020 2020 2020 2020 2020 2020   15             
│ │ │ │ +0006bf90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006bfa0: 0a7c 2020 2020 2020 2020 2020 303a 2031  .|          0: 1
│ │ │ │ +0006bfb0: 2033 2033 2020 3120 202e 2020 2020 2020   3 3  1  .      
│ │ │ │ +0006bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006bfd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0006bfe0: 2020 2031 3a20 2e20 2e20 3320 2039 2020     1: . . 3  9  
│ │ │ │ +0006bff0: 3920 2020 2020 2020 2020 2020 2020 2020  9               
│ │ │ │ +0006c000: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006c010: 2020 2020 2020 2020 2020 323a 202e 202e            2: . .
│ │ │ │ +0006c020: 202e 2020 2e20 2036 2020 2020 2020 2020   .  .  6        
│ │ │ │ +0006c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c040: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c060: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c090: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0006c0a0: 3131 203a 2042 6574 7469 5461 6c6c 7920  11 : BettiTally 
│ │ │ │ -0006c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c0d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ -0006c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132  ----------+.|i12
│ │ │ │ -0006c110: 203a 2072 696e 6720 4747 2020 2020 2020   : ring GG      
│ │ │ │ -0006c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c070: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +0006c080: 3120 3a20 4265 7474 6954 616c 6c79 2020  1 : BettiTally  
│ │ │ │ +0006c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c0b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0006c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006c0e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220  ---------+.|i12 
│ │ │ │ +0006c0f0: 3a20 7269 6e67 2047 4720 2020 2020 2020  : ring GG       
│ │ │ │ +0006c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c120: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  0006c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c140: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -0006c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c170: 2020 2020 2020 2020 7c0a 7c6f 3132 203d          |.|o12 =
│ │ │ │ -0006c180: 2052 3120 2020 2020 2020 2020 2020 2020   R1             
│ │ │ │ -0006c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c1a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006c1b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0006c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c1e0: 2020 2020 2020 7c0a 7c6f 3132 203a 2051        |.|o12 : Q
│ │ │ │ -0006c1f0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ -0006c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c210: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0006c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c150: 2020 2020 2020 207c 0a7c 6f31 3220 3d20         |.|o12 = 
│ │ │ │ +0006c160: 5231 2020 2020 2020 2020 2020 2020 2020  R1              
│ │ │ │ +0006c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c180: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006c190: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c1c0: 2020 2020 207c 0a7c 6f31 3220 3a20 5175       |.|o12 : Qu
│ │ │ │ +0006c1d0: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +0006c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c1f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │ +0006c200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c250: 2d2d 2d2d 2b0a 7c69 3133 203a 2061 7070  ----+.|i13 : app
│ │ │ │ -0006c260: 6c79 286c 656e 6774 6820 4747 2c20 692d  ly(length GG, i-
│ │ │ │ -0006c270: 3e70 7275 6e65 2048 485f 6920 4646 2920  >prune HH_i FF) 
│ │ │ │ -0006c280: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006c230: 2d2d 2d2b 0a7c 6931 3320 3a20 6170 706c  ---+.|i13 : appl
│ │ │ │ +0006c240: 7928 6c65 6e67 7468 2047 472c 2069 2d3e  y(length GG, i->
│ │ │ │ +0006c250: 7072 756e 6520 4848 5f69 2046 4629 2020  prune HH_i FF)  
│ │ │ │ +0006c260: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c2c0: 2020 7c0a 7c6f 3133 203d 207b 636f 6b65    |.|o13 = {coke
│ │ │ │ -0006c2d0: 726e 656c 207c 207a 2079 2078 207c 2c20  rnel | z y x |, 
│ │ │ │ -0006c2e0: 302c 2030 2c20 307d 2020 2020 2020 2020  0, 0, 0}        
│ │ │ │ -0006c2f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -0006c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c2a0: 207c 0a7c 6f31 3320 3d20 7b63 6f6b 6572   |.|o13 = {coker
│ │ │ │ +0006c2b0: 6e65 6c20 7c20 7a20 7920 7820 7c2c 2030  nel | z y x |, 0
│ │ │ │ +0006c2c0: 2c20 302c 2030 7d20 2020 2020 2020 2020  , 0, 0}         
│ │ │ │ +0006c2d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006c310: 0a7c 6f31 3320 3a20 4c69 7374 2020 2020  .|o13 : List    
│ │ │ │  0006c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c330: 7c0a 7c6f 3133 203a 204c 6973 7420 2020  |.|o13 : List   
│ │ │ │ -0006c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006c360: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ -0006c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0006c3a0: 0a43 6176 6561 740a 3d3d 3d3d 3d3d 0a0a  .Caveat.======..
│ │ │ │ -0006c3b0: 4620 6973 2061 7373 756d 6564 2074 6f20  F is assumed to 
│ │ │ │ -0006c3c0: 6265 2061 2068 6f6d 6f6c 6f67 6963 616c  be a homological
│ │ │ │ -0006c3d0: 2063 6f6d 706c 6578 2073 7461 7274 696e   complex startin
│ │ │ │ -0006c3e0: 6720 6672 6f6d 2046 5f30 2e20 5468 6520  g from F_0. The 
│ │ │ │ -0006c3f0: 6d61 7472 6978 2066 6620 6d75 7374 0a62  matrix ff must.b
│ │ │ │ -0006c400: 6520 3178 312e 0a0a 5365 6520 616c 736f  e 1x1...See also
│ │ │ │ -0006c410: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -0006c420: 6e6f 7465 2045 6973 656e 6275 6453 6861  note EisenbudSha
│ │ │ │ -0006c430: 6d61 7368 3a20 4569 7365 6e62 7564 5368  mash: EisenbudSh
│ │ │ │ -0006c440: 616d 6173 682c 202d 2d20 436f 6d70 7574  amash, -- Comput
│ │ │ │ -0006c450: 6573 2074 6865 2045 6973 656e 6275 642d  es the Eisenbud-
│ │ │ │ -0006c460: 5368 616d 6173 680a 2020 2020 436f 6d70  Shamash.    Comp
│ │ │ │ -0006c470: 6c65 780a 2020 2a20 2a6e 6f74 6520 6d61  lex.  * *note ma
│ │ │ │ -0006c480: 6b65 486f 6d6f 746f 7069 6573 3a20 6d61  keHomotopies: ma
│ │ │ │ -0006c490: 6b65 486f 6d6f 746f 7069 6573 2c20 2d2d  keHomotopies, --
│ │ │ │ -0006c4a0: 2072 6574 7572 6e73 2061 2073 7973 7465   returns a syste
│ │ │ │ -0006c4b0: 6d20 6f66 2068 6967 6865 720a 2020 2020  m of higher.    
│ │ │ │ -0006c4c0: 686f 6d6f 746f 7069 6573 0a0a 5761 7973  homotopies..Ways
│ │ │ │ -0006c4d0: 2074 6f20 7573 6520 5368 616d 6173 683a   to use Shamash:
│ │ │ │ -0006c4e0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -0006c4f0: 3d3d 3d3d 3d0a 0a20 202a 2022 5368 616d  =====..  * "Sham
│ │ │ │ -0006c500: 6173 6828 4d61 7472 6978 2c43 6f6d 706c  ash(Matrix,Compl
│ │ │ │ -0006c510: 6578 2c5a 5a29 220a 2020 2a20 2253 6861  ex,ZZ)".  * "Sha
│ │ │ │ -0006c520: 6d61 7368 2852 696e 672c 436f 6d70 6c65  mash(Ring,Comple
│ │ │ │ -0006c530: 782c 5a5a 2922 0a0a 466f 7220 7468 6520  x,ZZ)"..For the 
│ │ │ │ -0006c540: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d  programmer.=====
│ │ │ │ -0006c550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  =============..T
│ │ │ │ -0006c560: 6865 206f 626a 6563 7420 2a6e 6f74 6520  he object *note 
│ │ │ │ -0006c570: 5368 616d 6173 683a 2053 6861 6d61 7368  Shamash: Shamash
│ │ │ │ -0006c580: 2c20 6973 2061 202a 6e6f 7465 206d 6574  , is a *note met
│ │ │ │ -0006c590: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d  hod function:.(M
│ │ │ │ -0006c5a0: 6163 6175 6c61 7932 446f 6329 4d65 7468  acaulay2Doc)Meth
│ │ │ │ -0006c5b0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d  odFunction,...--
│ │ │ │ +0006c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006c340: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0006c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +0006c380: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a46  Caveat.======..F
│ │ │ │ +0006c390: 2069 7320 6173 7375 6d65 6420 746f 2062   is assumed to b
│ │ │ │ +0006c3a0: 6520 6120 686f 6d6f 6c6f 6769 6361 6c20  e a homological 
│ │ │ │ +0006c3b0: 636f 6d70 6c65 7820 7374 6172 7469 6e67  complex starting
│ │ │ │ +0006c3c0: 2066 726f 6d20 465f 302e 2054 6865 206d   from F_0. The m
│ │ │ │ +0006c3d0: 6174 7269 7820 6666 206d 7573 740a 6265  atrix ff must.be
│ │ │ │ +0006c3e0: 2031 7831 2e0a 0a53 6565 2061 6c73 6f0a   1x1...See also.
│ │ │ │ +0006c3f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e  ========..  * *n
│ │ │ │ +0006c400: 6f74 6520 4569 7365 6e62 7564 5368 616d  ote EisenbudSham
│ │ │ │ +0006c410: 6173 683a 2045 6973 656e 6275 6453 6861  ash: EisenbudSha
│ │ │ │ +0006c420: 6d61 7368 2c20 2d2d 2043 6f6d 7075 7465  mash, -- Compute
│ │ │ │ +0006c430: 7320 7468 6520 4569 7365 6e62 7564 2d53  s the Eisenbud-S
│ │ │ │ +0006c440: 6861 6d61 7368 0a20 2020 2043 6f6d 706c  hamash.    Compl
│ │ │ │ +0006c450: 6578 0a20 202a 202a 6e6f 7465 206d 616b  ex.  * *note mak
│ │ │ │ +0006c460: 6548 6f6d 6f74 6f70 6965 733a 206d 616b  eHomotopies: mak
│ │ │ │ +0006c470: 6548 6f6d 6f74 6f70 6965 732c 202d 2d20  eHomotopies, -- 
│ │ │ │ +0006c480: 7265 7475 726e 7320 6120 7379 7374 656d  returns a system
│ │ │ │ +0006c490: 206f 6620 6869 6768 6572 0a20 2020 2068   of higher.    h
│ │ │ │ +0006c4a0: 6f6d 6f74 6f70 6965 730a 0a57 6179 7320  omotopies..Ways 
│ │ │ │ +0006c4b0: 746f 2075 7365 2053 6861 6d61 7368 3a0a  to use Shamash:.
│ │ │ │ +0006c4c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006c4d0: 3d3d 3d3d 0a0a 2020 2a20 2253 6861 6d61  ====..  * "Shama
│ │ │ │ +0006c4e0: 7368 284d 6174 7269 782c 436f 6d70 6c65  sh(Matrix,Comple
│ │ │ │ +0006c4f0: 782c 5a5a 2922 0a20 202a 2022 5368 616d  x,ZZ)".  * "Sham
│ │ │ │ +0006c500: 6173 6828 5269 6e67 2c43 6f6d 706c 6578  ash(Ring,Complex
│ │ │ │ +0006c510: 2c5a 5a29 220a 0a46 6f72 2074 6865 2070  ,ZZ)"..For the p
│ │ │ │ +0006c520: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d  rogrammer.======
│ │ │ │ +0006c530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468  ============..Th
│ │ │ │ +0006c540: 6520 6f62 6a65 6374 202a 6e6f 7465 2053  e object *note S
│ │ │ │ +0006c550: 6861 6d61 7368 3a20 5368 616d 6173 682c  hamash: Shamash,
│ │ │ │ +0006c560: 2069 7320 6120 2a6e 6f74 6520 6d65 7468   is a *note meth
│ │ │ │ +0006c570: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61  od function:.(Ma
│ │ │ │ +0006c580: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ +0006c590: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d  dFunction,...---
│ │ │ │ +0006c5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006c5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ -0006c610: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ -0006c620: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ -0006c630: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ -0006c640: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ -0006c650: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ -0006c660: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ -0006c670: 6167 6573 2f0a 436f 6d70 6c65 7465 496e  ages/.CompleteIn
│ │ │ │ -0006c680: 7465 7273 6563 7469 6f6e 5265 736f 6c75  tersectionResolu
│ │ │ │ -0006c690: 7469 6f6e 732e 6d32 3a34 3736 303a 302e  tions.m2:4760:0.
│ │ │ │ -0006c6a0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574  ...File: Complet
│ │ │ │ -0006c6b0: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -0006c6c0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e  olutions.info, N
│ │ │ │ -0006c6d0: 6f64 653a 2073 706c 6974 7469 6e67 732c  ode: splittings,
│ │ │ │ -0006c6e0: 204e 6578 743a 2073 7461 626c 6548 6f6d   Next: stableHom
│ │ │ │ -0006c6f0: 2c20 5072 6576 3a20 5368 616d 6173 682c  , Prev: Shamash,
│ │ │ │ -0006c700: 2055 703a 2054 6f70 0a0a 7370 6c69 7474   Up: Top..splitt
│ │ │ │ -0006c710: 696e 6773 202d 2d20 636f 6d70 7574 6520  ings -- compute 
│ │ │ │ -0006c720: 7468 6520 7370 6c69 7474 696e 6773 206f  the splittings o
│ │ │ │ -0006c730: 6620 6120 7370 6c69 7420 7269 6768 7420  f a split right 
│ │ │ │ -0006c740: 6578 6163 7420 7365 7175 656e 6365 0a2a  exact sequence.*
│ │ │ │ +0006c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ +0006c5f0: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ +0006c600: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ +0006c610: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ +0006c620: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ +0006c630: 7932 2d31 2e32 362e 3036 2b64 732f 4d32  y2-1.26.06+ds/M2
│ │ │ │ +0006c640: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ +0006c650: 6765 732f 0a43 6f6d 706c 6574 6549 6e74  ges/.CompleteInt
│ │ │ │ +0006c660: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ +0006c670: 696f 6e73 2e6d 323a 3437 3630 3a30 2e0a  ions.m2:4760:0..
│ │ │ │ +0006c680: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465  ..File: Complete
│ │ │ │ +0006c690: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +0006c6a0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f  lutions.info, No
│ │ │ │ +0006c6b0: 6465 3a20 7370 6c69 7474 696e 6773 2c20  de: splittings, 
│ │ │ │ +0006c6c0: 4e65 7874 3a20 7374 6162 6c65 486f 6d2c  Next: stableHom,
│ │ │ │ +0006c6d0: 2050 7265 763a 2053 6861 6d61 7368 2c20   Prev: Shamash, 
│ │ │ │ +0006c6e0: 5570 3a20 546f 700a 0a73 706c 6974 7469  Up: Top..splitti
│ │ │ │ +0006c6f0: 6e67 7320 2d2d 2063 6f6d 7075 7465 2074  ngs -- compute t
│ │ │ │ +0006c700: 6865 2073 706c 6974 7469 6e67 7320 6f66  he splittings of
│ │ │ │ +0006c710: 2061 2073 706c 6974 2072 6967 6874 2065   a split right e
│ │ │ │ +0006c720: 7861 6374 2073 6571 7565 6e63 650a 2a2a  xact sequence.**
│ │ │ │ +0006c730: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0006c740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006c750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006c760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006c770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006c780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006c790: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ -0006c7a0: 0a20 2020 2020 2020 2078 203d 2073 706c  .        x = spl
│ │ │ │ -0006c7b0: 6974 7469 6e67 7328 612c 6229 0a20 202a  ittings(a,b).  *
│ │ │ │ -0006c7c0: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -0006c7d0: 2061 2c20 6120 2a6e 6f74 6520 6d61 7472   a, a *note matr
│ │ │ │ -0006c7e0: 6978 3a20 284d 6163 6175 6c61 7932 446f  ix: (Macaulay2Do
│ │ │ │ -0006c7f0: 6329 4d61 7472 6978 2c2c 206d 6170 7320  c)Matrix,, maps 
│ │ │ │ -0006c800: 696e 746f 2074 6865 206b 6572 6e65 6c20  into the kernel 
│ │ │ │ -0006c810: 6f66 2062 0a20 2020 2020 202a 2062 2c20  of b.      * b, 
│ │ │ │ -0006c820: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20  a *note matrix: 
│ │ │ │ -0006c830: 284d 6163 6175 6c61 7932 446f 6329 4d61  (Macaulay2Doc)Ma
│ │ │ │ -0006c840: 7472 6978 2c2c 2072 6570 7265 7365 6e74  trix,, represent
│ │ │ │ -0006c850: 696e 6720 6120 7375 726a 6563 7469 6f6e  ing a surjection
│ │ │ │ -0006c860: 0a20 2020 2020 2020 2066 726f 6d20 7461  .        from ta
│ │ │ │ -0006c870: 7267 6574 2061 2074 6f20 6120 6672 6565  rget a to a free
│ │ │ │ -0006c880: 206d 6f64 756c 650a 2020 2a20 4f75 7470   module.  * Outp
│ │ │ │ -0006c890: 7574 733a 0a20 2020 2020 202a 204c 2c20  uts:.      * L, 
│ │ │ │ -0006c8a0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d  a *note list: (M
│ │ │ │ -0006c8b0: 6163 6175 6c61 7932 446f 6329 4c69 7374  acaulay2Doc)List
│ │ │ │ -0006c8c0: 2c2c 204c 203d 205c 7b73 6967 6d61 2c74  ,, L = \{sigma,t
│ │ │ │ -0006c8d0: 6175 5c7d 2c20 7370 6c69 7474 696e 6773  au\}, splittings
│ │ │ │ -0006c8e0: 206f 660a 2020 2020 2020 2020 612c 6220   of.        a,b 
│ │ │ │ -0006c8f0: 7265 7370 6563 7469 7665 6c79 0a0a 4465  respectively..De
│ │ │ │ -0006c900: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -0006c910: 3d3d 3d3d 3d0a 0a41 7373 756d 696e 6720  =====..Assuming 
│ │ │ │ -0006c920: 7468 6174 2028 612c 6229 2061 7265 2074  that (a,b) are t
│ │ │ │ -0006c930: 6865 206d 6170 7320 6f66 2061 2072 6967  he maps of a rig
│ │ │ │ -0006c940: 6874 2065 7861 6374 2073 6571 7565 6e63  ht exact sequenc
│ │ │ │ -0006c950: 650a 0a24 305c 746f 2041 5c74 6f20 425c  e..$0\to A\to B\
│ │ │ │ -0006c960: 746f 2043 205c 746f 2030 240a 0a77 6974  to C \to 0$..wit
│ │ │ │ -0006c970: 6820 422c 2043 2066 7265 652c 2074 6865  h B, C free, the
│ │ │ │ -0006c980: 2073 6372 6970 7420 7072 6f64 7563 6573   script produces
│ │ │ │ -0006c990: 2061 2070 6169 7220 6f66 206d 6170 7320   a pair of maps 
│ │ │ │ -0006c9a0: 7369 676d 612c 2074 6175 2077 6974 6820  sigma, tau with 
│ │ │ │ -0006c9b0: 2474 6175 3a20 4320 5c74 6f0a 4224 2061  $tau: C \to.B$ a
│ │ │ │ -0006c9c0: 2073 706c 6974 7469 6e67 206f 6620 6220   splitting of b 
│ │ │ │ -0006c9d0: 616e 6420 2473 6967 6d61 3a20 4220 5c74  and $sigma: B \t
│ │ │ │ -0006c9e0: 6f20 4124 2061 2073 706c 6974 7469 6e67  o A$ a splitting
│ │ │ │ -0006c9f0: 206f 6620 613b 2074 6861 7420 6973 2c0a   of a; that is,.
│ │ │ │ -0006ca00: 0a24 612a 7369 676d 612b 7461 752a 6220  .$a*sigma+tau*b 
│ │ │ │ -0006ca10: 3d20 315f 4224 0a0a 2473 6967 6d61 2a61  = 1_B$..$sigma*a
│ │ │ │ -0006ca20: 203d 2031 5f41 240a 0a24 622a 7461 7520   = 1_A$..$b*tau 
│ │ │ │ -0006ca30: 3d20 315f 4324 0a0a 2b2d 2d2d 2d2d 2d2d  = 1_C$..+-------
│ │ │ │ +0006c770: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a  **..  * Usage: .
│ │ │ │ +0006c780: 2020 2020 2020 2020 7820 3d20 7370 6c69          x = spli
│ │ │ │ +0006c790: 7474 696e 6773 2861 2c62 290a 2020 2a20  ttings(a,b).  * 
│ │ │ │ +0006c7a0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +0006c7b0: 612c 2061 202a 6e6f 7465 206d 6174 7269  a, a *note matri
│ │ │ │ +0006c7c0: 783a 2028 4d61 6361 756c 6179 3244 6f63  x: (Macaulay2Doc
│ │ │ │ +0006c7d0: 294d 6174 7269 782c 2c20 6d61 7073 2069  )Matrix,, maps i
│ │ │ │ +0006c7e0: 6e74 6f20 7468 6520 6b65 726e 656c 206f  nto the kernel o
│ │ │ │ +0006c7f0: 6620 620a 2020 2020 2020 2a20 622c 2061  f b.      * b, a
│ │ │ │ +0006c800: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ +0006c810: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ +0006c820: 7269 782c 2c20 7265 7072 6573 656e 7469  rix,, representi
│ │ │ │ +0006c830: 6e67 2061 2073 7572 6a65 6374 696f 6e0a  ng a surjection.
│ │ │ │ +0006c840: 2020 2020 2020 2020 6672 6f6d 2074 6172          from tar
│ │ │ │ +0006c850: 6765 7420 6120 746f 2061 2066 7265 6520  get a to a free 
│ │ │ │ +0006c860: 6d6f 6475 6c65 0a20 202a 204f 7574 7075  module.  * Outpu
│ │ │ │ +0006c870: 7473 3a0a 2020 2020 2020 2a20 4c2c 2061  ts:.      * L, a
│ │ │ │ +0006c880: 202a 6e6f 7465 206c 6973 743a 2028 4d61   *note list: (Ma
│ │ │ │ +0006c890: 6361 756c 6179 3244 6f63 294c 6973 742c  caulay2Doc)List,
│ │ │ │ +0006c8a0: 2c20 4c20 3d20 5c7b 7369 676d 612c 7461  , L = \{sigma,ta
│ │ │ │ +0006c8b0: 755c 7d2c 2073 706c 6974 7469 6e67 7320  u\}, splittings 
│ │ │ │ +0006c8c0: 6f66 0a20 2020 2020 2020 2061 2c62 2072  of.        a,b r
│ │ │ │ +0006c8d0: 6573 7065 6374 6976 656c 790a 0a44 6573  espectively..Des
│ │ │ │ +0006c8e0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d  cription.=======
│ │ │ │ +0006c8f0: 3d3d 3d3d 0a0a 4173 7375 6d69 6e67 2074  ====..Assuming t
│ │ │ │ +0006c900: 6861 7420 2861 2c62 2920 6172 6520 7468  hat (a,b) are th
│ │ │ │ +0006c910: 6520 6d61 7073 206f 6620 6120 7269 6768  e maps of a righ
│ │ │ │ +0006c920: 7420 6578 6163 7420 7365 7175 656e 6365  t exact sequence
│ │ │ │ +0006c930: 0a0a 2430 5c74 6f20 415c 746f 2042 5c74  ..$0\to A\to B\t
│ │ │ │ +0006c940: 6f20 4320 5c74 6f20 3024 0a0a 7769 7468  o C \to 0$..with
│ │ │ │ +0006c950: 2042 2c20 4320 6672 6565 2c20 7468 6520   B, C free, the 
│ │ │ │ +0006c960: 7363 7269 7074 2070 726f 6475 6365 7320  script produces 
│ │ │ │ +0006c970: 6120 7061 6972 206f 6620 6d61 7073 2073  a pair of maps s
│ │ │ │ +0006c980: 6967 6d61 2c20 7461 7520 7769 7468 2024  igma, tau with $
│ │ │ │ +0006c990: 7461 753a 2043 205c 746f 0a42 2420 6120  tau: C \to.B$ a 
│ │ │ │ +0006c9a0: 7370 6c69 7474 696e 6720 6f66 2062 2061  splitting of b a
│ │ │ │ +0006c9b0: 6e64 2024 7369 676d 613a 2042 205c 746f  nd $sigma: B \to
│ │ │ │ +0006c9c0: 2041 2420 6120 7370 6c69 7474 696e 6720   A$ a splitting 
│ │ │ │ +0006c9d0: 6f66 2061 3b20 7468 6174 2069 732c 0a0a  of a; that is,..
│ │ │ │ +0006c9e0: 2461 2a73 6967 6d61 2b74 6175 2a62 203d  $a*sigma+tau*b =
│ │ │ │ +0006c9f0: 2031 5f42 240a 0a24 7369 676d 612a 6120   1_B$..$sigma*a 
│ │ │ │ +0006ca00: 3d20 315f 4124 0a0a 2462 2a74 6175 203d  = 1_A$..$b*tau =
│ │ │ │ +0006ca10: 2031 5f43 240a 0a2b 2d2d 2d2d 2d2d 2d2d   1_C$..+--------
│ │ │ │ +0006ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ca40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ca80: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a  ---+.|i1 : kk= Z
│ │ │ │ -0006ca90: 5a2f 3130 3120 2020 2020 2020 2020 2020  Z/101           
│ │ │ │ -0006caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ca60: 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20 5a5a  --+.|i1 : kk= ZZ
│ │ │ │ +0006ca70: 2f31 3031 2020 2020 2020 2020 2020 2020  /101            
│ │ │ │ +0006ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006caa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006cab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cad0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +0006cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb10: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -0006cb20: 6f31 203d 206b 6b20 2020 2020 2020 2020  o1 = kk         
│ │ │ │ +0006caf0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0006cb00: 3120 3d20 6b6b 2020 2020 2020 2020 2020  1 = kk          
│ │ │ │ +0006cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cb40: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0006cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cbb0: 2020 2020 2020 207c 0a7c 6f31 203a 2051         |.|o1 : Q
│ │ │ │ -0006cbc0: 756f 7469 656e 7452 696e 6720 2020 2020  uotientRing     
│ │ │ │ +0006cb90: 2020 2020 2020 7c0a 7c6f 3120 3a20 5175        |.|o1 : Qu
│ │ │ │ +0006cba0: 6f74 6965 6e74 5269 6e67 2020 2020 2020  otientRing      
│ │ │ │ +0006cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cc00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006cbe0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0006cbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006cc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cc50: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b  -+.|i2 : S = kk[
│ │ │ │ -0006cc60: 782c 792c 7a5d 2020 2020 2020 2020 2020  x,y,z]          
│ │ │ │ -0006cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cc30: 2b0a 7c69 3220 3a20 5320 3d20 6b6b 5b78  +.|i2 : S = kk[x
│ │ │ │ +0006cc40: 2c79 2c7a 5d20 2020 2020 2020 2020 2020  ,y,z]           
│ │ │ │ +0006cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cc70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  0006cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cc90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006cca0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cce0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ -0006ccf0: 203d 2053 2020 2020 2020 2020 2020 2020   = S            
│ │ │ │ +0006ccc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +0006ccd0: 3d20 5320 2020 2020 2020 2020 2020 2020  = S             
│ │ │ │ +0006cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cd10: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cd30: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cd80: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c       |.|o2 : Pol
│ │ │ │ -0006cd90: 796e 6f6d 6961 6c52 696e 6720 2020 2020  ynomialRing     
│ │ │ │ +0006cd60: 2020 2020 7c0a 7c6f 3220 3a20 506f 6c79      |.|o2 : Poly
│ │ │ │ +0006cd70: 6e6f 6d69 616c 5269 6e67 2020 2020 2020  nomialRing      
│ │ │ │ +0006cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cdd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0006cdb0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0006cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0006ce20: 0a7c 6933 203a 2073 6574 5261 6e64 6f6d  .|i3 : setRandom
│ │ │ │ -0006ce30: 5365 6564 2030 2020 2020 2020 2020 2020  Seed 0          
│ │ │ │ -0006ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ce60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0006ce70: 2d2d 2073 6574 7469 6e67 2072 616e 646f  -- setting rando
│ │ │ │ -0006ce80: 6d20 7365 6564 2074 6f20 3020 2020 2020  m seed to 0     
│ │ │ │ -0006ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +0006ce00: 7c69 3320 3a20 7365 7452 616e 646f 6d53  |i3 : setRandomS
│ │ │ │ +0006ce10: 6565 6420 3020 2020 2020 2020 2020 2020  eed 0           
│ │ │ │ +0006ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ce40: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +0006ce50: 2d20 7365 7474 696e 6720 7261 6e64 6f6d  - setting random
│ │ │ │ +0006ce60: 2073 6565 6420 746f 2030 2020 2020 2020   seed to 0      
│ │ │ │ +0006ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ce90: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0006cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ceb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cee0: 2020 2020 207c 0a7c 6f33 203d 2030 2020       |.|o3 = 0  
│ │ │ │  0006cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf00: 2020 2020 2020 7c0a 7c6f 3320 3d20 3020        |.|o3 = 0 
│ │ │ │ +0006cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cf50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0006cf30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0006cf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006cf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006cf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006cfa0: 2b0a 7c69 3420 3a20 7420 3d20 7261 6e64  +.|i4 : t = rand
│ │ │ │ -0006cfb0: 6f6d 2853 5e7b 323a 2d31 2c32 3a2d 327d  om(S^{2:-1,2:-2}
│ │ │ │ -0006cfc0: 2c20 535e 7b33 3a2d 312c 343a 2d32 7d29  , S^{3:-1,4:-2})
│ │ │ │ +0006cf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0006cf80: 0a7c 6934 203a 2074 203d 2072 616e 646f  .|i4 : t = rando
│ │ │ │ +0006cf90: 6d28 535e 7b32 3a2d 312c 323a 2d32 7d2c  m(S^{2:-1,2:-2},
│ │ │ │ +0006cfa0: 2053 5e7b 333a 2d31 2c34 3a2d 327d 2920   S^{3:-1,4:-2}) 
│ │ │ │ +0006cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006cfc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0006cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006cfe0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d030: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -0006d040: 3d20 7b31 7d20 7c20 3234 2020 2d33 3620  = {1} | 24  -36 
│ │ │ │ -0006d050: 2d33 3020 3339 782d 3433 792b 3435 7a20  -30 39x-43y+45z 
│ │ │ │ -0006d060: 2032 3178 2d31 3579 2d33 347a 2033 3478   21x-15y-34z 34x
│ │ │ │ -0006d070: 2d32 3879 2d34 387a 2020 3139 782d 3437  -28y-48z  19x-47
│ │ │ │ -0006d080: 792d 3437 7a20 7c7c 0a7c 2020 2020 207b  y-47z ||.|     {
│ │ │ │ -0006d090: 317d 207c 202d 3239 2031 3920 2031 3920  1} | -29 19  19 
│ │ │ │ -0006d0a0: 202d 3437 782b 3338 792b 3437 7a20 2d33   -47x+38y+47z -3
│ │ │ │ -0006d0b0: 3978 2b32 792b 3139 7a20 2d31 3878 2b31  9x+2y+19z -18x+1
│ │ │ │ -0006d0c0: 3679 2d31 367a 202d 3133 782b 3232 792b  6y-16z -13x+22y+
│ │ │ │ -0006d0d0: 377a 207c 7c0a 7c20 2020 2020 7b32 7d20  7z ||.|     {2} 
│ │ │ │ -0006d0e0: 7c20 3020 2020 3020 2020 3020 2020 2d31  | 0   0   0   -1
│ │ │ │ -0006d0f0: 3020 2020 2020 2020 2020 202d 3239 2020  0          -29  
│ │ │ │ -0006d100: 2020 2020 2020 202d 3820 2020 2020 2020         -8       
│ │ │ │ -0006d110: 2020 2020 2d32 3220 2020 2020 2020 2020      -22         
│ │ │ │ -0006d120: 7c7c 0a7c 2020 2020 207b 327d 207c 2030  ||.|     {2} | 0
│ │ │ │ -0006d130: 2020 2030 2020 2030 2020 202d 3239 2020     0   0   -29  
│ │ │ │ -0006d140: 2020 2020 2020 2020 2d32 3420 2020 2020          -24     
│ │ │ │ -0006d150: 2020 2020 2d33 3820 2020 2020 2020 2020      -38         
│ │ │ │ -0006d160: 202d 3136 2020 2020 2020 2020 207c 7c0a   -16         ||.
│ │ │ │ -0006d170: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006d010: 2020 2020 2020 2020 207c 0a7c 6f34 203d           |.|o4 =
│ │ │ │ +0006d020: 207b 317d 207c 2032 3420 202d 3336 202d   {1} | 24  -36 -
│ │ │ │ +0006d030: 3330 2033 3978 2d34 3379 2b34 357a 2020  30 39x-43y+45z  
│ │ │ │ +0006d040: 3231 782d 3135 792d 3334 7a20 3334 782d  21x-15y-34z 34x-
│ │ │ │ +0006d050: 3238 792d 3438 7a20 2031 3978 2d34 3779  28y-48z  19x-47y
│ │ │ │ +0006d060: 2d34 377a 207c 7c0a 7c20 2020 2020 7b31  -47z ||.|     {1
│ │ │ │ +0006d070: 7d20 7c20 2d32 3920 3139 2020 3139 2020  } | -29 19  19  
│ │ │ │ +0006d080: 2d34 3778 2b33 3879 2b34 377a 202d 3339  -47x+38y+47z -39
│ │ │ │ +0006d090: 782b 3279 2b31 397a 202d 3138 782b 3136  x+2y+19z -18x+16
│ │ │ │ +0006d0a0: 792d 3136 7a20 2d31 3378 2b32 3279 2b37  y-16z -13x+22y+7
│ │ │ │ +0006d0b0: 7a20 7c7c 0a7c 2020 2020 207b 327d 207c  z ||.|     {2} |
│ │ │ │ +0006d0c0: 2030 2020 2030 2020 2030 2020 202d 3130   0   0   0   -10
│ │ │ │ +0006d0d0: 2020 2020 2020 2020 2020 2d32 3920 2020            -29   
│ │ │ │ +0006d0e0: 2020 2020 2020 2d38 2020 2020 2020 2020        -8        
│ │ │ │ +0006d0f0: 2020 202d 3232 2020 2020 2020 2020 207c     -22         |
│ │ │ │ +0006d100: 7c0a 7c20 2020 2020 7b32 7d20 7c20 3020  |.|     {2} | 0 
│ │ │ │ +0006d110: 2020 3020 2020 3020 2020 2d32 3920 2020    0   0   -29   
│ │ │ │ +0006d120: 2020 2020 2020 202d 3234 2020 2020 2020         -24      
│ │ │ │ +0006d130: 2020 202d 3338 2020 2020 2020 2020 2020     -38          
│ │ │ │ +0006d140: 2d31 3620 2020 2020 2020 2020 7c7c 0a7c  -16         ||.|
│ │ │ │ +0006d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d1b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0006d1c0: 2020 2020 2020 2020 2020 2034 2020 2020             4    
│ │ │ │ -0006d1d0: 2020 3720 2020 2020 2020 2020 2020 2020    7             
│ │ │ │ -0006d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d200: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20          |.|o4 : 
│ │ │ │ -0006d210: 4d61 7472 6978 2053 2020 3c2d 2d20 5320  Matrix S  <-- S 
│ │ │ │ +0006d190: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006d1a0: 2020 2020 2020 2020 2020 3420 2020 2020            4     
│ │ │ │ +0006d1b0: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │ +0006d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d1e0: 2020 2020 2020 207c 0a7c 6f34 203a 204d         |.|o4 : M
│ │ │ │ +0006d1f0: 6174 7269 7820 5320 203c 2d2d 2053 2020  atrix S  <-- S  
│ │ │ │ +0006d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d250: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006d230: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d2a0: 2d2d 2b0a 7c69 3520 3a20 7373 203d 2073  --+.|i5 : ss = s
│ │ │ │ -0006d2b0: 706c 6974 7469 6e67 7328 7379 7a20 742c  plittings(syz t,
│ │ │ │ -0006d2c0: 2074 2920 2020 2020 2020 2020 2020 2020   t)             
│ │ │ │ -0006d2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d2e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006d2f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006d280: 2d2b 0a7c 6935 203a 2073 7320 3d20 7370  -+.|i5 : ss = sp
│ │ │ │ +0006d290: 6c69 7474 696e 6773 2873 797a 2074 2c20  littings(syz t, 
│ │ │ │ +0006d2a0: 7429 2020 2020 2020 2020 2020 2020 2020  t)              
│ │ │ │ +0006d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d2c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006d2d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +0006d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d330: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -0006d340: 3520 3d20 7b7b 317d 207c 2030 2030 2031  5 = {{1} | 0 0 1
│ │ │ │ -0006d350: 2030 2030 2030 2020 2030 2020 7c2c 207b   0 0 0   0  |, {
│ │ │ │ -0006d360: 317d 207c 202d 3237 2032 2020 3133 782d  1} | -27 2  13x-
│ │ │ │ -0006d370: 3130 792b 3433 7a20 3530 782d 3334 792d  10y+43z 50x-34y-
│ │ │ │ -0006d380: 3530 7a20 7c7d 2020 207c 0a7c 2020 2020  50z |}   |.|    
│ │ │ │ -0006d390: 2020 7b32 7d20 7c20 3020 3020 3020 3020    {2} | 0 0 0 0 
│ │ │ │ -0006d3a0: 3020 2d33 3120 2d36 207c 2020 7b31 7d20  0 -31 -6 |  {1} 
│ │ │ │ -0006d3b0: 7c20 2d34 2020 3335 2032 3278 2b33 3279  | -4  35 22x+32y
│ │ │ │ -0006d3c0: 2b34 337a 202d 3778 2d38 792d 3237 7a20  +43z -7x-8y-27z 
│ │ │ │ -0006d3d0: 207c 2020 2020 7c0a 7c20 2020 2020 207b   |    |.|      {
│ │ │ │ -0006d3e0: 327d 207c 2030 2030 2030 2030 2030 2032  2} | 0 0 0 0 0 2
│ │ │ │ -0006d3f0: 3920 2039 2020 7c20 207b 317d 207c 2030  9  9  |  {1} | 0
│ │ │ │ -0006d400: 2020 2030 2020 3020 2020 2020 2020 2020     0  0         
│ │ │ │ -0006d410: 2020 3020 2020 2020 2020 2020 2020 7c20    0           | 
│ │ │ │ -0006d420: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -0006d430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d440: 2020 2020 2020 7b32 7d20 7c20 3020 2020        {2} | 0   
│ │ │ │ -0006d450: 3020 202d 3235 2020 2020 2020 2020 2032  0  -25         2
│ │ │ │ -0006d460: 3620 2020 2020 2020 2020 207c 2020 2020  6          |    
│ │ │ │ -0006d470: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0006d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d490: 2020 207b 327d 207c 2030 2020 2030 2020     {2} | 0   0  
│ │ │ │ -0006d4a0: 3236 2020 2020 2020 2020 2020 2d32 2020  26          -2  
│ │ │ │ -0006d4b0: 2020 2020 2020 2020 7c20 2020 207c 0a7c          |    |.|
│ │ │ │ -0006d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d4e0: 7b32 7d20 7c20 3020 2020 3020 2030 2020  {2} | 0   0  0  
│ │ │ │ -0006d4f0: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ -0006d500: 2020 2020 207c 2020 2020 7c0a 7c20 2020       |    |.|   
│ │ │ │ -0006d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d520: 2020 2020 2020 2020 2020 2020 207b 327d               {2}
│ │ │ │ -0006d530: 207c 2030 2020 2030 2020 3020 2020 2020   | 0   0  0     
│ │ │ │ -0006d540: 2020 2020 2020 3020 2020 2020 2020 2020        0         
│ │ │ │ -0006d550: 2020 7c20 2020 207c 0a7c 2020 2020 2020    |    |.|      
│ │ │ │ +0006d310: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ +0006d320: 203d 207b 7b31 7d20 7c20 3020 3020 3120   = {{1} | 0 0 1 
│ │ │ │ +0006d330: 3020 3020 3020 2020 3020 207c 2c20 7b31  0 0 0   0  |, {1
│ │ │ │ +0006d340: 7d20 7c20 2d32 3720 3220 2031 3378 2d31  } | -27 2  13x-1
│ │ │ │ +0006d350: 3079 2b34 337a 2035 3078 2d33 3479 2d35  0y+43z 50x-34y-5
│ │ │ │ +0006d360: 307a 207c 7d20 2020 7c0a 7c20 2020 2020  0z |}   |.|     
│ │ │ │ +0006d370: 207b 327d 207c 2030 2030 2030 2030 2030   {2} | 0 0 0 0 0
│ │ │ │ +0006d380: 202d 3331 202d 3620 7c20 207b 317d 207c   -31 -6 |  {1} |
│ │ │ │ +0006d390: 202d 3420 2033 3520 3232 782b 3332 792b   -4  35 22x+32y+
│ │ │ │ +0006d3a0: 3433 7a20 2d37 782d 3879 2d32 377a 2020  43z -7x-8y-27z  
│ │ │ │ +0006d3b0: 7c20 2020 207c 0a7c 2020 2020 2020 7b32  |    |.|      {2
│ │ │ │ +0006d3c0: 7d20 7c20 3020 3020 3020 3020 3020 3239  } | 0 0 0 0 0 29
│ │ │ │ +0006d3d0: 2020 3920 207c 2020 7b31 7d20 7c20 3020    9  |  {1} | 0 
│ │ │ │ +0006d3e0: 2020 3020 2030 2020 2020 2020 2020 2020    0  0          
│ │ │ │ +0006d3f0: 2030 2020 2020 2020 2020 2020 207c 2020   0           |  
│ │ │ │ +0006d400: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +0006d410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d420: 2020 2020 207b 327d 207c 2030 2020 2030       {2} | 0   0
│ │ │ │ +0006d430: 2020 2d32 3520 2020 2020 2020 2020 3236    -25         26
│ │ │ │ +0006d440: 2020 2020 2020 2020 2020 7c20 2020 207c            |    |
│ │ │ │ +0006d450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0006d460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d470: 2020 7b32 7d20 7c20 3020 2020 3020 2032    {2} | 0   0  2
│ │ │ │ +0006d480: 3620 2020 2020 2020 2020 202d 3220 2020  6          -2   
│ │ │ │ +0006d490: 2020 2020 2020 207c 2020 2020 7c0a 7c20         |    |.| 
│ │ │ │ +0006d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d4b0: 2020 2020 2020 2020 2020 2020 2020 207b                 {
│ │ │ │ +0006d4c0: 327d 207c 2030 2020 2030 2020 3020 2020  2} | 0   0  0   
│ │ │ │ +0006d4d0: 2020 2020 2020 2020 3020 2020 2020 2020          0       
│ │ │ │ +0006d4e0: 2020 2020 7c20 2020 207c 0a7c 2020 2020      |    |.|    
│ │ │ │ +0006d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d500: 2020 2020 2020 2020 2020 2020 7b32 7d20              {2} 
│ │ │ │ +0006d510: 7c20 3020 2020 3020 2030 2020 2020 2020  | 0   0  0      
│ │ │ │ +0006d520: 2020 2020 2030 2020 2020 2020 2020 2020       0          
│ │ │ │ +0006d530: 207c 2020 2020 7c0a 7c20 2020 2020 2020   |    |.|       
│ │ │ │ +0006d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d580: 2020 207c 0a7c 6f35 203a 204c 6973 7420     |.|o5 : List 
│ │ │ │  0006d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d5a0: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374      |.|o5 : List
│ │ │ │ +0006d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d5f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +0006d5d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +0006d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -0006d640: 7c69 3620 3a20 7373 2f62 6574 7469 2020  |i6 : ss/betti  
│ │ │ │ +0006d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +0006d620: 6936 203a 2073 732f 6265 7474 6920 2020  i6 : ss/betti   
│ │ │ │ +0006d630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d660: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  0006d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d680: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006d680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d6d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -0006d6e0: 2020 2020 2020 2020 3020 3120 2020 2020          0 1     
│ │ │ │ -0006d6f0: 2020 2020 3020 3120 2020 2020 2020 2020      0 1         
│ │ │ │ -0006d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d720: 2020 2020 207c 0a7c 6f36 203d 207b 746f       |.|o6 = {to
│ │ │ │ -0006d730: 7461 6c3a 2033 2037 2c20 746f 7461 6c3a  tal: 3 7, total:
│ │ │ │ -0006d740: 2037 2034 7d20 2020 2020 2020 2020 2020   7 4}           
│ │ │ │ -0006d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d770: 2020 7c0a 7c20 2020 2020 2020 2020 2030    |.|          0
│ │ │ │ -0006d780: 3a20 2e20 3320 2020 2020 2030 3a20 2e20  : . 3      0: . 
│ │ │ │ -0006d790: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0006d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d7b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0006d7c0: 0a7c 2020 2020 2020 2020 2020 313a 2031  .|          1: 1
│ │ │ │ -0006d7d0: 2034 2020 2020 2020 313a 2033 2032 2020   4      1: 3 2  
│ │ │ │ -0006d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d800: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0006d810: 2020 2020 2020 2020 2032 3a20 3220 2e20           2: 2 . 
│ │ │ │ -0006d820: 2020 2020 2032 3a20 3420 2e20 2020 2020       2: 4 .     
│ │ │ │ -0006d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d6b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +0006d6c0: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ +0006d6d0: 2020 2030 2031 2020 2020 2020 2020 2020     0 1          
│ │ │ │ +0006d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d700: 2020 2020 7c0a 7c6f 3620 3d20 7b74 6f74      |.|o6 = {tot
│ │ │ │ +0006d710: 616c 3a20 3320 372c 2074 6f74 616c 3a20  al: 3 7, total: 
│ │ │ │ +0006d720: 3720 347d 2020 2020 2020 2020 2020 2020  7 4}            
│ │ │ │ +0006d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d750: 207c 0a7c 2020 2020 2020 2020 2020 303a   |.|          0:
│ │ │ │ +0006d760: 202e 2033 2020 2020 2020 303a 202e 2032   . 3      0: . 2
│ │ │ │ +0006d770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d790: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +0006d7a0: 7c20 2020 2020 2020 2020 2031 3a20 3120  |          1: 1 
│ │ │ │ +0006d7b0: 3420 2020 2020 2031 3a20 3320 3220 2020  4      1: 3 2   
│ │ │ │ +0006d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d7e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +0006d7f0: 2020 2020 2020 2020 323a 2032 202e 2020          2: 2 .  
│ │ │ │ +0006d800: 2020 2020 323a 2034 202e 2020 2020 2020      2: 4 .      
│ │ │ │ +0006d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d830: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  0006d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d850: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +0006d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d8a0: 2020 2020 2020 7c0a 7c6f 3620 3a20 4c69        |.|o6 : Li
│ │ │ │ -0006d8b0: 7374 2020 2020 2020 2020 2020 2020 2020  st              
│ │ │ │ +0006d880: 2020 2020 207c 0a7c 6f36 203a 204c 6973       |.|o6 : Lis
│ │ │ │ +0006d890: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +0006d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006d8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006d8f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │ +0006d8d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d    |.+-----------
│ │ │ │ +0006d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006d940: 2b0a 0a57 6179 7320 746f 2075 7365 2073  +..Ways to use s
│ │ │ │ -0006d950: 706c 6974 7469 6e67 733a 0a3d 3d3d 3d3d  plittings:.=====
│ │ │ │ -0006d960: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006d970: 3d3d 0a0a 2020 2a20 2273 706c 6974 7469  ==..  * "splitti
│ │ │ │ -0006d980: 6e67 7328 4d61 7472 6978 2c4d 6174 7269  ngs(Matrix,Matri
│ │ │ │ -0006d990: 7829 220a 0a46 6f72 2074 6865 2070 726f  x)"..For the pro
│ │ │ │ -0006d9a0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -0006d9b0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -0006d9c0: 6f62 6a65 6374 202a 6e6f 7465 2073 706c  object *note spl
│ │ │ │ -0006d9d0: 6974 7469 6e67 733a 2073 706c 6974 7469  ittings: splitti
│ │ │ │ -0006d9e0: 6e67 732c 2069 7320 6120 2a6e 6f74 6520  ngs, is a *note 
│ │ │ │ -0006d9f0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a  method function:
│ │ │ │ -0006da00: 0a28 4d61 6361 756c 6179 3244 6f63 294d  .(Macaulay2Doc)M
│ │ │ │ -0006da10: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ -0006da20: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +0006d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +0006d920: 0a0a 5761 7973 2074 6f20 7573 6520 7370  ..Ways to use sp
│ │ │ │ +0006d930: 6c69 7474 696e 6773 3a0a 3d3d 3d3d 3d3d  littings:.======
│ │ │ │ +0006d940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006d950: 3d0a 0a20 202a 2022 7370 6c69 7474 696e  =..  * "splittin
│ │ │ │ +0006d960: 6773 284d 6174 7269 782c 4d61 7472 6978  gs(Matrix,Matrix
│ │ │ │ +0006d970: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +0006d980: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +0006d990: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +0006d9a0: 626a 6563 7420 2a6e 6f74 6520 7370 6c69  bject *note spli
│ │ │ │ +0006d9b0: 7474 696e 6773 3a20 7370 6c69 7474 696e  ttings: splittin
│ │ │ │ +0006d9c0: 6773 2c20 6973 2061 202a 6e6f 7465 206d  gs, is a *note m
│ │ │ │ +0006d9d0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a  ethod function:.
│ │ │ │ +0006d9e0: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ +0006d9f0: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a  thodFunction,...
│ │ │ │ +0006da00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006da70: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -0006da80: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ -0006da90: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ -0006daa0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -0006dab0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64  aulay2-1.26.06+d
│ │ │ │ -0006dac0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -0006dad0: 6163 6b61 6765 732f 0a43 6f6d 706c 6574  ackages/.Complet
│ │ │ │ -0006dae0: 6549 6e74 6572 7365 6374 696f 6e52 6573  eIntersectionRes
│ │ │ │ -0006daf0: 6f6c 7574 696f 6e73 2e6d 323a 3339 3235  olutions.m2:3925
│ │ │ │ -0006db00: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70  :0....File: Comp
│ │ │ │ -0006db10: 6c65 7465 496e 7465 7273 6563 7469 6f6e  leteIntersection
│ │ │ │ -0006db20: 5265 736f 6c75 7469 6f6e 732e 696e 666f  Resolutions.info
│ │ │ │ -0006db30: 2c20 4e6f 6465 3a20 7374 6162 6c65 486f  , Node: stableHo
│ │ │ │ -0006db40: 6d2c 204e 6578 743a 2073 756d 5477 6f4d  m, Next: sumTwoM
│ │ │ │ -0006db50: 6f6e 6f6d 6961 6c73 2c20 5072 6576 3a20  onomials, Prev: 
│ │ │ │ -0006db60: 7370 6c69 7474 696e 6773 2c20 5570 3a20  splittings, Up: 
│ │ │ │ -0006db70: 546f 700a 0a73 7461 626c 6548 6f6d 202d  Top..stableHom -
│ │ │ │ -0006db80: 2d20 6d61 7020 6672 6f6d 2048 6f6d 284d  - map from Hom(M
│ │ │ │ -0006db90: 2c4e 2920 746f 2074 6865 2073 7461 626c  ,N) to the stabl
│ │ │ │ -0006dba0: 6520 486f 6d20 6d6f 6475 6c65 0a2a 2a2a  e Hom module.***
│ │ │ │ +0006da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ +0006da50: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ +0006da60: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ +0006da70: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ +0006da80: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ +0006da90: 756c 6179 322d 312e 3236 2e30 362b 6473  ulay2-1.26.06+ds
│ │ │ │ +0006daa0: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ +0006dab0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465  ckages/.Complete
│ │ │ │ +0006dac0: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ +0006dad0: 6c75 7469 6f6e 732e 6d32 3a33 3932 353a  lutions.m2:3925:
│ │ │ │ +0006dae0: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c  0....File: Compl
│ │ │ │ +0006daf0: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ +0006db00: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ +0006db10: 204e 6f64 653a 2073 7461 626c 6548 6f6d   Node: stableHom
│ │ │ │ +0006db20: 2c20 4e65 7874 3a20 7375 6d54 776f 4d6f  , Next: sumTwoMo
│ │ │ │ +0006db30: 6e6f 6d69 616c 732c 2050 7265 763a 2073  nomials, Prev: s
│ │ │ │ +0006db40: 706c 6974 7469 6e67 732c 2055 703a 2054  plittings, Up: T
│ │ │ │ +0006db50: 6f70 0a0a 7374 6162 6c65 486f 6d20 2d2d  op..stableHom --
│ │ │ │ +0006db60: 206d 6170 2066 726f 6d20 486f 6d28 4d2c   map from Hom(M,
│ │ │ │ +0006db70: 4e29 2074 6f20 7468 6520 7374 6162 6c65  N) to the stable
│ │ │ │ +0006db80: 2048 6f6d 206d 6f64 756c 650a 2a2a 2a2a   Hom module.****
│ │ │ │ +0006db90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0006dba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006dbb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006dbc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006dbd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006dbe0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -0006dbf0: 200a 2020 2020 2020 2020 7020 3d20 7374   .        p = st
│ │ │ │ -0006dc00: 6162 6c65 486f 6d28 4d2c 4e29 0a20 202a  ableHom(M,N).  *
│ │ │ │ -0006dc10: 2049 6e70 7574 733a 0a20 2020 2020 202a   Inputs:.      *
│ │ │ │ -0006dc20: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475   M, a *note modu
│ │ │ │ -0006dc30: 6c65 3a20 284d 6163 6175 6c61 7932 446f  le: (Macaulay2Do
│ │ │ │ -0006dc40: 6329 4d6f 6475 6c65 2c2c 200a 2020 2020  c)Module,, .    
│ │ │ │ -0006dc50: 2020 2a20 4e2c 2061 202a 6e6f 7465 206d    * N, a *note m
│ │ │ │ -0006dc60: 6f64 756c 653a 2028 4d61 6361 756c 6179  odule: (Macaulay
│ │ │ │ -0006dc70: 3244 6f63 294d 6f64 756c 652c 2c20 0a20  2Doc)Module,, . 
│ │ │ │ -0006dc80: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ -0006dc90: 2020 2a20 702c 2061 202a 6e6f 7465 206d    * p, a *note m
│ │ │ │ -0006dca0: 6174 7269 783a 2028 4d61 6361 756c 6179  atrix: (Macaulay
│ │ │ │ -0006dcb0: 3244 6f63 294d 6174 7269 782c 2c20 7072  2Doc)Matrix,, pr
│ │ │ │ -0006dcc0: 6f6a 6563 7469 6f6e 2066 726f 6d20 486f  ojection from Ho
│ │ │ │ -0006dcd0: 6d28 4d2c 4e29 2074 6f0a 2020 2020 2020  m(M,N) to.      
│ │ │ │ -0006dce0: 2020 7468 6520 7374 6162 6c65 2048 6f6d    the stable Hom
│ │ │ │ -0006dcf0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ -0006dd00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2073  =========..The s
│ │ │ │ -0006dd10: 7461 626c 6520 486f 6d20 6973 2048 6f6d  table Hom is Hom
│ │ │ │ -0006dd20: 284d 2c4e 292f 5420 7768 6572 6520 5420  (M,N)/T where T 
│ │ │ │ -0006dd30: 6973 2074 6865 2073 7562 6d6f 6475 6c65  is the submodule
│ │ │ │ -0006dd40: 206f 6620 686f 6d6f 6d6f 7270 6869 736d   of homomorphism
│ │ │ │ -0006dd50: 7320 7468 6174 0a66 6163 746f 7220 7468  s that.factor th
│ │ │ │ -0006dd60: 726f 7567 6820 6120 6672 6565 2063 6f76  rough a free cov
│ │ │ │ -0006dd70: 6572 206f 6620 4e20 286f 722c 2065 7175  er of N (or, equ
│ │ │ │ -0006dd80: 6976 616c 656e 746c 792c 2074 6872 6f75  ivalently, throu
│ │ │ │ -0006dd90: 6768 2061 6e79 2070 726f 6a65 6374 6976  gh any projectiv
│ │ │ │ -0006dda0: 6529 0a0a 5365 6520 616c 736f 0a3d 3d3d  e)..See also.===
│ │ │ │ -0006ddb0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -0006ddc0: 2069 7353 7461 626c 7954 7269 7669 616c   isStablyTrivial
│ │ │ │ -0006ddd0: 3a20 6973 5374 6162 6c79 5472 6976 6961  : isStablyTrivia
│ │ │ │ -0006dde0: 6c2c 202d 2d20 7265 7475 726e 7320 7472  l, -- returns tr
│ │ │ │ -0006ddf0: 7565 2069 6620 7468 6520 6d61 7020 676f  ue if the map go
│ │ │ │ -0006de00: 6573 2074 6f0a 2020 2020 3020 756e 6465  es to.    0 unde
│ │ │ │ -0006de10: 7220 7374 6162 6c65 486f 6d0a 0a57 6179  r stableHom..Way
│ │ │ │ -0006de20: 7320 746f 2075 7365 2073 7461 626c 6548  s to use stableH
│ │ │ │ -0006de30: 6f6d 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  om:.============
│ │ │ │ -0006de40: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0006de50: 2273 7461 626c 6548 6f6d 284d 6f64 756c  "stableHom(Modul
│ │ │ │ -0006de60: 652c 4d6f 6475 6c65 2922 0a0a 466f 7220  e,Module)"..For 
│ │ │ │ -0006de70: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ -0006de80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006de90: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ -0006dea0: 6f74 6520 7374 6162 6c65 486f 6d3a 2073  ote stableHom: s
│ │ │ │ -0006deb0: 7461 626c 6548 6f6d 2c20 6973 2061 202a  tableHom, is a *
│ │ │ │ -0006dec0: 6e6f 7465 206d 6574 686f 6420 6675 6e63  note method func
│ │ │ │ -0006ded0: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932  tion:.(Macaulay2
│ │ │ │ -0006dee0: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -0006def0: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d  on,...----------
│ │ │ │ +0006dbc0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20  ***..  * Usage: 
│ │ │ │ +0006dbd0: 0a20 2020 2020 2020 2070 203d 2073 7461  .        p = sta
│ │ │ │ +0006dbe0: 626c 6548 6f6d 284d 2c4e 290a 2020 2a20  bleHom(M,N).  * 
│ │ │ │ +0006dbf0: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +0006dc00: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c  M, a *note modul
│ │ │ │ +0006dc10: 653a 2028 4d61 6361 756c 6179 3244 6f63  e: (Macaulay2Doc
│ │ │ │ +0006dc20: 294d 6f64 756c 652c 2c20 0a20 2020 2020  )Module,, .     
│ │ │ │ +0006dc30: 202a 204e 2c20 6120 2a6e 6f74 6520 6d6f   * N, a *note mo
│ │ │ │ +0006dc40: 6475 6c65 3a20 284d 6163 6175 6c61 7932  dule: (Macaulay2
│ │ │ │ +0006dc50: 446f 6329 4d6f 6475 6c65 2c2c 200a 2020  Doc)Module,, .  
│ │ │ │ +0006dc60: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +0006dc70: 202a 2070 2c20 6120 2a6e 6f74 6520 6d61   * p, a *note ma
│ │ │ │ +0006dc80: 7472 6978 3a20 284d 6163 6175 6c61 7932  trix: (Macaulay2
│ │ │ │ +0006dc90: 446f 6329 4d61 7472 6978 2c2c 2070 726f  Doc)Matrix,, pro
│ │ │ │ +0006dca0: 6a65 6374 696f 6e20 6672 6f6d 2048 6f6d  jection from Hom
│ │ │ │ +0006dcb0: 284d 2c4e 2920 746f 0a20 2020 2020 2020  (M,N) to.       
│ │ │ │ +0006dcc0: 2074 6865 2073 7461 626c 6520 486f 6d0a   the stable Hom.
│ │ │ │ +0006dcd0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +0006dce0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 7374  ========..The st
│ │ │ │ +0006dcf0: 6162 6c65 2048 6f6d 2069 7320 486f 6d28  able Hom is Hom(
│ │ │ │ +0006dd00: 4d2c 4e29 2f54 2077 6865 7265 2054 2069  M,N)/T where T i
│ │ │ │ +0006dd10: 7320 7468 6520 7375 626d 6f64 756c 6520  s the submodule 
│ │ │ │ +0006dd20: 6f66 2068 6f6d 6f6d 6f72 7068 6973 6d73  of homomorphisms
│ │ │ │ +0006dd30: 2074 6861 740a 6661 6374 6f72 2074 6872   that.factor thr
│ │ │ │ +0006dd40: 6f75 6768 2061 2066 7265 6520 636f 7665  ough a free cove
│ │ │ │ +0006dd50: 7220 6f66 204e 2028 6f72 2c20 6571 7569  r of N (or, equi
│ │ │ │ +0006dd60: 7661 6c65 6e74 6c79 2c20 7468 726f 7567  valently, throug
│ │ │ │ +0006dd70: 6820 616e 7920 7072 6f6a 6563 7469 7665  h any projective
│ │ │ │ +0006dd80: 290a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  )..See also.====
│ │ │ │ +0006dd90: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ +0006dda0: 6973 5374 6162 6c79 5472 6976 6961 6c3a  isStablyTrivial:
│ │ │ │ +0006ddb0: 2069 7353 7461 626c 7954 7269 7669 616c   isStablyTrivial
│ │ │ │ +0006ddc0: 2c20 2d2d 2072 6574 7572 6e73 2074 7275  , -- returns tru
│ │ │ │ +0006ddd0: 6520 6966 2074 6865 206d 6170 2067 6f65  e if the map goe
│ │ │ │ +0006dde0: 7320 746f 0a20 2020 2030 2075 6e64 6572  s to.    0 under
│ │ │ │ +0006ddf0: 2073 7461 626c 6548 6f6d 0a0a 5761 7973   stableHom..Ways
│ │ │ │ +0006de00: 2074 6f20 7573 6520 7374 6162 6c65 486f   to use stableHo
│ │ │ │ +0006de10: 6d3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  m:.=============
│ │ │ │ +0006de20: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +0006de30: 7374 6162 6c65 486f 6d28 4d6f 6475 6c65  stableHom(Module
│ │ │ │ +0006de40: 2c4d 6f64 756c 6529 220a 0a46 6f72 2074  ,Module)"..For t
│ │ │ │ +0006de50: 6865 2070 726f 6772 616d 6d65 720a 3d3d  he programmer.==
│ │ │ │ +0006de60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006de70: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f  ..The object *no
│ │ │ │ +0006de80: 7465 2073 7461 626c 6548 6f6d 3a20 7374  te stableHom: st
│ │ │ │ +0006de90: 6162 6c65 486f 6d2c 2069 7320 6120 2a6e  ableHom, is a *n
│ │ │ │ +0006dea0: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ +0006deb0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244  ion:.(Macaulay2D
│ │ │ │ +0006dec0: 6f63 294d 6574 686f 6446 756e 6374 696f  oc)MethodFunctio
│ │ │ │ +0006ded0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d  n,...-----------
│ │ │ │ +0006dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006df00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006df10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006df20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006df30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006df40: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -0006df50: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -0006df60: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -0006df70: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -0006df80: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ -0006df90: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ -0006dfa0: 6179 322f 7061 636b 6167 6573 2f0a 436f  ay2/packages/.Co
│ │ │ │ -0006dfb0: 6d70 6c65 7465 496e 7465 7273 6563 7469  mpleteIntersecti
│ │ │ │ -0006dfc0: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32  onResolutions.m2
│ │ │ │ -0006dfd0: 3a34 3634 393a 302e 0a1f 0a46 696c 653a  :4649:0....File:
│ │ │ │ -0006dfe0: 2043 6f6d 706c 6574 6549 6e74 6572 7365   CompleteInterse
│ │ │ │ -0006dff0: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ -0006e000: 2e69 6e66 6f2c 204e 6f64 653a 2073 756d  .info, Node: sum
│ │ │ │ -0006e010: 5477 6f4d 6f6e 6f6d 6961 6c73 2c20 4e65  TwoMonomials, Ne
│ │ │ │ -0006e020: 7874 3a20 5461 7465 5265 736f 6c75 7469  xt: TateResoluti
│ │ │ │ -0006e030: 6f6e 2c20 5072 6576 3a20 7374 6162 6c65  on, Prev: stable
│ │ │ │ -0006e040: 486f 6d2c 2055 703a 2054 6f70 0a0a 7375  Hom, Up: Top..su
│ │ │ │ -0006e050: 6d54 776f 4d6f 6e6f 6d69 616c 7320 2d2d  mTwoMonomials --
│ │ │ │ -0006e060: 2074 616c 6c79 2074 6865 2073 6571 7565   tally the seque
│ │ │ │ -0006e070: 6e63 6573 206f 6620 4252 616e 6b73 2066  nces of BRanks f
│ │ │ │ -0006e080: 6f72 2063 6572 7461 696e 2065 7861 6d70  or certain examp
│ │ │ │ -0006e090: 6c65 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  les.************
│ │ │ │ +0006df20: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365  ----..The source
│ │ │ │ +0006df30: 206f 6620 7468 6973 2064 6f63 756d 656e   of this documen
│ │ │ │ +0006df40: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72  t is in./build/r
│ │ │ │ +0006df50: 6570 726f 6475 6369 626c 652d 7061 7468  eproducible-path
│ │ │ │ +0006df60: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e  /macaulay2-1.26.
│ │ │ │ +0006df70: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61  06+ds/M2/Macaula
│ │ │ │ +0006df80: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d  y2/packages/.Com
│ │ │ │ +0006df90: 706c 6574 6549 6e74 6572 7365 6374 696f  pleteIntersectio
│ │ │ │ +0006dfa0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a  nResolutions.m2:
│ │ │ │ +0006dfb0: 3436 3439 3a30 2e0a 1f0a 4669 6c65 3a20  4649:0....File: 
│ │ │ │ +0006dfc0: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +0006dfd0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +0006dfe0: 696e 666f 2c20 4e6f 6465 3a20 7375 6d54  info, Node: sumT
│ │ │ │ +0006dff0: 776f 4d6f 6e6f 6d69 616c 732c 204e 6578  woMonomials, Nex
│ │ │ │ +0006e000: 743a 2054 6174 6552 6573 6f6c 7574 696f  t: TateResolutio
│ │ │ │ +0006e010: 6e2c 2050 7265 763a 2073 7461 626c 6548  n, Prev: stableH
│ │ │ │ +0006e020: 6f6d 2c20 5570 3a20 546f 700a 0a73 756d  om, Up: Top..sum
│ │ │ │ +0006e030: 5477 6f4d 6f6e 6f6d 6961 6c73 202d 2d20  TwoMonomials -- 
│ │ │ │ +0006e040: 7461 6c6c 7920 7468 6520 7365 7175 656e  tally the sequen
│ │ │ │ +0006e050: 6365 7320 6f66 2042 5261 6e6b 7320 666f  ces of BRanks fo
│ │ │ │ +0006e060: 7220 6365 7274 6169 6e20 6578 616d 706c  r certain exampl
│ │ │ │ +0006e070: 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  es.*************
│ │ │ │ +0006e080: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0006e090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006e0a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006e0b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006e0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006e0d0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055  *********..  * U
│ │ │ │ -0006e0e0: 7361 6765 3a20 0a20 2020 2020 2020 2073  sage: .        s
│ │ │ │ -0006e0f0: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 2863  umTwoMonomials(c
│ │ │ │ -0006e100: 2c64 290a 2020 2a20 496e 7075 7473 3a0a  ,d).  * Inputs:.
│ │ │ │ -0006e110: 2020 2020 2020 2a20 632c 2061 6e20 2a6e        * c, an *n
│ │ │ │ -0006e120: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ -0006e130: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ -0006e140: 636f 6469 6d65 6e73 696f 6e20 696e 2077  codimension in w
│ │ │ │ -0006e150: 6869 6368 2074 6f20 776f 726b 0a20 2020  hich to work.   
│ │ │ │ -0006e160: 2020 202a 2064 2c20 616e 202a 6e6f 7465     * d, an *note
│ │ │ │ -0006e170: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ -0006e180: 6c61 7932 446f 6329 5a5a 2c2c 2064 6567  lay2Doc)ZZ,, deg
│ │ │ │ -0006e190: 7265 6520 6f66 2074 6865 206d 6f6e 6f6d  ree of the monom
│ │ │ │ -0006e1a0: 6961 6c73 2074 6f20 7461 6b65 0a20 202a  ials to take.  *
│ │ │ │ -0006e1b0: 204f 7574 7075 7473 3a0a 2020 2020 2020   Outputs:.      
│ │ │ │ -0006e1c0: 2a20 542c 2061 202a 6e6f 7465 2074 616c  * T, a *note tal
│ │ │ │ -0006e1d0: 6c79 3a20 284d 6163 6175 6c61 7932 446f  ly: (Macaulay2Do
│ │ │ │ -0006e1e0: 6329 5461 6c6c 792c 2c20 0a0a 4465 7363  c)Tally,, ..Desc
│ │ │ │ -0006e1f0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -0006e200: 3d3d 3d0a 0a74 616c 6c69 6573 2074 6865  ===..tallies the
│ │ │ │ -0006e210: 2073 6571 7565 6e63 6573 206f 6620 422d   sequences of B-
│ │ │ │ -0006e220: 7261 6e6b 7320 7468 6174 206f 6363 7572  ranks that occur
│ │ │ │ -0006e230: 2066 6f72 2073 756d 7320 6f66 2070 6169   for sums of pai
│ │ │ │ -0006e240: 7273 206f 6620 6d6f 6e6f 6d69 616c 7320  rs of monomials 
│ │ │ │ -0006e250: 696e 2052 0a3d 2053 2f28 642d 7468 2070  in R.= S/(d-th p
│ │ │ │ -0006e260: 6f77 6572 7320 6f66 2074 6865 2076 6172  owers of the var
│ │ │ │ -0006e270: 6961 626c 6573 292c 2077 6974 6820 6675  iables), with fu
│ │ │ │ -0006e280: 6c6c 2063 6f6d 706c 6578 6974 7920 283d  ll complexity (=
│ │ │ │ -0006e290: 6329 3b20 7468 6174 2069 732c 2066 6f72  c); that is, for
│ │ │ │ -0006e2a0: 2061 6e0a 6170 7072 6f70 7269 6174 6520   an.appropriate 
│ │ │ │ -0006e2b0: 7379 7a79 6779 204d 206f 6620 4d30 203d  syzygy M of M0 =
│ │ │ │ -0006e2c0: 2052 2f28 6d31 2b6d 3229 2077 6865 7265   R/(m1+m2) where
│ │ │ │ -0006e2d0: 206d 3120 616e 6420 6d32 2061 7265 206d   m1 and m2 are m
│ │ │ │ -0006e2e0: 6f6e 6f6d 6961 6c73 206f 6620 7468 650a  onomials of the.
│ │ │ │ -0006e2f0: 7361 6d65 2064 6567 7265 652e 0a0a 2b2d  same degree...+-
│ │ │ │ +0006e0b0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573  ********..  * Us
│ │ │ │ +0006e0c0: 6167 653a 200a 2020 2020 2020 2020 7375  age: .        su
│ │ │ │ +0006e0d0: 6d54 776f 4d6f 6e6f 6d69 616c 7328 632c  mTwoMonomials(c,
│ │ │ │ +0006e0e0: 6429 0a20 202a 2049 6e70 7574 733a 0a20  d).  * Inputs:. 
│ │ │ │ +0006e0f0: 2020 2020 202a 2063 2c20 616e 202a 6e6f       * c, an *no
│ │ │ │ +0006e100: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ +0006e110: 6175 6c61 7932 446f 6329 5a5a 2c2c 2063  aulay2Doc)ZZ,, c
│ │ │ │ +0006e120: 6f64 696d 656e 7369 6f6e 2069 6e20 7768  odimension in wh
│ │ │ │ +0006e130: 6963 6820 746f 2077 6f72 6b0a 2020 2020  ich to work.    
│ │ │ │ +0006e140: 2020 2a20 642c 2061 6e20 2a6e 6f74 6520    * d, an *note 
│ │ │ │ +0006e150: 696e 7465 6765 723a 2028 4d61 6361 756c  integer: (Macaul
│ │ │ │ +0006e160: 6179 3244 6f63 295a 5a2c 2c20 6465 6772  ay2Doc)ZZ,, degr
│ │ │ │ +0006e170: 6565 206f 6620 7468 6520 6d6f 6e6f 6d69  ee of the monomi
│ │ │ │ +0006e180: 616c 7320 746f 2074 616b 650a 2020 2a20  als to take.  * 
│ │ │ │ +0006e190: 4f75 7470 7574 733a 0a20 2020 2020 202a  Outputs:.      *
│ │ │ │ +0006e1a0: 2054 2c20 6120 2a6e 6f74 6520 7461 6c6c   T, a *note tall
│ │ │ │ +0006e1b0: 793a 2028 4d61 6361 756c 6179 3244 6f63  y: (Macaulay2Doc
│ │ │ │ +0006e1c0: 2954 616c 6c79 2c2c 200a 0a44 6573 6372  )Tally,, ..Descr
│ │ │ │ +0006e1d0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ +0006e1e0: 3d3d 0a0a 7461 6c6c 6965 7320 7468 6520  ==..tallies the 
│ │ │ │ +0006e1f0: 7365 7175 656e 6365 7320 6f66 2042 2d72  sequences of B-r
│ │ │ │ +0006e200: 616e 6b73 2074 6861 7420 6f63 6375 7220  anks that occur 
│ │ │ │ +0006e210: 666f 7220 7375 6d73 206f 6620 7061 6972  for sums of pair
│ │ │ │ +0006e220: 7320 6f66 206d 6f6e 6f6d 6961 6c73 2069  s of monomials i
│ │ │ │ +0006e230: 6e20 520a 3d20 532f 2864 2d74 6820 706f  n R.= S/(d-th po
│ │ │ │ +0006e240: 7765 7273 206f 6620 7468 6520 7661 7269  wers of the vari
│ │ │ │ +0006e250: 6162 6c65 7329 2c20 7769 7468 2066 756c  ables), with ful
│ │ │ │ +0006e260: 6c20 636f 6d70 6c65 7869 7479 2028 3d63  l complexity (=c
│ │ │ │ +0006e270: 293b 2074 6861 7420 6973 2c20 666f 7220  ); that is, for 
│ │ │ │ +0006e280: 616e 0a61 7070 726f 7072 6961 7465 2073  an.appropriate s
│ │ │ │ +0006e290: 797a 7967 7920 4d20 6f66 204d 3020 3d20  yzygy M of M0 = 
│ │ │ │ +0006e2a0: 522f 286d 312b 6d32 2920 7768 6572 6520  R/(m1+m2) where 
│ │ │ │ +0006e2b0: 6d31 2061 6e64 206d 3220 6172 6520 6d6f  m1 and m2 are mo
│ │ │ │ +0006e2c0: 6e6f 6d69 616c 7320 6f66 2074 6865 0a73  nomials of the.s
│ │ │ │ +0006e2d0: 616d 6520 6465 6772 6565 2e0a 0a2b 2d2d  ame degree...+--
│ │ │ │ +0006e2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006e2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e330: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365  ------+.|i1 : se
│ │ │ │ -0006e340: 7452 616e 646f 6d53 6565 6420 3020 2020  tRandomSeed 0   
│ │ │ │ -0006e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e370: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072  |.| -- setting r
│ │ │ │ -0006e380: 616e 646f 6d20 7365 6564 2074 6f20 3020  andom seed to 0 
│ │ │ │ +0006e310: 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 6574  -----+.|i1 : set
│ │ │ │ +0006e320: 5261 6e64 6f6d 5365 6564 2030 2020 2020  RandomSeed 0    
│ │ │ │ +0006e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006e350: 0a7c 202d 2d20 7365 7474 696e 6720 7261  .| -- setting ra
│ │ │ │ +0006e360: 6e64 6f6d 2073 6565 6420 746f 2030 2020  ndom seed to 0  
│ │ │ │ +0006e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e380: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  0006e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ +0006e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e3c0: 2020 207c 0a7c 6f31 203d 2030 2020 2020     |.|o1 = 0    
│ │ │ │  0006e3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e3e0: 2020 2020 7c0a 7c6f 3120 3d20 3020 2020      |.|o1 = 0   
│ │ │ │ -0006e3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e410: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e420: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -0006e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20  --------+.|i2 : 
│ │ │ │ -0006e460: 7375 6d54 776f 4d6f 6e6f 6d69 616c 7328  sumTwoMonomials(
│ │ │ │ -0006e470: 322c 3329 2020 2020 2020 2020 2020 2020  2,3)            
│ │ │ │ -0006e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e490: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e    |.| -- used 0.
│ │ │ │ -0006e4a0: 3534 3735 3131 7320 2863 7075 293b 2030  547511s (cpu); 0
│ │ │ │ -0006e4b0: 2e34 3130 3232 7320 2874 6872 6561 6429  .41022s (thread)
│ │ │ │ -0006e4c0: 3b20 3073 2028 6763 2920 2020 7c0a 7c32  ; 0s (gc)   |.|2
│ │ │ │ +0006e3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e3f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +0006e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006e430: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2073  -------+.|i2 : s
│ │ │ │ +0006e440: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 2832  umTwoMonomials(2
│ │ │ │ +0006e450: 2c33 2920 2020 2020 2020 2020 2020 2020  ,3)             
│ │ │ │ +0006e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e470: 207c 0a7c 202d 2d20 7573 6564 2030 2e39   |.| -- used 0.9
│ │ │ │ +0006e480: 3535 3933 3473 2028 6370 7529 3b20 302e  55934s (cpu); 0.
│ │ │ │ +0006e490: 3538 3839 3234 7320 2874 6872 6561 6429  588924s (thread)
│ │ │ │ +0006e4a0: 3b20 3073 2028 6763 2920 207c 0a7c 3220  ; 0s (gc)  |.|2 
│ │ │ │ +0006e4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e500: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b        |.|Tally{{
│ │ │ │ -0006e510: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ -0006e520: 3d3e 2033 7d20 2020 2020 2020 2020 2020  => 3}           
│ │ │ │ +0006e4e0: 2020 2020 207c 0a7c 5461 6c6c 797b 7b7b       |.|Tally{{{
│ │ │ │ +0006e4f0: 322c 2032 7d2c 207b 312c 2032 7d7d 203d  2, 2}, {1, 2}} =
│ │ │ │ +0006e500: 3e20 337d 2020 2020 2020 2020 2020 2020  > 3}            
│ │ │ │ +0006e510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006e520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0006e530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e540: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -0006e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e570: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d            |.| --
│ │ │ │ -0006e580: 2075 7365 6420 302e 3130 3638 3434 7320   used 0.106844s 
│ │ │ │ -0006e590: 2863 7075 293b 2030 2e31 3036 3431 7320  (cpu); 0.10641s 
│ │ │ │ -0006e5a0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -0006e5b0: 2920 2020 7c0a 7c33 2020 2020 2020 2020  )   |.|3        
│ │ │ │ -0006e5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e5e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -0006e5f0: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20  |Tally{{{2, 2}, 
│ │ │ │ -0006e600: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020  {1, 2}} => 1}   
│ │ │ │ +0006e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e550: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +0006e560: 7573 6564 2030 2e32 3136 3737 3373 2028  used 0.216773s (
│ │ │ │ +0006e570: 6370 7529 3b20 302e 3136 3434 3636 7320  cpu); 0.164466s 
│ │ │ │ +0006e580: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +0006e590: 2920 207c 0a7c 3320 2020 2020 2020 2020  )  |.|3         
│ │ │ │ +0006e5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e5c0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +0006e5d0: 5461 6c6c 797b 7b7b 322c 2032 7d2c 207b  Tally{{{2, 2}, {
│ │ │ │ +0006e5e0: 312c 2032 7d7d 203d 3e20 317d 2020 2020  1, 2}} => 1}    
│ │ │ │ +0006e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e600: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  0006e610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e620: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +0006e620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e660: 2020 7c0a 7c20 2d2d 2075 7365 6420 342e    |.| -- used 4.
│ │ │ │ -0006e670: 3035 3865 2d30 3673 2028 6370 7529 3b20  058e-06s (cpu); 
│ │ │ │ -0006e680: 332e 3438 3665 2d30 3673 2028 7468 7265  3.486e-06s (thre
│ │ │ │ -0006e690: 6164 293b 2030 7320 2867 6329 7c0a 7c34  ad); 0s (gc)|.|4
│ │ │ │ +0006e640: 207c 0a7c 202d 2d20 7573 6564 2034 2e35   |.| -- used 4.5
│ │ │ │ +0006e650: 3234 652d 3036 7320 2863 7075 293b 2033  24e-06s (cpu); 3
│ │ │ │ +0006e660: 2e30 3434 652d 3036 7320 2874 6872 6561  .044e-06s (threa
│ │ │ │ +0006e670: 6429 3b20 3073 2028 6763 297c 0a7c 3420  d); 0s (gc)|.|4 
│ │ │ │ +0006e680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006e6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e6b0: 2020 2020 207c 0a7c 5461 6c6c 797b 7d20       |.|Tally{} 
│ │ │ │  0006e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e6d0: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7d        |.|Tally{}
│ │ │ │ -0006e6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006e710: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ -0006e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565  ----------+..See
│ │ │ │ -0006e750: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ -0006e760: 2020 2a20 2a6e 6f74 6520 7477 6f4d 6f6e    * *note twoMon
│ │ │ │ -0006e770: 6f6d 6961 6c73 3a20 7477 6f4d 6f6e 6f6d  omials: twoMonom
│ │ │ │ -0006e780: 6961 6c73 2c20 2d2d 2074 616c 6c79 2074  ials, -- tally t
│ │ │ │ -0006e790: 6865 2073 6571 7565 6e63 6573 206f 6620  he sequences of 
│ │ │ │ -0006e7a0: 4252 616e 6b73 2066 6f72 0a20 2020 2063  BRanks for.    c
│ │ │ │ -0006e7b0: 6572 7461 696e 2065 7861 6d70 6c65 730a  ertain examples.
│ │ │ │ -0006e7c0: 0a57 6179 7320 746f 2075 7365 2073 756d  .Ways to use sum
│ │ │ │ -0006e7d0: 5477 6f4d 6f6e 6f6d 6961 6c73 3a0a 3d3d  TwoMonomials:.==
│ │ │ │ -0006e7e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006e7f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -0006e800: 2273 756d 5477 6f4d 6f6e 6f6d 6961 6c73  "sumTwoMonomials
│ │ │ │ -0006e810: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468  (ZZ,ZZ)"..For th
│ │ │ │ -0006e820: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d  e programmer.===
│ │ │ │ -0006e830: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ -0006e840: 0a54 6865 206f 626a 6563 7420 2a6e 6f74  .The object *not
│ │ │ │ -0006e850: 6520 7375 6d54 776f 4d6f 6e6f 6d69 616c  e sumTwoMonomial
│ │ │ │ -0006e860: 733a 2073 756d 5477 6f4d 6f6e 6f6d 6961  s: sumTwoMonomia
│ │ │ │ -0006e870: 6c73 2c20 6973 2061 202a 6e6f 7465 206d  ls, is a *note m
│ │ │ │ -0006e880: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a  ethod function:.
│ │ │ │ -0006e890: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -0006e8a0: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a  thodFunction,...
│ │ │ │ +0006e6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006e6e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +0006e6f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0006e700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006e710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006e720: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520  ---------+..See 
│ │ │ │ +0006e730: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20  also.========.. 
│ │ │ │ +0006e740: 202a 202a 6e6f 7465 2074 776f 4d6f 6e6f   * *note twoMono
│ │ │ │ +0006e750: 6d69 616c 733a 2074 776f 4d6f 6e6f 6d69  mials: twoMonomi
│ │ │ │ +0006e760: 616c 732c 202d 2d20 7461 6c6c 7920 7468  als, -- tally th
│ │ │ │ +0006e770: 6520 7365 7175 656e 6365 7320 6f66 2042  e sequences of B
│ │ │ │ +0006e780: 5261 6e6b 7320 666f 720a 2020 2020 6365  Ranks for.    ce
│ │ │ │ +0006e790: 7274 6169 6e20 6578 616d 706c 6573 0a0a  rtain examples..
│ │ │ │ +0006e7a0: 5761 7973 2074 6f20 7573 6520 7375 6d54  Ways to use sumT
│ │ │ │ +0006e7b0: 776f 4d6f 6e6f 6d69 616c 733a 0a3d 3d3d  woMonomials:.===
│ │ │ │ +0006e7c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006e7d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +0006e7e0: 7375 6d54 776f 4d6f 6e6f 6d69 616c 7328  sumTwoMonomials(
│ │ │ │ +0006e7f0: 5a5a 2c5a 5a29 220a 0a46 6f72 2074 6865  ZZ,ZZ)"..For the
│ │ │ │ +0006e800: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +0006e810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +0006e820: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +0006e830: 2073 756d 5477 6f4d 6f6e 6f6d 6961 6c73   sumTwoMonomials
│ │ │ │ +0006e840: 3a20 7375 6d54 776f 4d6f 6e6f 6d69 616c  : sumTwoMonomial
│ │ │ │ +0006e850: 732c 2069 7320 6120 2a6e 6f74 6520 6d65  s, is a *note me
│ │ │ │ +0006e860: 7468 6f64 2066 756e 6374 696f 6e3a 0a28  thod function:.(
│ │ │ │ +0006e870: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +0006e880: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │ +0006e890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006e8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006e8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -0006e900: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -0006e910: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ -0006e920: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ -0006e930: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -0006e940: 756c 6179 322d 312e 3236 2e30 362b 6473  ulay2-1.26.06+ds
│ │ │ │ -0006e950: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -0006e960: 636b 6167 6573 2f0a 436f 6d70 6c65 7465  ckages/.Complete
│ │ │ │ -0006e970: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ -0006e980: 6c75 7469 6f6e 732e 6d32 3a34 3531 323a  lutions.m2:4512:
│ │ │ │ -0006e990: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c  0....File: Compl
│ │ │ │ -0006e9a0: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -0006e9b0: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ -0006e9c0: 204e 6f64 653a 2054 6174 6552 6573 6f6c   Node: TateResol
│ │ │ │ -0006e9d0: 7574 696f 6e2c 204e 6578 743a 2074 656e  ution, Next: ten
│ │ │ │ -0006e9e0: 736f 7257 6974 6843 6f6d 706f 6e65 6e74  sorWithComponent
│ │ │ │ -0006e9f0: 732c 2050 7265 763a 2073 756d 5477 6f4d  s, Prev: sumTwoM
│ │ │ │ -0006ea00: 6f6e 6f6d 6961 6c73 2c20 5570 3a20 546f  onomials, Up: To
│ │ │ │ -0006ea10: 700a 0a54 6174 6552 6573 6f6c 7574 696f  p..TateResolutio
│ │ │ │ -0006ea20: 6e20 2d2d 2054 6174 6552 6573 6f6c 7574  n -- TateResolut
│ │ │ │ -0006ea30: 696f 6e20 6f66 2061 206d 6f64 756c 6520  ion of a module 
│ │ │ │ -0006ea40: 6f76 6572 2061 6e20 6578 7465 7269 6f72  over an exterior
│ │ │ │ -0006ea50: 2061 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a   algebra.*******
│ │ │ │ +0006e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0006e8e0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +0006e8f0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +0006e900: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0006e910: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0006e920: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ +0006e930: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +0006e940: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ +0006e950: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +0006e960: 7574 696f 6e73 2e6d 323a 3435 3132 3a30  utions.m2:4512:0
│ │ │ │ +0006e970: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ +0006e980: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +0006e990: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ +0006e9a0: 4e6f 6465 3a20 5461 7465 5265 736f 6c75  Node: TateResolu
│ │ │ │ +0006e9b0: 7469 6f6e 2c20 4e65 7874 3a20 7465 6e73  tion, Next: tens
│ │ │ │ +0006e9c0: 6f72 5769 7468 436f 6d70 6f6e 656e 7473  orWithComponents
│ │ │ │ +0006e9d0: 2c20 5072 6576 3a20 7375 6d54 776f 4d6f  , Prev: sumTwoMo
│ │ │ │ +0006e9e0: 6e6f 6d69 616c 732c 2055 703a 2054 6f70  nomials, Up: Top
│ │ │ │ +0006e9f0: 0a0a 5461 7465 5265 736f 6c75 7469 6f6e  ..TateResolution
│ │ │ │ +0006ea00: 202d 2d20 5461 7465 5265 736f 6c75 7469   -- TateResoluti
│ │ │ │ +0006ea10: 6f6e 206f 6620 6120 6d6f 6475 6c65 206f  on of a module o
│ │ │ │ +0006ea20: 7665 7220 616e 2065 7874 6572 696f 7220  ver an exterior 
│ │ │ │ +0006ea30: 616c 6765 6272 610a 2a2a 2a2a 2a2a 2a2a  algebra.********
│ │ │ │ +0006ea40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0006ea50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006ea60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006ea70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006ea80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006ea90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -0006eaa0: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ -0006eab0: 2020 2020 4620 3d20 5461 7465 5265 736f      F = TateReso
│ │ │ │ -0006eac0: 6c75 7469 6f6e 284d 2c6c 6f77 6572 2c75  lution(M,lower,u
│ │ │ │ -0006ead0: 7070 6572 290a 2020 2a20 496e 7075 7473  pper).  * Inputs
│ │ │ │ -0006eae0: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a  :.      * M, a *
│ │ │ │ -0006eaf0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61  note module: (Ma
│ │ │ │ -0006eb00: 6361 756c 6179 3244 6f63 294d 6f64 756c  caulay2Doc)Modul
│ │ │ │ -0006eb10: 652c 2c20 0a20 2020 2020 202a 206c 6f77  e,, .      * low
│ │ │ │ -0006eb20: 6572 2c20 616e 202a 6e6f 7465 2069 6e74  er, an *note int
│ │ │ │ -0006eb30: 6567 6572 3a20 284d 6163 6175 6c61 7932  eger: (Macaulay2
│ │ │ │ -0006eb40: 446f 6329 5a5a 2c2c 200a 2020 2020 2020  Doc)ZZ,, .      
│ │ │ │ -0006eb50: 2a20 7570 7065 722c 2061 6e20 2a6e 6f74  * upper, an *not
│ │ │ │ -0006eb60: 6520 696e 7465 6765 723a 2028 4d61 6361  e integer: (Maca
│ │ │ │ -0006eb70: 756c 6179 3244 6f63 295a 5a2c 2c20 6c6f  ulay2Doc)ZZ,, lo
│ │ │ │ -0006eb80: 7765 7220 616e 6420 7570 7065 7220 626f  wer and upper bo
│ │ │ │ -0006eb90: 756e 6473 2066 6f72 0a20 2020 2020 2020  unds for.       
│ │ │ │ -0006eba0: 2074 6865 2072 6573 6f6c 7574 696f 6e0a   the resolution.
│ │ │ │ -0006ebb0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -0006ebc0: 2020 202a 2046 2c20 6120 2a6e 6f74 6520     * F, a *note 
│ │ │ │ -0006ebd0: 636f 6d70 6c65 783a 2028 436f 6d70 6c65  complex: (Comple
│ │ │ │ -0006ebe0: 7865 7329 436f 6d70 6c65 782c 2c20 0a0a  xes)Complex,, ..
│ │ │ │ -0006ebf0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -0006ec00: 3d3d 3d3d 3d3d 3d0a 0a46 6f72 6d73 2061  =======..Forms a
│ │ │ │ -0006ec10: 6e20 696e 7465 7276 616c 2c20 6c6f 7765  n interval, lowe
│ │ │ │ -0006ec20: 722e 2e75 7070 6572 2c20 6f66 2061 2064  r..upper, of a d
│ │ │ │ -0006ec30: 6f75 626c 7920 696e 6669 6e69 7465 2066  oubly infinite f
│ │ │ │ -0006ec40: 7265 6520 7265 736f 6c75 7469 6f6e 206f  ree resolution o
│ │ │ │ -0006ec50: 6620 6120 610a 436f 6865 6e2d 4d61 6361  f a a.Cohen-Maca
│ │ │ │ -0006ec60: 756c 6179 206d 6f64 756c 6520 6f76 6572  ulay module over
│ │ │ │ -0006ec70: 2061 2047 6f72 656e 7374 6569 6e20 7269   a Gorenstein ri
│ │ │ │ -0006ec80: 6e67 2c20 7375 6368 2061 7320 616e 7920  ng, such as any 
│ │ │ │ -0006ec90: 6d6f 6475 6c65 206f 7665 7220 616e 0a65  module over an.e
│ │ │ │ -0006eca0: 7874 6572 696f 7220 616c 6765 6272 6120  xterior algebra 
│ │ │ │ -0006ecb0: 2861 6374 7561 6c6c 792c 2061 6e79 206d  (actually, any m
│ │ │ │ -0006ecc0: 6f64 756c 6520 6f76 6572 2061 6e79 2072  odule over any r
│ │ │ │ -0006ecd0: 696e 672e 290a 0a2b 2d2d 2d2d 2d2d 2d2d  ing.)..+--------
│ │ │ │ +0006ea70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20  *************.. 
│ │ │ │ +0006ea80: 202a 2055 7361 6765 3a20 0a20 2020 2020   * Usage: .     
│ │ │ │ +0006ea90: 2020 2046 203d 2054 6174 6552 6573 6f6c     F = TateResol
│ │ │ │ +0006eaa0: 7574 696f 6e28 4d2c 6c6f 7765 722c 7570  ution(M,lower,up
│ │ │ │ +0006eab0: 7065 7229 0a20 202a 2049 6e70 7574 733a  per).  * Inputs:
│ │ │ │ +0006eac0: 0a20 2020 2020 202a 204d 2c20 6120 2a6e  .      * M, a *n
│ │ │ │ +0006ead0: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163  ote module: (Mac
│ │ │ │ +0006eae0: 6175 6c61 7932 446f 6329 4d6f 6475 6c65  aulay2Doc)Module
│ │ │ │ +0006eaf0: 2c2c 200a 2020 2020 2020 2a20 6c6f 7765  ,, .      * lowe
│ │ │ │ +0006eb00: 722c 2061 6e20 2a6e 6f74 6520 696e 7465  r, an *note inte
│ │ │ │ +0006eb10: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ +0006eb20: 6f63 295a 5a2c 2c20 0a20 2020 2020 202a  oc)ZZ,, .      *
│ │ │ │ +0006eb30: 2075 7070 6572 2c20 616e 202a 6e6f 7465   upper, an *note
│ │ │ │ +0006eb40: 2069 6e74 6567 6572 3a20 284d 6163 6175   integer: (Macau
│ │ │ │ +0006eb50: 6c61 7932 446f 6329 5a5a 2c2c 206c 6f77  lay2Doc)ZZ,, low
│ │ │ │ +0006eb60: 6572 2061 6e64 2075 7070 6572 2062 6f75  er and upper bou
│ │ │ │ +0006eb70: 6e64 7320 666f 720a 2020 2020 2020 2020  nds for.        
│ │ │ │ +0006eb80: 7468 6520 7265 736f 6c75 7469 6f6e 0a20  the resolution. 
│ │ │ │ +0006eb90: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +0006eba0: 2020 2a20 462c 2061 202a 6e6f 7465 2063    * F, a *note c
│ │ │ │ +0006ebb0: 6f6d 706c 6578 3a20 2843 6f6d 706c 6578  omplex: (Complex
│ │ │ │ +0006ebc0: 6573 2943 6f6d 706c 6578 2c2c 200a 0a44  es)Complex,, ..D
│ │ │ │ +0006ebd0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +0006ebe0: 3d3d 3d3d 3d3d 0a0a 466f 726d 7320 616e  ======..Forms an
│ │ │ │ +0006ebf0: 2069 6e74 6572 7661 6c2c 206c 6f77 6572   interval, lower
│ │ │ │ +0006ec00: 2e2e 7570 7065 722c 206f 6620 6120 646f  ..upper, of a do
│ │ │ │ +0006ec10: 7562 6c79 2069 6e66 696e 6974 6520 6672  ubly infinite fr
│ │ │ │ +0006ec20: 6565 2072 6573 6f6c 7574 696f 6e20 6f66  ee resolution of
│ │ │ │ +0006ec30: 2061 2061 0a43 6f68 656e 2d4d 6163 6175   a a.Cohen-Macau
│ │ │ │ +0006ec40: 6c61 7920 6d6f 6475 6c65 206f 7665 7220  lay module over 
│ │ │ │ +0006ec50: 6120 476f 7265 6e73 7465 696e 2072 696e  a Gorenstein rin
│ │ │ │ +0006ec60: 672c 2073 7563 6820 6173 2061 6e79 206d  g, such as any m
│ │ │ │ +0006ec70: 6f64 756c 6520 6f76 6572 2061 6e0a 6578  odule over an.ex
│ │ │ │ +0006ec80: 7465 7269 6f72 2061 6c67 6562 7261 2028  terior algebra (
│ │ │ │ +0006ec90: 6163 7475 616c 6c79 2c20 616e 7920 6d6f  actually, any mo
│ │ │ │ +0006eca0: 6475 6c65 206f 7665 7220 616e 7920 7269  dule over any ri
│ │ │ │ +0006ecb0: 6e67 2e29 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d  ng.)..+---------
│ │ │ │ +0006ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ed20: 2d2d 2d2d 2d2b 0a7c 6931 203a 2045 203d  -----+.|i1 : E =
│ │ │ │ -0006ed30: 205a 5a2f 3130 315b 612c 622c 632c 2053   ZZ/101[a,b,c, S
│ │ │ │ -0006ed40: 6b65 7743 6f6d 6d75 7461 7469 7665 3d3e  kewCommutative=>
│ │ │ │ -0006ed50: 7472 7565 5d20 2020 2020 2020 2020 2020  true]           
│ │ │ │ +0006ed00: 2d2d 2d2d 2b0a 7c69 3120 3a20 4520 3d20  ----+.|i1 : E = 
│ │ │ │ +0006ed10: 5a5a 2f31 3031 5b61 2c62 2c63 2c20 536b  ZZ/101[a,b,c, Sk
│ │ │ │ +0006ed20: 6577 436f 6d6d 7574 6174 6976 653d 3e74  ewCommutative=>t
│ │ │ │ +0006ed30: 7275 655d 2020 2020 2020 2020 2020 2020  rue]            
│ │ │ │ +0006ed40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ed50: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006ed60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ed70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ed70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ed80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ed90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006eda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006eda0: 2020 2020 7c0a 7c6f 3120 3d20 4520 2020      |.|o1 = E   
│ │ │ │  0006edb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006edc0: 2020 2020 207c 0a7c 6f31 203d 2045 2020       |.|o1 = E  
│ │ │ │ +0006edc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006edf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ee10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ee50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ee60: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c       |.|o1 : Pol
│ │ │ │ -0006ee70: 796e 6f6d 6961 6c52 696e 672c 2033 2073  ynomialRing, 3 s
│ │ │ │ -0006ee80: 6b65 7720 636f 6d6d 7574 6174 6976 6520  kew commutative 
│ │ │ │ -0006ee90: 7661 7269 6162 6c65 2873 2920 2020 2020  variable(s)     
│ │ │ │ -0006eea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006eeb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006ee40: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79      |.|o1 : Poly
│ │ │ │ +0006ee50: 6e6f 6d69 616c 5269 6e67 2c20 3320 736b  nomialRing, 3 sk
│ │ │ │ +0006ee60: 6577 2063 6f6d 6d75 7461 7469 7665 2076  ew commutative v
│ │ │ │ +0006ee70: 6172 6961 626c 6528 7329 2020 2020 2020  ariable(s)      
│ │ │ │ +0006ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ee90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ef00: 2d2d 2d2d 2d2b 0a7c 6932 203a 204d 203d  -----+.|i2 : M =
│ │ │ │ -0006ef10: 2063 6f6b 6572 206d 6170 2845 5e32 2c20   coker map(E^2, 
│ │ │ │ -0006ef20: 455e 7b2d 317d 2c20 6d61 7472 6978 2261  E^{-1}, matrix"a
│ │ │ │ -0006ef30: 623b 6263 2229 2020 2020 2020 2020 2020  b;bc")          
│ │ │ │ +0006eee0: 2d2d 2d2d 2b0a 7c69 3220 3a20 4d20 3d20  ----+.|i2 : M = 
│ │ │ │ +0006eef0: 636f 6b65 7220 6d61 7028 455e 322c 2045  coker map(E^2, E
│ │ │ │ +0006ef00: 5e7b 2d31 7d2c 206d 6174 7269 7822 6162  ^{-1}, matrix"ab
│ │ │ │ +0006ef10: 3b62 6322 2920 2020 2020 2020 2020 2020  ;bc")           
│ │ │ │ +0006ef20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006ef30: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006ef40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ef50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ef80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006ef90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006efa0: 2020 2020 207c 0a7c 6f32 203d 2063 6f6b       |.|o2 = cok
│ │ │ │ -0006efb0: 6572 6e65 6c20 7c20 6162 207c 2020 2020  ernel | ab |    
│ │ │ │ +0006ef80: 2020 2020 7c0a 7c6f 3220 3d20 636f 6b65      |.|o2 = coke
│ │ │ │ +0006ef90: 726e 656c 207c 2061 6220 7c20 2020 2020  rnel | ab |     
│ │ │ │ +0006efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006efe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006eff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006f000: 2020 2020 2020 7c20 6263 207c 2020 2020        | bc |    
│ │ │ │ +0006efd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006efe0: 2020 2020 207c 2062 6320 7c20 2020 2020       | bc |     
│ │ │ │ +0006eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f020: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006f030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f070: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f090: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0006f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f0b0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0006f0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f0e0: 2020 2020 207c 0a7c 6f32 203a 2045 2d6d       |.|o2 : E-m
│ │ │ │ -0006f0f0: 6f64 756c 652c 2071 756f 7469 656e 7420  odule, quotient 
│ │ │ │ -0006f100: 6f66 2045 2020 2020 2020 2020 2020 2020  of E            
│ │ │ │ -0006f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f130: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f0c0: 2020 2020 7c0a 7c6f 3220 3a20 452d 6d6f      |.|o2 : E-mo
│ │ │ │ +0006f0d0: 6475 6c65 2c20 7175 6f74 6965 6e74 206f  dule, quotient o
│ │ │ │ +0006f0e0: 6620 4520 2020 2020 2020 2020 2020 2020  f E             
│ │ │ │ +0006f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f110: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f180: 2d2d 2d2d 2d2b 0a7c 6933 203a 2070 7265  -----+.|i3 : pre
│ │ │ │ -0006f190: 7365 6e74 6174 696f 6e20 4d20 2020 2020  sentation M     
│ │ │ │ +0006f160: 2d2d 2d2d 2b0a 7c69 3320 3a20 7072 6573  ----+.|i3 : pres
│ │ │ │ +0006f170: 656e 7461 7469 6f6e 204d 2020 2020 2020  entation M      
│ │ │ │ +0006f180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f1b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f1d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f220: 2020 2020 207c 0a7c 6f33 203d 207c 2061       |.|o3 = | a
│ │ │ │ -0006f230: 6220 7c20 2020 2020 2020 2020 2020 2020  b |             
│ │ │ │ +0006f200: 2020 2020 7c0a 7c6f 3320 3d20 7c20 6162      |.|o3 = | ab
│ │ │ │ +0006f210: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0006f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f270: 2020 2020 207c 0a7c 2020 2020 207c 2062       |.|     | b
│ │ │ │ -0006f280: 6320 7c20 2020 2020 2020 2020 2020 2020  c |             
│ │ │ │ +0006f250: 2020 2020 7c0a 7c20 2020 2020 7c20 6263      |.|     | bc
│ │ │ │ +0006f260: 207c 2020 2020 2020 2020 2020 2020 2020   |              
│ │ │ │ +0006f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f2a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006f2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f2c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f310: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -0006f320: 2020 2020 2032 2020 2020 2020 3120 2020       2      1   
│ │ │ │ +0006f2f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006f300: 2020 2020 3220 2020 2020 2031 2020 2020      2      1    
│ │ │ │ +0006f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f360: 2020 2020 207c 0a7c 6f33 203a 204d 6174       |.|o3 : Mat
│ │ │ │ -0006f370: 7269 7820 4520 203c 2d2d 2045 2020 2020  rix E  <-- E    
│ │ │ │ +0006f340: 2020 2020 7c0a 7c6f 3320 3a20 4d61 7472      |.|o3 : Matr
│ │ │ │ +0006f350: 6978 2045 2020 3c2d 2d20 4520 2020 2020  ix E  <-- E     
│ │ │ │ +0006f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f3b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006f390: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f400: 2d2d 2d2d 2d2b 0a7c 6934 203a 2054 6174  -----+.|i4 : Tat
│ │ │ │ -0006f410: 6552 6573 6f6c 7574 696f 6e28 4d2c 2d32  eResolution(M,-2
│ │ │ │ -0006f420: 2c37 2920 2020 2020 2020 2020 2020 2020  ,7)             
│ │ │ │ -0006f430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f3e0: 2d2d 2d2d 2b0a 7c69 3420 3a20 5461 7465  ----+.|i4 : Tate
│ │ │ │ +0006f3f0: 5265 736f 6c75 7469 6f6e 284d 2c2d 322c  Resolution(M,-2,
│ │ │ │ +0006f400: 3729 2020 2020 2020 2020 2020 2020 2020  7)              
│ │ │ │ +0006f410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f430: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006f440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f4a0: 2020 2020 207c 0a7c 2020 2020 2020 3920       |.|      9 
│ │ │ │ -0006f4b0: 2020 2020 2035 2020 2020 2020 3220 2020       5      2   
│ │ │ │ -0006f4c0: 2020 2031 2020 2020 2020 3220 2020 2020     1      2     
│ │ │ │ -0006f4d0: 2034 2020 2020 2020 3720 2020 2020 2031   4      7      1
│ │ │ │ -0006f4e0: 3120 2020 2020 2031 3620 2020 2020 2032  1      16      2
│ │ │ │ -0006f4f0: 3220 2020 207c 0a7c 6f34 203d 2045 2020  2    |.|o4 = E  
│ │ │ │ -0006f500: 3c2d 2d20 4520 203c 2d2d 2045 2020 3c2d  <-- E  <-- E  <-
│ │ │ │ -0006f510: 2d20 4520 203c 2d2d 2045 2020 3c2d 2d20  - E  <-- E  <-- 
│ │ │ │ -0006f520: 4520 203c 2d2d 2045 2020 3c2d 2d20 4520  E  <-- E  <-- E 
│ │ │ │ -0006f530: 2020 3c2d 2d20 4520 2020 3c2d 2d20 4520    <-- E   <-- E 
│ │ │ │ -0006f540: 2020 3c2d 2d7c 0a7c 2020 2020 2020 2020    <--|.|        
│ │ │ │ +0006f480: 2020 2020 7c0a 7c20 2020 2020 2039 2020      |.|      9  
│ │ │ │ +0006f490: 2020 2020 3520 2020 2020 2032 2020 2020      5      2    
│ │ │ │ +0006f4a0: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ +0006f4b0: 3420 2020 2020 2037 2020 2020 2020 3131  4      7      11
│ │ │ │ +0006f4c0: 2020 2020 2020 3136 2020 2020 2020 3232        16      22
│ │ │ │ +0006f4d0: 2020 2020 7c0a 7c6f 3420 3d20 4520 203c      |.|o4 = E  <
│ │ │ │ +0006f4e0: 2d2d 2045 2020 3c2d 2d20 4520 203c 2d2d  -- E  <-- E  <--
│ │ │ │ +0006f4f0: 2045 2020 3c2d 2d20 4520 203c 2d2d 2045   E  <-- E  <-- E
│ │ │ │ +0006f500: 2020 3c2d 2d20 4520 203c 2d2d 2045 2020    <-- E  <-- E  
│ │ │ │ +0006f510: 203c 2d2d 2045 2020 203c 2d2d 2045 2020   <-- E   <-- E  
│ │ │ │ +0006f520: 203c 2d2d 7c0a 7c20 2020 2020 2020 2020   <--|.|         
│ │ │ │ +0006f530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f590: 2020 2020 207c 0a7c 2020 2020 202d 3220       |.|     -2 
│ │ │ │ -0006f5a0: 2020 2020 2d31 2020 2020 2030 2020 2020      -1     0    
│ │ │ │ -0006f5b0: 2020 3120 2020 2020 2032 2020 2020 2020    1      2      
│ │ │ │ -0006f5c0: 3320 2020 2020 2034 2020 2020 2020 3520  3      4      5 
│ │ │ │ -0006f5d0: 2020 2020 2020 3620 2020 2020 2020 3720        6       7 
│ │ │ │ -0006f5e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f570: 2020 2020 7c0a 7c20 2020 2020 2d32 2020      |.|     -2  
│ │ │ │ +0006f580: 2020 202d 3120 2020 2020 3020 2020 2020     -1     0     
│ │ │ │ +0006f590: 2031 2020 2020 2020 3220 2020 2020 2033   1      2      3
│ │ │ │ +0006f5a0: 2020 2020 2020 3420 2020 2020 2035 2020        4      5  
│ │ │ │ +0006f5b0: 2020 2020 2036 2020 2020 2020 2037 2020       6       7  
│ │ │ │ +0006f5c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +0006f5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f630: 2020 2020 207c 0a7c 6f34 203a 2043 6f6d       |.|o4 : Com
│ │ │ │ -0006f640: 706c 6578 2020 2020 2020 2020 2020 2020  plex            
│ │ │ │ +0006f610: 2020 2020 7c0a 7c6f 3420 3a20 436f 6d70      |.|o4 : Comp
│ │ │ │ +0006f620: 6c65 7820 2020 2020 2020 2020 2020 2020  lex             
│ │ │ │ +0006f630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f680: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │ +0006f660: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d      |.|---------
│ │ │ │ +0006f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f6d0: 2d2d 2d2d 2d7c 0a7c 3020 2020 2020 2020  -----|.|0       
│ │ │ │ +0006f6b0: 2d2d 2d2d 7c0a 7c30 2020 2020 2020 2020  ----|.|0        
│ │ │ │ +0006f6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f700: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  0006f710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f720: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0006f720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0006f750: 2020 2020 7c0a 7c38 2020 2020 2020 2020      |.|8        
│ │ │ │  0006f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f770: 2020 2020 207c 0a7c 3820 2020 2020 2020       |.|8       
│ │ │ │ +0006f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0006f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0006f7c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +0006f7a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +0006f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f810: 2d2d 2d2d 2d2b 0a0a 4361 7665 6174 0a3d  -----+..Caveat.=
│ │ │ │ -0006f820: 3d3d 3d3d 3d0a 0a49 6e20 6120 7072 6576  =====..In a prev
│ │ │ │ -0006f830: 696f 7573 2076 6572 7369 6f6e 206f 6620  ious version of 
│ │ │ │ -0006f840: 7468 6973 2073 6372 6970 742c 2074 6869  this script, thi
│ │ │ │ -0006f850: 7320 636f 6d6d 616e 6420 7265 7475 726e  s command return
│ │ │ │ -0006f860: 6564 2061 2062 6574 7469 2074 6162 6c65  ed a betti table
│ │ │ │ -0006f870: 3b20 6e6f 770a 7573 6520 2262 6574 7469  ; now.use "betti
│ │ │ │ -0006f880: 2054 6174 6552 6573 6f6c 7574 696f 6e22   TateResolution"
│ │ │ │ -0006f890: 2069 6e73 7465 6164 2e0a 0a57 6179 7320   instead...Ways 
│ │ │ │ -0006f8a0: 746f 2075 7365 2054 6174 6552 6573 6f6c  to use TateResol
│ │ │ │ -0006f8b0: 7574 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d  ution:.=========
│ │ │ │ -0006f8c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006f8d0: 3d3d 0a0a 2020 2a20 2254 6174 6552 6573  ==..  * "TateRes
│ │ │ │ -0006f8e0: 6f6c 7574 696f 6e28 4d6f 6475 6c65 2922  olution(Module)"
│ │ │ │ -0006f8f0: 0a20 202a 2022 5461 7465 5265 736f 6c75  .  * "TateResolu
│ │ │ │ -0006f900: 7469 6f6e 284d 6f64 756c 652c 5a5a 2922  tion(Module,ZZ)"
│ │ │ │ -0006f910: 0a20 202a 2022 5461 7465 5265 736f 6c75  .  * "TateResolu
│ │ │ │ -0006f920: 7469 6f6e 284d 6f64 756c 652c 5a5a 2c5a  tion(Module,ZZ,Z
│ │ │ │ -0006f930: 5a29 220a 0a46 6f72 2074 6865 2070 726f  Z)"..For the pro
│ │ │ │ -0006f940: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -0006f950: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -0006f960: 6f62 6a65 6374 202a 6e6f 7465 2054 6174  object *note Tat
│ │ │ │ -0006f970: 6552 6573 6f6c 7574 696f 6e3a 2054 6174  eResolution: Tat
│ │ │ │ -0006f980: 6552 6573 6f6c 7574 696f 6e2c 2069 7320  eResolution, is 
│ │ │ │ -0006f990: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ -0006f9a0: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ -0006f9b0: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ -0006f9c0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +0006f7f0: 2d2d 2d2d 2b0a 0a43 6176 6561 740a 3d3d  ----+..Caveat.==
│ │ │ │ +0006f800: 3d3d 3d3d 0a0a 496e 2061 2070 7265 7669  ====..In a previ
│ │ │ │ +0006f810: 6f75 7320 7665 7273 696f 6e20 6f66 2074  ous version of t
│ │ │ │ +0006f820: 6869 7320 7363 7269 7074 2c20 7468 6973  his script, this
│ │ │ │ +0006f830: 2063 6f6d 6d61 6e64 2072 6574 7572 6e65   command returne
│ │ │ │ +0006f840: 6420 6120 6265 7474 6920 7461 626c 653b  d a betti table;
│ │ │ │ +0006f850: 206e 6f77 0a75 7365 2022 6265 7474 6920   now.use "betti 
│ │ │ │ +0006f860: 5461 7465 5265 736f 6c75 7469 6f6e 2220  TateResolution" 
│ │ │ │ +0006f870: 696e 7374 6561 642e 0a0a 5761 7973 2074  instead...Ways t
│ │ │ │ +0006f880: 6f20 7573 6520 5461 7465 5265 736f 6c75  o use TateResolu
│ │ │ │ +0006f890: 7469 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  tion:.==========
│ │ │ │ +0006f8a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006f8b0: 3d0a 0a20 202a 2022 5461 7465 5265 736f  =..  * "TateReso
│ │ │ │ +0006f8c0: 6c75 7469 6f6e 284d 6f64 756c 6529 220a  lution(Module)".
│ │ │ │ +0006f8d0: 2020 2a20 2254 6174 6552 6573 6f6c 7574    * "TateResolut
│ │ │ │ +0006f8e0: 696f 6e28 4d6f 6475 6c65 2c5a 5a29 220a  ion(Module,ZZ)".
│ │ │ │ +0006f8f0: 2020 2a20 2254 6174 6552 6573 6f6c 7574    * "TateResolut
│ │ │ │ +0006f900: 696f 6e28 4d6f 6475 6c65 2c5a 5a2c 5a5a  ion(Module,ZZ,ZZ
│ │ │ │ +0006f910: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +0006f920: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +0006f930: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +0006f940: 626a 6563 7420 2a6e 6f74 6520 5461 7465  bject *note Tate
│ │ │ │ +0006f950: 5265 736f 6c75 7469 6f6e 3a20 5461 7465  Resolution: Tate
│ │ │ │ +0006f960: 5265 736f 6c75 7469 6f6e 2c20 6973 2061  Resolution, is a
│ │ │ │ +0006f970: 202a 6e6f 7465 206d 6574 686f 6420 6675   *note method fu
│ │ │ │ +0006f980: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61  nction:.(Macaula
│ │ │ │ +0006f990: 7932 446f 6329 4d65 7468 6f64 4675 6e63  y2Doc)MethodFunc
│ │ │ │ +0006f9a0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  tion,...--------
│ │ │ │ +0006f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006fa10: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -0006fa20: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -0006fa30: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -0006fa40: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -0006fa50: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -0006fa60: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ -0006fa70: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -0006fa80: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ -0006fa90: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ -0006faa0: 2e6d 323a 3334 3939 3a30 2e0a 1f0a 4669  .m2:3499:0....Fi
│ │ │ │ -0006fab0: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ -0006fac0: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ -0006fad0: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ -0006fae0: 7465 6e73 6f72 5769 7468 436f 6d70 6f6e  tensorWithCompon
│ │ │ │ -0006faf0: 656e 7473 2c20 4e65 7874 3a20 746f 4172  ents, Next: toAr
│ │ │ │ -0006fb00: 7261 792c 2050 7265 763a 2054 6174 6552  ray, Prev: TateR
│ │ │ │ -0006fb10: 6573 6f6c 7574 696f 6e2c 2055 703a 2054  esolution, Up: T
│ │ │ │ -0006fb20: 6f70 0a0a 7465 6e73 6f72 5769 7468 436f  op..tensorWithCo
│ │ │ │ -0006fb30: 6d70 6f6e 656e 7473 202d 2d20 666f 726d  mponents -- form
│ │ │ │ -0006fb40: 7320 7468 6520 7465 6e73 6f72 2070 726f  s the tensor pro
│ │ │ │ -0006fb50: 6475 6374 2c20 7072 6573 6572 7669 6e67  duct, preserving
│ │ │ │ -0006fb60: 2064 6972 6563 7420 7375 6d20 696e 666f   direct sum info
│ │ │ │ -0006fb70: 726d 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a  rmation.********
│ │ │ │ +0006f9f0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ +0006fa00: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ +0006fa10: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ +0006fa20: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ +0006fa30: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ +0006fa40: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ +0006fa50: 756c 6179 322f 7061 636b 6167 6573 2f0a  ulay2/packages/.
│ │ │ │ +0006fa60: 436f 6d70 6c65 7465 496e 7465 7273 6563  CompleteIntersec
│ │ │ │ +0006fa70: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e  tionResolutions.
│ │ │ │ +0006fa80: 6d32 3a33 3439 393a 302e 0a1f 0a46 696c  m2:3499:0....Fil
│ │ │ │ +0006fa90: 653a 2043 6f6d 706c 6574 6549 6e74 6572  e: CompleteInter
│ │ │ │ +0006faa0: 7365 6374 696f 6e52 6573 6f6c 7574 696f  sectionResolutio
│ │ │ │ +0006fab0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2074  ns.info, Node: t
│ │ │ │ +0006fac0: 656e 736f 7257 6974 6843 6f6d 706f 6e65  ensorWithCompone
│ │ │ │ +0006fad0: 6e74 732c 204e 6578 743a 2074 6f41 7272  nts, Next: toArr
│ │ │ │ +0006fae0: 6179 2c20 5072 6576 3a20 5461 7465 5265  ay, Prev: TateRe
│ │ │ │ +0006faf0: 736f 6c75 7469 6f6e 2c20 5570 3a20 546f  solution, Up: To
│ │ │ │ +0006fb00: 700a 0a74 656e 736f 7257 6974 6843 6f6d  p..tensorWithCom
│ │ │ │ +0006fb10: 706f 6e65 6e74 7320 2d2d 2066 6f72 6d73  ponents -- forms
│ │ │ │ +0006fb20: 2074 6865 2074 656e 736f 7220 7072 6f64   the tensor prod
│ │ │ │ +0006fb30: 7563 742c 2070 7265 7365 7276 696e 6720  uct, preserving 
│ │ │ │ +0006fb40: 6469 7265 6374 2073 756d 2069 6e66 6f72  direct sum infor
│ │ │ │ +0006fb50: 6d61 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a  mation.*********
│ │ │ │ +0006fb60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +0006fb70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006fb80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  0006fb90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006fba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006fbb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -0006fbc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ -0006fbd0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ -0006fbe0: 2054 203d 2074 656e 736f 7228 4d2c 4e29   T = tensor(M,N)
│ │ │ │ -0006fbf0: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -0006fc00: 2020 202a 204d 2c20 6120 2a6e 6f74 6520     * M, a *note 
│ │ │ │ -0006fc10: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61  module: (Macaula
│ │ │ │ -0006fc20: 7932 446f 6329 4d6f 6475 6c65 2c2c 200a  y2Doc)Module,, .
│ │ │ │ -0006fc30: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f        * N, a *no
│ │ │ │ -0006fc40: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ -0006fc50: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ -0006fc60: 2c20 0a20 202a 204f 7574 7075 7473 3a0a  , .  * Outputs:.
│ │ │ │ -0006fc70: 2020 2020 2020 2a20 542c 2061 202a 6e6f        * T, a *no
│ │ │ │ -0006fc80: 7465 206d 6f64 756c 653a 2028 4d61 6361  te module: (Maca
│ │ │ │ -0006fc90: 756c 6179 3244 6f63 294d 6f64 756c 652c  ulay2Doc)Module,
│ │ │ │ -0006fca0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a  , ..Description.
│ │ │ │ -0006fcb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620  ===========..If 
│ │ │ │ -0006fcc0: 4d20 616e 642f 6f72 204e 2061 7265 2064  M and/or N are d
│ │ │ │ -0006fcd0: 6972 6563 7420 7375 6d20 6d6f 6475 6c65  irect sum module
│ │ │ │ -0006fce0: 7320 2869 7344 6972 6563 7453 756d 204d  s (isDirectSum M
│ │ │ │ -0006fcf0: 203d 3d20 7472 7565 2920 7468 656e 2054   == true) then T
│ │ │ │ -0006fd00: 2069 7320 7468 650a 6469 7265 6374 2073   is the.direct s
│ │ │ │ -0006fd10: 756d 206f 6620 7468 6520 7465 6e73 6f72  um of the tensor
│ │ │ │ -0006fd20: 2070 726f 6475 6374 7320 6265 7477 6565   products betwee
│ │ │ │ -0006fd30: 6e20 7468 6520 636f 6d70 6f6e 656e 7473  n the components
│ │ │ │ -0006fd40: 2e20 5468 6973 2053 484f 554c 4420 6265  . This SHOULD be
│ │ │ │ -0006fd50: 2062 7569 6c74 0a69 6e74 6f20 4d2a 2a4e   built.into M**N
│ │ │ │ -0006fd60: 2c20 6275 7420 6973 6e27 7420 6173 206f  , but isn't as o
│ │ │ │ -0006fd70: 6620 4d32 2c20 762e 2031 2e37 0a0a 5365  f M2, v. 1.7..Se
│ │ │ │ -0006fd80: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ -0006fd90: 0a20 202a 202a 6e6f 7465 2048 6f6d 5769  .  * *note HomWi
│ │ │ │ -0006fda0: 7468 436f 6d70 6f6e 656e 7473 3a20 486f  thComponents: Ho
│ │ │ │ -0006fdb0: 6d57 6974 6843 6f6d 706f 6e65 6e74 732c  mWithComponents,
│ │ │ │ -0006fdc0: 202d 2d20 636f 6d70 7574 6573 2048 6f6d   -- computes Hom
│ │ │ │ -0006fdd0: 2c20 7072 6573 6572 7669 6e67 0a20 2020  , preserving.   
│ │ │ │ -0006fde0: 2064 6972 6563 7420 7375 6d20 696e 666f   direct sum info
│ │ │ │ -0006fdf0: 726d 6174 696f 6e0a 2020 2a20 2a6e 6f74  rmation.  * *not
│ │ │ │ -0006fe00: 6520 6475 616c 5769 7468 436f 6d70 6f6e  e dualWithCompon
│ │ │ │ -0006fe10: 656e 7473 3a20 6475 616c 5769 7468 436f  ents: dualWithCo
│ │ │ │ -0006fe20: 6d70 6f6e 656e 7473 2c20 2d2d 2064 7561  mponents, -- dua
│ │ │ │ -0006fe30: 6c20 6d6f 6475 6c65 2070 7265 7365 7276  l module preserv
│ │ │ │ -0006fe40: 696e 670a 2020 2020 6469 7265 6374 2073  ing.    direct s
│ │ │ │ -0006fe50: 756d 2069 6e66 6f72 6d61 7469 6f6e 0a0a  um information..
│ │ │ │ -0006fe60: 5761 7973 2074 6f20 7573 6520 7465 6e73  Ways to use tens
│ │ │ │ -0006fe70: 6f72 5769 7468 436f 6d70 6f6e 656e 7473  orWithComponents
│ │ │ │ -0006fe80: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ -0006fe90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -0006fea0: 3d3d 3d0a 0a20 202a 2022 7465 6e73 6f72  ===..  * "tensor
│ │ │ │ -0006feb0: 5769 7468 436f 6d70 6f6e 656e 7473 284d  WithComponents(M
│ │ │ │ -0006fec0: 6f64 756c 652c 4d6f 6475 6c65 2922 0a0a  odule,Module)"..
│ │ │ │ -0006fed0: 466f 7220 7468 6520 7072 6f67 7261 6d6d  For the programm
│ │ │ │ -0006fee0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  er.=============
│ │ │ │ -0006fef0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563  =====..The objec
│ │ │ │ -0006ff00: 7420 2a6e 6f74 6520 7465 6e73 6f72 5769  t *note tensorWi
│ │ │ │ -0006ff10: 7468 436f 6d70 6f6e 656e 7473 3a20 7465  thComponents: te
│ │ │ │ -0006ff20: 6e73 6f72 5769 7468 436f 6d70 6f6e 656e  nsorWithComponen
│ │ │ │ -0006ff30: 7473 2c20 6973 2061 202a 6e6f 7465 206d  ts, is a *note m
│ │ │ │ -0006ff40: 6574 686f 640a 6675 6e63 7469 6f6e 3a20  ethod.function: 
│ │ │ │ -0006ff50: 284d 6163 6175 6c61 7932 446f 6329 4d65  (Macaulay2Doc)Me
│ │ │ │ -0006ff60: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a  thodFunction,...
│ │ │ │ +0006fba0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +0006fbb0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +0006fbc0: 5420 3d20 7465 6e73 6f72 284d 2c4e 290a  T = tensor(M,N).
│ │ │ │ +0006fbd0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +0006fbe0: 2020 2a20 4d2c 2061 202a 6e6f 7465 206d    * M, a *note m
│ │ │ │ +0006fbf0: 6f64 756c 653a 2028 4d61 6361 756c 6179  odule: (Macaulay
│ │ │ │ +0006fc00: 3244 6f63 294d 6f64 756c 652c 2c20 0a20  2Doc)Module,, . 
│ │ │ │ +0006fc10: 2020 2020 202a 204e 2c20 6120 2a6e 6f74       * N, a *not
│ │ │ │ +0006fc20: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +0006fc30: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +0006fc40: 200a 2020 2a20 4f75 7470 7574 733a 0a20   .  * Outputs:. 
│ │ │ │ +0006fc50: 2020 2020 202a 2054 2c20 6120 2a6e 6f74       * T, a *not
│ │ │ │ +0006fc60: 6520 6d6f 6475 6c65 3a20 284d 6163 6175  e module: (Macau
│ │ │ │ +0006fc70: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c  lay2Doc)Module,,
│ │ │ │ +0006fc80: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d   ..Description.=
│ │ │ │ +0006fc90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 204d  ==========..If M
│ │ │ │ +0006fca0: 2061 6e64 2f6f 7220 4e20 6172 6520 6469   and/or N are di
│ │ │ │ +0006fcb0: 7265 6374 2073 756d 206d 6f64 756c 6573  rect sum modules
│ │ │ │ +0006fcc0: 2028 6973 4469 7265 6374 5375 6d20 4d20   (isDirectSum M 
│ │ │ │ +0006fcd0: 3d3d 2074 7275 6529 2074 6865 6e20 5420  == true) then T 
│ │ │ │ +0006fce0: 6973 2074 6865 0a64 6972 6563 7420 7375  is the.direct su
│ │ │ │ +0006fcf0: 6d20 6f66 2074 6865 2074 656e 736f 7220  m of the tensor 
│ │ │ │ +0006fd00: 7072 6f64 7563 7473 2062 6574 7765 656e  products between
│ │ │ │ +0006fd10: 2074 6865 2063 6f6d 706f 6e65 6e74 732e   the components.
│ │ │ │ +0006fd20: 2054 6869 7320 5348 4f55 4c44 2062 6520   This SHOULD be 
│ │ │ │ +0006fd30: 6275 696c 740a 696e 746f 204d 2a2a 4e2c  built.into M**N,
│ │ │ │ +0006fd40: 2062 7574 2069 736e 2774 2061 7320 6f66   but isn't as of
│ │ │ │ +0006fd50: 204d 322c 2076 2e20 312e 370a 0a53 6565   M2, v. 1.7..See
│ │ │ │ +0006fd60: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a   also.========..
│ │ │ │ +0006fd70: 2020 2a20 2a6e 6f74 6520 486f 6d57 6974    * *note HomWit
│ │ │ │ +0006fd80: 6843 6f6d 706f 6e65 6e74 733a 2048 6f6d  hComponents: Hom
│ │ │ │ +0006fd90: 5769 7468 436f 6d70 6f6e 656e 7473 2c20  WithComponents, 
│ │ │ │ +0006fda0: 2d2d 2063 6f6d 7075 7465 7320 486f 6d2c  -- computes Hom,
│ │ │ │ +0006fdb0: 2070 7265 7365 7276 696e 670a 2020 2020   preserving.    
│ │ │ │ +0006fdc0: 6469 7265 6374 2073 756d 2069 6e66 6f72  direct sum infor
│ │ │ │ +0006fdd0: 6d61 7469 6f6e 0a20 202a 202a 6e6f 7465  mation.  * *note
│ │ │ │ +0006fde0: 2064 7561 6c57 6974 6843 6f6d 706f 6e65   dualWithCompone
│ │ │ │ +0006fdf0: 6e74 733a 2064 7561 6c57 6974 6843 6f6d  nts: dualWithCom
│ │ │ │ +0006fe00: 706f 6e65 6e74 732c 202d 2d20 6475 616c  ponents, -- dual
│ │ │ │ +0006fe10: 206d 6f64 756c 6520 7072 6573 6572 7669   module preservi
│ │ │ │ +0006fe20: 6e67 0a20 2020 2064 6972 6563 7420 7375  ng.    direct su
│ │ │ │ +0006fe30: 6d20 696e 666f 726d 6174 696f 6e0a 0a57  m information..W
│ │ │ │ +0006fe40: 6179 7320 746f 2075 7365 2074 656e 736f  ays to use tenso
│ │ │ │ +0006fe50: 7257 6974 6843 6f6d 706f 6e65 6e74 733a  rWithComponents:
│ │ │ │ +0006fe60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ +0006fe70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +0006fe80: 3d3d 0a0a 2020 2a20 2274 656e 736f 7257  ==..  * "tensorW
│ │ │ │ +0006fe90: 6974 6843 6f6d 706f 6e65 6e74 7328 4d6f  ithComponents(Mo
│ │ │ │ +0006fea0: 6475 6c65 2c4d 6f64 756c 6529 220a 0a46  dule,Module)"..F
│ │ │ │ +0006feb0: 6f72 2074 6865 2070 726f 6772 616d 6d65  or the programme
│ │ │ │ +0006fec0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  r.==============
│ │ │ │ +0006fed0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374  ====..The object
│ │ │ │ +0006fee0: 202a 6e6f 7465 2074 656e 736f 7257 6974   *note tensorWit
│ │ │ │ +0006fef0: 6843 6f6d 706f 6e65 6e74 733a 2074 656e  hComponents: ten
│ │ │ │ +0006ff00: 736f 7257 6974 6843 6f6d 706f 6e65 6e74  sorWithComponent
│ │ │ │ +0006ff10: 732c 2069 7320 6120 2a6e 6f74 6520 6d65  s, is a *note me
│ │ │ │ +0006ff20: 7468 6f64 0a66 756e 6374 696f 6e3a 2028  thod.function: (
│ │ │ │ +0006ff30: 4d61 6361 756c 6179 3244 6f63 294d 6574  Macaulay2Doc)Met
│ │ │ │ +0006ff40: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d  hodFunction,...-
│ │ │ │ +0006ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0006ff60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ff70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0006ff80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ff90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ffa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0006ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a  ---------------.
│ │ │ │ -0006ffc0: 0a54 6865 2073 6f75 7263 6520 6f66 2074  .The source of t
│ │ │ │ -0006ffd0: 6869 7320 646f 6375 6d65 6e74 2069 7320  his document is 
│ │ │ │ -0006ffe0: 696e 0a2f 6275 696c 642f 7265 7072 6f64  in./build/reprod
│ │ │ │ -0006fff0: 7563 6962 6c65 2d70 6174 682f 6d61 6361  ucible-path/maca
│ │ │ │ -00070000: 756c 6179 322d 312e 3236 2e30 362b 6473  ulay2-1.26.06+ds
│ │ │ │ -00070010: 2f4d 322f 4d61 6361 756c 6179 322f 7061  /M2/Macaulay2/pa
│ │ │ │ -00070020: 636b 6167 6573 2f0a 436f 6d70 6c65 7465  ckages/.Complete
│ │ │ │ -00070030: 496e 7465 7273 6563 7469 6f6e 5265 736f  IntersectionReso
│ │ │ │ -00070040: 6c75 7469 6f6e 732e 6d32 3a32 3636 373a  lutions.m2:2667:
│ │ │ │ -00070050: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c  0....File: Compl
│ │ │ │ -00070060: 6574 6549 6e74 6572 7365 6374 696f 6e52  eteIntersectionR
│ │ │ │ -00070070: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c  esolutions.info,
│ │ │ │ -00070080: 204e 6f64 653a 2074 6f41 7272 6179 2c20   Node: toArray, 
│ │ │ │ -00070090: 4e65 7874 3a20 7477 6f4d 6f6e 6f6d 6961  Next: twoMonomia
│ │ │ │ -000700a0: 6c73 2c20 5072 6576 3a20 7465 6e73 6f72  ls, Prev: tensor
│ │ │ │ -000700b0: 5769 7468 436f 6d70 6f6e 656e 7473 2c20  WithComponents, 
│ │ │ │ -000700c0: 5570 3a20 546f 700a 0a74 6f41 7272 6179  Up: Top..toArray
│ │ │ │ -000700d0: 202d 2d20 6d61 6b65 7320 616e 2061 7272   -- makes an arr
│ │ │ │ -000700e0: 6179 2066 726f 6d20 6120 4c69 7374 206f  ay from a List o
│ │ │ │ -000700f0: 7220 6672 6f6d 2061 2073 696e 676c 6520  r from a single 
│ │ │ │ -00070100: 696e 7465 6765 720a 2a2a 2a2a 2a2a 2a2a  integer.********
│ │ │ │ +0006ff90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +0006ffa0: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +0006ffb0: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +0006ffc0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +0006ffd0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +0006ffe0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ +0006fff0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +00070000: 6b61 6765 732f 0a43 6f6d 706c 6574 6549  kages/.CompleteI
│ │ │ │ +00070010: 6e74 6572 7365 6374 696f 6e52 6573 6f6c  ntersectionResol
│ │ │ │ +00070020: 7574 696f 6e73 2e6d 323a 3236 3637 3a30  utions.m2:2667:0
│ │ │ │ +00070030: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65  ....File: Comple
│ │ │ │ +00070040: 7465 496e 7465 7273 6563 7469 6f6e 5265  teIntersectionRe
│ │ │ │ +00070050: 736f 6c75 7469 6f6e 732e 696e 666f 2c20  solutions.info, 
│ │ │ │ +00070060: 4e6f 6465 3a20 746f 4172 7261 792c 204e  Node: toArray, N
│ │ │ │ +00070070: 6578 743a 2074 776f 4d6f 6e6f 6d69 616c  ext: twoMonomial
│ │ │ │ +00070080: 732c 2050 7265 763a 2074 656e 736f 7257  s, Prev: tensorW
│ │ │ │ +00070090: 6974 6843 6f6d 706f 6e65 6e74 732c 2055  ithComponents, U
│ │ │ │ +000700a0: 703a 2054 6f70 0a0a 746f 4172 7261 7920  p: Top..toArray 
│ │ │ │ +000700b0: 2d2d 206d 616b 6573 2061 6e20 6172 7261  -- makes an arra
│ │ │ │ +000700c0: 7920 6672 6f6d 2061 204c 6973 7420 6f72  y from a List or
│ │ │ │ +000700d0: 2066 726f 6d20 6120 7369 6e67 6c65 2069   from a single i
│ │ │ │ +000700e0: 6e74 6567 6572 0a2a 2a2a 2a2a 2a2a 2a2a  nteger.*********
│ │ │ │ +000700f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00070100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00070110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00070120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00070130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00070140: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167  ******..  * Usag
│ │ │ │ -00070150: 653a 200a 2020 2020 2020 2020 6172 7220  e: .        arr 
│ │ │ │ -00070160: 3d20 746f 4172 7261 7920 4c0a 2020 2020  = toArray L.    
│ │ │ │ -00070170: 2020 2020 6172 7220 3d20 746f 4172 7261      arr = toArra
│ │ │ │ -00070180: 7920 6e0a 2020 2a20 496e 7075 7473 3a0a  y n.  * Inputs:.
│ │ │ │ -00070190: 2020 2020 2020 2a20 4c2c 2061 202a 6e6f        * L, a *no
│ │ │ │ -000701a0: 7465 206c 6973 743a 2028 4d61 6361 756c  te list: (Macaul
│ │ │ │ -000701b0: 6179 3244 6f63 294c 6973 742c 2c20 0a20  ay2Doc)List,, . 
│ │ │ │ -000701c0: 2020 2020 202a 206e 2c20 616e 202a 6e6f       * n, an *no
│ │ │ │ -000701d0: 7465 2069 6e74 6567 6572 3a20 284d 6163  te integer: (Mac
│ │ │ │ -000701e0: 6175 6c61 7932 446f 6329 5a5a 2c2c 200a  aulay2Doc)ZZ,, .
│ │ │ │ -000701f0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00070200: 2020 202a 2061 7272 2c20 616e 202a 6e6f     * arr, an *no
│ │ │ │ -00070210: 7465 2061 7272 6179 3a20 284d 6163 6175  te array: (Macau
│ │ │ │ -00070220: 6c61 7932 446f 6329 4172 7261 792c 2c20  lay2Doc)Array,, 
│ │ │ │ -00070230: 0a0a 5761 7973 2074 6f20 7573 6520 746f  ..Ways to use to
│ │ │ │ -00070240: 4172 7261 793a 0a3d 3d3d 3d3d 3d3d 3d3d  Array:.=========
│ │ │ │ -00070250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00070260: 2022 746f 4172 7261 7928 4c69 7374 2922   "toArray(List)"
│ │ │ │ -00070270: 0a20 202a 2022 746f 4172 7261 7928 5a5a  .  * "toArray(ZZ
│ │ │ │ -00070280: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ -00070290: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ -000702a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ -000702b0: 626a 6563 7420 2a6e 6f74 6520 746f 4172  bject *note toAr
│ │ │ │ -000702c0: 7261 793a 2074 6f41 7272 6179 2c20 6973  ray: toArray, is
│ │ │ │ -000702d0: 2061 202a 6e6f 7465 206d 6574 686f 6420   a *note method 
│ │ │ │ -000702e0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175  function:.(Macau
│ │ │ │ -000702f0: 6c61 7932 446f 6329 4d65 7468 6f64 4675  lay2Doc)MethodFu
│ │ │ │ -00070300: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d  nction,...------
│ │ │ │ +00070120: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765  *****..  * Usage
│ │ │ │ +00070130: 3a20 0a20 2020 2020 2020 2061 7272 203d  : .        arr =
│ │ │ │ +00070140: 2074 6f41 7272 6179 204c 0a20 2020 2020   toArray L.     
│ │ │ │ +00070150: 2020 2061 7272 203d 2074 6f41 7272 6179     arr = toArray
│ │ │ │ +00070160: 206e 0a20 202a 2049 6e70 7574 733a 0a20   n.  * Inputs:. 
│ │ │ │ +00070170: 2020 2020 202a 204c 2c20 6120 2a6e 6f74       * L, a *not
│ │ │ │ +00070180: 6520 6c69 7374 3a20 284d 6163 6175 6c61  e list: (Macaula
│ │ │ │ +00070190: 7932 446f 6329 4c69 7374 2c2c 200a 2020  y2Doc)List,, .  
│ │ │ │ +000701a0: 2020 2020 2a20 6e2c 2061 6e20 2a6e 6f74      * n, an *not
│ │ │ │ +000701b0: 6520 696e 7465 6765 723a 2028 4d61 6361  e integer: (Maca
│ │ │ │ +000701c0: 756c 6179 3244 6f63 295a 5a2c 2c20 0a20  ulay2Doc)ZZ,, . 
│ │ │ │ +000701d0: 202a 204f 7574 7075 7473 3a0a 2020 2020   * Outputs:.    
│ │ │ │ +000701e0: 2020 2a20 6172 722c 2061 6e20 2a6e 6f74    * arr, an *not
│ │ │ │ +000701f0: 6520 6172 7261 793a 2028 4d61 6361 756c  e array: (Macaul
│ │ │ │ +00070200: 6179 3244 6f63 2941 7272 6179 2c2c 200a  ay2Doc)Array,, .
│ │ │ │ +00070210: 0a57 6179 7320 746f 2075 7365 2074 6f41  .Ways to use toA
│ │ │ │ +00070220: 7272 6179 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d  rray:.==========
│ │ │ │ +00070230: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00070240: 2274 6f41 7272 6179 284c 6973 7429 220a  "toArray(List)".
│ │ │ │ +00070250: 2020 2a20 2274 6f41 7272 6179 285a 5a29    * "toArray(ZZ)
│ │ │ │ +00070260: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ +00070270: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ +00070280: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ +00070290: 6a65 6374 202a 6e6f 7465 2074 6f41 7272  ject *note toArr
│ │ │ │ +000702a0: 6179 3a20 746f 4172 7261 792c 2069 7320  ay: toArray, is 
│ │ │ │ +000702b0: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066  a *note method f
│ │ │ │ +000702c0: 756e 6374 696f 6e3a 0a28 4d61 6361 756c  unction:.(Macaul
│ │ │ │ +000702d0: 6179 3244 6f63 294d 6574 686f 6446 756e  ay2Doc)MethodFun
│ │ │ │ +000702e0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d  ction,...-------
│ │ │ │ +000702f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070350: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073  ---------..The s
│ │ │ │ -00070360: 6f75 7263 6520 6f66 2074 6869 7320 646f  ource of this do
│ │ │ │ -00070370: 6375 6d65 6e74 2069 7320 696e 0a2f 6275  cument is in./bu
│ │ │ │ -00070380: 696c 642f 7265 7072 6f64 7563 6962 6c65  ild/reproducible
│ │ │ │ -00070390: 2d70 6174 682f 6d61 6361 756c 6179 322d  -path/macaulay2-
│ │ │ │ -000703a0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61  1.26.06+ds/M2/Ma
│ │ │ │ -000703b0: 6361 756c 6179 322f 7061 636b 6167 6573  caulay2/packages
│ │ │ │ -000703c0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273  /.CompleteInters
│ │ │ │ -000703d0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e  ectionResolution
│ │ │ │ -000703e0: 732e 6d32 3a33 3934 333a 302e 0a1f 0a46  s.m2:3943:0....F
│ │ │ │ -000703f0: 696c 653a 2043 6f6d 706c 6574 6549 6e74  ile: CompleteInt
│ │ │ │ -00070400: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -00070410: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a  ions.info, Node:
│ │ │ │ -00070420: 2074 776f 4d6f 6e6f 6d69 616c 732c 2050   twoMonomials, P
│ │ │ │ -00070430: 7265 763a 2074 6f41 7272 6179 2c20 5570  rev: toArray, Up
│ │ │ │ -00070440: 3a20 546f 700a 0a74 776f 4d6f 6e6f 6d69  : Top..twoMonomi
│ │ │ │ -00070450: 616c 7320 2d2d 2074 616c 6c79 2074 6865  als -- tally the
│ │ │ │ -00070460: 2073 6571 7565 6e63 6573 206f 6620 4252   sequences of BR
│ │ │ │ -00070470: 616e 6b73 2066 6f72 2063 6572 7461 696e  anks for certain
│ │ │ │ -00070480: 2065 7861 6d70 6c65 730a 2a2a 2a2a 2a2a   examples.******
│ │ │ │ +00070330: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ +00070340: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ +00070350: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ +00070360: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ +00070370: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ +00070380: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ +00070390: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ +000703a0: 0a43 6f6d 706c 6574 6549 6e74 6572 7365  .CompleteInterse
│ │ │ │ +000703b0: 6374 696f 6e52 6573 6f6c 7574 696f 6e73  ctionResolutions
│ │ │ │ +000703c0: 2e6d 323a 3339 3433 3a30 2e0a 1f0a 4669  .m2:3943:0....Fi
│ │ │ │ +000703d0: 6c65 3a20 436f 6d70 6c65 7465 496e 7465  le: CompleteInte
│ │ │ │ +000703e0: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +000703f0: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20  ons.info, Node: 
│ │ │ │ +00070400: 7477 6f4d 6f6e 6f6d 6961 6c73 2c20 5072  twoMonomials, Pr
│ │ │ │ +00070410: 6576 3a20 746f 4172 7261 792c 2055 703a  ev: toArray, Up:
│ │ │ │ +00070420: 2054 6f70 0a0a 7477 6f4d 6f6e 6f6d 6961   Top..twoMonomia
│ │ │ │ +00070430: 6c73 202d 2d20 7461 6c6c 7920 7468 6520  ls -- tally the 
│ │ │ │ +00070440: 7365 7175 656e 6365 7320 6f66 2042 5261  sequences of BRa
│ │ │ │ +00070450: 6e6b 7320 666f 7220 6365 7274 6169 6e20  nks for certain 
│ │ │ │ +00070460: 6578 616d 706c 6573 0a2a 2a2a 2a2a 2a2a  examples.*******
│ │ │ │ +00070470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00070480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00070490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000704a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000704b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000704c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020  ************..  
│ │ │ │ -000704d0: 2a20 5573 6167 653a 200a 2020 2020 2020  * Usage: .      
│ │ │ │ -000704e0: 2020 5420 3d20 5477 6f4d 6f6e 6f6d 6961    T = TwoMonomia
│ │ │ │ -000704f0: 6c73 2863 2c64 290a 2020 2a20 496e 7075  ls(c,d).  * Inpu
│ │ │ │ -00070500: 7473 3a0a 2020 2020 2020 2a20 632c 2061  ts:.      * c, a
│ │ │ │ -00070510: 6e20 2a6e 6f74 6520 696e 7465 6765 723a  n *note integer:
│ │ │ │ -00070520: 2028 4d61 6361 756c 6179 3244 6f63 295a   (Macaulay2Doc)Z
│ │ │ │ -00070530: 5a2c 2c20 636f 6469 6d65 6e73 696f 6e20  Z,, codimension 
│ │ │ │ -00070540: 696e 2077 6869 6368 2074 6f20 776f 726b  in which to work
│ │ │ │ -00070550: 0a20 2020 2020 202a 2064 2c20 616e 202a  .      * d, an *
│ │ │ │ -00070560: 6e6f 7465 2069 6e74 6567 6572 3a20 284d  note integer: (M
│ │ │ │ -00070570: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c  acaulay2Doc)ZZ,,
│ │ │ │ -00070580: 2064 6567 7265 6520 6f66 2074 6865 206d   degree of the m
│ │ │ │ -00070590: 6f6e 6f6d 6961 6c73 2074 6f20 7461 6b65  onomials to take
│ │ │ │ -000705a0: 0a20 202a 202a 6e6f 7465 204f 7074 696f  .  * *note Optio
│ │ │ │ -000705b0: 6e61 6c20 696e 7075 7473 3a20 284d 6163  nal inputs: (Mac
│ │ │ │ -000705c0: 6175 6c61 7932 446f 6329 7573 696e 6720  aulay2Doc)using 
│ │ │ │ -000705d0: 6675 6e63 7469 6f6e 7320 7769 7468 206f  functions with o
│ │ │ │ -000705e0: 7074 696f 6e61 6c20 696e 7075 7473 2c3a  ptional inputs,:
│ │ │ │ -000705f0: 0a20 2020 2020 202a 204f 7074 696d 6973  .      * Optimis
│ │ │ │ -00070600: 6d20 3d3e 202e 2e2e 2c20 6465 6661 756c  m => ..., defaul
│ │ │ │ -00070610: 7420 7661 6c75 6520 300a 2020 2a20 4f75  t value 0.  * Ou
│ │ │ │ -00070620: 7470 7574 733a 0a20 2020 2020 202a 2054  tputs:.      * T
│ │ │ │ -00070630: 2c20 6120 2a6e 6f74 6520 7461 6c6c 793a  , a *note tally:
│ │ │ │ -00070640: 2028 4d61 6361 756c 6179 3244 6f63 2954   (Macaulay2Doc)T
│ │ │ │ -00070650: 616c 6c79 2c2c 200a 0a44 6573 6372 6970  ally,, ..Descrip
│ │ │ │ -00070660: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00070670: 0a0a 7461 6c6c 6965 7320 7468 6520 7365  ..tallies the se
│ │ │ │ -00070680: 7175 656e 6365 7320 6f66 2042 2d72 616e  quences of B-ran
│ │ │ │ -00070690: 6b73 2074 6861 7420 6f63 6375 7220 666f  ks that occur fo
│ │ │ │ -000706a0: 7220 6964 6561 6c73 2067 656e 6572 6174  r ideals generat
│ │ │ │ -000706b0: 6564 2062 7920 7061 6972 7320 6f66 0a6d  ed by pairs of.m
│ │ │ │ -000706c0: 6f6e 6f6d 6961 6c73 2069 6e20 5220 3d20  onomials in R = 
│ │ │ │ -000706d0: 532f 2864 2d74 6820 706f 7765 7273 206f  S/(d-th powers o
│ │ │ │ -000706e0: 6620 7468 6520 7661 7269 6162 6c65 7329  f the variables)
│ │ │ │ -000706f0: 2c20 7769 7468 2066 756c 6c20 636f 6d70  , with full comp
│ │ │ │ -00070700: 6c65 7869 7479 2028 3d63 293b 0a74 6861  lexity (=c);.tha
│ │ │ │ -00070710: 7420 6973 2c20 666f 7220 616e 2061 7070  t is, for an app
│ │ │ │ -00070720: 726f 7072 6961 7465 2073 797a 7967 7920  ropriate syzygy 
│ │ │ │ -00070730: 4d20 6f66 204d 3020 3d20 522f 286d 312c  M of M0 = R/(m1,
│ │ │ │ -00070740: 206d 3229 2077 6865 7265 206d 3120 616e   m2) where m1 an
│ │ │ │ -00070750: 6420 6d32 2061 7265 0a6d 6f6e 6f6d 6961  d m2 are.monomia
│ │ │ │ -00070760: 6c73 206f 6620 7468 6520 7361 6d65 2064  ls of the same d
│ │ │ │ -00070770: 6567 7265 652e 0a0a 2b2d 2d2d 2d2d 2d2d  egree...+-------
│ │ │ │ -00070780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000707a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -000707b0: 7c69 3120 3a20 7365 7452 616e 646f 6d53  |i1 : setRandomS
│ │ │ │ -000707c0: 6565 6420 3020 2020 2020 2020 2020 2020  eed 0           
│ │ │ │ -000707d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000707e0: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574        |.| -- set
│ │ │ │ -000707f0: 7469 6e67 2072 616e 646f 6d20 7365 6564  ting random seed
│ │ │ │ -00070800: 2074 6f20 3020 2020 2020 2020 2020 2020   to 0           
│ │ │ │ -00070810: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070820: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -00070830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000704a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a  ***********..  *
│ │ │ │ +000704b0: 2055 7361 6765 3a20 0a20 2020 2020 2020   Usage: .       
│ │ │ │ +000704c0: 2054 203d 2054 776f 4d6f 6e6f 6d69 616c   T = TwoMonomial
│ │ │ │ +000704d0: 7328 632c 6429 0a20 202a 2049 6e70 7574  s(c,d).  * Input
│ │ │ │ +000704e0: 733a 0a20 2020 2020 202a 2063 2c20 616e  s:.      * c, an
│ │ │ │ +000704f0: 202a 6e6f 7465 2069 6e74 6567 6572 3a20   *note integer: 
│ │ │ │ +00070500: 284d 6163 6175 6c61 7932 446f 6329 5a5a  (Macaulay2Doc)ZZ
│ │ │ │ +00070510: 2c2c 2063 6f64 696d 656e 7369 6f6e 2069  ,, codimension i
│ │ │ │ +00070520: 6e20 7768 6963 6820 746f 2077 6f72 6b0a  n which to work.
│ │ │ │ +00070530: 2020 2020 2020 2a20 642c 2061 6e20 2a6e        * d, an *n
│ │ │ │ +00070540: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ +00070550: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ +00070560: 6465 6772 6565 206f 6620 7468 6520 6d6f  degree of the mo
│ │ │ │ +00070570: 6e6f 6d69 616c 7320 746f 2074 616b 650a  nomials to take.
│ │ │ │ +00070580: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e    * *note Option
│ │ │ │ +00070590: 616c 2069 6e70 7574 733a 2028 4d61 6361  al inputs: (Maca
│ │ │ │ +000705a0: 756c 6179 3244 6f63 2975 7369 6e67 2066  ulay2Doc)using f
│ │ │ │ +000705b0: 756e 6374 696f 6e73 2077 6974 6820 6f70  unctions with op
│ │ │ │ +000705c0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a  tional inputs,:.
│ │ │ │ +000705d0: 2020 2020 2020 2a20 4f70 7469 6d69 736d        * Optimism
│ │ │ │ +000705e0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +000705f0: 2076 616c 7565 2030 0a20 202a 204f 7574   value 0.  * Out
│ │ │ │ +00070600: 7075 7473 3a0a 2020 2020 2020 2a20 542c  puts:.      * T,
│ │ │ │ +00070610: 2061 202a 6e6f 7465 2074 616c 6c79 3a20   a *note tally: 
│ │ │ │ +00070620: 284d 6163 6175 6c61 7932 446f 6329 5461  (Macaulay2Doc)Ta
│ │ │ │ +00070630: 6c6c 792c 2c20 0a0a 4465 7363 7269 7074  lly,, ..Descript
│ │ │ │ +00070640: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ +00070650: 0a74 616c 6c69 6573 2074 6865 2073 6571  .tallies the seq
│ │ │ │ +00070660: 7565 6e63 6573 206f 6620 422d 7261 6e6b  uences of B-rank
│ │ │ │ +00070670: 7320 7468 6174 206f 6363 7572 2066 6f72  s that occur for
│ │ │ │ +00070680: 2069 6465 616c 7320 6765 6e65 7261 7465   ideals generate
│ │ │ │ +00070690: 6420 6279 2070 6169 7273 206f 660a 6d6f  d by pairs of.mo
│ │ │ │ +000706a0: 6e6f 6d69 616c 7320 696e 2052 203d 2053  nomials in R = S
│ │ │ │ +000706b0: 2f28 642d 7468 2070 6f77 6572 7320 6f66  /(d-th powers of
│ │ │ │ +000706c0: 2074 6865 2076 6172 6961 626c 6573 292c   the variables),
│ │ │ │ +000706d0: 2077 6974 6820 6675 6c6c 2063 6f6d 706c   with full compl
│ │ │ │ +000706e0: 6578 6974 7920 283d 6329 3b0a 7468 6174  exity (=c);.that
│ │ │ │ +000706f0: 2069 732c 2066 6f72 2061 6e20 6170 7072   is, for an appr
│ │ │ │ +00070700: 6f70 7269 6174 6520 7379 7a79 6779 204d  opriate syzygy M
│ │ │ │ +00070710: 206f 6620 4d30 203d 2052 2f28 6d31 2c20   of M0 = R/(m1, 
│ │ │ │ +00070720: 6d32 2920 7768 6572 6520 6d31 2061 6e64  m2) where m1 and
│ │ │ │ +00070730: 206d 3220 6172 650a 6d6f 6e6f 6d69 616c   m2 are.monomial
│ │ │ │ +00070740: 7320 6f66 2074 6865 2073 616d 6520 6465  s of the same de
│ │ │ │ +00070750: 6772 6565 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  gree...+--------
│ │ │ │ +00070760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ +00070790: 6931 203a 2073 6574 5261 6e64 6f6d 5365  i1 : setRandomSe
│ │ │ │ +000707a0: 6564 2030 2020 2020 2020 2020 2020 2020  ed 0            
│ │ │ │ +000707b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000707c0: 2020 2020 207c 0a7c 202d 2d20 7365 7474       |.| -- sett
│ │ │ │ +000707d0: 696e 6720 7261 6e64 6f6d 2073 6565 6420  ing random seed 
│ │ │ │ +000707e0: 746f 2030 2020 2020 2020 2020 2020 2020  to 0            
│ │ │ │ +000707f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00070800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070830: 2020 2020 207c 0a7c 6f31 203d 2030 2020       |.|o1 = 0  
│ │ │ │  00070840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070850: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020        |.|o1 = 0 
│ │ │ │ -00070860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070880: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070890: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ -000708a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000708b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000708c0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7477  ------+.|i2 : tw
│ │ │ │ -000708d0: 6f4d 6f6e 6f6d 6961 6c73 2832 2c33 2920  oMonomials(2,3) 
│ │ │ │ -000708e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000708f0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070900: 7c20 2d2d 2075 7365 6420 312e 3034 3038  | -- used 1.0408
│ │ │ │ -00070910: 3973 2028 6370 7529 3b20 302e 3833 3934  9s (cpu); 0.8394
│ │ │ │ -00070920: 3837 7320 2874 6872 6561 6429 3b20 3073  87s (thread); 0s
│ │ │ │ -00070930: 2028 6763 2920 7c0a 7c32 2020 2020 2020   (gc) |.|2      
│ │ │ │ -00070940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070960: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070970: 7c54 616c 6c79 7b7b 7b31 2c20 317d 7d20  |Tally{{{1, 1}} 
│ │ │ │ -00070980: 3d3e 2032 2020 2020 2020 2020 7d20 2020  => 2        }   
│ │ │ │ -00070990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000709a0: 2020 2020 2020 7c0a 7c20 2020 2020 207b        |.|      {
│ │ │ │ -000709b0: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ -000709c0: 3d3e 2034 2020 2020 2020 2020 2020 2020  => 4            
│ │ │ │ -000709d0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000709e0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ -000709f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070a10: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00070a20: 6420 302e 3535 3236 3739 7320 2863 7075  d 0.552679s (cpu
│ │ │ │ -00070a30: 293b 2030 2e33 3935 3239 3373 2028 7468  ); 0.395293s (th
│ │ │ │ -00070a40: 7265 6164 293b 2030 7320 2867 6329 7c0a  read); 0s (gc)|.
│ │ │ │ -00070a50: 7c33 2020 2020 2020 2020 2020 2020 2020  |3              
│ │ │ │ -00070a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070a80: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b        |.|Tally{{
│ │ │ │ -00070a90: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20  {2, 2}, {1, 2}} 
│ │ │ │ -00070aa0: 3d3e 2032 7d20 2020 2020 2020 2020 2020  => 2}           
│ │ │ │ -00070ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070ac0: 7c20 2020 2020 207b 7b33 2c20 337d 2c20  |      {{3, 3}, 
│ │ │ │ -00070ad0: 7b32 2c20 337d 7d20 3d3e 2031 2020 2020  {2, 3}} => 1    
│ │ │ │ +00070850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070860: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ +00070870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000708a0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2074 776f  -----+.|i2 : two
│ │ │ │ +000708b0: 4d6f 6e6f 6d69 616c 7328 322c 3329 2020  Monomials(2,3)  
│ │ │ │ +000708c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000708d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000708e0: 202d 2d20 7573 6564 2031 2e33 3335 3131   -- used 1.33511
│ │ │ │ +000708f0: 7320 2863 7075 293b 2030 2e39 3232 3630  s (cpu); 0.92260
│ │ │ │ +00070900: 3473 2028 7468 7265 6164 293b 2030 7320  4s (thread); 0s 
│ │ │ │ +00070910: 2867 6329 207c 0a7c 3220 2020 2020 2020  (gc) |.|2       
│ │ │ │ +00070920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070940: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00070950: 5461 6c6c 797b 7b7b 312c 2031 7d7d 203d  Tally{{{1, 1}} =
│ │ │ │ +00070960: 3e20 3220 2020 2020 2020 207d 2020 2020  > 2        }    
│ │ │ │ +00070970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070980: 2020 2020 207c 0a7c 2020 2020 2020 7b7b       |.|      {{
│ │ │ │ +00070990: 322c 2032 7d2c 207b 312c 2032 7d7d 203d  2, 2}, {1, 2}} =
│ │ │ │ +000709a0: 3e20 3420 2020 2020 2020 2020 2020 2020  > 4             
│ │ │ │ +000709b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000709c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000709d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000709e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000709f0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00070a00: 2030 2e38 3339 3835 3873 2028 6370 7529   0.839858s (cpu)
│ │ │ │ +00070a10: 3b20 302e 3536 3839 3937 7320 2874 6872  ; 0.568997s (thr
│ │ │ │ +00070a20: 6561 6429 3b20 3073 2028 6763 297c 0a7c  ead); 0s (gc)|.|
│ │ │ │ +00070a30: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00070a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070a60: 2020 2020 207c 0a7c 5461 6c6c 797b 7b7b       |.|Tally{{{
│ │ │ │ +00070a70: 322c 2032 7d2c 207b 312c 2032 7d7d 203d  2, 2}, {1, 2}} =
│ │ │ │ +00070a80: 3e20 327d 2020 2020 2020 2020 2020 2020  > 2}            
│ │ │ │ +00070a90: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00070aa0: 2020 2020 2020 7b7b 332c 2033 7d2c 207b        {{3, 3}, {
│ │ │ │ +00070ab0: 322c 2033 7d7d 203d 3e20 3120 2020 2020  2, 3}} => 1     
│ │ │ │ +00070ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070ad0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00070ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00070b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070b20: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070b30: 7c20 2d2d 2075 7365 6420 302e 3130 3430  | -- used 0.1040
│ │ │ │ -00070b40: 3773 2028 6370 7529 3b20 302e 3130 3337  7s (cpu); 0.1037
│ │ │ │ -00070b50: 3934 7320 2874 6872 6561 6429 3b20 3073  94s (thread); 0s
│ │ │ │ -00070b60: 2028 6763 2920 7c0a 7c34 2020 2020 2020   (gc) |.|4      
│ │ │ │ -00070b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070b90: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00070ba0: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20  |Tally{{{2, 2}, 
│ │ │ │ -00070bb0: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020  {1, 2}} => 1}   
│ │ │ │ -00070bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00070bd0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ -00070be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ -00070c10: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d  .See also.======
│ │ │ │ -00070c20: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7477  ==..  * *note tw
│ │ │ │ -00070c30: 6f4d 6f6e 6f6d 6961 6c73 3a20 7477 6f4d  oMonomials: twoM
│ │ │ │ -00070c40: 6f6e 6f6d 6961 6c73 2c20 2d2d 2074 616c  onomials, -- tal
│ │ │ │ -00070c50: 6c79 2074 6865 2073 6571 7565 6e63 6573  ly the sequences
│ │ │ │ -00070c60: 206f 6620 4252 616e 6b73 2066 6f72 0a20   of BRanks for. 
│ │ │ │ -00070c70: 2020 2063 6572 7461 696e 2065 7861 6d70     certain examp
│ │ │ │ -00070c80: 6c65 730a 0a57 6179 7320 746f 2075 7365  les..Ways to use
│ │ │ │ -00070c90: 2074 776f 4d6f 6e6f 6d69 616c 733a 0a3d   twoMonomials:.=
│ │ │ │ -00070ca0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00070cb0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2274  ========..  * "t
│ │ │ │ -00070cc0: 776f 4d6f 6e6f 6d69 616c 7328 5a5a 2c5a  woMonomials(ZZ,Z
│ │ │ │ -00070cd0: 5a29 220a 0a46 6f72 2074 6865 2070 726f  Z)"..For the pro
│ │ │ │ -00070ce0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d  grammer.========
│ │ │ │ -00070cf0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520  ==========..The 
│ │ │ │ -00070d00: 6f62 6a65 6374 202a 6e6f 7465 2074 776f  object *note two
│ │ │ │ -00070d10: 4d6f 6e6f 6d69 616c 733a 2074 776f 4d6f  Monomials: twoMo
│ │ │ │ -00070d20: 6e6f 6d69 616c 732c 2069 7320 6120 2a6e  nomials, is a *n
│ │ │ │ -00070d30: 6f74 6520 6d65 7468 6f64 2066 756e 6374  ote method funct
│ │ │ │ -00070d40: 696f 6e20 7769 7468 0a6f 7074 696f 6e73  ion with.options
│ │ │ │ -00070d50: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00070d60: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769  MethodFunctionWi
│ │ │ │ -00070d70: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d  thOptions,...---
│ │ │ │ +00070af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070b00: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00070b10: 202d 2d20 7573 6564 2030 2e32 3031 3832   -- used 0.20182
│ │ │ │ +00070b20: 3173 2028 6370 7529 3b20 302e 3139 3030  1s (cpu); 0.1900
│ │ │ │ +00070b30: 3634 7320 2874 6872 6561 6429 3b20 3073  64s (thread); 0s
│ │ │ │ +00070b40: 2028 6763 297c 0a7c 3420 2020 2020 2020   (gc)|.|4       
│ │ │ │ +00070b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070b70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00070b80: 5461 6c6c 797b 7b7b 322c 2032 7d2c 207b  Tally{{{2, 2}, {
│ │ │ │ +00070b90: 312c 2032 7d7d 203d 3e20 317d 2020 2020  1, 2}} => 1}    
│ │ │ │ +00070ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00070bb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00070bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ +00070bf0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d  See also.=======
│ │ │ │ +00070c00: 3d0a 0a20 202a 202a 6e6f 7465 2074 776f  =..  * *note two
│ │ │ │ +00070c10: 4d6f 6e6f 6d69 616c 733a 2074 776f 4d6f  Monomials: twoMo
│ │ │ │ +00070c20: 6e6f 6d69 616c 732c 202d 2d20 7461 6c6c  nomials, -- tall
│ │ │ │ +00070c30: 7920 7468 6520 7365 7175 656e 6365 7320  y the sequences 
│ │ │ │ +00070c40: 6f66 2042 5261 6e6b 7320 666f 720a 2020  of BRanks for.  
│ │ │ │ +00070c50: 2020 6365 7274 6169 6e20 6578 616d 706c    certain exampl
│ │ │ │ +00070c60: 6573 0a0a 5761 7973 2074 6f20 7573 6520  es..Ways to use 
│ │ │ │ +00070c70: 7477 6f4d 6f6e 6f6d 6961 6c73 3a0a 3d3d  twoMonomials:.==
│ │ │ │ +00070c80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00070c90: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7477  =======..  * "tw
│ │ │ │ +00070ca0: 6f4d 6f6e 6f6d 6961 6c73 285a 5a2c 5a5a  oMonomials(ZZ,ZZ
│ │ │ │ +00070cb0: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00070cc0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00070cd0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00070ce0: 626a 6563 7420 2a6e 6f74 6520 7477 6f4d  bject *note twoM
│ │ │ │ +00070cf0: 6f6e 6f6d 6961 6c73 3a20 7477 6f4d 6f6e  onomials: twoMon
│ │ │ │ +00070d00: 6f6d 6961 6c73 2c20 6973 2061 202a 6e6f  omials, is a *no
│ │ │ │ +00070d10: 7465 206d 6574 686f 6420 6675 6e63 7469  te method functi
│ │ │ │ +00070d20: 6f6e 2077 6974 680a 6f70 7469 6f6e 733a  on with.options:
│ │ │ │ +00070d30: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ +00070d40: 6574 686f 6446 756e 6374 696f 6e57 6974  ethodFunctionWit
│ │ │ │ +00070d50: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d  hOptions,...----
│ │ │ │ +00070d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00070d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00070d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00070dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468  ------------..Th
│ │ │ │ -00070dd0: 6520 736f 7572 6365 206f 6620 7468 6973  e source of this
│ │ │ │ -00070de0: 2064 6f63 756d 656e 7420 6973 2069 6e0a   document is in.
│ │ │ │ -00070df0: 2f62 7569 6c64 2f72 6570 726f 6475 6369  /build/reproduci
│ │ │ │ -00070e00: 626c 652d 7061 7468 2f6d 6163 6175 6c61  ble-path/macaula
│ │ │ │ -00070e10: 7932 2d31 2e32 362e 3036 2b64 732f 4d32  y2-1.26.06+ds/M2
│ │ │ │ -00070e20: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61  /Macaulay2/packa
│ │ │ │ -00070e30: 6765 732f 0a43 6f6d 706c 6574 6549 6e74  ges/.CompleteInt
│ │ │ │ -00070e40: 6572 7365 6374 696f 6e52 6573 6f6c 7574  ersectionResolut
│ │ │ │ -00070e50: 696f 6e73 2e6d 323a 3435 3434 3a30 2e0a  ions.m2:4544:0..
│ │ │ │ -00070e60: 1f0a 5461 6720 5461 626c 653a 0a4e 6f64  ..Tag Table:.Nod
│ │ │ │ -00070e70: 653a 2054 6f70 7f33 3136 0a4e 6f64 653a  e: Top.316.Node:
│ │ │ │ -00070e80: 2041 5261 6e6b 737f 3336 3338 320a 4e6f   ARanks.36382.No
│ │ │ │ -00070e90: 6465 3a20 4175 676d 656e 7461 7469 6f6e  de: Augmentation
│ │ │ │ -00070ea0: 7f33 3739 3438 0a4e 6f64 653a 2042 4747  .37948.Node: BGG
│ │ │ │ -00070eb0: 4c7f 3339 3136 310a 4e6f 6465 3a20 624d  L.39161.Node: bM
│ │ │ │ -00070ec0: 6170 737f 3432 3539 380a 4e6f 6465 3a20  aps.42598.Node: 
│ │ │ │ -00070ed0: 4252 616e 6b73 7f34 3431 3030 0a4e 6f64  BRanks.44100.Nod
│ │ │ │ -00070ee0: 653a 2043 6865 636b 7f35 3136 3934 0a4e  e: Check.51694.N
│ │ │ │ -00070ef0: 6f64 653a 2063 6f6d 706c 6578 6974 797f  ode: complexity.
│ │ │ │ -00070f00: 3533 3434 330a 4e6f 6465 3a20 636f 7379  53443.Node: cosy
│ │ │ │ -00070f10: 7a79 6779 5265 737f 3539 3636 360a 4e6f  zygyRes.59666.No
│ │ │ │ -00070f20: 6465 3a20 644d 6170 737f 3632 3434 350a  de: dMaps.62445.
│ │ │ │ -00070f30: 4e6f 6465 3a20 6475 616c 5769 7468 436f  Node: dualWithCo
│ │ │ │ -00070f40: 6d70 6f6e 656e 7473 7f36 3339 3631 0a4e  mponents.63961.N
│ │ │ │ -00070f50: 6f64 653a 2045 6973 656e 6275 6453 6861  ode: EisenbudSha
│ │ │ │ -00070f60: 6d61 7368 7f36 3533 3431 0a4e 6f64 653a  mash.65341.Node:
│ │ │ │ -00070f70: 2045 6973 656e 6275 6453 6861 6d61 7368   EisenbudShamash
│ │ │ │ -00070f80: 546f 7461 6c7f 3831 3038 300a 4e6f 6465  Total.81080.Node
│ │ │ │ -00070f90: 3a20 6576 656e 4578 744d 6f64 756c 657f  : evenExtModule.
│ │ │ │ -00070fa0: 3936 3938 360a 4e6f 6465 3a20 6578 706f  96986.Node: expo
│ │ │ │ -00070fb0: 7f31 3033 3239 340a 4e6f 6465 3a20 6578  .103294.Node: ex
│ │ │ │ -00070fc0: 7465 7269 6f72 4578 744d 6f64 756c 657f  teriorExtModule.
│ │ │ │ -00070fd0: 3130 3731 3031 0a4e 6f64 653a 2065 7874  107101.Node: ext
│ │ │ │ -00070fe0: 6572 696f 7248 6f6d 6f6c 6f67 794d 6f64  eriorHomologyMod
│ │ │ │ -00070ff0: 756c 657f 3131 3935 3431 0a4e 6f64 653a  ule.119541.Node:
│ │ │ │ -00071000: 2065 7874 6572 696f 7254 6f72 4d6f 6475   exteriorTorModu
│ │ │ │ -00071010: 6c65 7f31 3231 3230 370a 4e6f 6465 3a20  le.121207.Node: 
│ │ │ │ -00071020: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69  extIsOnePolynomi
│ │ │ │ -00071030: 616c 7f31 3332 3032 370a 4e6f 6465 3a20  al.132027.Node: 
│ │ │ │ -00071040: 4578 744d 6f64 756c 657f 3133 3537 3335  ExtModule.135735
│ │ │ │ -00071050: 0a4e 6f64 653a 2045 7874 4d6f 6475 6c65  .Node: ExtModule
│ │ │ │ -00071060: 4461 7461 7f31 3437 3736 320a 4e6f 6465  Data.147762.Node
│ │ │ │ -00071070: 3a20 6578 7456 7343 6f68 6f6d 6f6c 6f67  : extVsCohomolog
│ │ │ │ -00071080: 797f 3135 3432 3032 0a4e 6f64 653a 2066  y.154202.Node: f
│ │ │ │ -00071090: 696e 6974 6542 6574 7469 4e75 6d62 6572  initeBettiNumber
│ │ │ │ -000710a0: 737f 3136 3030 3434 0a4e 6f64 653a 2066  s.160044.Node: f
│ │ │ │ -000710b0: 7265 6545 7874 6572 696f 7253 756d 6d61  reeExteriorSumma
│ │ │ │ -000710c0: 6e64 7f31 3635 3836 320a 4e6f 6465 3a20  nd.165862.Node: 
│ │ │ │ -000710d0: 4772 6164 696e 677f 3136 3837 3135 0a4e  Grading.168715.N
│ │ │ │ -000710e0: 6f64 653a 2068 667f 3137 3030 3730 0a4e  ode: hf.170070.N
│ │ │ │ -000710f0: 6f64 653a 2068 664d 6f64 756c 6541 7345  ode: hfModuleAsE
│ │ │ │ -00071100: 7874 7f31 3730 3935 320a 4e6f 6465 3a20  xt.170952.Node: 
│ │ │ │ -00071110: 6869 6768 5379 7a79 6779 7f31 3734 3838  highSyzygy.17488
│ │ │ │ -00071120: 310a 4e6f 6465 3a20 684d 6170 737f 3138  1.Node: hMaps.18
│ │ │ │ -00071130: 3233 3831 0a4e 6f64 653a 2048 6f6d 5769  2381.Node: HomWi
│ │ │ │ -00071140: 7468 436f 6d70 6f6e 656e 7473 7f31 3833  thComponents.183
│ │ │ │ -00071150: 3937 350a 4e6f 6465 3a20 696e 6669 6e69  975.Node: infini
│ │ │ │ -00071160: 7465 4265 7474 694e 756d 6265 7273 7f31  teBettiNumbers.1
│ │ │ │ -00071170: 3835 3338 360a 4e6f 6465 3a20 6973 4c69  85386.Node: isLi
│ │ │ │ -00071180: 6e65 6172 7f31 3931 3332 370a 4e6f 6465  near.191327.Node
│ │ │ │ -00071190: 3a20 6973 5175 6173 6952 6567 756c 6172  : isQuasiRegular
│ │ │ │ -000711a0: 7f31 3932 3334 390a 4e6f 6465 3a20 6973  .192349.Node: is
│ │ │ │ -000711b0: 5374 6162 6c79 5472 6976 6961 6c7f 3139  StablyTrivial.19
│ │ │ │ -000711c0: 3535 3836 0a4e 6f64 653a 206b 6f73 7a75  5586.Node: koszu
│ │ │ │ -000711d0: 6c45 7874 656e 7369 6f6e 7f32 3033 3033  lExtension.20303
│ │ │ │ -000711e0: 390a 4e6f 6465 3a20 4c61 7965 7265 647f  9.Node: Layered.
│ │ │ │ -000711f0: 3230 3435 3934 0a4e 6f64 653a 206c 6179  204594.Node: lay
│ │ │ │ -00071200: 6572 6564 5265 736f 6c75 7469 6f6e 7f32  eredResolution.2
│ │ │ │ -00071210: 3035 3936 320a 4e6f 6465 3a20 4c69 6674  05962.Node: Lift
│ │ │ │ -00071220: 7f32 3238 3632 320a 4e6f 6465 3a20 6d61  .228622.Node: ma
│ │ │ │ -00071230: 6b65 4669 6e69 7465 5265 736f 6c75 7469  keFiniteResoluti
│ │ │ │ -00071240: 6f6e 7f32 3239 3634 350a 4e6f 6465 3a20  on.229645.Node: 
│ │ │ │ -00071250: 6d61 6b65 4669 6e69 7465 5265 736f 6c75  makeFiniteResolu
│ │ │ │ -00071260: 7469 6f6e 436f 6469 6d32 7f32 3530 3338  tionCodim2.25038
│ │ │ │ -00071270: 320a 4e6f 6465 3a20 6d61 6b65 486f 6d6f  2.Node: makeHomo
│ │ │ │ -00071280: 746f 7069 6573 7f32 3539 3437 300a 4e6f  topies.259470.No
│ │ │ │ -00071290: 6465 3a20 6d61 6b65 486f 6d6f 746f 7069  de: makeHomotopi
│ │ │ │ -000712a0: 6573 317f 3334 3732 3331 0a4e 6f64 653a  es1.347231.Node:
│ │ │ │ -000712b0: 206d 616b 6548 6f6d 6f74 6f70 6965 734f   makeHomotopiesO
│ │ │ │ -000712c0: 6e48 6f6d 6f6c 6f67 797f 3334 3838 3635  nHomology.348865
│ │ │ │ -000712d0: 0a4e 6f64 653a 206d 616b 654d 6f64 756c  .Node: makeModul
│ │ │ │ -000712e0: 657f 3335 3035 3130 0a4e 6f64 653a 206d  e.350510.Node: m
│ │ │ │ -000712f0: 616b 6554 7f33 3631 3230 380a 4e6f 6465  akeT.361208.Node
│ │ │ │ -00071300: 3a20 6d61 7472 6978 4661 6374 6f72 697a  : matrixFactoriz
│ │ │ │ -00071310: 6174 696f 6e7f 3336 3534 3037 0a4e 6f64  ation.365407.Nod
│ │ │ │ -00071320: 653a 206d 6642 6f75 6e64 7f33 3734 3235  e: mfBound.37425
│ │ │ │ -00071330: 390a 4e6f 6465 3a20 6d6f 6475 6c65 4173  9.Node: moduleAs
│ │ │ │ -00071340: 4578 747f 3337 3631 3531 0a4e 6f64 653a  Ext.376151.Node:
│ │ │ │ -00071350: 206e 6577 4578 747f 3338 3233 3939 0a4e   newExt.382399.N
│ │ │ │ -00071360: 6f64 653a 206f 6464 4578 744d 6f64 756c  ode: oddExtModul
│ │ │ │ -00071370: 657f 3431 3039 3835 0a4e 6f64 653a 204f  e.410985.Node: O
│ │ │ │ -00071380: 7074 696d 6973 6d7f 3431 3732 3735 0a4e  ptimism.417275.N
│ │ │ │ -00071390: 6f64 653a 204f 7574 5269 6e67 7f34 3138  ode: OutRing.418
│ │ │ │ -000713a0: 3830 310a 4e6f 6465 3a20 7073 694d 6170  801.Node: psiMap
│ │ │ │ -000713b0: 737f 3432 3031 3438 0a4e 6f64 653a 2072  s.420148.Node: r
│ │ │ │ -000713c0: 6567 756c 6172 6974 7953 6571 7565 6e63  egularitySequenc
│ │ │ │ -000713d0: 657f 3432 3136 3838 0a4e 6f64 653a 2053  e.421688.Node: S
│ │ │ │ -000713e0: 327f 3432 3532 3433 0a4e 6f64 653a 2053  2.425243.Node: S
│ │ │ │ -000713f0: 6861 6d61 7368 7f34 3336 3332 360a 4e6f  hamash.436326.No
│ │ │ │ -00071400: 6465 3a20 7370 6c69 7474 696e 6773 7f34  de: splittings.4
│ │ │ │ -00071410: 3434 3036 350a 4e6f 6465 3a20 7374 6162  44065.Node: stab
│ │ │ │ -00071420: 6c65 486f 6d7f 3434 3932 3834 0a4e 6f64  leHom.449284.Nod
│ │ │ │ -00071430: 653a 2073 756d 5477 6f4d 6f6e 6f6d 6961  e: sumTwoMonomia
│ │ │ │ -00071440: 6c73 7f34 3530 3532 310a 4e6f 6465 3a20  ls.450521.Node: 
│ │ │ │ -00071450: 5461 7465 5265 736f 6c75 7469 6f6e 7f34  TateResolution.4
│ │ │ │ -00071460: 3533 3031 310a 4e6f 6465 3a20 7465 6e73  53011.Node: tens
│ │ │ │ -00071470: 6f72 5769 7468 436f 6d70 6f6e 656e 7473  orWithComponents
│ │ │ │ -00071480: 7f34 3537 3338 380a 4e6f 6465 3a20 746f  .457388.Node: to
│ │ │ │ -00071490: 4172 7261 797f 3435 3838 3335 0a4e 6f64  Array.458835.Nod
│ │ │ │ -000714a0: 653a 2074 776f 4d6f 6e6f 6d69 616c 737f  e: twoMonomials.
│ │ │ │ -000714b0: 3435 3937 3537 0a1f 0a45 6e64 2054 6167  459757...End Tag
│ │ │ │ -000714c0: 2054 6162 6c65 0a                         Table.
│ │ │ │ +00070da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00070db0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00070dc0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00070dd0: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00070de0: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00070df0: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ +00070e00: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00070e10: 6573 2f0a 436f 6d70 6c65 7465 496e 7465  es/.CompleteInte
│ │ │ │ +00070e20: 7273 6563 7469 6f6e 5265 736f 6c75 7469  rsectionResoluti
│ │ │ │ +00070e30: 6f6e 732e 6d32 3a34 3534 343a 302e 0a1f  ons.m2:4544:0...
│ │ │ │ +00070e40: 0a54 6167 2054 6162 6c65 3a0a 4e6f 6465  .Tag Table:.Node
│ │ │ │ +00070e50: 3a20 546f 707f 3331 360a 4e6f 6465 3a20  : Top.316.Node: 
│ │ │ │ +00070e60: 4152 616e 6b73 7f33 3633 3832 0a4e 6f64  ARanks.36382.Nod
│ │ │ │ +00070e70: 653a 2041 7567 6d65 6e74 6174 696f 6e7f  e: Augmentation.
│ │ │ │ +00070e80: 3337 3934 380a 4e6f 6465 3a20 4247 474c  37948.Node: BGGL
│ │ │ │ +00070e90: 7f33 3931 3631 0a4e 6f64 653a 2062 4d61  .39161.Node: bMa
│ │ │ │ +00070ea0: 7073 7f34 3235 3938 0a4e 6f64 653a 2042  ps.42598.Node: B
│ │ │ │ +00070eb0: 5261 6e6b 737f 3434 3130 300a 4e6f 6465  Ranks.44100.Node
│ │ │ │ +00070ec0: 3a20 4368 6563 6b7f 3531 3639 340a 4e6f  : Check.51694.No
│ │ │ │ +00070ed0: 6465 3a20 636f 6d70 6c65 7869 7479 7f35  de: complexity.5
│ │ │ │ +00070ee0: 3334 3433 0a4e 6f64 653a 2063 6f73 797a  3443.Node: cosyz
│ │ │ │ +00070ef0: 7967 7952 6573 7f35 3936 3636 0a4e 6f64  ygyRes.59666.Nod
│ │ │ │ +00070f00: 653a 2064 4d61 7073 7f36 3234 3435 0a4e  e: dMaps.62445.N
│ │ │ │ +00070f10: 6f64 653a 2064 7561 6c57 6974 6843 6f6d  ode: dualWithCom
│ │ │ │ +00070f20: 706f 6e65 6e74 737f 3633 3936 310a 4e6f  ponents.63961.No
│ │ │ │ +00070f30: 6465 3a20 4569 7365 6e62 7564 5368 616d  de: EisenbudSham
│ │ │ │ +00070f40: 6173 687f 3635 3334 310a 4e6f 6465 3a20  ash.65341.Node: 
│ │ │ │ +00070f50: 4569 7365 6e62 7564 5368 616d 6173 6854  EisenbudShamashT
│ │ │ │ +00070f60: 6f74 616c 7f38 3130 3437 0a4e 6f64 653a  otal.81047.Node:
│ │ │ │ +00070f70: 2065 7665 6e45 7874 4d6f 6475 6c65 7f39   evenExtModule.9
│ │ │ │ +00070f80: 3639 3533 0a4e 6f64 653a 2065 7870 6f7f  6953.Node: expo.
│ │ │ │ +00070f90: 3130 3332 3631 0a4e 6f64 653a 2065 7874  103261.Node: ext
│ │ │ │ +00070fa0: 6572 696f 7245 7874 4d6f 6475 6c65 7f31  eriorExtModule.1
│ │ │ │ +00070fb0: 3037 3036 380a 4e6f 6465 3a20 6578 7465  07068.Node: exte
│ │ │ │ +00070fc0: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475  riorHomologyModu
│ │ │ │ +00070fd0: 6c65 7f31 3139 3530 380a 4e6f 6465 3a20  le.119508.Node: 
│ │ │ │ +00070fe0: 6578 7465 7269 6f72 546f 724d 6f64 756c  exteriorTorModul
│ │ │ │ +00070ff0: 657f 3132 3131 3734 0a4e 6f64 653a 2065  e.121174.Node: e
│ │ │ │ +00071000: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961  xtIsOnePolynomia
│ │ │ │ +00071010: 6c7f 3133 3139 3934 0a4e 6f64 653a 2045  l.131994.Node: E
│ │ │ │ +00071020: 7874 4d6f 6475 6c65 7f31 3335 3730 320a  xtModule.135702.
│ │ │ │ +00071030: 4e6f 6465 3a20 4578 744d 6f64 756c 6544  Node: ExtModuleD
│ │ │ │ +00071040: 6174 617f 3134 3737 3239 0a4e 6f64 653a  ata.147729.Node:
│ │ │ │ +00071050: 2065 7874 5673 436f 686f 6d6f 6c6f 6779   extVsCohomology
│ │ │ │ +00071060: 7f31 3534 3136 390a 4e6f 6465 3a20 6669  .154169.Node: fi
│ │ │ │ +00071070: 6e69 7465 4265 7474 694e 756d 6265 7273  niteBettiNumbers
│ │ │ │ +00071080: 7f31 3630 3031 310a 4e6f 6465 3a20 6672  .160011.Node: fr
│ │ │ │ +00071090: 6565 4578 7465 7269 6f72 5375 6d6d 616e  eeExteriorSumman
│ │ │ │ +000710a0: 647f 3136 3538 3239 0a4e 6f64 653a 2047  d.165829.Node: G
│ │ │ │ +000710b0: 7261 6469 6e67 7f31 3638 3638 320a 4e6f  rading.168682.No
│ │ │ │ +000710c0: 6465 3a20 6866 7f31 3730 3033 370a 4e6f  de: hf.170037.No
│ │ │ │ +000710d0: 6465 3a20 6866 4d6f 6475 6c65 4173 4578  de: hfModuleAsEx
│ │ │ │ +000710e0: 747f 3137 3039 3139 0a4e 6f64 653a 2068  t.170919.Node: h
│ │ │ │ +000710f0: 6967 6853 797a 7967 797f 3137 3438 3438  ighSyzygy.174848
│ │ │ │ +00071100: 0a4e 6f64 653a 2068 4d61 7073 7f31 3832  .Node: hMaps.182
│ │ │ │ +00071110: 3334 380a 4e6f 6465 3a20 486f 6d57 6974  348.Node: HomWit
│ │ │ │ +00071120: 6843 6f6d 706f 6e65 6e74 737f 3138 3339  hComponents.1839
│ │ │ │ +00071130: 3432 0a4e 6f64 653a 2069 6e66 696e 6974  42.Node: infinit
│ │ │ │ +00071140: 6542 6574 7469 4e75 6d62 6572 737f 3138  eBettiNumbers.18
│ │ │ │ +00071150: 3533 3533 0a4e 6f64 653a 2069 734c 696e  5353.Node: isLin
│ │ │ │ +00071160: 6561 727f 3139 3132 3934 0a4e 6f64 653a  ear.191294.Node:
│ │ │ │ +00071170: 2069 7351 7561 7369 5265 6775 6c61 727f   isQuasiRegular.
│ │ │ │ +00071180: 3139 3233 3136 0a4e 6f64 653a 2069 7353  192316.Node: isS
│ │ │ │ +00071190: 7461 626c 7954 7269 7669 616c 7f31 3935  tablyTrivial.195
│ │ │ │ +000711a0: 3535 330a 4e6f 6465 3a20 6b6f 737a 756c  553.Node: koszul
│ │ │ │ +000711b0: 4578 7465 6e73 696f 6e7f 3230 3330 3036  Extension.203006
│ │ │ │ +000711c0: 0a4e 6f64 653a 204c 6179 6572 6564 7f32  .Node: Layered.2
│ │ │ │ +000711d0: 3034 3536 310a 4e6f 6465 3a20 6c61 7965  04561.Node: laye
│ │ │ │ +000711e0: 7265 6452 6573 6f6c 7574 696f 6e7f 3230  redResolution.20
│ │ │ │ +000711f0: 3539 3239 0a4e 6f64 653a 204c 6966 747f  5929.Node: Lift.
│ │ │ │ +00071200: 3232 3835 3839 0a4e 6f64 653a 206d 616b  228589.Node: mak
│ │ │ │ +00071210: 6546 696e 6974 6552 6573 6f6c 7574 696f  eFiniteResolutio
│ │ │ │ +00071220: 6e7f 3232 3936 3132 0a4e 6f64 653a 206d  n.229612.Node: m
│ │ │ │ +00071230: 616b 6546 696e 6974 6552 6573 6f6c 7574  akeFiniteResolut
│ │ │ │ +00071240: 696f 6e43 6f64 696d 327f 3235 3033 3439  ionCodim2.250349
│ │ │ │ +00071250: 0a4e 6f64 653a 206d 616b 6548 6f6d 6f74  .Node: makeHomot
│ │ │ │ +00071260: 6f70 6965 737f 3235 3934 3337 0a4e 6f64  opies.259437.Nod
│ │ │ │ +00071270: 653a 206d 616b 6548 6f6d 6f74 6f70 6965  e: makeHomotopie
│ │ │ │ +00071280: 7331 7f33 3437 3139 380a 4e6f 6465 3a20  s1.347198.Node: 
│ │ │ │ +00071290: 6d61 6b65 486f 6d6f 746f 7069 6573 4f6e  makeHomotopiesOn
│ │ │ │ +000712a0: 486f 6d6f 6c6f 6779 7f33 3438 3833 320a  Homology.348832.
│ │ │ │ +000712b0: 4e6f 6465 3a20 6d61 6b65 4d6f 6475 6c65  Node: makeModule
│ │ │ │ +000712c0: 7f33 3530 3437 370a 4e6f 6465 3a20 6d61  .350477.Node: ma
│ │ │ │ +000712d0: 6b65 547f 3336 3131 3735 0a4e 6f64 653a  keT.361175.Node:
│ │ │ │ +000712e0: 206d 6174 7269 7846 6163 746f 7269 7a61   matrixFactoriza
│ │ │ │ +000712f0: 7469 6f6e 7f33 3635 3337 340a 4e6f 6465  tion.365374.Node
│ │ │ │ +00071300: 3a20 6d66 426f 756e 647f 3337 3432 3236  : mfBound.374226
│ │ │ │ +00071310: 0a4e 6f64 653a 206d 6f64 756c 6541 7345  .Node: moduleAsE
│ │ │ │ +00071320: 7874 7f33 3736 3131 380a 4e6f 6465 3a20  xt.376118.Node: 
│ │ │ │ +00071330: 6e65 7745 7874 7f33 3832 3336 360a 4e6f  newExt.382366.No
│ │ │ │ +00071340: 6465 3a20 6f64 6445 7874 4d6f 6475 6c65  de: oddExtModule
│ │ │ │ +00071350: 7f34 3130 3935 320a 4e6f 6465 3a20 4f70  .410952.Node: Op
│ │ │ │ +00071360: 7469 6d69 736d 7f34 3137 3234 320a 4e6f  timism.417242.No
│ │ │ │ +00071370: 6465 3a20 4f75 7452 696e 677f 3431 3837  de: OutRing.4187
│ │ │ │ +00071380: 3638 0a4e 6f64 653a 2070 7369 4d61 7073  68.Node: psiMaps
│ │ │ │ +00071390: 7f34 3230 3131 350a 4e6f 6465 3a20 7265  .420115.Node: re
│ │ │ │ +000713a0: 6775 6c61 7269 7479 5365 7175 656e 6365  gularitySequence
│ │ │ │ +000713b0: 7f34 3231 3635 350a 4e6f 6465 3a20 5332  .421655.Node: S2
│ │ │ │ +000713c0: 7f34 3235 3231 300a 4e6f 6465 3a20 5368  .425210.Node: Sh
│ │ │ │ +000713d0: 616d 6173 687f 3433 3632 3933 0a4e 6f64  amash.436293.Nod
│ │ │ │ +000713e0: 653a 2073 706c 6974 7469 6e67 737f 3434  e: splittings.44
│ │ │ │ +000713f0: 3430 3332 0a4e 6f64 653a 2073 7461 626c  4032.Node: stabl
│ │ │ │ +00071400: 6548 6f6d 7f34 3439 3235 310a 4e6f 6465  eHom.449251.Node
│ │ │ │ +00071410: 3a20 7375 6d54 776f 4d6f 6e6f 6d69 616c  : sumTwoMonomial
│ │ │ │ +00071420: 737f 3435 3034 3838 0a4e 6f64 653a 2054  s.450488.Node: T
│ │ │ │ +00071430: 6174 6552 6573 6f6c 7574 696f 6e7f 3435  ateResolution.45
│ │ │ │ +00071440: 3239 3738 0a4e 6f64 653a 2074 656e 736f  2978.Node: tenso
│ │ │ │ +00071450: 7257 6974 6843 6f6d 706f 6e65 6e74 737f  rWithComponents.
│ │ │ │ +00071460: 3435 3733 3535 0a4e 6f64 653a 2074 6f41  457355.Node: toA
│ │ │ │ +00071470: 7272 6179 7f34 3538 3830 320a 4e6f 6465  rray.458802.Node
│ │ │ │ +00071480: 3a20 7477 6f4d 6f6e 6f6d 6961 6c73 7f34  : twoMonomials.4
│ │ │ │ +00071490: 3539 3732 340a 1f0a 456e 6420 5461 6720  59724...End Tag 
│ │ │ │ +000714a0: 5461 626c 650a                           Table.
│ │ ├── ./usr/share/info/ConnectionMatrices.info.gz
│ │ │ ├── ConnectionMatrices.info
│ │ │ │ @@ -2993,30 +2993,30 @@
│ │ │ │  0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0000bb20: 3920 3a20 656c 6170 7365 6454 696d 6520  9 : elapsedTime 
│ │ │ │  0000bb30: 4120 3d20 7066 6166 6669 616e 5379 7374  A = pfaffianSyst
│ │ │ │  0000bb40: 656d 2049 3b20 2020 2020 2020 2020 2020  em I;           
│ │ │ │  0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0000bb70: 2d2d 2034 2e33 3936 3431 7320 656c 6170  -- 4.39641s elap
│ │ │ │ +0000bb70: 2d2d 2033 2e33 3834 3231 7320 656c 6170  -- 3.38421s elap
│ │ │ │  0000bb80: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  0000bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0000bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │  0000bc10: 3130 203a 2065 6c61 7073 6564 5469 6d65  10 : elapsedTime
│ │ │ │  0000bc20: 2061 7373 6572 7420 6973 496e 7465 6772   assert isIntegr
│ │ │ │  0000bc30: 6162 6c65 2041 2020 2020 2020 2020 2020  able A          
│ │ │ │  0000bc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -0000bc60: 2d2d 2035 2e34 3638 3832 7320 656c 6170  -- 5.46882s elap
│ │ │ │ +0000bc60: 2d2d 2034 2e34 3337 3038 7320 656c 6170  -- 4.43708s elap
│ │ │ │  0000bc70: 7365 6420 2020 2020 2020 2020 2020 2020  sed             
│ │ │ │  0000bc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bca0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0000bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5137,15 +5137,15 @@
│ │ │ │  00014100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00014120: 7c69 3134 203a 2065 6c61 7073 6564 5469  |i14 : elapsedTi
│ │ │ │  00014130: 6d65 2067 203d 2067 6175 6765 4d61 7472  me g = gaugeMatr
│ │ │ │  00014140: 6978 2849 2c20 4229 3b20 2020 2020 2020  ix(I, B);       
│ │ │ │  00014150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014160: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00014170: 7c20 2d2d 202e 3630 3639 3734 7320 656c  | -- .606974s el
│ │ │ │ +00014170: 7c20 2d2d 202e 3633 3636 3537 7320 656c  | -- .636657s el
│ │ │ │  00014180: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00014190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000141a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000141b0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000141c0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000141d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000141e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5167,30 +5167,30 @@
│ │ │ │  000142e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000142f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  00014300: 7c69 3135 203a 2065 6c61 7073 6564 5469  |i15 : elapsedTi
│ │ │ │  00014310: 6d65 2041 3120 3d20 6761 7567 6554 7261  me A1 = gaugeTra
│ │ │ │  00014320: 6e73 666f 726d 2867 2c20 4129 3b20 2020  nsform(g, A);   
│ │ │ │  00014330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014340: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00014350: 7c20 2d2d 2031 2e33 3030 3333 7320 656c  | -- 1.30033s el
│ │ │ │ +00014350: 7c20 2d2d 2031 2e32 3533 3036 7320 656c  | -- 1.25306s el
│ │ │ │  00014360: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00014370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014390: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  000143a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000143b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000143c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000143d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000143e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │  000143f0: 7c69 3136 203a 2065 6c61 7073 6564 5469  |i16 : elapsedTi
│ │ │ │  00014400: 6d65 2061 7373 6572 7420 6973 496e 7465  me assert isInte
│ │ │ │  00014410: 6772 6162 6c65 2041 3120 2020 2020 2020  grable A1       
│ │ │ │  00014420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014430: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00014440: 7c20 2d2d 202e 3835 3739 3433 7320 656c  | -- .857943s el
│ │ │ │ +00014440: 7c20 2d2d 202e 3936 3938 3839 7320 656c  | -- .969889s el
│ │ │ │  00014450: 6170 7365 6420 2020 2020 2020 2020 2020  apsed           
│ │ │ │  00014460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014480: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │  00014490: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  000144a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000144b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -5608,31 +5608,31 @@
│ │ │ │  00015e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015e90: 2d2d 2d2b 0a7c 6931 3920 3a20 656c 6170  ---+.|i19 : elap
│ │ │ │  00015ea0: 7365 6454 696d 6520 4132 203d 2067 6175  sedTime A2 = gau
│ │ │ │  00015eb0: 6765 5472 616e 7366 6f72 6d28 6368 616e  geTransform(chan
│ │ │ │  00015ec0: 6765 4570 732c 2041 3129 3b20 2020 2020  geEps, A1);     
│ │ │ │  00015ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ee0: 2020 207c 0a7c 202d 2d20 2e33 3933 3538     |.| -- .39358
│ │ │ │ -00015ef0: 3773 2065 6c61 7073 6564 2020 2020 2020  7s elapsed      
│ │ │ │ +00015ee0: 2020 207c 0a7c 202d 2d20 2e33 3839 3638     |.| -- .38968
│ │ │ │ +00015ef0: 3873 2065 6c61 7073 6564 2020 2020 2020  8s elapsed      
│ │ │ │  00015f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015f30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00015f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015f80: 2d2d 2d2b 0a7c 6932 3020 3a20 656c 6170  ---+.|i20 : elap
│ │ │ │  00015f90: 7365 6454 696d 6520 6173 7365 7274 2069  sedTime assert i
│ │ │ │  00015fa0: 7349 6e74 6567 7261 626c 6520 4132 2020  sIntegrable A2  
│ │ │ │  00015fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015fd0: 2020 207c 0a7c 202d 2d20 312e 3036 3132     |.| -- 1.0612
│ │ │ │ -00015fe0: 3473 2065 6c61 7073 6564 2020 2020 2020  4s elapsed      
│ │ │ │ +00015fd0: 2020 207c 0a7c 202d 2d20 2e38 3530 3539     |.| -- .85059
│ │ │ │ +00015fe0: 3573 2065 6c61 7073 6564 2020 2020 2020  5s elapsed      
│ │ │ │  00015ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016020: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00016030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -6018,30 +6018,30 @@
│ │ │ │  00017810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │  00017830: 203a 2065 6c61 7073 6564 5469 6d65 2041   : elapsedTime A
│ │ │ │  00017840: 203d 2070 6661 6666 6961 6e53 7973 7465   = pfaffianSyste
│ │ │ │  00017850: 6d20 493b 2020 2020 2020 2020 2020 2020  m I;            
│ │ │ │  00017860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017870: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00017880: 2d20 2e33 3239 3332 7320 656c 6170 7365  - .32932s elapse
│ │ │ │ -00017890: 6420 2020 2020 2020 2020 2020 2020 2020  d               
│ │ │ │ +00017880: 2d20 2e33 3236 3430 3373 2065 6c61 7073  - .326403s elaps
│ │ │ │ +00017890: 6564 2020 2020 2020 2020 2020 2020 2020  ed              
│ │ │ │  000178a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000178b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000178c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  000178d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000178e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000178f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │  00017920: 203a 2065 6c61 7073 6564 5469 6d65 2061   : elapsedTime a
│ │ │ │  00017930: 7373 6572 7420 6973 496e 7465 6772 6162  ssert isIntegrab
│ │ │ │  00017940: 6c65 2041 2020 2020 2020 2020 2020 2020  le A            
│ │ │ │  00017950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017960: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00017970: 2d20 2e31 3638 3736 3873 2065 6c61 7073  - .168768s elaps
│ │ │ │ +00017970: 2d20 2e32 3238 3231 3673 2065 6c61 7073  - .228216s elaps
│ │ │ │  00017980: 6564 2020 2020 2020 2020 2020 2020 2020  ed              
│ │ │ │  00017990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000179a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000179b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  000179c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000179d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000179e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ ├── ./usr/share/info/Cremona.info.gz
│ │ │ ├── Cremona.info
│ │ │ │ @@ -147,18 +147,18 @@
│ │ │ │  00000920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00000930: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2074  -------+.|i2 : t
│ │ │ │  00000940: 696d 6520 7068 6920 3d20 746f 4d61 7020  ime phi = toMap 
│ │ │ │  00000950: 6d69 6e6f 7273 2833 2c6d 6174 7269 787b  minors(3,matrix{
│ │ │ │  00000960: 7b74 5f30 2e2e 745f 347d 2c7b 745f 312e  {t_0..t_4},{t_1.
│ │ │ │  00000970: 2e74 5f35 7d2c 7b74 5f32 2e2e 745f 367d  .t_5},{t_2..t_6}
│ │ │ │  00000980: 7d29 2020 2020 207c 0a7c 202d 2d20 7573  })     |.| -- us
│ │ │ │ -00000990: 6564 2030 2e30 3035 3538 3539 3373 2028  ed 0.00558593s (
│ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3535 3831 3132  cpu); 0.00558112
│ │ │ │ -000009b0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ -000009c0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +00000990: 6564 2030 2e30 3035 3636 3938 3473 2028  ed 0.00566984s (
│ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3536 3637 3373  cpu); 0.0056673s
│ │ │ │ +000009b0: 2028 7468 7265 6164 293b 2030 7320 2867   (thread); 0s (g
│ │ │ │ +000009c0: 6329 2020 2020 2020 2020 2020 2020 2020  c)              
│ │ │ │  000009d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000009e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000009f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00000a20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00000a30: 2020 2020 2020 5a5a 2020 2020 2020 2020        ZZ        
│ │ │ │ @@ -322,16 +322,16 @@
│ │ │ │  00001410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001420: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074  -------+.|i3 : t
│ │ │ │  00001430: 696d 6520 4a20 3d20 6b65 726e 656c 2870  ime J = kernel(p
│ │ │ │  00001440: 6869 2c32 2920 2020 2020 2020 2020 2020  hi,2)           
│ │ │ │  00001450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001470: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001480: 6564 2030 2e30 3539 3136 3433 7320 2863  ed 0.0591643s (c
│ │ │ │ -00001490: 7075 293b 2030 2e30 3539 3137 3432 7320  pu); 0.0591742s 
│ │ │ │ +00001480: 6564 2030 2e30 3536 3730 3733 7320 2863  ed 0.0567073s (c
│ │ │ │ +00001490: 7075 293b 2030 2e30 3536 3730 3834 7320  pu); 0.0567084s 
│ │ │ │  000014a0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │  000014b0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000014d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000014e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000014f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -387,18 +387,18 @@
│ │ │ │  00001820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001830: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074  -------+.|i4 : t
│ │ │ │  00001840: 696d 6520 6465 6772 6565 4d61 7020 7068  ime degreeMap ph
│ │ │ │  00001850: 6920 2020 2020 2020 2020 2020 2020 2020  i               
│ │ │ │  00001860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001880: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001890: 6564 2030 2e30 3335 3431 3336 7320 2863  ed 0.0354136s (c
│ │ │ │ -000018a0: 7075 293b 2030 2e30 3335 3432 3033 7320  pu); 0.0354203s 
│ │ │ │ -000018b0: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -000018c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00001890: 6564 2030 2e30 3337 3238 3438 7320 2863  ed 0.0372848s (c
│ │ │ │ +000018a0: 7075 293b 2030 2e30 3337 3135 3773 2028  pu); 0.037157s (
│ │ │ │ +000018b0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ +000018c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000018d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000018e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000018f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001920: 2020 2020 2020 207c 0a7c 6f34 203d 2031         |.|o4 = 1
│ │ │ │  00001930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -412,16 +412,16 @@
│ │ │ │  000019b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000019c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074  -------+.|i5 : t
│ │ │ │  000019d0: 696d 6520 7072 6f6a 6563 7469 7665 4465  ime projectiveDe
│ │ │ │  000019e0: 6772 6565 7320 7068 6920 2020 2020 2020  grees phi       
│ │ │ │  000019f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a10: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001a20: 6564 2030 2e36 3833 3933 3873 2028 6370  ed 0.683938s (cp
│ │ │ │ -00001a30: 7529 3b20 302e 3533 3731 3236 7320 2874  u); 0.537126s (t
│ │ │ │ +00001a20: 6564 2030 2e36 3333 3139 3673 2028 6370  ed 0.633196s (cp
│ │ │ │ +00001a30: 7529 3b20 302e 3531 3339 3837 7320 2874  u); 0.513987s (t
│ │ │ │  00001a40: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00001a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a60: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -447,18 +447,18 @@
│ │ │ │  00001be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001bf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2074  -------+.|i6 : t
│ │ │ │  00001c00: 696d 6520 7072 6f6a 6563 7469 7665 4465  ime projectiveDe
│ │ │ │  00001c10: 6772 6565 7328 7068 692c 4e75 6d44 6567  grees(phi,NumDeg
│ │ │ │  00001c20: 7265 6573 3d3e 3029 2020 2020 2020 2020  rees=>0)        
│ │ │ │  00001c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001c40: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001c50: 6564 2030 2e32 3233 3032 3373 2028 6370  ed 0.223023s (cp
│ │ │ │ -00001c60: 7529 3b20 302e 3133 3734 3333 7320 2874  u); 0.137433s (t
│ │ │ │ -00001c70: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -00001c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00001c50: 6564 2030 2e30 3931 3435 3234 7320 2863  ed 0.0914524s (c
│ │ │ │ +00001c60: 7075 293b 2030 2e30 3931 3436 3235 7320  pu); 0.0914625s 
│ │ │ │ +00001c70: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00001c80: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00001c90: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ce0: 2020 2020 2020 207c 0a7c 6f36 203d 207b         |.|o6 = {
│ │ │ │  00001cf0: 357d 2020 2020 2020 2020 2020 2020 2020  5}              
│ │ │ │ @@ -478,20 +478,20 @@
│ │ │ │  00001dd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00001de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00001e20: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2074  -------+.|i7 : t
│ │ │ │  00001e30: 696d 6520 7068 6920 3d20 746f 4d61 7028  ime phi = toMap(
│ │ │ │ -00001e40: 7068 6920 2020 2020 2020 2020 2020 2020  phi             
│ │ │ │ +00001e40: 7068 692c 2020 2020 2020 2020 2020 2020  phi,            
│ │ │ │  00001e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001e70: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00001e80: 6564 2030 2e30 3033 3534 3130 3973 2028  ed 0.00354109s (
│ │ │ │ -00001e90: 6370 7520 2020 2020 2020 2020 2020 2020  cpu             
│ │ │ │ +00001e80: 6564 2030 2e30 3034 3138 3933 7320 2863  ed 0.0041893s (c
│ │ │ │ +00001e90: 7075 293b 2020 2020 2020 2020 2020 2020  pu);            
│ │ │ │  00001ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00001ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00001f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -548,130 +548,130 @@
│ │ │ │  00002230: 2020 2020 2020 207c 0a7c 6f37 203a 2052         |.|o7 : R
│ │ │ │  00002240: 696e 674d 6170 202d 2d2d 2d2d 2d5b 7420  ingMap ------[t 
│ │ │ │  00002250: 2e2e 7420 2020 2020 2020 2020 2020 2020  ..t             
│ │ │ │  00002260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002280: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002290: 2020 2020 2020 2033 3030 3030 3720 2030         300007  0
│ │ │ │ -000022a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000022a0: 2020 2036 2020 2020 2020 2020 2020 2020     6            
│ │ │ │  000022b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000022c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000022d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000022e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000022f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002320: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 2c44 6f6d 696e  -------|.|,Domin
│ │ │ │ -00002380: 616e 743d 3e4a 2920 2020 2020 2020 2020  ant=>J)         
│ │ │ │ +00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 446f 6d69 6e61  -------|.|Domina
│ │ │ │ +00002380: 6e74 3d3e 4a29 2020 2020 2020 2020 2020  nt=>J)          
│ │ │ │  00002390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000023a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000023b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000023c0: 2020 2020 2020 207c 0a7c 293b 2030 2e30         |.|); 0.0
│ │ │ │ -000023d0: 3033 3534 3233 3573 2028 7468 7265 6164  0354235s (thread
│ │ │ │ -000023e0: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │ +000023c0: 2020 2020 2020 207c 0a7c 2030 2e30 3034         |.| 0.004
│ │ │ │ +000023d0: 3139 3234 3273 2028 7468 7265 6164 293b  19242s (thread);
│ │ │ │ +000023e0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  000023f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002410: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002460: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002490: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +00002490: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │  000024a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000024b0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000024c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000024d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000024e0: 2020 2020 2020 2d2d 2d2d 2d2d 5b78 202e        ------[x .
│ │ │ │ -000024f0: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │ +000024e0: 2020 2020 202d 2d2d 2d2d 2d5b 7820 2e2e       ------[x ..
│ │ │ │ +000024f0: 7820 5d20 2020 2020 2020 2020 2020 2020  x ]             
│ │ │ │  00002500: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002530: 2020 2020 2020 3330 3030 3037 2020 3020        300007  0 
│ │ │ │ -00002540: 2020 3920 2020 2020 2020 2020 2020 2020    9             
│ │ │ │ -00002550: 2020 2020 2020 207c 0a7c 202d 2d2d 2d2d         |.| -----
│ │ │ │ +00002530: 2020 2020 2033 3030 3030 3720 2030 2020       300007  0  
│ │ │ │ +00002540: 2039 2020 2020 2020 2020 2020 2020 2020   9              
│ │ │ │ +00002550: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000025a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2028 7820 7820  -------|.| (x x 
│ │ │ │ -000025b0: 202d 2078 2078 2020 2b20 7820 7820 2c20   - x x  + x x , 
│ │ │ │ -000025c0: 7820 7820 202d 2078 2078 2020 2b20 7820  x x  - x x  + x 
│ │ │ │ -000025d0: 7820 2c20 7820 7820 202d 2078 2078 2020  x , x x  - x x  
│ │ │ │ -000025e0: 2b20 7820 7820 2c20 7820 7820 202d 2078  + x x , x x  - x
│ │ │ │ -000025f0: 2078 2020 2b20 787c 0a7c 2020 2036 2037   x  + x|.|   6 7
│ │ │ │ -00002600: 2020 2020 3520 3820 2020 2034 2039 2020      5 8    4 9  
│ │ │ │ -00002610: 2033 2037 2020 2020 3220 3820 2020 2031   3 7    2 8    1
│ │ │ │ -00002620: 2039 2020 2033 2035 2020 2020 3220 3620   9   3 5    2 6 
│ │ │ │ -00002630: 2020 2030 2039 2020 2033 2034 2020 2020     0 9   3 4    
│ │ │ │ -00002640: 3120 3620 2020 207c 0a7c 2020 2020 2020  1 6    |.|      
│ │ │ │ +000025a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2878 2078 2020  -------|.|(x x  
│ │ │ │ +000025b0: 2d20 7820 7820 202b 2078 2078 202c 2078  - x x  + x x , x
│ │ │ │ +000025c0: 2078 2020 2d20 7820 7820 202b 2078 2078   x  - x x  + x x
│ │ │ │ +000025d0: 202c 2078 2078 2020 2d20 7820 7820 202b   , x x  - x x  +
│ │ │ │ +000025e0: 2078 2078 202c 2078 2078 2020 2d20 7820   x x , x x  - x 
│ │ │ │ +000025f0: 7820 202b 2078 207c 0a7c 2020 3620 3720  x  + x |.|  6 7 
│ │ │ │ +00002600: 2020 2035 2038 2020 2020 3420 3920 2020     5 8    4 9   
│ │ │ │ +00002610: 3320 3720 2020 2032 2038 2020 2020 3120  3 7    2 8    1 
│ │ │ │ +00002620: 3920 2020 3320 3520 2020 2032 2036 2020  9   3 5    2 6  
│ │ │ │ +00002630: 2020 3020 3920 2020 3320 3420 2020 2031    0 9   3 4    1
│ │ │ │ +00002640: 2036 2020 2020 307c 0a7c 2020 2020 2020   6    0|.|      
│ │ │ │  00002650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002690: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000026a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000026b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000026c0: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │ +000026c0: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │  000026d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000026e0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000026f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002710: 2020 2020 2020 2020 2020 2020 2d2d 2d2d              ----
│ │ │ │ -00002720: 2d2d 5b78 202e 2e78 205d 2020 2020 2020  --[x ..x ]      
│ │ │ │ +00002710: 2020 2020 2020 2020 2020 202d 2d2d 2d2d             -----
│ │ │ │ +00002720: 2d5b 7820 2e2e 7820 5d20 2020 2020 2020  -[x ..x ]       
│ │ │ │  00002730: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002760: 2020 2020 2020 2020 2020 2020 3330 3030              3000
│ │ │ │ -00002770: 3037 2020 3020 2020 3920 2020 2020 2020  07  0   9       
│ │ │ │ -00002780: 2020 2020 2020 207c 0a7c 205d 203c 2d2d         |.| ] <--
│ │ │ │ -00002790: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   ---------------
│ │ │ │ +00002760: 2020 2020 2020 2020 2020 2033 3030 3030             30000
│ │ │ │ +00002770: 3720 2030 2020 2039 2020 2020 2020 2020  7  0   9        
│ │ │ │ +00002780: 2020 2020 2020 207c 0a7c 5d20 3c2d 2d20         |.|] <-- 
│ │ │ │ +00002790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000027a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000027b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000027c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000027d0: 2d2d 2d2d 2d2d 2d7c 0a7c 3620 2020 2020  -------|.|6     
│ │ │ │ -000027e0: 2028 7820 7820 202d 2078 2078 2020 2b20   (x x  - x x  + 
│ │ │ │ -000027f0: 7820 7820 2c20 7820 7820 202d 2078 2078  x x , x x  - x x
│ │ │ │ -00002800: 2020 2b20 7820 7820 2c20 7820 7820 202d    + x x , x x  -
│ │ │ │ -00002810: 2078 2078 2020 2b20 7820 7820 2c20 7820   x x  + x x , x 
│ │ │ │ -00002820: 7820 202d 2078 207c 0a7c 2020 2020 2020  x  - x |.|      
│ │ │ │ -00002830: 2020 2036 2037 2020 2020 3520 3820 2020     6 7    5 8   
│ │ │ │ -00002840: 2034 2039 2020 2033 2037 2020 2020 3220   4 9   3 7    2 
│ │ │ │ -00002850: 3820 2020 2031 2039 2020 2033 2035 2020  8    1 9   3 5  
│ │ │ │ -00002860: 2020 3220 3620 2020 2030 2039 2020 2033    2 6    0 9   3
│ │ │ │ -00002870: 2034 2020 2020 317c 0a7c 2d2d 2d2d 2d2d   4    1|.|------
│ │ │ │ +000027d0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │ +000027e0: 2878 2078 2020 2d20 7820 7820 202b 2078  (x x  - x x  + x
│ │ │ │ +000027f0: 2078 202c 2078 2078 2020 2d20 7820 7820   x , x x  - x x 
│ │ │ │ +00002800: 202b 2078 2078 202c 2078 2078 2020 2d20   + x x , x x  - 
│ │ │ │ +00002810: 7820 7820 202b 2078 2078 202c 2078 2078  x x  + x x , x x
│ │ │ │ +00002820: 2020 2d20 7820 787c 0a7c 2020 2020 2020    - x x|.|      
│ │ │ │ +00002830: 2020 3620 3720 2020 2035 2038 2020 2020    6 7    5 8    
│ │ │ │ +00002840: 3420 3920 2020 3320 3720 2020 2032 2038  4 9   3 7    2 8
│ │ │ │ +00002850: 2020 2020 3120 3920 2020 3320 3520 2020      1 9   3 5   
│ │ │ │ +00002860: 2032 2036 2020 2020 3020 3920 2020 3320   2 6    0 9   3 
│ │ │ │ +00002870: 3420 2020 2031 207c 0a7c 2d2d 2d2d 2d2d  4    1 |.|------
│ │ │ │  00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000028c0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2020  -------|.|      
│ │ │ │  000028d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000028e0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -000028f0: 2020 2020 2020 2020 2032 2020 2020 3220           2    2 
│ │ │ │ -00002900: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +000028e0: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +000028f0: 2020 2020 2020 2020 3220 2020 2032 2020          2    2  
│ │ │ │ +00002900: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00002910: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002930: 2d2d 2c20 7b2d 2074 2020 2b20 3274 2074  --, {- t  + 2t t
│ │ │ │ -00002940: 2074 2020 2d20 7420 7420 202d 2074 2074   t  - t t  - t t
│ │ │ │ -00002950: 2020 2b20 7420 7420 7420 2c20 2d20 7420    + t t t , - t 
│ │ │ │ -00002960: 7420 202b 2020 207c 0a7c 2078 202c 2078  t  +   |.| x , x
│ │ │ │ -00002970: 2078 2020 2d20 7820 7820 202b 2078 2078   x  - x x  + x x
│ │ │ │ -00002980: 2029 2020 2020 2020 3220 2020 2020 3120   )      2     1 
│ │ │ │ -00002990: 3220 3320 2020 2030 2033 2020 2020 3120  2 3    0 3    1 
│ │ │ │ -000029a0: 3420 2020 2030 2032 2034 2020 2020 2032  4    0 2 4     2
│ │ │ │ -000029b0: 2033 2020 2020 207c 0a7c 3020 3820 2020   3     |.|0 8   
│ │ │ │ -000029c0: 3220 3420 2020 2031 2035 2020 2020 3020  2 4    1 5    0 
│ │ │ │ -000029d0: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +00002930: 2d2c 207b 2d20 7420 202b 2032 7420 7420  -, {- t  + 2t t 
│ │ │ │ +00002940: 7420 202d 2074 2074 2020 2d20 7420 7420  t  - t t  - t t 
│ │ │ │ +00002950: 202b 2074 2074 2074 202c 202d 2074 2074   + t t t , - t t
│ │ │ │ +00002960: 2020 2b20 2020 207c 0a7c 7820 2c20 7820    +    |.|x , x 
│ │ │ │ +00002970: 7820 202d 2078 2078 2020 2b20 7820 7820  x  - x x  + x x 
│ │ │ │ +00002980: 2920 2020 2020 2032 2020 2020 2031 2032  )      2     1 2
│ │ │ │ +00002990: 2033 2020 2020 3020 3320 2020 2031 2034   3    0 3    1 4
│ │ │ │ +000029a0: 2020 2020 3020 3220 3420 2020 2020 3220      0 2 4     2 
│ │ │ │ +000029b0: 3320 2020 2020 207c 0a7c 2038 2020 2032  3      |.| 8   2
│ │ │ │ +000029c0: 2034 2020 2020 3120 3520 2020 2030 2037   4    1 5    0 7
│ │ │ │ +000029d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000029e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000029f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a00: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -688,25 +688,25 @@
│ │ │ │  00002af0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00002b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b40: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00002b60: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020  --------        
│ │ │ │ +00002b60: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020  -------         
│ │ │ │  00002b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002b90: 2020 2020 2020 207c 0a7c 7820 202b 2078         |.|x  + x
│ │ │ │ -00002ba0: 2078 202c 2078 2078 2020 2d20 7820 7820   x , x x  - x x 
│ │ │ │ -00002bb0: 202b 2078 2078 2029 2020 2020 2020 2020   + x x )        
│ │ │ │ +00002b90: 2020 2020 2020 207c 0a7c 2020 2b20 7820         |.|  + x 
│ │ │ │ +00002ba0: 7820 2c20 7820 7820 202d 2078 2078 2020  x , x x  - x x  
│ │ │ │ +00002bb0: 2b20 7820 7820 2920 2020 2020 2020 2020  + x x )         
│ │ │ │  00002bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00002be0: 2020 2020 2020 207c 0a7c 2036 2020 2020         |.| 6    
│ │ │ │ -00002bf0: 3020 3820 2020 3220 3420 2020 2031 2035  0 8   2 4    1 5
│ │ │ │ -00002c00: 2020 2020 3020 3720 2020 2020 2020 2020      0 7         
│ │ │ │ +00002be0: 2020 2020 2020 207c 0a7c 3620 2020 2030         |.|6    0
│ │ │ │ +00002bf0: 2038 2020 2032 2034 2020 2020 3120 3520   8   2 4    1 5 
│ │ │ │ +00002c00: 2020 2030 2037 2020 2020 2020 2020 2020     0 7          
│ │ │ │  00002c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00002c30: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d         |.|------
│ │ │ │  00002c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00002c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ @@ -832,16 +832,16 @@
│ │ │ │  000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074  -------+.|i8 : t
│ │ │ │  00003410: 696d 6520 7073 6920 3d20 696e 7665 7273  ime psi = invers
│ │ │ │  00003420: 654d 6170 2070 6869 2020 2020 2020 2020  eMap phi        
│ │ │ │  00003430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00003450: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00003460: 6564 2030 2e37 3433 3138 3373 2028 6370  ed 0.743183s (cp
│ │ │ │ -00003470: 7529 3b20 302e 3636 3231 3035 7320 2874  u); 0.662105s (t
│ │ │ │ +00003460: 6564 2030 2e36 3537 3035 3273 2028 6370  ed 0.657052s (cp
│ │ │ │ +00003470: 7529 3b20 302e 3534 3139 3539 7320 2874  u); 0.541959s (t
│ │ │ │  00003480: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  00003490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000034b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000034e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1117,18 +1117,18 @@
│ │ │ │  000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000045d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074  -------+.|i9 : t
│ │ │ │  000045e0: 696d 6520 6973 496e 7665 7273 654d 6170  ime isInverseMap
│ │ │ │  000045f0: 2870 6869 2c70 7369 2920 2020 2020 2020  (phi,psi)       
│ │ │ │  00004600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004620: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00004630: 6564 2030 2e30 3136 3435 3673 2028 6370  ed 0.016456s (cp
│ │ │ │ -00004640: 7529 3b20 302e 3031 3634 3537 3973 2028  u); 0.0164579s (
│ │ │ │ -00004650: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │ -00004660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00004630: 6564 2030 2e30 3131 3932 3339 7320 2863  ed 0.0119239s (c
│ │ │ │ +00004640: 7075 293b 2030 2e30 3131 3932 3836 7320  pu); 0.0119286s 
│ │ │ │ +00004650: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ +00004660: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00004670: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00004680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000046c0: 2020 2020 2020 207c 0a7c 6f39 203d 2074         |.|o9 = t
│ │ │ │  000046d0: 7275 6520 2020 2020 2020 2020 2020 2020  rue             
│ │ │ │ @@ -1142,16 +1142,16 @@
│ │ │ │  00004750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00004760: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20  -------+.|i10 : 
│ │ │ │  00004770: 7469 6d65 2064 6567 7265 654d 6170 2070  time degreeMap p
│ │ │ │  00004780: 7369 2020 2020 2020 2020 2020 2020 2020  si              
│ │ │ │  00004790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000047a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000047b0: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -000047c0: 6564 2030 2e33 3736 3530 3773 2028 6370  ed 0.376507s (cp
│ │ │ │ -000047d0: 7529 3b20 302e 3237 3433 3137 7320 2874  u); 0.274317s (t
│ │ │ │ +000047c0: 6564 2030 2e31 3939 3630 3373 2028 6370  ed 0.199603s (cp
│ │ │ │ +000047d0: 7529 3b20 302e 3139 3936 3131 7320 2874  u); 0.199611s (t
│ │ │ │  000047e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │  000047f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004800: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00004810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -1167,17 +1167,17 @@
│ │ │ │  000048e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20  -------+.|i11 : 
│ │ │ │  00004900: 7469 6d65 2070 726f 6a65 6374 6976 6544  time projectiveD
│ │ │ │  00004910: 6567 7265 6573 2070 7369 2020 2020 2020  egrees psi      
│ │ │ │  00004920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004940: 2020 2020 2020 207c 0a7c 202d 2d20 7573         |.| -- us
│ │ │ │ -00004950: 6564 2035 2e36 3532 3231 7320 2863 7075  ed 5.65221s (cpu
│ │ │ │ -00004960: 293b 2034 2e37 3931 3539 7320 2874 6872  ); 4.79159s (thr
│ │ │ │ -00004970: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ +00004950: 6564 2037 2e35 3238 3338 7320 2863 7075  ed 7.52838s (cpu
│ │ │ │ +00004960: 293b 2036 2e38 3631 3373 2028 7468 7265  ); 6.8613s (thre
│ │ │ │ +00004970: 6164 293b 2030 7320 2867 6329 2020 2020  ad); 0s (gc)    
│ │ │ │  00004980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004990: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  000049a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000049e0: 2020 2020 2020 207c 0a7c 6f31 3120 3d20         |.|o11 = 
│ │ │ │ @@ -1214,17 +1214,17 @@
│ │ │ │  00004bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00004be0: 0a7c 6931 3220 3a20 7469 6d65 2070 6869  .|i12 : time phi
│ │ │ │  00004bf0: 203d 2072 6174 696f 6e61 6c4d 6170 206d   = rationalMap m
│ │ │ │  00004c00: 696e 6f72 7328 332c 6d61 7472 6978 7b7b  inors(3,matrix{{
│ │ │ │  00004c10: 745f 302e 2e74 5f34 7d2c 7b74 5f31 2e2e  t_0..t_4},{t_1..
│ │ │ │  00004c20: 745f 357d 2c7b 745f 322e 2e74 5f36 207c  t_5},{t_2..t_6 |
│ │ │ │  00004c30: 0a7c 202d 2d20 7573 6564 2030 2e30 3032  .| -- used 0.002
│ │ │ │ -00004c40: 3234 3034 3973 2028 6370 7529 3b20 302e  24049s (cpu); 0.
│ │ │ │ -00004c50: 3030 3232 3431 3536 7320 2874 6872 6561  00224156s (threa
│ │ │ │ -00004c60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00004c40: 3735 3230 3773 2028 6370 7529 3b20 302e  75207s (cpu); 0.
│ │ │ │ +00004c50: 3030 3237 3538 3273 2028 7468 7265 6164  0027582s (thread
│ │ │ │ +00004c60: 293b 2030 7320 2867 6329 2020 2020 2020  ); 0s (gc)      
│ │ │ │  00004c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004c80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00004c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00004cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00004cd0: 0a7c 6f31 3220 3d20 2d2d 2072 6174 696f  .|o12 = -- ratio
│ │ │ │ @@ -1493,18 +1493,18 @@
│ │ │ │  00005d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00005d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  00005d60: 0a7c 6931 3320 3a20 7469 6d65 2070 6869  .|i13 : time phi
│ │ │ │  00005d70: 203d 2072 6174 696f 6e61 6c4d 6170 2870   = rationalMap(p
│ │ │ │  00005d80: 6869 2c44 6f6d 696e 616e 743d 3e32 2920  hi,Dominant=>2) 
│ │ │ │  00005d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3532  .| -- used 0.052
│ │ │ │ -00005dc0: 3338 3737 7320 2863 7075 293b 2030 2e30  3877s (cpu); 0.0
│ │ │ │ -00005dd0: 3532 3339 3733 7320 2874 6872 6561 6429  523973s (thread)
│ │ │ │ -00005de0: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3735  .| -- used 0.075
│ │ │ │ +00005dc0: 3738 3938 7320 2863 7075 293b 2030 2e30  7898s (cpu); 0.0
│ │ │ │ +00005dd0: 3735 3739 3173 2028 7468 7265 6164 293b  75791s (thread);
│ │ │ │ +00005de0: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  00005df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00005e00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00005e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00005e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00005e50: 0a7c 6f31 3320 3d20 2d2d 2072 6174 696f  .|o13 = -- ratio
│ │ │ │ @@ -2153,17 +2153,17 @@
│ │ │ │  00008680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00008690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  000086a0: 0a7c 6931 3420 3a20 7469 6d65 2070 6869  .|i14 : time phi
│ │ │ │  000086b0: 5e28 2d31 2920 2020 2020 2020 2020 2020  ^(-1)           
│ │ │ │  000086c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000086d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000086e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000086f0: 0a7c 202d 2d20 7573 6564 2030 2e35 3836  .| -- used 0.586
│ │ │ │ -00008700: 3234 3173 2028 6370 7529 3b20 302e 3434  241s (cpu); 0.44
│ │ │ │ -00008710: 3634 3031 7320 2874 6872 6561 6429 3b20  6401s (thread); 
│ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e36 3939  .| -- used 0.699
│ │ │ │ +00008700: 3235 3473 2028 6370 7529 3b20 302e 3539  254s (cpu); 0.59
│ │ │ │ +00008710: 3138 3338 7320 2874 6872 6561 6429 3b20  1838s (thread); 
│ │ │ │  00008720: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  00008730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  00008740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00008750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00008780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2708,17 +2708,17 @@
│ │ │ │  0000a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000a950: 0a7c 6931 3520 3a20 7469 6d65 2064 6567  .|i15 : time deg
│ │ │ │  0000a960: 7265 6573 2070 6869 5e28 2d31 2920 2020  rees phi^(-1)   
│ │ │ │  0000a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000a990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e36 3234  .| -- used 0.624
│ │ │ │ -0000a9b0: 3830 3973 2028 6370 7529 3b20 302e 3436  809s (cpu); 0.46
│ │ │ │ -0000a9c0: 3030 3832 7320 2874 6872 6561 6429 3b20  0082s (thread); 
│ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e33 3634  .| -- used 0.364
│ │ │ │ +0000a9b0: 3033 3373 2028 6370 7529 3b20 302e 3331  033s (cpu); 0.31
│ │ │ │ +0000a9c0: 3237 3033 7320 2874 6872 6561 6429 3b20  2703s (thread); 
│ │ │ │  0000a9d0: 3073 2028 6763 2920 2020 2020 2020 2020  0s (gc)         
│ │ │ │  0000a9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000a9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000aa30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2743,18 +2743,18 @@
│ │ │ │  0000ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000ab80: 0a7c 6931 3620 3a20 7469 6d65 2064 6567  .|i16 : time deg
│ │ │ │  0000ab90: 7265 6573 2070 6869 2020 2020 2020 2020  rees phi        
│ │ │ │  0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000abc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3235  .| -- used 0.025
│ │ │ │ -0000abe0: 3737 3536 7320 2863 7075 293b 2030 2e30  7756s (cpu); 0.0
│ │ │ │ -0000abf0: 3235 3232 3638 7320 2874 6872 6561 6429  252268s (thread)
│ │ │ │ -0000ac00: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3835  .| -- used 0.085
│ │ │ │ +0000abe0: 3034 3273 2028 6370 7529 3b20 302e 3032  042s (cpu); 0.02
│ │ │ │ +0000abf0: 3638 3039 3473 2028 7468 7265 6164 293b  68094s (thread);
│ │ │ │ +0000ac00: 2030 7320 2867 6329 2020 2020 2020 2020   0s (gc)        
│ │ │ │  0000ac10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ac20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ac60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ac70: 0a7c 6f31 3620 3d20 7b31 2c20 332c 2039  .|o16 = {1, 3, 9
│ │ │ │ @@ -2779,16 +2779,16 @@
│ │ │ │  0000ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000adb0: 0a7c 6931 3720 3a20 7469 6d65 2064 6573  .|i17 : time des
│ │ │ │  0000adc0: 6372 6962 6520 7068 6920 2020 2020 2020  cribe phi       
│ │ │ │  0000add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000adf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3034  .| -- used 0.004
│ │ │ │ -0000ae10: 3539 3032 3373 2028 6370 7529 3b20 302e  59023s (cpu); 0.
│ │ │ │ -0000ae20: 3030 3435 3930 3732 7320 2874 6872 6561  00459072s (threa
│ │ │ │ +0000ae10: 3533 3337 3273 2028 6370 7529 3b20 302e  53372s (cpu); 0.
│ │ │ │ +0000ae20: 3030 3435 3431 3039 7320 2874 6872 6561  00454109s (threa
│ │ │ │  0000ae30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0000ae40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000ae50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000ae90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2844,16 +2844,16 @@
│ │ │ │  0000b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b1c0: 0a7c 6931 3820 3a20 7469 6d65 2064 6573  .|i18 : time des
│ │ │ │  0000b1d0: 6372 6962 6520 7068 695e 282d 3129 2020  cribe phi^(-1)  
│ │ │ │  0000b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3133  .| -- used 0.013
│ │ │ │ -0000b220: 3830 3435 7320 2863 7075 293b 2030 2e30  8045s (cpu); 0.0
│ │ │ │ -0000b230: 3133 3830 3739 7320 2874 6872 6561 6429  138079s (thread)
│ │ │ │ +0000b220: 3036 3938 7320 2863 7075 293b 2030 2e30  0698s (cpu); 0.0
│ │ │ │ +0000b230: 3133 3037 3838 7320 2874 6872 6561 6429  130788s (thread)
│ │ │ │  0000b240: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0000b250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2924,16 +2924,16 @@
│ │ │ │  0000b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b6c0: 0a7c 6931 3920 3a20 7469 6d65 2028 662c  .|i19 : time (f,
│ │ │ │  0000b6d0: 6729 203d 2067 7261 7068 2070 6869 5e2d  g) = graph phi^-
│ │ │ │  0000b6e0: 313b 2066 3b20 2020 2020 2020 2020 2020  1; f;           
│ │ │ │  0000b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3133  .| -- used 0.013
│ │ │ │ -0000b720: 3231 3636 7320 2863 7075 293b 2030 2e30  2166s (cpu); 0.0
│ │ │ │ -0000b730: 3133 3139 3036 7320 2874 6872 6561 6429  131906s (thread)
│ │ │ │ +0000b720: 3234 3733 7320 2863 7075 293b 2030 2e30  2473s (cpu); 0.0
│ │ │ │ +0000b730: 3133 3235 3437 7320 2874 6872 6561 6429  132547s (thread)
│ │ │ │  0000b740: 3b20 3073 2028 6763 2920 2020 2020 2020  ; 0s (gc)       
│ │ │ │  0000b750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b7a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2958,17 +2958,17 @@
│ │ │ │  0000b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000b8f0: 0a7c 6932 3120 3a20 7469 6d65 2064 6567  .|i21 : time deg
│ │ │ │  0000b900: 7265 6573 2066 2020 2020 2020 2020 2020  rees f          
│ │ │ │  0000b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000b940: 0a7c 202d 2d20 7573 6564 2031 2e36 3033  .| -- used 1.603
│ │ │ │ -0000b950: 3732 7320 2863 7075 293b 2031 2e31 3831  72s (cpu); 1.181
│ │ │ │ -0000b960: 3134 7320 2874 6872 6561 6429 3b20 3073  14s (thread); 0s
│ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e35 3137  .| -- used 1.517
│ │ │ │ +0000b950: 3939 7320 2863 7075 293b 2031 2e31 3435  99s (cpu); 1.145
│ │ │ │ +0000b960: 3039 7320 2874 6872 6561 6429 3b20 3073  09s (thread); 0s
│ │ │ │  0000b970: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │  0000b980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000b990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000b9d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -2993,17 +2993,17 @@
│ │ │ │  0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000bb20: 0a7c 6932 3220 3a20 7469 6d65 2064 6567  .|i22 : time deg
│ │ │ │  0000bb30: 7265 6520 6620 2020 2020 2020 2020 2020  ree f           
│ │ │ │  0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bb60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0000bb70: 0a7c 202d 2d20 7573 6564 2032 2e30 3136  .| -- used 2.016
│ │ │ │ -0000bb80: 3865 2d30 3573 2028 6370 7529 3b20 312e  8e-05s (cpu); 1.
│ │ │ │ -0000bb90: 3936 3036 652d 3035 7320 2874 6872 6561  9606e-05s (threa
│ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2031 2e37 3736  .| -- used 1.776
│ │ │ │ +0000bb80: 3165 2d30 3573 2028 6370 7529 3b20 312e  1e-05s (cpu); 1.
│ │ │ │ +0000bb90: 3732 3838 652d 3035 7320 2874 6872 6561  7288e-05s (threa
│ │ │ │  0000bba0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0000bbb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bc00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -3019,16 +3019,16 @@
│ │ │ │  0000bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │  0000bcb0: 0a7c 6932 3320 3a20 7469 6d65 2064 6573  .|i23 : time des
│ │ │ │  0000bcc0: 6372 6962 6520 6620 2020 2020 2020 2020  cribe f         
│ │ │ │  0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bcf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bd00: 0a7c 202d 2d20 7573 6564 2030 2e30 3031  .| -- used 0.001
│ │ │ │ -0000bd10: 3731 3638 3673 2028 6370 7529 3b20 302e  71686s (cpu); 0.
│ │ │ │ -0000bd20: 3030 3137 3137 3935 7320 2874 6872 6561  00171795s (threa
│ │ │ │ +0000bd10: 3737 3034 3773 2028 6370 7529 3b20 302e  77047s (cpu); 0.
│ │ │ │ +0000bd20: 3030 3137 3734 3832 7320 2874 6872 6561  00177482s (threa
│ │ │ │  0000bd30: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │  0000bd40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │  0000bd50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  0000bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0000bd90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ @@ -4676,16 +4676,16 @@
│ │ │ │  00012430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00012440: 2b0a 7c69 3420 3a20 7469 6d65 2070 7369  +.|i4 : time psi
│ │ │ │  00012450: 203d 2061 6273 7472 6163 7452 6174 696f   = abstractRatio
│ │ │ │  00012460: 6e61 6c4d 6170 2850 342c 5035 2c66 2920  nalMap(P4,P5,f) 
│ │ │ │  00012470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012490: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030  |.| -- used 0.00
│ │ │ │ -000124a0: 3034 3736 3832 3573 2028 6370 7529 3b20  0476825s (cpu); 
│ │ │ │ -000124b0: 302e 3030 3034 3732 3530 3673 2028 7468  0.000472506s (th
│ │ │ │ +000124a0: 3035 3138 3137 3173 2028 6370 7529 3b20  0518171s (cpu); 
│ │ │ │ +000124b0: 302e 3030 3035 3131 3237 3373 2028 7468  0.000511273s (th
│ │ │ │  000124c0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │  000124d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000124e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │  000124f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4746,18 +4746,18 @@
│ │ │ │  00012890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000128a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000128b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │  000128c0: 6935 203a 2074 696d 6520 7072 6f6a 6563  i5 : time projec
│ │ │ │  000128d0: 7469 7665 4465 6772 6565 7328 7073 692c  tiveDegrees(psi,
│ │ │ │  000128e0: 3329 2020 2020 2020 2020 2020 2020 2020  3)              
│ │ │ │  000128f0: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -00012900: 2d20 7573 6564 2030 2e31 3139 3473 2028  - used 0.1194s (
│ │ │ │ -00012910: 6370 7529 3b20 302e 3131 3934 3233 7320  cpu); 0.119423s 
│ │ │ │ -00012920: 2874 6872 6561 6429 3b20 3073 2028 6763  (thread); 0s (gc
│ │ │ │ -00012930: 2920 2020 2020 2020 207c 0a7c 2020 2020  )        |.|    
│ │ │ │ +00012900: 2d20 7573 6564 2030 2e31 3530 3232 3773  - used 0.150227s
│ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3135 3032 3334   (cpu); 0.150234
│ │ │ │ +00012920: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +00012930: 6763 2920 2020 2020 207c 0a7c 2020 2020  gc)      |.|    
│ │ │ │  00012940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012970: 2020 2020 2020 207c 0a7c 6f35 203d 2032         |.|o5 = 2
│ │ │ │  00012980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000129a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -4766,16 +4766,16 @@
│ │ │ │  000129d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000129e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000129f0: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520  ---+.|i6 : time 
│ │ │ │  00012a00: 7261 7469 6f6e 616c 4d61 7020 7073 6920  rationalMap psi 
│ │ │ │  00012a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e35   |.| -- used 0.5
│ │ │ │ -00012a40: 3332 3736 3973 2028 6370 7529 3b20 302e  32769s (cpu); 0.
│ │ │ │ -00012a50: 3430 3139 3233 7320 2874 6872 6561 6429  401923s (thread)
│ │ │ │ +00012a40: 3935 3439 3973 2028 6370 7529 3b20 302e  95499s (cpu); 0.
│ │ │ │ +00012a50: 3437 3532 3933 7320 2874 6872 6561 6429  475293s (thread)
│ │ │ │  00012a60: 3b20 3073 2028 6763 2920 2020 2020 207c  ; 0s (gc)      |
│ │ │ │  00012a70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │  00012a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00012aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  00012ab0: 6f36 203d 202d 2d20 7261 7469 6f6e 616c  o6 = -- rational
│ │ │ │  00012ac0: 206d 6170 202d 2d20 2020 2020 2020 2020   map --         
│ │ │ │ @@ -5189,16 +5189,16 @@
│ │ │ │  00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a  --------+.|i14 :
│ │ │ │  00014460: 2074 696d 6520 5420 3d20 6162 7374 7261   time T = abstra
│ │ │ │  00014470: 6374 5261 7469 6f6e 616c 4d61 7028 492c  ctRationalMap(I,
│ │ │ │  00014480: 224f 4144 5022 2920 2020 2020 2020 2020  "OADP")         
│ │ │ │  00014490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144a0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -000144b0: 6420 302e 3033 3833 3137 3673 2028 6370  d 0.0383176s (cp
│ │ │ │ -000144c0: 7529 3b20 302e 3033 3833 3230 3273 2028  u); 0.0383202s (
│ │ │ │ +000144b0: 6420 302e 3035 3036 3436 3673 2028 6370  d 0.0506466s (cp
│ │ │ │ +000144c0: 7529 3b20 302e 3035 3036 3435 3773 2028  u); 0.0506457s (
│ │ │ │  000144d0: 7468 7265 6164 293b 2030 7320 2867 6329  thread); 0s (gc)
│ │ │ │  000144e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000144f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  00014500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ @@ -5261,48252 +5261,48253 @@
│ │ │ │  000148c0: 7468 6520 6162 7374 7261 6374 206d 6170  the abstract map
│ │ │ │  000148d0: 2054 2063 616e 2062 6520 6f62 7461 696e   T can be obtain
│ │ │ │  000148e0: 6564 2062 7920 7468 650a 666f 6c6c 6f77  ed by the.follow
│ │ │ │  000148f0: 696e 6720 636f 6d6d 616e 643a 0a0a 2b2d  ing command:..+-
│ │ │ │  00014900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014930: 2d2b 0a7c 6931 3520 3a20 7469 6d65 2070  -+.|i15 : time p
│ │ │ │ -00014940: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -00014950: 2854 2c32 2920 2020 2020 2020 2020 2020  (T,2)           
│ │ │ │ -00014960: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -00014970: 6420 322e 3138 3636 3373 2028 6370 7529  d 2.18663s (cpu)
│ │ │ │ -00014980: 3b20 312e 3538 3437 7320 2874 6872 6561  ; 1.5847s (threa
│ │ │ │ -00014990: 6429 3b20 3073 2028 6763 297c 0a7c 2020  d); 0s (gc)|.|  
│ │ │ │ -000149a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014930: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520  --+.|i15 : time 
│ │ │ │ +00014940: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ +00014950: 7328 542c 3229 2020 2020 2020 2020 2020  s(T,2)          
│ │ │ │ +00014960: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +00014970: 7365 6420 332e 3037 3731 3473 2028 6370  sed 3.07714s (cp
│ │ │ │ +00014980: 7529 3b20 322e 3230 3137 3373 2028 7468  u); 2.20173s (th
│ │ │ │ +00014990: 7265 6164 293b 2030 7320 2867 6329 7c0a  read); 0s (gc)|.
│ │ │ │ +000149a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000149b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000149d0: 7c0a 7c6f 3135 203d 2033 2020 2020 2020  |.|o15 = 3      
│ │ │ │ +000149d0: 2020 2020 7c0a 7c6f 3135 203d 2033 2020      |.|o15 = 3  
│ │ │ │  000149e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000149f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014a00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00014a00: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00014a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520  ----------+..We 
│ │ │ │ -00014a40: 7665 7269 6679 2074 6861 7420 7468 6520  verify that the 
│ │ │ │ -00014a50: 636f 6d70 6f73 6974 696f 6e20 6f66 2054  composition of T
│ │ │ │ -00014a60: 2077 6974 6820 6974 7365 6c66 2069 7320   with itself is 
│ │ │ │ -00014a70: 6465 6669 6e65 6420 6279 206c 696e 6561  defined by linea
│ │ │ │ -00014a80: 7220 666f 726d 733a 0a0a 2b2d 2d2d 2d2d  r forms:..+-----
│ │ │ │ -00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014a40: 2b0a 0a57 6520 7665 7269 6679 2074 6861  +..We verify tha
│ │ │ │ +00014a50: 7420 7468 6520 636f 6d70 6f73 6974 696f  t the compositio
│ │ │ │ +00014a60: 6e20 6f66 2054 2077 6974 6820 6974 7365  n of T with itse
│ │ │ │ +00014a70: 6c66 2069 7320 6465 6669 6e65 6420 6279  lf is defined by
│ │ │ │ +00014a80: 206c 696e 6561 7220 666f 726d 733a 0a0a   linear forms:..
│ │ │ │ +00014a90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │  00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014ac0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 7469  -----+.|i16 : ti
│ │ │ │ -00014ad0: 6d65 2054 3220 3d20 5420 2a20 5420 2020  me T2 = T * T   
│ │ │ │ -00014ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +00014ad0: 3620 3a20 7469 6d65 2054 3220 3d20 5420  6 : time T2 = T 
│ │ │ │ +00014ae0: 2a20 5420 2020 2020 2020 2020 2020 2020  * T             
│ │ │ │  00014af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b00: 2020 7c0a 7c20 2d2d 2075 7365 6420 332e    |.| -- used 3.
│ │ │ │ -00014b10: 3030 3037 652d 3035 7320 2863 7075 293b  0007e-05s (cpu);
│ │ │ │ -00014b20: 2032 2e39 3730 3565 2d30 3573 2028 7468   2.9705e-05s (th
│ │ │ │ -00014b30: 7265 6164 293b 2030 7320 2867 6329 207c  read); 0s (gc) |
│ │ │ │ -00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014b00: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ +00014b10: 7365 6420 332e 3438 3036 652d 3035 7320  sed 3.4806e-05s 
│ │ │ │ +00014b20: 2863 7075 293b 2033 2e32 3532 3365 2d30  (cpu); 3.2523e-0
│ │ │ │ +00014b30: 3573 2028 7468 7265 6164 293b 2030 7320  5s (thread); 0s 
│ │ │ │ +00014b40: 2867 6329 207c 0a7c 2020 2020 2020 2020  (gc) |.|        
│ │ │ │  00014b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00014b80: 3136 203d 202d 2d20 7261 7469 6f6e 616c  16 = -- rational
│ │ │ │ -00014b90: 206d 6170 202d 2d20 2020 2020 2020 2020   map --         
│ │ │ │ +00014b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014b80: 2020 7c0a 7c6f 3136 203d 202d 2d20 7261    |.|o16 = -- ra
│ │ │ │ +00014b90: 7469 6f6e 616c 206d 6170 202d 2d20 2020  tional map --   
│ │ │ │  00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bb0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bd0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │ +00014bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00014bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00014bd0: 2020 2020 2020 205a 5a20 2020 2020 2020         ZZ       
│ │ │ │  00014be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014bf0: 2020 2020 2020 7c0a 7c20 2020 2020 2073        |.|      s
│ │ │ │ -00014c00: 6f75 7263 653a 2050 726f 6a28 2d2d 2d2d  ource: Proj(----
│ │ │ │ -00014c10: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ -00014c20: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ -00014c30: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00014c40: 2020 2020 2020 2020 2036 3535 3231 2020           65521  
│ │ │ │ -00014c50: 3020 2020 3120 2020 3220 2020 3320 2020  0   1   2   3   
│ │ │ │ -00014c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014c70: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ -00014c80: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +00014bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00014c00: 2020 2020 2073 6f75 7263 653a 2050 726f       source: Pro
│ │ │ │ +00014c10: 6a28 2d2d 2d2d 2d5b 7820 2c20 7820 2c20  j(-----[x , x , 
│ │ │ │ +00014c20: 7820 2c20 7820 5d29 2020 2020 2020 2020  x , x ])        
│ │ │ │ +00014c30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00014c40: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ +00014c50: 3535 3231 2020 3020 2020 3120 2020 3220  5521  0   1   2 
│ │ │ │ +00014c60: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +00014c70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00014c80: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │  00014c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00014cb0: 2020 2020 2020 7461 7267 6574 3a20 5072        target: Pr
│ │ │ │ -00014cc0: 6f6a 282d 2d2d 2d2d 5b78 202c 2078 202c  oj(-----[x , x ,
│ │ │ │ -00014cd0: 2078 202c 2078 205d 2920 2020 2020 2020   x , x ])       
│ │ │ │ -00014ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │ -00014cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014d00: 3635 3532 3120 2030 2020 2031 2020 2032  65521  0   1   2
│ │ │ │ -00014d10: 2020 2033 2020 2020 2020 2020 2020 2020     3            
│ │ │ │ -00014d20: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00014d30: 6465 6669 6e69 6e67 2066 6f72 6d73 3a20  defining forms: 
│ │ │ │ -00014d40: 6769 7665 6e20 6279 2061 2066 756e 6374  given by a funct
│ │ │ │ -00014d50: 696f 6e20 2020 2020 2020 2020 2020 2020  ion             
│ │ │ │ -00014d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │ +00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014cb0: 2020 207c 0a7c 2020 2020 2020 7461 7267     |.|      targ
│ │ │ │ +00014cc0: 6574 3a20 5072 6f6a 282d 2d2d 2d2d 5b78  et: Proj(-----[x
│ │ │ │ +00014cd0: 202c 2078 202c 2078 202c 2078 205d 2920   , x , x , x ]) 
│ │ │ │ +00014ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014cf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020  |.|             
│ │ │ │ +00014d00: 2020 2020 2020 3635 3532 3120 2030 2020        65521  0  
│ │ │ │ +00014d10: 2031 2020 2032 2020 2033 2020 2020 2020   1   2   3      
│ │ │ │ +00014d20: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00014d30: 2020 2020 2020 6465 6669 6e69 6e67 2066        defining f
│ │ │ │ +00014d40: 6f72 6d73 3a20 6769 7665 6e20 6279 2061  orms: given by a
│ │ │ │ +00014d50: 2066 756e 6374 696f 6e20 2020 2020 2020   function       
│ │ │ │ +00014d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020            |.|   
│ │ │ │  00014d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014da0: 207c 0a7c 6f31 3620 3a20 4162 7374 7261   |.|o16 : Abstra
│ │ │ │ -00014db0: 6374 5261 7469 6f6e 616c 4d61 7020 2872  ctRationalMap (r
│ │ │ │ -00014dc0: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d  ational map from
│ │ │ │ -00014dd0: 2050 505e 3320 746f 2050 505e 3329 7c0a   PP^3 to PP^3)|.
│ │ │ │ -00014de0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  +---------------
│ │ │ │ +00014da0: 2020 2020 2020 207c 0a7c 6f31 3620 3a20         |.|o16 : 
│ │ │ │ +00014db0: 4162 7374 7261 6374 5261 7469 6f6e 616c  AbstractRational
│ │ │ │ +00014dc0: 4d61 7020 2872 6174 696f 6e61 6c20 6d61  Map (rational ma
│ │ │ │ +00014dd0: 7020 6672 6f6d 2050 505e 3320 746f 2050  p from PP^3 to P
│ │ │ │ +00014de0: 505e 3329 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d  P^3)|.+---------
│ │ │ │  00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00014e20: 3720 3a20 7469 6d65 2070 726f 6a65 6374  7 : time project
│ │ │ │ -00014e30: 6976 6544 6567 7265 6573 2854 322c 3229  iveDegrees(T2,2)
│ │ │ │ -00014e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014e50: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075          |.| -- u
│ │ │ │ -00014e60: 7365 6420 332e 3637 3932 3873 2028 6370  sed 3.67928s (cp
│ │ │ │ -00014e70: 7529 3b20 322e 3630 3139 7320 2874 6872  u); 2.6019s (thr
│ │ │ │ -00014e80: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -00014e90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014e20: 2d2b 0a7c 6931 3720 3a20 7469 6d65 2070  -+.|i17 : time p
│ │ │ │ +00014e30: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ +00014e40: 2854 322c 3229 2020 2020 2020 2020 2020  (T2,2)          
│ │ │ │ +00014e50: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00014e60: 7c20 2d2d 2075 7365 6420 342e 3833 3436  | -- used 4.8346
│ │ │ │ +00014e70: 3773 2028 6370 7529 3b20 332e 3532 3339  7s (cpu); 3.5239
│ │ │ │ +00014e80: 3973 2028 7468 7265 6164 293b 2030 7320  9s (thread); 0s 
│ │ │ │ +00014e90: 2867 6329 2020 2020 2020 207c 0a7c 2020  (gc)       |.|  
│ │ │ │  00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014ed0: 2020 7c0a 7c6f 3137 203d 2031 2020 2020    |.|o17 = 1    
│ │ │ │ -00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014ed0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d          |.|o17 =
│ │ │ │ +00014ee0: 2031 2020 2020 2020 2020 2020 2020 2020   1              
│ │ │ │  00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00014f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00014f10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00014f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00014f10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00014f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57  ------------+..W
│ │ │ │ -00014f50: 6520 7665 7269 6679 2074 6861 7420 7468  e verify that th
│ │ │ │ -00014f60: 6520 636f 6d70 6f73 6974 696f 6e20 6f66  e composition of
│ │ │ │ -00014f70: 2054 2077 6974 6820 6974 7365 6c66 206c   T with itself l
│ │ │ │ -00014f80: 6561 7665 7320 6120 7261 6e64 6f6d 2070  eaves a random p
│ │ │ │ -00014f90: 6f69 6e74 2066 6978 6564 3a0a 0a2b 2d2d  oint fixed:..+--
│ │ │ │ -00014fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00014f50: 2d2d 2b0a 0a57 6520 7665 7269 6679 2074  --+..We verify t
│ │ │ │ +00014f60: 6861 7420 7468 6520 636f 6d70 6f73 6974  hat the composit
│ │ │ │ +00014f70: 696f 6e20 6f66 2054 2077 6974 6820 6974  ion of T with it
│ │ │ │ +00014f80: 7365 6c66 206c 6561 7665 7320 6120 7261  self leaves a ra
│ │ │ │ +00014f90: 6e64 6f6d 2070 6f69 6e74 2066 6978 6564  ndom point fixed
│ │ │ │ +00014fa0: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :..+------------
│ │ │ │  00014fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00014fc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a  --------+.|i18 :
│ │ │ │ -00014fd0: 2070 203d 2061 7070 6c79 2833 2c69 2d3e   p = apply(3,i->
│ │ │ │ -00014fe0: 7261 6e64 6f6d 285a 5a2f 3635 3532 3129  random(ZZ/65521)
│ │ │ │ -00014ff0: 297c 7b31 7d7c 0a7c 2020 2020 2020 2020  )|{1}|.|        
│ │ │ │ +00014fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00014fd0: 7c69 3138 203a 2070 203d 2061 7070 6c79  |i18 : p = apply
│ │ │ │ +00014fe0: 2833 2c69 2d3e 7261 6e64 6f6d 285a 5a2f  (3,i->random(ZZ/
│ │ │ │ +00014ff0: 3635 3532 3129 297c 7b31 7d7c 0a7c 2020  65521))|{1}|.|  
│ │ │ │  00015000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015020: 2020 7c0a 7c6f 3138 203d 207b 2d36 3634    |.|o18 = {-664
│ │ │ │ -00015030: 382c 202d 3233 3339 362c 202d 3132 3331  8, -23396, -1231
│ │ │ │ -00015040: 312c 2031 7d20 2020 2020 2020 2020 207c  1, 1}          |
│ │ │ │ -00015050: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015020: 2020 2020 2020 2020 7c0a 7c6f 3138 203d          |.|o18 =
│ │ │ │ +00015030: 207b 2d36 3634 382c 202d 3233 3339 362c   {-6648, -23396,
│ │ │ │ +00015040: 202d 3132 3331 312c 2031 7d20 2020 2020   -12311, 1}     
│ │ │ │ +00015050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00015060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00015080: 3138 203a 204c 6973 7420 2020 2020 2020  18 : List       
│ │ │ │ +00015070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015080: 2020 7c0a 7c6f 3138 203a 204c 6973 7420    |.|o18 : List 
│ │ │ │  00015090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000150a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000150b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000150b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000150c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000150d0: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2071  ------+.|i19 : q
│ │ │ │ -000150e0: 203d 2054 2070 2020 2020 2020 2020 2020   = T p          
│ │ │ │ +000150d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000150e0: 3139 203a 2071 203d 2054 2070 2020 2020  19 : q = T p    
│ │ │ │  000150f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015100: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015100: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00015110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015130: 7c0a 7c6f 3139 203d 207b 2d39 3633 342c  |.|o19 = {-9634,
│ │ │ │ -00015140: 2032 3037 3034 2c20 2d32 3530 3134 2c20   20704, -25014, 
│ │ │ │ -00015150: 317d 2020 2020 2020 2020 2020 207c 0a7c  1}           |.|
│ │ │ │ -00015160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015130: 2020 2020 2020 7c0a 7c6f 3139 203d 207b        |.|o19 = {
│ │ │ │ +00015140: 2d39 3633 342c 2032 3037 3034 2c20 2d32  -9634, 20704, -2
│ │ │ │ +00015150: 3530 3134 2c20 317d 2020 2020 2020 2020  5014, 1}        
│ │ │ │ +00015160: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015180: 2020 2020 2020 2020 2020 7c0a 7c6f 3139            |.|o19
│ │ │ │ -00015190: 203a 204c 6973 7420 2020 2020 2020 2020   : List         
│ │ │ │ +00015180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015190: 7c0a 7c6f 3139 203a 204c 6973 7420 2020  |.|o19 : List   
│ │ │ │  000151a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000151b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │ +000151b0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │  000151c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000151d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000151e0: 2d2d 2d2d 2b0a 7c69 3230 203a 2054 2071  ----+.|i20 : T q
│ │ │ │ -000151f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230  ----------+.|i20
│ │ │ │ +000151f0: 203a 2054 2071 2020 2020 2020 2020 2020   : T q          
│ │ │ │  00015200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015210: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015210: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00015220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015230: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00015240: 7c6f 3230 203d 207b 2d36 3634 382c 202d  |o20 = {-6648, -
│ │ │ │ -00015250: 3233 3339 362c 202d 3132 3331 312c 2031  23396, -12311, 1
│ │ │ │ -00015260: 7d20 2020 2020 2020 2020 207c 0a7c 2020  }          |.|  
│ │ │ │ -00015270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015240: 2020 2020 7c0a 7c6f 3230 203d 207b 2d36      |.|o20 = {-6
│ │ │ │ +00015250: 3634 382c 202d 3233 3339 362c 202d 3132  648, -23396, -12
│ │ │ │ +00015260: 3331 312c 2031 7d20 2020 2020 2020 2020  311, 1}         
│ │ │ │ +00015270: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00015280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015290: 2020 2020 2020 2020 7c0a 7c6f 3230 203a          |.|o20 :
│ │ │ │ -000152a0: 204c 6973 7420 2020 2020 2020 2020 2020   List           
│ │ │ │ +00015290: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +000152a0: 7c6f 3230 203a 204c 6973 7420 2020 2020  |o20 : List     
│ │ │ │  000152b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000152c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000152c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  000152d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000152e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000152f0: 2d2d 2b0a 0a57 6520 6e6f 7720 636f 6d70  --+..We now comp
│ │ │ │ -00015300: 7574 6520 7468 6520 636f 6e63 7265 7465  ute the concrete
│ │ │ │ -00015310: 2072 6174 696f 6e61 6c20 6d61 7020 636f   rational map co
│ │ │ │ -00015320: 7272 6573 706f 6e64 696e 6720 746f 2054  rresponding to T
│ │ │ │ -00015330: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  :..+------------
│ │ │ │ +000152f0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6520 6e6f  --------+..We no
│ │ │ │ +00015300: 7720 636f 6d70 7574 6520 7468 6520 636f  w compute the co
│ │ │ │ +00015310: 6e63 7265 7465 2072 6174 696f 6e61 6c20  ncrete rational 
│ │ │ │ +00015320: 6d61 7020 636f 7272 6573 706f 6e64 696e  map correspondin
│ │ │ │ +00015330: 6720 746f 2054 3a0a 0a2b 2d2d 2d2d 2d2d  g to T:..+------
│ │ │ │  00015340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015370: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 7469  -----+.|i21 : ti
│ │ │ │ -00015380: 6d65 2066 203d 2072 6174 696f 6e61 6c4d  me f = rationalM
│ │ │ │ -00015390: 6170 2054 2020 2020 2020 2020 2020 2020  ap T            
│ │ │ │ +00015370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932  -----------+.|i2
│ │ │ │ +00015380: 3120 3a20 7469 6d65 2066 203d 2072 6174  1 : time f = rat
│ │ │ │ +00015390: 696f 6e61 6c4d 6170 2054 2020 2020 2020  ionalMap T      
│ │ │ │  000153a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000153b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ -000153c0: 7573 6564 2033 2e31 3236 3536 7320 2863  used 3.12656s (c
│ │ │ │ -000153d0: 7075 293b 2032 2e32 3137 7320 2874 6872  pu); 2.217s (thr
│ │ │ │ -000153e0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -000153f0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000153b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000153c0: 0a7c 202d 2d20 7573 6564 2034 2e30 3136  .| -- used 4.016
│ │ │ │ +000153d0: 3334 7320 2863 7075 293b 2032 2e39 3235  34s (cpu); 2.925
│ │ │ │ +000153e0: 3232 7320 2874 6872 6561 6429 3b20 3073  22s (thread); 0s
│ │ │ │ +000153f0: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +00015400: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015440: 207c 0a7c 6f32 3120 3d20 2d2d 2072 6174   |.|o21 = -- rat
│ │ │ │ -00015450: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020  ional map --    
│ │ │ │ -00015460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015440: 2020 2020 2020 207c 0a7c 6f32 3120 3d20         |.|o21 = 
│ │ │ │ +00015450: 2d2d 2072 6174 696f 6e61 6c20 6d61 7020  -- rational map 
│ │ │ │ +00015460: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │  00015470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 205a 5a20               ZZ 
│ │ │ │ -000154a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015480: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00015490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000154a0: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │  000154b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000154c0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000154d0: 2020 736f 7572 6365 3a20 5072 6f6a 282d    source: Proj(-
│ │ │ │ -000154e0: 2d2d 2d2d 5b78 202c 2078 202c 2078 202c  ----[x , x , x ,
│ │ │ │ -000154f0: 2078 205d 2920 2020 2020 2020 2020 2020   x ])           
│ │ │ │ -00015500: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015520: 2020 2036 3535 3231 2020 3020 2020 3120     65521  0   1 
│ │ │ │ -00015530: 2020 3220 2020 3320 2020 2020 2020 2020    2   3         
│ │ │ │ +000154c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000154d0: 0a7c 2020 2020 2020 736f 7572 6365 3a20  .|      source: 
│ │ │ │ +000154e0: 5072 6f6a 282d 2d2d 2d2d 5b78 202c 2078  Proj(-----[x , x
│ │ │ │ +000154f0: 202c 2078 202c 2078 205d 2920 2020 2020   , x , x ])     
│ │ │ │ +00015500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015510: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015520: 2020 2020 2020 2020 2036 3535 3231 2020           65521  
│ │ │ │ +00015530: 3020 2020 3120 2020 3220 2020 3320 2020  0   1   2   3   
│ │ │ │  00015540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015550: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00015560: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ -00015570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015550: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015560: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +00015570: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │  00015580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015590: 2020 2020 207c 0a7c 2020 2020 2020 7461       |.|      ta
│ │ │ │ -000155a0: 7267 6574 3a20 5072 6f6a 282d 2d2d 2d2d  rget: Proj(-----
│ │ │ │ -000155b0: 5b78 202c 2078 202c 2078 202c 2078 205d  [x , x , x , x ]
│ │ │ │ -000155c0: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -000155d0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000155e0: 2020 2020 2020 2020 2020 2020 2020 2036                 6
│ │ │ │ -000155f0: 3535 3231 2020 3020 2020 3120 2020 3220  5521  0   1   2 
│ │ │ │ -00015600: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ -00015610: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015620: 2020 2020 2020 6465 6669 6e69 6e67 2066        defining f
│ │ │ │ -00015630: 6f72 6d73 3a20 7b20 2020 2020 2020 2020  orms: {         
│ │ │ │ +00015590: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000155a0: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ +000155b0: 282d 2d2d 2d2d 5b78 202c 2078 202c 2078  (-----[x , x , x
│ │ │ │ +000155c0: 202c 2078 205d 2920 2020 2020 2020 2020   , x ])         
│ │ │ │ +000155d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000155e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000155f0: 2020 2020 2036 3535 3231 2020 3020 2020       65521  0   
│ │ │ │ +00015600: 3120 2020 3220 2020 3320 2020 2020 2020  1   2   3       
│ │ │ │ +00015610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015620: 2020 207c 0a7c 2020 2020 2020 6465 6669     |.|      defi
│ │ │ │ +00015630: 6e69 6e67 2066 6f72 6d73 3a20 7b20 2020  ning forms: {   
│ │ │ │  00015640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015660: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -00015670: 2020 2020 2020 2020 2020 2020 2020 3320                3 
│ │ │ │ -00015680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015690: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000156a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000156b0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -000156c0: 2078 2020 2d20 3332 3735 3978 2078 2078   x  - 32759x x x
│ │ │ │ -000156d0: 2020 2b20 3332 3736 3078 2078 202c 2020    + 32760x x ,  
│ │ │ │ -000156e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -000156f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015700: 2020 2020 2020 3120 2020 2020 2020 2020        1         
│ │ │ │ -00015710: 3020 3120 3220 2020 2020 2020 2020 3020  0 1 2         0 
│ │ │ │ -00015720: 3320 2020 2020 2020 2020 2020 207c 0a7c  3            |.|
│ │ │ │ -00015730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015660: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015680: 2020 2020 3320 2020 2020 2020 2020 2020      3           
│ │ │ │ +00015690: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000156a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000156b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000156c0: 2020 2020 202d 2078 2020 2d20 3332 3735       - x  - 3275
│ │ │ │ +000156d0: 3978 2078 2078 2020 2b20 3332 3736 3078  9x x x  + 32760x
│ │ │ │ +000156e0: 2078 202c 2020 2020 2020 2020 2020 207c   x ,           |
│ │ │ │ +000156f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015700: 2020 2020 2020 2020 2020 2020 3120 2020              1   
│ │ │ │ +00015710: 2020 2020 2020 3020 3120 3220 2020 2020        0 1 2     
│ │ │ │ +00015720: 2020 2020 3020 3320 2020 2020 2020 2020      0 3         
│ │ │ │ +00015730: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015770: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015770: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00015780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015790: 2032 2020 2020 2020 2020 3220 2020 2020   2        2     
│ │ │ │ -000157a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000157b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000157c0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ -000157d0: 3237 3630 7820 7820 202b 2078 2078 2020  2760x x  + x x  
│ │ │ │ -000157e0: 2b20 3332 3736 3078 2078 2078 202c 2020  + 32760x x x ,  
│ │ │ │ -000157f0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015810: 2020 2020 2020 2020 2031 2032 2020 2020           1 2    
│ │ │ │ -00015820: 3020 3220 2020 2020 2020 2020 3020 3120  0 2         0 1 
│ │ │ │ -00015830: 3320 2020 2020 2020 2020 2020 207c 0a7c  3            |.|
│ │ │ │ -00015840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015790: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000157a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000157b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000157c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000157d0: 2020 2020 2033 3237 3630 7820 7820 202b       32760x x  +
│ │ │ │ +000157e0: 2078 2078 2020 2b20 3332 3736 3078 2078   x x  + 32760x x
│ │ │ │ +000157f0: 2078 202c 2020 2020 2020 2020 2020 207c   x ,           |
│ │ │ │ +00015800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015810: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00015820: 2032 2020 2020 3020 3220 2020 2020 2020   2    0 2       
│ │ │ │ +00015830: 2020 3020 3120 3320 2020 2020 2020 2020    0 1 3         
│ │ │ │ +00015840: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015880: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015880: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00015890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158a0: 2020 2020 2032 2020 2020 3220 2020 2020       2    2     
│ │ │ │ -000158b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000158c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000158d0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ -000158e0: 2033 3237 3630 7820 7820 202d 2078 2078   32760x x  - x x
│ │ │ │ -000158f0: 2020 2d20 3332 3736 3078 2078 2078 202c    - 32760x x x ,
│ │ │ │ -00015900: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015920: 2020 2020 2020 2020 2020 2031 2032 2020             1 2  
│ │ │ │ -00015930: 2020 3120 3320 2020 2020 2020 2020 3020    1 3         0 
│ │ │ │ -00015940: 3220 3320 2020 2020 2020 2020 207c 0a7c  2 3          |.|
│ │ │ │ -00015950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000158a0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000158b0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +000158c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000158d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000158e0: 2020 2020 202d 2033 3237 3630 7820 7820       - 32760x x 
│ │ │ │ +000158f0: 202d 2078 2078 2020 2d20 3332 3736 3078   - x x  - 32760x
│ │ │ │ +00015900: 2078 2078 202c 2020 2020 2020 2020 207c   x x ,         |
│ │ │ │ +00015910: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015930: 2031 2032 2020 2020 3120 3320 2020 2020   1 2    1 3     
│ │ │ │ +00015940: 2020 2020 3020 3220 3320 2020 2020 2020      0 2 3       
│ │ │ │ +00015950: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  00015960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015990: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ -000159a0: 2020 2020 2020 2020 2020 2020 3320 2020              3   
│ │ │ │ -000159b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000159c0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -000159d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000159e0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -000159f0: 2020 2b20 3332 3735 3978 2078 2078 2020    + 32759x x x  
│ │ │ │ -00015a00: 2d20 3332 3736 3078 2078 2020 2020 2020  - 32760x x      
│ │ │ │ -00015a10: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a30: 2020 2020 3220 2020 2020 2020 2020 3120      2         1 
│ │ │ │ -00015a40: 3220 3320 2020 2020 2020 2020 3020 3320  2 3         0 3 
│ │ │ │ -00015a50: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015a70: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +00015990: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000159a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000159b0: 2020 3320 2020 2020 2020 2020 2020 2020    3             
│ │ │ │ +000159c0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000159d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000159e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000159f0: 2020 2020 2078 2020 2b20 3332 3735 3978       x  + 32759x
│ │ │ │ +00015a00: 2078 2078 2020 2d20 3332 3736 3078 2078   x x  - 32760x x
│ │ │ │ +00015a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015a20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00015a30: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00015a40: 2020 2020 3120 3220 3320 2020 2020 2020      1 2 3       
│ │ │ │ +00015a50: 2020 3020 3320 2020 2020 2020 2020 2020    0 3           
│ │ │ │ +00015a60: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00015a70: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │  00015a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015aa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015aa0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00015ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015ae0: 2020 2020 207c 0a7c 6f32 3120 3a20 5261       |.|o21 : Ra
│ │ │ │ -00015af0: 7469 6f6e 616c 4d61 7020 2863 7562 6963  tionalMap (cubic
│ │ │ │ -00015b00: 2072 6174 696f 6e61 6c20 6d61 7020 6672   rational map fr
│ │ │ │ -00015b10: 6f6d 2050 505e 3320 746f 2050 505e 3329  om PP^3 to PP^3)
│ │ │ │ -00015b20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -00015b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015ae0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00015af0: 3120 3a20 5261 7469 6f6e 616c 4d61 7020  1 : RationalMap 
│ │ │ │ +00015b00: 2863 7562 6963 2072 6174 696f 6e61 6c20  (cubic rational 
│ │ │ │ +00015b10: 6d61 7020 6672 6f6d 2050 505e 3320 746f  map from PP^3 to
│ │ │ │ +00015b20: 2050 505e 3329 2020 2020 2020 2020 207c   PP^3)         |
│ │ │ │ +00015b30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00015b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00015b70: 6932 3220 3a20 6465 7363 7269 6265 2066  i22 : describe f
│ │ │ │ -00015b80: 2120 2020 2020 2020 2020 2020 2020 2020  !               
│ │ │ │ +00015b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015b70: 2d2d 2d2b 0a7c 6932 3220 3a20 6465 7363  ---+.|i22 : desc
│ │ │ │ +00015b80: 7269 6265 2066 2120 2020 2020 2020 2020  ribe f!         
│ │ │ │  00015b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00015bb0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │  00015bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015bf0: 2020 2020 207c 0a7c 6f32 3220 3d20 7261       |.|o22 = ra
│ │ │ │ -00015c00: 7469 6f6e 616c 206d 6170 2064 6566 696e  tional map defin
│ │ │ │ -00015c10: 6564 2062 7920 666f 726d 7320 6f66 2064  ed by forms of d
│ │ │ │ -00015c20: 6567 7265 6520 3320 2020 2020 2020 2020  egree 3         
│ │ │ │ -00015c30: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015c40: 2020 736f 7572 6365 2076 6172 6965 7479    source variety
│ │ │ │ -00015c50: 3a20 5050 5e33 2020 2020 2020 2020 2020  : PP^3          
│ │ │ │ +00015bf0: 2020 2020 2020 2020 2020 207c 0a7c 6f32             |.|o2
│ │ │ │ +00015c00: 3220 3d20 7261 7469 6f6e 616c 206d 6170  2 = rational map
│ │ │ │ +00015c10: 2064 6566 696e 6564 2062 7920 666f 726d   defined by form
│ │ │ │ +00015c20: 7320 6f66 2064 6567 7265 6520 3320 2020  s of degree 3   
│ │ │ │ +00015c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015c40: 0a7c 2020 2020 2020 736f 7572 6365 2076  .|      source v
│ │ │ │ +00015c50: 6172 6965 7479 3a20 5050 5e33 2020 2020  ariety: PP^3    
│ │ │ │  00015c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015c70: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015c80: 2020 2020 2020 7461 7267 6574 2076 6172        target var
│ │ │ │ -00015c90: 6965 7479 3a20 5050 5e33 2020 2020 2020  iety: PP^3      
│ │ │ │ +00015c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015c80: 2020 207c 0a7c 2020 2020 2020 7461 7267     |.|      targ
│ │ │ │ +00015c90: 6574 2076 6172 6965 7479 3a20 5050 5e33  et variety: PP^3
│ │ │ │  00015ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015cc0: 207c 0a7c 2020 2020 2020 646f 6d69 6e61   |.|      domina
│ │ │ │ -00015cd0: 6e63 653a 2074 7275 6520 2020 2020 2020  nce: true       
│ │ │ │ +00015cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015cd0: 646f 6d69 6e61 6e63 653a 2074 7275 6520  dominance: true 
│ │ │ │  00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00015cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015d00: 2020 2020 207c 0a7c 2020 2020 2020 6269       |.|      bi
│ │ │ │ -00015d10: 7261 7469 6f6e 616c 6974 793a 2074 7275  rationality: tru
│ │ │ │ -00015d20: 6520 2874 6865 2069 6e76 6572 7365 206d  e (the inverse m
│ │ │ │ -00015d30: 6170 2069 7320 616c 7265 6164 7920 6361  ap is already ca
│ │ │ │ -00015d40: 6c63 756c 6174 6564 297c 0a7c 2020 2020  lculated)|.|    
│ │ │ │ -00015d50: 2020 7072 6f6a 6563 7469 7665 2064 6567    projective deg
│ │ │ │ -00015d60: 7265 6573 3a20 7b31 2c20 332c 2033 2c20  rees: {1, 3, 3, 
│ │ │ │ -00015d70: 317d 2020 2020 2020 2020 2020 2020 2020  1}              
│ │ │ │ -00015d80: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00015d90: 2020 2020 2020 6e75 6d62 6572 206f 6620        number of 
│ │ │ │ -00015da0: 6d69 6e69 6d61 6c20 7265 7072 6573 656e  minimal represen
│ │ │ │ -00015db0: 7461 7469 7665 733a 2031 2020 2020 2020  tatives: 1      
│ │ │ │ +00015d00: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00015d10: 2020 2020 6269 7261 7469 6f6e 616c 6974      birationalit
│ │ │ │ +00015d20: 793a 2074 7275 6520 2874 6865 2069 6e76  y: true (the inv
│ │ │ │ +00015d30: 6572 7365 206d 6170 2069 7320 616c 7265  erse map is alre
│ │ │ │ +00015d40: 6164 7920 6361 6c63 756c 6174 6564 297c  ady calculated)|
│ │ │ │ +00015d50: 0a7c 2020 2020 2020 7072 6f6a 6563 7469  .|      projecti
│ │ │ │ +00015d60: 7665 2064 6567 7265 6573 3a20 7b31 2c20  ve degrees: {1, 
│ │ │ │ +00015d70: 332c 2033 2c20 317d 2020 2020 2020 2020  3, 3, 1}        
│ │ │ │ +00015d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015d90: 2020 207c 0a7c 2020 2020 2020 6e75 6d62     |.|      numb
│ │ │ │ +00015da0: 6572 206f 6620 6d69 6e69 6d61 6c20 7265  er of minimal re
│ │ │ │ +00015db0: 7072 6573 656e 7461 7469 7665 733a 2031  presentatives: 1
│ │ │ │  00015dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015dd0: 207c 0a7c 2020 2020 2020 6469 6d65 6e73   |.|      dimens
│ │ │ │ -00015de0: 696f 6e20 6261 7365 206c 6f63 7573 3a20  ion base locus: 
│ │ │ │ -00015df0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00015dd0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00015de0: 6469 6d65 6e73 696f 6e20 6261 7365 206c  dimension base l
│ │ │ │ +00015df0: 6f63 7573 3a20 3120 2020 2020 2020 2020  ocus: 1         
│ │ │ │  00015e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e10: 2020 2020 207c 0a7c 2020 2020 2020 6465       |.|      de
│ │ │ │ -00015e20: 6772 6565 2062 6173 6520 6c6f 6375 733a  gree base locus:
│ │ │ │ -00015e30: 2036 2020 2020 2020 2020 2020 2020 2020   6              
│ │ │ │ +00015e10: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00015e20: 2020 2020 6465 6772 6565 2062 6173 6520      degree base 
│ │ │ │ +00015e30: 6c6f 6375 733a 2036 2020 2020 2020 2020  locus: 6        
│ │ │ │  00015e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e50: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ -00015e60: 2020 636f 6566 6669 6369 656e 7420 7269    coefficient ri
│ │ │ │ -00015e70: 6e67 3a20 5a5a 2f36 3535 3231 2020 2020  ng: ZZ/65521    
│ │ │ │ -00015e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00015e90: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00015ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00015e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00015e60: 0a7c 2020 2020 2020 636f 6566 6669 6369  .|      coeffici
│ │ │ │ +00015e70: 656e 7420 7269 6e67 3a20 5a5a 2f36 3535  ent ring: ZZ/655
│ │ │ │ +00015e80: 3231 2020 2020 2020 2020 2020 2020 2020  21              
│ │ │ │ +00015e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00015ea0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00015eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00015ee0: 2d2b 0a0a 4361 7665 6174 0a3d 3d3d 3d3d  -+..Caveat.=====
│ │ │ │ -00015ef0: 3d0a 0a54 6869 7320 6973 2075 6e64 6572  =..This is under
│ │ │ │ -00015f00: 2064 6576 656c 6f70 6d65 6e74 2079 6574   development yet
│ │ │ │ -00015f10: 2e0a 0a57 6179 7320 746f 2075 7365 2061  ...Ways to use a
│ │ │ │ -00015f20: 6273 7472 6163 7452 6174 696f 6e61 6c4d  bstractRationalM
│ │ │ │ -00015f30: 6170 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ap:.============
│ │ │ │ +00015ee0: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174  -------+..Caveat
│ │ │ │ +00015ef0: 0a3d 3d3d 3d3d 3d0a 0a54 6869 7320 6973  .======..This is
│ │ │ │ +00015f00: 2075 6e64 6572 2064 6576 656c 6f70 6d65   under developme
│ │ │ │ +00015f10: 6e74 2079 6574 2e0a 0a57 6179 7320 746f  nt yet...Ways to
│ │ │ │ +00015f20: 2075 7365 2061 6273 7472 6163 7452 6174   use abstractRat
│ │ │ │ +00015f30: 696f 6e61 6c4d 6170 3a0a 3d3d 3d3d 3d3d  ionalMap:.======
│ │ │ │  00015f40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00015f50: 3d3d 3d3d 0a0a 2020 2a20 2261 6273 7472  ====..  * "abstr
│ │ │ │ -00015f60: 6163 7452 6174 696f 6e61 6c4d 6170 2850  actRationalMap(P
│ │ │ │ -00015f70: 6f6c 796e 6f6d 6961 6c52 696e 672c 506f  olynomialRing,Po
│ │ │ │ -00015f80: 6c79 6e6f 6d69 616c 5269 6e67 2c46 756e  lynomialRing,Fun
│ │ │ │ -00015f90: 6374 696f 6e43 6c6f 7375 7265 2922 0a20  ctionClosure)". 
│ │ │ │ -00015fa0: 202a 2022 6162 7374 7261 6374 5261 7469   * "abstractRati
│ │ │ │ -00015fb0: 6f6e 616c 4d61 7028 506f 6c79 6e6f 6d69  onalMap(Polynomi
│ │ │ │ -00015fc0: 616c 5269 6e67 2c50 6f6c 796e 6f6d 6961  alRing,Polynomia
│ │ │ │ -00015fd0: 6c52 696e 672c 4675 6e63 7469 6f6e 436c  lRing,FunctionCl
│ │ │ │ -00015fe0: 6f73 7572 652c 5a5a 2922 0a20 202a 2022  osure,ZZ)".  * "
│ │ │ │ -00015ff0: 6162 7374 7261 6374 5261 7469 6f6e 616c  abstractRational
│ │ │ │ -00016000: 4d61 7028 5261 7469 6f6e 616c 4d61 7029  Map(RationalMap)
│ │ │ │ -00016010: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00016020: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00016030: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -00016040: 6a65 6374 202a 6e6f 7465 2061 6273 7472  ject *note abstr
│ │ │ │ -00016050: 6163 7452 6174 696f 6e61 6c4d 6170 3a20  actRationalMap: 
│ │ │ │ -00016060: 6162 7374 7261 6374 5261 7469 6f6e 616c  abstractRational
│ │ │ │ -00016070: 4d61 702c 2069 7320 6120 2a6e 6f74 6520  Map, is a *note 
│ │ │ │ -00016080: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a  method.function:
│ │ │ │ -00016090: 2028 4d61 6361 756c 6179 3244 6f63 294d   (Macaulay2Doc)M
│ │ │ │ -000160a0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a  ethodFunction,..
│ │ │ │ -000160b0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +00015f50: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00015f60: 2261 6273 7472 6163 7452 6174 696f 6e61  "abstractRationa
│ │ │ │ +00015f70: 6c4d 6170 2850 6f6c 796e 6f6d 6961 6c52  lMap(PolynomialR
│ │ │ │ +00015f80: 696e 672c 506f 6c79 6e6f 6d69 616c 5269  ing,PolynomialRi
│ │ │ │ +00015f90: 6e67 2c46 756e 6374 696f 6e43 6c6f 7375  ng,FunctionClosu
│ │ │ │ +00015fa0: 7265 2922 0a20 202a 2022 6162 7374 7261  re)".  * "abstra
│ │ │ │ +00015fb0: 6374 5261 7469 6f6e 616c 4d61 7028 506f  ctRationalMap(Po
│ │ │ │ +00015fc0: 6c79 6e6f 6d69 616c 5269 6e67 2c50 6f6c  lynomialRing,Pol
│ │ │ │ +00015fd0: 796e 6f6d 6961 6c52 696e 672c 4675 6e63  ynomialRing,Func
│ │ │ │ +00015fe0: 7469 6f6e 436c 6f73 7572 652c 5a5a 2922  tionClosure,ZZ)"
│ │ │ │ +00015ff0: 0a20 202a 2022 6162 7374 7261 6374 5261  .  * "abstractRa
│ │ │ │ +00016000: 7469 6f6e 616c 4d61 7028 5261 7469 6f6e  tionalMap(Ration
│ │ │ │ +00016010: 616c 4d61 7029 220a 0a46 6f72 2074 6865  alMap)"..For the
│ │ │ │ +00016020: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +00016030: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00016040: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00016050: 2061 6273 7472 6163 7452 6174 696f 6e61   abstractRationa
│ │ │ │ +00016060: 6c4d 6170 3a20 6162 7374 7261 6374 5261  lMap: abstractRa
│ │ │ │ +00016070: 7469 6f6e 616c 4d61 702c 2069 7320 6120  tionalMap, is a 
│ │ │ │ +00016080: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e  *note method.fun
│ │ │ │ +00016090: 6374 696f 6e3a 2028 4d61 6361 756c 6179  ction: (Macaulay
│ │ │ │ +000160a0: 3244 6f63 294d 6574 686f 6446 756e 6374  2Doc)MethodFunct
│ │ │ │ +000160b0: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ion,...---------
│ │ │ │  000160c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000160d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000160e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016100: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -00016110: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ -00016120: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ -00016130: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -00016140: 6175 6c61 7932 2d31 2e32 362e 3036 2b64  aulay2-1.26.06+d
│ │ │ │ -00016150: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -00016160: 6163 6b61 6765 732f 4372 656d 6f6e 612f  ackages/Cremona/
│ │ │ │ -00016170: 0a64 6f63 756d 656e 7461 7469 6f6e 2e6d  .documentation.m
│ │ │ │ -00016180: 323a 3131 3032 3a30 2e0a 1f0a 4669 6c65  2:1102:0....File
│ │ │ │ -00016190: 3a20 4372 656d 6f6e 612e 696e 666f 2c20  : Cremona.info, 
│ │ │ │ -000161a0: 4e6f 6465 3a20 6170 7072 6f78 696d 6174  Node: approximat
│ │ │ │ -000161b0: 6549 6e76 6572 7365 4d61 702c 204e 6578  eInverseMap, Nex
│ │ │ │ -000161c0: 743a 2042 6c6f 7755 7053 7472 6174 6567  t: BlowUpStrateg
│ │ │ │ -000161d0: 792c 2050 7265 763a 2061 6273 7472 6163  y, Prev: abstrac
│ │ │ │ -000161e0: 7452 6174 696f 6e61 6c4d 6170 2c20 5570  tRationalMap, Up
│ │ │ │ -000161f0: 3a20 546f 700a 0a61 7070 726f 7869 6d61  : Top..approxima
│ │ │ │ -00016200: 7465 496e 7665 7273 654d 6170 202d 2d20  teInverseMap -- 
│ │ │ │ -00016210: 7261 6e64 6f6d 206d 6170 2072 656c 6174  random map relat
│ │ │ │ -00016220: 6564 2074 6f20 7468 6520 696e 7665 7273  ed to the invers
│ │ │ │ -00016230: 6520 6f66 2061 2062 6972 6174 696f 6e61  e of a birationa
│ │ │ │ -00016240: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a  l map.**********
│ │ │ │ +00016100: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +00016110: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +00016120: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00016130: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00016140: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00016150: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ +00016160: 6c61 7932 2f70 6163 6b61 6765 732f 4372  lay2/packages/Cr
│ │ │ │ +00016170: 656d 6f6e 612f 0a64 6f63 756d 656e 7461  emona/.documenta
│ │ │ │ +00016180: 7469 6f6e 2e6d 323a 3131 3032 3a30 2e0a  tion.m2:1102:0..
│ │ │ │ +00016190: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ +000161a0: 696e 666f 2c20 4e6f 6465 3a20 6170 7072  info, Node: appr
│ │ │ │ +000161b0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +000161c0: 702c 204e 6578 743a 2042 6c6f 7755 7053  p, Next: BlowUpS
│ │ │ │ +000161d0: 7472 6174 6567 792c 2050 7265 763a 2061  trategy, Prev: a
│ │ │ │ +000161e0: 6273 7472 6163 7452 6174 696f 6e61 6c4d  bstractRationalM
│ │ │ │ +000161f0: 6170 2c20 5570 3a20 546f 700a 0a61 7070  ap, Up: Top..app
│ │ │ │ +00016200: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ +00016210: 6170 202d 2d20 7261 6e64 6f6d 206d 6170  ap -- random map
│ │ │ │ +00016220: 2072 656c 6174 6564 2074 6f20 7468 6520   related to the 
│ │ │ │ +00016230: 696e 7665 7273 6520 6f66 2061 2062 6972  inverse of a bir
│ │ │ │ +00016240: 6174 696f 6e61 6c20 6d61 700a 2a2a 2a2a  ational map.****
│ │ │ │  00016250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016260: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016270: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00016280: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00016290: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -000162a0: 200a 2020 2020 2020 2020 6170 7072 6f78   .        approx
│ │ │ │ -000162b0: 696d 6174 6549 6e76 6572 7365 4d61 7020  imateInverseMap 
│ │ │ │ -000162c0: 7068 6920 0a20 2020 2020 2020 2061 7070  phi .        app
│ │ │ │ -000162d0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ -000162e0: 6170 2870 6869 2c64 290a 2020 2a20 496e  ap(phi,d).  * In
│ │ │ │ -000162f0: 7075 7473 3a0a 2020 2020 2020 2a20 7068  puts:.      * ph
│ │ │ │ -00016300: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ -00016310: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ -00016320: 6c4d 6170 2c2c 2061 2062 6972 6174 696f  lMap,, a biratio
│ │ │ │ -00016330: 6e61 6c20 6d61 700a 2020 2020 2020 2a20  nal map.      * 
│ │ │ │ -00016340: 642c 2061 6e20 2a6e 6f74 6520 696e 7465  d, an *note inte
│ │ │ │ -00016350: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ -00016360: 6f63 295a 5a2c 2c20 6f70 7469 6f6e 616c  oc)ZZ,, optional
│ │ │ │ -00016370: 2c20 6275 7420 6974 2073 686f 756c 6420  , but it should 
│ │ │ │ -00016380: 6265 2074 6865 0a20 2020 2020 2020 2064  be the.        d
│ │ │ │ -00016390: 6567 7265 6520 6f66 2074 6865 2066 6f72  egree of the for
│ │ │ │ -000163a0: 6d73 2064 6566 696e 696e 6720 7468 6520  ms defining the 
│ │ │ │ -000163b0: 696e 7665 7273 6520 6f66 2070 6869 0a20  inverse of phi. 
│ │ │ │ -000163c0: 202a 202a 6e6f 7465 204f 7074 696f 6e61   * *note Optiona
│ │ │ │ -000163d0: 6c20 696e 7075 7473 3a20 284d 6163 6175  l inputs: (Macau
│ │ │ │ -000163e0: 6c61 7932 446f 6329 7573 696e 6720 6675  lay2Doc)using fu
│ │ │ │ -000163f0: 6e63 7469 6f6e 7320 7769 7468 206f 7074  nctions with opt
│ │ │ │ -00016400: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20  ional inputs,:. 
│ │ │ │ -00016410: 2020 2020 202a 202a 6e6f 7465 2043 6572       * *note Cer
│ │ │ │ -00016420: 7469 6679 3a20 4365 7274 6966 792c 203d  tify: Certify, =
│ │ │ │ -00016430: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076  > ..., default v
│ │ │ │ -00016440: 616c 7565 2066 616c 7365 2c20 7768 6574  alue false, whet
│ │ │ │ -00016450: 6865 7220 746f 2065 6e73 7572 650a 2020  her to ensure.  
│ │ │ │ -00016460: 2020 2020 2020 636f 7272 6563 746e 6573        correctnes
│ │ │ │ -00016470: 7320 6f66 206f 7574 7075 740a 2020 2020  s of output.    
│ │ │ │ -00016480: 2020 2a20 2a6e 6f74 6520 436f 6469 6d42    * *note CodimB
│ │ │ │ -00016490: 7349 6e76 3a20 436f 6469 6d42 7349 6e76  sInv: CodimBsInv
│ │ │ │ -000164a0: 2c20 3d3e 202e 2e2e 2c20 6465 6661 756c  , => ..., defaul
│ │ │ │ -000164b0: 7420 7661 6c75 6520 6e75 6c6c 2c20 0a20  t value null, . 
│ │ │ │ -000164c0: 2020 2020 202a 202a 6e6f 7465 2056 6572       * *note Ver
│ │ │ │ -000164d0: 626f 7365 3a20 696e 7665 7273 654d 6170  bose: inverseMap
│ │ │ │ -000164e0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56  _lp_pd_pd_pd_cmV
│ │ │ │ -000164f0: 6572 626f 7365 3d3e 5f70 645f 7064 5f70  erbose=>_pd_pd_p
│ │ │ │ -00016500: 645f 7270 2c20 3d3e 202e 2e2e 2c0a 2020  d_rp, => ...,.  
│ │ │ │ -00016510: 2020 2020 2020 6465 6661 756c 7420 7661        default va
│ │ │ │ -00016520: 6c75 6520 7472 7565 2c0a 2020 2a20 4f75  lue true,.  * Ou
│ │ │ │ -00016530: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ -00016540: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -00016550: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -00016560: 2c2c 2061 2072 616e 646f 6d20 7261 7469  ,, a random rati
│ │ │ │ -00016570: 6f6e 616c 206d 6170 2077 6869 6368 2069  onal map which i
│ │ │ │ -00016580: 6e20 736f 6d65 0a20 2020 2020 2020 2073  n some.        s
│ │ │ │ -00016590: 656e 7365 2069 7320 7265 6c61 7465 6420  ense is related 
│ │ │ │ -000165a0: 746f 2074 6865 2069 6e76 6572 7365 206f  to the inverse o
│ │ │ │ -000165b0: 6620 7068 6920 2865 2e67 2e2c 2074 6865  f phi (e.g., the
│ │ │ │ -000165c0: 7920 7368 6f75 6c64 2068 6176 6520 7468  y should have th
│ │ │ │ -000165d0: 6520 7361 6d65 0a20 2020 2020 2020 2062  e same.        b
│ │ │ │ -000165e0: 6173 6520 6c6f 6375 7329 0a0a 4465 7363  ase locus)..Desc
│ │ │ │ -000165f0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d  ription.========
│ │ │ │ -00016600: 3d3d 3d0a 0a54 6865 2061 6c67 6f72 6974  ===..The algorit
│ │ │ │ -00016610: 686d 2069 7320 746f 2074 7279 2074 6f20  hm is to try to 
│ │ │ │ -00016620: 636f 6e73 7472 7563 7420 7468 6520 6964  construct the id
│ │ │ │ -00016630: 6561 6c20 6f66 2074 6865 2062 6173 6520  eal of the base 
│ │ │ │ -00016640: 6c6f 6375 7320 6f66 2074 6865 2069 6e76  locus of the inv
│ │ │ │ -00016650: 6572 7365 0a62 7920 6c6f 6f6b 696e 6720  erse.by looking 
│ │ │ │ -00016660: 666f 7220 7468 6520 696d 6167 6573 2076  for the images v
│ │ │ │ -00016670: 6961 2070 6869 206f 6620 7261 6e64 6f6d  ia phi of random
│ │ │ │ -00016680: 206c 696e 6561 7220 7365 6374 696f 6e73   linear sections
│ │ │ │ -00016690: 206f 6620 7468 6520 736f 7572 6365 0a76   of the source.v
│ │ │ │ -000166a0: 6172 6965 7479 2e20 4765 6e65 7261 6c6c  ariety. Generall
│ │ │ │ -000166b0: 792c 206f 6e65 2063 616e 2073 7065 6564  y, one can speed
│ │ │ │ -000166c0: 2075 7020 7468 6520 7072 6f63 6573 7320   up the process 
│ │ │ │ -000166d0: 6279 2070 6173 7369 6e67 2074 6872 6f75  by passing throu
│ │ │ │ -000166e0: 6768 2074 6865 206f 7074 696f 6e0a 2a6e  gh the option.*n
│ │ │ │ -000166f0: 6f74 6520 436f 6469 6d42 7349 6e76 3a20  ote CodimBsInv: 
│ │ │ │ -00016700: 436f 6469 6d42 7349 6e76 2c20 6120 676f  CodimBsInv, a go
│ │ │ │ -00016710: 6f64 206c 6f77 6572 2062 6f75 6e64 2066  od lower bound f
│ │ │ │ -00016720: 6f72 2074 6865 2063 6f64 696d 656e 7369  or the codimensi
│ │ │ │ -00016730: 6f6e 206f 6620 7468 6973 0a62 6173 6520  on of this.base 
│ │ │ │ -00016740: 6c6f 6375 732e 0a0a 2b2d 2d2d 2d2d 2d2d  locus...+-------
│ │ │ │ +00016290: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +000162a0: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +000162b0: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +000162c0: 7365 4d61 7020 7068 6920 0a20 2020 2020  seMap phi .     
│ │ │ │ +000162d0: 2020 2061 7070 726f 7869 6d61 7465 496e     approximateIn
│ │ │ │ +000162e0: 7665 7273 654d 6170 2870 6869 2c64 290a  verseMap(phi,d).
│ │ │ │ +000162f0: 2020 2a20 496e 7075 7473 3a0a 2020 2020    * Inputs:.    
│ │ │ │ +00016300: 2020 2a20 7068 692c 2061 202a 6e6f 7465    * phi, a *note
│ │ │ │ +00016310: 2072 6174 696f 6e61 6c20 6d61 703a 2052   rational map: R
│ │ │ │ +00016320: 6174 696f 6e61 6c4d 6170 2c2c 2061 2062  ationalMap,, a b
│ │ │ │ +00016330: 6972 6174 696f 6e61 6c20 6d61 700a 2020  irational map.  
│ │ │ │ +00016340: 2020 2020 2a20 642c 2061 6e20 2a6e 6f74      * d, an *not
│ │ │ │ +00016350: 6520 696e 7465 6765 723a 2028 4d61 6361  e integer: (Maca
│ │ │ │ +00016360: 756c 6179 3244 6f63 295a 5a2c 2c20 6f70  ulay2Doc)ZZ,, op
│ │ │ │ +00016370: 7469 6f6e 616c 2c20 6275 7420 6974 2073  tional, but it s
│ │ │ │ +00016380: 686f 756c 6420 6265 2074 6865 0a20 2020  hould be the.   
│ │ │ │ +00016390: 2020 2020 2064 6567 7265 6520 6f66 2074       degree of t
│ │ │ │ +000163a0: 6865 2066 6f72 6d73 2064 6566 696e 696e  he forms definin
│ │ │ │ +000163b0: 6720 7468 6520 696e 7665 7273 6520 6f66  g the inverse of
│ │ │ │ +000163c0: 2070 6869 0a20 202a 202a 6e6f 7465 204f   phi.  * *note O
│ │ │ │ +000163d0: 7074 696f 6e61 6c20 696e 7075 7473 3a20  ptional inputs: 
│ │ │ │ +000163e0: 284d 6163 6175 6c61 7932 446f 6329 7573  (Macaulay2Doc)us
│ │ │ │ +000163f0: 696e 6720 6675 6e63 7469 6f6e 7320 7769  ing functions wi
│ │ │ │ +00016400: 7468 206f 7074 696f 6e61 6c20 696e 7075  th optional inpu
│ │ │ │ +00016410: 7473 2c3a 0a20 2020 2020 202a 202a 6e6f  ts,:.      * *no
│ │ │ │ +00016420: 7465 2043 6572 7469 6679 3a20 4365 7274  te Certify: Cert
│ │ │ │ +00016430: 6966 792c 203d 3e20 2e2e 2e2c 2064 6566  ify, => ..., def
│ │ │ │ +00016440: 6175 6c74 2076 616c 7565 2066 616c 7365  ault value false
│ │ │ │ +00016450: 2c20 7768 6574 6865 7220 746f 2065 6e73  , whether to ens
│ │ │ │ +00016460: 7572 650a 2020 2020 2020 2020 636f 7272  ure.        corr
│ │ │ │ +00016470: 6563 746e 6573 7320 6f66 206f 7574 7075  ectness of outpu
│ │ │ │ +00016480: 740a 2020 2020 2020 2a20 2a6e 6f74 6520  t.      * *note 
│ │ │ │ +00016490: 436f 6469 6d42 7349 6e76 3a20 436f 6469  CodimBsInv: Codi
│ │ │ │ +000164a0: 6d42 7349 6e76 2c20 3d3e 202e 2e2e 2c20  mBsInv, => ..., 
│ │ │ │ +000164b0: 6465 6661 756c 7420 7661 6c75 6520 6e75  default value nu
│ │ │ │ +000164c0: 6c6c 2c20 0a20 2020 2020 202a 202a 6e6f  ll, .      * *no
│ │ │ │ +000164d0: 7465 2056 6572 626f 7365 3a20 696e 7665  te Verbose: inve
│ │ │ │ +000164e0: 7273 654d 6170 5f6c 705f 7064 5f70 645f  rseMap_lp_pd_pd_
│ │ │ │ +000164f0: 7064 5f63 6d56 6572 626f 7365 3d3e 5f70  pd_cmVerbose=>_p
│ │ │ │ +00016500: 645f 7064 5f70 645f 7270 2c20 3d3e 202e  d_pd_pd_rp, => .
│ │ │ │ +00016510: 2e2e 2c0a 2020 2020 2020 2020 6465 6661  ..,.        defa
│ │ │ │ +00016520: 756c 7420 7661 6c75 6520 7472 7565 2c0a  ult value true,.
│ │ │ │ +00016530: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ +00016540: 2020 202a 2061 202a 6e6f 7465 2072 6174     * a *note rat
│ │ │ │ +00016550: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +00016560: 6e61 6c4d 6170 2c2c 2061 2072 616e 646f  nalMap,, a rando
│ │ │ │ +00016570: 6d20 7261 7469 6f6e 616c 206d 6170 2077  m rational map w
│ │ │ │ +00016580: 6869 6368 2069 6e20 736f 6d65 0a20 2020  hich in some.   
│ │ │ │ +00016590: 2020 2020 2073 656e 7365 2069 7320 7265       sense is re
│ │ │ │ +000165a0: 6c61 7465 6420 746f 2074 6865 2069 6e76  lated to the inv
│ │ │ │ +000165b0: 6572 7365 206f 6620 7068 6920 2865 2e67  erse of phi (e.g
│ │ │ │ +000165c0: 2e2c 2074 6865 7920 7368 6f75 6c64 2068  ., they should h
│ │ │ │ +000165d0: 6176 6520 7468 6520 7361 6d65 0a20 2020  ave the same.   
│ │ │ │ +000165e0: 2020 2020 2062 6173 6520 6c6f 6375 7329       base locus)
│ │ │ │ +000165f0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d  ..Description.==
│ │ │ │ +00016600: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2061  =========..The a
│ │ │ │ +00016610: 6c67 6f72 6974 686d 2069 7320 746f 2074  lgorithm is to t
│ │ │ │ +00016620: 7279 2074 6f20 636f 6e73 7472 7563 7420  ry to construct 
│ │ │ │ +00016630: 7468 6520 6964 6561 6c20 6f66 2074 6865  the ideal of the
│ │ │ │ +00016640: 2062 6173 6520 6c6f 6375 7320 6f66 2074   base locus of t
│ │ │ │ +00016650: 6865 2069 6e76 6572 7365 0a62 7920 6c6f  he inverse.by lo
│ │ │ │ +00016660: 6f6b 696e 6720 666f 7220 7468 6520 696d  oking for the im
│ │ │ │ +00016670: 6167 6573 2076 6961 2070 6869 206f 6620  ages via phi of 
│ │ │ │ +00016680: 7261 6e64 6f6d 206c 696e 6561 7220 7365  random linear se
│ │ │ │ +00016690: 6374 696f 6e73 206f 6620 7468 6520 736f  ctions of the so
│ │ │ │ +000166a0: 7572 6365 0a76 6172 6965 7479 2e20 4765  urce.variety. Ge
│ │ │ │ +000166b0: 6e65 7261 6c6c 792c 206f 6e65 2063 616e  nerally, one can
│ │ │ │ +000166c0: 2073 7065 6564 2075 7020 7468 6520 7072   speed up the pr
│ │ │ │ +000166d0: 6f63 6573 7320 6279 2070 6173 7369 6e67  ocess by passing
│ │ │ │ +000166e0: 2074 6872 6f75 6768 2074 6865 206f 7074   through the opt
│ │ │ │ +000166f0: 696f 6e0a 2a6e 6f74 6520 436f 6469 6d42  ion.*note CodimB
│ │ │ │ +00016700: 7349 6e76 3a20 436f 6469 6d42 7349 6e76  sInv: CodimBsInv
│ │ │ │ +00016710: 2c20 6120 676f 6f64 206c 6f77 6572 2062  , a good lower b
│ │ │ │ +00016720: 6f75 6e64 2066 6f72 2074 6865 2063 6f64  ound for the cod
│ │ │ │ +00016730: 696d 656e 7369 6f6e 206f 6620 7468 6973  imension of this
│ │ │ │ +00016740: 0a62 6173 6520 6c6f 6375 732e 0a0a 2b2d  .base locus...+-
│ │ │ │  00016750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016790: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5038  ------+.|i1 : P8
│ │ │ │ -000167a0: 203d 205a 5a2f 3937 5b74 5f30 2e2e 745f   = ZZ/97[t_0..t_
│ │ │ │ -000167b0: 385d 3b20 2020 2020 2020 2020 2020 2020  8];             
│ │ │ │ +00016790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +000167a0: 3120 3a20 5038 203d 205a 5a2f 3937 5b74  1 : P8 = ZZ/97[t
│ │ │ │ +000167b0: 5f30 2e2e 745f 385d 3b20 2020 2020 2020  _0..t_8];       
│ │ │ │  000167c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000167d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000167e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +000167e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  000167f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00016820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00016830: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7068  ------+.|i2 : ph
│ │ │ │ -00016840: 6920 3d20 696e 7665 7273 654d 6170 2072  i = inverseMap r
│ │ │ │ -00016850: 6174 696f 6e61 6c4d 6170 2874 7269 6d28  ationalMap(trim(
│ │ │ │ -00016860: 6d69 6e6f 7273 2832 2c67 656e 6572 6963  minors(2,generic
│ │ │ │ -00016870: 4d61 7472 6978 2850 382c 3320 2020 2020  Matrix(P8,3     
│ │ │ │ -00016880: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00016840: 3220 3a20 7068 6920 3d20 696e 7665 7273  2 : phi = invers
│ │ │ │ +00016850: 654d 6170 2072 6174 696f 6e61 6c4d 6170  eMap rationalMap
│ │ │ │ +00016860: 2874 7269 6d28 6d69 6e6f 7273 2832 2c67  (trim(minors(2,g
│ │ │ │ +00016870: 656e 6572 6963 4d61 7472 6978 2850 382c  enericMatrix(P8,
│ │ │ │ +00016880: 3320 2020 2020 2020 2020 2020 7c0a 7c20  3           |.| 
│ │ │ │  00016890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000168a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000168b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000168c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000168d0: 2020 2020 2020 7c0a 7c6f 3220 3d20 2d2d        |.|o2 = --
│ │ │ │ -000168e0: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ -000168f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000168d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000168e0: 3220 3d20 2d2d 2072 6174 696f 6e61 6c20  2 = -- rational 
│ │ │ │ +000168f0: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │  00016900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016920: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016920: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016940: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ -00016950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016940: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +00016950: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │  00016960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016970: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ -00016980: 7572 6365 3a20 7375 6276 6172 6965 7479  urce: subvariety
│ │ │ │ -00016990: 206f 6620 5072 6f6a 282d 2d5b 7820 2c20   of Proj(--[x , 
│ │ │ │ -000169a0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -000169b0: 7820 2c20 7820 2c20 7820 2c20 2020 2020  x , x , x ,     
│ │ │ │ -000169c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016970: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016980: 2020 2020 736f 7572 6365 3a20 7375 6276      source: subv
│ │ │ │ +00016990: 6172 6965 7479 206f 6620 5072 6f6a 282d  ariety of Proj(-
│ │ │ │ +000169a0: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ +000169b0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +000169c0: 2c20 2020 2020 2020 2020 2020 7c0a 7c20  ,           |.| 
│ │ │ │  000169d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000169e0: 2020 2020 2020 2020 2039 3720 2030 2020           97  0  
│ │ │ │ -000169f0: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -00016a00: 2035 2020 2036 2020 2037 2020 2020 2020   5   6   7      
│ │ │ │ -00016a10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016a20: 2020 2020 2020 7b20 2020 2020 2020 2020        {         
│ │ │ │ +000169e0: 2020 2020 2020 2020 2020 2020 2020 2039                 9
│ │ │ │ +000169f0: 3720 2030 2020 2031 2020 2032 2020 2033  7  0   1   2   3
│ │ │ │ +00016a00: 2020 2034 2020 2035 2020 2036 2020 2037     4   5   6   7
│ │ │ │ +00016a10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016a20: 2020 2020 2020 2020 2020 2020 7b20 2020              {   
│ │ │ │  00016a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016a60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016a70: 2020 2020 2020 2020 3220 3220 2020 2020          2 2     
│ │ │ │ -00016a80: 2032 2020 2020 2020 2020 2020 3220 3220   2          2 2 
│ │ │ │ -00016a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016a60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016a70: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00016a80: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ +00016a90: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │  00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ab0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016ac0: 2020 2020 2020 2078 2078 2020 2b20 3134         x x  + 14
│ │ │ │ -00016ad0: 7820 7820 7820 202b 2032 3678 2078 2020  x x x  + 26x x  
│ │ │ │ -00016ae0: 2d20 3278 2078 2078 2078 2020 2d20 3134  - 2x x x x  - 14
│ │ │ │ -00016af0: 7820 7820 7820 7820 202b 2020 2020 2020  x x x x  +      
│ │ │ │ -00016b00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016b10: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -00016b20: 2031 2032 2033 2020 2020 2020 3120 3320   1 2 3      1 3 
│ │ │ │ -00016b30: 2020 2020 3020 3120 3220 3420 2020 2020      0 1 2 4     
│ │ │ │ -00016b40: 2030 2031 2033 2034 2020 2020 2020 2020   0 1 3 4        
│ │ │ │ -00016b50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016b60: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +00016ab0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016ac0: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ +00016ad0: 2020 2b20 3134 7820 7820 7820 202b 2032    + 14x x x  + 2
│ │ │ │ +00016ae0: 3678 2078 2020 2d20 3278 2078 2078 2078  6x x  - 2x x x x
│ │ │ │ +00016af0: 2020 2d20 3134 7820 7820 7820 7820 202b    - 14x x x x  +
│ │ │ │ +00016b00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016b10: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +00016b20: 3220 2020 2020 2031 2032 2033 2020 2020  2      1 2 3    
│ │ │ │ +00016b30: 2020 3120 3320 2020 2020 3020 3120 3220    1 3     0 1 2 
│ │ │ │ +00016b40: 3420 2020 2020 2030 2031 2033 2034 2020  4      0 1 3 4  
│ │ │ │ +00016b50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016b60: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │  00016b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ba0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016bb0: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ -00016bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bc0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │  00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016bf0: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ -00016c00: 7267 6574 3a20 5072 6f6a 282d 2d5b 7420  rget: Proj(--[t 
│ │ │ │ -00016c10: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ -00016c20: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ -00016c30: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ -00016c40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016c50: 2020 2020 2020 2020 2020 2039 3720 2030             97  0
│ │ │ │ -00016c60: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ -00016c70: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ -00016c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016c90: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ -00016ca0: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ -00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016bf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016c00: 2020 2020 7461 7267 6574 3a20 5072 6f6a      target: Proj
│ │ │ │ +00016c10: 282d 2d5b 7420 2c20 7420 2c20 7420 2c20  (--[t , t , t , 
│ │ │ │ +00016c20: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ +00016c30: 7420 2c20 7420 5d29 2020 2020 2020 2020  t , t ])        
│ │ │ │ +00016c40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016c60: 2039 3720 2030 2020 2031 2020 2032 2020   97  0   1   2  
│ │ │ │ +00016c70: 2033 2020 2034 2020 2035 2020 2036 2020   3   4   5   6  
│ │ │ │ +00016c80: 2037 2020 2038 2020 2020 2020 2020 2020   7   8          
│ │ │ │ +00016c90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016ca0: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ +00016cb0: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │  00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ce0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016cf0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -00016d00: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ -00016d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ce0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016d00: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ +00016d10: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  00016d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016d30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d50: 3520 3720 2020 2034 2038 2020 2020 2020  5 7    4 8      
│ │ │ │ +00016d50: 2020 2020 2020 3520 3720 2020 2034 2038        5 7    4 8
│ │ │ │  00016d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016d80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016d80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016dd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016de0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -00016df0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ -00016e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016df0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ +00016e00: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  00016e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016e20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e40: 3520 3620 2020 2031 2038 2020 2020 2020  5 6    1 8      
│ │ │ │ +00016e40: 2020 2020 2020 3520 3620 2020 2031 2038        5 6    1 8
│ │ │ │  00016e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016e70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016e70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016ec0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016ed0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -00016ee0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ -00016ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ec0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016ee0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ +00016ef0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  00016f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016f10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f30: 3420 3620 2020 2031 2037 2020 2020 2020  4 6    1 7      
│ │ │ │ +00016f30: 2020 2020 2020 3420 3620 2020 2031 2037        4 6    1 7
│ │ │ │  00016f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016f60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00016f60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00016f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00016fb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -00016fd0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ -00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00016fd0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ +00016fe0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017000: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017000: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017020: 3320 3720 2020 2032 2038 2020 2020 2020  3 7    2 8      
│ │ │ │ +00017020: 2020 2020 2020 3320 3720 2020 2032 2038        3 7    2 8
│ │ │ │  00017030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017050: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000170b0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -000170c0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ -000170d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000170b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000170c0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ +000170d0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  000170e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000170f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000170f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017110: 3320 3620 2020 2030 2038 2020 2020 2020  3 6    0 8      
│ │ │ │ +00017110: 2020 2020 2020 3320 3620 2020 2030 2038        3 6    0 8
│ │ │ │  00017120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017140: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017190: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -000171a0: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -000171b0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ -000171c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000171a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000171b0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ +000171c0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  000171d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000171e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000171e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000171f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017200: 3220 3620 2020 2030 2037 2020 2020 2020  2 6    0 7      
│ │ │ │ +00017200: 2020 2020 2020 3220 3620 2020 2030 2037        2 6    0 7
│ │ │ │  00017210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017230: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017230: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017280: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017290: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -000172a0: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ -000172b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017280: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000172a0: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ +000172b0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  000172c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000172d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000172e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000172f0: 3320 3420 2020 2032 2035 2020 2020 2020  3 4    2 5      
│ │ │ │ +000172f0: 2020 2020 2020 3320 3420 2020 2032 2035        3 4    2 5
│ │ │ │  00017300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017320: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017380: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -00017390: 2078 2020 2d20 7820 7820 2c20 2020 2020   x  - x x ,     
│ │ │ │ -000173a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017370: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017390: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │ +000173a0: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  000173b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000173c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000173c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000173d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000173e0: 3120 3320 2020 2030 2035 2020 2020 2020  1 3    0 5      
│ │ │ │ +000173e0: 2020 2020 2020 3120 3320 2020 2030 2035        1 3    0 5
│ │ │ │  000173f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017410: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017410: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017460: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017470: 2020 2020 2020 2020 2020 2020 2020 2078                 x
│ │ │ │ -00017480: 2078 2020 2d20 7820 7820 2020 2020 2020   x  - x x       
│ │ │ │ +00017460: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017480: 2020 2020 2078 2078 2020 2d20 7820 7820       x x  - x x 
│ │ │ │  00017490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000174a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000174b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000174c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000174d0: 3120 3220 2020 2030 2034 2020 2020 2020  1 2    0 4      
│ │ │ │ +000174d0: 2020 2020 2020 3120 3220 2020 2030 2034        1 2    0 4
│ │ │ │  000174e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000174f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017500: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017510: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ -00017520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017520: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │  00017530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017550: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017550: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000175a0: 2020 2020 2020 7c0a 7c6f 3220 3a20 5261        |.|o2 : Ra
│ │ │ │ -000175b0: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ -000175c0: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ -000175d0: 7020 6672 6f6d 2068 7970 6572 7375 7266  p from hypersurf
│ │ │ │ -000175e0: 6163 6520 696e 2050 505e 3920 2020 2020  ace in PP^9     
│ │ │ │ -000175f0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +000175a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +000175b0: 3220 3a20 5261 7469 6f6e 616c 4d61 7020  2 : RationalMap 
│ │ │ │ +000175c0: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ +000175d0: 6e61 6c20 6d61 7020 6672 6f6d 2068 7970  nal map from hyp
│ │ │ │ +000175e0: 6572 7375 7266 6163 6520 696e 2050 505e  ersurface in PP^
│ │ │ │ +000175f0: 3920 2020 2020 2020 2020 2020 7c0a 7c2d  9           |.|-
│ │ │ │  00017600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00017630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00017640: 2d2d 2d2d 2d2d 7c0a 7c2c 3329 292b 7261  ------|.|,3))+ra
│ │ │ │ -00017650: 6e64 6f6d 2832 2c50 3829 292c 446f 6d69  ndom(2,P8)),Domi
│ │ │ │ -00017660: 6e61 6e74 3d3e 7472 7565 2920 2020 2020  nant=>true)     
│ │ │ │ -00017670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2c  ------------|.|,
│ │ │ │ +00017650: 3329 292b 7261 6e64 6f6d 2832 2c50 3829  3))+random(2,P8)
│ │ │ │ +00017660: 292c 446f 6d69 6e61 6e74 3d3e 7472 7565  ),Dominant=>true
│ │ │ │ +00017670: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │  00017680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017690: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000176a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000176b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000176c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000176d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000176e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000176e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000176f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017730: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017780: 2020 2020 2020 7c0a 7c20 7820 2c20 7820        |.| x , x 
│ │ │ │ -00017790: 5d29 2064 6566 696e 6564 2062 7920 2020  ]) defined by   
│ │ │ │ -000177a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017780: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017790: 7820 2c20 7820 5d29 2064 6566 696e 6564  x , x ]) defined
│ │ │ │ +000177a0: 2062 7920 2020 2020 2020 2020 2020 2020   by             
│ │ │ │  000177b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000177c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000177d0: 2020 2020 2020 7c0a 7c20 2038 2020 2039        |.|  8   9
│ │ │ │ -000177e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000177d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000177e0: 2038 2020 2039 2020 2020 2020 2020 2020   8   9          
│ │ │ │  000177f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017820: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017820: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017870: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -00017880: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00017890: 2020 3220 3220 2020 2020 2020 2020 2032    2 2          2
│ │ │ │ -000178a0: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ -000178b0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000178c0: 2020 2020 2020 7c0a 7c31 3178 2078 2078        |.|11x x x
│ │ │ │ -000178d0: 2078 2020 2b20 3139 7820 7820 7820 202b   x  + 19x x x  +
│ │ │ │ -000178e0: 2078 2078 2020 2d20 3131 7820 7820 7820   x x  - 11x x x 
│ │ │ │ -000178f0: 202b 2033 3878 2078 2020 2d20 3134 7820   + 38x x  - 14x 
│ │ │ │ -00017900: 7820 7820 7820 202d 2031 3178 2078 2078  x x x  - 11x x x
│ │ │ │ -00017910: 2020 2b20 2020 7c0a 7c20 2020 3120 3220    +   |.|   1 2 
│ │ │ │ -00017920: 3320 3420 2020 2020 2031 2033 2034 2020  3 4      1 3 4  
│ │ │ │ -00017930: 2020 3020 3420 2020 2020 2030 2033 2034    0 4      0 3 4
│ │ │ │ -00017940: 2020 2020 2020 3320 3420 2020 2020 2030        3 4      0
│ │ │ │ -00017950: 2031 2032 2035 2020 2020 2020 3120 3220   1 2 5      1 2 
│ │ │ │ -00017960: 3520 2020 2020 7c0a 7c20 2020 2020 2020  5     |.|       
│ │ │ │ +00017870: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00017880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00017890: 2032 2020 2020 2020 3220 3220 2020 2020   2      2 2     
│ │ │ │ +000178a0: 2020 2020 2032 2020 2020 2020 3220 3220       2      2 2 
│ │ │ │ +000178b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000178c0: 2020 2020 3220 2020 2020 2020 7c0a 7c31      2       |.|1
│ │ │ │ +000178d0: 3178 2078 2078 2078 2020 2b20 3139 7820  1x x x x  + 19x 
│ │ │ │ +000178e0: 7820 7820 202b 2078 2078 2020 2d20 3131  x x  + x x  - 11
│ │ │ │ +000178f0: 7820 7820 7820 202b 2033 3878 2078 2020  x x x  + 38x x  
│ │ │ │ +00017900: 2d20 3134 7820 7820 7820 7820 202d 2031  - 14x x x x  - 1
│ │ │ │ +00017910: 3178 2078 2078 2020 2b20 2020 7c0a 7c20  1x x x  +   |.| 
│ │ │ │ +00017920: 2020 3120 3220 3320 3420 2020 2020 2031    1 2 3 4      1
│ │ │ │ +00017930: 2033 2034 2020 2020 3020 3420 2020 2020   3 4    0 4     
│ │ │ │ +00017940: 2030 2033 2034 2020 2020 2020 3320 3420   0 3 4      3 4 
│ │ │ │ +00017950: 2020 2020 2030 2031 2032 2035 2020 2020       0 1 2 5    
│ │ │ │ +00017960: 2020 3120 3220 3520 2020 2020 7c0a 7c20    1 2 5     |.| 
│ │ │ │  00017970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000179a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000179b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000179b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000179c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000179d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000179e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000179f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017a50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017aa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017b90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017c80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017cd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017d20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017d70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017d70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017e10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017e60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017eb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017f50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017fa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00017ff0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00017ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018040: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00018040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018090: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00018090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000180a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000180b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000180c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000180d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000180e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000180e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000180f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018130: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00018130: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018180: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00018180: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000181d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000181d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000181e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000181f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018220: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00018220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018270: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00018270: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000182c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +000182c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  000182d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000182f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018310: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00018310: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018360: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +00018360: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  00018370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000183b0: 2020 2020 2020 7c0a 7c20 746f 2050 505e        |.| to PP^
│ │ │ │ -000183c0: 3829 2020 2020 2020 2020 2020 2020 2020  8)              
│ │ │ │ +000183b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +000183c0: 746f 2050 505e 3829 2020 2020 2020 2020  to PP^8)        
│ │ │ │  000183d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000183f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018400: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00018400: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  00018410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018450: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00018450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  00018460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018470: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00018470: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00018480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000184a0: 2020 2020 2020 7c0a 7c34 3578 2078 2078        |.|45x x x
│ │ │ │ -000184b0: 2078 2020 2d20 3139 7820 7820 7820 7820   x  - 19x x x x 
│ │ │ │ -000184c0: 202b 2031 3478 2078 2078 2020 2b20 3131   + 14x x x  + 11
│ │ │ │ -000184d0: 7820 7820 7820 7820 202d 2031 3978 2078  x x x x  - 19x x
│ │ │ │ -000184e0: 2078 2078 2020 2b20 3231 7820 7820 7820   x x  + 21x x x 
│ │ │ │ -000184f0: 7820 202b 2020 7c0a 7c20 2020 3020 3120  x  +  |.|   0 1 
│ │ │ │ -00018500: 3320 3520 2020 2020 2031 2032 2033 2035  3 5      1 2 3 5
│ │ │ │ -00018510: 2020 2020 2020 3020 3420 3520 2020 2020        0 4 5     
│ │ │ │ -00018520: 2030 2032 2034 2035 2020 2020 2020 3020   0 2 4 5      0 
│ │ │ │ -00018530: 3320 3420 3520 2020 2020 2032 2033 2034  3 4 5      2 3 4
│ │ │ │ -00018540: 2035 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   5    |.|-------
│ │ │ │ +000184a0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +000184b0: 3578 2078 2078 2078 2020 2d20 3139 7820  5x x x x  - 19x 
│ │ │ │ +000184c0: 7820 7820 7820 202b 2031 3478 2078 2078  x x x  + 14x x x
│ │ │ │ +000184d0: 2020 2b20 3131 7820 7820 7820 7820 202d    + 11x x x x  -
│ │ │ │ +000184e0: 2031 3978 2078 2078 2078 2020 2b20 3231   19x x x x  + 21
│ │ │ │ +000184f0: 7820 7820 7820 7820 202b 2020 7c0a 7c20  x x x x  +  |.| 
│ │ │ │ +00018500: 2020 3020 3120 3320 3520 2020 2020 2031    0 1 3 5      1
│ │ │ │ +00018510: 2032 2033 2035 2020 2020 2020 3020 3420   2 3 5      0 4 
│ │ │ │ +00018520: 3520 2020 2020 2030 2032 2034 2035 2020  5      0 2 4 5  
│ │ │ │ +00018530: 2020 2020 3020 3320 3420 3520 2020 2020      0 3 4 5     
│ │ │ │ +00018540: 2032 2033 2034 2035 2020 2020 7c0a 7c2d   2 3 4 5    |.|-
│ │ │ │  00018550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018590: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 3220  ------|.|   2 2 
│ │ │ │ -000185a0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -000185b0: 3220 3220 2020 2020 2020 2032 2020 2020  2 2        2    
│ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000185d0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -000185e0: 2020 2020 2020 7c0a 7c32 3678 2078 2020        |.|26x x  
│ │ │ │ -000185f0: 2b20 3139 7820 7820 7820 202b 2033 3878  + 19x x x  + 38x
│ │ │ │ -00018600: 2078 2020 2d20 3330 7820 7820 7820 202d   x  - 30x x x  -
│ │ │ │ -00018610: 2031 3178 2078 2078 2078 2020 2d20 3131   11x x x x  - 11
│ │ │ │ -00018620: 7820 7820 7820 202b 2033 3078 2078 2078  x x x  + 30x x x
│ │ │ │ -00018630: 2078 2020 2b20 7c0a 7c20 2020 3020 3520   x  + |.|   0 5 
│ │ │ │ -00018640: 2020 2020 2030 2032 2035 2020 2020 2020       0 2 5      
│ │ │ │ -00018650: 3220 3520 2020 2020 2031 2032 2036 2020  2 5      1 2 6  
│ │ │ │ -00018660: 2020 2020 3120 3220 3320 3620 2020 2020      1 2 3 6     
│ │ │ │ -00018670: 2031 2033 2036 2020 2020 2020 3020 3220   1 3 6      0 2 
│ │ │ │ -00018680: 3420 3620 2020 7c0a 7c2d 2d2d 2d2d 2d2d  4 6   |.|-------
│ │ │ │ +00018590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +000185a0: 2020 3220 3220 2020 2020 2020 2020 2032    2 2          2
│ │ │ │ +000185b0: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ +000185c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000185d0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000185e0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +000185f0: 3678 2078 2020 2b20 3139 7820 7820 7820  6x x  + 19x x x 
│ │ │ │ +00018600: 202b 2033 3878 2078 2020 2d20 3330 7820   + 38x x  - 30x 
│ │ │ │ +00018610: 7820 7820 202d 2031 3178 2078 2078 2078  x x  - 11x x x x
│ │ │ │ +00018620: 2020 2d20 3131 7820 7820 7820 202b 2033    - 11x x x  + 3
│ │ │ │ +00018630: 3078 2078 2078 2078 2020 2b20 7c0a 7c20  0x x x x  + |.| 
│ │ │ │ +00018640: 2020 3020 3520 2020 2020 2030 2032 2035    0 5      0 2 5
│ │ │ │ +00018650: 2020 2020 2020 3220 3520 2020 2020 2031        2 5      1
│ │ │ │ +00018660: 2032 2036 2020 2020 2020 3120 3220 3320   2 6      1 2 3 
│ │ │ │ +00018670: 3620 2020 2020 2031 2033 2036 2020 2020  6      1 3 6    
│ │ │ │ +00018680: 2020 3020 3220 3420 3620 2020 7c0a 7c2d    0 2 4 6   |.|-
│ │ │ │  00018690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000186c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000186d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +000186d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  000186e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000186f0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000186f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00018700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018720: 2020 2020 2020 7c0a 7c32 3278 2078 2078        |.|22x x x
│ │ │ │ -00018730: 2078 2020 2b20 3131 7820 7820 7820 7820   x  + 11x x x x 
│ │ │ │ -00018740: 202d 2032 3278 2078 2078 2020 2b20 3130   - 22x x x  + 10
│ │ │ │ -00018750: 7820 7820 7820 7820 202b 2031 3178 2078  x x x x  + 11x x
│ │ │ │ -00018760: 2078 2078 2020 2d20 3432 7820 7820 7820   x x  - 42x x x 
│ │ │ │ -00018770: 7820 202d 2020 7c0a 7c20 2020 3120 3220  x  -  |.|   1 2 
│ │ │ │ -00018780: 3420 3620 2020 2020 2030 2033 2034 2036  4 6      0 3 4 6
│ │ │ │ -00018790: 2020 2020 2020 3020 3420 3620 2020 2020        0 4 6     
│ │ │ │ -000187a0: 2031 2032 2035 2036 2020 2020 2020 3020   1 2 5 6      0 
│ │ │ │ -000187b0: 3320 3520 3620 2020 2020 2031 2033 2035  3 5 6      1 3 5
│ │ │ │ -000187c0: 2036 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   6    |.|-------
│ │ │ │ +00018720: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +00018730: 3278 2078 2078 2078 2020 2b20 3131 7820  2x x x x  + 11x 
│ │ │ │ +00018740: 7820 7820 7820 202d 2032 3278 2078 2078  x x x  - 22x x x
│ │ │ │ +00018750: 2020 2b20 3130 7820 7820 7820 7820 202b    + 10x x x x  +
│ │ │ │ +00018760: 2031 3178 2078 2078 2078 2020 2d20 3432   11x x x x  - 42
│ │ │ │ +00018770: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ +00018780: 2020 3120 3220 3420 3620 2020 2020 2030    1 2 4 6      0
│ │ │ │ +00018790: 2033 2034 2036 2020 2020 2020 3020 3420   3 4 6      0 4 
│ │ │ │ +000187a0: 3620 2020 2020 2031 2032 2035 2036 2020  6      1 2 5 6  
│ │ │ │ +000187b0: 2020 2020 3020 3320 3520 3620 2020 2020      0 3 5 6     
│ │ │ │ +000187c0: 2031 2033 2035 2036 2020 2020 7c0a 7c2d   1 3 5 6    |.|-
│ │ │ │  000187d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000187f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018810: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -00018820: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00018830: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ -00018840: 2032 2020 2020 2032 2032 2020 2020 2020   2     2 2      
│ │ │ │ -00018850: 2020 2020 3220 2020 2020 2032 2032 2020      2      2 2  
│ │ │ │ -00018860: 2020 2020 2020 7c0a 7c31 3078 2078 2078        |.|10x x x
│ │ │ │ -00018870: 2078 2020 2b20 3432 7820 7820 7820 202d   x  + 42x x x  -
│ │ │ │ -00018880: 2033 3878 2078 2020 2d20 3337 7820 7820   38x x  - 37x x 
│ │ │ │ -00018890: 7820 202b 2037 7820 7820 202b 2032 3878  x  + 7x x  + 28x
│ │ │ │ -000188a0: 2078 2078 2020 2d20 3338 7820 7820 202d   x x  - 38x x  -
│ │ │ │ -000188b0: 2020 2020 2020 7c0a 7c20 2020 3020 3420        |.|   0 4 
│ │ │ │ -000188c0: 3520 3620 2020 2020 2030 2035 2036 2020  5 6      0 5 6  
│ │ │ │ -000188d0: 2020 2020 3220 3620 2020 2020 2032 2033      2 6      2 3
│ │ │ │ -000188e0: 2036 2020 2020 2033 2036 2020 2020 2020   6     3 6      
│ │ │ │ -000188f0: 3220 3420 3620 2020 2020 2034 2036 2020  2 4 6      4 6  
│ │ │ │ -00018900: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00018810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00018820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018830: 2032 2020 2020 2020 2020 3220 3220 2020   2        2 2   
│ │ │ │ +00018840: 2020 2020 2020 2032 2020 2020 2032 2032         2     2 2
│ │ │ │ +00018850: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00018860: 2032 2032 2020 2020 2020 2020 7c0a 7c31   2 2        |.|1
│ │ │ │ +00018870: 3078 2078 2078 2078 2020 2b20 3432 7820  0x x x x  + 42x 
│ │ │ │ +00018880: 7820 7820 202d 2033 3878 2078 2020 2d20  x x  - 38x x  - 
│ │ │ │ +00018890: 3337 7820 7820 7820 202b 2037 7820 7820  37x x x  + 7x x 
│ │ │ │ +000188a0: 202b 2032 3878 2078 2078 2020 2d20 3338   + 28x x x  - 38
│ │ │ │ +000188b0: 7820 7820 202d 2020 2020 2020 7c0a 7c20  x x  -      |.| 
│ │ │ │ +000188c0: 2020 3020 3420 3520 3620 2020 2020 2030    0 4 5 6      0
│ │ │ │ +000188d0: 2035 2036 2020 2020 2020 3220 3620 2020   5 6      2 6   
│ │ │ │ +000188e0: 2020 2032 2033 2036 2020 2020 2033 2036     2 3 6     3 6
│ │ │ │ +000188f0: 2020 2020 2020 3220 3420 3620 2020 2020        2 4 6     
│ │ │ │ +00018900: 2034 2036 2020 2020 2020 2020 7c0a 7c2d   4 6        |.|-
│ │ │ │  00018910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018950: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2032  ------|.|      2
│ │ │ │ -00018960: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -00018970: 2020 2020 2032 2020 2020 2020 3220 3220       2      2 2 
│ │ │ │ -00018980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018990: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -000189a0: 2020 2020 2020 7c0a 7c33 7820 7820 7820        |.|3x x x 
│ │ │ │ -000189b0: 202d 2032 3978 2078 2078 2020 2d20 3437   - 29x x x  - 47
│ │ │ │ -000189c0: 7820 7820 7820 202b 2033 3678 2078 2020  x x x  + 36x x  
│ │ │ │ -000189d0: 2b20 3330 7820 7820 7820 7820 202d 2032  + 30x x x x  - 2
│ │ │ │ -000189e0: 3278 2078 2078 2020 2d20 3230 7820 7820  2x x x  - 20x x 
│ │ │ │ -000189f0: 7820 7820 202d 7c0a 7c20 2032 2035 2036  x x  -|.|  2 5 6
│ │ │ │ -00018a00: 2020 2020 2020 3320 3520 3620 2020 2020        3 5 6     
│ │ │ │ -00018a10: 2034 2035 2036 2020 2020 2020 3520 3620   4 5 6      5 6 
│ │ │ │ -00018a20: 2020 2020 2030 2031 2032 2037 2020 2020       0 1 2 7    
│ │ │ │ -00018a30: 2020 3120 3220 3720 2020 2020 2031 2032    1 2 7      1 2
│ │ │ │ -00018a40: 2033 2037 2020 7c0a 7c2d 2d2d 2d2d 2d2d   3 7  |.|-------
│ │ │ │ +00018950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00018960: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00018970: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ +00018980: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +00018990: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000189a0: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +000189b0: 7820 7820 7820 202d 2032 3978 2078 2078  x x x  - 29x x x
│ │ │ │ +000189c0: 2020 2d20 3437 7820 7820 7820 202b 2033    - 47x x x  + 3
│ │ │ │ +000189d0: 3678 2078 2020 2b20 3330 7820 7820 7820  6x x  + 30x x x 
│ │ │ │ +000189e0: 7820 202d 2032 3278 2078 2078 2020 2d20  x  - 22x x x  - 
│ │ │ │ +000189f0: 3230 7820 7820 7820 7820 202d 7c0a 7c20  20x x x x  -|.| 
│ │ │ │ +00018a00: 2032 2035 2036 2020 2020 2020 3320 3520   2 5 6      3 5 
│ │ │ │ +00018a10: 3620 2020 2020 2034 2035 2036 2020 2020  6      4 5 6    
│ │ │ │ +00018a20: 2020 3520 3620 2020 2020 2030 2031 2032    5 6      0 1 2
│ │ │ │ +00018a30: 2037 2020 2020 2020 3120 3220 3720 2020   7      1 2 7   
│ │ │ │ +00018a40: 2020 2031 2032 2033 2037 2020 7c0a 7c2d     1 2 3 7  |.|-
│ │ │ │  00018a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018a90: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220  ------|.|     2 
│ │ │ │ -00018aa0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00018a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00018aa0: 2020 2020 3220 2020 2020 2020 2032 2020      2        2  
│ │ │ │  00018ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ac0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00018ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ae0: 2020 2020 2020 7c0a 7c34 3178 2078 2078        |.|41x x x
│ │ │ │ -00018af0: 2020 2d20 3330 7820 7820 7820 202b 2032    - 30x x x  + 2
│ │ │ │ -00018b00: 3278 2078 2078 2078 2020 2b20 3230 7820  2x x x x  + 20x 
│ │ │ │ -00018b10: 7820 7820 7820 202d 2034 7820 7820 7820  x x x  - 4x x x 
│ │ │ │ -00018b20: 202b 2032 3678 2078 2078 2078 2020 2b20   + 26x x x x  + 
│ │ │ │ -00018b30: 2020 2020 2020 7c0a 7c20 2020 3120 3320        |.|   1 3 
│ │ │ │ -00018b40: 3720 2020 2020 2030 2034 2037 2020 2020  7      0 4 7    
│ │ │ │ -00018b50: 2020 3020 3120 3420 3720 2020 2020 2030    0 1 4 7      0
│ │ │ │ -00018b60: 2033 2034 2037 2020 2020 2033 2034 2037   3 4 7     3 4 7
│ │ │ │ -00018b70: 2020 2020 2020 3120 3220 3520 3720 2020        1 2 5 7   
│ │ │ │ -00018b80: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00018ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018ad0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00018ae0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +00018af0: 3178 2078 2078 2020 2d20 3330 7820 7820  1x x x  - 30x x 
│ │ │ │ +00018b00: 7820 202b 2032 3278 2078 2078 2078 2020  x  + 22x x x x  
│ │ │ │ +00018b10: 2b20 3230 7820 7820 7820 7820 202d 2034  + 20x x x x  - 4
│ │ │ │ +00018b20: 7820 7820 7820 202b 2032 3678 2078 2078  x x x  + 26x x x
│ │ │ │ +00018b30: 2078 2020 2b20 2020 2020 2020 7c0a 7c20   x  +       |.| 
│ │ │ │ +00018b40: 2020 3120 3320 3720 2020 2020 2030 2034    1 3 7      0 4
│ │ │ │ +00018b50: 2037 2020 2020 2020 3020 3120 3420 3720   7      0 1 4 7 
│ │ │ │ +00018b60: 2020 2020 2030 2033 2034 2037 2020 2020       0 3 4 7    
│ │ │ │ +00018b70: 2033 2034 2037 2020 2020 2020 3120 3220   3 4 7      1 2 
│ │ │ │ +00018b80: 3520 3720 2020 2020 2020 2020 7c0a 7c2d  5 7         |.|-
│ │ │ │  00018b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018bd0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00018bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  00018be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018c10: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00018c20: 2020 2020 2020 7c0a 7c34 3178 2078 2078        |.|41x x x
│ │ │ │ -00018c30: 2078 2020 2d20 3238 7820 7820 7820 7820   x  - 28x x x x 
│ │ │ │ -00018c40: 202b 2034 7820 7820 7820 7820 202d 2032   + 4x x x x  - 2
│ │ │ │ -00018c50: 3678 2078 2078 2078 2020 2b20 3132 7820  6x x x x  + 12x 
│ │ │ │ -00018c60: 7820 7820 7820 202b 2032 3878 2078 2078  x x x  + 28x x x
│ │ │ │ -00018c70: 2020 2d20 2020 7c0a 7c20 2020 3020 3320    -   |.|   0 3 
│ │ │ │ -00018c80: 3520 3720 2020 2020 2031 2033 2035 2037  5 7      1 3 5 7
│ │ │ │ -00018c90: 2020 2020 2032 2033 2035 2037 2020 2020       2 3 5 7    
│ │ │ │ -00018ca0: 2020 3020 3420 3520 3720 2020 2020 2033    0 4 5 7      3
│ │ │ │ -00018cb0: 2034 2035 2037 2020 2020 2020 3020 3520   4 5 7      0 5 
│ │ │ │ -00018cc0: 3720 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d  7     |.|-------
│ │ │ │ +00018c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018c20: 2020 2020 3220 2020 2020 2020 7c0a 7c34      2       |.|4
│ │ │ │ +00018c30: 3178 2078 2078 2078 2020 2d20 3238 7820  1x x x x  - 28x 
│ │ │ │ +00018c40: 7820 7820 7820 202b 2034 7820 7820 7820  x x x  + 4x x x 
│ │ │ │ +00018c50: 7820 202d 2032 3678 2078 2078 2078 2020  x  - 26x x x x  
│ │ │ │ +00018c60: 2b20 3132 7820 7820 7820 7820 202b 2032  + 12x x x x  + 2
│ │ │ │ +00018c70: 3878 2078 2078 2020 2d20 2020 7c0a 7c20  8x x x  -   |.| 
│ │ │ │ +00018c80: 2020 3020 3320 3520 3720 2020 2020 2031    0 3 5 7      1
│ │ │ │ +00018c90: 2033 2035 2037 2020 2020 2032 2033 2035   3 5 7     2 3 5
│ │ │ │ +00018ca0: 2037 2020 2020 2020 3020 3420 3520 3720   7      0 4 5 7 
│ │ │ │ +00018cb0: 2020 2020 2033 2034 2035 2037 2020 2020       3 4 5 7    
│ │ │ │ +00018cc0: 2020 3020 3520 3720 2020 2020 7c0a 7c2d    0 5 7     |.|-
│ │ │ │  00018cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018d10: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220  ------|.|     2 
│ │ │ │ -00018d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00018d20: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00018d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018d50: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00018d60: 2020 2020 2020 7c0a 7c31 3278 2078 2078        |.|12x x x
│ │ │ │ -00018d70: 2020 2d20 3231 7820 7820 7820 7820 202d    - 21x x x x  -
│ │ │ │ -00018d80: 2032 3878 2078 2078 2078 2020 2b20 3337   28x x x x  + 37
│ │ │ │ -00018d90: 7820 7820 7820 7820 202b 2032 3178 2078  x x x x  + 21x x
│ │ │ │ -00018da0: 2078 2078 2020 2d20 3131 7820 7820 7820   x x  - 11x x x 
│ │ │ │ -00018db0: 202d 2020 2020 7c0a 7c20 2020 3220 3520   -    |.|   2 5 
│ │ │ │ -00018dc0: 3720 2020 2020 2030 2032 2036 2037 2020  7      0 2 6 7  
│ │ │ │ -00018dd0: 2020 2020 3120 3220 3620 3720 2020 2020      1 2 6 7     
│ │ │ │ -00018de0: 2030 2033 2036 2037 2020 2020 2020 3220   0 3 6 7      2 
│ │ │ │ -00018df0: 3320 3620 3720 2020 2020 2033 2036 2037  3 6 7      3 6 7
│ │ │ │ -00018e00: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00018d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00018d60: 2032 2020 2020 2020 2020 2020 7c0a 7c31   2          |.|1
│ │ │ │ +00018d70: 3278 2078 2078 2020 2d20 3231 7820 7820  2x x x  - 21x x 
│ │ │ │ +00018d80: 7820 7820 202d 2032 3878 2078 2078 2078  x x  - 28x x x x
│ │ │ │ +00018d90: 2020 2b20 3337 7820 7820 7820 7820 202b    + 37x x x x  +
│ │ │ │ +00018da0: 2032 3178 2078 2078 2078 2020 2d20 3131   21x x x x  - 11
│ │ │ │ +00018db0: 7820 7820 7820 202d 2020 2020 7c0a 7c20  x x x  -    |.| 
│ │ │ │ +00018dc0: 2020 3220 3520 3720 2020 2020 2030 2032    2 5 7      0 2
│ │ │ │ +00018dd0: 2036 2037 2020 2020 2020 3120 3220 3620   6 7      1 2 6 
│ │ │ │ +00018de0: 3720 2020 2020 2030 2033 2036 2037 2020  7      0 3 6 7  
│ │ │ │ +00018df0: 2020 2020 3220 3320 3620 3720 2020 2020      2 3 6 7     
│ │ │ │ +00018e00: 2033 2036 2037 2020 2020 2020 7c0a 7c2d   3 6 7      |.|-
│ │ │ │  00018e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018e50: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00018e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  00018e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00018e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00018ea0: 2020 2020 2020 7c0a 7c32 3878 2078 2078        |.|28x x x
│ │ │ │ -00018eb0: 2078 2020 2d20 3231 7820 7820 7820 7820   x  - 21x x x x 
│ │ │ │ -00018ec0: 202b 2033 7820 7820 7820 7820 202b 2034   + 3x x x x  + 4
│ │ │ │ -00018ed0: 3778 2078 2078 2078 2020 2b20 3438 7820  7x x x x  + 48x 
│ │ │ │ -00018ee0: 7820 7820 7820 202b 2032 7820 7820 7820  x x x  + 2x x x 
│ │ │ │ -00018ef0: 7820 202d 2020 7c0a 7c20 2020 3020 3420  x  -  |.|   0 4 
│ │ │ │ -00018f00: 3620 3720 2020 2020 2031 2034 2036 2037  6 7      1 4 6 7
│ │ │ │ -00018f10: 2020 2020 2030 2035 2036 2037 2020 2020       0 5 6 7    
│ │ │ │ -00018f20: 2020 3120 3520 3620 3720 2020 2020 2032    1 5 6 7      2
│ │ │ │ -00018f30: 2035 2036 2037 2020 2020 2033 2035 2036   5 6 7     3 5 6
│ │ │ │ -00018f40: 2037 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   7    |.|-------
│ │ │ │ +00018ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +00018eb0: 3878 2078 2078 2078 2020 2d20 3231 7820  8x x x x  - 21x 
│ │ │ │ +00018ec0: 7820 7820 7820 202b 2033 7820 7820 7820  x x x  + 3x x x 
│ │ │ │ +00018ed0: 7820 202b 2034 3778 2078 2078 2078 2020  x  + 47x x x x  
│ │ │ │ +00018ee0: 2b20 3438 7820 7820 7820 7820 202b 2032  + 48x x x x  + 2
│ │ │ │ +00018ef0: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ +00018f00: 2020 3020 3420 3620 3720 2020 2020 2031    0 4 6 7      1
│ │ │ │ +00018f10: 2034 2036 2037 2020 2020 2030 2035 2036   4 6 7     0 5 6
│ │ │ │ +00018f20: 2037 2020 2020 2020 3120 3520 3620 3720   7      1 5 6 7 
│ │ │ │ +00018f30: 2020 2020 2032 2035 2036 2037 2020 2020       2 5 6 7    
│ │ │ │ +00018f40: 2033 2035 2036 2037 2020 2020 7c0a 7c2d   3 5 6 7    |.|-
│ │ │ │  00018f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00018f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00018f90: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -00018fa0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00018fb0: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ -00018fc0: 2032 2020 2020 2020 3220 3220 2020 2020   2      2 2     
│ │ │ │ -00018fd0: 2020 2020 2032 2020 2020 2020 3220 3220       2      2 2 
│ │ │ │ -00018fe0: 2020 2020 2020 7c0a 7c34 3578 2078 2078        |.|45x x x
│ │ │ │ -00018ff0: 2078 2020 2d20 3333 7820 7820 7820 202d   x  - 33x x x  -
│ │ │ │ -00019000: 2033 3878 2078 2020 2b20 3238 7820 7820   38x x  + 28x x 
│ │ │ │ -00019010: 7820 202d 2033 3878 2078 2020 2d20 3231  x  - 38x x  - 21
│ │ │ │ -00019020: 7820 7820 7820 202b 2034 3878 2078 2020  x x x  + 48x x  
│ │ │ │ -00019030: 2d20 2020 2020 7c0a 7c20 2020 3420 3520  -     |.|   4 5 
│ │ │ │ -00019040: 3620 3720 2020 2020 2035 2036 2037 2020  6 7      5 6 7  
│ │ │ │ -00019050: 2020 2020 3020 3720 2020 2020 2030 2031      0 7      0 1
│ │ │ │ -00019060: 2037 2020 2020 2020 3120 3720 2020 2020   7      1 7     
│ │ │ │ -00019070: 2030 2033 2037 2020 2020 2020 3320 3720   0 3 7      3 7 
│ │ │ │ -00019080: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00018f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00018fa0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00018fb0: 2020 2020 2020 2020 2020 3220 3220 2020            2 2   
│ │ │ │ +00018fc0: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ +00018fd0: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ +00018fe0: 2020 3220 3220 2020 2020 2020 7c0a 7c34    2 2       |.|4
│ │ │ │ +00018ff0: 3578 2078 2078 2078 2020 2d20 3333 7820  5x x x x  - 33x 
│ │ │ │ +00019000: 7820 7820 202d 2033 3878 2078 2020 2b20  x x  - 38x x  + 
│ │ │ │ +00019010: 3238 7820 7820 7820 202d 2033 3878 2078  28x x x  - 38x x
│ │ │ │ +00019020: 2020 2d20 3231 7820 7820 7820 202b 2034    - 21x x x  + 4
│ │ │ │ +00019030: 3878 2078 2020 2d20 2020 2020 7c0a 7c20  8x x  -     |.| 
│ │ │ │ +00019040: 2020 3420 3520 3620 3720 2020 2020 2035    4 5 6 7      5
│ │ │ │ +00019050: 2036 2037 2020 2020 2020 3020 3720 2020   6 7      0 7   
│ │ │ │ +00019060: 2020 2030 2031 2037 2020 2020 2020 3120     0 1 7      1 
│ │ │ │ +00019070: 3720 2020 2020 2030 2033 2037 2020 2020  7      0 3 7    
│ │ │ │ +00019080: 2020 3320 3720 2020 2020 2020 7c0a 7c2d    3 7       |.|-
│ │ │ │  00019090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000190a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000190b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000190c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000190d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -000190e0: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ -000190f0: 2020 2020 2020 3220 2020 2020 2032 2032        2      2 2
│ │ │ │ -00019100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019110: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ -00019120: 3220 2020 2020 7c0a 7c34 3878 2078 2078  2     |.|48x x x
│ │ │ │ -00019130: 2020 2b20 3435 7820 7820 7820 202d 2034    + 45x x x  - 4
│ │ │ │ -00019140: 3778 2078 2078 2020 2d20 3438 7820 7820  7x x x  - 48x x 
│ │ │ │ -00019150: 202b 2031 3178 2078 2078 2078 2020 2d20   + 11x x x x  - 
│ │ │ │ -00019160: 3130 7820 7820 7820 202b 2032 3078 2078  10x x x  + 20x x
│ │ │ │ -00019170: 2078 2020 2b20 7c0a 7c20 2020 3020 3520   x  + |.|   0 5 
│ │ │ │ -00019180: 3720 2020 2020 2031 2035 2037 2020 2020  7      1 5 7    
│ │ │ │ -00019190: 2020 3320 3520 3720 2020 2020 2035 2037    3 5 7      5 7
│ │ │ │ -000191a0: 2020 2020 2020 3020 3120 3220 3820 2020        0 1 2 8   
│ │ │ │ -000191b0: 2020 2031 2032 2038 2020 2020 2020 3120     1 2 8      1 
│ │ │ │ -000191c0: 3220 3820 2020 7c0a 7c2d 2d2d 2d2d 2d2d  2 8   |.|-------
│ │ │ │ +000190d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +000190e0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +000190f0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ +00019100: 2020 2032 2032 2020 2020 2020 2020 2020     2 2          
│ │ │ │ +00019110: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00019120: 2020 2020 2020 3220 2020 2020 7c0a 7c34        2     |.|4
│ │ │ │ +00019130: 3878 2078 2078 2020 2b20 3435 7820 7820  8x x x  + 45x x 
│ │ │ │ +00019140: 7820 202d 2034 3778 2078 2078 2020 2d20  x  - 47x x x  - 
│ │ │ │ +00019150: 3438 7820 7820 202b 2031 3178 2078 2078  48x x  + 11x x x
│ │ │ │ +00019160: 2078 2020 2d20 3130 7820 7820 7820 202b   x  - 10x x x  +
│ │ │ │ +00019170: 2032 3078 2078 2078 2020 2b20 7c0a 7c20   20x x x  + |.| 
│ │ │ │ +00019180: 2020 3020 3520 3720 2020 2020 2031 2035    0 5 7      1 5
│ │ │ │ +00019190: 2037 2020 2020 2020 3320 3520 3720 2020   7      3 5 7   
│ │ │ │ +000191a0: 2020 2035 2037 2020 2020 2020 3020 3120     5 7      0 1 
│ │ │ │ +000191b0: 3220 3820 2020 2020 2031 2032 2038 2020  2 8      1 2 8  
│ │ │ │ +000191c0: 2020 2020 3120 3220 3820 2020 7c0a 7c2d      1 2 8   |.|-
│ │ │ │  000191d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000191e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000191f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019210: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -00019220: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00019210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00019220: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00019230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019240: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00019240: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00019250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019260: 2020 2020 2020 7c0a 7c31 3178 2078 2078        |.|11x x x
│ │ │ │ -00019270: 2078 2020 2b20 3432 7820 7820 7820 202b   x  + 42x x x  +
│ │ │ │ -00019280: 2034 3178 2078 2078 2078 2020 2d20 3131   41x x x x  - 11
│ │ │ │ -00019290: 7820 7820 7820 202b 2031 3078 2078 2078  x x x  + 10x x x
│ │ │ │ -000192a0: 2078 2020 2d20 3230 7820 7820 7820 7820   x  - 20x x x x 
│ │ │ │ -000192b0: 202d 2020 2020 7c0a 7c20 2020 3020 3120   -    |.|   0 1 
│ │ │ │ -000192c0: 3320 3820 2020 2020 2031 2033 2038 2020  3 8      1 3 8  
│ │ │ │ -000192d0: 2020 2020 3120 3220 3320 3820 2020 2020      1 2 3 8     
│ │ │ │ -000192e0: 2030 2034 2038 2020 2020 2020 3020 3120   0 4 8      0 1 
│ │ │ │ -000192f0: 3420 3820 2020 2020 2030 2032 2034 2038  4 8      0 2 4 8
│ │ │ │ -00019300: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00019260: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +00019270: 3178 2078 2078 2078 2020 2b20 3432 7820  1x x x x  + 42x 
│ │ │ │ +00019280: 7820 7820 202b 2034 3178 2078 2078 2078  x x  + 41x x x x
│ │ │ │ +00019290: 2020 2d20 3131 7820 7820 7820 202b 2031    - 11x x x  + 1
│ │ │ │ +000192a0: 3078 2078 2078 2078 2020 2d20 3230 7820  0x x x x  - 20x 
│ │ │ │ +000192b0: 7820 7820 7820 202d 2020 2020 7c0a 7c20  x x x  -    |.| 
│ │ │ │ +000192c0: 2020 3020 3120 3320 3820 2020 2020 2031    0 1 3 8      1
│ │ │ │ +000192d0: 2033 2038 2020 2020 2020 3120 3220 3320   3 8      1 2 3 
│ │ │ │ +000192e0: 3820 2020 2020 2030 2034 2038 2020 2020  8      0 4 8    
│ │ │ │ +000192f0: 2020 3020 3120 3420 3820 2020 2020 2030    0 1 4 8      0
│ │ │ │ +00019300: 2032 2034 2038 2020 2020 2020 7c0a 7c2d   2 4 8      |.|-
│ │ │ │  00019310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019350: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00019350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  00019360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019380: 2020 2020 3220 2020 2020 2020 2020 2032      2          2
│ │ │ │ -00019390: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -000193a0: 2020 2020 2020 7c0a 7c32 3678 2078 2078        |.|26x x x
│ │ │ │ -000193b0: 2078 2020 2b20 3238 7820 7820 7820 7820   x  + 28x x x x 
│ │ │ │ -000193c0: 202b 2034 7820 7820 7820 7820 202b 2032   + 4x x x x  + 2
│ │ │ │ -000193d0: 3678 2078 2078 2020 2d20 3132 7820 7820  6x x x  - 12x x 
│ │ │ │ -000193e0: 7820 202d 2031 3178 2078 2078 2020 2d20  x  - 11x x x  - 
│ │ │ │ -000193f0: 2020 2020 2020 7c0a 7c20 2020 3120 3220        |.|   1 2 
│ │ │ │ -00019400: 3420 3820 2020 2020 2031 2033 2034 2038  4 8      1 3 4 8
│ │ │ │ -00019410: 2020 2020 2032 2033 2034 2038 2020 2020       2 3 4 8    
│ │ │ │ -00019420: 2020 3020 3420 3820 2020 2020 2033 2034    0 4 8      3 4
│ │ │ │ -00019430: 2038 2020 2020 2020 3020 3520 3820 2020   8      0 5 8   
│ │ │ │ -00019440: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00019380: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00019390: 2020 2020 2032 2020 2020 2020 2020 3220       2        2 
│ │ │ │ +000193a0: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +000193b0: 3678 2078 2078 2078 2020 2b20 3238 7820  6x x x x  + 28x 
│ │ │ │ +000193c0: 7820 7820 7820 202b 2034 7820 7820 7820  x x x  + 4x x x 
│ │ │ │ +000193d0: 7820 202b 2032 3678 2078 2078 2020 2d20  x  + 26x x x  - 
│ │ │ │ +000193e0: 3132 7820 7820 7820 202d 2031 3178 2078  12x x x  - 11x x
│ │ │ │ +000193f0: 2078 2020 2d20 2020 2020 2020 7c0a 7c20   x  -       |.| 
│ │ │ │ +00019400: 2020 3120 3220 3420 3820 2020 2020 2031    1 2 4 8      1
│ │ │ │ +00019410: 2033 2034 2038 2020 2020 2032 2033 2034   3 4 8     2 3 4
│ │ │ │ +00019420: 2038 2020 2020 2020 3020 3420 3820 2020   8      0 4 8   
│ │ │ │ +00019430: 2020 2033 2034 2038 2020 2020 2020 3020     3 4 8      0 
│ │ │ │ +00019440: 3520 3820 2020 2020 2020 2020 7c0a 7c2d  5 8         |.|-
│ │ │ │  00019450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019490: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00019490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  000194a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000194b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000194b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  000194c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000194d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000194e0: 2020 2020 2020 7c0a 7c34 3278 2078 2078        |.|42x x x
│ │ │ │ -000194f0: 2078 2020 2d20 3431 7820 7820 7820 7820   x  - 41x x x x 
│ │ │ │ -00019500: 202d 2034 7820 7820 7820 202d 2032 3878   - 4x x x  - 28x
│ │ │ │ -00019510: 2078 2078 2078 2020 2b20 3132 7820 7820   x x x  + 12x x 
│ │ │ │ -00019520: 7820 7820 202b 2033 3778 2078 2078 2078  x x  + 37x x x x
│ │ │ │ -00019530: 2020 2b20 2020 7c0a 7c20 2020 3020 3120    +   |.|   0 1 
│ │ │ │ -00019540: 3520 3820 2020 2020 2030 2032 2035 2038  5 8      0 2 5 8
│ │ │ │ -00019550: 2020 2020 2032 2035 2038 2020 2020 2020       2 5 8      
│ │ │ │ -00019560: 3020 3420 3520 3820 2020 2020 2032 2034  0 4 5 8      2 4
│ │ │ │ -00019570: 2035 2038 2020 2020 2020 3020 3220 3620   5 8      0 2 6 
│ │ │ │ -00019580: 3820 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d  8     |.|-------
│ │ │ │ +000194e0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +000194f0: 3278 2078 2078 2078 2020 2d20 3431 7820  2x x x x  - 41x 
│ │ │ │ +00019500: 7820 7820 7820 202d 2034 7820 7820 7820  x x x  - 4x x x 
│ │ │ │ +00019510: 202d 2032 3878 2078 2078 2078 2020 2b20   - 28x x x x  + 
│ │ │ │ +00019520: 3132 7820 7820 7820 7820 202b 2033 3778  12x x x x  + 37x
│ │ │ │ +00019530: 2078 2078 2078 2020 2b20 2020 7c0a 7c20   x x x  +   |.| 
│ │ │ │ +00019540: 2020 3020 3120 3520 3820 2020 2020 2030    0 1 5 8      0
│ │ │ │ +00019550: 2032 2035 2038 2020 2020 2032 2035 2038   2 5 8     2 5 8
│ │ │ │ +00019560: 2020 2020 2020 3020 3420 3520 3820 2020        0 4 5 8   
│ │ │ │ +00019570: 2020 2032 2034 2035 2038 2020 2020 2020     2 4 5 8      
│ │ │ │ +00019580: 3020 3220 3620 3820 2020 2020 7c0a 7c2d  0 2 6 8     |.|-
│ │ │ │  00019590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000195a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000195b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000195c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000195d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -000195e0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000195d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +000195e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  000195f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019620: 2020 2020 2020 7c0a 7c33 7820 7820 7820        |.|3x x x 
│ │ │ │ -00019630: 7820 202d 2032 3178 2078 2078 2020 2d20  x  - 21x x x  - 
│ │ │ │ -00019640: 3134 7820 7820 7820 7820 202b 2032 3978  14x x x x  + 29x
│ │ │ │ -00019650: 2078 2078 2078 2020 2b20 3131 7820 7820   x x x  + 11x x 
│ │ │ │ -00019660: 7820 7820 202b 2034 3778 2078 2078 2078  x x  + 47x x x x
│ │ │ │ -00019670: 2020 2d20 2020 7c0a 7c20 2031 2032 2036    -   |.|  1 2 6
│ │ │ │ -00019680: 2038 2020 2020 2020 3220 3620 3820 2020   8      2 6 8   
│ │ │ │ -00019690: 2020 2030 2033 2036 2038 2020 2020 2020     0 3 6 8      
│ │ │ │ -000196a0: 3120 3320 3620 3820 2020 2020 2032 2033  1 3 6 8      2 3
│ │ │ │ -000196b0: 2036 2038 2020 2020 2020 3120 3420 3620   6 8      1 4 6 
│ │ │ │ -000196c0: 3820 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d  8     |.|-------
│ │ │ │ +00019620: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +00019630: 7820 7820 7820 7820 202d 2032 3178 2078  x x x x  - 21x x
│ │ │ │ +00019640: 2078 2020 2d20 3134 7820 7820 7820 7820   x  - 14x x x x 
│ │ │ │ +00019650: 202b 2032 3978 2078 2078 2078 2020 2b20   + 29x x x x  + 
│ │ │ │ +00019660: 3131 7820 7820 7820 7820 202b 2034 3778  11x x x x  + 47x
│ │ │ │ +00019670: 2078 2078 2078 2020 2d20 2020 7c0a 7c20   x x x  -   |.| 
│ │ │ │ +00019680: 2031 2032 2036 2038 2020 2020 2020 3220   1 2 6 8      2 
│ │ │ │ +00019690: 3620 3820 2020 2020 2030 2033 2036 2038  6 8      0 3 6 8
│ │ │ │ +000196a0: 2020 2020 2020 3120 3320 3620 3820 2020        1 3 6 8   
│ │ │ │ +000196b0: 2020 2032 2033 2036 2038 2020 2020 2020     2 3 6 8      
│ │ │ │ +000196c0: 3120 3420 3620 3820 2020 2020 7c0a 7c2d  1 4 6 8     |.|-
│ │ │ │  000196d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000196e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000196f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019710: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00019710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  00019720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019730: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00019730: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  00019740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019760: 2020 2020 2020 7c0a 7c34 3878 2078 2078        |.|48x x x
│ │ │ │ -00019770: 2078 2020 2d20 3278 2078 2078 2078 2020   x  - 2x x x x  
│ │ │ │ -00019780: 2b20 3435 7820 7820 7820 202b 2032 3978  + 45x x x  + 29x
│ │ │ │ -00019790: 2078 2078 2078 2020 2b20 3235 7820 7820   x x x  + 25x x 
│ │ │ │ -000197a0: 7820 7820 202b 2033 3378 2078 2078 2078  x x  + 33x x x x
│ │ │ │ -000197b0: 2020 2d20 2020 7c0a 7c20 2020 3220 3420    -   |.|   2 4 
│ │ │ │ -000197c0: 3620 3820 2020 2020 3320 3420 3620 3820  6 8     3 4 6 8 
│ │ │ │ -000197d0: 2020 2020 2034 2036 2038 2020 2020 2020       4 6 8      
│ │ │ │ -000197e0: 3020 3520 3620 3820 2020 2020 2031 2035  0 5 6 8      1 5
│ │ │ │ -000197f0: 2036 2038 2020 2020 2020 3420 3520 3620   6 8      4 5 6 
│ │ │ │ -00019800: 3820 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d  8     |.|-------
│ │ │ │ +00019760: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +00019770: 3878 2078 2078 2078 2020 2d20 3278 2078  8x x x x  - 2x x
│ │ │ │ +00019780: 2078 2078 2020 2b20 3435 7820 7820 7820   x x  + 45x x x 
│ │ │ │ +00019790: 202b 2032 3978 2078 2078 2078 2020 2b20   + 29x x x x  + 
│ │ │ │ +000197a0: 3235 7820 7820 7820 7820 202b 2033 3378  25x x x x  + 33x
│ │ │ │ +000197b0: 2078 2078 2078 2020 2d20 2020 7c0a 7c20   x x x  -   |.| 
│ │ │ │ +000197c0: 2020 3220 3420 3620 3820 2020 2020 3320    2 4 6 8     3 
│ │ │ │ +000197d0: 3420 3620 3820 2020 2020 2034 2036 2038  4 6 8      4 6 8
│ │ │ │ +000197e0: 2020 2020 2020 3020 3520 3620 3820 2020        0 5 6 8   
│ │ │ │ +000197f0: 2020 2031 2035 2036 2038 2020 2020 2020     1 5 6 8      
│ │ │ │ +00019800: 3420 3520 3620 3820 2020 2020 7c0a 7c2d  4 5 6 8     |.|-
│ │ │ │  00019810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019850: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ -00019860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019870: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00019850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00019860: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00019870: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00019880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000198a0: 2020 2020 2020 7c0a 7c33 3778 2078 2078        |.|37x x x
│ │ │ │ -000198b0: 2020 2d20 3378 2078 2078 2078 2020 2d20    - 3x x x x  - 
│ │ │ │ -000198c0: 3437 7820 7820 7820 202b 2032 3178 2078  47x x x  + 21x x
│ │ │ │ -000198d0: 2078 2078 2020 2b20 3131 7820 7820 7820   x x  + 11x x x 
│ │ │ │ -000198e0: 7820 202b 2078 2078 2078 2078 2020 2b20  x  + x x x x  + 
│ │ │ │ -000198f0: 2020 2020 2020 7c0a 7c20 2020 3020 3720        |.|   0 7 
│ │ │ │ -00019900: 3820 2020 2020 3020 3120 3720 3820 2020  8     0 1 7 8   
│ │ │ │ -00019910: 2020 2031 2037 2038 2020 2020 2020 3020     1 7 8      0 
│ │ │ │ -00019920: 3220 3720 3820 2020 2020 2030 2033 2037  2 7 8      0 3 7
│ │ │ │ -00019930: 2038 2020 2020 3220 3320 3720 3820 2020   8    2 3 7 8   
│ │ │ │ -00019940: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +000198a0: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +000198b0: 3778 2078 2078 2020 2d20 3378 2078 2078  7x x x  - 3x x x
│ │ │ │ +000198c0: 2078 2020 2d20 3437 7820 7820 7820 202b   x  - 47x x x  +
│ │ │ │ +000198d0: 2032 3178 2078 2078 2078 2020 2b20 3131   21x x x x  + 11
│ │ │ │ +000198e0: 7820 7820 7820 7820 202b 2078 2078 2078  x x x x  + x x x
│ │ │ │ +000198f0: 2078 2020 2b20 2020 2020 2020 7c0a 7c20   x  +       |.| 
│ │ │ │ +00019900: 2020 3020 3720 3820 2020 2020 3020 3120    0 7 8     0 1 
│ │ │ │ +00019910: 3720 3820 2020 2020 2031 2037 2038 2020  7 8      1 7 8  
│ │ │ │ +00019920: 2020 2020 3020 3220 3720 3820 2020 2020      0 2 7 8     
│ │ │ │ +00019930: 2030 2033 2037 2038 2020 2020 3220 3320   0 3 7 8    2 3 
│ │ │ │ +00019940: 3720 3820 2020 2020 2020 2020 7c0a 7c2d  7 8         |.|-
│ │ │ │  00019950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019990: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +00019990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  000199a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000199b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000199c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000199d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000199e0: 2020 2020 2020 7c0a 7c34 3878 2078 2078        |.|48x x x
│ │ │ │ -000199f0: 2078 2020 2d20 3435 7820 7820 7820 7820   x  - 45x x x x 
│ │ │ │ -00019a00: 202b 2034 3778 2078 2078 2078 2020 2d20   + 47x x x x  - 
│ │ │ │ -00019a10: 3278 2078 2078 2078 2020 2b20 3333 7820  2x x x x  + 33x 
│ │ │ │ -00019a20: 7820 7820 7820 202b 2034 3778 2078 2078  x x x  + 47x x x
│ │ │ │ -00019a30: 2078 2020 2d20 7c0a 7c20 2020 3020 3420   x  - |.|   0 4 
│ │ │ │ -00019a40: 3720 3820 2020 2020 2031 2034 2037 2038  7 8      1 4 7 8
│ │ │ │ -00019a50: 2020 2020 2020 3320 3420 3720 3820 2020        3 4 7 8   
│ │ │ │ -00019a60: 2020 3020 3520 3720 3820 2020 2020 2031    0 5 7 8      1
│ │ │ │ -00019a70: 2035 2037 2038 2020 2020 2020 3220 3520   5 7 8      2 5 
│ │ │ │ -00019a80: 3720 3820 2020 7c0a 7c2d 2d2d 2d2d 2d2d  7 8   |.|-------
│ │ │ │ +000199e0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +000199f0: 3878 2078 2078 2078 2020 2d20 3435 7820  8x x x x  - 45x 
│ │ │ │ +00019a00: 7820 7820 7820 202b 2034 3778 2078 2078  x x x  + 47x x x
│ │ │ │ +00019a10: 2078 2020 2d20 3278 2078 2078 2078 2020   x  - 2x x x x  
│ │ │ │ +00019a20: 2b20 3333 7820 7820 7820 7820 202b 2034  + 33x x x x  + 4
│ │ │ │ +00019a30: 3778 2078 2078 2078 2020 2d20 7c0a 7c20  7x x x x  - |.| 
│ │ │ │ +00019a40: 2020 3020 3420 3720 3820 2020 2020 2031    0 4 7 8      1
│ │ │ │ +00019a50: 2034 2037 2038 2020 2020 2020 3320 3420   4 7 8      3 4 
│ │ │ │ +00019a60: 3720 3820 2020 2020 3020 3520 3720 3820  7 8     0 5 7 8 
│ │ │ │ +00019a70: 2020 2020 2031 2035 2037 2038 2020 2020       1 5 7 8    
│ │ │ │ +00019a80: 2020 3220 3520 3720 3820 2020 7c0a 7c2d    2 5 7 8   |.|-
│ │ │ │  00019a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019ad0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -00019ae0: 2020 2020 2020 3220 3220 2020 2020 2020        2 2       
│ │ │ │ -00019af0: 2020 2032 2020 2020 2020 3220 3220 2020     2      2 2   
│ │ │ │ -00019b00: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ -00019b10: 3220 2020 2020 2020 2020 3220 2020 2020  2         2     
│ │ │ │ -00019b20: 2020 2020 2032 7c0a 7c78 2078 2078 2078       2|.|x x x x
│ │ │ │ -00019b30: 2020 2b20 3778 2078 2020 2d20 3239 7820    + 7x x  - 29x 
│ │ │ │ -00019b40: 7820 7820 202b 2033 3678 2078 2020 2d20  x x  + 36x x  - 
│ │ │ │ -00019b50: 3131 7820 7820 7820 202b 2034 3878 2078  11x x x  + 48x x
│ │ │ │ -00019b60: 2020 2b20 3278 2078 2078 2020 2d20 3333    + 2x x x  - 33
│ │ │ │ -00019b70: 7820 7820 7820 7c0a 7c20 3420 3520 3720  x x x |.| 4 5 7 
│ │ │ │ -00019b80: 3820 2020 2020 3020 3820 2020 2020 2030  8     0 8      0
│ │ │ │ -00019b90: 2031 2038 2020 2020 2020 3120 3820 2020   1 8      1 8   
│ │ │ │ -00019ba0: 2020 2030 2032 2038 2020 2020 2020 3220     0 2 8      2 
│ │ │ │ -00019bb0: 3820 2020 2020 3020 3420 3820 2020 2020  8     0 4 8     
│ │ │ │ -00019bc0: 2031 2034 2038 7c0a 7c2d 2d2d 2d2d 2d2d   1 4 8|.|-------
│ │ │ │ +00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00019ae0: 2020 2020 2020 2020 2020 2020 3220 3220              2 2 
│ │ │ │ +00019af0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00019b00: 3220 3220 2020 2020 2020 2020 2032 2020  2 2          2  
│ │ │ │ +00019b10: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ +00019b20: 3220 2020 2020 2020 2020 2032 7c0a 7c78  2          2|.|x
│ │ │ │ +00019b30: 2078 2078 2078 2020 2b20 3778 2078 2020   x x x  + 7x x  
│ │ │ │ +00019b40: 2d20 3239 7820 7820 7820 202b 2033 3678  - 29x x x  + 36x
│ │ │ │ +00019b50: 2078 2020 2d20 3131 7820 7820 7820 202b   x  - 11x x x  +
│ │ │ │ +00019b60: 2034 3878 2078 2020 2b20 3278 2078 2078   48x x  + 2x x x
│ │ │ │ +00019b70: 2020 2d20 3333 7820 7820 7820 7c0a 7c20    - 33x x x |.| 
│ │ │ │ +00019b80: 3420 3520 3720 3820 2020 2020 3020 3820  4 5 7 8     0 8 
│ │ │ │ +00019b90: 2020 2020 2030 2031 2038 2020 2020 2020       0 1 8      
│ │ │ │ +00019ba0: 3120 3820 2020 2020 2030 2032 2038 2020  1 8      0 2 8  
│ │ │ │ +00019bb0: 2020 2020 3220 3820 2020 2020 3020 3420      2 8     0 4 
│ │ │ │ +00019bc0: 3820 2020 2020 2031 2034 2038 7c0a 7c2d  8      1 4 8|.|-
│ │ │ │  00019bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019c10: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -00019c20: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ -00019c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +00019c20: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ +00019c30: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00019c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019c60: 2020 2020 2020 7c0a 7c2d 2034 3778 2078        |.|- 47x x
│ │ │ │ -00019c70: 2078 2020 2d20 3438 7820 7820 202b 2034   x  - 48x x  + 4
│ │ │ │ -00019c80: 3878 2078 2078 2078 2020 2d20 3438 7820  8x x x x  - 48x 
│ │ │ │ -00019c90: 7820 7820 7820 202d 2034 3878 2078 2078  x x x  - 48x x x
│ │ │ │ -00019ca0: 2078 2020 2b20 3438 7820 7820 7820 7820   x  + 48x x x x 
│ │ │ │ -00019cb0: 202b 2020 2020 7c0a 7c20 2020 2020 3220   +    |.|     2 
│ │ │ │ -00019cc0: 3420 3820 2020 2020 2034 2038 2020 2020  4 8      4 8    
│ │ │ │ -00019cd0: 2020 3320 3420 3620 3920 2020 2020 2032    3 4 6 9      2
│ │ │ │ -00019ce0: 2035 2036 2039 2020 2020 2020 3120 3320   5 6 9      1 3 
│ │ │ │ -00019cf0: 3720 3920 2020 2020 2030 2035 2037 2039  7 9      0 5 7 9
│ │ │ │ -00019d00: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +00019c60: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00019c70: 2034 3778 2078 2078 2020 2d20 3438 7820   47x x x  - 48x 
│ │ │ │ +00019c80: 7820 202b 2034 3878 2078 2078 2078 2020  x  + 48x x x x  
│ │ │ │ +00019c90: 2d20 3438 7820 7820 7820 7820 202d 2034  - 48x x x x  - 4
│ │ │ │ +00019ca0: 3878 2078 2078 2078 2020 2b20 3438 7820  8x x x x  + 48x 
│ │ │ │ +00019cb0: 7820 7820 7820 202b 2020 2020 7c0a 7c20  x x x  +    |.| 
│ │ │ │ +00019cc0: 2020 2020 3220 3420 3820 2020 2020 2034      2 4 8      4
│ │ │ │ +00019cd0: 2038 2020 2020 2020 3320 3420 3620 3920   8      3 4 6 9 
│ │ │ │ +00019ce0: 2020 2020 2032 2035 2036 2039 2020 2020       2 5 6 9    
│ │ │ │ +00019cf0: 2020 3120 3320 3720 3920 2020 2020 2030    1 3 7 9      0
│ │ │ │ +00019d00: 2035 2037 2039 2020 2020 2020 7c0a 7c2d   5 7 9      |.|-
│ │ │ │  00019d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019d50: 2d2d 2d2d 2d2d 7c0a 7c34 3878 2078 2078  ------|.|48x x x
│ │ │ │ -00019d60: 2078 2020 2d20 3438 7820 7820 7820 7820   x  - 48x x x x 
│ │ │ │ -00019d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ +00019d60: 3878 2078 2078 2078 2020 2d20 3438 7820  8x x x x  - 48x 
│ │ │ │ +00019d70: 7820 7820 7820 2020 2020 2020 2020 2020  x x x           
│ │ │ │  00019d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019da0: 2020 2020 2020 7c0a 7c20 2020 3120 3220        |.|   1 2 
│ │ │ │ -00019db0: 3820 3920 2020 2020 2030 2034 2038 2039  8 9      0 4 8 9
│ │ │ │ -00019dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +00019db0: 2020 3120 3220 3820 3920 2020 2020 2030    1 2 8 9      0
│ │ │ │ +00019dc0: 2034 2038 2039 2020 2020 2020 2020 2020   4 8 9          
│ │ │ │  00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00019de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019df0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +00019df0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  00019e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00019e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00019e40: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7469  ------+.|i3 : ti
│ │ │ │ -00019e50: 6d65 2070 7369 203d 2061 7070 726f 7869  me psi = approxi
│ │ │ │ -00019e60: 6d61 7465 496e 7665 7273 654d 6170 2070  mateInverseMap p
│ │ │ │ -00019e70: 6869 2020 2020 2020 2020 2020 2020 2020  hi              
│ │ │ │ +00019e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +00019e50: 3320 3a20 7469 6d65 2070 7369 203d 2061  3 : time psi = a
│ │ │ │ +00019e60: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ +00019e70: 654d 6170 2070 6869 2020 2020 2020 2020  eMap phi        
│ │ │ │  00019e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019e90: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -00019ea0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -00019eb0: 703a 2073 7465 7020 3120 6f66 2031 3020  p: step 1 of 10 
│ │ │ │ -00019ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019e90: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00019ea0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +00019eb0: 6572 7365 4d61 703a 2073 7465 7020 3120  erseMap: step 1 
│ │ │ │ +00019ec0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019ee0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -00019ef0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -00019f00: 703a 2073 7465 7020 3220 6f66 2031 3020  p: step 2 of 10 
│ │ │ │ -00019f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00019ef0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +00019f00: 6572 7365 4d61 703a 2073 7465 7020 3220  erseMap: step 2 
│ │ │ │ +00019f10: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  00019f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f30: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -00019f40: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -00019f50: 703a 2073 7465 7020 3320 6f66 2031 3020  p: step 3 of 10 
│ │ │ │ -00019f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019f30: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00019f40: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +00019f50: 6572 7365 4d61 703a 2073 7465 7020 3320  erseMap: step 3 
│ │ │ │ +00019f60: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  00019f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019f80: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -00019f90: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -00019fa0: 703a 2073 7465 7020 3420 6f66 2031 3020  p: step 4 of 10 
│ │ │ │ -00019fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019f80: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00019f90: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +00019fa0: 6572 7365 4d61 703a 2073 7465 7020 3420  erseMap: step 4 
│ │ │ │ +00019fb0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  00019fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00019fd0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -00019fe0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -00019ff0: 703a 2073 7465 7020 3520 6f66 2031 3020  p: step 5 of 10 
│ │ │ │ -0001a000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00019fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +00019fe0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +00019ff0: 6572 7365 4d61 703a 2073 7465 7020 3520  erseMap: step 5 
│ │ │ │ +0001a000: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  0001a010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a020: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -0001a030: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -0001a040: 703a 2073 7465 7020 3620 6f66 2031 3020  p: step 6 of 10 
│ │ │ │ -0001a050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a020: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001a030: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +0001a040: 6572 7365 4d61 703a 2073 7465 7020 3620  erseMap: step 6 
│ │ │ │ +0001a050: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  0001a060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a070: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -0001a080: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -0001a090: 703a 2073 7465 7020 3720 6f66 2031 3020  p: step 7 of 10 
│ │ │ │ -0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a070: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001a080: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +0001a090: 6572 7365 4d61 703a 2073 7465 7020 3720  erseMap: step 7 
│ │ │ │ +0001a0a0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a0c0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -0001a0d0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -0001a0e0: 703a 2073 7465 7020 3820 6f66 2031 3020  p: step 8 of 10 
│ │ │ │ -0001a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001a0d0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +0001a0e0: 6572 7365 4d61 703a 2073 7465 7020 3820  erseMap: step 8 
│ │ │ │ +0001a0f0: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  0001a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a110: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -0001a120: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -0001a130: 703a 2073 7465 7020 3920 6f66 2031 3020  p: step 9 of 10 
│ │ │ │ -0001a140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a110: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001a120: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +0001a130: 6572 7365 4d61 703a 2073 7465 7020 3920  erseMap: step 9 
│ │ │ │ +0001a140: 6f66 2031 3020 2020 2020 2020 2020 2020  of 10           
│ │ │ │  0001a150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a160: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -0001a170: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -0001a180: 703a 2073 7465 7020 3130 206f 6620 3130  p: step 10 of 10
│ │ │ │ -0001a190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a160: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001a170: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +0001a180: 6572 7365 4d61 703a 2073 7465 7020 3130  erseMap: step 10
│ │ │ │ +0001a190: 206f 6620 3130 2020 2020 2020 2020 2020   of 10          
│ │ │ │  0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a1b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -0001a1c0: 6420 302e 3535 3931 3734 7320 2863 7075  d 0.559174s (cpu
│ │ │ │ -0001a1d0: 293b 2030 2e33 3734 3838 3773 2028 7468  ); 0.374887s (th
│ │ │ │ -0001a1e0: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ -0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a200: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a1b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a1c0: 2d2d 2075 7365 6420 302e 3539 3639 3138  -- used 0.596918
│ │ │ │ +0001a1d0: 7320 2863 7075 293b 2030 2e34 3037 3837  s (cpu); 0.40787
│ │ │ │ +0001a1e0: 3873 2028 7468 7265 6164 293b 2030 7320  8s (thread); 0s 
│ │ │ │ +0001a1f0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +0001a200: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a250: 2020 2020 2020 7c0a 7c6f 3320 3d20 2d2d        |.|o3 = --
│ │ │ │ -0001a260: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │ -0001a270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a250: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001a260: 3320 3d20 2d2d 2072 6174 696f 6e61 6c20  3 = -- rational 
│ │ │ │ +0001a270: 6d61 7020 2d2d 2020 2020 2020 2020 2020  map --          
│ │ │ │  0001a280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a2a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a2b0: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │ -0001a2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a2a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a2c0: 205a 5a20 2020 2020 2020 2020 2020 2020   ZZ             
│ │ │ │  0001a2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a2f0: 2020 2020 2020 7c0a 7c20 2020 2020 736f        |.|     so
│ │ │ │ -0001a300: 7572 6365 3a20 5072 6f6a 282d 2d5b 7420  urce: Proj(--[t 
│ │ │ │ -0001a310: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ -0001a320: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ -0001a330: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ -0001a340: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a350: 2020 2020 2020 2020 2020 2039 3720 2030             97  0
│ │ │ │ -0001a360: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ -0001a370: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ -0001a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a390: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a2f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a300: 2020 2020 736f 7572 6365 3a20 5072 6f6a      source: Proj
│ │ │ │ +0001a310: 282d 2d5b 7420 2c20 7420 2c20 7420 2c20  (--[t , t , t , 
│ │ │ │ +0001a320: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ +0001a330: 7420 2c20 7420 5d29 2020 2020 2020 2020  t , t ])        
│ │ │ │ +0001a340: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a360: 2039 3720 2030 2020 2031 2020 2032 2020   97  0   1   2  
│ │ │ │ +0001a370: 2033 2020 2034 2020 2035 2020 2036 2020   3   4   5   6  
│ │ │ │ +0001a380: 2037 2020 2038 2020 2020 2020 2020 2020   7   8          
│ │ │ │ +0001a390: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a3b0: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │ -0001a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a3b0: 2020 2020 2020 2020 2020 2020 2020 205a                 Z
│ │ │ │ +0001a3c0: 5a20 2020 2020 2020 2020 2020 2020 2020  Z               
│ │ │ │  0001a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a3e0: 2020 2020 2020 7c0a 7c20 2020 2020 7461        |.|     ta
│ │ │ │ -0001a3f0: 7267 6574 3a20 7375 6276 6172 6965 7479  rget: subvariety
│ │ │ │ -0001a400: 206f 6620 5072 6f6a 282d 2d5b 7820 2c20   of Proj(--[x , 
│ │ │ │ -0001a410: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -0001a420: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -0001a430: 7820 5d29 2020 7c0a 7c20 2020 2020 2020  x ])  |.|       
│ │ │ │ +0001a3e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a3f0: 2020 2020 7461 7267 6574 3a20 7375 6276      target: subv
│ │ │ │ +0001a400: 6172 6965 7479 206f 6620 5072 6f6a 282d  ariety of Proj(-
│ │ │ │ +0001a410: 2d5b 7820 2c20 7820 2c20 7820 2c20 7820  -[x , x , x , x 
│ │ │ │ +0001a420: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +0001a430: 2c20 7820 2c20 7820 5d29 2020 7c0a 7c20  , x , x ])  |.| 
│ │ │ │  0001a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a450: 2020 2020 2020 2020 2039 3720 2030 2020           97  0  
│ │ │ │ -0001a460: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ -0001a470: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ -0001a480: 2039 2020 2020 7c0a 7c20 2020 2020 2020   9    |.|       
│ │ │ │ -0001a490: 2020 2020 2020 7b20 2020 2020 2020 2020        {         
│ │ │ │ +0001a450: 2020 2020 2020 2020 2020 2020 2020 2039                 9
│ │ │ │ +0001a460: 3720 2030 2020 2031 2020 2032 2020 2033  7  0   1   2   3
│ │ │ │ +0001a470: 2020 2034 2020 2035 2020 2036 2020 2037     4   5   6   7
│ │ │ │ +0001a480: 2020 2038 2020 2039 2020 2020 7c0a 7c20     8   9    |.| 
│ │ │ │ +0001a490: 2020 2020 2020 2020 2020 2020 7b20 2020              {   
│ │ │ │  0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a4d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a4e0: 2020 2020 2020 2020 3220 3220 2020 2020          2 2     
│ │ │ │ -0001a4f0: 2032 2020 2020 2020 2020 2020 3220 3220   2          2 2 
│ │ │ │ -0001a500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a4d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a4e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0001a4f0: 3220 2020 2020 2032 2020 2020 2020 2020  2      2        
│ │ │ │ +0001a500: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │  0001a510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a520: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a530: 2020 2020 2020 2078 2078 2020 2b20 3134         x x  + 14
│ │ │ │ -0001a540: 7820 7820 7820 202b 2032 3678 2078 2020  x x x  + 26x x  
│ │ │ │ -0001a550: 2d20 3278 2078 2078 2078 2020 2d20 3134  - 2x x x x  - 14
│ │ │ │ -0001a560: 7820 7820 7820 7820 202b 2031 3178 2078  x x x x  + 11x x
│ │ │ │ -0001a570: 2078 2078 2020 7c0a 7c20 2020 2020 2020   x x  |.|       
│ │ │ │ -0001a580: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ -0001a590: 2031 2032 2033 2020 2020 2020 3120 3320   1 2 3      1 3 
│ │ │ │ -0001a5a0: 2020 2020 3020 3120 3220 3420 2020 2020      0 1 2 4     
│ │ │ │ -0001a5b0: 2030 2031 2033 2034 2020 2020 2020 3120   0 1 3 4      1 
│ │ │ │ -0001a5c0: 3220 3320 3420 7c0a 7c20 2020 2020 2020  2 3 4 |.|       
│ │ │ │ -0001a5d0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │ +0001a520: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a530: 2020 2020 2020 2020 2020 2020 2078 2078               x x
│ │ │ │ +0001a540: 2020 2b20 3134 7820 7820 7820 202b 2032    + 14x x x  + 2
│ │ │ │ +0001a550: 3678 2078 2020 2d20 3278 2078 2078 2078  6x x  - 2x x x x
│ │ │ │ +0001a560: 2020 2d20 3134 7820 7820 7820 7820 202b    - 14x x x x  +
│ │ │ │ +0001a570: 2031 3178 2078 2078 2078 2020 7c0a 7c20   11x x x x  |.| 
│ │ │ │ +0001a580: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ +0001a590: 3220 2020 2020 2031 2032 2033 2020 2020  2      1 2 3    
│ │ │ │ +0001a5a0: 2020 3120 3320 2020 2020 3020 3120 3220    1 3     0 1 2 
│ │ │ │ +0001a5b0: 3420 2020 2020 2030 2031 2033 2034 2020  4      0 1 3 4  
│ │ │ │ +0001a5c0: 2020 2020 3120 3220 3320 3420 7c0a 7c20      1 2 3 4 |.| 
│ │ │ │ +0001a5d0: 2020 2020 2020 2020 2020 2020 7d20 2020              }   
│ │ │ │  0001a5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a610: 2020 2020 2020 7c0a 7c20 2020 2020 6465        |.|     de
│ │ │ │ -0001a620: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │ -0001a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a610: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a620: 2020 2020 6465 6669 6e69 6e67 2066 6f72      defining for
│ │ │ │ +0001a630: 6d73 3a20 7b20 2020 2020 2020 2020 2020  ms: {           
│ │ │ │  0001a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a660: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a670: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001a680: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001a690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a660: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a680: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001a690: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001a6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a6d0: 3520 3720 2020 2034 2038 2020 2020 2020  5 7    4 8      
│ │ │ │ +0001a6d0: 2020 2020 2020 3520 3720 2020 2034 2038        5 7    4 8
│ │ │ │  0001a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a700: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a750: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a760: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001a770: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001a780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a770: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001a780: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001a790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7c0: 3220 3720 2020 2031 2038 2020 2020 2020  2 7    1 8      
│ │ │ │ +0001a7c0: 2020 2020 2020 3220 3720 2020 2031 2038        2 7    1 8
│ │ │ │  0001a7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a7f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a840: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a850: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001a860: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001a870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a840: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a860: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001a870: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001a880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a890: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a890: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8b0: 3520 3620 2020 2033 2038 2020 2020 2020  5 6    3 8      
│ │ │ │ +0001a8b0: 2020 2020 2020 3520 3620 2020 2033 2038        5 6    3 8
│ │ │ │  0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a8e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a930: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001a940: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001a950: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a930: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001a950: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001a960: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a980: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a980: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9a0: 3420 3620 2020 2033 2037 2020 2020 2020  4 6    3 7      
│ │ │ │ +0001a9a0: 2020 2020 2020 3420 3620 2020 2033 2037        4 6    3 7
│ │ │ │  0001a9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001a9d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001a9d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001aa30: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001aa40: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001aa40: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001aa50: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001aa70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aa90: 3220 3620 2020 2030 2038 2020 2020 2020  2 6    0 8      
│ │ │ │ +0001aa90: 2020 2020 2020 3220 3620 2020 2030 2038        2 6    0 8
│ │ │ │  0001aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aac0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001aac0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aaf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ab00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ab20: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001ab30: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ab20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ab30: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001ab40: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ab60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ab70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ab80: 3120 3620 2020 2030 2037 2020 2020 2020  1 6    0 7      
│ │ │ │ +0001ab80: 2020 2020 2020 3120 3620 2020 2030 2037        1 6    0 7
│ │ │ │  0001ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001abb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001abb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ac10: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001ac20: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001ac30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ac10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ac20: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001ac30: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001ac40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ac50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ac60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ac70: 3220 3420 2020 2031 2035 2020 2020 2020  2 4    1 5      
│ │ │ │ +0001ac70: 2020 2020 2020 3220 3420 2020 2031 2035        2 4    1 5
│ │ │ │  0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aca0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001aca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001acb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001acc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001acd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ace0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001acf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001ad00: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001ad10: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001acf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ad10: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001ad20: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ad40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ad50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad60: 3220 3320 2020 2030 2035 2020 2020 2020  2 3    0 5      
│ │ │ │ +0001ad60: 2020 2020 2020 3220 3320 2020 2030 2035        2 3    0 5
│ │ │ │  0001ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ad90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ad90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ade0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001adf0: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001ae00: 2074 2020 2d20 7420 7420 2c20 2020 2020   t  - t t ,     
│ │ │ │ -0001ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ade0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001adf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ae00: 2020 2020 2074 2074 2020 2d20 7420 7420       t t  - t t 
│ │ │ │ +0001ae10: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ae30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae50: 3120 3320 2020 2030 2034 2020 2020 2020  1 3    0 4      
│ │ │ │ +0001ae50: 2020 2020 2020 3120 3320 2020 2030 2034        1 3    0 4
│ │ │ │  0001ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ae80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ae80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001aec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aed0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001aed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001aee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001aef0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0001af00: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0001af10: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0001af20: 2020 2020 3220 7c0a 7c20 2020 2020 2020      2 |.|       
│ │ │ │ -0001af30: 2020 2020 2020 2020 2020 2020 2020 2074                 t
│ │ │ │ -0001af40: 2020 2b20 3331 7420 7420 202d 2032 3574    + 31t t  - 25t
│ │ │ │ -0001af50: 2020 2b20 3774 2074 2020 2b20 3374 2074    + 7t t  + 3t t
│ │ │ │ -0001af60: 2020 2b20 3231 7420 202b 2033 7420 7420    + 21t  + 3t t 
│ │ │ │ -0001af70: 202d 2074 2020 7c0a 7c20 2020 2020 2020   - t  |.|       
│ │ │ │ +0001aef0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0001af00: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0001af10: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0001af20: 2020 2020 2020 2020 2020 3220 7c0a 7c20            2 |.| 
│ │ │ │ +0001af30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001af40: 2020 2020 2074 2020 2b20 3331 7420 7420       t  + 31t t 
│ │ │ │ +0001af50: 202d 2032 3574 2020 2b20 3774 2074 2020   - 25t  + 7t t  
│ │ │ │ +0001af60: 2b20 3374 2074 2020 2b20 3231 7420 202b  + 3t t  + 21t  +
│ │ │ │ +0001af70: 2033 7420 7420 202d 2074 2020 7c0a 7c20   3t t  - t  |.| 
│ │ │ │  0001af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001af90: 3020 2020 2020 2030 2031 2020 2020 2020  0      0 1      
│ │ │ │ -0001afa0: 3120 2020 2020 3020 3220 2020 2020 3120  1     0 2     1 
│ │ │ │ -0001afb0: 3220 2020 2020 2032 2020 2020 2030 2033  2      2     0 3
│ │ │ │ -0001afc0: 2020 2020 3320 7c0a 7c20 2020 2020 2020      3 |.|       
│ │ │ │ -0001afd0: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │ -0001afe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001af90: 2020 2020 2020 3020 2020 2020 2030 2031        0      0 1
│ │ │ │ +0001afa0: 2020 2020 2020 3120 2020 2020 3020 3220        1     0 2 
│ │ │ │ +0001afb0: 2020 2020 3120 3220 2020 2020 2032 2020      1 2      2  
│ │ │ │ +0001afc0: 2020 2030 2033 2020 2020 3320 7c0a 7c20     0 3    3 |.| 
│ │ │ │ +0001afd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001afe0: 2020 2020 7d20 2020 2020 2020 2020 2020      }           
│ │ │ │  0001aff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b010: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b010: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b060: 2020 2020 2020 7c0a 7c6f 3320 3a20 5261        |.|o3 : Ra
│ │ │ │ -0001b070: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ -0001b080: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ -0001b090: 7020 6672 6f6d 2050 505e 3820 746f 2068  p from PP^8 to h
│ │ │ │ -0001b0a0: 7970 6572 7375 7266 6163 6520 696e 2050  ypersurface in P
│ │ │ │ -0001b0b0: 505e 3929 2020 7c0a 7c2d 2d2d 2d2d 2d2d  P^9)  |.|-------
│ │ │ │ +0001b060: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001b070: 3320 3a20 5261 7469 6f6e 616c 4d61 7020  3 : RationalMap 
│ │ │ │ +0001b080: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ +0001b090: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ +0001b0a0: 3820 746f 2068 7970 6572 7375 7266 6163  8 to hypersurfac
│ │ │ │ +0001b0b0: 6520 696e 2050 505e 3929 2020 7c0a 7c2d  e in PP^9)  |.|-
│ │ │ │  0001b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001b100: 2d2d 2d2d 2d2d 7c0a 7c64 6566 696e 6564  ------|.|defined
│ │ │ │ -0001b110: 2062 7920 2020 2020 2020 2020 2020 2020   by             
│ │ │ │ +0001b100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c64  ------------|.|d
│ │ │ │ +0001b110: 6566 696e 6564 2062 7920 2020 2020 2020  efined by       
│ │ │ │  0001b120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b150: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b150: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b1a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b1f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ -0001b200: 2032 2020 2020 2020 3220 3220 2020 2020   2      2 2     
│ │ │ │ -0001b210: 2020 2020 2032 2020 2020 2020 3220 3220       2      2 2 
│ │ │ │ -0001b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b230: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0001b240: 2020 2020 2020 7c0a 7c20 2b20 3139 7820        |.| + 19x 
│ │ │ │ -0001b250: 7820 7820 202b 2078 2078 2020 2d20 3131  x x  + x x  - 11
│ │ │ │ -0001b260: 7820 7820 7820 202b 2033 3878 2078 2020  x x x  + 38x x  
│ │ │ │ -0001b270: 2d20 3134 7820 7820 7820 7820 202d 2031  - 14x x x x  - 1
│ │ │ │ -0001b280: 3178 2078 2078 2020 2b20 3435 7820 7820  1x x x  + 45x x 
│ │ │ │ -0001b290: 7820 7820 202d 7c0a 7c20 2020 2020 2031  x x  -|.|      1
│ │ │ │ -0001b2a0: 2033 2034 2020 2020 3020 3420 2020 2020   3 4    0 4     
│ │ │ │ -0001b2b0: 2030 2033 2034 2020 2020 2020 3320 3420   0 3 4      3 4 
│ │ │ │ -0001b2c0: 2020 2020 2030 2031 2032 2035 2020 2020       0 1 2 5    
│ │ │ │ -0001b2d0: 2020 3120 3220 3520 2020 2020 2030 2031    1 2 5      0 1
│ │ │ │ -0001b2e0: 2033 2035 2020 7c0a 7c20 2020 2020 2020   3 5  |.|       
│ │ │ │ +0001b1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b200: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ +0001b210: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ +0001b220: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ +0001b230: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0001b240: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001b250: 2b20 3139 7820 7820 7820 202b 2078 2078  + 19x x x  + x x
│ │ │ │ +0001b260: 2020 2d20 3131 7820 7820 7820 202b 2033    - 11x x x  + 3
│ │ │ │ +0001b270: 3878 2078 2020 2d20 3134 7820 7820 7820  8x x  - 14x x x 
│ │ │ │ +0001b280: 7820 202d 2031 3178 2078 2078 2020 2b20  x  - 11x x x  + 
│ │ │ │ +0001b290: 3435 7820 7820 7820 7820 202d 7c0a 7c20  45x x x x  -|.| 
│ │ │ │ +0001b2a0: 2020 2020 2031 2033 2034 2020 2020 3020       1 3 4    0 
│ │ │ │ +0001b2b0: 3420 2020 2020 2030 2033 2034 2020 2020  4      0 3 4    
│ │ │ │ +0001b2c0: 2020 3320 3420 2020 2020 2030 2031 2032    3 4      0 1 2
│ │ │ │ +0001b2d0: 2035 2020 2020 2020 3120 3220 3520 2020   5      1 2 5   
│ │ │ │ +0001b2e0: 2020 2030 2031 2033 2035 2020 7c0a 7c20     0 1 3 5  |.| 
│ │ │ │  0001b2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b330: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b330: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b380: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b380: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b3d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b420: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b420: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b470: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b470: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b4c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b510: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b510: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b560: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b560: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b5b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b600: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b600: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b650: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b650: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b6a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b6f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b740: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b740: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b790: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b790: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b7e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b7e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b830: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b830: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b880: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b880: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b8d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b920: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b920: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b970: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001b9c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001b9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ba10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ba20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ba60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ba60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ba70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001baa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bab0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bab0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bb00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bb50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bb50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bba0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bbf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bbf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc10: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0001bc10: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  0001bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bc40: 2020 2020 2020 7c0a 7c20 2b20 3474 2074        |.| + 4t t
│ │ │ │ -0001bc50: 2020 2b20 3339 7420 7420 202d 2032 3274    + 39t t  - 22t
│ │ │ │ -0001bc60: 2074 2020 2b20 3134 7420 202d 2074 2074   t  + 14t  - t t
│ │ │ │ -0001bc70: 2020 2d20 3674 2074 2020 2d20 3431 7420    - 6t t  - 41t 
│ │ │ │ -0001bc80: 7420 202b 2034 3274 2074 2020 2b20 3233  t  + 42t t  + 23
│ │ │ │ -0001bc90: 7420 7420 202b 7c0a 7c20 2020 2020 3020  t t  +|.|     0 
│ │ │ │ -0001bca0: 3420 2020 2020 2031 2034 2020 2020 2020  4      1 4      
│ │ │ │ -0001bcb0: 3320 3420 2020 2020 2034 2020 2020 3020  3 4      4    0 
│ │ │ │ -0001bcc0: 3520 2020 2020 3120 3520 2020 2020 2032  5     1 5      2
│ │ │ │ -0001bcd0: 2035 2020 2020 2020 3320 3520 2020 2020   5      3 5     
│ │ │ │ -0001bce0: 2034 2035 2020 7c0a 7c2d 2d2d 2d2d 2d2d   4 5  |.|-------
│ │ │ │ +0001bc40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001bc50: 2b20 3474 2074 2020 2b20 3339 7420 7420  + 4t t  + 39t t 
│ │ │ │ +0001bc60: 202d 2032 3274 2074 2020 2b20 3134 7420   - 22t t  + 14t 
│ │ │ │ +0001bc70: 202d 2074 2074 2020 2d20 3674 2074 2020   - t t  - 6t t  
│ │ │ │ +0001bc80: 2d20 3431 7420 7420 202b 2034 3274 2074  - 41t t  + 42t t
│ │ │ │ +0001bc90: 2020 2b20 3233 7420 7420 202b 7c0a 7c20    + 23t t  +|.| 
│ │ │ │ +0001bca0: 2020 2020 3020 3420 2020 2020 2031 2034      0 4      1 4
│ │ │ │ +0001bcb0: 2020 2020 2020 3320 3420 2020 2020 2034        3 4      4
│ │ │ │ +0001bcc0: 2020 2020 3020 3520 2020 2020 3120 3520      0 5     1 5 
│ │ │ │ +0001bcd0: 2020 2020 2032 2035 2020 2020 2020 3320       2 5      3 
│ │ │ │ +0001bce0: 3520 2020 2020 2034 2035 2020 7c0a 7c2d  5      4 5  |.|-
│ │ │ │  0001bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001bd30: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0001bd40: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001bd40: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  0001bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bd70: 2020 2020 2020 2020 2020 2032 2032 2020             2 2  
│ │ │ │ -0001bd80: 2020 2020 2020 7c0a 7c31 3978 2078 2078        |.|19x x x
│ │ │ │ -0001bd90: 2078 2020 2b20 3134 7820 7820 7820 202b   x  + 14x x x  +
│ │ │ │ -0001bda0: 2031 3178 2078 2078 2078 2020 2d20 3139   11x x x x  - 19
│ │ │ │ -0001bdb0: 7820 7820 7820 7820 202b 2032 3178 2078  x x x x  + 21x x
│ │ │ │ -0001bdc0: 2078 2078 2020 2b20 3236 7820 7820 202b   x x  + 26x x  +
│ │ │ │ -0001bdd0: 2031 3978 2020 7c0a 7c20 2020 3120 3220   19x  |.|   1 2 
│ │ │ │ -0001bde0: 3320 3520 2020 2020 2030 2034 2035 2020  3 5      0 4 5  
│ │ │ │ -0001bdf0: 2020 2020 3020 3220 3420 3520 2020 2020      0 2 4 5     
│ │ │ │ -0001be00: 2030 2033 2034 2035 2020 2020 2020 3220   0 3 4 5      2 
│ │ │ │ -0001be10: 3320 3420 3520 2020 2020 2030 2035 2020  3 4 5      0 5  
│ │ │ │ -0001be20: 2020 2020 3020 7c0a 7c20 2020 2020 2020      0 |.|       
│ │ │ │ +0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001bd80: 2032 2032 2020 2020 2020 2020 7c0a 7c31   2 2        |.|1
│ │ │ │ +0001bd90: 3978 2078 2078 2078 2020 2b20 3134 7820  9x x x x  + 14x 
│ │ │ │ +0001bda0: 7820 7820 202b 2031 3178 2078 2078 2078  x x  + 11x x x x
│ │ │ │ +0001bdb0: 2020 2d20 3139 7820 7820 7820 7820 202b    - 19x x x x  +
│ │ │ │ +0001bdc0: 2032 3178 2078 2078 2078 2020 2b20 3236   21x x x x  + 26
│ │ │ │ +0001bdd0: 7820 7820 202b 2031 3978 2020 7c0a 7c20  x x  + 19x  |.| 
│ │ │ │ +0001bde0: 2020 3120 3220 3320 3520 2020 2020 2030    1 2 3 5      0
│ │ │ │ +0001bdf0: 2034 2035 2020 2020 2020 3020 3220 3420   4 5      0 2 4 
│ │ │ │ +0001be00: 3520 2020 2020 2030 2033 2034 2035 2020  5      0 3 4 5  
│ │ │ │ +0001be10: 2020 2020 3220 3320 3420 3520 2020 2020      2 3 4 5     
│ │ │ │ +0001be20: 2030 2035 2020 2020 2020 3020 7c0a 7c20   0 5      0 |.| 
│ │ │ │  0001be30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001be70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001be70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001be80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bec0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bec0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bf10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bf60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bf60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001bfb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001bfb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001bfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c000: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c050: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c050: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c0a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c0f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c140: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c140: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c190: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c190: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c1e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c1e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c230: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c230: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c280: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c280: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c2d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c320: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c320: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c370: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c370: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c3c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c410: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c410: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c460: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c460: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c4b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c500: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c500: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c550: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c550: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c5a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c5f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c640: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c640: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c690: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c690: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c6e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c6e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c730: 2020 2020 2020 7c0a 7c20 2020 3220 2020        |.|   2   
│ │ │ │ -0001c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c750: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0001c730: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001c740: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001c750: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0001c760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c780: 2020 2020 2020 7c0a 7c32 3174 2020 2b20        |.|21t  + 
│ │ │ │ -0001c790: 3234 7420 7420 202d 2038 7420 7420 202d  24t t  - 8t t  -
│ │ │ │ -0001c7a0: 2032 3174 2020 2b20 3431 7420 7420 202b   21t  + 41t t  +
│ │ │ │ -0001c7b0: 2031 3374 2074 2020 2b20 3135 7420 7420   13t t  + 15t t 
│ │ │ │ -0001c7c0: 202d 2032 3274 2074 2020 2b20 3338 7420   - 22t t  + 38t 
│ │ │ │ -0001c7d0: 7420 202d 2020 7c0a 7c20 2020 3520 2020  t  -  |.|   5   
│ │ │ │ -0001c7e0: 2020 2030 2036 2020 2020 2033 2036 2020     0 6     3 6  
│ │ │ │ -0001c7f0: 2020 2020 3620 2020 2020 2030 2037 2020      6      0 7  
│ │ │ │ -0001c800: 2020 2020 3120 3720 2020 2020 2033 2037      1 7      3 7
│ │ │ │ -0001c810: 2020 2020 2020 3420 3720 2020 2020 2036        4 7      6
│ │ │ │ -0001c820: 2037 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   7    |.|-------
│ │ │ │ +0001c780: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +0001c790: 3174 2020 2b20 3234 7420 7420 202d 2038  1t  + 24t t  - 8
│ │ │ │ +0001c7a0: 7420 7420 202d 2032 3174 2020 2b20 3431  t t  - 21t  + 41
│ │ │ │ +0001c7b0: 7420 7420 202b 2031 3374 2074 2020 2b20  t t  + 13t t  + 
│ │ │ │ +0001c7c0: 3135 7420 7420 202d 2032 3274 2074 2020  15t t  - 22t t  
│ │ │ │ +0001c7d0: 2b20 3338 7420 7420 202d 2020 7c0a 7c20  + 38t t  -  |.| 
│ │ │ │ +0001c7e0: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ +0001c7f0: 2033 2036 2020 2020 2020 3620 2020 2020   3 6      6     
│ │ │ │ +0001c800: 2030 2037 2020 2020 2020 3120 3720 2020   0 7      1 7   
│ │ │ │ +0001c810: 2020 2033 2037 2020 2020 2020 3420 3720     3 7      4 7 
│ │ │ │ +0001c820: 2020 2020 2036 2037 2020 2020 7c0a 7c2d       6 7    |.|-
│ │ │ │  0001c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001c870: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ -0001c880: 2020 2032 2032 2020 2020 2020 2020 3220     2 2        2 
│ │ │ │ -0001c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0001c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001c880: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ +0001c890: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0001c8a0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c8c0: 2020 2020 2020 7c0a 7c78 2078 2020 2b20        |.|x x  + 
│ │ │ │ -0001c8d0: 3338 7820 7820 202d 2033 3078 2078 2078  38x x  - 30x x x
│ │ │ │ -0001c8e0: 2020 2d20 3131 7820 7820 7820 7820 202d    - 11x x x x  -
│ │ │ │ -0001c8f0: 2031 3178 2078 2078 2020 2b20 3330 7820   11x x x  + 30x 
│ │ │ │ -0001c900: 7820 7820 7820 202b 2032 3278 2078 2078  x x x  + 22x x x
│ │ │ │ -0001c910: 2078 2020 2b20 7c0a 7c20 3220 3520 2020   x  + |.| 2 5   
│ │ │ │ -0001c920: 2020 2032 2035 2020 2020 2020 3120 3220     2 5      1 2 
│ │ │ │ -0001c930: 3620 2020 2020 2031 2032 2033 2036 2020  6      1 2 3 6  
│ │ │ │ -0001c940: 2020 2020 3120 3320 3620 2020 2020 2030      1 3 6      0
│ │ │ │ -0001c950: 2032 2034 2036 2020 2020 2020 3120 3220   2 4 6      1 2 
│ │ │ │ -0001c960: 3420 3620 2020 7c0a 7c20 2020 2020 2020  4 6   |.|       
│ │ │ │ +0001c8c0: 2020 2020 2020 2020 2020 2020 7c0a 7c78              |.|x
│ │ │ │ +0001c8d0: 2078 2020 2b20 3338 7820 7820 202d 2033   x  + 38x x  - 3
│ │ │ │ +0001c8e0: 3078 2078 2078 2020 2d20 3131 7820 7820  0x x x  - 11x x 
│ │ │ │ +0001c8f0: 7820 7820 202d 2031 3178 2078 2078 2020  x x  - 11x x x  
│ │ │ │ +0001c900: 2b20 3330 7820 7820 7820 7820 202b 2032  + 30x x x x  + 2
│ │ │ │ +0001c910: 3278 2078 2078 2078 2020 2b20 7c0a 7c20  2x x x x  + |.| 
│ │ │ │ +0001c920: 3220 3520 2020 2020 2032 2035 2020 2020  2 5      2 5    
│ │ │ │ +0001c930: 2020 3120 3220 3620 2020 2020 2031 2032    1 2 6      1 2
│ │ │ │ +0001c940: 2033 2036 2020 2020 2020 3120 3320 3620   3 6      1 3 6 
│ │ │ │ +0001c950: 2020 2020 2030 2032 2034 2036 2020 2020       0 2 4 6    
│ │ │ │ +0001c960: 2020 3120 3220 3420 3620 2020 7c0a 7c20    1 2 4 6   |.| 
│ │ │ │  0001c970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001c9b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001c9b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ca00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ca10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ca50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ca50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ca60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001caa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001caa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001caf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001caf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cb40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cb90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cb90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cbe0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cbe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cc80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cc80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ccd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ccd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ccf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cd20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cd70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cd70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cdc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cdc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ce10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ce20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ce60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ce60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ceb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001ceb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf00: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cf00: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cf10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cf50: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cf50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cf60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cfa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cfa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001cfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001cff0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001cff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d040: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d040: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d090: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d090: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d0e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d0e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d130: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d130: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d180: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d180: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d1d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d1d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d220: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d220: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d270: 2020 2020 2020 7c0a 7c20 2020 3220 2020        |.|   2   
│ │ │ │ -0001d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d270: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001d280: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0001d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d2c0: 2020 2020 2020 7c0a 7c34 3574 2020 2d20        |.|45t  - 
│ │ │ │ -0001d2d0: 3435 7420 7420 202b 2032 3074 2074 2020  45t t  + 20t t  
│ │ │ │ -0001d2e0: 2b20 3434 7420 7420 202d 2034 3074 2074  + 44t t  - 40t t
│ │ │ │ -0001d2f0: 2020 2d20 3232 7420 7420 202b 2033 3774    - 22t t  + 37t
│ │ │ │ -0001d300: 2074 2020 2b20 3232 7420 7420 202b 2032   t  + 22t t  + 2
│ │ │ │ -0001d310: 3874 2074 2020 7c0a 7c20 2020 3720 2020  8t t  |.|   7   
│ │ │ │ -0001d320: 2020 2030 2038 2020 2020 2020 3120 3820     0 8      1 8 
│ │ │ │ -0001d330: 2020 2020 2032 2038 2020 2020 2020 3320       2 8      3 
│ │ │ │ -0001d340: 3820 2020 2020 2034 2038 2020 2020 2020  8      4 8      
│ │ │ │ -0001d350: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ -0001d360: 2020 3720 3820 7c0a 7c2d 2d2d 2d2d 2d2d    7 8 |.|-------
│ │ │ │ +0001d2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +0001d2d0: 3574 2020 2d20 3435 7420 7420 202b 2032  5t  - 45t t  + 2
│ │ │ │ +0001d2e0: 3074 2074 2020 2b20 3434 7420 7420 202d  0t t  + 44t t  -
│ │ │ │ +0001d2f0: 2034 3074 2074 2020 2d20 3232 7420 7420   40t t  - 22t t 
│ │ │ │ +0001d300: 202b 2033 3774 2074 2020 2b20 3232 7420   + 37t t  + 22t 
│ │ │ │ +0001d310: 7420 202b 2032 3874 2074 2020 7c0a 7c20  t  + 28t t  |.| 
│ │ │ │ +0001d320: 2020 3720 2020 2020 2030 2038 2020 2020    7      0 8    
│ │ │ │ +0001d330: 2020 3120 3820 2020 2020 2032 2038 2020    1 8      2 8  
│ │ │ │ +0001d340: 2020 2020 3320 3820 2020 2020 2034 2038      3 8      4 8
│ │ │ │ +0001d350: 2020 2020 2020 3520 3820 2020 2020 2036        5 8      6
│ │ │ │ +0001d360: 2038 2020 2020 2020 3720 3820 7c0a 7c2d   8      7 8 |.|-
│ │ │ │  0001d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001d3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001d3b0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0001d3c0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0001d3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001d3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001d3d0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0001d3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d400: 2020 2020 2020 7c0a 7c31 3178 2078 2078        |.|11x x x
│ │ │ │ -0001d410: 2078 2020 2d20 3232 7820 7820 7820 202b   x  - 22x x x  +
│ │ │ │ -0001d420: 2031 3078 2078 2078 2078 2020 2b20 3131   10x x x x  + 11
│ │ │ │ -0001d430: 7820 7820 7820 7820 202d 2034 3278 2078  x x x x  - 42x x
│ │ │ │ -0001d440: 2078 2078 2020 2d20 3130 7820 7820 7820   x x  - 10x x x 
│ │ │ │ -0001d450: 7820 202b 2020 7c0a 7c20 2020 3020 3320  x  +  |.|   0 3 
│ │ │ │ -0001d460: 3420 3620 2020 2020 2030 2034 2036 2020  4 6      0 4 6  
│ │ │ │ -0001d470: 2020 2020 3120 3220 3520 3620 2020 2020      1 2 5 6     
│ │ │ │ -0001d480: 2030 2033 2035 2036 2020 2020 2020 3120   0 3 5 6      1 
│ │ │ │ -0001d490: 3320 3520 3620 2020 2020 2030 2034 2035  3 5 6      0 4 5
│ │ │ │ -0001d4a0: 2036 2020 2020 7c0a 7c20 2020 2020 2020   6    |.|       
│ │ │ │ +0001d400: 2020 2020 2020 2020 2020 2020 7c0a 7c31              |.|1
│ │ │ │ +0001d410: 3178 2078 2078 2078 2020 2d20 3232 7820  1x x x x  - 22x 
│ │ │ │ +0001d420: 7820 7820 202b 2031 3078 2078 2078 2078  x x  + 10x x x x
│ │ │ │ +0001d430: 2020 2b20 3131 7820 7820 7820 7820 202d    + 11x x x x  -
│ │ │ │ +0001d440: 2034 3278 2078 2078 2078 2020 2d20 3130   42x x x x  - 10
│ │ │ │ +0001d450: 7820 7820 7820 7820 202b 2020 7c0a 7c20  x x x x  +  |.| 
│ │ │ │ +0001d460: 2020 3020 3320 3420 3620 2020 2020 2030    0 3 4 6      0
│ │ │ │ +0001d470: 2034 2036 2020 2020 2020 3120 3220 3520   4 6      1 2 5 
│ │ │ │ +0001d480: 3620 2020 2020 2030 2033 2035 2036 2020  6      0 3 5 6  
│ │ │ │ +0001d490: 2020 2020 3120 3320 3520 3620 2020 2020      1 3 5 6     
│ │ │ │ +0001d4a0: 2030 2034 2035 2036 2020 2020 7c0a 7c20   0 4 5 6    |.| 
│ │ │ │  0001d4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d4f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d4f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d540: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d540: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d590: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d590: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d5e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d630: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d630: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d680: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d680: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d6d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d6d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d720: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d720: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d770: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d7c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d7c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d810: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d810: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d860: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d860: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d8b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d8b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d900: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d900: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d950: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d950: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001d9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001d9f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001d9f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da40: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001da40: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001da50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001da80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001da90: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001da90: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001daa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dae0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001dae0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001daf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db30: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001db30: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001db80: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001db80: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001db90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dbd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001dbd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001dbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc20: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001dc20: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001dc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dc70: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001dc70: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dcc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001dcc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001dcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd10: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001dd10: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001dd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dd60: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001dd60: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001dd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ddb0: 2020 2020 2020 7c0a 7c20 2020 2032 2020        |.|    2  
│ │ │ │ -0001ddc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ddb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001ddc0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0001ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001dde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de00: 2020 2020 2020 7c0a 7c2b 2032 7420 2020        |.|+ 2t   
│ │ │ │ -0001de10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001de00: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ +0001de10: 2032 7420 2020 2020 2020 2020 2020 2020   2t             
│ │ │ │  0001de20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001de50: 2020 2020 2020 7c0a 7c20 2020 2038 2020        |.|    8  
│ │ │ │ -0001de60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001de50: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001de60: 2020 2038 2020 2020 2020 2020 2020 2020     8            
│ │ │ │  0001de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001dea0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001dea0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │  0001deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001def0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220  ------|.|     2 
│ │ │ │ -0001df00: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ -0001df10: 2020 2020 3220 2020 2020 3220 3220 2020      2     2 2   
│ │ │ │ -0001df20: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ -0001df30: 3220 2020 2020 2020 2020 3220 2020 2020  2         2     
│ │ │ │ -0001df40: 2020 2020 2032 7c0a 7c34 3278 2078 2078       2|.|42x x x
│ │ │ │ -0001df50: 2020 2d20 3338 7820 7820 202d 2033 3778    - 38x x  - 37x
│ │ │ │ -0001df60: 2078 2078 2020 2b20 3778 2078 2020 2b20   x x  + 7x x  + 
│ │ │ │ -0001df70: 3238 7820 7820 7820 202d 2033 3878 2078  28x x x  - 38x x
│ │ │ │ -0001df80: 2020 2d20 3378 2078 2078 2020 2d20 3239    - 3x x x  - 29
│ │ │ │ -0001df90: 7820 7820 7820 7c0a 7c20 2020 3020 3520  x x x |.|   0 5 
│ │ │ │ -0001dfa0: 3620 2020 2020 2032 2036 2020 2020 2020  6      2 6      
│ │ │ │ -0001dfb0: 3220 3320 3620 2020 2020 3320 3620 2020  2 3 6     3 6   
│ │ │ │ -0001dfc0: 2020 2032 2034 2036 2020 2020 2020 3420     2 4 6      4 
│ │ │ │ -0001dfd0: 3620 2020 2020 3220 3520 3620 2020 2020  6     2 5 6     
│ │ │ │ -0001dfe0: 2033 2035 2036 7c0a 7c2d 2d2d 2d2d 2d2d   3 5 6|.|-------
│ │ │ │ +0001def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001df00: 2020 2020 3220 2020 2020 2020 2032 2032      2        2 2
│ │ │ │ +0001df10: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0001df20: 3220 3220 2020 2020 2020 2020 2032 2020  2 2          2  
│ │ │ │ +0001df30: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │ +0001df40: 3220 2020 2020 2020 2020 2032 7c0a 7c34  2          2|.|4
│ │ │ │ +0001df50: 3278 2078 2078 2020 2d20 3338 7820 7820  2x x x  - 38x x 
│ │ │ │ +0001df60: 202d 2033 3778 2078 2078 2020 2b20 3778   - 37x x x  + 7x
│ │ │ │ +0001df70: 2078 2020 2b20 3238 7820 7820 7820 202d   x  + 28x x x  -
│ │ │ │ +0001df80: 2033 3878 2078 2020 2d20 3378 2078 2078   38x x  - 3x x x
│ │ │ │ +0001df90: 2020 2d20 3239 7820 7820 7820 7c0a 7c20    - 29x x x |.| 
│ │ │ │ +0001dfa0: 2020 3020 3520 3620 2020 2020 2032 2036    0 5 6      2 6
│ │ │ │ +0001dfb0: 2020 2020 2020 3220 3320 3620 2020 2020        2 3 6     
│ │ │ │ +0001dfc0: 3320 3620 2020 2020 2032 2034 2036 2020  3 6      2 4 6  
│ │ │ │ +0001dfd0: 2020 2020 3420 3620 2020 2020 3220 3520      4 6     2 5 
│ │ │ │ +0001dfe0: 3620 2020 2020 2033 2035 2036 7c0a 7c2d  6      3 5 6|.|-
│ │ │ │  0001dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e030: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0001e040: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ -0001e050: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0001e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e070: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -0001e080: 2020 2020 2020 7c0a 7c2d 2034 3778 2078        |.|- 47x x
│ │ │ │ -0001e090: 2078 2020 2b20 3336 7820 7820 202b 2033   x  + 36x x  + 3
│ │ │ │ -0001e0a0: 3078 2078 2078 2078 2020 2d20 3232 7820  0x x x x  - 22x 
│ │ │ │ -0001e0b0: 7820 7820 202d 2032 3078 2078 2078 2078  x x  - 20x x x x
│ │ │ │ -0001e0c0: 2020 2d20 3431 7820 7820 7820 202d 2020    - 41x x x  -  
│ │ │ │ -0001e0d0: 2020 2020 2020 7c0a 7c20 2020 2020 3420        |.|     4 
│ │ │ │ -0001e0e0: 3520 3620 2020 2020 2035 2036 2020 2020  5 6      5 6    
│ │ │ │ -0001e0f0: 2020 3020 3120 3220 3720 2020 2020 2031    0 1 2 7      1
│ │ │ │ -0001e100: 2032 2037 2020 2020 2020 3120 3220 3320   2 7      1 2 3 
│ │ │ │ -0001e110: 3720 2020 2020 2031 2033 2037 2020 2020  7      1 3 7    
│ │ │ │ -0001e120: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001e040: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ +0001e050: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0001e060: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0001e070: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0001e080: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001e090: 2034 3778 2078 2078 2020 2b20 3336 7820   47x x x  + 36x 
│ │ │ │ +0001e0a0: 7820 202b 2033 3078 2078 2078 2078 2020  x  + 30x x x x  
│ │ │ │ +0001e0b0: 2d20 3232 7820 7820 7820 202d 2032 3078  - 22x x x  - 20x
│ │ │ │ +0001e0c0: 2078 2078 2078 2020 2d20 3431 7820 7820   x x x  - 41x x 
│ │ │ │ +0001e0d0: 7820 202d 2020 2020 2020 2020 7c0a 7c20  x  -        |.| 
│ │ │ │ +0001e0e0: 2020 2020 3420 3520 3620 2020 2020 2035      4 5 6      5
│ │ │ │ +0001e0f0: 2036 2020 2020 2020 3020 3120 3220 3720   6      0 1 2 7 
│ │ │ │ +0001e100: 2020 2020 2031 2032 2037 2020 2020 2020       1 2 7      
│ │ │ │ +0001e110: 3120 3220 3320 3720 2020 2020 2031 2033  1 2 3 7      1 3
│ │ │ │ +0001e120: 2037 2020 2020 2020 2020 2020 7c0a 7c2d   7          |.|-
│ │ │ │  0001e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e170: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ -0001e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001e180: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0001e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1a0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001e1a0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0001e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e1c0: 2020 2020 2020 7c0a 7c33 3078 2078 2078        |.|30x x x
│ │ │ │ -0001e1d0: 2020 2b20 3232 7820 7820 7820 7820 202b    + 22x x x x  +
│ │ │ │ -0001e1e0: 2032 3078 2078 2078 2078 2020 2d20 3478   20x x x x  - 4x
│ │ │ │ -0001e1f0: 2078 2078 2020 2b20 3236 7820 7820 7820   x x  + 26x x x 
│ │ │ │ -0001e200: 7820 202b 2034 3178 2078 2078 2078 2020  x  + 41x x x x  
│ │ │ │ -0001e210: 2d20 2020 2020 7c0a 7c20 2020 3020 3420  -     |.|   0 4 
│ │ │ │ -0001e220: 3720 2020 2020 2030 2031 2034 2037 2020  7      0 1 4 7  
│ │ │ │ -0001e230: 2020 2020 3020 3320 3420 3720 2020 2020      0 3 4 7     
│ │ │ │ -0001e240: 3320 3420 3720 2020 2020 2031 2032 2035  3 4 7      1 2 5
│ │ │ │ -0001e250: 2037 2020 2020 2020 3020 3320 3520 3720   7      0 3 5 7 
│ │ │ │ -0001e260: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001e1c0: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +0001e1d0: 3078 2078 2078 2020 2b20 3232 7820 7820  0x x x  + 22x x 
│ │ │ │ +0001e1e0: 7820 7820 202b 2032 3078 2078 2078 2078  x x  + 20x x x x
│ │ │ │ +0001e1f0: 2020 2d20 3478 2078 2078 2020 2b20 3236    - 4x x x  + 26
│ │ │ │ +0001e200: 7820 7820 7820 7820 202b 2034 3178 2078  x x x x  + 41x x
│ │ │ │ +0001e210: 2078 2078 2020 2d20 2020 2020 7c0a 7c20   x x  -     |.| 
│ │ │ │ +0001e220: 2020 3020 3420 3720 2020 2020 2030 2031    0 4 7      0 1
│ │ │ │ +0001e230: 2034 2037 2020 2020 2020 3020 3320 3420   4 7      0 3 4 
│ │ │ │ +0001e240: 3720 2020 2020 3320 3420 3720 2020 2020  7     3 4 7     
│ │ │ │ +0001e250: 2031 2032 2035 2037 2020 2020 2020 3020   1 2 5 7      0 
│ │ │ │ +0001e260: 3320 3520 3720 2020 2020 2020 7c0a 7c2d  3 5 7       |.|-
│ │ │ │  0001e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e2b0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  0001e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e2f0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ -0001e300: 2020 2020 2020 7c0a 7c32 3878 2078 2078        |.|28x x x
│ │ │ │ -0001e310: 2078 2020 2b20 3478 2078 2078 2078 2020   x  + 4x x x x  
│ │ │ │ -0001e320: 2d20 3236 7820 7820 7820 7820 202b 2031  - 26x x x x  + 1
│ │ │ │ -0001e330: 3278 2078 2078 2078 2020 2b20 3238 7820  2x x x x  + 28x 
│ │ │ │ -0001e340: 7820 7820 202d 2031 3278 2078 2078 2020  x x  - 12x x x  
│ │ │ │ -0001e350: 2d20 2020 2020 7c0a 7c20 2020 3120 3320  -     |.|   1 3 
│ │ │ │ -0001e360: 3520 3720 2020 2020 3220 3320 3520 3720  5 7     2 3 5 7 
│ │ │ │ -0001e370: 2020 2020 2030 2034 2035 2037 2020 2020       0 4 5 7    
│ │ │ │ -0001e380: 2020 3320 3420 3520 3720 2020 2020 2030    3 4 5 7      0
│ │ │ │ -0001e390: 2035 2037 2020 2020 2020 3220 3520 3720   5 7      2 5 7 
│ │ │ │ -0001e3a0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001e2f0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001e300: 2020 3220 2020 2020 2020 2020 7c0a 7c32    2         |.|2
│ │ │ │ +0001e310: 3878 2078 2078 2078 2020 2b20 3478 2078  8x x x x  + 4x x
│ │ │ │ +0001e320: 2078 2078 2020 2d20 3236 7820 7820 7820   x x  - 26x x x 
│ │ │ │ +0001e330: 7820 202b 2031 3278 2078 2078 2078 2020  x  + 12x x x x  
│ │ │ │ +0001e340: 2b20 3238 7820 7820 7820 202d 2031 3278  + 28x x x  - 12x
│ │ │ │ +0001e350: 2078 2078 2020 2d20 2020 2020 7c0a 7c20   x x  -     |.| 
│ │ │ │ +0001e360: 2020 3120 3320 3520 3720 2020 2020 3220    1 3 5 7     2 
│ │ │ │ +0001e370: 3320 3520 3720 2020 2020 2030 2034 2035  3 5 7      0 4 5
│ │ │ │ +0001e380: 2037 2020 2020 2020 3320 3420 3520 3720   7      3 4 5 7 
│ │ │ │ +0001e390: 2020 2020 2030 2035 2037 2020 2020 2020       0 5 7      
│ │ │ │ +0001e3a0: 3220 3520 3720 2020 2020 2020 7c0a 7c2d  2 5 7       |.|-
│ │ │ │  0001e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e3f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  0001e400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e430: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0001e440: 2020 2020 2020 7c0a 7c32 3178 2078 2078        |.|21x x x
│ │ │ │ -0001e450: 2078 2020 2d20 3238 7820 7820 7820 7820   x  - 28x x x x 
│ │ │ │ -0001e460: 202b 2033 3778 2078 2078 2078 2020 2b20   + 37x x x x  + 
│ │ │ │ -0001e470: 3231 7820 7820 7820 7820 202d 2031 3178  21x x x x  - 11x
│ │ │ │ -0001e480: 2078 2078 2020 2d20 3238 7820 7820 7820   x x  - 28x x x 
│ │ │ │ -0001e490: 7820 202d 2020 7c0a 7c20 2020 3020 3220  x  -  |.|   0 2 
│ │ │ │ -0001e4a0: 3620 3720 2020 2020 2031 2032 2036 2037  6 7      1 2 6 7
│ │ │ │ -0001e4b0: 2020 2020 2020 3020 3320 3620 3720 2020        0 3 6 7   
│ │ │ │ -0001e4c0: 2020 2032 2033 2036 2037 2020 2020 2020     2 3 6 7      
│ │ │ │ -0001e4d0: 3320 3620 3720 2020 2020 2030 2034 2036  3 6 7      0 4 6
│ │ │ │ -0001e4e0: 2037 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   7    |.|-------
│ │ │ │ +0001e430: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0001e440: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +0001e450: 3178 2078 2078 2078 2020 2d20 3238 7820  1x x x x  - 28x 
│ │ │ │ +0001e460: 7820 7820 7820 202b 2033 3778 2078 2078  x x x  + 37x x x
│ │ │ │ +0001e470: 2078 2020 2b20 3231 7820 7820 7820 7820   x  + 21x x x x 
│ │ │ │ +0001e480: 202d 2031 3178 2078 2078 2020 2d20 3238   - 11x x x  - 28
│ │ │ │ +0001e490: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ +0001e4a0: 2020 3020 3220 3620 3720 2020 2020 2031    0 2 6 7      1
│ │ │ │ +0001e4b0: 2032 2036 2037 2020 2020 2020 3020 3320   2 6 7      0 3 
│ │ │ │ +0001e4c0: 3620 3720 2020 2020 2032 2033 2036 2037  6 7      2 3 6 7
│ │ │ │ +0001e4d0: 2020 2020 2020 3320 3620 3720 2020 2020        3 6 7     
│ │ │ │ +0001e4e0: 2030 2034 2036 2037 2020 2020 7c0a 7c2d   0 4 6 7    |.|-
│ │ │ │  0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e530: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  0001e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e580: 2020 2020 2020 7c0a 7c32 3178 2078 2078        |.|21x x x
│ │ │ │ -0001e590: 2078 2020 2b20 3378 2078 2078 2078 2020   x  + 3x x x x  
│ │ │ │ -0001e5a0: 2b20 3437 7820 7820 7820 7820 202b 2034  + 47x x x x  + 4
│ │ │ │ -0001e5b0: 3878 2078 2078 2078 2020 2b20 3278 2078  8x x x x  + 2x x
│ │ │ │ -0001e5c0: 2078 2078 2020 2d20 3435 7820 7820 7820   x x  - 45x x x 
│ │ │ │ -0001e5d0: 7820 202d 2020 7c0a 7c20 2020 3120 3420  x  -  |.|   1 4 
│ │ │ │ -0001e5e0: 3620 3720 2020 2020 3020 3520 3620 3720  6 7     0 5 6 7 
│ │ │ │ -0001e5f0: 2020 2020 2031 2035 2036 2037 2020 2020       1 5 6 7    
│ │ │ │ -0001e600: 2020 3220 3520 3620 3720 2020 2020 3320    2 5 6 7     3 
│ │ │ │ -0001e610: 3520 3620 3720 2020 2020 2034 2035 2036  5 6 7      4 5 6
│ │ │ │ -0001e620: 2037 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   7    |.|-------
│ │ │ │ +0001e580: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +0001e590: 3178 2078 2078 2078 2020 2b20 3378 2078  1x x x x  + 3x x
│ │ │ │ +0001e5a0: 2078 2078 2020 2b20 3437 7820 7820 7820   x x  + 47x x x 
│ │ │ │ +0001e5b0: 7820 202b 2034 3878 2078 2078 2078 2020  x  + 48x x x x  
│ │ │ │ +0001e5c0: 2b20 3278 2078 2078 2078 2020 2d20 3435  + 2x x x x  - 45
│ │ │ │ +0001e5d0: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ +0001e5e0: 2020 3120 3420 3620 3720 2020 2020 3020    1 4 6 7     0 
│ │ │ │ +0001e5f0: 3520 3620 3720 2020 2020 2031 2035 2036  5 6 7      1 5 6
│ │ │ │ +0001e600: 2037 2020 2020 2020 3220 3520 3620 3720   7      2 5 6 7 
│ │ │ │ +0001e610: 2020 2020 3320 3520 3620 3720 2020 2020      3 5 6 7     
│ │ │ │ +0001e620: 2034 2035 2036 2037 2020 2020 7c0a 7c2d   4 5 6 7    |.|-
│ │ │ │  0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e670: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ -0001e680: 2020 2020 2020 2032 2032 2020 2020 2020         2 2      
│ │ │ │ -0001e690: 2020 2020 3220 2020 2020 2032 2032 2020      2      2 2  
│ │ │ │ -0001e6a0: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ -0001e6b0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ -0001e6c0: 2020 2020 2020 7c0a 7c33 3378 2078 2078        |.|33x x x
│ │ │ │ -0001e6d0: 2020 2d20 3338 7820 7820 202b 2032 3878    - 38x x  + 28x
│ │ │ │ -0001e6e0: 2078 2078 2020 2d20 3338 7820 7820 202d   x x  - 38x x  -
│ │ │ │ -0001e6f0: 2032 3178 2078 2078 2020 2b20 3438 7820   21x x x  + 48x 
│ │ │ │ -0001e700: 7820 202d 2034 3878 2078 2078 2020 2b20  x  - 48x x x  + 
│ │ │ │ -0001e710: 2020 2020 2020 7c0a 7c20 2020 3520 3620        |.|   5 6 
│ │ │ │ -0001e720: 3720 2020 2020 2030 2037 2020 2020 2020  7      0 7      
│ │ │ │ -0001e730: 3020 3120 3720 2020 2020 2031 2037 2020  0 1 7      1 7  
│ │ │ │ -0001e740: 2020 2020 3020 3320 3720 2020 2020 2033      0 3 7      3
│ │ │ │ -0001e750: 2037 2020 2020 2020 3020 3520 3720 2020   7      0 5 7   
│ │ │ │ -0001e760: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001e680: 2020 3220 2020 2020 2020 2020 2032 2032    2          2 2
│ │ │ │ +0001e690: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0001e6a0: 2032 2032 2020 2020 2020 2020 2020 3220   2 2          2 
│ │ │ │ +0001e6b0: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ +0001e6c0: 2020 3220 2020 2020 2020 2020 7c0a 7c33    2         |.|3
│ │ │ │ +0001e6d0: 3378 2078 2078 2020 2d20 3338 7820 7820  3x x x  - 38x x 
│ │ │ │ +0001e6e0: 202b 2032 3878 2078 2078 2020 2d20 3338   + 28x x x  - 38
│ │ │ │ +0001e6f0: 7820 7820 202d 2032 3178 2078 2078 2020  x x  - 21x x x  
│ │ │ │ +0001e700: 2b20 3438 7820 7820 202d 2034 3878 2078  + 48x x  - 48x x
│ │ │ │ +0001e710: 2078 2020 2b20 2020 2020 2020 7c0a 7c20   x  +       |.| 
│ │ │ │ +0001e720: 2020 3520 3620 3720 2020 2020 2030 2037    5 6 7      0 7
│ │ │ │ +0001e730: 2020 2020 2020 3020 3120 3720 2020 2020        0 1 7     
│ │ │ │ +0001e740: 2031 2037 2020 2020 2020 3020 3320 3720   1 7      0 3 7 
│ │ │ │ +0001e750: 2020 2020 2033 2037 2020 2020 2020 3020       3 7      0 
│ │ │ │ +0001e760: 3520 3720 2020 2020 2020 2020 7c0a 7c2d  5 7         |.|-
│ │ │ │  0001e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e7b0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0001e7c0: 3220 2020 2020 2020 2020 2032 2020 2020  2          2    
│ │ │ │ -0001e7d0: 2020 3220 3220 2020 2020 2020 2020 2020    2 2           
│ │ │ │ -0001e7e0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -0001e7f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0001e800: 2020 2020 2020 7c0a 7c34 3578 2078 2078        |.|45x x x
│ │ │ │ -0001e810: 2020 2d20 3437 7820 7820 7820 202d 2034    - 47x x x  - 4
│ │ │ │ -0001e820: 3878 2078 2020 2b20 3131 7820 7820 7820  8x x  + 11x x x 
│ │ │ │ -0001e830: 7820 202d 2031 3078 2078 2078 2020 2b20  x  - 10x x x  + 
│ │ │ │ -0001e840: 3230 7820 7820 7820 202b 2031 3178 2078  20x x x  + 11x x
│ │ │ │ -0001e850: 2078 2078 2020 7c0a 7c20 2020 3120 3520   x x  |.|   1 5 
│ │ │ │ -0001e860: 3720 2020 2020 2033 2035 2037 2020 2020  7      3 5 7    
│ │ │ │ -0001e870: 2020 3520 3720 2020 2020 2030 2031 2032    5 7      0 1 2
│ │ │ │ -0001e880: 2038 2020 2020 2020 3120 3220 3820 2020   8      1 2 8   
│ │ │ │ -0001e890: 2020 2031 2032 2038 2020 2020 2020 3020     1 2 8      0 
│ │ │ │ -0001e8a0: 3120 3320 3820 7c0a 7c2d 2d2d 2d2d 2d2d  1 3 8 |.|-------
│ │ │ │ +0001e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001e7c0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0001e7d0: 2032 2020 2020 2020 3220 3220 2020 2020   2      2 2     
│ │ │ │ +0001e7e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0001e7f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0001e800: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +0001e810: 3578 2078 2078 2020 2d20 3437 7820 7820  5x x x  - 47x x 
│ │ │ │ +0001e820: 7820 202d 2034 3878 2078 2020 2b20 3131  x  - 48x x  + 11
│ │ │ │ +0001e830: 7820 7820 7820 7820 202d 2031 3078 2078  x x x x  - 10x x
│ │ │ │ +0001e840: 2078 2020 2b20 3230 7820 7820 7820 202b   x  + 20x x x  +
│ │ │ │ +0001e850: 2031 3178 2078 2078 2078 2020 7c0a 7c20   11x x x x  |.| 
│ │ │ │ +0001e860: 2020 3120 3520 3720 2020 2020 2033 2035    1 5 7      3 5
│ │ │ │ +0001e870: 2037 2020 2020 2020 3520 3720 2020 2020   7      5 7     
│ │ │ │ +0001e880: 2030 2031 2032 2038 2020 2020 2020 3120   0 1 2 8      1 
│ │ │ │ +0001e890: 3220 3820 2020 2020 2031 2032 2038 2020  2 8      1 2 8  
│ │ │ │ +0001e8a0: 2020 2020 3020 3120 3320 3820 7c0a 7c2d      0 1 3 8 |.|-
│ │ │ │  0001e8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001e8f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220  ------|.|     2 
│ │ │ │ -0001e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e910: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0001e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001e900: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0001e910: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  0001e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001e940: 2020 2020 2020 7c0a 7c2b 2034 3278 2078        |.|+ 42x x
│ │ │ │ -0001e950: 2078 2020 2b20 3431 7820 7820 7820 7820   x  + 41x x x x 
│ │ │ │ -0001e960: 202d 2031 3178 2078 2078 2020 2b20 3130   - 11x x x  + 10
│ │ │ │ -0001e970: 7820 7820 7820 7820 202d 2032 3078 2078  x x x x  - 20x x
│ │ │ │ -0001e980: 2078 2078 2020 2d20 3236 7820 7820 7820   x x  - 26x x x 
│ │ │ │ -0001e990: 7820 202b 2020 7c0a 7c20 2020 2020 3120  x  +  |.|     1 
│ │ │ │ -0001e9a0: 3320 3820 2020 2020 2031 2032 2033 2038  3 8      1 2 3 8
│ │ │ │ -0001e9b0: 2020 2020 2020 3020 3420 3820 2020 2020        0 4 8     
│ │ │ │ -0001e9c0: 2030 2031 2034 2038 2020 2020 2020 3020   0 1 4 8      0 
│ │ │ │ -0001e9d0: 3220 3420 3820 2020 2020 2031 2032 2034  2 4 8      1 2 4
│ │ │ │ -0001e9e0: 2038 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   8    |.|-------
│ │ │ │ +0001e940: 2020 2020 2020 2020 2020 2020 7c0a 7c2b              |.|+
│ │ │ │ +0001e950: 2034 3278 2078 2078 2020 2b20 3431 7820   42x x x  + 41x 
│ │ │ │ +0001e960: 7820 7820 7820 202d 2031 3178 2078 2078  x x x  - 11x x x
│ │ │ │ +0001e970: 2020 2b20 3130 7820 7820 7820 7820 202d    + 10x x x x  -
│ │ │ │ +0001e980: 2032 3078 2078 2078 2078 2020 2d20 3236   20x x x x  - 26
│ │ │ │ +0001e990: 7820 7820 7820 7820 202b 2020 7c0a 7c20  x x x x  +  |.| 
│ │ │ │ +0001e9a0: 2020 2020 3120 3320 3820 2020 2020 2031      1 3 8      1
│ │ │ │ +0001e9b0: 2032 2033 2038 2020 2020 2020 3020 3420   2 3 8      0 4 
│ │ │ │ +0001e9c0: 3820 2020 2020 2030 2031 2034 2038 2020  8      0 1 4 8  
│ │ │ │ +0001e9d0: 2020 2020 3020 3220 3420 3820 2020 2020      0 2 4 8     
│ │ │ │ +0001e9e0: 2031 2032 2034 2038 2020 2020 7c0a 7c2d   1 2 4 8    |.|-
│ │ │ │  0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ea30: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  0001ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea50: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0001ea60: 2020 3220 2020 2020 2020 2032 2020 2020    2        2    
│ │ │ │ -0001ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ea80: 2020 2020 2020 7c0a 7c32 3878 2078 2078        |.|28x x x
│ │ │ │ -0001ea90: 2078 2020 2b20 3478 2078 2078 2078 2020   x  + 4x x x x  
│ │ │ │ -0001eaa0: 2b20 3236 7820 7820 7820 202d 2031 3278  + 26x x x  - 12x
│ │ │ │ -0001eab0: 2078 2078 2020 2d20 3131 7820 7820 7820   x x  - 11x x x 
│ │ │ │ -0001eac0: 202d 2034 3278 2078 2078 2078 2020 2d20   - 42x x x x  - 
│ │ │ │ -0001ead0: 2020 2020 2020 7c0a 7c20 2020 3120 3320        |.|   1 3 
│ │ │ │ -0001eae0: 3420 3820 2020 2020 3220 3320 3420 3820  4 8     2 3 4 8 
│ │ │ │ -0001eaf0: 2020 2020 2030 2034 2038 2020 2020 2020       0 4 8      
│ │ │ │ -0001eb00: 3320 3420 3820 2020 2020 2030 2035 2038  3 4 8      0 5 8
│ │ │ │ -0001eb10: 2020 2020 2020 3020 3120 3520 3820 2020        0 1 5 8   
│ │ │ │ -0001eb20: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001ea50: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0001ea60: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0001ea70: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0001ea80: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +0001ea90: 3878 2078 2078 2078 2020 2b20 3478 2078  8x x x x  + 4x x
│ │ │ │ +0001eaa0: 2078 2078 2020 2b20 3236 7820 7820 7820   x x  + 26x x x 
│ │ │ │ +0001eab0: 202d 2031 3278 2078 2078 2020 2d20 3131   - 12x x x  - 11
│ │ │ │ +0001eac0: 7820 7820 7820 202d 2034 3278 2078 2078  x x x  - 42x x x
│ │ │ │ +0001ead0: 2078 2020 2d20 2020 2020 2020 7c0a 7c20   x  -       |.| 
│ │ │ │ +0001eae0: 2020 3120 3320 3420 3820 2020 2020 3220    1 3 4 8     2 
│ │ │ │ +0001eaf0: 3320 3420 3820 2020 2020 2030 2034 2038  3 4 8      0 4 8
│ │ │ │ +0001eb00: 2020 2020 2020 3320 3420 3820 2020 2020        3 4 8     
│ │ │ │ +0001eb10: 2030 2035 2038 2020 2020 2020 3020 3120   0 5 8      0 1 
│ │ │ │ +0001eb20: 3520 3820 2020 2020 2020 2020 7c0a 7c2d  5 8         |.|-
│ │ │ │  0001eb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001eb70: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0001eb80: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0001eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001eb80: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0001eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ebc0: 2020 2020 2020 7c0a 7c34 3178 2078 2078        |.|41x x x
│ │ │ │ -0001ebd0: 2078 2020 2d20 3478 2078 2078 2020 2d20   x  - 4x x x  - 
│ │ │ │ -0001ebe0: 3238 7820 7820 7820 7820 202b 2031 3278  28x x x x  + 12x
│ │ │ │ -0001ebf0: 2078 2078 2078 2020 2b20 3337 7820 7820   x x x  + 37x x 
│ │ │ │ -0001ec00: 7820 7820 202b 2033 7820 7820 7820 7820  x x  + 3x x x x 
│ │ │ │ -0001ec10: 202d 2020 2020 7c0a 7c20 2020 3020 3220   -    |.|   0 2 
│ │ │ │ -0001ec20: 3520 3820 2020 2020 3220 3520 3820 2020  5 8     2 5 8   
│ │ │ │ -0001ec30: 2020 2030 2034 2035 2038 2020 2020 2020     0 4 5 8      
│ │ │ │ -0001ec40: 3220 3420 3520 3820 2020 2020 2030 2032  2 4 5 8      0 2
│ │ │ │ -0001ec50: 2036 2038 2020 2020 2031 2032 2036 2038   6 8     1 2 6 8
│ │ │ │ -0001ec60: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001ebc0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +0001ebd0: 3178 2078 2078 2078 2020 2d20 3478 2078  1x x x x  - 4x x
│ │ │ │ +0001ebe0: 2078 2020 2d20 3238 7820 7820 7820 7820   x  - 28x x x x 
│ │ │ │ +0001ebf0: 202b 2031 3278 2078 2078 2078 2020 2b20   + 12x x x x  + 
│ │ │ │ +0001ec00: 3337 7820 7820 7820 7820 202b 2033 7820  37x x x x  + 3x 
│ │ │ │ +0001ec10: 7820 7820 7820 202d 2020 2020 7c0a 7c20  x x x  -    |.| 
│ │ │ │ +0001ec20: 2020 3020 3220 3520 3820 2020 2020 3220    0 2 5 8     2 
│ │ │ │ +0001ec30: 3520 3820 2020 2020 2030 2034 2035 2038  5 8      0 4 5 8
│ │ │ │ +0001ec40: 2020 2020 2020 3220 3420 3520 3820 2020        2 4 5 8   
│ │ │ │ +0001ec50: 2020 2030 2032 2036 2038 2020 2020 2031     0 2 6 8     1
│ │ │ │ +0001ec60: 2032 2036 2038 2020 2020 2020 7c0a 7c2d   2 6 8      |.|-
│ │ │ │  0001ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ecb0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020  ------|.|   2   
│ │ │ │ -0001ecc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001ecc0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ed00: 2020 2020 2020 7c0a 7c32 3178 2078 2078        |.|21x x x
│ │ │ │ -0001ed10: 2020 2d20 3134 7820 7820 7820 7820 202b    - 14x x x x  +
│ │ │ │ -0001ed20: 2032 3978 2078 2078 2078 2020 2b20 3131   29x x x x  + 11
│ │ │ │ -0001ed30: 7820 7820 7820 7820 202b 2034 3778 2078  x x x x  + 47x x
│ │ │ │ -0001ed40: 2078 2078 2020 2d20 3438 7820 7820 7820   x x  - 48x x x 
│ │ │ │ -0001ed50: 7820 202d 2020 7c0a 7c20 2020 3220 3620  x  -  |.|   2 6 
│ │ │ │ -0001ed60: 3820 2020 2020 2030 2033 2036 2038 2020  8      0 3 6 8  
│ │ │ │ -0001ed70: 2020 2020 3120 3320 3620 3820 2020 2020      1 3 6 8     
│ │ │ │ -0001ed80: 2032 2033 2036 2038 2020 2020 2020 3120   2 3 6 8      1 
│ │ │ │ -0001ed90: 3420 3620 3820 2020 2020 2032 2034 2036  4 6 8      2 4 6
│ │ │ │ -0001eda0: 2038 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   8    |.|-------
│ │ │ │ +0001ed00: 2020 2020 2020 2020 2020 2020 7c0a 7c32              |.|2
│ │ │ │ +0001ed10: 3178 2078 2078 2020 2d20 3134 7820 7820  1x x x  - 14x x 
│ │ │ │ +0001ed20: 7820 7820 202b 2032 3978 2078 2078 2078  x x  + 29x x x x
│ │ │ │ +0001ed30: 2020 2b20 3131 7820 7820 7820 7820 202b    + 11x x x x  +
│ │ │ │ +0001ed40: 2034 3778 2078 2078 2078 2020 2d20 3438   47x x x x  - 48
│ │ │ │ +0001ed50: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ +0001ed60: 2020 3220 3620 3820 2020 2020 2030 2033    2 6 8      0 3
│ │ │ │ +0001ed70: 2036 2038 2020 2020 2020 3120 3320 3620   6 8      1 3 6 
│ │ │ │ +0001ed80: 3820 2020 2020 2032 2033 2036 2038 2020  8      2 3 6 8  
│ │ │ │ +0001ed90: 2020 2020 3120 3420 3620 3820 2020 2020      1 4 6 8     
│ │ │ │ +0001eda0: 2032 2034 2036 2038 2020 2020 7c0a 7c2d   2 4 6 8    |.|-
│ │ │ │  0001edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001edf0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0001ee00: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0001edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001ee00: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0001ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ee30: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0001ee40: 2020 2020 2020 7c0a 7c32 7820 7820 7820        |.|2x x x 
│ │ │ │ -0001ee50: 7820 202b 2034 3578 2078 2078 2020 2b20  x  + 45x x x  + 
│ │ │ │ -0001ee60: 3239 7820 7820 7820 7820 202b 2032 3578  29x x x x  + 25x
│ │ │ │ -0001ee70: 2078 2078 2078 2020 2b20 3333 7820 7820   x x x  + 33x x 
│ │ │ │ -0001ee80: 7820 7820 202d 2033 3778 2078 2078 2020  x x  - 37x x x  
│ │ │ │ -0001ee90: 2d20 2020 2020 7c0a 7c20 2033 2034 2036  -     |.|  3 4 6
│ │ │ │ -0001eea0: 2038 2020 2020 2020 3420 3620 3820 2020   8      4 6 8   
│ │ │ │ -0001eeb0: 2020 2030 2035 2036 2038 2020 2020 2020     0 5 6 8      
│ │ │ │ -0001eec0: 3120 3520 3620 3820 2020 2020 2034 2035  1 5 6 8      4 5
│ │ │ │ -0001eed0: 2036 2038 2020 2020 2020 3020 3720 3820   6 8      0 7 8 
│ │ │ │ -0001eee0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ee40: 3220 2020 2020 2020 2020 2020 7c0a 7c32  2           |.|2
│ │ │ │ +0001ee50: 7820 7820 7820 7820 202b 2034 3578 2078  x x x x  + 45x x
│ │ │ │ +0001ee60: 2078 2020 2b20 3239 7820 7820 7820 7820   x  + 29x x x x 
│ │ │ │ +0001ee70: 202b 2032 3578 2078 2078 2078 2020 2b20   + 25x x x x  + 
│ │ │ │ +0001ee80: 3333 7820 7820 7820 7820 202d 2033 3778  33x x x x  - 37x
│ │ │ │ +0001ee90: 2078 2078 2020 2d20 2020 2020 7c0a 7c20   x x  -     |.| 
│ │ │ │ +0001eea0: 2033 2034 2036 2038 2020 2020 2020 3420   3 4 6 8      4 
│ │ │ │ +0001eeb0: 3620 3820 2020 2020 2030 2035 2036 2038  6 8      0 5 6 8
│ │ │ │ +0001eec0: 2020 2020 2020 3120 3520 3620 3820 2020        1 5 6 8   
│ │ │ │ +0001eed0: 2020 2034 2035 2036 2038 2020 2020 2020     4 5 6 8      
│ │ │ │ +0001eee0: 3020 3720 3820 2020 2020 2020 7c0a 7c2d  0 7 8       |.|-
│ │ │ │  0001eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001ef30: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ -0001ef40: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0001ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001ef40: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ef80: 2020 2020 2020 7c0a 7c33 7820 7820 7820        |.|3x x x 
│ │ │ │ -0001ef90: 7820 202d 2034 3778 2078 2078 2020 2b20  x  - 47x x x  + 
│ │ │ │ -0001efa0: 3231 7820 7820 7820 7820 202b 2031 3178  21x x x x  + 11x
│ │ │ │ -0001efb0: 2078 2078 2078 2020 2b20 7820 7820 7820   x x x  + x x x 
│ │ │ │ -0001efc0: 7820 202b 2034 3878 2078 2078 2078 2020  x  + 48x x x x  
│ │ │ │ -0001efd0: 2d20 2020 2020 7c0a 7c20 2030 2031 2037  -     |.|  0 1 7
│ │ │ │ -0001efe0: 2038 2020 2020 2020 3120 3720 3820 2020   8      1 7 8   
│ │ │ │ -0001eff0: 2020 2030 2032 2037 2038 2020 2020 2020     0 2 7 8      
│ │ │ │ -0001f000: 3020 3320 3720 3820 2020 2032 2033 2037  0 3 7 8    2 3 7
│ │ │ │ -0001f010: 2038 2020 2020 2020 3020 3420 3720 3820   8      0 4 7 8 
│ │ │ │ -0001f020: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d        |.|-------
│ │ │ │ +0001ef80: 2020 2020 2020 2020 2020 2020 7c0a 7c33              |.|3
│ │ │ │ +0001ef90: 7820 7820 7820 7820 202d 2034 3778 2078  x x x x  - 47x x
│ │ │ │ +0001efa0: 2078 2020 2b20 3231 7820 7820 7820 7820   x  + 21x x x x 
│ │ │ │ +0001efb0: 202b 2031 3178 2078 2078 2078 2020 2b20   + 11x x x x  + 
│ │ │ │ +0001efc0: 7820 7820 7820 7820 202b 2034 3878 2078  x x x x  + 48x x
│ │ │ │ +0001efd0: 2078 2078 2020 2d20 2020 2020 7c0a 7c20   x x  -     |.| 
│ │ │ │ +0001efe0: 2030 2031 2037 2038 2020 2020 2020 3120   0 1 7 8      1 
│ │ │ │ +0001eff0: 3720 3820 2020 2020 2030 2032 2037 2038  7 8      0 2 7 8
│ │ │ │ +0001f000: 2020 2020 2020 3020 3320 3720 3820 2020        0 3 7 8   
│ │ │ │ +0001f010: 2032 2033 2037 2038 2020 2020 2020 3020   2 3 7 8      0 
│ │ │ │ +0001f020: 3420 3720 3820 2020 2020 2020 7c0a 7c2d  4 7 8       |.|-
│ │ │ │  0001f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f070: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020  ------|.|       
│ │ │ │ +0001f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │  0001f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f0c0: 2020 2020 2020 7c0a 7c34 3578 2078 2078        |.|45x x x
│ │ │ │ -0001f0d0: 2078 2020 2b20 3437 7820 7820 7820 7820   x  + 47x x x x 
│ │ │ │ -0001f0e0: 202d 2032 7820 7820 7820 7820 202b 2033   - 2x x x x  + 3
│ │ │ │ -0001f0f0: 3378 2078 2078 2078 2020 2b20 3437 7820  3x x x x  + 47x 
│ │ │ │ -0001f100: 7820 7820 7820 202d 2078 2078 2078 2078  x x x  - x x x x
│ │ │ │ -0001f110: 2020 2b20 2020 7c0a 7c20 2020 3120 3420    +   |.|   1 4 
│ │ │ │ -0001f120: 3720 3820 2020 2020 2033 2034 2037 2038  7 8      3 4 7 8
│ │ │ │ -0001f130: 2020 2020 2030 2035 2037 2038 2020 2020       0 5 7 8    
│ │ │ │ -0001f140: 2020 3120 3520 3720 3820 2020 2020 2032    1 5 7 8      2
│ │ │ │ -0001f150: 2035 2037 2038 2020 2020 3420 3520 3720   5 7 8    4 5 7 
│ │ │ │ -0001f160: 3820 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d  8     |.|-------
│ │ │ │ +0001f0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c34              |.|4
│ │ │ │ +0001f0d0: 3578 2078 2078 2078 2020 2b20 3437 7820  5x x x x  + 47x 
│ │ │ │ +0001f0e0: 7820 7820 7820 202d 2032 7820 7820 7820  x x x  - 2x x x 
│ │ │ │ +0001f0f0: 7820 202b 2033 3378 2078 2078 2078 2020  x  + 33x x x x  
│ │ │ │ +0001f100: 2b20 3437 7820 7820 7820 7820 202d 2078  + 47x x x x  - x
│ │ │ │ +0001f110: 2078 2078 2078 2020 2b20 2020 7c0a 7c20   x x x  +   |.| 
│ │ │ │ +0001f120: 2020 3120 3420 3720 3820 2020 2020 2033    1 4 7 8      3
│ │ │ │ +0001f130: 2034 2037 2038 2020 2020 2030 2035 2037   4 7 8     0 5 7
│ │ │ │ +0001f140: 2038 2020 2020 2020 3120 3520 3720 3820   8      1 5 7 8 
│ │ │ │ +0001f150: 2020 2020 2032 2035 2037 2038 2020 2020       2 5 7 8    
│ │ │ │ +0001f160: 3420 3520 3720 3820 2020 2020 7c0a 7c2d  4 5 7 8     |.|-
│ │ │ │  0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f1b0: 2d2d 2d2d 2d2d 7c0a 7c20 2032 2032 2020  ------|.|  2 2  
│ │ │ │ -0001f1c0: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ -0001f1d0: 2032 2020 2020 2020 2020 2020 3220 2020   2          2   
│ │ │ │ -0001f1e0: 2020 2032 2032 2020 2020 2020 2020 2032     2 2         2
│ │ │ │ -0001f1f0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0001f200: 2020 2020 2032 7c0a 7c37 7820 7820 202d       2|.|7x x  -
│ │ │ │ -0001f210: 2032 3978 2078 2078 2020 2b20 3336 7820   29x x x  + 36x 
│ │ │ │ -0001f220: 7820 202d 2031 3178 2078 2078 2020 2b20  x  - 11x x x  + 
│ │ │ │ -0001f230: 3438 7820 7820 202b 2032 7820 7820 7820  48x x  + 2x x x 
│ │ │ │ -0001f240: 202d 2033 3378 2078 2078 2020 2d20 3437   - 33x x x  - 47
│ │ │ │ -0001f250: 7820 7820 7820 7c0a 7c20 2030 2038 2020  x x x |.|  0 8  
│ │ │ │ -0001f260: 2020 2020 3020 3120 3820 2020 2020 2031      0 1 8      1
│ │ │ │ -0001f270: 2038 2020 2020 2020 3020 3220 3820 2020   8      0 2 8   
│ │ │ │ -0001f280: 2020 2032 2038 2020 2020 2030 2034 2038     2 8     0 4 8
│ │ │ │ -0001f290: 2020 2020 2020 3120 3420 3820 2020 2020        1 4 8     
│ │ │ │ -0001f2a0: 2032 2034 2038 7c0a 7c2d 2d2d 2d2d 2d2d   2 4 8|.|-------
│ │ │ │ +0001f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001f1c0: 2032 2032 2020 2020 2020 2020 2020 3220   2 2          2 
│ │ │ │ +0001f1d0: 2020 2020 2032 2032 2020 2020 2020 2020       2 2        
│ │ │ │ +0001f1e0: 2020 3220 2020 2020 2032 2032 2020 2020    2      2 2    
│ │ │ │ +0001f1f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0001f200: 3220 2020 2020 2020 2020 2032 7c0a 7c37  2          2|.|7
│ │ │ │ +0001f210: 7820 7820 202d 2032 3978 2078 2078 2020  x x  - 29x x x  
│ │ │ │ +0001f220: 2b20 3336 7820 7820 202d 2031 3178 2078  + 36x x  - 11x x
│ │ │ │ +0001f230: 2078 2020 2b20 3438 7820 7820 202b 2032   x  + 48x x  + 2
│ │ │ │ +0001f240: 7820 7820 7820 202d 2033 3378 2078 2078  x x x  - 33x x x
│ │ │ │ +0001f250: 2020 2d20 3437 7820 7820 7820 7c0a 7c20    - 47x x x |.| 
│ │ │ │ +0001f260: 2030 2038 2020 2020 2020 3020 3120 3820   0 8      0 1 8 
│ │ │ │ +0001f270: 2020 2020 2031 2038 2020 2020 2020 3020       1 8      0 
│ │ │ │ +0001f280: 3220 3820 2020 2020 2032 2038 2020 2020  2 8      2 8    
│ │ │ │ +0001f290: 2030 2034 2038 2020 2020 2020 3120 3420   0 4 8      1 4 
│ │ │ │ +0001f2a0: 3820 2020 2020 2032 2034 2038 7c0a 7c2d  8      2 4 8|.|-
│ │ │ │  0001f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f2f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220  ------|.|     2 
│ │ │ │ -0001f300: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0001f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20  ------------|.| 
│ │ │ │ +0001f300: 2020 2020 3220 3220 2020 2020 2020 2020      2 2         
│ │ │ │  0001f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f340: 2020 2020 2020 7c0a 7c2d 2034 3878 2078        |.|- 48x x
│ │ │ │ -0001f350: 2020 2b20 3438 7820 7820 7820 7820 202d    + 48x x x x  -
│ │ │ │ -0001f360: 2034 3878 2078 2078 2078 2020 2d20 3438   48x x x x  - 48
│ │ │ │ -0001f370: 7820 7820 7820 7820 202b 2034 3878 2078  x x x x  + 48x x
│ │ │ │ -0001f380: 2078 2078 2020 2b20 3438 7820 7820 7820   x x  + 48x x x 
│ │ │ │ -0001f390: 7820 202d 2020 7c0a 7c20 2020 2020 3420  x  -  |.|     4 
│ │ │ │ -0001f3a0: 3820 2020 2020 2033 2034 2036 2039 2020  8      3 4 6 9  
│ │ │ │ -0001f3b0: 2020 2020 3220 3520 3620 3920 2020 2020      2 5 6 9     
│ │ │ │ -0001f3c0: 2031 2033 2037 2039 2020 2020 2020 3020   1 3 7 9      0 
│ │ │ │ -0001f3d0: 3520 3720 3920 2020 2020 2031 2032 2038  5 7 9      1 2 8
│ │ │ │ -0001f3e0: 2039 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d   9    |.|-------
│ │ │ │ +0001f340: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001f350: 2034 3878 2078 2020 2b20 3438 7820 7820   48x x  + 48x x 
│ │ │ │ +0001f360: 7820 7820 202d 2034 3878 2078 2078 2078  x x  - 48x x x x
│ │ │ │ +0001f370: 2020 2d20 3438 7820 7820 7820 7820 202b    - 48x x x x  +
│ │ │ │ +0001f380: 2034 3878 2078 2078 2078 2020 2b20 3438   48x x x x  + 48
│ │ │ │ +0001f390: 7820 7820 7820 7820 202d 2020 7c0a 7c20  x x x x  -  |.| 
│ │ │ │ +0001f3a0: 2020 2020 3420 3820 2020 2020 2033 2034      4 8      3 4
│ │ │ │ +0001f3b0: 2036 2039 2020 2020 2020 3220 3520 3620   6 9      2 5 6 
│ │ │ │ +0001f3c0: 3920 2020 2020 2031 2033 2037 2039 2020  9      1 3 7 9  
│ │ │ │ +0001f3d0: 2020 2020 3020 3520 3720 3920 2020 2020      0 5 7 9     
│ │ │ │ +0001f3e0: 2031 2032 2038 2039 2020 2020 7c0a 7c2d   1 2 8 9    |.|-
│ │ │ │  0001f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f430: 2d2d 2d2d 2d2d 7c0a 7c34 3878 2078 2078  ------|.|48x x x
│ │ │ │ -0001f440: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +0001f430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34  ------------|.|4
│ │ │ │ +0001f440: 3878 2078 2078 2078 2020 2020 2020 2020  8x x x x        
│ │ │ │  0001f450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f480: 2020 2020 2020 7c0a 7c20 2020 3020 3420        |.|   0 4 
│ │ │ │ -0001f490: 3820 3920 2020 2020 2020 2020 2020 2020  8 9             
│ │ │ │ +0001f480: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f490: 2020 3020 3420 3820 3920 2020 2020 2020    0 4 8 9       
│ │ │ │  0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f4d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001f4d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0001f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f520: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6173  ------+.|i4 : as
│ │ │ │ -0001f530: 7365 7274 2870 6869 202a 2070 7369 203d  sert(phi * psi =
│ │ │ │ -0001f540: 3d20 3120 616e 6420 7073 6920 2a20 7068  = 1 and psi * ph
│ │ │ │ -0001f550: 6920 3d3d 2031 2920 2020 2020 2020 2020  i == 1)         
│ │ │ │ +0001f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001f530: 3420 3a20 6173 7365 7274 2870 6869 202a  4 : assert(phi *
│ │ │ │ +0001f540: 2070 7369 203d 3d20 3120 616e 6420 7073   psi == 1 and ps
│ │ │ │ +0001f550: 6920 2a20 7068 6920 3d3d 2031 2920 2020  i * phi == 1)   
│ │ │ │  0001f560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f570: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001f570: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0001f580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f5c0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7469  ------+.|i5 : ti
│ │ │ │ -0001f5d0: 6d65 2070 7369 2720 3d20 6170 7072 6f78  me psi' = approx
│ │ │ │ -0001f5e0: 696d 6174 6549 6e76 6572 7365 4d61 7028  imateInverseMap(
│ │ │ │ -0001f5f0: 7068 692c 436f 6469 6d42 7349 6e76 3d3e  phi,CodimBsInv=>
│ │ │ │ -0001f600: 3529 3b20 2020 2020 2020 2020 2020 2020  5);             
│ │ │ │ -0001f610: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -0001f620: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -0001f630: 703a 2073 7465 7020 3120 6f66 2033 2020  p: step 1 of 3  
│ │ │ │ -0001f640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001f5d0: 3520 3a20 7469 6d65 2070 7369 2720 3d20  5 : time psi' = 
│ │ │ │ +0001f5e0: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +0001f5f0: 7365 4d61 7028 7068 692c 436f 6469 6d42  seMap(phi,CodimB
│ │ │ │ +0001f600: 7349 6e76 3d3e 3529 3b20 2020 2020 2020  sInv=>5);       
│ │ │ │ +0001f610: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001f620: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +0001f630: 6572 7365 4d61 703a 2073 7465 7020 3120  erseMap: step 1 
│ │ │ │ +0001f640: 6f66 2033 2020 2020 2020 2020 2020 2020  of 3            
│ │ │ │  0001f650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f660: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -0001f670: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -0001f680: 703a 2073 7465 7020 3220 6f66 2033 2020  p: step 2 of 3  
│ │ │ │ -0001f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f660: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001f670: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +0001f680: 6572 7365 4d61 703a 2073 7465 7020 3220  erseMap: step 2 
│ │ │ │ +0001f690: 6f66 2033 2020 2020 2020 2020 2020 2020  of 3            
│ │ │ │  0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f6b0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072        |.|-- appr
│ │ │ │ -0001f6c0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ -0001f6d0: 703a 2073 7465 7020 3320 6f66 2033 2020  p: step 3 of 3  
│ │ │ │ -0001f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d              |.|-
│ │ │ │ +0001f6c0: 2d20 6170 7072 6f78 696d 6174 6549 6e76  - approximateInv
│ │ │ │ +0001f6d0: 6572 7365 4d61 703a 2073 7465 7020 3320  erseMap: step 3 
│ │ │ │ +0001f6e0: 6f66 2033 2020 2020 2020 2020 2020 2020  of 3            
│ │ │ │  0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f700: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365        |.| -- use
│ │ │ │ -0001f710: 6420 302e 3336 3739 3339 7320 2863 7075  d 0.367939s (cpu
│ │ │ │ -0001f720: 293b 2030 2e32 3830 3934 3273 2028 7468  ); 0.280942s (th
│ │ │ │ -0001f730: 7265 6164 293b 2030 7320 2867 6329 2020  read); 0s (gc)  
│ │ │ │ -0001f740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f750: 2020 2020 2020 7c0a 7c20 2020 2020 2020        |.|       
│ │ │ │ +0001f700: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ +0001f710: 2d2d 2075 7365 6420 302e 3334 3337 3738  -- used 0.343778
│ │ │ │ +0001f720: 7320 2863 7075 293b 2030 2e32 3336 3739  s (cpu); 0.23679
│ │ │ │ +0001f730: 3273 2028 7468 7265 6164 293b 2030 7320  2s (thread); 0s 
│ │ │ │ +0001f740: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ +0001f750: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │  0001f760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f7a0: 2020 2020 2020 7c0a 7c6f 3520 3a20 5261        |.|o5 : Ra
│ │ │ │ -0001f7b0: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ -0001f7c0: 6174 6963 2072 6174 696f 6e61 6c20 6d61  atic rational ma
│ │ │ │ -0001f7d0: 7020 6672 6f6d 2050 505e 3820 746f 2068  p from PP^8 to h
│ │ │ │ -0001f7e0: 7970 6572 7375 7266 6163 6520 696e 2050  ypersurface in P
│ │ │ │ -0001f7f0: 505e 3929 2020 7c0a 2b2d 2d2d 2d2d 2d2d  P^9)  |.+-------
│ │ │ │ +0001f7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ +0001f7b0: 3520 3a20 5261 7469 6f6e 616c 4d61 7020  5 : RationalMap 
│ │ │ │ +0001f7c0: 2871 7561 6472 6174 6963 2072 6174 696f  (quadratic ratio
│ │ │ │ +0001f7d0: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e  nal map from PP^
│ │ │ │ +0001f7e0: 3820 746f 2068 7970 6572 7375 7266 6163  8 to hypersurfac
│ │ │ │ +0001f7f0: 6520 696e 2050 505e 3929 2020 7c0a 2b2d  e in PP^9)  |.+-
│ │ │ │  0001f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f840: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 6173  ------+.|i6 : as
│ │ │ │ -0001f850: 7365 7274 2870 7369 203d 3d20 7073 6927  sert(psi == psi'
│ │ │ │ -0001f860: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +0001f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69  ------------+.|i
│ │ │ │ +0001f850: 3620 3a20 6173 7365 7274 2870 7369 203d  6 : assert(psi =
│ │ │ │ +0001f860: 3d20 7073 6927 2920 2020 2020 2020 2020  = psi')         
│ │ │ │  0001f870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001f880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f890: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │ +0001f890: 2020 2020 2020 2020 2020 2020 7c0a 2b2d              |.+-
│ │ │ │  0001f8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f8e0: 2d2d 2d2d 2d2d 2b0a 0a41 206d 6f72 6520  ------+..A more 
│ │ │ │ -0001f8f0: 636f 6d70 6c69 6361 7465 6420 6578 616d  complicated exam
│ │ │ │ -0001f900: 706c 6520 6973 2074 6865 2066 6f6c 6c6f  ple is the follo
│ │ │ │ -0001f910: 7769 6e67 2028 6865 7265 202a 6e6f 7465  wing (here *note
│ │ │ │ -0001f920: 2069 6e76 6572 7365 4d61 703a 2069 6e76   inverseMap: inv
│ │ │ │ -0001f930: 6572 7365 4d61 702c 0a74 616b 6573 2061  erseMap,.takes a
│ │ │ │ -0001f940: 206c 6f74 206f 6620 7469 6d65 2129 2e0a   lot of time!)..
│ │ │ │ -0001f950: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0001f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41  ------------+..A
│ │ │ │ +0001f8f0: 206d 6f72 6520 636f 6d70 6c69 6361 7465   more complicate
│ │ │ │ +0001f900: 6420 6578 616d 706c 6520 6973 2074 6865  d example is the
│ │ │ │ +0001f910: 2066 6f6c 6c6f 7769 6e67 2028 6865 7265   following (here
│ │ │ │ +0001f920: 202a 6e6f 7465 2069 6e76 6572 7365 4d61   *note inverseMa
│ │ │ │ +0001f930: 703a 2069 6e76 6572 7365 4d61 702c 0a74  p: inverseMap,.t
│ │ │ │ +0001f940: 616b 6573 2061 206c 6f74 206f 6620 7469  akes a lot of ti
│ │ │ │ +0001f950: 6d65 2129 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  me!)...+--------
│ │ │ │  0001f960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0001f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0001f9a0: 0a7c 6937 203a 2070 6869 203d 2072 6174  .|i7 : phi = rat
│ │ │ │ -0001f9b0: 696f 6e61 6c4d 6170 206d 6170 2850 382c  ionalMap map(P8,
│ │ │ │ -0001f9c0: 5a5a 2f39 375b 785f 302e 2e20 2020 2020  ZZ/97[x_0..     
│ │ │ │ -0001f9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001f9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001f9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0001f9a0: 2d2d 2d2d 2d2b 0a7c 6937 203a 2070 6869  -----+.|i7 : phi
│ │ │ │ +0001f9b0: 203d 2072 6174 696f 6e61 6c4d 6170 206d   = rationalMap m
│ │ │ │ +0001f9c0: 6170 2850 382c 5a5a 2f39 375b 785f 302e  ap(P8,ZZ/97[x_0.
│ │ │ │ +0001f9d0: 2e20 2020 2020 2020 2020 2020 2020 2020  .               
│ │ │ │ +0001f9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001f9f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fa40: 0a7c 6f37 203d 202d 2d20 7261 7469 6f6e  .|o7 = -- ration
│ │ │ │ -0001fa50: 616c 206d 6170 202d 2d20 2020 2020 2020  al map --       
│ │ │ │ +0001fa30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fa40: 2020 2020 207c 0a7c 6f37 203d 202d 2d20       |.|o7 = -- 
│ │ │ │ +0001fa50: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │  0001fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fa80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fa90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001faa0: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │ +0001fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fa90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001faa0: 2020 2020 2020 2020 2020 5a5a 2020 2020            ZZ    
│ │ │ │  0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fae0: 0a7c 2020 2020 2073 6f75 7263 653a 2050  .|     source: P
│ │ │ │ -0001faf0: 726f 6a28 2d2d 5b74 202c 2074 202c 2074  roj(--[t , t , t
│ │ │ │ -0001fb00: 202c 2074 202c 2074 202c 2020 2020 2020   , t , t ,      
│ │ │ │ +0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fae0: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ +0001faf0: 7263 653a 2050 726f 6a28 2d2d 5b74 202c  rce: Proj(--[t ,
│ │ │ │ +0001fb00: 2074 202c 2074 202c 2074 202c 2074 202c   t , t , t , t ,
│ │ │ │  0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fb30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001fb40: 2020 2020 3937 2020 3020 2020 3120 2020      97  0   1   
│ │ │ │ -0001fb50: 3220 2020 3320 2020 3420 2020 2020 2020  2   3   4       
│ │ │ │ +0001fb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fb40: 2020 2020 2020 2020 2020 3937 2020 3020            97  0 
│ │ │ │ +0001fb50: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │  0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fb70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fb80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001fb70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fb80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001fb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fba0: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │ +0001fba0: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │  0001fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fbc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fbd0: 0a7c 2020 2020 2074 6172 6765 743a 2073  .|     target: s
│ │ │ │ -0001fbe0: 7562 7661 7269 6574 7920 6f66 2050 726f  ubvariety of Pro
│ │ │ │ -0001fbf0: 6a28 2d2d 5b78 202c 2078 2020 2020 2020  j(--[x , x      
│ │ │ │ +0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fbd0: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ +0001fbe0: 6765 743a 2073 7562 7661 7269 6574 7920  get: subvariety 
│ │ │ │ +0001fbf0: 6f66 2050 726f 6a28 2d2d 5b78 202c 2078  of Proj(--[x , x
│ │ │ │  0001fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fc20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc40: 2020 3937 2020 3020 2020 3120 2020 2020    97  0   1     
│ │ │ │ -0001fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fc60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fc70: 0a7c 2020 2020 2020 2020 2020 2020 207b  .|             {
│ │ │ │ -0001fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc40: 2020 2020 2020 2020 3937 2020 3020 2020          97  0   
│ │ │ │ +0001fc50: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0001fc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fc70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fc80: 2020 2020 207b 2020 2020 2020 2020 2020       {          
│ │ │ │  0001fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fcb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fcc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001fcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fcc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fce0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0001fce0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0001fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fd10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001fd20: 7820 7820 202d 2038 7820 7820 202b 2032  x x  - 8x x  + 2
│ │ │ │ -0001fd30: 3578 2020 2d20 3235 7820 7820 2020 2020  5x  - 25x x     
│ │ │ │ -0001fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fd50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fd60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001fd70: 2031 2033 2020 2020 2032 2033 2020 2020   1 3     2 3    
│ │ │ │ -0001fd80: 2020 3320 2020 2020 2032 2020 2020 2020    3      2      
│ │ │ │ +0001fd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fd10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fd20: 2020 2020 2020 7820 7820 202d 2038 7820        x x  - 8x 
│ │ │ │ +0001fd30: 7820 202b 2032 3578 2020 2d20 3235 7820  x  + 25x  - 25x 
│ │ │ │ +0001fd40: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +0001fd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fd60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fd70: 2020 2020 2020 2031 2033 2020 2020 2032         1 3     2
│ │ │ │ +0001fd80: 2033 2020 2020 2020 3320 2020 2020 2032   3      3      2
│ │ │ │  0001fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fda0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fdb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fdb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fdf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fe00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001fe10: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0001fe20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0001fdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fe00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fe10: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0001fe20: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0001fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fe50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001fe60: 7820 202b 2031 3778 2078 2020 2d20 3134  x  + 17x x  - 14
│ │ │ │ -0001fe70: 7820 202d 2031 3378 2078 2020 2020 2020  x  - 13x x      
│ │ │ │ +0001fe40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fe50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fe60: 2020 2020 2020 7820 202b 2031 3778 2078        x  + 17x x
│ │ │ │ +0001fe70: 2020 2d20 3134 7820 202d 2031 3378 2078    - 14x  - 13x x
│ │ │ │  0001fe80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fe90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fea0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001feb0: 2032 2020 2020 2020 3220 3320 2020 2020   2      2 3     
│ │ │ │ -0001fec0: 2033 2020 2020 2020 3220 3420 2020 2020   3      2 4     
│ │ │ │ -0001fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001fee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001fef0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001fe90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001feb0: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ +0001fec0: 3320 2020 2020 2033 2020 2020 2020 3220  3      3      2 
│ │ │ │ +0001fed0: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +0001fee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001fef0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0001ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001ff40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0001ff30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ff40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0001ff50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff60: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0001ff60: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  0001ff70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ff80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001ff90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001ffa0: 7820 7820 202d 2034 3078 2078 2020 2b20  x x  - 40x x  + 
│ │ │ │ -0001ffb0: 3238 7820 202d 2078 2078 2020 2020 2020  28x  - x x      
│ │ │ │ +0001ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ff90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001ffa0: 2020 2020 2020 7820 7820 202d 2034 3078        x x  - 40x
│ │ │ │ +0001ffb0: 2078 2020 2b20 3238 7820 202d 2078 2078   x  + 28x  - x x
│ │ │ │  0001ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0001ffd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0001ffe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0001fff0: 2031 2032 2020 2020 2020 3220 3320 2020   1 2      2 3   
│ │ │ │ -00020000: 2020 2033 2020 2020 3020 3420 2020 2020     3    0 4     
│ │ │ │ -00020010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020030: 0a7c 2020 2020 2020 2020 2020 2020 207d  .|             }
│ │ │ │ -00020040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0001ffe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0001fff0: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +00020000: 3220 3320 2020 2020 2033 2020 2020 3020  2 3      3    0 
│ │ │ │ +00020010: 3420 2020 2020 2020 2020 2020 2020 2020  4               
│ │ │ │ +00020020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020030: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020040: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │  00020050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020080: 0a7c 2020 2020 2064 6566 696e 696e 6720  .|     defining 
│ │ │ │ -00020090: 666f 726d 733a 207b 2020 2020 2020 2020  forms: {        
│ │ │ │ +00020070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020080: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ +00020090: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │  000200a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000200b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000200c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000200d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000200e0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000200c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000200d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000200e0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  000200f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020120: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020130: 2020 2020 2020 2020 7420 202b 2074 2074          t  + t t
│ │ │ │ -00020140: 2020 2b20 7420 7420 202b 2020 2020 2020    + t t  +      
│ │ │ │ +00020110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020130: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020140: 202b 2074 2074 2020 2b20 7420 7420 202b   + t t  + t t  +
│ │ │ │  00020150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020170: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020180: 2020 2020 2020 2020 2034 2020 2020 3020           4    0 
│ │ │ │ -00020190: 3520 2020 2031 2035 2020 2020 2020 2020  5    1 5        
│ │ │ │ +00020160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020170: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020180: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ +00020190: 2020 2020 3020 3520 2020 2031 2035 2020      0 5    1 5  
│ │ │ │  000201a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000201b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000201c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000201b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000201c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000201d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000201f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020210: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020270: 2020 2020 2020 2020 7420 7420 202b 2034          t t  + 4
│ │ │ │ -00020280: 7420 7420 202b 2033 3974 2020 2020 2020  t t  + 39t      
│ │ │ │ +00020250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020270: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020280: 7420 202b 2034 7420 7420 202b 2033 3974  t  + 4t t  + 39t
│ │ │ │  00020290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000202a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000202b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000202c0: 2020 2020 2020 2020 2033 2034 2020 2020           3 4    
│ │ │ │ -000202d0: 2030 2035 2020 2020 2020 3120 2020 2020   0 5      1     
│ │ │ │ -000202e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000202f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020300: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000202a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000202b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000202c0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +000202d0: 2034 2020 2020 2030 2035 2020 2020 2020   4     0 5      
│ │ │ │ +000202e0: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +000202f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000203a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000203b0: 2020 2020 2020 2020 7420 7420 202d 2035          t t  - 5
│ │ │ │ -000203c0: 7420 7420 202d 2034 3074 2020 2020 2020  t t  - 40t      
│ │ │ │ +00020390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000203a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000203b0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +000203c0: 7420 202d 2035 7420 7420 202d 2034 3074  t  - 5t t  - 40t
│ │ │ │  000203d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000203e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000203f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020400: 2020 2020 2020 2020 2032 2034 2020 2020           2 4    
│ │ │ │ -00020410: 2030 2035 2020 2020 2020 3120 2020 2020   0 5      1     
│ │ │ │ -00020420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020440: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000203e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000203f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020400: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00020410: 2034 2020 2020 2030 2035 2020 2020 2020   4     0 5      
│ │ │ │ +00020420: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +00020430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020490: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000204a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000204b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000204c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000204d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000204e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000204f0: 2020 2020 2020 2020 7420 7420 202d 2074          t t  - t
│ │ │ │ -00020500: 2074 2020 2d20 3874 2074 2020 2020 2020   t  - 8t t      
│ │ │ │ +000204d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000204e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000204f0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020500: 7420 202d 2074 2074 2020 2d20 3874 2074  t  - t t  - 8t t
│ │ │ │  00020510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020530: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020540: 2020 2020 2020 2020 2030 2034 2020 2020           0 4    
│ │ │ │ -00020550: 3020 3520 2020 2020 3120 3520 2020 2020  0 5     1 5     
│ │ │ │ -00020560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020580: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020530: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020540: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00020550: 2034 2020 2020 3020 3520 2020 2020 3120   4    0 5     1 
│ │ │ │ +00020560: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ +00020570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020580: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000205a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000205b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000205c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000205d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000205c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000205d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000205e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000205f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020610: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020620: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020630: 2020 2020 2020 2020 7420 7420 202d 2074          t t  - t
│ │ │ │ -00020640: 2074 2020 2d20 7420 7420 2020 2020 2020   t  - t t       
│ │ │ │ +00020610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020620: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020630: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020640: 7420 202d 2074 2074 2020 2d20 7420 7420  t  - t t  - t t 
│ │ │ │  00020650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020680: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -00020690: 3020 3520 2020 2031 2035 2020 2020 2020  0 5    1 5      
│ │ │ │ +00020660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020670: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020680: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00020690: 2033 2020 2020 3020 3520 2020 2031 2035   3    0 5    1 5
│ │ │ │  000206a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000206b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000206c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000206b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000206c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000206d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000206e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000206f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020710: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020710: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020770: 2020 2020 2020 2020 7420 7420 202d 2037          t t  - 7
│ │ │ │ -00020780: 7420 7420 202b 2074 2074 2020 2020 2020  t t  + t t      
│ │ │ │ +00020750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020760: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020770: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020780: 7420 202d 2037 7420 7420 202b 2074 2074  t  - 7t t  + t t
│ │ │ │  00020790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000207a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000207b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000207c0: 2020 2020 2020 2020 2031 2033 2020 2020           1 3    
│ │ │ │ -000207d0: 2031 2035 2020 2020 3120 3620 2020 2020   1 5    1 6     
│ │ │ │ -000207e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000207f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000207a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000207b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000207c0: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +000207d0: 2033 2020 2020 2031 2035 2020 2020 3120   3     1 5    1 
│ │ │ │ +000207e0: 3620 2020 2020 2020 2020 2020 2020 2020  6               
│ │ │ │ +000207f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020800: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020850: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020850: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000208a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000208b0: 2020 2020 2020 2020 7420 7420 202d 2034          t t  - 4
│ │ │ │ -000208c0: 3674 2074 2020 2d20 3339 7420 2020 2020  6t t  - 39t     
│ │ │ │ -000208d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000208e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000208f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020900: 2020 2020 2020 2020 2030 2033 2020 2020           0 3    
│ │ │ │ -00020910: 2020 3020 3520 2020 2020 2020 2020 2020    0 5           
│ │ │ │ +00020890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000208a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000208b0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +000208c0: 7420 202d 2034 3674 2074 2020 2d20 3339  t  - 46t t  - 39
│ │ │ │ +000208d0: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +000208e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000208f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020900: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00020910: 2033 2020 2020 2020 3020 3520 2020 2020   3      0 5     
│ │ │ │  00020920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020940: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000209a0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00020980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000209a0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  000209b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000209c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000209d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000209e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000209f0: 2020 2020 2020 2020 7420 202d 2034 3674          t  - 46t
│ │ │ │ -00020a00: 2074 2020 2d20 3333 7420 7420 2020 2020   t  - 33t t     
│ │ │ │ -00020a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020a40: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00020a50: 3120 3420 2020 2020 2030 2020 2020 2020  1 4      0      
│ │ │ │ +000209d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000209e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000209f0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020a00: 202d 2034 3674 2074 2020 2d20 3333 7420   - 46t t  - 33t 
│ │ │ │ +00020a10: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00020a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020a40: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00020a50: 2020 2020 2020 3120 3420 2020 2020 2030        1 4      0
│ │ │ │  00020a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020a80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020a80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020ad0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020ae0: 2020 2020 2020 2020 7420 7420 202b 2036          t t  + 6
│ │ │ │ -00020af0: 7420 7420 202b 2035 7420 7420 2020 2020  t t  + 5t t     
│ │ │ │ -00020b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020b20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020b30: 2020 2020 2020 2020 2031 2032 2020 2020           1 2    
│ │ │ │ -00020b40: 2031 2035 2020 2020 2030 2020 2020 2020   1 5     0      
│ │ │ │ +00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020ad0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020ae0: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020af0: 7420 202b 2036 7420 7420 202b 2035 7420  t  + 6t t  + 5t 
│ │ │ │ +00020b00: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00020b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020b20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020b30: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00020b40: 2032 2020 2020 2031 2035 2020 2020 2030   2     1 5     0
│ │ │ │  00020b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020b70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020bc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020c10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020c20: 2020 2020 2020 2020 7420 7420 202b 2074          t t  + t
│ │ │ │ -00020c30: 2074 2020 2b20 3574 2074 2020 2020 2020   t  + 5t t      
│ │ │ │ +00020c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020c10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020c20: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020c30: 7420 202b 2074 2074 2020 2b20 3574 2074  t  + t t  + 5t t
│ │ │ │  00020c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020c50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020c60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020c70: 2020 2020 2020 2020 2030 2032 2020 2020           0 2    
│ │ │ │ -00020c80: 3120 3420 2020 2020 3020 3520 2020 2020  1 4     0 5     
│ │ │ │ -00020c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020c60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020c70: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00020c80: 2032 2020 2020 3120 3420 2020 2020 3020   2    1 4     0 
│ │ │ │ +00020c90: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ +00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020cb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020cf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020d00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020d10: 2020 2020 2020 2020 7420 7420 202d 2033          t t  - 3
│ │ │ │ -00020d20: 3974 2074 2020 2b20 3430 7420 2020 2020  9t t  + 40t     
│ │ │ │ -00020d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020d50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020d60: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ -00020d70: 2020 3120 3420 2020 2020 2020 2020 2020    1 4           
│ │ │ │ +00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020d10: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020d20: 7420 202d 2033 3974 2074 2020 2b20 3430  t  - 39t t  + 40
│ │ │ │ +00020d30: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00020d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020d50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020d60: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00020d70: 2031 2020 2020 2020 3120 3420 2020 2020   1      1 4     
│ │ │ │  00020d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020d90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020da0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020da0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020df0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020e00: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00020de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020df0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020e00: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00020e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020e40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020e50: 2020 2020 2020 2020 7420 202b 2031 3274          t  + 12t
│ │ │ │ -00020e60: 2074 2020 2b20 3230 7420 7420 2020 2020   t  + 20t t     
│ │ │ │ -00020e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020e80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020ea0: 2020 2020 2020 2020 2030 2020 2020 2020           0      
│ │ │ │ -00020eb0: 3120 3420 2020 2020 2030 2020 2020 2020  1 4      0      
│ │ │ │ +00020e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020e40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020e50: 2020 2020 2020 2020 2020 2020 2020 7420                t 
│ │ │ │ +00020e60: 202b 2031 3274 2074 2020 2b20 3230 7420   + 12t t  + 20t 
│ │ │ │ +00020e70: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00020e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020e90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020ea0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00020eb0: 2020 2020 2020 3120 3420 2020 2020 2030        1 4      0
│ │ │ │  00020ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020ed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00020ef0: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │ +00020ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020ee0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00020ef0: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │  00020f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020f20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020f30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00020f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020f30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00020f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00020f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020f70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020f80: 0a7c 6f37 203a 2052 6174 696f 6e61 6c4d  .|o7 : RationalM
│ │ │ │ -00020f90: 6170 2028 7175 6164 7261 7469 6320 7261  ap (quadratic ra
│ │ │ │ -00020fa0: 7469 6f6e 616c 206d 6170 2020 2020 2020  tional map      
│ │ │ │ +00020f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020f80: 2020 2020 207c 0a7c 6f37 203a 2052 6174       |.|o7 : Rat
│ │ │ │ +00020f90: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ +00020fa0: 7469 6320 7261 7469 6f6e 616c 206d 6170  tic rational map
│ │ │ │  00020fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00020fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00020fd0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00020fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00020fd0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00020fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00020ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00021000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00021010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00021020: 0a7c 785f 3131 5d2f 6964 6561 6c28 785f  .|x_11]/ideal(x_
│ │ │ │ -00021030: 312a 785f 332d 382a 785f 322a 785f 332b  1*x_3-8*x_2*x_3+
│ │ │ │ -00021040: 3235 2a78 5f33 5e32 2d32 352a 785f 322a  25*x_3^2-25*x_2*
│ │ │ │ -00021050: 785f 342d 3232 2a78 5f33 2a78 5f34 2b78  x_4-22*x_3*x_4+x
│ │ │ │ -00021060: 5f30 2a78 5f35 2b31 332a 785f 322a 787c  _0*x_5+13*x_2*x|
│ │ │ │ -00021070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00021020: 2d2d 2d2d 2d7c 0a7c 785f 3131 5d2f 6964  -----|.|x_11]/id
│ │ │ │ +00021030: 6561 6c28 785f 312a 785f 332d 382a 785f  eal(x_1*x_3-8*x_
│ │ │ │ +00021040: 322a 785f 332b 3235 2a78 5f33 5e32 2d32  2*x_3+25*x_3^2-2
│ │ │ │ +00021050: 352a 785f 322a 785f 342d 3232 2a78 5f33  5*x_2*x_4-22*x_3
│ │ │ │ +00021060: 2a78 5f34 2b78 5f30 2a78 5f35 2b31 332a  *x_4+x_0*x_5+13*
│ │ │ │ +00021070: 785f 322a 787c 0a7c 2020 2020 2020 2020  x_2*x|.|        
│ │ │ │  00021080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000210a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000210b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000210c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000210b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000210c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000210d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000210e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000210f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021110: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021150: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021160: 0a7c 7420 2c20 7420 2c20 7420 2c20 7420  .|t , t , t , t 
│ │ │ │ -00021170: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +00021150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021160: 2020 2020 207c 0a7c 7420 2c20 7420 2c20       |.|t , t , 
│ │ │ │ +00021170: 7420 2c20 7420 5d29 2020 2020 2020 2020  t , t ])        
│ │ │ │  00021180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000211b0: 0a7c 2035 2020 2036 2020 2037 2020 2038  .| 5   6   7   8
│ │ │ │ -000211c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000211a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000211b0: 2020 2020 207c 0a7c 2035 2020 2036 2020       |.| 5   6  
│ │ │ │ +000211c0: 2037 2020 2038 2020 2020 2020 2020 2020   7   8          
│ │ │ │  000211d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000211e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000211f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021200: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000211f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021250: 0a7c 2c20 7820 2c20 7820 2c20 7820 2c20  .|, x , x , x , 
│ │ │ │ -00021260: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ -00021270: 7820 2c20 7820 202c 2078 2020 5d29 2064  x , x  , x  ]) d
│ │ │ │ -00021280: 6566 696e 6564 2062 7920 2020 2020 2020  efined by       
│ │ │ │ -00021290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000212a0: 0a7c 2020 2032 2020 2033 2020 2034 2020  .|   2   3   4  
│ │ │ │ -000212b0: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ -000212c0: 2039 2020 2031 3020 2020 3131 2020 2020   9   10   11    
│ │ │ │ -000212d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000212e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000212f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021250: 2020 2020 207c 0a7c 2c20 7820 2c20 7820       |.|, x , x 
│ │ │ │ +00021260: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ +00021270: 2c20 7820 2c20 7820 2c20 7820 202c 2078  , x , x , x  , x
│ │ │ │ +00021280: 2020 5d29 2064 6566 696e 6564 2062 7920    ]) defined by 
│ │ │ │ +00021290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000212a0: 2020 2020 207c 0a7c 2020 2032 2020 2033       |.|   2   3
│ │ │ │ +000212b0: 2020 2034 2020 2035 2020 2036 2020 2037     4   5   6   7
│ │ │ │ +000212c0: 2020 2038 2020 2039 2020 2031 3020 2020     8   9   10   
│ │ │ │ +000212d0: 3131 2020 2020 2020 2020 2020 2020 2020  11              
│ │ │ │ +000212e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000212f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021330: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021340: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021390: 0a7c 2020 2d20 3232 7820 7820 202b 2078  .|  - 22x x  + x
│ │ │ │ -000213a0: 2078 2020 2b20 3133 7820 7820 202b 2034   x  + 13x x  + 4
│ │ │ │ -000213b0: 3178 2078 2020 2d20 7820 7820 202b 2031  1x x  - x x  + 1
│ │ │ │ -000213c0: 3278 2078 2020 2b20 3235 7820 7820 202b  2x x  + 25x x  +
│ │ │ │ -000213d0: 2032 3578 2078 2020 2b20 3233 7820 787c   25x x  + 23x x|
│ │ │ │ -000213e0: 0a7c 3420 2020 2020 2033 2034 2020 2020  .|4      3 4    
│ │ │ │ -000213f0: 3020 3520 2020 2020 2032 2035 2020 2020  0 5      2 5    
│ │ │ │ -00021400: 2020 3320 3520 2020 2030 2036 2020 2020    3 5    0 6    
│ │ │ │ -00021410: 2020 3220 3620 2020 2020 2031 2037 2020    2 6      1 7  
│ │ │ │ -00021420: 2020 2020 3320 3720 2020 2020 2035 207c      3 7      5 |
│ │ │ │ -00021430: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021390: 2020 2020 207c 0a7c 2020 2d20 3232 7820       |.|  - 22x 
│ │ │ │ +000213a0: 7820 202b 2078 2078 2020 2b20 3133 7820  x  + x x  + 13x 
│ │ │ │ +000213b0: 7820 202b 2034 3178 2078 2020 2d20 7820  x  + 41x x  - x 
│ │ │ │ +000213c0: 7820 202b 2031 3278 2078 2020 2b20 3235  x  + 12x x  + 25
│ │ │ │ +000213d0: 7820 7820 202b 2032 3578 2078 2020 2b20  x x  + 25x x  + 
│ │ │ │ +000213e0: 3233 7820 787c 0a7c 3420 2020 2020 2033  23x x|.|4      3
│ │ │ │ +000213f0: 2034 2020 2020 3020 3520 2020 2020 2032   4    0 5      2
│ │ │ │ +00021400: 2035 2020 2020 2020 3320 3520 2020 2030   5      3 5    0
│ │ │ │ +00021410: 2036 2020 2020 2020 3220 3620 2020 2020   6      2 6     
│ │ │ │ +00021420: 2031 2037 2020 2020 2020 3320 3720 2020   1 7      3 7   
│ │ │ │ +00021430: 2020 2035 207c 0a7c 2020 2020 2020 2020     5 |.|        
│ │ │ │  00021440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000214b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000214c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000214d0: 0a7c 202b 2033 3478 2078 2020 2b20 3434  .| + 34x x  + 44
│ │ │ │ -000214e0: 7820 7820 202d 2033 3078 2078 2020 2b20  x x  - 30x x  + 
│ │ │ │ -000214f0: 3237 7820 7820 202b 2033 3178 2078 2020  27x x  + 31x x  
│ │ │ │ -00021500: 2d20 3336 7820 7820 202d 2078 2078 2020  - 36x x  - x x  
│ │ │ │ -00021510: 2b20 3133 7820 7820 202b 2038 7820 787c  + 13x x  + 8x x|
│ │ │ │ -00021520: 0a7c 2020 2020 2020 3320 3420 2020 2020  .|      3 4     
│ │ │ │ -00021530: 2030 2035 2020 2020 2020 3220 3520 2020   0 5      2 5   
│ │ │ │ -00021540: 2020 2033 2035 2020 2020 2020 3220 3620     3 5      2 6 
│ │ │ │ -00021550: 2020 2020 2033 2036 2020 2020 3020 3720       3 6    0 7 
│ │ │ │ -00021560: 2020 2020 2031 2037 2020 2020 2033 207c       1 7     3 |
│ │ │ │ -00021570: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000214c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000214d0: 2020 2020 207c 0a7c 202b 2033 3478 2078       |.| + 34x x
│ │ │ │ +000214e0: 2020 2b20 3434 7820 7820 202d 2033 3078    + 44x x  - 30x
│ │ │ │ +000214f0: 2078 2020 2b20 3237 7820 7820 202b 2033   x  + 27x x  + 3
│ │ │ │ +00021500: 3178 2078 2020 2d20 3336 7820 7820 202d  1x x  - 36x x  -
│ │ │ │ +00021510: 2078 2078 2020 2b20 3133 7820 7820 202b   x x  + 13x x  +
│ │ │ │ +00021520: 2038 7820 787c 0a7c 2020 2020 2020 3320   8x x|.|      3 
│ │ │ │ +00021530: 3420 2020 2020 2030 2035 2020 2020 2020  4      0 5      
│ │ │ │ +00021540: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ +00021550: 2020 3220 3620 2020 2020 2033 2036 2020    2 6      3 6  
│ │ │ │ +00021560: 2020 3020 3720 2020 2020 2031 2037 2020    0 7      1 7  
│ │ │ │ +00021570: 2020 2033 207c 0a7c 2020 2020 2020 2020     3 |.|        
│ │ │ │  00021580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000215b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000215c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000215b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000215c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000215d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000215f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021610: 0a7c 202b 2035 7820 7820 202d 2031 3678  .| + 5x x  - 16x
│ │ │ │ -00021620: 2078 2020 2b20 3578 2078 2020 2d20 3336   x  + 5x x  - 36
│ │ │ │ -00021630: 7820 7820 202b 2033 3778 2078 2020 2b20  x x  + 37x x  + 
│ │ │ │ -00021640: 3438 7820 7820 202d 2035 7820 7820 202d  48x x  - 5x x  -
│ │ │ │ -00021650: 2035 7820 7820 202b 2078 2078 2020 2b7c   5x x  + x x  +|
│ │ │ │ -00021660: 0a7c 2020 2020 2032 2034 2020 2020 2020  .|     2 4      
│ │ │ │ -00021670: 3320 3420 2020 2020 3020 3520 2020 2020  3 4     0 5     
│ │ │ │ -00021680: 2032 2035 2020 2020 2020 3320 3520 2020   2 5      3 5   
│ │ │ │ -00021690: 2020 2032 2036 2020 2020 2031 2037 2020     2 6     1 7  
│ │ │ │ -000216a0: 2020 2033 2037 2020 2020 3520 3720 207c     3 7    5 7  |
│ │ │ │ -000216b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021610: 2020 2020 207c 0a7c 202b 2035 7820 7820       |.| + 5x x 
│ │ │ │ +00021620: 202d 2031 3678 2078 2020 2b20 3578 2078   - 16x x  + 5x x
│ │ │ │ +00021630: 2020 2d20 3336 7820 7820 202b 2033 3778    - 36x x  + 37x
│ │ │ │ +00021640: 2078 2020 2b20 3438 7820 7820 202d 2035   x  + 48x x  - 5
│ │ │ │ +00021650: 7820 7820 202d 2035 7820 7820 202b 2078  x x  - 5x x  + x
│ │ │ │ +00021660: 2078 2020 2b7c 0a7c 2020 2020 2032 2034   x  +|.|     2 4
│ │ │ │ +00021670: 2020 2020 2020 3320 3420 2020 2020 3020        3 4     0 
│ │ │ │ +00021680: 3520 2020 2020 2032 2035 2020 2020 2020  5      2 5      
│ │ │ │ +00021690: 3320 3520 2020 2020 2032 2036 2020 2020  3 5      2 6    
│ │ │ │ +000216a0: 2031 2037 2020 2020 2033 2037 2020 2020   1 7     3 7    
│ │ │ │ +000216b0: 3520 3720 207c 0a7c 2020 2020 2020 2020  5 7  |.|        
│ │ │ │  000216c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000216e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000216f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021700: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000216f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021750: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00021760: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00021770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021750: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021770: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00021780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021790: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000217a0: 0a7c 3335 7420 7420 202b 2031 3074 2074  .|35t t  + 10t t
│ │ │ │ -000217b0: 2020 2b20 3235 7420 7420 202d 2035 7420    + 25t t  - 5t 
│ │ │ │ -000217c0: 202d 2031 3474 2074 2020 2d20 3134 7420   - 14t t  - 14t 
│ │ │ │ -000217d0: 7420 202d 2035 7420 7420 202d 2031 3374  t  - 5t t  - 13t
│ │ │ │ -000217e0: 2074 2020 2b20 3337 7420 7420 202b 207c   t  + 37t t  + |
│ │ │ │ -000217f0: 0a7c 2020 2032 2035 2020 2020 2020 3320  .|   2 5      3 
│ │ │ │ -00021800: 3520 2020 2020 2034 2035 2020 2020 2035  5      4 5     5
│ │ │ │ -00021810: 2020 2020 2020 3020 3620 2020 2020 2031        0 6      1
│ │ │ │ -00021820: 2036 2020 2020 2032 2036 2020 2020 2020   6     2 6      
│ │ │ │ -00021830: 3420 3620 2020 2020 2035 2036 2020 207c  4 6      5 6   |
│ │ │ │ -00021840: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000217a0: 2020 2020 207c 0a7c 3335 7420 7420 202b       |.|35t t  +
│ │ │ │ +000217b0: 2031 3074 2074 2020 2b20 3235 7420 7420   10t t  + 25t t 
│ │ │ │ +000217c0: 202d 2035 7420 202d 2031 3474 2074 2020   - 5t  - 14t t  
│ │ │ │ +000217d0: 2d20 3134 7420 7420 202d 2035 7420 7420  - 14t t  - 5t t 
│ │ │ │ +000217e0: 202d 2031 3374 2074 2020 2b20 3337 7420   - 13t t  + 37t 
│ │ │ │ +000217f0: 7420 202b 207c 0a7c 2020 2032 2035 2020  t  + |.|   2 5  
│ │ │ │ +00021800: 2020 2020 3320 3520 2020 2020 2034 2035      3 5      4 5
│ │ │ │ +00021810: 2020 2020 2035 2020 2020 2020 3020 3620       5      0 6 
│ │ │ │ +00021820: 2020 2020 2031 2036 2020 2020 2032 2036       1 6     2 6
│ │ │ │ +00021830: 2020 2020 2020 3420 3620 2020 2020 2035        4 6      5
│ │ │ │ +00021840: 2036 2020 207c 0a7c 2020 2020 2020 2020   6   |.|        
│ │ │ │  00021850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021880: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021890: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000218a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000218b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  000218c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000218d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000218e0: 0a7c 7420 202d 2034 3074 2074 2020 2b20  .|t  - 40t t  + 
│ │ │ │ -000218f0: 3430 7420 7420 202b 2032 3674 2074 2020  40t t  + 26t t  
│ │ │ │ -00021900: 2d20 3230 7420 202b 2034 3174 2074 2020  - 20t  + 41t t  
│ │ │ │ -00021910: 2b20 3336 7420 7420 202d 2032 3274 2074  + 36t t  - 22t t
│ │ │ │ -00021920: 2020 2b20 3336 7420 7420 202d 2033 307c    + 36t t  - 30|
│ │ │ │ -00021930: 0a7c 2035 2020 2020 2020 3220 3520 2020  .| 5      2 5   
│ │ │ │ -00021940: 2020 2033 2035 2020 2020 2020 3420 3520     3 5      4 5 
│ │ │ │ -00021950: 2020 2020 2035 2020 2020 2020 3020 3620       5      0 6 
│ │ │ │ -00021960: 2020 2020 2031 2036 2020 2020 2020 3220       1 6      2 
│ │ │ │ -00021970: 3620 2020 2020 2034 2036 2020 2020 207c  6      4 6     |
│ │ │ │ -00021980: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000218d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000218e0: 2020 2020 207c 0a7c 7420 202d 2034 3074       |.|t  - 40t
│ │ │ │ +000218f0: 2074 2020 2b20 3430 7420 7420 202b 2032   t  + 40t t  + 2
│ │ │ │ +00021900: 3674 2074 2020 2d20 3230 7420 202b 2034  6t t  - 20t  + 4
│ │ │ │ +00021910: 3174 2074 2020 2b20 3336 7420 7420 202d  1t t  + 36t t  -
│ │ │ │ +00021920: 2032 3274 2074 2020 2b20 3336 7420 7420   22t t  + 36t t 
│ │ │ │ +00021930: 202d 2033 307c 0a7c 2035 2020 2020 2020   - 30|.| 5      
│ │ │ │ +00021940: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ +00021950: 2020 3420 3520 2020 2020 2035 2020 2020    4 5      5    
│ │ │ │ +00021960: 2020 3020 3620 2020 2020 2031 2036 2020    0 6      1 6  
│ │ │ │ +00021970: 2020 2020 3220 3620 2020 2020 2034 2036      2 6      4 6
│ │ │ │ +00021980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000219b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000219d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000219c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000219d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000219e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000219f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +000219f0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  00021a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021a20: 0a7c 7420 202b 2031 3274 2074 2020 2b20  .|t  + 12t t  + 
│ │ │ │ -00021a30: 3437 7420 7420 202b 2033 3774 2074 2020  47t t  + 37t t  
│ │ │ │ -00021a40: 2b20 3235 7420 202d 2032 3774 2074 2020  + 25t  - 27t t  
│ │ │ │ -00021a50: 2d20 3232 7420 7420 202b 2032 3774 2074  - 22t t  + 27t t
│ │ │ │ -00021a60: 2020 2d20 3233 7420 7420 202b 2035 747c    - 23t t  + 5t|
│ │ │ │ -00021a70: 0a7c 2035 2020 2020 2020 3220 3520 2020  .| 5      2 5   
│ │ │ │ -00021a80: 2020 2033 2035 2020 2020 2020 3420 3520     3 5      4 5 
│ │ │ │ -00021a90: 2020 2020 2035 2020 2020 2020 3020 3620       5      0 6 
│ │ │ │ -00021aa0: 2020 2020 2031 2036 2020 2020 2020 3220       1 6      2 
│ │ │ │ -00021ab0: 3620 2020 2020 2034 2036 2020 2020 207c  6      4 6     |
│ │ │ │ -00021ac0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021a20: 2020 2020 207c 0a7c 7420 202b 2031 3274       |.|t  + 12t
│ │ │ │ +00021a30: 2074 2020 2b20 3437 7420 7420 202b 2033   t  + 47t t  + 3
│ │ │ │ +00021a40: 3774 2074 2020 2b20 3235 7420 202d 2032  7t t  + 25t  - 2
│ │ │ │ +00021a50: 3774 2074 2020 2d20 3232 7420 7420 202b  7t t  - 22t t  +
│ │ │ │ +00021a60: 2032 3774 2074 2020 2d20 3233 7420 7420   27t t  - 23t t 
│ │ │ │ +00021a70: 202b 2035 747c 0a7c 2035 2020 2020 2020   + 5t|.| 5      
│ │ │ │ +00021a80: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ +00021a90: 2020 3420 3520 2020 2020 2035 2020 2020    4 5      5    
│ │ │ │ +00021aa0: 2020 3020 3620 2020 2020 2031 2036 2020    0 6      1 6  
│ │ │ │ +00021ab0: 2020 2020 3220 3620 2020 2020 2034 2036      2 6      4 6
│ │ │ │ +00021ac0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021b10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021b10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b30: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00021b30: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00021b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021b50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021b60: 0a7c 202d 2033 3574 2074 2020 2d20 3130  .| - 35t t  - 10
│ │ │ │ -00021b70: 7420 7420 202d 2033 3374 2074 2020 2b20  t t  - 33t t  + 
│ │ │ │ -00021b80: 3574 2020 2b20 3135 7420 7420 202b 2031  5t  + 15t t  + 1
│ │ │ │ -00021b90: 3574 2074 2020 2b20 3574 2074 2020 2b20  5t t  + 5t t  + 
│ │ │ │ -00021ba0: 3135 7420 7420 202d 2033 3874 2074 207c  15t t  - 38t t |
│ │ │ │ -00021bb0: 0a7c 2020 2020 2020 3220 3520 2020 2020  .|      2 5     
│ │ │ │ -00021bc0: 2033 2035 2020 2020 2020 3420 3520 2020   3 5      4 5   
│ │ │ │ -00021bd0: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ -00021be0: 2020 3120 3620 2020 2020 3220 3620 2020    1 6     2 6   
│ │ │ │ -00021bf0: 2020 2034 2036 2020 2020 2020 3520 367c     4 6      5 6|
│ │ │ │ -00021c00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021b60: 2020 2020 207c 0a7c 202d 2033 3574 2074       |.| - 35t t
│ │ │ │ +00021b70: 2020 2d20 3130 7420 7420 202d 2033 3374    - 10t t  - 33t
│ │ │ │ +00021b80: 2074 2020 2b20 3574 2020 2b20 3135 7420   t  + 5t  + 15t 
│ │ │ │ +00021b90: 7420 202b 2031 3574 2074 2020 2b20 3574  t  + 15t t  + 5t
│ │ │ │ +00021ba0: 2074 2020 2b20 3135 7420 7420 202d 2033   t  + 15t t  - 3
│ │ │ │ +00021bb0: 3874 2074 207c 0a7c 2020 2020 2020 3220  8t t |.|      2 
│ │ │ │ +00021bc0: 3520 2020 2020 2033 2035 2020 2020 2020  5      3 5      
│ │ │ │ +00021bd0: 3420 3520 2020 2020 3520 2020 2020 2030  4 5     5      0
│ │ │ │ +00021be0: 2036 2020 2020 2020 3120 3620 2020 2020   6      1 6     
│ │ │ │ +00021bf0: 3220 3620 2020 2020 2034 2036 2020 2020  2 6      4 6    
│ │ │ │ +00021c00: 2020 3520 367c 0a7c 2020 2020 2020 2020    5 6|.|        
│ │ │ │  00021c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021c50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021c50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c70: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00021c70: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00021c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021c90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021ca0: 0a7c 2d20 3335 7420 7420 202d 2031 3074  .|- 35t t  - 10t
│ │ │ │ -00021cb0: 2074 2020 2d20 3333 7420 7420 202b 2035   t  - 33t t  + 5
│ │ │ │ -00021cc0: 7420 202b 2031 3474 2074 2020 2b20 3134  t  + 14t t  + 14
│ │ │ │ -00021cd0: 7420 7420 202b 2035 7420 7420 202b 2031  t t  + 5t t  + 1
│ │ │ │ -00021ce0: 3474 2074 2020 2d20 3331 7420 7420 207c  4t t  - 31t t  |
│ │ │ │ -00021cf0: 0a7c 2020 2020 2032 2035 2020 2020 2020  .|     2 5      
│ │ │ │ -00021d00: 3320 3520 2020 2020 2034 2035 2020 2020  3 5      4 5    
│ │ │ │ -00021d10: 2035 2020 2020 2020 3020 3620 2020 2020   5      0 6     
│ │ │ │ -00021d20: 2031 2036 2020 2020 2032 2036 2020 2020   1 6     2 6    
│ │ │ │ -00021d30: 2020 3420 3620 2020 2020 2035 2036 207c    4 6      5 6 |
│ │ │ │ -00021d40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021ca0: 2020 2020 207c 0a7c 2d20 3335 7420 7420       |.|- 35t t 
│ │ │ │ +00021cb0: 202d 2031 3074 2074 2020 2d20 3333 7420   - 10t t  - 33t 
│ │ │ │ +00021cc0: 7420 202b 2035 7420 202b 2031 3474 2074  t  + 5t  + 14t t
│ │ │ │ +00021cd0: 2020 2b20 3134 7420 7420 202b 2035 7420    + 14t t  + 5t 
│ │ │ │ +00021ce0: 7420 202b 2031 3474 2074 2020 2d20 3331  t  + 14t t  - 31
│ │ │ │ +00021cf0: 7420 7420 207c 0a7c 2020 2020 2032 2035  t t  |.|     2 5
│ │ │ │ +00021d00: 2020 2020 2020 3320 3520 2020 2020 2034        3 5      4
│ │ │ │ +00021d10: 2035 2020 2020 2035 2020 2020 2020 3020   5     5      0 
│ │ │ │ +00021d20: 3620 2020 2020 2031 2036 2020 2020 2032  6      1 6     2
│ │ │ │ +00021d30: 2036 2020 2020 2020 3420 3620 2020 2020   6      4 6     
│ │ │ │ +00021d40: 2035 2036 207c 0a7c 2020 2020 2020 2020   5 6 |.|        
│ │ │ │  00021d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021d90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00021da0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00021d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021d90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00021da0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00021db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021de0: 0a7c 202b 2074 2074 2020 2d20 3774 2074  .| + t t  - 7t t
│ │ │ │ -00021df0: 2020 2b20 3274 2020 2d20 7420 7420 2c20    + 2t  - t t , 
│ │ │ │ -00021e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021de0: 2020 2020 207c 0a7c 202b 2074 2074 2020       |.| + t t  
│ │ │ │ +00021df0: 2d20 3774 2074 2020 2b20 3274 2020 2d20  - 7t t  + 2t  - 
│ │ │ │ +00021e00: 7420 7420 2c20 2020 2020 2020 2020 2020  t t ,           
│ │ │ │  00021e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021e30: 0a7c 2020 2020 3420 3620 2020 2020 3520  .|    4 6     5 
│ │ │ │ -00021e40: 3620 2020 2020 3620 2020 2033 2037 2020  6     6    3 7  
│ │ │ │ -00021e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021e30: 2020 2020 207c 0a7c 2020 2020 3420 3620       |.|    4 6 
│ │ │ │ +00021e40: 2020 2020 3520 3620 2020 2020 3620 2020      5 6     6   
│ │ │ │ +00021e50: 2033 2037 2020 2020 2020 2020 2020 2020   3 7            
│ │ │ │  00021e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021e80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021e80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021ed0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021ed0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00021ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021ef0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00021ef0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00021f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00021f10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00021f20: 0a7c 2074 2020 2d20 3433 7420 7420 202d  .| t  - 43t t  -
│ │ │ │ -00021f30: 2034 3174 2074 2020 2d20 3236 7420 7420   41t t  - 26t t 
│ │ │ │ -00021f40: 202d 2032 3874 2020 2d20 3335 7420 7420   - 28t  - 35t t 
│ │ │ │ -00021f50: 202d 2033 3674 2074 2020 2b20 3230 7420   - 36t t  + 20t 
│ │ │ │ -00021f60: 7420 202d 2033 3674 2074 2020 2b20 397c  t  - 36t t  + 9|
│ │ │ │ -00021f70: 0a7c 3120 3520 2020 2020 2032 2035 2020  .|1 5      2 5  
│ │ │ │ -00021f80: 2020 2020 3320 3520 2020 2020 2034 2035      3 5      4 5
│ │ │ │ -00021f90: 2020 2020 2020 3520 2020 2020 2030 2036        5      0 6
│ │ │ │ -00021fa0: 2020 2020 2020 3120 3620 2020 2020 2032        1 6      2
│ │ │ │ -00021fb0: 2036 2020 2020 2020 3420 3620 2020 207c   6      4 6    |
│ │ │ │ -00021fc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00021f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00021f20: 2020 2020 207c 0a7c 2074 2020 2d20 3433       |.| t  - 43
│ │ │ │ +00021f30: 7420 7420 202d 2034 3174 2074 2020 2d20  t t  - 41t t  - 
│ │ │ │ +00021f40: 3236 7420 7420 202d 2032 3874 2020 2d20  26t t  - 28t  - 
│ │ │ │ +00021f50: 3335 7420 7420 202d 2033 3674 2074 2020  35t t  - 36t t  
│ │ │ │ +00021f60: 2b20 3230 7420 7420 202d 2033 3674 2074  + 20t t  - 36t t
│ │ │ │ +00021f70: 2020 2b20 397c 0a7c 3120 3520 2020 2020    + 9|.|1 5     
│ │ │ │ +00021f80: 2032 2035 2020 2020 2020 3320 3520 2020   2 5      3 5   
│ │ │ │ +00021f90: 2020 2034 2035 2020 2020 2020 3520 2020     4 5      5   
│ │ │ │ +00021fa0: 2020 2030 2036 2020 2020 2020 3120 3620     0 6      1 6 
│ │ │ │ +00021fb0: 2020 2020 2032 2036 2020 2020 2020 3420       2 6      4 
│ │ │ │ +00021fc0: 3620 2020 207c 0a7c 2020 2020 2020 2020  6    |.|        
│ │ │ │  00021fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00021ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022010: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022030: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00022040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022060: 0a7c 2020 2d20 3435 7420 7420 202d 2033  .|  - 45t t  - 3
│ │ │ │ -00022070: 3974 2074 2020 2d20 3339 7420 7420 202d  9t t  - 39t t  -
│ │ │ │ -00022080: 2034 3674 2074 2020 2d20 3239 7420 202d   46t t  - 29t  -
│ │ │ │ -00022090: 2034 3874 2074 2020 2d20 3338 7420 7420   48t t  - 38t t 
│ │ │ │ -000220a0: 202d 2033 3074 2074 2020 2b20 3139 747c   - 30t t  + 19t|
│ │ │ │ -000220b0: 0a7c 3520 2020 2020 2031 2035 2020 2020  .|5      1 5    
│ │ │ │ -000220c0: 2020 3220 3520 2020 2020 2033 2035 2020    2 5      3 5  
│ │ │ │ -000220d0: 2020 2020 3420 3520 2020 2020 2035 2020      4 5      5  
│ │ │ │ -000220e0: 2020 2020 3020 3620 2020 2020 2031 2036      0 6      1 6
│ │ │ │ -000220f0: 2020 2020 2020 3220 3620 2020 2020 207c        2 6      |
│ │ │ │ -00022100: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022040: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00022050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022060: 2020 2020 207c 0a7c 2020 2d20 3435 7420       |.|  - 45t 
│ │ │ │ +00022070: 7420 202d 2033 3974 2074 2020 2d20 3339  t  - 39t t  - 39
│ │ │ │ +00022080: 7420 7420 202d 2034 3674 2074 2020 2d20  t t  - 46t t  - 
│ │ │ │ +00022090: 3239 7420 202d 2034 3874 2074 2020 2d20  29t  - 48t t  - 
│ │ │ │ +000220a0: 3338 7420 7420 202d 2033 3074 2074 2020  38t t  - 30t t  
│ │ │ │ +000220b0: 2b20 3139 747c 0a7c 3520 2020 2020 2031  + 19t|.|5      1
│ │ │ │ +000220c0: 2035 2020 2020 2020 3220 3520 2020 2020   5      2 5     
│ │ │ │ +000220d0: 2033 2035 2020 2020 2020 3420 3520 2020   3 5      4 5   
│ │ │ │ +000220e0: 2020 2035 2020 2020 2020 3020 3620 2020     5      0 6   
│ │ │ │ +000220f0: 2020 2031 2036 2020 2020 2020 3220 3620     1 6      2 6 
│ │ │ │ +00022100: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022150: 0a7c 2020 2d20 3274 2074 2020 2d20 7420  .|  - 2t t  - t 
│ │ │ │ -00022160: 7420 202d 2074 2074 2020 2b20 3574 2074  t  - t t  + 5t t
│ │ │ │ -00022170: 2020 2b20 7420 7420 202d 2032 7420 7420    + t t  - 2t t 
│ │ │ │ -00022180: 202d 2037 7420 7420 202b 2032 7420 7420   - 7t t  + 2t t 
│ │ │ │ -00022190: 202d 2032 7420 7420 202b 2033 7420 747c   - 2t t  + 3t t|
│ │ │ │ -000221a0: 0a7c 3620 2020 2020 3120 3620 2020 2034  .|6     1 6    4
│ │ │ │ -000221b0: 2036 2020 2020 3520 3620 2020 2020 3020   6    5 6     0 
│ │ │ │ -000221c0: 3720 2020 2031 2037 2020 2020 2032 2037  7    1 7     2 7
│ │ │ │ -000221d0: 2020 2020 2035 2037 2020 2020 2036 2037       5 7     6 7
│ │ │ │ -000221e0: 2020 2020 2031 2038 2020 2020 2037 207c       1 8     7 |
│ │ │ │ -000221f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022150: 2020 2020 207c 0a7c 2020 2d20 3274 2074       |.|  - 2t t
│ │ │ │ +00022160: 2020 2d20 7420 7420 202d 2074 2074 2020    - t t  - t t  
│ │ │ │ +00022170: 2b20 3574 2074 2020 2b20 7420 7420 202d  + 5t t  + t t  -
│ │ │ │ +00022180: 2032 7420 7420 202d 2037 7420 7420 202b   2t t  - 7t t  +
│ │ │ │ +00022190: 2032 7420 7420 202d 2032 7420 7420 202b   2t t  - 2t t  +
│ │ │ │ +000221a0: 2033 7420 747c 0a7c 3620 2020 2020 3120   3t t|.|6     1 
│ │ │ │ +000221b0: 3620 2020 2034 2036 2020 2020 3520 3620  6    4 6    5 6 
│ │ │ │ +000221c0: 2020 2020 3020 3720 2020 2031 2037 2020      0 7    1 7  
│ │ │ │ +000221d0: 2020 2032 2037 2020 2020 2035 2037 2020     2 7     5 7  
│ │ │ │ +000221e0: 2020 2036 2037 2020 2020 2031 2038 2020     6 7     1 8  
│ │ │ │ +000221f0: 2020 2037 207c 0a7c 2020 2020 2020 2020     7 |.|        
│ │ │ │  00022200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022240: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022260: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -00022270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022290: 0a7c 202b 2033 3274 2074 2020 2d20 3230  .| + 32t t  - 20
│ │ │ │ -000222a0: 7420 7420 202d 2034 3774 2074 2020 2d20  t t  - 47t t  - 
│ │ │ │ -000222b0: 3337 7420 7420 202d 2032 3574 2020 2b20  37t t  - 25t  + 
│ │ │ │ -000222c0: 3139 7420 7420 202b 2032 3274 2074 2020  19t t  + 22t t  
│ │ │ │ -000222d0: 2d20 3235 7420 7420 202b 2032 3574 207c  - 25t t  + 25t |
│ │ │ │ -000222e0: 0a7c 2020 2020 2020 3120 3520 2020 2020  .|      1 5     
│ │ │ │ -000222f0: 2032 2035 2020 2020 2020 3320 3520 2020   2 5      3 5   
│ │ │ │ -00022300: 2020 2034 2035 2020 2020 2020 3520 2020     4 5      5   
│ │ │ │ -00022310: 2020 2030 2036 2020 2020 2020 3120 3620     0 6      1 6 
│ │ │ │ -00022320: 2020 2020 2032 2036 2020 2020 2020 347c       2 6      4|
│ │ │ │ -00022330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022270: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00022280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022290: 2020 2020 207c 0a7c 202b 2033 3274 2074       |.| + 32t t
│ │ │ │ +000222a0: 2020 2d20 3230 7420 7420 202d 2034 3774    - 20t t  - 47t
│ │ │ │ +000222b0: 2074 2020 2d20 3337 7420 7420 202d 2032   t  - 37t t  - 2
│ │ │ │ +000222c0: 3574 2020 2b20 3139 7420 7420 202b 2032  5t  + 19t t  + 2
│ │ │ │ +000222d0: 3274 2074 2020 2d20 3235 7420 7420 202b  2t t  - 25t t  +
│ │ │ │ +000222e0: 2032 3574 207c 0a7c 2020 2020 2020 3120   25t |.|      1 
│ │ │ │ +000222f0: 3520 2020 2020 2032 2035 2020 2020 2020  5      2 5      
│ │ │ │ +00022300: 3320 3520 2020 2020 2034 2035 2020 2020  3 5      4 5    
│ │ │ │ +00022310: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ +00022320: 2020 3120 3620 2020 2020 2032 2036 2020    1 6      2 6  
│ │ │ │ +00022330: 2020 2020 347c 0a7c 2020 2020 2020 2020      4|.|        
│ │ │ │  00022340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022380: 0a7c 2074 2020 2d20 3337 7420 7420 202d  .| t  - 37t t  -
│ │ │ │ -00022390: 2033 3974 2074 2020 2b20 3139 7420 7420   39t t  + 19t t 
│ │ │ │ -000223a0: 202d 2033 3974 2074 2020 2d20 3338 7420   - 39t t  - 38t 
│ │ │ │ -000223b0: 7420 202b 2033 3974 2074 2020 2b20 3139  t  + 39t t  + 19
│ │ │ │ -000223c0: 7420 7420 202b 2031 3874 2074 2020 2d7c  t t  + 18t t  -|
│ │ │ │ -000223d0: 0a7c 3120 3520 2020 2020 2030 2036 2020  .|1 5      0 6  
│ │ │ │ -000223e0: 2020 2020 3120 3620 2020 2020 2034 2036      1 6      4 6
│ │ │ │ -000223f0: 2020 2020 2020 3520 3620 2020 2020 2030        5 6      0
│ │ │ │ -00022400: 2037 2020 2020 2020 3120 3720 2020 2020   7      1 7     
│ │ │ │ -00022410: 2032 2037 2020 2020 2020 3520 3720 207c   2 7      5 7  |
│ │ │ │ -00022420: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022380: 2020 2020 207c 0a7c 2074 2020 2d20 3337       |.| t  - 37
│ │ │ │ +00022390: 7420 7420 202d 2033 3974 2074 2020 2b20  t t  - 39t t  + 
│ │ │ │ +000223a0: 3139 7420 7420 202d 2033 3974 2074 2020  19t t  - 39t t  
│ │ │ │ +000223b0: 2d20 3338 7420 7420 202b 2033 3974 2074  - 38t t  + 39t t
│ │ │ │ +000223c0: 2020 2b20 3139 7420 7420 202b 2031 3874    + 19t t  + 18t
│ │ │ │ +000223d0: 2074 2020 2d7c 0a7c 3120 3520 2020 2020   t  -|.|1 5     
│ │ │ │ +000223e0: 2030 2036 2020 2020 2020 3120 3620 2020   0 6      1 6   
│ │ │ │ +000223f0: 2020 2034 2036 2020 2020 2020 3520 3620     4 6      5 6 
│ │ │ │ +00022400: 2020 2020 2030 2037 2020 2020 2020 3120       0 7      1 
│ │ │ │ +00022410: 3720 2020 2020 2032 2037 2020 2020 2020  7      2 7      
│ │ │ │ +00022420: 3520 3720 207c 0a7c 2020 2020 2020 2020  5 7  |.|        
│ │ │ │  00022430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022490: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000224a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000224b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000224c0: 0a7c 2020 2b20 3237 7420 7420 202d 2038  .|  + 27t t  - 8
│ │ │ │ -000224d0: 7420 7420 202b 2033 3774 2074 2020 2b20  t t  + 37t t  + 
│ │ │ │ -000224e0: 3238 7420 7420 202b 2033 3074 2020 2d20  28t t  + 30t  - 
│ │ │ │ -000224f0: 3436 7420 7420 202b 2032 3474 2074 2020  46t t  + 24t t  
│ │ │ │ -00022500: 2d20 3430 7420 7420 202b 2032 3574 207c  - 40t t  + 25t |
│ │ │ │ -00022510: 0a7c 3520 2020 2020 2031 2035 2020 2020  .|5      1 5    
│ │ │ │ -00022520: 2032 2035 2020 2020 2020 3320 3520 2020   2 5      3 5   
│ │ │ │ -00022530: 2020 2034 2035 2020 2020 2020 3520 2020     4 5      5   
│ │ │ │ -00022540: 2020 2030 2036 2020 2020 2020 3120 3620     0 6      1 6 
│ │ │ │ -00022550: 2020 2020 2032 2036 2020 2020 2020 347c       2 6      4|
│ │ │ │ -00022560: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224a0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000224b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000224c0: 2020 2020 207c 0a7c 2020 2b20 3237 7420       |.|  + 27t 
│ │ │ │ +000224d0: 7420 202d 2038 7420 7420 202b 2033 3774  t  - 8t t  + 37t
│ │ │ │ +000224e0: 2074 2020 2b20 3238 7420 7420 202b 2033   t  + 28t t  + 3
│ │ │ │ +000224f0: 3074 2020 2d20 3436 7420 7420 202b 2032  0t  - 46t t  + 2
│ │ │ │ +00022500: 3474 2074 2020 2d20 3430 7420 7420 202b  4t t  - 40t t  +
│ │ │ │ +00022510: 2032 3574 207c 0a7c 3520 2020 2020 2031   25t |.|5      1
│ │ │ │ +00022520: 2035 2020 2020 2032 2035 2020 2020 2020   5     2 5      
│ │ │ │ +00022530: 3320 3520 2020 2020 2034 2035 2020 2020  3 5      4 5    
│ │ │ │ +00022540: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ +00022550: 2020 3120 3620 2020 2020 2032 2036 2020    1 6      2 6  
│ │ │ │ +00022560: 2020 2020 347c 0a7c 2020 2020 2020 2020      4|.|        
│ │ │ │  00022570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000225b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000225a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000225b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000225c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000225e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000225f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022600: 0a7c 6672 6f6d 2050 505e 3820 746f 2038  .|from PP^8 to 8
│ │ │ │ -00022610: 2d64 696d 656e 7369 6f6e 616c 2073 7562  -dimensional sub
│ │ │ │ -00022620: 7661 7269 6574 7920 6f66 2050 505e 3131  variety of PP^11
│ │ │ │ -00022630: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ -00022640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022650: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022600: 2020 2020 207c 0a7c 6672 6f6d 2050 505e       |.|from PP^
│ │ │ │ +00022610: 3820 746f 2038 2d64 696d 656e 7369 6f6e  8 to 8-dimension
│ │ │ │ +00022620: 616c 2073 7562 7661 7269 6574 7920 6f66  al subvariety of
│ │ │ │ +00022630: 2050 505e 3131 2920 2020 2020 2020 2020   PP^11)         
│ │ │ │ +00022640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022650: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00022660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00022680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000226a0: 0a7c 5f35 2b34 312a 785f 332a 785f 352d  .|_5+41*x_3*x_5-
│ │ │ │ -000226b0: 785f 302a 785f 362b 3132 2a78 5f32 2a78  x_0*x_6+12*x_2*x
│ │ │ │ -000226c0: 5f36 2b32 352a 785f 312a 785f 3720 2020  _6+25*x_1*x_7   
│ │ │ │ -000226d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000226e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000226f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000226a0: 2d2d 2d2d 2d7c 0a7c 5f35 2b34 312a 785f  -----|.|_5+41*x_
│ │ │ │ +000226b0: 332a 785f 352d 785f 302a 785f 362b 3132  3*x_5-x_0*x_6+12
│ │ │ │ +000226c0: 2a78 5f32 2a78 5f36 2b32 352a 785f 312a  *x_2*x_6+25*x_1*
│ │ │ │ +000226d0: 785f 3720 2020 2020 2020 2020 2020 2020  x_7             
│ │ │ │ +000226e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000226f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022740: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022790: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022790: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000227a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000227b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000227c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000227d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000227e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000227d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000227e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000227f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022830: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022830: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022870: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022880: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022880: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000228c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000228d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000228c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000228d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000228e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000228f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022920: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022920: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022960: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022970: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022970: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000229b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000229c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000229b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000229c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000229d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000229f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022a10: 0a7c 2020 2d20 3378 2078 2020 2b20 3278  .|  - 3x x  + 2x
│ │ │ │ -00022a20: 2078 2020 2b20 3131 7820 7820 202d 2033   x  + 11x x  - 3
│ │ │ │ -00022a30: 3778 2078 2020 2d20 3233 7820 7820 2020  7x x  - 23x x   
│ │ │ │ -00022a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022a60: 0a7c 3720 2020 2020 3620 3720 2020 2020  .|7     6 7     
│ │ │ │ -00022a70: 3020 3820 2020 2020 2031 2038 2020 2020  0 8      1 8    
│ │ │ │ -00022a80: 2020 3320 3820 2020 2020 2034 2020 2020    3 8      4    
│ │ │ │ -00022a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022ab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a10: 2020 2020 207c 0a7c 2020 2d20 3378 2078       |.|  - 3x x
│ │ │ │ +00022a20: 2020 2b20 3278 2078 2020 2b20 3131 7820    + 2x x  + 11x 
│ │ │ │ +00022a30: 7820 202d 2033 3778 2078 2020 2d20 3233  x  - 37x x  - 23
│ │ │ │ +00022a40: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +00022a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022a60: 2020 2020 207c 0a7c 3720 2020 2020 3620       |.|7     6 
│ │ │ │ +00022a70: 3720 2020 2020 3020 3820 2020 2020 2031  7     0 8      1
│ │ │ │ +00022a80: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ +00022a90: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ab0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022af0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022b00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022b00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022b50: 0a7c 2020 2b20 3978 2078 2020 2b20 3436  .|  + 9x x  + 46
│ │ │ │ -00022b60: 7820 7820 202b 2034 3178 2078 2020 2d20  x x  + 41x x  - 
│ │ │ │ -00022b70: 3778 2078 2020 2d20 3334 7820 7820 2020  7x x  - 34x x   
│ │ │ │ -00022b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022ba0: 0a7c 3720 2020 2020 3520 3720 2020 2020  .|7     5 7     
│ │ │ │ -00022bb0: 2036 2037 2020 2020 2020 3020 3820 2020   6 7      0 8   
│ │ │ │ -00022bc0: 2020 3120 3820 2020 2020 2033 2020 2020    1 8      3    
│ │ │ │ -00022bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022bf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022b50: 2020 2020 207c 0a7c 2020 2b20 3978 2078       |.|  + 9x x
│ │ │ │ +00022b60: 2020 2b20 3436 7820 7820 202b 2034 3178    + 46x x  + 41x
│ │ │ │ +00022b70: 2078 2020 2d20 3778 2078 2020 2d20 3334   x  - 7x x  - 34
│ │ │ │ +00022b80: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +00022b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ba0: 2020 2020 207c 0a7c 3720 2020 2020 3520       |.|7     5 
│ │ │ │ +00022bb0: 3720 2020 2020 2036 2037 2020 2020 2020  7      6 7      
│ │ │ │ +00022bc0: 3020 3820 2020 2020 3120 3820 2020 2020  0 8     1 8     
│ │ │ │ +00022bd0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00022be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022bf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022c40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022c90: 0a7c 2032 3078 2078 2020 2b20 3130 7820  .| 20x x  + 10x 
│ │ │ │ -00022ca0: 7820 202b 2033 3478 2078 2020 2b20 3431  x  + 34x x  + 41
│ │ │ │ -00022cb0: 7820 7820 202d 2078 2078 2020 2b20 2020  x x  - x x  +   
│ │ │ │ -00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022ce0: 0a7c 2020 2020 3620 3720 2020 2020 2030  .|    6 7      0
│ │ │ │ -00022cf0: 2038 2020 2020 2020 3120 3820 2020 2020   8      1 8     
│ │ │ │ -00022d00: 2033 2038 2020 2020 3420 3820 2020 2020   3 8    4 8     
│ │ │ │ -00022d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022d30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022c90: 2020 2020 207c 0a7c 2032 3078 2078 2020       |.| 20x x  
│ │ │ │ +00022ca0: 2b20 3130 7820 7820 202b 2033 3478 2078  + 10x x  + 34x x
│ │ │ │ +00022cb0: 2020 2b20 3431 7820 7820 202d 2078 2078    + 41x x  - x x
│ │ │ │ +00022cc0: 2020 2b20 2020 2020 2020 2020 2020 2020    +             
│ │ │ │ +00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ce0: 2020 2020 207c 0a7c 2020 2020 3620 3720       |.|    6 7 
│ │ │ │ +00022cf0: 2020 2020 2030 2038 2020 2020 2020 3120       0 8      1 
│ │ │ │ +00022d00: 3820 2020 2020 2033 2038 2020 2020 3420  8      3 8    4 
│ │ │ │ +00022d10: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │ +00022d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022d80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022d80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022dd0: 0a7c 2020 2032 2020 2020 2020 2020 2020  .|   2          
│ │ │ │ +00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022dd0: 2020 2020 207c 0a7c 2020 2032 2020 2020       |.|   2    
│ │ │ │  00022de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022e20: 0a7c 3232 7420 202d 2033 3174 2074 2020  .|22t  - 31t t  
│ │ │ │ -00022e30: 2b20 3236 7420 7420 202b 2031 3274 2074  + 26t t  + 12t t
│ │ │ │ -00022e40: 2020 2d20 3435 7420 7420 202d 2020 2020    - 45t t  -    
│ │ │ │ -00022e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022e70: 0a7c 2020 2036 2020 2020 2020 3320 3720  .|   6      3 7 
│ │ │ │ -00022e80: 2020 2020 2034 2037 2020 2020 2020 3520       4 7      5 
│ │ │ │ -00022e90: 3720 2020 2020 2036 2037 2020 2020 2020  7      6 7      
│ │ │ │ +00022e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e20: 2020 2020 207c 0a7c 3232 7420 202d 2033       |.|22t  - 3
│ │ │ │ +00022e30: 3174 2074 2020 2b20 3236 7420 7420 202b  1t t  + 26t t  +
│ │ │ │ +00022e40: 2031 3274 2074 2020 2d20 3435 7420 7420   12t t  - 45t t 
│ │ │ │ +00022e50: 202d 2020 2020 2020 2020 2020 2020 2020   -              
│ │ │ │ +00022e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022e70: 2020 2020 207c 0a7c 2020 2036 2020 2020       |.|   6    
│ │ │ │ +00022e80: 2020 3320 3720 2020 2020 2034 2037 2020    3 7      4 7  
│ │ │ │ +00022e90: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │  00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022eb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00022ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022f10: 0a7c 2020 2020 2020 2020 2020 3220 2020  .|          2   
│ │ │ │ -00022f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00022f20: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00022f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00022f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022f60: 0a7c 7420 7420 202d 2031 3374 2020 2d20  .|t t  - 13t  - 
│ │ │ │ -00022f70: 3235 7420 7420 202b 2035 7420 7420 202d  25t t  + 5t t  -
│ │ │ │ -00022f80: 2033 3574 2074 2020 2b20 3130 7420 2020   35t t  + 10t   
│ │ │ │ -00022f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00022fb0: 0a7c 2035 2036 2020 2020 2020 3620 2020  .| 5 6      6   
│ │ │ │ -00022fc0: 2020 2033 2037 2020 2020 2034 2037 2020     3 7     4 7  
│ │ │ │ -00022fd0: 2020 2020 3520 3720 2020 2020 2020 2020      5 7         
│ │ │ │ +00022f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022f60: 2020 2020 207c 0a7c 7420 7420 202d 2031       |.|t t  - 1
│ │ │ │ +00022f70: 3374 2020 2d20 3235 7420 7420 202b 2035  3t  - 25t t  + 5
│ │ │ │ +00022f80: 7420 7420 202d 2033 3574 2074 2020 2b20  t t  - 35t t  + 
│ │ │ │ +00022f90: 3130 7420 2020 2020 2020 2020 2020 2020  10t             
│ │ │ │ +00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00022fb0: 2020 2020 207c 0a7c 2035 2036 2020 2020       |.| 5 6    
│ │ │ │ +00022fc0: 2020 3620 2020 2020 2033 2037 2020 2020    6      3 7    
│ │ │ │ +00022fd0: 2034 2037 2020 2020 2020 3520 3720 2020   4 7      5 7   
│ │ │ │  00022fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00022ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023000: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00022ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023050: 0a7c 2020 2020 2020 2020 2032 2020 2020  .|         2    
│ │ │ │ -00023060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023060: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  00023070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000230a0: 0a7c 2074 2020 2d20 3133 7420 202d 2033  .| t  - 13t  - 3
│ │ │ │ -000230b0: 3974 2074 2020 2d20 3239 7420 7420 202b  9t t  - 29t t  +
│ │ │ │ -000230c0: 2039 7420 7420 202b 2033 3974 2020 2020   9t t  + 39t    
│ │ │ │ -000230d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000230e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000230f0: 0a7c 3520 3620 2020 2020 2036 2020 2020  .|5 6      6    
│ │ │ │ -00023100: 2020 3320 3720 2020 2020 2034 2037 2020    3 7      4 7  
│ │ │ │ -00023110: 2020 2035 2037 2020 2020 2020 3620 2020     5 7      6   
│ │ │ │ -00023120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000230a0: 2020 2020 207c 0a7c 2074 2020 2d20 3133       |.| t  - 13
│ │ │ │ +000230b0: 7420 202d 2033 3974 2074 2020 2d20 3239  t  - 39t t  - 29
│ │ │ │ +000230c0: 7420 7420 202b 2039 7420 7420 202b 2033  t t  + 9t t  + 3
│ │ │ │ +000230d0: 3974 2020 2020 2020 2020 2020 2020 2020  9t              
│ │ │ │ +000230e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000230f0: 2020 2020 207c 0a7c 3520 3620 2020 2020       |.|5 6     
│ │ │ │ +00023100: 2036 2020 2020 2020 3320 3720 2020 2020   6      3 7     
│ │ │ │ +00023110: 2034 2037 2020 2020 2035 2037 2020 2020   4 7     5 7    
│ │ │ │ +00023120: 2020 3620 2020 2020 2020 2020 2020 2020    6             
│ │ │ │ +00023130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023140: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023190: 0a7c 2020 2020 2020 3220 2020 2020 2020  .|      2       
│ │ │ │ +00023180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023190: 2020 2020 207c 0a7c 2020 2020 2020 3220       |.|      2 
│ │ │ │  000231a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000231c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000231d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000231e0: 0a7c 202d 2032 3274 2020 2b20 3331 7420  .| - 22t  + 31t 
│ │ │ │ -000231f0: 7420 202d 2032 3574 2074 2020 2d20 3139  t  - 25t t  - 19
│ │ │ │ -00023200: 7420 7420 202b 2034 3774 2074 2020 2020  t t  + 47t t    
│ │ │ │ -00023210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023230: 0a7c 2020 2020 2020 3620 2020 2020 2033  .|      6      3
│ │ │ │ -00023240: 2037 2020 2020 2020 3420 3720 2020 2020   7      4 7     
│ │ │ │ -00023250: 2035 2037 2020 2020 2020 3620 3720 2020   5 7      6 7   
│ │ │ │ -00023260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023280: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000231d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000231e0: 2020 2020 207c 0a7c 202d 2032 3274 2020       |.| - 22t  
│ │ │ │ +000231f0: 2b20 3331 7420 7420 202d 2032 3574 2074  + 31t t  - 25t t
│ │ │ │ +00023200: 2020 2d20 3139 7420 7420 202b 2034 3774    - 19t t  + 47t
│ │ │ │ +00023210: 2074 2020 2020 2020 2020 2020 2020 2020   t              
│ │ │ │ +00023220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023230: 2020 2020 207c 0a7c 2020 2020 2020 3620       |.|      6 
│ │ │ │ +00023240: 2020 2020 2033 2037 2020 2020 2020 3420       3 7      4 
│ │ │ │ +00023250: 3720 2020 2020 2035 2037 2020 2020 2020  7      5 7      
│ │ │ │ +00023260: 3620 3720 2020 2020 2020 2020 2020 2020  6 7             
│ │ │ │ +00023270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023280: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000232c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000232d0: 0a7c 2020 2020 2032 2020 2020 2020 2020  .|     2        
│ │ │ │ +000232c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000232d0: 2020 2020 207c 0a7c 2020 2020 2032 2020       |.|     2  
│ │ │ │  000232e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000232f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023320: 0a7c 2d20 3234 7420 202b 2033 3274 2074  .|- 24t  + 32t t
│ │ │ │ -00023330: 2020 2d20 3235 7420 7420 202d 2031 3974    - 25t t  - 19t
│ │ │ │ -00023340: 2074 2020 2b20 3437 7420 7420 2020 2020   t  + 47t t     
│ │ │ │ -00023350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023370: 0a7c 2020 2020 2036 2020 2020 2020 3320  .|     6      3 
│ │ │ │ -00023380: 3720 2020 2020 2034 2037 2020 2020 2020  7      4 7      
│ │ │ │ -00023390: 3520 3720 2020 2020 2036 2037 2020 2020  5 7      6 7    
│ │ │ │ -000233a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000233b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000233c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023320: 2020 2020 207c 0a7c 2d20 3234 7420 202b       |.|- 24t  +
│ │ │ │ +00023330: 2033 3274 2074 2020 2d20 3235 7420 7420   32t t  - 25t t 
│ │ │ │ +00023340: 202d 2031 3974 2074 2020 2b20 3437 7420   - 19t t  + 47t 
│ │ │ │ +00023350: 7420 2020 2020 2020 2020 2020 2020 2020  t               
│ │ │ │ +00023360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023370: 2020 2020 207c 0a7c 2020 2020 2036 2020       |.|     6  
│ │ │ │ +00023380: 2020 2020 3320 3720 2020 2020 2034 2037      3 7      4 7
│ │ │ │ +00023390: 2020 2020 2020 3520 3720 2020 2020 2036        5 7      6
│ │ │ │ +000233a0: 2037 2020 2020 2020 2020 2020 2020 2020   7              
│ │ │ │ +000233b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000233c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000233d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000233e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000233f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023400: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023410: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023410: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023460: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000234b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000234a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000234b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000234c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000234d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000234e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000234f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023500: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000234f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023500: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023550: 0a7c 2020 2020 2020 2020 2020 3220 2020  .|          2   
│ │ │ │ -00023560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023560: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  00023570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000235a0: 0a7c 7420 7420 202b 2031 3574 2020 2b20  .|t t  + 15t  + 
│ │ │ │ -000235b0: 3236 7420 7420 202d 2035 7420 7420 202b  26t t  - 5t t  +
│ │ │ │ -000235c0: 2033 3574 2074 2020 2d20 3130 7420 2020   35t t  - 10t   
│ │ │ │ -000235d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000235e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000235f0: 0a7c 2035 2036 2020 2020 2020 3620 2020  .| 5 6      6   
│ │ │ │ -00023600: 2020 2033 2037 2020 2020 2034 2037 2020     3 7     4 7  
│ │ │ │ -00023610: 2020 2020 3520 3720 2020 2020 2020 2020      5 7         
│ │ │ │ +00023590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000235a0: 2020 2020 207c 0a7c 7420 7420 202b 2031       |.|t t  + 1
│ │ │ │ +000235b0: 3574 2020 2b20 3236 7420 7420 202d 2035  5t  + 26t t  - 5
│ │ │ │ +000235c0: 7420 7420 202b 2033 3574 2074 2020 2d20  t t  + 35t t  - 
│ │ │ │ +000235d0: 3130 7420 2020 2020 2020 2020 2020 2020  10t             
│ │ │ │ +000235e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000235f0: 2020 2020 207c 0a7c 2035 2036 2020 2020       |.| 5 6    
│ │ │ │ +00023600: 2020 3620 2020 2020 2033 2037 2020 2020    6      3 7    
│ │ │ │ +00023610: 2034 2037 2020 2020 2020 3520 3720 2020   4 7      5 7   
│ │ │ │  00023620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023640: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023640: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023680: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023690: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000236a0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00023680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023690: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000236a0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  000236b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000236c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000236d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000236e0: 0a7c 2074 2020 2d20 3434 7420 7420 202d  .| t  - 44t t  -
│ │ │ │ -000236f0: 2034 3774 2020 2d20 3336 7420 7420 202d   47t  - 36t t  -
│ │ │ │ -00023700: 2034 3674 2074 2020 2b20 7420 7420 2020   46t t  + t t   
│ │ │ │ -00023710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023730: 0a7c 3420 3620 2020 2020 2035 2036 2020  .|4 6      5 6  
│ │ │ │ -00023740: 2020 2020 3620 2020 2020 2030 2037 2020      6      0 7  
│ │ │ │ -00023750: 2020 2020 3120 3720 2020 2032 2020 2020      1 7    2    
│ │ │ │ -00023760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023780: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000236d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000236e0: 2020 2020 207c 0a7c 2074 2020 2d20 3434       |.| t  - 44
│ │ │ │ +000236f0: 7420 7420 202d 2034 3774 2020 2d20 3336  t t  - 47t  - 36
│ │ │ │ +00023700: 7420 7420 202d 2034 3674 2074 2020 2b20  t t  - 46t t  + 
│ │ │ │ +00023710: 7420 7420 2020 2020 2020 2020 2020 2020  t t             
│ │ │ │ +00023720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023730: 2020 2020 207c 0a7c 3420 3620 2020 2020       |.|4 6     
│ │ │ │ +00023740: 2035 2036 2020 2020 2020 3620 2020 2020   5 6      6     
│ │ │ │ +00023750: 2030 2037 2020 2020 2020 3120 3720 2020   0 7      1 7   
│ │ │ │ +00023760: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00023770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023780: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000237c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000237d0: 0a7c 202c 2020 2020 2020 2020 2020 2020  .| ,            
│ │ │ │ +000237c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000237d0: 2020 2020 207c 0a7c 202c 2020 2020 2020       |.| ,      
│ │ │ │  000237e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000237f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023820: 0a7c 3820 2020 2020 2020 2020 2020 2020  .|8             
│ │ │ │ +00023810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023820: 2020 2020 207c 0a7c 3820 2020 2020 2020       |.|8       
│ │ │ │  00023830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023870: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023870: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000238b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000238c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000238d0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000238b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000238c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000238d0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  000238e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000238f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023910: 0a7c 7420 202d 2035 7420 7420 202b 2031  .|t  - 5t t  + 1
│ │ │ │ -00023920: 3374 2020 2b20 3574 2074 2020 2b20 7420  3t  + 5t t  + t 
│ │ │ │ -00023930: 7420 202b 2033 3974 2074 2020 2b20 2020  t  + 39t t  +   
│ │ │ │ -00023940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023960: 0a7c 2036 2020 2020 2035 2036 2020 2020  .| 6     5 6    
│ │ │ │ -00023970: 2020 3620 2020 2020 3020 3720 2020 2031    6     0 7    1
│ │ │ │ -00023980: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ -00023990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000239b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023910: 2020 2020 207c 0a7c 7420 202d 2035 7420       |.|t  - 5t 
│ │ │ │ +00023920: 7420 202b 2031 3374 2020 2b20 3574 2074  t  + 13t  + 5t t
│ │ │ │ +00023930: 2020 2b20 7420 7420 202b 2033 3974 2074    + t t  + 39t t
│ │ │ │ +00023940: 2020 2b20 2020 2020 2020 2020 2020 2020    +             
│ │ │ │ +00023950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023960: 2020 2020 207c 0a7c 2036 2020 2020 2035       |.| 6     5
│ │ │ │ +00023970: 2036 2020 2020 2020 3620 2020 2020 3020   6      6     0 
│ │ │ │ +00023980: 3720 2020 2031 2037 2020 2020 2020 3320  7    1 7      3 
│ │ │ │ +00023990: 3720 2020 2020 2020 2020 2020 2020 2020  7               
│ │ │ │ +000239a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000239b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000239c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000239e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000239f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023a00: 0a7c 2031 3974 2074 2020 2b20 3139 7420  .| 19t t  + 19t 
│ │ │ │ -00023a10: 7420 202b 2032 3074 2074 202c 2020 2020  t  + 20t t ,    
│ │ │ │ -00023a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000239f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a00: 2020 2020 207c 0a7c 2031 3974 2074 2020       |.| 19t t  
│ │ │ │ +00023a10: 2b20 3139 7420 7420 202b 2032 3074 2074  + 19t t  + 20t t
│ │ │ │ +00023a20: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  00023a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023a50: 0a7c 2020 2020 3620 3720 2020 2020 2031  .|    6 7      1
│ │ │ │ -00023a60: 2038 2020 2020 2020 3720 3820 2020 2020   8      7 8     
│ │ │ │ -00023a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023a50: 2020 2020 207c 0a7c 2020 2020 3620 3720       |.|    6 7 
│ │ │ │ +00023a60: 2020 2020 2031 2038 2020 2020 2020 3720       1 8      7 
│ │ │ │ +00023a70: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │  00023a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023aa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023aa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023af0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00023b00: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00023ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023af0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00023b00: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00023b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023b40: 0a7c 7420 202b 2031 3674 2074 2020 2d20  .|t  + 16t t  - 
│ │ │ │ -00023b50: 3335 7420 202b 2032 3974 2074 2020 2b20  35t  + 29t t  + 
│ │ │ │ -00023b60: 3132 7420 7420 202d 2033 3574 2020 2020  12t t  - 35t    
│ │ │ │ -00023b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023b90: 0a7c 2036 2020 2020 2020 3520 3620 2020  .| 6      5 6   
│ │ │ │ -00023ba0: 2020 2036 2020 2020 2020 3020 3720 2020     6      0 7   
│ │ │ │ -00023bb0: 2020 2031 2037 2020 2020 2020 3220 2020     1 7      2   
│ │ │ │ -00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023be0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00023b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b40: 2020 2020 207c 0a7c 7420 202b 2031 3674       |.|t  + 16t
│ │ │ │ +00023b50: 2074 2020 2d20 3335 7420 202b 2032 3974   t  - 35t  + 29t
│ │ │ │ +00023b60: 2074 2020 2b20 3132 7420 7420 202d 2033   t  + 12t t  - 3
│ │ │ │ +00023b70: 3574 2020 2020 2020 2020 2020 2020 2020  5t              
│ │ │ │ +00023b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023b90: 2020 2020 207c 0a7c 2036 2020 2020 2020       |.| 6      
│ │ │ │ +00023ba0: 3520 3620 2020 2020 2036 2020 2020 2020  5 6      6      
│ │ │ │ +00023bb0: 3020 3720 2020 2020 2031 2037 2020 2020  0 7      1 7    
│ │ │ │ +00023bc0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023be0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00023bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00023c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00023c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00023c30: 0a7c 2b32 352a 785f 332a 785f 372b 3233  .|+25*x_3*x_7+23
│ │ │ │ -00023c40: 2a78 5f35 2a78 5f37 2d33 2a78 5f36 2a78  *x_5*x_7-3*x_6*x
│ │ │ │ -00023c50: 5f37 2b32 2a78 5f30 2a78 5f38 2b31 312a  _7+2*x_0*x_8+11*
│ │ │ │ -00023c60: 785f 312a 785f 382d 3337 2a78 5f33 2a78  x_1*x_8-37*x_3*x
│ │ │ │ -00023c70: 5f38 2d32 332a 785f 342a 7820 2020 207c  _8-23*x_4*x    |
│ │ │ │ -00023c80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00023c30: 2d2d 2d2d 2d7c 0a7c 2b32 352a 785f 332a  -----|.|+25*x_3*
│ │ │ │ +00023c40: 785f 372b 3233 2a78 5f35 2a78 5f37 2d33  x_7+23*x_5*x_7-3
│ │ │ │ +00023c50: 2a78 5f36 2a78 5f37 2b32 2a78 5f30 2a78  *x_6*x_7+2*x_0*x
│ │ │ │ +00023c60: 5f38 2b31 312a 785f 312a 785f 382d 3337  _8+11*x_1*x_8-37
│ │ │ │ +00023c70: 2a78 5f33 2a78 5f38 2d32 332a 785f 342a  *x_3*x_8-23*x_4*
│ │ │ │ +00023c80: 7820 2020 207c 0a7c 2020 2020 2020 2020  x    |.|        
│ │ │ │  00023c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023cd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023d20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023d70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023d70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023dc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023e10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023e60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023e60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023eb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023eb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023f00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023f50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023f50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00023f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00023f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00023f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00023fa0: 0a7c 2020 2d20 3333 7820 7820 202b 2038  .|  - 33x x  + 8
│ │ │ │ -00023fb0: 7820 7820 202b 2031 3078 2078 2020 2d20  x x  + 10x x  - 
│ │ │ │ -00023fc0: 3235 7820 7820 202d 2039 7820 7820 202b  25x x  - 9x x  +
│ │ │ │ -00023fd0: 2033 7820 7820 202b 2032 3478 2078 2020   3x x  + 24x x  
│ │ │ │ -00023fe0: 2d20 3237 7820 7820 202d 2020 2020 207c  - 27x x  -     |
│ │ │ │ -00023ff0: 0a7c 3820 2020 2020 2036 2038 2020 2020  .|8      6 8    
│ │ │ │ -00024000: 2030 2039 2020 2020 2020 3120 3920 2020   0 9      1 9   
│ │ │ │ -00024010: 2020 2032 2039 2020 2020 2033 2039 2020     2 9     3 9  
│ │ │ │ -00024020: 2020 2034 2039 2020 2020 2020 3520 3920     4 9      5 9 
│ │ │ │ -00024030: 2020 2020 2036 2039 2020 2020 2020 207c       6 9       |
│ │ │ │ -00024040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00023f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00023fa0: 2020 2020 207c 0a7c 2020 2d20 3333 7820       |.|  - 33x 
│ │ │ │ +00023fb0: 7820 202b 2038 7820 7820 202b 2031 3078  x  + 8x x  + 10x
│ │ │ │ +00023fc0: 2078 2020 2d20 3235 7820 7820 202d 2039   x  - 25x x  - 9
│ │ │ │ +00023fd0: 7820 7820 202b 2033 7820 7820 202b 2032  x x  + 3x x  + 2
│ │ │ │ +00023fe0: 3478 2078 2020 2d20 3237 7820 7820 202d  4x x  - 27x x  -
│ │ │ │ +00023ff0: 2020 2020 207c 0a7c 3820 2020 2020 2036       |.|8      6
│ │ │ │ +00024000: 2038 2020 2020 2030 2039 2020 2020 2020   8     0 9      
│ │ │ │ +00024010: 3120 3920 2020 2020 2032 2039 2020 2020  1 9      2 9    
│ │ │ │ +00024020: 2033 2039 2020 2020 2034 2039 2020 2020   3 9     4 9    
│ │ │ │ +00024030: 2020 3520 3920 2020 2020 2036 2039 2020    5 9      6 9  
│ │ │ │ +00024040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000240a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000240b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000240c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000240d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000240e0: 0a7c 2020 2d20 3978 2078 2020 2d20 3436  .|  - 9x x  - 46
│ │ │ │ -000240f0: 7820 7820 202d 2031 3778 2078 2020 2b20  x x  - 17x x  + 
│ │ │ │ -00024100: 3332 7820 7820 202d 2038 7820 7820 202d  32x x  - 8x x  -
│ │ │ │ -00024110: 2033 3578 2078 2020 2d20 3436 7820 7820   35x x  - 46x x 
│ │ │ │ -00024120: 202b 2032 3678 2078 2020 2b20 2020 207c   + 26x x  +    |
│ │ │ │ -00024130: 0a7c 3820 2020 2020 3420 3820 2020 2020  .|8     4 8     
│ │ │ │ -00024140: 2036 2038 2020 2020 2020 3020 3920 2020   6 8      0 9   
│ │ │ │ -00024150: 2020 2031 2039 2020 2020 2032 2039 2020     1 9     2 9  
│ │ │ │ -00024160: 2020 2020 3320 3920 2020 2020 2034 2039      3 9      4 9
│ │ │ │ -00024170: 2020 2020 2020 3520 3920 2020 2020 207c        5 9      |
│ │ │ │ -00024180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000240d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000240e0: 2020 2020 207c 0a7c 2020 2d20 3978 2078       |.|  - 9x x
│ │ │ │ +000240f0: 2020 2d20 3436 7820 7820 202d 2031 3778    - 46x x  - 17x
│ │ │ │ +00024100: 2078 2020 2b20 3332 7820 7820 202d 2038   x  + 32x x  - 8
│ │ │ │ +00024110: 7820 7820 202d 2033 3578 2078 2020 2d20  x x  - 35x x  - 
│ │ │ │ +00024120: 3436 7820 7820 202b 2032 3678 2078 2020  46x x  + 26x x  
│ │ │ │ +00024130: 2b20 2020 207c 0a7c 3820 2020 2020 3420  +    |.|8     4 
│ │ │ │ +00024140: 3820 2020 2020 2036 2038 2020 2020 2020  8      6 8      
│ │ │ │ +00024150: 3020 3920 2020 2020 2031 2039 2020 2020  0 9      1 9    
│ │ │ │ +00024160: 2032 2039 2020 2020 2020 3320 3920 2020   2 9      3 9   
│ │ │ │ +00024170: 2020 2034 2039 2020 2020 2020 3520 3920     4 9      5 9 
│ │ │ │ +00024180: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000241c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000241d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000241c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000241d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000241e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000241f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024220: 0a7c 2078 2078 2020 2b20 3430 7820 7820  .| x x  + 40x x 
│ │ │ │ -00024230: 202d 2033 3278 2078 2020 2b20 3578 2078   - 32x x  + 5x x
│ │ │ │ -00024240: 2020 2d20 3131 7820 7820 202d 2032 3078    - 11x x  - 20x
│ │ │ │ -00024250: 2078 2020 2b20 3435 7820 7820 202d 2031   x  + 45x x  - 1
│ │ │ │ -00024260: 3478 2078 2020 2d20 3235 7820 2020 207c  4x x  - 25x    |
│ │ │ │ -00024270: 0a7c 2020 3620 3820 2020 2020 2030 2039  .|  6 8      0 9
│ │ │ │ -00024280: 2020 2020 2020 3120 3920 2020 2020 3220        1 9     2 
│ │ │ │ -00024290: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ -000242a0: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ -000242b0: 2020 3620 3920 2020 2020 2020 2020 207c    6 9          |
│ │ │ │ -000242c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024220: 2020 2020 207c 0a7c 2078 2078 2020 2b20       |.| x x  + 
│ │ │ │ +00024230: 3430 7820 7820 202d 2033 3278 2078 2020  40x x  - 32x x  
│ │ │ │ +00024240: 2b20 3578 2078 2020 2d20 3131 7820 7820  + 5x x  - 11x x 
│ │ │ │ +00024250: 202d 2032 3078 2078 2020 2b20 3435 7820   - 20x x  + 45x 
│ │ │ │ +00024260: 7820 202d 2031 3478 2078 2020 2d20 3235  x  - 14x x  - 25
│ │ │ │ +00024270: 7820 2020 207c 0a7c 2020 3620 3820 2020  x    |.|  6 8   
│ │ │ │ +00024280: 2020 2030 2039 2020 2020 2020 3120 3920     0 9      1 9 
│ │ │ │ +00024290: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ +000242a0: 2020 2020 2020 3420 3920 2020 2020 2035        4 9      5
│ │ │ │ +000242b0: 2039 2020 2020 2020 3620 3920 2020 2020   9      6 9     
│ │ │ │ +000242c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000242d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000242e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000242f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024310: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000243b0: 0a7c 3436 7420 7420 202b 2033 3774 2074  .|46t t  + 37t t
│ │ │ │ -000243c0: 2020 2b20 3238 7420 7420 202b 2033 3374    + 28t t  + 33t
│ │ │ │ -000243d0: 2074 202c 2020 2020 2020 2020 2020 2020   t ,            
│ │ │ │ +000243a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000243b0: 2020 2020 207c 0a7c 3436 7420 7420 202b       |.|46t t  +
│ │ │ │ +000243c0: 2033 3774 2074 2020 2b20 3238 7420 7420   37t t  + 28t t 
│ │ │ │ +000243d0: 202b 2033 3374 2074 202c 2020 2020 2020   + 33t t ,      
│ │ │ │  000243e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000243f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024400: 0a7c 2020 2033 2038 2020 2020 2020 3420  .|   3 8      4 
│ │ │ │ -00024410: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ -00024420: 3620 3820 2020 2020 2020 2020 2020 2020  6 8             
│ │ │ │ +000243f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024400: 2020 2020 207c 0a7c 2020 2033 2038 2020       |.|   3 8  
│ │ │ │ +00024410: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ +00024420: 2020 2020 2020 3620 3820 2020 2020 2020        6 8       
│ │ │ │  00024430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000244a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000244b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000244d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000244e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000244f0: 0a7c 2074 2020 2b20 3131 7420 7420 202b  .| t  + 11t t  +
│ │ │ │ -00024500: 2034 3674 2074 2020 2b20 3239 7420 7420   46t t  + 29t t 
│ │ │ │ -00024510: 202b 2032 3874 2074 202c 2020 2020 2020   + 28t t ,      
│ │ │ │ +000244e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000244f0: 2020 2020 207c 0a7c 2074 2020 2b20 3131       |.| t  + 11
│ │ │ │ +00024500: 7420 7420 202b 2034 3674 2074 2020 2b20  t t  + 46t t  + 
│ │ │ │ +00024510: 3239 7420 7420 202b 2032 3874 2074 202c  29t t  + 28t t ,
│ │ │ │  00024520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024540: 0a7c 3620 3720 2020 2020 2033 2038 2020  .|6 7      3 8  
│ │ │ │ -00024550: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ -00024560: 2020 2020 2020 3620 3820 2020 2020 2020        6 8       
│ │ │ │ +00024530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024540: 2020 2020 207c 0a7c 3620 3720 2020 2020       |.|6 7     
│ │ │ │ +00024550: 2033 2038 2020 2020 2020 3420 3820 2020   3 8      4 8   
│ │ │ │ +00024560: 2020 2035 2038 2020 2020 2020 3620 3820     5 8      6 8 
│ │ │ │  00024570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024590: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000245a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000245c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000245d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000245e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000245d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000245e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000245f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024630: 0a7c 7420 202b 2033 3674 2074 2020 2b20  .|t  + 36t t  + 
│ │ │ │ -00024640: 3133 7420 7420 202b 2032 3674 2074 2020  13t t  + 26t t  
│ │ │ │ -00024650: 2b20 3337 7420 7420 2c20 2020 2020 2020  + 37t t ,       
│ │ │ │ +00024620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024630: 2020 2020 207c 0a7c 7420 202b 2033 3674       |.|t  + 36t
│ │ │ │ +00024640: 2074 2020 2b20 3133 7420 7420 202b 2032   t  + 13t t  + 2
│ │ │ │ +00024650: 3674 2074 2020 2b20 3337 7420 7420 2c20  6t t  + 37t t , 
│ │ │ │  00024660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024680: 0a7c 2037 2020 2020 2020 3320 3820 2020  .| 7      3 8   
│ │ │ │ -00024690: 2020 2034 2038 2020 2020 2020 3520 3820     4 8      5 8 
│ │ │ │ -000246a0: 2020 2020 2036 2038 2020 2020 2020 2020       6 8        
│ │ │ │ +00024670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024680: 2020 2020 207c 0a7c 2037 2020 2020 2020       |.| 7      
│ │ │ │ +00024690: 3320 3820 2020 2020 2034 2038 2020 2020  3 8      4 8    
│ │ │ │ +000246a0: 2020 3520 3820 2020 2020 2036 2038 2020    5 8      6 8  
│ │ │ │  000246b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000246c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000246d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000246c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000246d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000246e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000246f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024720: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024720: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024770: 0a7c 202b 2034 3674 2074 2020 2d20 3336  .| + 46t t  - 36
│ │ │ │ -00024780: 7420 7420 202d 2033 3574 2074 2020 2d20  t t  - 35t t  - 
│ │ │ │ -00024790: 3331 7420 7420 2c20 2020 2020 2020 2020  31t t ,         
│ │ │ │ +00024760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024770: 2020 2020 207c 0a7c 202b 2034 3674 2074       |.| + 46t t
│ │ │ │ +00024780: 2020 2d20 3336 7420 7420 202d 2033 3574    - 36t t  - 35t
│ │ │ │ +00024790: 2074 2020 2d20 3331 7420 7420 2c20 2020   t  - 31t t ,   
│ │ │ │  000247a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000247b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000247c0: 0a7c 2020 2020 2020 3320 3820 2020 2020  .|      3 8     
│ │ │ │ -000247d0: 2034 2038 2020 2020 2020 3520 3820 2020   4 8      5 8   
│ │ │ │ -000247e0: 2020 2036 2038 2020 2020 2020 2020 2020     6 8          
│ │ │ │ +000247b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000247c0: 2020 2020 207c 0a7c 2020 2020 2020 3320       |.|      3 
│ │ │ │ +000247d0: 3820 2020 2020 2034 2038 2020 2020 2020  8      4 8      
│ │ │ │ +000247e0: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │  000247f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024810: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024860: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000248b0: 0a7c 2b20 3436 7420 7420 202d 2033 3674  .|+ 46t t  - 36t
│ │ │ │ -000248c0: 2074 2020 2d20 3335 7420 7420 202d 2033   t  - 35t t  - 3
│ │ │ │ -000248d0: 3174 2074 202c 2020 2020 2020 2020 2020  1t t ,          
│ │ │ │ +000248a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000248b0: 2020 2020 207c 0a7c 2b20 3436 7420 7420       |.|+ 46t t 
│ │ │ │ +000248c0: 202d 2033 3674 2074 2020 2d20 3335 7420   - 36t t  - 35t 
│ │ │ │ +000248d0: 7420 202d 2033 3174 2074 202c 2020 2020  t  - 31t t ,    
│ │ │ │  000248e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000248f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024900: 0a7c 2020 2020 2033 2038 2020 2020 2020  .|     3 8      
│ │ │ │ -00024910: 3420 3820 2020 2020 2035 2038 2020 2020  4 8      5 8    
│ │ │ │ -00024920: 2020 3620 3820 2020 2020 2020 2020 2020    6 8           
│ │ │ │ +000248f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024900: 2020 2020 207c 0a7c 2020 2020 2033 2038       |.|     3 8
│ │ │ │ +00024910: 2020 2020 2020 3420 3820 2020 2020 2035        4 8      5
│ │ │ │ +00024920: 2038 2020 2020 2020 3620 3820 2020 2020   8      6 8     
│ │ │ │  00024930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024950: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000249a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000249a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000249b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000249d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000249e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000249f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000249e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000249f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024a40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024a90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024a90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024ae0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024b30: 0a7c 2074 2020 2d20 3130 7420 7420 202d  .| t  - 10t t  -
│ │ │ │ -00024b40: 2034 3674 2074 2020 2b20 3437 7420 7420   46t t  + 47t t 
│ │ │ │ -00024b50: 202d 2032 3574 2074 202c 2020 2020 2020   - 25t t ,      
│ │ │ │ +00024b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b30: 2020 2020 207c 0a7c 2074 2020 2d20 3130       |.| t  - 10
│ │ │ │ +00024b40: 7420 7420 202d 2034 3674 2074 2020 2b20  t t  - 46t t  + 
│ │ │ │ +00024b50: 3437 7420 7420 202d 2032 3574 2074 202c  47t t  - 25t t ,
│ │ │ │  00024b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024b80: 0a7c 3620 3720 2020 2020 2033 2038 2020  .|6 7      3 8  
│ │ │ │ -00024b90: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ -00024ba0: 2020 2020 2020 3620 3820 2020 2020 2020        6 8       
│ │ │ │ +00024b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024b80: 2020 2020 207c 0a7c 3620 3720 2020 2020       |.|6 7     
│ │ │ │ +00024b90: 2033 2038 2020 2020 2020 3420 3820 2020   3 8      4 8   
│ │ │ │ +00024ba0: 2020 2035 2038 2020 2020 2020 3620 3820     5 8      6 8 
│ │ │ │  00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024bd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024bd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024c70: 0a7c 2020 2d20 3434 7420 7420 202b 2034  .|  - 44t t  + 4
│ │ │ │ -00024c80: 3874 2074 2020 2d20 3134 7420 7420 202b  8t t  - 14t t  +
│ │ │ │ -00024c90: 2034 7420 7420 202d 2033 3674 2074 2020   4t t  - 36t t  
│ │ │ │ -00024ca0: 2d20 3436 7420 7420 202b 2034 3774 2074  - 46t t  + 47t t
│ │ │ │ -00024cb0: 2020 2d20 3334 7420 7420 2020 2020 207c    - 34t t      |
│ │ │ │ -00024cc0: 0a7c 3720 2020 2020 2033 2037 2020 2020  .|7      3 7    
│ │ │ │ -00024cd0: 2020 3420 3720 2020 2020 2035 2037 2020    4 7      5 7  
│ │ │ │ -00024ce0: 2020 2036 2037 2020 2020 2020 3020 3820     6 7      0 8 
│ │ │ │ -00024cf0: 2020 2020 2031 2038 2020 2020 2020 3220       1 8      2 
│ │ │ │ -00024d00: 3820 2020 2020 2033 2038 2020 2020 207c  8      3 8     |
│ │ │ │ -00024d10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024c70: 2020 2020 207c 0a7c 2020 2d20 3434 7420       |.|  - 44t 
│ │ │ │ +00024c80: 7420 202b 2034 3874 2074 2020 2d20 3134  t  + 48t t  - 14
│ │ │ │ +00024c90: 7420 7420 202b 2034 7420 7420 202d 2033  t t  + 4t t  - 3
│ │ │ │ +00024ca0: 3674 2074 2020 2d20 3436 7420 7420 202b  6t t  - 46t t  +
│ │ │ │ +00024cb0: 2034 3774 2074 2020 2d20 3334 7420 7420   47t t  - 34t t 
│ │ │ │ +00024cc0: 2020 2020 207c 0a7c 3720 2020 2020 2033       |.|7      3
│ │ │ │ +00024cd0: 2037 2020 2020 2020 3420 3720 2020 2020   7      4 7     
│ │ │ │ +00024ce0: 2035 2037 2020 2020 2036 2037 2020 2020   5 7     6 7    
│ │ │ │ +00024cf0: 2020 3020 3820 2020 2020 2031 2038 2020    0 8      1 8  
│ │ │ │ +00024d00: 2020 2020 3220 3820 2020 2020 2033 2038      2 8      3 8
│ │ │ │ +00024d10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024d60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024db0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024e00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024e50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024e50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024ea0: 0a7c 2032 3874 2074 2020 2d20 3974 2074  .| 28t t  - 9t t
│ │ │ │ -00024eb0: 2020 2d20 3339 7420 7420 202b 2034 7420    - 39t t  + 4t 
│ │ │ │ -00024ec0: 7420 202b 2074 2074 2020 2d20 3336 7420  t  + t t  - 36t 
│ │ │ │ -00024ed0: 7420 202d 2031 3474 2074 2020 2d20 3236  t  - 14t t  - 26
│ │ │ │ -00024ee0: 7420 7420 202d 2033 3774 2020 2020 207c  t t  - 37t     |
│ │ │ │ -00024ef0: 0a7c 2020 2020 3420 3720 2020 2020 3520  .|    4 7     5 
│ │ │ │ -00024f00: 3720 2020 2020 2036 2037 2020 2020 2030  7      6 7     0
│ │ │ │ -00024f10: 2038 2020 2020 3120 3820 2020 2020 2033   8    1 8      3
│ │ │ │ -00024f20: 2038 2020 2020 2020 3420 3820 2020 2020   8      4 8     
│ │ │ │ -00024f30: 2035 2038 2020 2020 2020 3620 2020 207c   5 8      6    |
│ │ │ │ -00024f40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024ea0: 2020 2020 207c 0a7c 2032 3874 2074 2020       |.| 28t t  
│ │ │ │ +00024eb0: 2d20 3974 2074 2020 2d20 3339 7420 7420  - 9t t  - 39t t 
│ │ │ │ +00024ec0: 202b 2034 7420 7420 202b 2074 2074 2020   + 4t t  + t t  
│ │ │ │ +00024ed0: 2d20 3336 7420 7420 202d 2031 3474 2074  - 36t t  - 14t t
│ │ │ │ +00024ee0: 2020 2d20 3236 7420 7420 202d 2033 3774    - 26t t  - 37t
│ │ │ │ +00024ef0: 2020 2020 207c 0a7c 2020 2020 3420 3720       |.|    4 7 
│ │ │ │ +00024f00: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ +00024f10: 2020 2020 2030 2038 2020 2020 3120 3820       0 8    1 8 
│ │ │ │ +00024f20: 2020 2020 2033 2038 2020 2020 2020 3420       3 8      4 
│ │ │ │ +00024f30: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ +00024f40: 3620 2020 207c 0a7c 2020 2020 2020 2020  6    |.|        
│ │ │ │  00024f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024f90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024f90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00024fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00024fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00024fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025030: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025030: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025070: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025080: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025080: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000250b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000250c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000250d0: 0a7c 7420 202d 2038 7420 7420 202d 2031  .|t  - 8t t  - 1
│ │ │ │ -000250e0: 3874 2074 2020 2b20 3432 7420 7420 202d  8t t  + 42t t  -
│ │ │ │ -000250f0: 2031 3274 2074 2020 2d20 3674 2074 2020   12t t  - 6t t  
│ │ │ │ -00025100: 2b20 3132 7420 7420 202d 2031 3574 2074  + 12t t  - 15t t
│ │ │ │ -00025110: 2020 2b20 3974 2074 2020 2b20 2020 207c    + 9t t  +    |
│ │ │ │ -00025120: 0a7c 2037 2020 2020 2033 2037 2020 2020  .| 7     3 7    
│ │ │ │ -00025130: 2020 3420 3720 2020 2020 2035 2037 2020    4 7      5 7  
│ │ │ │ -00025140: 2020 2020 3620 3720 2020 2020 3020 3820      6 7     0 8 
│ │ │ │ -00025150: 2020 2020 2031 2038 2020 2020 2020 3320       1 8      3 
│ │ │ │ -00025160: 3820 2020 2020 3420 3820 2020 2020 207c  8     4 8      |
│ │ │ │ -00025170: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000250c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000250d0: 2020 2020 207c 0a7c 7420 202d 2038 7420       |.|t  - 8t 
│ │ │ │ +000250e0: 7420 202d 2031 3874 2074 2020 2b20 3432  t  - 18t t  + 42
│ │ │ │ +000250f0: 7420 7420 202d 2031 3274 2074 2020 2d20  t t  - 12t t  - 
│ │ │ │ +00025100: 3674 2074 2020 2b20 3132 7420 7420 202d  6t t  + 12t t  -
│ │ │ │ +00025110: 2031 3574 2074 2020 2b20 3974 2074 2020   15t t  + 9t t  
│ │ │ │ +00025120: 2b20 2020 207c 0a7c 2037 2020 2020 2033  +    |.| 7     3
│ │ │ │ +00025130: 2037 2020 2020 2020 3420 3720 2020 2020   7      4 7     
│ │ │ │ +00025140: 2035 2037 2020 2020 2020 3620 3720 2020   5 7      6 7   
│ │ │ │ +00025150: 2020 3020 3820 2020 2020 2031 2038 2020    0 8      1 8  
│ │ │ │ +00025160: 2020 2020 3320 3820 2020 2020 3420 3820      3 8     4 8 
│ │ │ │ +00025170: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00025180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00025190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000251a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000251c0: 0a7c 5f38 2d33 332a 785f 362a 785f 382b  .|_8-33*x_6*x_8+
│ │ │ │ -000251d0: 382a 785f 302a 785f 392b 3130 2a78 5f31  8*x_0*x_9+10*x_1
│ │ │ │ -000251e0: 2a78 5f39 2d32 352a 785f 322a 785f 392d  *x_9-25*x_2*x_9-
│ │ │ │ -000251f0: 392a 785f 332a 785f 392b 332a 785f 342a  9*x_3*x_9+3*x_4*
│ │ │ │ -00025200: 785f 392b 3234 2a78 5f35 2a78 5f20 207c  x_9+24*x_5*x_  |
│ │ │ │ -00025210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000251c0: 2d2d 2d2d 2d7c 0a7c 5f38 2d33 332a 785f  -----|.|_8-33*x_
│ │ │ │ +000251d0: 362a 785f 382b 382a 785f 302a 785f 392b  6*x_8+8*x_0*x_9+
│ │ │ │ +000251e0: 3130 2a78 5f31 2a78 5f39 2d32 352a 785f  10*x_1*x_9-25*x_
│ │ │ │ +000251f0: 322a 785f 392d 392a 785f 332a 785f 392b  2*x_9-9*x_3*x_9+
│ │ │ │ +00025200: 332a 785f 342a 785f 392b 3234 2a78 5f35  3*x_4*x_9+24*x_5
│ │ │ │ +00025210: 2a78 5f20 207c 0a7c 2020 2020 2020 2020  *x_  |.|        
│ │ │ │  00025220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000252b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000252a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000252b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000252c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000252d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000252e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000252f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025300: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000253a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000253a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000253b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000253d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000253e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000253f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000253e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000253f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025440: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025490: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000254a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000254c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000254e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000254d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000254e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000254f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025520: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025530: 0a7c 3578 2078 2020 202b 2032 3878 2078  .|5x x   + 28x x
│ │ │ │ -00025540: 2020 202b 2033 3778 2078 2020 202b 2039     + 37x x   + 9
│ │ │ │ -00025550: 7820 7820 2020 2b20 3237 7820 7820 2020  x x   + 27x x   
│ │ │ │ -00025560: 2d20 3235 7820 7820 2020 2b20 3978 2078  - 25x x   + 9x x
│ │ │ │ -00025570: 2020 202b 2032 3778 2078 2020 2020 207c     + 27x x     |
│ │ │ │ -00025580: 0a7c 2020 3020 3130 2020 2020 2020 3120  .|  0 10      1 
│ │ │ │ -00025590: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ -000255a0: 2034 2031 3020 2020 2020 2036 2031 3020   4 10      6 10 
│ │ │ │ -000255b0: 2020 2020 2030 2031 3120 2020 2020 3220       0 11     2 
│ │ │ │ -000255c0: 3131 2020 2020 2020 3420 3131 2020 207c  11      4 11   |
│ │ │ │ -000255d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025530: 2020 2020 207c 0a7c 3578 2078 2020 202b       |.|5x x   +
│ │ │ │ +00025540: 2032 3878 2078 2020 202b 2033 3778 2078   28x x   + 37x x
│ │ │ │ +00025550: 2020 202b 2039 7820 7820 2020 2b20 3237     + 9x x   + 27
│ │ │ │ +00025560: 7820 7820 2020 2d20 3235 7820 7820 2020  x x   - 25x x   
│ │ │ │ +00025570: 2b20 3978 2078 2020 202b 2032 3778 2078  + 9x x   + 27x x
│ │ │ │ +00025580: 2020 2020 207c 0a7c 2020 3020 3130 2020       |.|  0 10  
│ │ │ │ +00025590: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ +000255a0: 3130 2020 2020 2034 2031 3020 2020 2020  10     4 10     
│ │ │ │ +000255b0: 2036 2031 3020 2020 2020 2030 2031 3120   6 10      0 11 
│ │ │ │ +000255c0: 2020 2020 3220 3131 2020 2020 2020 3420      2 11      4 
│ │ │ │ +000255d0: 3131 2020 207c 0a7c 2020 2020 2020 2020  11   |.|        
│ │ │ │  000255e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000255f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025610: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025620: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025620: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025670: 0a7c 2031 3778 2078 2020 2b20 3135 7820  .| 17x x  + 15x 
│ │ │ │ -00025680: 7820 2020 2b20 3335 7820 7820 2020 2b20  x   + 35x x   + 
│ │ │ │ -00025690: 3334 7820 7820 2020 2b20 3230 7820 7820  34x x   + 20x x 
│ │ │ │ -000256a0: 2020 2b20 3134 7820 7820 2020 2b20 3336    + 14x x   + 36
│ │ │ │ -000256b0: 7820 7820 2020 2b20 3335 7820 7820 207c  x x   + 35x x  |
│ │ │ │ -000256c0: 0a7c 2020 2020 3620 3920 2020 2020 2030  .|    6 9      0
│ │ │ │ -000256d0: 2031 3020 2020 2020 2031 2031 3020 2020   10      1 10   
│ │ │ │ -000256e0: 2020 2032 2031 3020 2020 2020 2034 2031     2 10      4 1
│ │ │ │ -000256f0: 3020 2020 2020 2030 2031 3120 2020 2020  0      0 11     
│ │ │ │ -00025700: 2031 2031 3120 2020 2020 2032 2020 207c   1 11      2   |
│ │ │ │ -00025710: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025670: 2020 2020 207c 0a7c 2031 3778 2078 2020       |.| 17x x  
│ │ │ │ +00025680: 2b20 3135 7820 7820 2020 2b20 3335 7820  + 15x x   + 35x 
│ │ │ │ +00025690: 7820 2020 2b20 3334 7820 7820 2020 2b20  x   + 34x x   + 
│ │ │ │ +000256a0: 3230 7820 7820 2020 2b20 3134 7820 7820  20x x   + 14x x 
│ │ │ │ +000256b0: 2020 2b20 3336 7820 7820 2020 2b20 3335    + 36x x   + 35
│ │ │ │ +000256c0: 7820 7820 207c 0a7c 2020 2020 3620 3920  x x  |.|    6 9 
│ │ │ │ +000256d0: 2020 2020 2030 2031 3020 2020 2020 2031       0 10      1
│ │ │ │ +000256e0: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ +000256f0: 2020 2034 2031 3020 2020 2020 2030 2031     4 10      0 1
│ │ │ │ +00025700: 3120 2020 2020 2031 2031 3120 2020 2020  1      1 11     
│ │ │ │ +00025710: 2032 2020 207c 0a7c 2020 2020 2020 2020   2   |.|        
│ │ │ │  00025720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025760: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025760: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000257a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000257b0: 0a7c 2078 2020 202b 2034 3578 2078 2020  .| x   + 45x x  
│ │ │ │ -000257c0: 202d 2034 3178 2078 2020 202d 2034 3678   - 41x x   - 46x
│ │ │ │ -000257d0: 2078 2020 202b 2038 7820 7820 2020 2d20   x   + 8x x   - 
│ │ │ │ -000257e0: 3238 7820 7820 2020 2b20 3131 7820 7820  28x x   + 11x x 
│ │ │ │ -000257f0: 2020 2b20 3134 7820 7820 2020 2d20 207c    + 14x x   -  |
│ │ │ │ -00025800: 0a7c 3020 3130 2020 2020 2020 3120 3130  .|0 10      1 10
│ │ │ │ -00025810: 2020 2020 2020 3220 3130 2020 2020 2020        2 10      
│ │ │ │ -00025820: 3420 3130 2020 2020 2036 2031 3020 2020  4 10     6 10   
│ │ │ │ -00025830: 2020 2030 2031 3120 2020 2020 2032 2031     0 11      2 1
│ │ │ │ -00025840: 3120 2020 2020 2034 2031 3120 2020 207c  1      4 11    |
│ │ │ │ -00025850: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000257a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000257b0: 2020 2020 207c 0a7c 2078 2020 202b 2034       |.| x   + 4
│ │ │ │ +000257c0: 3578 2078 2020 202d 2034 3178 2078 2020  5x x   - 41x x  
│ │ │ │ +000257d0: 202d 2034 3678 2078 2020 202b 2038 7820   - 46x x   + 8x 
│ │ │ │ +000257e0: 7820 2020 2d20 3238 7820 7820 2020 2b20  x   - 28x x   + 
│ │ │ │ +000257f0: 3131 7820 7820 2020 2b20 3134 7820 7820  11x x   + 14x x 
│ │ │ │ +00025800: 2020 2d20 207c 0a7c 3020 3130 2020 2020    -  |.|0 10    
│ │ │ │ +00025810: 2020 3120 3130 2020 2020 2020 3220 3130    1 10      2 10
│ │ │ │ +00025820: 2020 2020 2020 3420 3130 2020 2020 2036        4 10     6
│ │ │ │ +00025830: 2031 3020 2020 2020 2030 2031 3120 2020   10      0 11   
│ │ │ │ +00025840: 2020 2032 2031 3120 2020 2020 2034 2031     2 11      4 1
│ │ │ │ +00025850: 3120 2020 207c 0a7c 2020 2020 2020 2020  1    |.|        
│ │ │ │  00025860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025890: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000258a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000258a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000258b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000258c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000258d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000258e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000258f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000258e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000258f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025940: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025990: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025990: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000259a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000259c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000259d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000259e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000259d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000259e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000259f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025a70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025a80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025a80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025ad0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025ad0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025b20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025b20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025b60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025b70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025b70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025bb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025bc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025bc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025c10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025c10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025c50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025c60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025c60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025cb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025cb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025cf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025d00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025d00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025d50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025d50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025d90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025da0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025da0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025de0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025df0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025df0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025e30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025e40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025e40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025e80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025e90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025e90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025ed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025ee0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025f20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025f30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025f30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025f70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025f80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025f80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00025fc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00025fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00025fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00025fd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00025fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00025ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026020: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000260a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000260b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000260c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000260b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000260c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000260d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000260e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000260f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026110: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026150: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026160: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000261b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000261a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000261b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000261c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000261d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000261e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000261f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026200: 0a7c 2d20 3234 7420 7420 202d 2031 3274  .|- 24t t  - 12t
│ │ │ │ -00026210: 2074 2020 2d20 3437 7420 7420 202b 2034   t  - 47t t  + 4
│ │ │ │ -00026220: 3774 2074 202c 2020 2020 2020 2020 2020  7t t ,          
│ │ │ │ +000261f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026200: 2020 2020 207c 0a7c 2d20 3234 7420 7420       |.|- 24t t 
│ │ │ │ +00026210: 202d 2031 3274 2074 2020 2d20 3437 7420   - 12t t  - 47t 
│ │ │ │ +00026220: 7420 202b 2034 3774 2074 202c 2020 2020  t  + 47t t ,    
│ │ │ │  00026230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026250: 0a7c 2020 2020 2034 2038 2020 2020 2020  .|     4 8      
│ │ │ │ -00026260: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ -00026270: 2020 3720 3820 2020 2020 2020 2020 2020    7 8           
│ │ │ │ +00026240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026250: 2020 2020 207c 0a7c 2020 2020 2034 2038       |.|     4 8
│ │ │ │ +00026260: 2020 2020 2020 3520 3820 2020 2020 2036        5 8      6
│ │ │ │ +00026270: 2038 2020 2020 2020 3720 3820 2020 2020   8      7 8     
│ │ │ │  00026280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000262a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000262a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000262b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000262c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000262d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000262e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000262f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000262e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000262f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026330: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026340: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026390: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000263a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000263b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000263c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000263d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000263e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000263d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000263e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000263f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026430: 0a7c 7420 2c20 2020 2020 2020 2020 2020  .|t ,           
│ │ │ │ +00026420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026430: 2020 2020 207c 0a7c 7420 2c20 2020 2020       |.|t ,     
│ │ │ │  00026440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026480: 0a7c 2038 2020 2020 2020 2020 2020 2020  .| 8            
│ │ │ │ +00026470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026480: 2020 2020 207c 0a7c 2038 2020 2020 2020       |.| 8      
│ │ │ │  00026490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000264c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000264d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000264c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000264d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000264e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000264f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026520: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026520: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026570: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000265a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000265b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000265c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000265b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000265c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000265d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000265e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000265f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026610: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026610: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026650: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026660: 0a7c 2032 3074 2074 2020 2d20 3330 7420  .| 20t t  - 30t 
│ │ │ │ -00026670: 7420 202b 2034 7420 7420 2020 2020 2020  t  + 4t t       
│ │ │ │ +00026650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026660: 2020 2020 207c 0a7c 2032 3074 2074 2020       |.| 20t t  
│ │ │ │ +00026670: 2d20 3330 7420 7420 202b 2034 7420 7420  - 30t t  + 4t t 
│ │ │ │  00026680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000266b0: 0a7c 2020 2020 3520 3820 2020 2020 2036  .|    5 8      6
│ │ │ │ -000266c0: 2038 2020 2020 2037 2038 2020 2020 2020   8     7 8      
│ │ │ │ +000266a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000266b0: 2020 2020 207c 0a7c 2020 2020 3520 3820       |.|    5 8 
│ │ │ │ +000266c0: 2020 2020 2036 2038 2020 2020 2037 2038       6 8     7 8
│ │ │ │  000266d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000266e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000266f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026700: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000266f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026700: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00026710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00026750: 0a7c 392d 3237 2a78 5f36 2a78 5f39 2d35  .|9-27*x_6*x_9-5
│ │ │ │ -00026760: 2a78 5f30 2a78 5f31 302b 3238 2a78 5f31  *x_0*x_10+28*x_1
│ │ │ │ -00026770: 2a78 5f31 302b 3337 2a78 5f32 2a78 5f31  *x_10+37*x_2*x_1
│ │ │ │ -00026780: 302b 392a 785f 342a 785f 3130 2b32 372a  0+9*x_4*x_10+27*
│ │ │ │ -00026790: 785f 362a 785f 3130 2d32 352a 785f 307c  x_6*x_10-25*x_0|
│ │ │ │ -000267a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026750: 2d2d 2d2d 2d7c 0a7c 392d 3237 2a78 5f36  -----|.|9-27*x_6
│ │ │ │ +00026760: 2a78 5f39 2d35 2a78 5f30 2a78 5f31 302b  *x_9-5*x_0*x_10+
│ │ │ │ +00026770: 3238 2a78 5f31 2a78 5f31 302b 3337 2a78  28*x_1*x_10+37*x
│ │ │ │ +00026780: 5f32 2a78 5f31 302b 392a 785f 342a 785f  _2*x_10+9*x_4*x_
│ │ │ │ +00026790: 3130 2b32 372a 785f 362a 785f 3130 2d32  10+27*x_6*x_10-2
│ │ │ │ +000267a0: 352a 785f 307c 0a7c 2020 2020 2020 2020  5*x_0|.|        
│ │ │ │  000267b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000267c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000267d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000267e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000267f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000267f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026840: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026840: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026880: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026890: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026890: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000268a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000268b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000268c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000268d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000268e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000268d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000268e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000268f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026930: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026980: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000269a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000269b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000269c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000269d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000269c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000269d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000269e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000269f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026a20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026a60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026a70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026a70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026ac0: 0a7c 2d20 3237 7820 7820 202c 2020 2020  .|- 27x x  ,    
│ │ │ │ -00026ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ac0: 2020 2020 207c 0a7c 2d20 3237 7820 7820       |.|- 27x x 
│ │ │ │ +00026ad0: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │  00026ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026b10: 0a7c 2020 2020 2035 2031 3120 2020 2020  .|     5 11     
│ │ │ │ -00026b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b10: 2020 2020 207c 0a7c 2020 2020 2035 2031       |.|     5 1
│ │ │ │ +00026b20: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │  00026b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026b50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026b60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026b60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ba0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026bb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026bf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026c00: 0a7c 2020 202d 2031 3778 2078 2020 2c20  .|   - 17x x  , 
│ │ │ │ -00026c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c00: 2020 2020 207c 0a7c 2020 202d 2031 3778       |.|   - 17x
│ │ │ │ +00026c10: 2078 2020 2c20 2020 2020 2020 2020 2020   x  ,           
│ │ │ │  00026c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026c50: 0a7c 3131 2020 2020 2020 3420 3131 2020  .|11      4 11  
│ │ │ │ -00026c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026c50: 2020 2020 207c 0a7c 3131 2020 2020 2020       |.|11      
│ │ │ │ +00026c60: 3420 3131 2020 2020 2020 2020 2020 2020  4 11            
│ │ │ │  00026c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026c90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026ca0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026ca0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026ce0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026cf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00026ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00026d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026d40: 0a7c 2038 7820 7820 2020 2020 2020 2020  .| 8x x         
│ │ │ │ +00026d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d40: 2020 2020 207c 0a7c 2038 7820 7820 2020       |.| 8x x   
│ │ │ │  00026d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026d80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026d90: 0a7c 2020 2035 2031 3120 2020 2020 2020  .|   5 11       
│ │ │ │ +00026d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026d90: 2020 2020 207c 0a7c 2020 2035 2031 3120       |.|   5 11 
│ │ │ │  00026da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00026dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00026dd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00026de0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00026de0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00026df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00026e30: 0a7c 2a78 5f31 312b 392a 785f 322a 785f  .|*x_11+9*x_2*x_
│ │ │ │ -00026e40: 3131 2b32 372a 785f 342a 785f 3131 2d32  11+27*x_4*x_11-2
│ │ │ │ -00026e50: 372a 785f 352a 785f 3131 2c78 5f32 5e32  7*x_5*x_11,x_2^2
│ │ │ │ -00026e60: 2b31 372a 785f 322a 785f 332d 3134 2a78  +17*x_2*x_3-14*x
│ │ │ │ -00026e70: 5f33 5e32 2d31 332a 785f 322a 785f 347c  _3^2-13*x_2*x_4|
│ │ │ │ -00026e80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00026e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026e30: 2d2d 2d2d 2d7c 0a7c 2a78 5f31 312b 392a  -----|.|*x_11+9*
│ │ │ │ +00026e40: 785f 322a 785f 3131 2b32 372a 785f 342a  x_2*x_11+27*x_4*
│ │ │ │ +00026e50: 785f 3131 2d32 372a 785f 352a 785f 3131  x_11-27*x_5*x_11
│ │ │ │ +00026e60: 2c78 5f32 5e32 2b31 372a 785f 322a 785f  ,x_2^2+17*x_2*x_
│ │ │ │ +00026e70: 332d 3134 2a78 5f33 5e32 2d31 332a 785f  3-14*x_3^2-13*x_
│ │ │ │ +00026e80: 322a 785f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  2*x_4|.|--------
│ │ │ │  00026e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00026ed0: 0a7c 2b33 342a 785f 332a 785f 342b 3434  .|+34*x_3*x_4+44
│ │ │ │ -00026ee0: 2a78 5f30 2a78 5f35 2d33 302a 785f 322a  *x_0*x_5-30*x_2*
│ │ │ │ -00026ef0: 785f 352b 3237 2a78 5f33 2a78 5f35 2b33  x_5+27*x_3*x_5+3
│ │ │ │ -00026f00: 312a 785f 322a 785f 362d 3336 2a78 5f33  1*x_2*x_6-36*x_3
│ │ │ │ -00026f10: 2a78 5f36 2d78 5f30 2a78 5f37 2b31 337c  *x_6-x_0*x_7+13|
│ │ │ │ -00026f20: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00026ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026ed0: 2d2d 2d2d 2d7c 0a7c 2b33 342a 785f 332a  -----|.|+34*x_3*
│ │ │ │ +00026ee0: 785f 342b 3434 2a78 5f30 2a78 5f35 2d33  x_4+44*x_0*x_5-3
│ │ │ │ +00026ef0: 302a 785f 322a 785f 352b 3237 2a78 5f33  0*x_2*x_5+27*x_3
│ │ │ │ +00026f00: 2a78 5f35 2b33 312a 785f 322a 785f 362d  *x_5+31*x_2*x_6-
│ │ │ │ +00026f10: 3336 2a78 5f33 2a78 5f36 2d78 5f30 2a78  36*x_3*x_6-x_0*x
│ │ │ │ +00026f20: 5f37 2b31 337c 0a7c 2d2d 2d2d 2d2d 2d2d  _7+13|.|--------
│ │ │ │  00026f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00026f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00026f70: 0a7c 2a78 5f31 2a78 5f37 2b38 2a78 5f33  .|*x_1*x_7+8*x_3
│ │ │ │ -00026f80: 2a78 5f37 2b39 2a78 5f35 2a78 5f37 2b34  *x_7+9*x_5*x_7+4
│ │ │ │ -00026f90: 362a 785f 362a 785f 372b 3431 2a78 5f30  6*x_6*x_7+41*x_0
│ │ │ │ -00026fa0: 2a78 5f38 2d37 2a78 5f31 2a78 5f38 2d33  *x_8-7*x_1*x_8-3
│ │ │ │ -00026fb0: 342a 785f 332a 785f 382d 392a 785f 347c  4*x_3*x_8-9*x_4|
│ │ │ │ -00026fc0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00026f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00026f70: 2d2d 2d2d 2d7c 0a7c 2a78 5f31 2a78 5f37  -----|.|*x_1*x_7
│ │ │ │ +00026f80: 2b38 2a78 5f33 2a78 5f37 2b39 2a78 5f35  +8*x_3*x_7+9*x_5
│ │ │ │ +00026f90: 2a78 5f37 2b34 362a 785f 362a 785f 372b  *x_7+46*x_6*x_7+
│ │ │ │ +00026fa0: 3431 2a78 5f30 2a78 5f38 2d37 2a78 5f31  41*x_0*x_8-7*x_1
│ │ │ │ +00026fb0: 2a78 5f38 2d33 342a 785f 332a 785f 382d  *x_8-34*x_3*x_8-
│ │ │ │ +00026fc0: 392a 785f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  9*x_4|.|--------
│ │ │ │  00026fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00026ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027010: 0a7c 2a78 5f38 2d34 362a 785f 362a 785f  .|*x_8-46*x_6*x_
│ │ │ │ -00027020: 382d 3137 2a78 5f30 2a78 5f39 2b33 322a  8-17*x_0*x_9+32*
│ │ │ │ -00027030: 785f 312a 785f 392d 382a 785f 322a 785f  x_1*x_9-8*x_2*x_
│ │ │ │ -00027040: 392d 3335 2a78 5f33 2a78 5f39 2d34 362a  9-35*x_3*x_9-46*
│ │ │ │ -00027050: 785f 342a 785f 392b 3236 2a78 5f35 2a7c  x_4*x_9+26*x_5*|
│ │ │ │ -00027060: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027010: 2d2d 2d2d 2d7c 0a7c 2a78 5f38 2d34 362a  -----|.|*x_8-46*
│ │ │ │ +00027020: 785f 362a 785f 382d 3137 2a78 5f30 2a78  x_6*x_8-17*x_0*x
│ │ │ │ +00027030: 5f39 2b33 322a 785f 312a 785f 392d 382a  _9+32*x_1*x_9-8*
│ │ │ │ +00027040: 785f 322a 785f 392d 3335 2a78 5f33 2a78  x_2*x_9-35*x_3*x
│ │ │ │ +00027050: 5f39 2d34 362a 785f 342a 785f 392b 3236  _9-46*x_4*x_9+26
│ │ │ │ +00027060: 2a78 5f35 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  *x_5*|.|--------
│ │ │ │  00027070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000270b0: 0a7c 785f 392b 3137 2a78 5f36 2a78 5f39  .|x_9+17*x_6*x_9
│ │ │ │ -000270c0: 2b31 352a 785f 302a 785f 3130 2b33 352a  +15*x_0*x_10+35*
│ │ │ │ -000270d0: 785f 312a 785f 3130 2b33 342a 785f 322a  x_1*x_10+34*x_2*
│ │ │ │ -000270e0: 785f 3130 2b32 302a 785f 342a 785f 3130  x_10+20*x_4*x_10
│ │ │ │ -000270f0: 2b31 342a 785f 302a 785f 3131 2b33 367c  +14*x_0*x_11+36|
│ │ │ │ -00027100: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000270a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000270b0: 2d2d 2d2d 2d7c 0a7c 785f 392b 3137 2a78  -----|.|x_9+17*x
│ │ │ │ +000270c0: 5f36 2a78 5f39 2b31 352a 785f 302a 785f  _6*x_9+15*x_0*x_
│ │ │ │ +000270d0: 3130 2b33 352a 785f 312a 785f 3130 2b33  10+35*x_1*x_10+3
│ │ │ │ +000270e0: 342a 785f 322a 785f 3130 2b32 302a 785f  4*x_2*x_10+20*x_
│ │ │ │ +000270f0: 342a 785f 3130 2b31 342a 785f 302a 785f  4*x_10+14*x_0*x_
│ │ │ │ +00027100: 3131 2b33 367c 0a7c 2d2d 2d2d 2d2d 2d2d  11+36|.|--------
│ │ │ │  00027110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027150: 0a7c 2a78 5f31 2a78 5f31 312b 3335 2a78  .|*x_1*x_11+35*x
│ │ │ │ -00027160: 5f32 2a78 5f31 312d 3137 2a78 5f34 2a78  _2*x_11-17*x_4*x
│ │ │ │ -00027170: 5f31 312c 785f 312a 785f 322d 3430 2a78  _11,x_1*x_2-40*x
│ │ │ │ -00027180: 5f32 2a78 5f33 2b32 382a 785f 335e 322d  _2*x_3+28*x_3^2-
│ │ │ │ -00027190: 785f 302a 785f 342b 352a 785f 322a 787c  x_0*x_4+5*x_2*x|
│ │ │ │ -000271a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027150: 2d2d 2d2d 2d7c 0a7c 2a78 5f31 2a78 5f31  -----|.|*x_1*x_1
│ │ │ │ +00027160: 312b 3335 2a78 5f32 2a78 5f31 312d 3137  1+35*x_2*x_11-17
│ │ │ │ +00027170: 2a78 5f34 2a78 5f31 312c 785f 312a 785f  *x_4*x_11,x_1*x_
│ │ │ │ +00027180: 322d 3430 2a78 5f32 2a78 5f33 2b32 382a  2-40*x_2*x_3+28*
│ │ │ │ +00027190: 785f 335e 322d 785f 302a 785f 342b 352a  x_3^2-x_0*x_4+5*
│ │ │ │ +000271a0: 785f 322a 787c 0a7c 2d2d 2d2d 2d2d 2d2d  x_2*x|.|--------
│ │ │ │  000271b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000271c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000271d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000271e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000271f0: 0a7c 5f34 2d31 362a 785f 332a 785f 342b  .|_4-16*x_3*x_4+
│ │ │ │ -00027200: 352a 785f 302a 785f 352d 3336 2a78 5f32  5*x_0*x_5-36*x_2
│ │ │ │ -00027210: 2a78 5f35 2b33 372a 785f 332a 785f 352b  *x_5+37*x_3*x_5+
│ │ │ │ -00027220: 3438 2a78 5f32 2a78 5f36 2d35 2a78 5f31  48*x_2*x_6-5*x_1
│ │ │ │ -00027230: 2a78 5f37 2d35 2a78 5f33 2a78 5f37 2b7c  *x_7-5*x_3*x_7+|
│ │ │ │ -00027240: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000271e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000271f0: 2d2d 2d2d 2d7c 0a7c 5f34 2d31 362a 785f  -----|.|_4-16*x_
│ │ │ │ +00027200: 332a 785f 342b 352a 785f 302a 785f 352d  3*x_4+5*x_0*x_5-
│ │ │ │ +00027210: 3336 2a78 5f32 2a78 5f35 2b33 372a 785f  36*x_2*x_5+37*x_
│ │ │ │ +00027220: 332a 785f 352b 3438 2a78 5f32 2a78 5f36  3*x_5+48*x_2*x_6
│ │ │ │ +00027230: 2d35 2a78 5f31 2a78 5f37 2d35 2a78 5f33  -5*x_1*x_7-5*x_3
│ │ │ │ +00027240: 2a78 5f37 2b7c 0a7c 2d2d 2d2d 2d2d 2d2d  *x_7+|.|--------
│ │ │ │  00027250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027290: 0a7c 785f 352a 785f 372b 3230 2a78 5f36  .|x_5*x_7+20*x_6
│ │ │ │ -000272a0: 2a78 5f37 2b31 302a 785f 302a 785f 382b  *x_7+10*x_0*x_8+
│ │ │ │ -000272b0: 3334 2a78 5f31 2a78 5f38 2b34 312a 785f  34*x_1*x_8+41*x_
│ │ │ │ -000272c0: 332a 785f 382d 785f 342a 785f 382b 785f  3*x_8-x_4*x_8+x_
│ │ │ │ -000272d0: 362a 785f 382b 3430 2a78 5f30 2a78 5f7c  6*x_8+40*x_0*x_|
│ │ │ │ -000272e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027290: 2d2d 2d2d 2d7c 0a7c 785f 352a 785f 372b  -----|.|x_5*x_7+
│ │ │ │ +000272a0: 3230 2a78 5f36 2a78 5f37 2b31 302a 785f  20*x_6*x_7+10*x_
│ │ │ │ +000272b0: 302a 785f 382b 3334 2a78 5f31 2a78 5f38  0*x_8+34*x_1*x_8
│ │ │ │ +000272c0: 2b34 312a 785f 332a 785f 382d 785f 342a  +41*x_3*x_8-x_4*
│ │ │ │ +000272d0: 785f 382b 785f 362a 785f 382b 3430 2a78  x_8+x_6*x_8+40*x
│ │ │ │ +000272e0: 5f30 2a78 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d  _0*x_|.|--------
│ │ │ │  000272f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027330: 0a7c 392d 3332 2a78 5f31 2a78 5f39 2b35  .|9-32*x_1*x_9+5
│ │ │ │ -00027340: 2a78 5f32 2a78 5f39 2d31 312a 785f 332a  *x_2*x_9-11*x_3*
│ │ │ │ -00027350: 785f 392d 3230 2a78 5f34 2a78 5f39 2b34  x_9-20*x_4*x_9+4
│ │ │ │ -00027360: 352a 785f 352a 785f 392d 3134 2a78 5f36  5*x_5*x_9-14*x_6
│ │ │ │ -00027370: 2a78 5f39 2d32 352a 785f 302a 785f 207c  *x_9-25*x_0*x_ |
│ │ │ │ -00027380: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027330: 2d2d 2d2d 2d7c 0a7c 392d 3332 2a78 5f31  -----|.|9-32*x_1
│ │ │ │ +00027340: 2a78 5f39 2b35 2a78 5f32 2a78 5f39 2d31  *x_9+5*x_2*x_9-1
│ │ │ │ +00027350: 312a 785f 332a 785f 392d 3230 2a78 5f34  1*x_3*x_9-20*x_4
│ │ │ │ +00027360: 2a78 5f39 2b34 352a 785f 352a 785f 392d  *x_9+45*x_5*x_9-
│ │ │ │ +00027370: 3134 2a78 5f36 2a78 5f39 2d32 352a 785f  14*x_6*x_9-25*x_
│ │ │ │ +00027380: 302a 785f 207c 0a7c 2d2d 2d2d 2d2d 2d2d  0*x_ |.|--------
│ │ │ │  00027390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000273b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000273d0: 0a7c 3130 2b34 352a 785f 312a 785f 3130  .|10+45*x_1*x_10
│ │ │ │ -000273e0: 2d34 312a 785f 322a 785f 3130 2d34 362a  -41*x_2*x_10-46*
│ │ │ │ -000273f0: 785f 342a 785f 3130 2b38 2a78 5f36 2a78  x_4*x_10+8*x_6*x
│ │ │ │ -00027400: 5f31 302d 3238 2a78 5f30 2a78 5f31 312b  _10-28*x_0*x_11+
│ │ │ │ -00027410: 3131 2a78 5f32 2a78 5f31 312b 3134 2a7c  11*x_2*x_11+14*|
│ │ │ │ -00027420: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000273c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000273d0: 2d2d 2d2d 2d7c 0a7c 3130 2b34 352a 785f  -----|.|10+45*x_
│ │ │ │ +000273e0: 312a 785f 3130 2d34 312a 785f 322a 785f  1*x_10-41*x_2*x_
│ │ │ │ +000273f0: 3130 2d34 362a 785f 342a 785f 3130 2b38  10-46*x_4*x_10+8
│ │ │ │ +00027400: 2a78 5f36 2a78 5f31 302d 3238 2a78 5f30  *x_6*x_10-28*x_0
│ │ │ │ +00027410: 2a78 5f31 312b 3131 2a78 5f32 2a78 5f31  *x_11+11*x_2*x_1
│ │ │ │ +00027420: 312b 3134 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  1+14*|.|--------
│ │ │ │  00027430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027470: 0a7c 785f 342a 785f 3131 2d38 2a78 5f35  .|x_4*x_11-8*x_5
│ │ │ │ -00027480: 2a78 5f31 3129 2c7b 745f 345e 322b 745f  *x_11),{t_4^2+t_
│ │ │ │ -00027490: 302a 745f 352b 745f 312a 745f 352b 3335  0*t_5+t_1*t_5+35
│ │ │ │ -000274a0: 2a74 5f32 2a74 5f35 2b31 302a 745f 332a  *t_2*t_5+10*t_3*
│ │ │ │ -000274b0: 745f 352b 3235 2a74 5f34 2a74 5f35 2d7c  t_5+25*t_4*t_5-|
│ │ │ │ -000274c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027470: 2d2d 2d2d 2d7c 0a7c 785f 342a 785f 3131  -----|.|x_4*x_11
│ │ │ │ +00027480: 2d38 2a78 5f35 2a78 5f31 3129 2c7b 745f  -8*x_5*x_11),{t_
│ │ │ │ +00027490: 345e 322b 745f 302a 745f 352b 745f 312a  4^2+t_0*t_5+t_1*
│ │ │ │ +000274a0: 745f 352b 3335 2a74 5f32 2a74 5f35 2b31  t_5+35*t_2*t_5+1
│ │ │ │ +000274b0: 302a 745f 332a 745f 352b 3235 2a74 5f34  0*t_3*t_5+25*t_4
│ │ │ │ +000274c0: 2a74 5f35 2d7c 0a7c 2d2d 2d2d 2d2d 2d2d  *t_5-|.|--------
│ │ │ │  000274d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000274e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000274f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027510: 0a7c 352a 745f 355e 322d 3134 2a74 5f30  .|5*t_5^2-14*t_0
│ │ │ │ -00027520: 2a74 5f36 2d31 342a 745f 312a 745f 362d  *t_6-14*t_1*t_6-
│ │ │ │ -00027530: 352a 745f 322a 745f 362d 3133 2a74 5f34  5*t_2*t_6-13*t_4
│ │ │ │ -00027540: 2a74 5f36 2b33 372a 745f 352a 745f 362b  *t_6+37*t_5*t_6+
│ │ │ │ -00027550: 3232 2a74 5f36 5e32 2d33 312a 745f 337c  22*t_6^2-31*t_3|
│ │ │ │ -00027560: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027510: 2d2d 2d2d 2d7c 0a7c 352a 745f 355e 322d  -----|.|5*t_5^2-
│ │ │ │ +00027520: 3134 2a74 5f30 2a74 5f36 2d31 342a 745f  14*t_0*t_6-14*t_
│ │ │ │ +00027530: 312a 745f 362d 352a 745f 322a 745f 362d  1*t_6-5*t_2*t_6-
│ │ │ │ +00027540: 3133 2a74 5f34 2a74 5f36 2b33 372a 745f  13*t_4*t_6+37*t_
│ │ │ │ +00027550: 352a 745f 362b 3232 2a74 5f36 5e32 2d33  5*t_6+22*t_6^2-3
│ │ │ │ +00027560: 312a 745f 337c 0a7c 2d2d 2d2d 2d2d 2d2d  1*t_3|.|--------
│ │ │ │  00027570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000275b0: 0a7c 2a74 5f37 2b32 362a 745f 342a 745f  .|*t_7+26*t_4*t_
│ │ │ │ -000275c0: 372b 3132 2a74 5f35 2a74 5f37 2d34 352a  7+12*t_5*t_7-45*
│ │ │ │ -000275d0: 745f 362a 745f 372d 3436 2a74 5f33 2a74  t_6*t_7-46*t_3*t
│ │ │ │ -000275e0: 5f38 2b33 372a 745f 342a 745f 382b 3238  _8+37*t_4*t_8+28
│ │ │ │ -000275f0: 2a74 5f35 2a74 5f38 2b33 332a 745f 367c  *t_5*t_8+33*t_6|
│ │ │ │ -00027600: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000275b0: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2b32 362a  -----|.|*t_7+26*
│ │ │ │ +000275c0: 745f 342a 745f 372b 3132 2a74 5f35 2a74  t_4*t_7+12*t_5*t
│ │ │ │ +000275d0: 5f37 2d34 352a 745f 362a 745f 372d 3436  _7-45*t_6*t_7-46
│ │ │ │ +000275e0: 2a74 5f33 2a74 5f38 2b33 372a 745f 342a  *t_3*t_8+37*t_4*
│ │ │ │ +000275f0: 745f 382b 3238 2a74 5f35 2a74 5f38 2b33  t_8+28*t_5*t_8+3
│ │ │ │ +00027600: 332a 745f 367c 0a7c 2d2d 2d2d 2d2d 2d2d  3*t_6|.|--------
│ │ │ │  00027610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027650: 0a7c 2a74 5f38 2c74 5f33 2a74 5f34 2b34  .|*t_8,t_3*t_4+4
│ │ │ │ -00027660: 2a74 5f30 2a74 5f35 2b33 392a 745f 312a  *t_0*t_5+39*t_1*
│ │ │ │ -00027670: 745f 352d 3430 2a74 5f32 2a74 5f35 2b34  t_5-40*t_2*t_5+4
│ │ │ │ -00027680: 302a 745f 332a 745f 352b 3236 2a74 5f34  0*t_3*t_5+26*t_4
│ │ │ │ -00027690: 2a74 5f35 2d32 302a 745f 355e 322b 207c  *t_5-20*t_5^2+ |
│ │ │ │ -000276a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027650: 2d2d 2d2d 2d7c 0a7c 2a74 5f38 2c74 5f33  -----|.|*t_8,t_3
│ │ │ │ +00027660: 2a74 5f34 2b34 2a74 5f30 2a74 5f35 2b33  *t_4+4*t_0*t_5+3
│ │ │ │ +00027670: 392a 745f 312a 745f 352d 3430 2a74 5f32  9*t_1*t_5-40*t_2
│ │ │ │ +00027680: 2a74 5f35 2b34 302a 745f 332a 745f 352b  *t_5+40*t_3*t_5+
│ │ │ │ +00027690: 3236 2a74 5f34 2a74 5f35 2d32 302a 745f  26*t_4*t_5-20*t_
│ │ │ │ +000276a0: 355e 322b 207c 0a7c 2d2d 2d2d 2d2d 2d2d  5^2+ |.|--------
│ │ │ │  000276b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000276c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000276e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000276f0: 0a7c 3431 2a74 5f30 2a74 5f36 2b33 362a  .|41*t_0*t_6+36*
│ │ │ │ -00027700: 745f 312a 745f 362d 3232 2a74 5f32 2a74  t_1*t_6-22*t_2*t
│ │ │ │ -00027710: 5f36 2b33 362a 745f 342a 745f 362d 3330  _6+36*t_4*t_6-30
│ │ │ │ -00027720: 2a74 5f35 2a74 5f36 2d31 332a 745f 365e  *t_5*t_6-13*t_6^
│ │ │ │ -00027730: 322d 3235 2a74 5f33 2a74 5f37 2b35 2a7c  2-25*t_3*t_7+5*|
│ │ │ │ -00027740: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000276e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000276f0: 2d2d 2d2d 2d7c 0a7c 3431 2a74 5f30 2a74  -----|.|41*t_0*t
│ │ │ │ +00027700: 5f36 2b33 362a 745f 312a 745f 362d 3232  _6+36*t_1*t_6-22
│ │ │ │ +00027710: 2a74 5f32 2a74 5f36 2b33 362a 745f 342a  *t_2*t_6+36*t_4*
│ │ │ │ +00027720: 745f 362d 3330 2a74 5f35 2a74 5f36 2d31  t_6-30*t_5*t_6-1
│ │ │ │ +00027730: 332a 745f 365e 322d 3235 2a74 5f33 2a74  3*t_6^2-25*t_3*t
│ │ │ │ +00027740: 5f37 2b35 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  _7+5*|.|--------
│ │ │ │  00027750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027790: 0a7c 745f 342a 745f 372d 3335 2a74 5f35  .|t_4*t_7-35*t_5
│ │ │ │ -000277a0: 2a74 5f37 2b31 302a 745f 362a 745f 372b  *t_7+10*t_6*t_7+
│ │ │ │ -000277b0: 3131 2a74 5f33 2a74 5f38 2b34 362a 745f  11*t_3*t_8+46*t_
│ │ │ │ -000277c0: 342a 745f 382b 3239 2a74 5f35 2a74 5f38  4*t_8+29*t_5*t_8
│ │ │ │ -000277d0: 2b32 382a 745f 362a 745f 382c 745f 327c  +28*t_6*t_8,t_2|
│ │ │ │ -000277e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027790: 2d2d 2d2d 2d7c 0a7c 745f 342a 745f 372d  -----|.|t_4*t_7-
│ │ │ │ +000277a0: 3335 2a74 5f35 2a74 5f37 2b31 302a 745f  35*t_5*t_7+10*t_
│ │ │ │ +000277b0: 362a 745f 372b 3131 2a74 5f33 2a74 5f38  6*t_7+11*t_3*t_8
│ │ │ │ +000277c0: 2b34 362a 745f 342a 745f 382b 3239 2a74  +46*t_4*t_8+29*t
│ │ │ │ +000277d0: 5f35 2a74 5f38 2b32 382a 745f 362a 745f  _5*t_8+28*t_6*t_
│ │ │ │ +000277e0: 382c 745f 327c 0a7c 2d2d 2d2d 2d2d 2d2d  8,t_2|.|--------
│ │ │ │  000277f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027830: 0a7c 2a74 5f34 2d35 2a74 5f30 2a74 5f35  .|*t_4-5*t_0*t_5
│ │ │ │ -00027840: 2d34 302a 745f 312a 745f 352b 3132 2a74  -40*t_1*t_5+12*t
│ │ │ │ -00027850: 5f32 2a74 5f35 2b34 372a 745f 332a 745f  _2*t_5+47*t_3*t_
│ │ │ │ -00027860: 352b 3337 2a74 5f34 2a74 5f35 2b32 352a  5+37*t_4*t_5+25*
│ │ │ │ -00027870: 745f 355e 322d 3237 2a74 5f30 2a74 5f7c  t_5^2-27*t_0*t_|
│ │ │ │ -00027880: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027830: 2d2d 2d2d 2d7c 0a7c 2a74 5f34 2d35 2a74  -----|.|*t_4-5*t
│ │ │ │ +00027840: 5f30 2a74 5f35 2d34 302a 745f 312a 745f  _0*t_5-40*t_1*t_
│ │ │ │ +00027850: 352b 3132 2a74 5f32 2a74 5f35 2b34 372a  5+12*t_2*t_5+47*
│ │ │ │ +00027860: 745f 332a 745f 352b 3337 2a74 5f34 2a74  t_3*t_5+37*t_4*t
│ │ │ │ +00027870: 5f35 2b32 352a 745f 355e 322d 3237 2a74  _5+25*t_5^2-27*t
│ │ │ │ +00027880: 5f30 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d  _0*t_|.|--------
│ │ │ │  00027890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000278a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000278b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000278c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000278d0: 0a7c 362d 3232 2a74 5f31 2a74 5f36 2b32  .|6-22*t_1*t_6+2
│ │ │ │ -000278e0: 372a 745f 322a 745f 362d 3233 2a74 5f34  7*t_2*t_6-23*t_4
│ │ │ │ -000278f0: 2a74 5f36 2b35 2a74 5f35 2a74 5f36 2d31  *t_6+5*t_5*t_6-1
│ │ │ │ -00027900: 332a 745f 365e 322d 3339 2a74 5f33 2a74  3*t_6^2-39*t_3*t
│ │ │ │ -00027910: 5f37 2d32 392a 745f 342a 745f 372b 397c  _7-29*t_4*t_7+9|
│ │ │ │ -00027920: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000278c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000278d0: 2d2d 2d2d 2d7c 0a7c 362d 3232 2a74 5f31  -----|.|6-22*t_1
│ │ │ │ +000278e0: 2a74 5f36 2b32 372a 745f 322a 745f 362d  *t_6+27*t_2*t_6-
│ │ │ │ +000278f0: 3233 2a74 5f34 2a74 5f36 2b35 2a74 5f35  23*t_4*t_6+5*t_5
│ │ │ │ +00027900: 2a74 5f36 2d31 332a 745f 365e 322d 3339  *t_6-13*t_6^2-39
│ │ │ │ +00027910: 2a74 5f33 2a74 5f37 2d32 392a 745f 342a  *t_3*t_7-29*t_4*
│ │ │ │ +00027920: 745f 372b 397c 0a7c 2d2d 2d2d 2d2d 2d2d  t_7+9|.|--------
│ │ │ │  00027930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027970: 0a7c 2a74 5f35 2a74 5f37 2b33 392a 745f  .|*t_5*t_7+39*t_
│ │ │ │ -00027980: 362a 745f 372b 3336 2a74 5f33 2a74 5f38  6*t_7+36*t_3*t_8
│ │ │ │ -00027990: 2b31 332a 745f 342a 745f 382b 3236 2a74  +13*t_4*t_8+26*t
│ │ │ │ -000279a0: 5f35 2a74 5f38 2b33 372a 745f 362a 745f  _5*t_8+37*t_6*t_
│ │ │ │ -000279b0: 382c 745f 302a 745f 342d 745f 302a 747c  8,t_0*t_4-t_0*t|
│ │ │ │ -000279c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027970: 2d2d 2d2d 2d7c 0a7c 2a74 5f35 2a74 5f37  -----|.|*t_5*t_7
│ │ │ │ +00027980: 2b33 392a 745f 362a 745f 372b 3336 2a74  +39*t_6*t_7+36*t
│ │ │ │ +00027990: 5f33 2a74 5f38 2b31 332a 745f 342a 745f  _3*t_8+13*t_4*t_
│ │ │ │ +000279a0: 382b 3236 2a74 5f35 2a74 5f38 2b33 372a  8+26*t_5*t_8+37*
│ │ │ │ +000279b0: 745f 362a 745f 382c 745f 302a 745f 342d  t_6*t_8,t_0*t_4-
│ │ │ │ +000279c0: 745f 302a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  t_0*t|.|--------
│ │ │ │  000279d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000279e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000279f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027a10: 0a7c 5f35 2d38 2a74 5f31 2a74 5f35 2d33  .|_5-8*t_1*t_5-3
│ │ │ │ -00027a20: 352a 745f 322a 745f 352d 3130 2a74 5f33  5*t_2*t_5-10*t_3
│ │ │ │ -00027a30: 2a74 5f35 2d33 332a 745f 342a 745f 352b  *t_5-33*t_4*t_5+
│ │ │ │ -00027a40: 352a 745f 355e 322b 3135 2a74 5f30 2a74  5*t_5^2+15*t_0*t
│ │ │ │ -00027a50: 5f36 2b31 352a 745f 312a 745f 362b 357c  _6+15*t_1*t_6+5|
│ │ │ │ -00027a60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027a10: 2d2d 2d2d 2d7c 0a7c 5f35 2d38 2a74 5f31  -----|.|_5-8*t_1
│ │ │ │ +00027a20: 2a74 5f35 2d33 352a 745f 322a 745f 352d  *t_5-35*t_2*t_5-
│ │ │ │ +00027a30: 3130 2a74 5f33 2a74 5f35 2d33 332a 745f  10*t_3*t_5-33*t_
│ │ │ │ +00027a40: 342a 745f 352b 352a 745f 355e 322b 3135  4*t_5+5*t_5^2+15
│ │ │ │ +00027a50: 2a74 5f30 2a74 5f36 2b31 352a 745f 312a  *t_0*t_6+15*t_1*
│ │ │ │ +00027a60: 745f 362b 357c 0a7c 2d2d 2d2d 2d2d 2d2d  t_6+5|.|--------
│ │ │ │  00027a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027ab0: 0a7c 2a74 5f32 2a74 5f36 2b31 352a 745f  .|*t_2*t_6+15*t_
│ │ │ │ -00027ac0: 342a 745f 362d 3338 2a74 5f35 2a74 5f36  4*t_6-38*t_5*t_6
│ │ │ │ -00027ad0: 2d32 322a 745f 365e 322b 3331 2a74 5f33  -22*t_6^2+31*t_3
│ │ │ │ -00027ae0: 2a74 5f37 2d32 352a 745f 342a 745f 372d  *t_7-25*t_4*t_7-
│ │ │ │ -00027af0: 3139 2a74 5f35 2a74 5f37 2b34 372a 747c  19*t_5*t_7+47*t|
│ │ │ │ -00027b00: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027ab0: 2d2d 2d2d 2d7c 0a7c 2a74 5f32 2a74 5f36  -----|.|*t_2*t_6
│ │ │ │ +00027ac0: 2b31 352a 745f 342a 745f 362d 3338 2a74  +15*t_4*t_6-38*t
│ │ │ │ +00027ad0: 5f35 2a74 5f36 2d32 322a 745f 365e 322b  _5*t_6-22*t_6^2+
│ │ │ │ +00027ae0: 3331 2a74 5f33 2a74 5f37 2d32 352a 745f  31*t_3*t_7-25*t_
│ │ │ │ +00027af0: 342a 745f 372d 3139 2a74 5f35 2a74 5f37  4*t_7-19*t_5*t_7
│ │ │ │ +00027b00: 2b34 372a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  +47*t|.|--------
│ │ │ │  00027b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027b50: 0a7c 5f36 2a74 5f37 2b34 362a 745f 332a  .|_6*t_7+46*t_3*
│ │ │ │ -00027b60: 745f 382d 3336 2a74 5f34 2a74 5f38 2d33  t_8-36*t_4*t_8-3
│ │ │ │ -00027b70: 352a 745f 352a 745f 382d 3331 2a74 5f36  5*t_5*t_8-31*t_6
│ │ │ │ -00027b80: 2a74 5f38 2c74 5f32 2a74 5f33 2d74 5f30  *t_8,t_2*t_3-t_0
│ │ │ │ -00027b90: 2a74 5f35 2d74 5f31 2a74 5f35 2d33 357c  *t_5-t_1*t_5-35|
│ │ │ │ -00027ba0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027b50: 2d2d 2d2d 2d7c 0a7c 5f36 2a74 5f37 2b34  -----|.|_6*t_7+4
│ │ │ │ +00027b60: 362a 745f 332a 745f 382d 3336 2a74 5f34  6*t_3*t_8-36*t_4
│ │ │ │ +00027b70: 2a74 5f38 2d33 352a 745f 352a 745f 382d  *t_8-35*t_5*t_8-
│ │ │ │ +00027b80: 3331 2a74 5f36 2a74 5f38 2c74 5f32 2a74  31*t_6*t_8,t_2*t
│ │ │ │ +00027b90: 5f33 2d74 5f30 2a74 5f35 2d74 5f31 2a74  _3-t_0*t_5-t_1*t
│ │ │ │ +00027ba0: 5f35 2d33 357c 0a7c 2d2d 2d2d 2d2d 2d2d  _5-35|.|--------
│ │ │ │  00027bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027bf0: 0a7c 2a74 5f32 2a74 5f35 2d31 302a 745f  .|*t_2*t_5-10*t_
│ │ │ │ -00027c00: 332a 745f 352d 3333 2a74 5f34 2a74 5f35  3*t_5-33*t_4*t_5
│ │ │ │ -00027c10: 2b35 2a74 5f35 5e32 2b31 342a 745f 302a  +5*t_5^2+14*t_0*
│ │ │ │ -00027c20: 745f 362b 3134 2a74 5f31 2a74 5f36 2b35  t_6+14*t_1*t_6+5
│ │ │ │ -00027c30: 2a74 5f32 2a74 5f36 2b31 342a 745f 347c  *t_2*t_6+14*t_4|
│ │ │ │ -00027c40: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027bf0: 2d2d 2d2d 2d7c 0a7c 2a74 5f32 2a74 5f35  -----|.|*t_2*t_5
│ │ │ │ +00027c00: 2d31 302a 745f 332a 745f 352d 3333 2a74  -10*t_3*t_5-33*t
│ │ │ │ +00027c10: 5f34 2a74 5f35 2b35 2a74 5f35 5e32 2b31  _4*t_5+5*t_5^2+1
│ │ │ │ +00027c20: 342a 745f 302a 745f 362b 3134 2a74 5f31  4*t_0*t_6+14*t_1
│ │ │ │ +00027c30: 2a74 5f36 2b35 2a74 5f32 2a74 5f36 2b31  *t_6+5*t_2*t_6+1
│ │ │ │ +00027c40: 342a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  4*t_4|.|--------
│ │ │ │  00027c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027c90: 0a7c 2a74 5f36 2d33 312a 745f 352a 745f  .|*t_6-31*t_5*t_
│ │ │ │ -00027ca0: 362d 3234 2a74 5f36 5e32 2b33 322a 745f  6-24*t_6^2+32*t_
│ │ │ │ -00027cb0: 332a 745f 372d 3235 2a74 5f34 2a74 5f37  3*t_7-25*t_4*t_7
│ │ │ │ -00027cc0: 2d31 392a 745f 352a 745f 372b 3437 2a74  -19*t_5*t_7+47*t
│ │ │ │ -00027cd0: 5f36 2a74 5f37 2b34 362a 745f 332a 747c  _6*t_7+46*t_3*t|
│ │ │ │ -00027ce0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027c90: 2d2d 2d2d 2d7c 0a7c 2a74 5f36 2d33 312a  -----|.|*t_6-31*
│ │ │ │ +00027ca0: 745f 352a 745f 362d 3234 2a74 5f36 5e32  t_5*t_6-24*t_6^2
│ │ │ │ +00027cb0: 2b33 322a 745f 332a 745f 372d 3235 2a74  +32*t_3*t_7-25*t
│ │ │ │ +00027cc0: 5f34 2a74 5f37 2d31 392a 745f 352a 745f  _4*t_7-19*t_5*t_
│ │ │ │ +00027cd0: 372b 3437 2a74 5f36 2a74 5f37 2b34 362a  7+47*t_6*t_7+46*
│ │ │ │ +00027ce0: 745f 332a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  t_3*t|.|--------
│ │ │ │  00027cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027d30: 0a7c 5f38 2d33 362a 745f 342a 745f 382d  .|_8-36*t_4*t_8-
│ │ │ │ -00027d40: 3335 2a74 5f35 2a74 5f38 2d33 312a 745f  35*t_5*t_8-31*t_
│ │ │ │ -00027d50: 362a 745f 382c 745f 312a 745f 332d 372a  6*t_8,t_1*t_3-7*
│ │ │ │ -00027d60: 745f 312a 745f 352b 745f 312a 745f 362b  t_1*t_5+t_1*t_6+
│ │ │ │ -00027d70: 745f 342a 745f 362d 372a 745f 352a 747c  t_4*t_6-7*t_5*t|
│ │ │ │ -00027d80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027d30: 2d2d 2d2d 2d7c 0a7c 5f38 2d33 362a 745f  -----|.|_8-36*t_
│ │ │ │ +00027d40: 342a 745f 382d 3335 2a74 5f35 2a74 5f38  4*t_8-35*t_5*t_8
│ │ │ │ +00027d50: 2d33 312a 745f 362a 745f 382c 745f 312a  -31*t_6*t_8,t_1*
│ │ │ │ +00027d60: 745f 332d 372a 745f 312a 745f 352b 745f  t_3-7*t_1*t_5+t_
│ │ │ │ +00027d70: 312a 745f 362b 745f 342a 745f 362d 372a  1*t_6+t_4*t_6-7*
│ │ │ │ +00027d80: 745f 352a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  t_5*t|.|--------
│ │ │ │  00027d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027dd0: 0a7c 5f36 2b32 2a74 5f36 5e32 2d74 5f33  .|_6+2*t_6^2-t_3
│ │ │ │ -00027de0: 2a74 5f37 2c74 5f30 2a74 5f33 2d34 362a  *t_7,t_0*t_3-46*
│ │ │ │ -00027df0: 745f 302a 745f 352d 3339 2a74 5f31 2a74  t_0*t_5-39*t_1*t
│ │ │ │ -00027e00: 5f35 2d34 332a 745f 322a 745f 352d 3431  _5-43*t_2*t_5-41
│ │ │ │ -00027e10: 2a74 5f33 2a74 5f35 2d32 362a 745f 347c  *t_3*t_5-26*t_4|
│ │ │ │ -00027e20: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027dd0: 2d2d 2d2d 2d7c 0a7c 5f36 2b32 2a74 5f36  -----|.|_6+2*t_6
│ │ │ │ +00027de0: 5e32 2d74 5f33 2a74 5f37 2c74 5f30 2a74  ^2-t_3*t_7,t_0*t
│ │ │ │ +00027df0: 5f33 2d34 362a 745f 302a 745f 352d 3339  _3-46*t_0*t_5-39
│ │ │ │ +00027e00: 2a74 5f31 2a74 5f35 2d34 332a 745f 322a  *t_1*t_5-43*t_2*
│ │ │ │ +00027e10: 745f 352d 3431 2a74 5f33 2a74 5f35 2d32  t_5-41*t_3*t_5-2
│ │ │ │ +00027e20: 362a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  6*t_4|.|--------
│ │ │ │  00027e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027e70: 0a7c 2a74 5f35 2d32 382a 745f 355e 322d  .|*t_5-28*t_5^2-
│ │ │ │ -00027e80: 3335 2a74 5f30 2a74 5f36 2d33 362a 745f  35*t_0*t_6-36*t_
│ │ │ │ -00027e90: 312a 745f 362b 3230 2a74 5f32 2a74 5f36  1*t_6+20*t_2*t_6
│ │ │ │ -00027ea0: 2d33 362a 745f 342a 745f 362b 392a 745f  -36*t_4*t_6+9*t_
│ │ │ │ -00027eb0: 352a 745f 362b 3135 2a74 5f36 5e32 2b7c  5*t_6+15*t_6^2+|
│ │ │ │ -00027ec0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027e70: 2d2d 2d2d 2d7c 0a7c 2a74 5f35 2d32 382a  -----|.|*t_5-28*
│ │ │ │ +00027e80: 745f 355e 322d 3335 2a74 5f30 2a74 5f36  t_5^2-35*t_0*t_6
│ │ │ │ +00027e90: 2d33 362a 745f 312a 745f 362b 3230 2a74  -36*t_1*t_6+20*t
│ │ │ │ +00027ea0: 5f32 2a74 5f36 2d33 362a 745f 342a 745f  _2*t_6-36*t_4*t_
│ │ │ │ +00027eb0: 362b 392a 745f 352a 745f 362b 3135 2a74  6+9*t_5*t_6+15*t
│ │ │ │ +00027ec0: 5f36 5e32 2b7c 0a7c 2d2d 2d2d 2d2d 2d2d  _6^2+|.|--------
│ │ │ │  00027ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027f10: 0a7c 3236 2a74 5f33 2a74 5f37 2d35 2a74  .|26*t_3*t_7-5*t
│ │ │ │ -00027f20: 5f34 2a74 5f37 2b33 352a 745f 352a 745f  _4*t_7+35*t_5*t_
│ │ │ │ -00027f30: 372d 3130 2a74 5f36 2a74 5f37 2d31 302a  7-10*t_6*t_7-10*
│ │ │ │ -00027f40: 745f 332a 745f 382d 3436 2a74 5f34 2a74  t_3*t_8-46*t_4*t
│ │ │ │ -00027f50: 5f38 2b34 372a 745f 352a 745f 382d 207c  _8+47*t_5*t_8- |
│ │ │ │ -00027f60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027f10: 2d2d 2d2d 2d7c 0a7c 3236 2a74 5f33 2a74  -----|.|26*t_3*t
│ │ │ │ +00027f20: 5f37 2d35 2a74 5f34 2a74 5f37 2b33 352a  _7-5*t_4*t_7+35*
│ │ │ │ +00027f30: 745f 352a 745f 372d 3130 2a74 5f36 2a74  t_5*t_7-10*t_6*t
│ │ │ │ +00027f40: 5f37 2d31 302a 745f 332a 745f 382d 3436  _7-10*t_3*t_8-46
│ │ │ │ +00027f50: 2a74 5f34 2a74 5f38 2b34 372a 745f 352a  *t_4*t_8+47*t_5*
│ │ │ │ +00027f60: 745f 382d 207c 0a7c 2d2d 2d2d 2d2d 2d2d  t_8- |.|--------
│ │ │ │  00027f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00027f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00027fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00027fb0: 0a7c 3235 2a74 5f36 2a74 5f38 2c74 5f32  .|25*t_6*t_8,t_2
│ │ │ │ -00027fc0: 5e32 2d34 362a 745f 312a 745f 342d 3333  ^2-46*t_1*t_4-33
│ │ │ │ -00027fd0: 2a74 5f30 2a74 5f35 2d34 352a 745f 312a  *t_0*t_5-45*t_1*
│ │ │ │ -00027fe0: 745f 352d 3339 2a74 5f32 2a74 5f35 2d33  t_5-39*t_2*t_5-3
│ │ │ │ -00027ff0: 392a 745f 332a 745f 352d 3436 2a74 5f7c  9*t_3*t_5-46*t_|
│ │ │ │ -00028000: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00027fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00027fb0: 2d2d 2d2d 2d7c 0a7c 3235 2a74 5f36 2a74  -----|.|25*t_6*t
│ │ │ │ +00027fc0: 5f38 2c74 5f32 5e32 2d34 362a 745f 312a  _8,t_2^2-46*t_1*
│ │ │ │ +00027fd0: 745f 342d 3333 2a74 5f30 2a74 5f35 2d34  t_4-33*t_0*t_5-4
│ │ │ │ +00027fe0: 352a 745f 312a 745f 352d 3339 2a74 5f32  5*t_1*t_5-39*t_2
│ │ │ │ +00027ff0: 2a74 5f35 2d33 392a 745f 332a 745f 352d  *t_5-39*t_3*t_5-
│ │ │ │ +00028000: 3436 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d  46*t_|.|--------
│ │ │ │  00028010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028050: 0a7c 342a 745f 352d 3239 2a74 5f35 5e32  .|4*t_5-29*t_5^2
│ │ │ │ -00028060: 2d34 382a 745f 302a 745f 362d 3338 2a74  -48*t_0*t_6-38*t
│ │ │ │ -00028070: 5f31 2a74 5f36 2d33 302a 745f 322a 745f  _1*t_6-30*t_2*t_
│ │ │ │ -00028080: 362b 3139 2a74 5f34 2a74 5f36 2d34 342a  6+19*t_4*t_6-44*
│ │ │ │ -00028090: 745f 352a 745f 362d 3437 2a74 5f36 5e7c  t_5*t_6-47*t_6^|
│ │ │ │ -000280a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00028040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028050: 2d2d 2d2d 2d7c 0a7c 342a 745f 352d 3239  -----|.|4*t_5-29
│ │ │ │ +00028060: 2a74 5f35 5e32 2d34 382a 745f 302a 745f  *t_5^2-48*t_0*t_
│ │ │ │ +00028070: 362d 3338 2a74 5f31 2a74 5f36 2d33 302a  6-38*t_1*t_6-30*
│ │ │ │ +00028080: 745f 322a 745f 362b 3139 2a74 5f34 2a74  t_2*t_6+19*t_4*t
│ │ │ │ +00028090: 5f36 2d34 342a 745f 352a 745f 362d 3437  _6-44*t_5*t_6-47
│ │ │ │ +000280a0: 2a74 5f36 5e7c 0a7c 2d2d 2d2d 2d2d 2d2d  *t_6^|.|--------
│ │ │ │  000280b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000280d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000280e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000280f0: 0a7c 322d 3336 2a74 5f30 2a74 5f37 2d34  .|2-36*t_0*t_7-4
│ │ │ │ -00028100: 362a 745f 312a 745f 372b 745f 322a 745f  6*t_1*t_7+t_2*t_
│ │ │ │ -00028110: 372d 3434 2a74 5f33 2a74 5f37 2b34 382a  7-44*t_3*t_7+48*
│ │ │ │ -00028120: 745f 342a 745f 372d 3134 2a74 5f35 2a74  t_4*t_7-14*t_5*t
│ │ │ │ -00028130: 5f37 2b34 2a74 5f36 2a74 5f37 2d33 367c  _7+4*t_6*t_7-36|
│ │ │ │ -00028140: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000280e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000280f0: 2d2d 2d2d 2d7c 0a7c 322d 3336 2a74 5f30  -----|.|2-36*t_0
│ │ │ │ +00028100: 2a74 5f37 2d34 362a 745f 312a 745f 372b  *t_7-46*t_1*t_7+
│ │ │ │ +00028110: 745f 322a 745f 372d 3434 2a74 5f33 2a74  t_2*t_7-44*t_3*t
│ │ │ │ +00028120: 5f37 2b34 382a 745f 342a 745f 372d 3134  _7+48*t_4*t_7-14
│ │ │ │ +00028130: 2a74 5f35 2a74 5f37 2b34 2a74 5f36 2a74  *t_5*t_7+4*t_6*t
│ │ │ │ +00028140: 5f37 2d33 367c 0a7c 2d2d 2d2d 2d2d 2d2d  _7-36|.|--------
│ │ │ │  00028150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028190: 0a7c 2a74 5f30 2a74 5f38 2d34 362a 745f  .|*t_0*t_8-46*t_
│ │ │ │ -000281a0: 312a 745f 382b 3437 2a74 5f32 2a74 5f38  1*t_8+47*t_2*t_8
│ │ │ │ -000281b0: 2d33 342a 745f 332a 745f 382d 3234 2a74  -34*t_3*t_8-24*t
│ │ │ │ -000281c0: 5f34 2a74 5f38 2d31 322a 745f 352a 745f  _4*t_8-12*t_5*t_
│ │ │ │ -000281d0: 382d 3437 2a74 5f36 2a74 5f38 2b34 377c  8-47*t_6*t_8+47|
│ │ │ │ -000281e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00028180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028190: 2d2d 2d2d 2d7c 0a7c 2a74 5f30 2a74 5f38  -----|.|*t_0*t_8
│ │ │ │ +000281a0: 2d34 362a 745f 312a 745f 382b 3437 2a74  -46*t_1*t_8+47*t
│ │ │ │ +000281b0: 5f32 2a74 5f38 2d33 342a 745f 332a 745f  _2*t_8-34*t_3*t_
│ │ │ │ +000281c0: 382d 3234 2a74 5f34 2a74 5f38 2d31 322a  8-24*t_4*t_8-12*
│ │ │ │ +000281d0: 745f 352a 745f 382d 3437 2a74 5f36 2a74  t_5*t_8-47*t_6*t
│ │ │ │ +000281e0: 5f38 2b34 377c 0a7c 2d2d 2d2d 2d2d 2d2d  _8+47|.|--------
│ │ │ │  000281f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028230: 0a7c 2a74 5f37 2a74 5f38 2c74 5f31 2a74  .|*t_7*t_8,t_1*t
│ │ │ │ -00028240: 5f32 2b36 2a74 5f31 2a74 5f35 2b35 2a74  _2+6*t_1*t_5+5*t
│ │ │ │ -00028250: 5f30 2a74 5f36 2d32 2a74 5f31 2a74 5f36  _0*t_6-2*t_1*t_6
│ │ │ │ -00028260: 2d74 5f34 2a74 5f36 2d74 5f35 2a74 5f36  -t_4*t_6-t_5*t_6
│ │ │ │ -00028270: 2b35 2a74 5f30 2a74 5f37 2b74 5f31 2a7c  +5*t_0*t_7+t_1*|
│ │ │ │ -00028280: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00028220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028230: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2a74 5f38  -----|.|*t_7*t_8
│ │ │ │ +00028240: 2c74 5f31 2a74 5f32 2b36 2a74 5f31 2a74  ,t_1*t_2+6*t_1*t
│ │ │ │ +00028250: 5f35 2b35 2a74 5f30 2a74 5f36 2d32 2a74  _5+5*t_0*t_6-2*t
│ │ │ │ +00028260: 5f31 2a74 5f36 2d74 5f34 2a74 5f36 2d74  _1*t_6-t_4*t_6-t
│ │ │ │ +00028270: 5f35 2a74 5f36 2b35 2a74 5f30 2a74 5f37  _5*t_6+5*t_0*t_7
│ │ │ │ +00028280: 2b74 5f31 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  +t_1*|.|--------
│ │ │ │  00028290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000282a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000282b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000282d0: 0a7c 745f 372d 322a 745f 322a 745f 372d  .|t_7-2*t_2*t_7-
│ │ │ │ -000282e0: 372a 745f 352a 745f 372b 322a 745f 362a  7*t_5*t_7+2*t_6*
│ │ │ │ -000282f0: 745f 372d 322a 745f 312a 745f 382b 332a  t_7-2*t_1*t_8+3*
│ │ │ │ -00028300: 745f 372a 745f 382c 745f 302a 745f 322b  t_7*t_8,t_0*t_2+
│ │ │ │ -00028310: 745f 312a 745f 342b 352a 745f 302a 747c  t_1*t_4+5*t_0*t|
│ │ │ │ -00028320: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000282c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000282d0: 2d2d 2d2d 2d7c 0a7c 745f 372d 322a 745f  -----|.|t_7-2*t_
│ │ │ │ +000282e0: 322a 745f 372d 372a 745f 352a 745f 372b  2*t_7-7*t_5*t_7+
│ │ │ │ +000282f0: 322a 745f 362a 745f 372d 322a 745f 312a  2*t_6*t_7-2*t_1*
│ │ │ │ +00028300: 745f 382b 332a 745f 372a 745f 382c 745f  t_8+3*t_7*t_8,t_
│ │ │ │ +00028310: 302a 745f 322b 745f 312a 745f 342b 352a  0*t_2+t_1*t_4+5*
│ │ │ │ +00028320: 745f 302a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  t_0*t|.|--------
│ │ │ │  00028330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028370: 0a7c 5f35 2b33 322a 745f 312a 745f 352d  .|_5+32*t_1*t_5-
│ │ │ │ -00028380: 3230 2a74 5f32 2a74 5f35 2d34 372a 745f  20*t_2*t_5-47*t_
│ │ │ │ -00028390: 332a 745f 352d 3337 2a74 5f34 2a74 5f35  3*t_5-37*t_4*t_5
│ │ │ │ -000283a0: 2d32 352a 745f 355e 322b 3139 2a74 5f30  -25*t_5^2+19*t_0
│ │ │ │ -000283b0: 2a74 5f36 2b32 322a 745f 312a 745f 367c  *t_6+22*t_1*t_6|
│ │ │ │ -000283c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00028360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028370: 2d2d 2d2d 2d7c 0a7c 5f35 2b33 322a 745f  -----|.|_5+32*t_
│ │ │ │ +00028380: 312a 745f 352d 3230 2a74 5f32 2a74 5f35  1*t_5-20*t_2*t_5
│ │ │ │ +00028390: 2d34 372a 745f 332a 745f 352d 3337 2a74  -47*t_3*t_5-37*t
│ │ │ │ +000283a0: 5f34 2a74 5f35 2d32 352a 745f 355e 322b  _4*t_5-25*t_5^2+
│ │ │ │ +000283b0: 3139 2a74 5f30 2a74 5f36 2b32 322a 745f  19*t_0*t_6+22*t_
│ │ │ │ +000283c0: 312a 745f 367c 0a7c 2d2d 2d2d 2d2d 2d2d  1*t_6|.|--------
│ │ │ │  000283d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000283e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000283f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028410: 0a7c 2d32 352a 745f 322a 745f 362b 3235  .|-25*t_2*t_6+25
│ │ │ │ -00028420: 2a74 5f34 2a74 5f36 2d35 2a74 5f35 2a74  *t_4*t_6-5*t_5*t
│ │ │ │ -00028430: 5f36 2b31 332a 745f 365e 322b 352a 745f  _6+13*t_6^2+5*t_
│ │ │ │ -00028440: 302a 745f 372b 745f 312a 745f 372b 3339  0*t_7+t_1*t_7+39
│ │ │ │ -00028450: 2a74 5f33 2a74 5f37 2b32 382a 745f 347c  *t_3*t_7+28*t_4|
│ │ │ │ -00028460: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00028400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028410: 2d2d 2d2d 2d7c 0a7c 2d32 352a 745f 322a  -----|.|-25*t_2*
│ │ │ │ +00028420: 745f 362b 3235 2a74 5f34 2a74 5f36 2d35  t_6+25*t_4*t_6-5
│ │ │ │ +00028430: 2a74 5f35 2a74 5f36 2b31 332a 745f 365e  *t_5*t_6+13*t_6^
│ │ │ │ +00028440: 322b 352a 745f 302a 745f 372b 745f 312a  2+5*t_0*t_7+t_1*
│ │ │ │ +00028450: 745f 372b 3339 2a74 5f33 2a74 5f37 2b32  t_7+39*t_3*t_7+2
│ │ │ │ +00028460: 382a 745f 347c 0a7c 2d2d 2d2d 2d2d 2d2d  8*t_4|.|--------
│ │ │ │  00028470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000284a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000284b0: 0a7c 2a74 5f37 2d39 2a74 5f35 2a74 5f37  .|*t_7-9*t_5*t_7
│ │ │ │ -000284c0: 2d33 392a 745f 362a 745f 372b 342a 745f  -39*t_6*t_7+4*t_
│ │ │ │ -000284d0: 302a 745f 382b 745f 312a 745f 382d 3336  0*t_8+t_1*t_8-36
│ │ │ │ -000284e0: 2a74 5f33 2a74 5f38 2d31 342a 745f 342a  *t_3*t_8-14*t_4*
│ │ │ │ -000284f0: 745f 382d 3236 2a74 5f35 2a74 5f38 2d7c  t_8-26*t_5*t_8-|
│ │ │ │ -00028500: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000284a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000284b0: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2d39 2a74  -----|.|*t_7-9*t
│ │ │ │ +000284c0: 5f35 2a74 5f37 2d33 392a 745f 362a 745f  _5*t_7-39*t_6*t_
│ │ │ │ +000284d0: 372b 342a 745f 302a 745f 382b 745f 312a  7+4*t_0*t_8+t_1*
│ │ │ │ +000284e0: 745f 382d 3336 2a74 5f33 2a74 5f38 2d31  t_8-36*t_3*t_8-1
│ │ │ │ +000284f0: 342a 745f 342a 745f 382d 3236 2a74 5f35  4*t_4*t_8-26*t_5
│ │ │ │ +00028500: 2a74 5f38 2d7c 0a7c 2d2d 2d2d 2d2d 2d2d  *t_8-|.|--------
│ │ │ │  00028510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028550: 0a7c 3337 2a74 5f36 2a74 5f38 2c74 5f30  .|37*t_6*t_8,t_0
│ │ │ │ -00028560: 2a74 5f31 2d33 392a 745f 312a 745f 342b  *t_1-39*t_1*t_4+
│ │ │ │ -00028570: 3430 2a74 5f31 2a74 5f35 2d33 372a 745f  40*t_1*t_5-37*t_
│ │ │ │ -00028580: 302a 745f 362d 3339 2a74 5f31 2a74 5f36  0*t_6-39*t_1*t_6
│ │ │ │ -00028590: 2b31 392a 745f 342a 745f 362d 3339 2a7c  +19*t_4*t_6-39*|
│ │ │ │ -000285a0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00028540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028550: 2d2d 2d2d 2d7c 0a7c 3337 2a74 5f36 2a74  -----|.|37*t_6*t
│ │ │ │ +00028560: 5f38 2c74 5f30 2a74 5f31 2d33 392a 745f  _8,t_0*t_1-39*t_
│ │ │ │ +00028570: 312a 745f 342b 3430 2a74 5f31 2a74 5f35  1*t_4+40*t_1*t_5
│ │ │ │ +00028580: 2d33 372a 745f 302a 745f 362d 3339 2a74  -37*t_0*t_6-39*t
│ │ │ │ +00028590: 5f31 2a74 5f36 2b31 392a 745f 342a 745f  _1*t_6+19*t_4*t_
│ │ │ │ +000285a0: 362d 3339 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  6-39*|.|--------
│ │ │ │  000285b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000285c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000285d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000285e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000285f0: 0a7c 745f 352a 745f 362d 3338 2a74 5f30  .|t_5*t_6-38*t_0
│ │ │ │ -00028600: 2a74 5f37 2b33 392a 745f 312a 745f 372b  *t_7+39*t_1*t_7+
│ │ │ │ -00028610: 3139 2a74 5f32 2a74 5f37 2b31 382a 745f  19*t_2*t_7+18*t_
│ │ │ │ -00028620: 352a 745f 372d 3139 2a74 5f36 2a74 5f37  5*t_7-19*t_6*t_7
│ │ │ │ -00028630: 2b31 392a 745f 312a 745f 382b 3230 2a7c  +19*t_1*t_8+20*|
│ │ │ │ -00028640: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000285e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000285f0: 2d2d 2d2d 2d7c 0a7c 745f 352a 745f 362d  -----|.|t_5*t_6-
│ │ │ │ +00028600: 3338 2a74 5f30 2a74 5f37 2b33 392a 745f  38*t_0*t_7+39*t_
│ │ │ │ +00028610: 312a 745f 372b 3139 2a74 5f32 2a74 5f37  1*t_7+19*t_2*t_7
│ │ │ │ +00028620: 2b31 382a 745f 352a 745f 372d 3139 2a74  +18*t_5*t_7-19*t
│ │ │ │ +00028630: 5f36 2a74 5f37 2b31 392a 745f 312a 745f  _6*t_7+19*t_1*t_
│ │ │ │ +00028640: 382b 3230 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d  8+20*|.|--------
│ │ │ │  00028650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028690: 0a7c 745f 372a 745f 382c 745f 305e 322b  .|t_7*t_8,t_0^2+
│ │ │ │ -000286a0: 3132 2a74 5f31 2a74 5f34 2b32 302a 745f  12*t_1*t_4+20*t_
│ │ │ │ -000286b0: 302a 745f 352b 3237 2a74 5f31 2a74 5f35  0*t_5+27*t_1*t_5
│ │ │ │ -000286c0: 2d38 2a74 5f32 2a74 5f35 2b33 372a 745f  -8*t_2*t_5+37*t_
│ │ │ │ -000286d0: 332a 745f 352b 3238 2a74 5f34 2a74 5f7c  3*t_5+28*t_4*t_|
│ │ │ │ -000286e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00028680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028690: 2d2d 2d2d 2d7c 0a7c 745f 372a 745f 382c  -----|.|t_7*t_8,
│ │ │ │ +000286a0: 745f 305e 322b 3132 2a74 5f31 2a74 5f34  t_0^2+12*t_1*t_4
│ │ │ │ +000286b0: 2b32 302a 745f 302a 745f 352b 3237 2a74  +20*t_0*t_5+27*t
│ │ │ │ +000286c0: 5f31 2a74 5f35 2d38 2a74 5f32 2a74 5f35  _1*t_5-8*t_2*t_5
│ │ │ │ +000286d0: 2b33 372a 745f 332a 745f 352b 3238 2a74  +37*t_3*t_5+28*t
│ │ │ │ +000286e0: 5f34 2a74 5f7c 0a7c 2d2d 2d2d 2d2d 2d2d  _4*t_|.|--------
│ │ │ │  000286f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028730: 0a7c 352b 3330 2a74 5f35 5e32 2d34 362a  .|5+30*t_5^2-46*
│ │ │ │ -00028740: 745f 302a 745f 362b 3234 2a74 5f31 2a74  t_0*t_6+24*t_1*t
│ │ │ │ -00028750: 5f36 2d34 302a 745f 322a 745f 362b 3235  _6-40*t_2*t_6+25
│ │ │ │ -00028760: 2a74 5f34 2a74 5f36 2b31 362a 745f 352a  *t_4*t_6+16*t_5*
│ │ │ │ -00028770: 745f 362d 3335 2a74 5f36 5e32 2b32 397c  t_6-35*t_6^2+29|
│ │ │ │ -00028780: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028730: 2d2d 2d2d 2d7c 0a7c 352b 3330 2a74 5f35  -----|.|5+30*t_5
│ │ │ │ +00028740: 5e32 2d34 362a 745f 302a 745f 362b 3234  ^2-46*t_0*t_6+24
│ │ │ │ +00028750: 2a74 5f31 2a74 5f36 2d34 302a 745f 322a  *t_1*t_6-40*t_2*
│ │ │ │ +00028760: 745f 362b 3235 2a74 5f34 2a74 5f36 2b31  t_6+25*t_4*t_6+1
│ │ │ │ +00028770: 362a 745f 352a 745f 362d 3335 2a74 5f36  6*t_5*t_6-35*t_6
│ │ │ │ +00028780: 5e32 2b32 397c 0a7c 2d2d 2d2d 2d2d 2d2d  ^2+29|.|--------
│ │ │ │  00028790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000287a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000287b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000287c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -000287d0: 0a7c 2a74 5f30 2a74 5f37 2b31 322a 745f  .|*t_0*t_7+12*t_
│ │ │ │ -000287e0: 312a 745f 372d 3335 2a74 5f32 2a74 5f37  1*t_7-35*t_2*t_7
│ │ │ │ -000287f0: 2d38 2a74 5f33 2a74 5f37 2d31 382a 745f  -8*t_3*t_7-18*t_
│ │ │ │ -00028800: 342a 745f 372b 3432 2a74 5f35 2a74 5f37  4*t_7+42*t_5*t_7
│ │ │ │ -00028810: 2d31 322a 745f 362a 745f 372d 362a 747c  -12*t_6*t_7-6*t|
│ │ │ │ -00028820: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +000287c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000287d0: 2d2d 2d2d 2d7c 0a7c 2a74 5f30 2a74 5f37  -----|.|*t_0*t_7
│ │ │ │ +000287e0: 2b31 322a 745f 312a 745f 372d 3335 2a74  +12*t_1*t_7-35*t
│ │ │ │ +000287f0: 5f32 2a74 5f37 2d38 2a74 5f33 2a74 5f37  _2*t_7-8*t_3*t_7
│ │ │ │ +00028800: 2d31 382a 745f 342a 745f 372b 3432 2a74  -18*t_4*t_7+42*t
│ │ │ │ +00028810: 5f35 2a74 5f37 2d31 322a 745f 362a 745f  _5*t_7-12*t_6*t_
│ │ │ │ +00028820: 372d 362a 747c 0a7c 2d2d 2d2d 2d2d 2d2d  7-6*t|.|--------
│ │ │ │  00028830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00028850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00028870: 0a7c 5f30 2a74 5f38 2b31 322a 745f 312a  .|_0*t_8+12*t_1*
│ │ │ │ -00028880: 745f 382d 3135 2a74 5f33 2a74 5f38 2b39  t_8-15*t_3*t_8+9
│ │ │ │ -00028890: 2a74 5f34 2a74 5f38 2b32 302a 745f 352a  *t_4*t_8+20*t_5*
│ │ │ │ -000288a0: 745f 382d 3330 2a74 5f36 2a74 5f38 2b34  t_8-30*t_6*t_8+4
│ │ │ │ -000288b0: 2a74 5f37 2a74 5f38 7d29 2020 2020 207c  *t_7*t_8})     |
│ │ │ │ -000288c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00028860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028870: 2d2d 2d2d 2d7c 0a7c 5f30 2a74 5f38 2b31  -----|.|_0*t_8+1
│ │ │ │ +00028880: 322a 745f 312a 745f 382d 3135 2a74 5f33  2*t_1*t_8-15*t_3
│ │ │ │ +00028890: 2a74 5f38 2b39 2a74 5f34 2a74 5f38 2b32  *t_8+9*t_4*t_8+2
│ │ │ │ +000288a0: 302a 745f 352a 745f 382d 3330 2a74 5f36  0*t_5*t_8-30*t_6
│ │ │ │ +000288b0: 2a74 5f38 2b34 2a74 5f37 2a74 5f38 7d29  *t_8+4*t_7*t_8})
│ │ │ │ +000288c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000288d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000288e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000288f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00028910: 0a7c 6938 203a 202d 2d20 7769 7468 6f75  .|i8 : -- withou
│ │ │ │ -00028920: 7420 7468 6520 6f70 7469 6f6e 2027 436f  t the option 'Co
│ │ │ │ -00028930: 6469 6d42 7349 6e76 3d3e 3427 2c20 6974  dimBsInv=>4', it
│ │ │ │ -00028940: 2074 616b 6573 2061 626f 7574 2020 2020   takes about    
│ │ │ │ -00028950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028960: 0a7c 2020 2020 2074 696d 6520 7073 693d  .|     time psi=
│ │ │ │ -00028970: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ -00028980: 7365 4d61 7028 7068 692c 436f 6469 6d42  seMap(phi,CodimB
│ │ │ │ -00028990: 7349 6e76 3d3e 3429 2020 2020 2020 2020  sInv=>4)        
│ │ │ │ -000289a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000289b0: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ -000289c0: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ -000289d0: 2031 206f 6620 3320 2020 2020 2020 2020   1 of 3         
│ │ │ │ +00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00028910: 2d2d 2d2d 2d2b 0a7c 6938 203a 202d 2d20  -----+.|i8 : -- 
│ │ │ │ +00028920: 7769 7468 6f75 7420 7468 6520 6f70 7469  without the opti
│ │ │ │ +00028930: 6f6e 2027 436f 6469 6d42 7349 6e76 3d3e  on 'CodimBsInv=>
│ │ │ │ +00028940: 3427 2c20 6974 2074 616b 6573 2061 626f  4', it takes abo
│ │ │ │ +00028950: 7574 2020 2020 2020 2020 2020 2020 2020  ut              
│ │ │ │ +00028960: 2020 2020 207c 0a7c 2020 2020 2074 696d       |.|     tim
│ │ │ │ +00028970: 6520 7073 693d 6170 7072 6f78 696d 6174  e psi=approximat
│ │ │ │ +00028980: 6549 6e76 6572 7365 4d61 7028 7068 692c  eInverseMap(phi,
│ │ │ │ +00028990: 436f 6469 6d42 7349 6e76 3d3e 3429 2020  CodimBsInv=>4)  
│ │ │ │ +000289a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000289b0: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ +000289c0: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ +000289d0: 3a20 7374 6570 2031 206f 6620 3320 2020  : step 1 of 3   
│ │ │ │  000289e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000289f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028a00: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ -00028a10: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ -00028a20: 2032 206f 6620 3320 2020 2020 2020 2020   2 of 3         
│ │ │ │ +000289f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a00: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ +00028a10: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ +00028a20: 3a20 7374 6570 2032 206f 6620 3320 2020  : step 2 of 3   
│ │ │ │  00028a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028a50: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ -00028a60: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ -00028a70: 2033 206f 6620 3320 2020 2020 2020 2020   3 of 3         
│ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028a50: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ +00028a60: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ +00028a70: 3a20 7374 6570 2033 206f 6620 3320 2020  : step 3 of 3   
│ │ │ │  00028a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028a90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028aa0: 0a7c 202d 2d20 7573 6564 2032 2e37 3631  .| -- used 2.761
│ │ │ │ -00028ab0: 3373 2028 6370 7529 3b20 312e 3839 3330  3s (cpu); 1.8930
│ │ │ │ -00028ac0: 3373 2028 7468 7265 6164 293b 2030 7320  3s (thread); 0s 
│ │ │ │ -00028ad0: 2867 6329 2020 2020 2020 2020 2020 2020  (gc)            
│ │ │ │ -00028ae0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028af0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028aa0: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00028ab0: 2032 2e38 3637 3336 7320 2863 7075 293b   2.86736s (cpu);
│ │ │ │ +00028ac0: 2032 2e33 3133 3235 7320 2874 6872 6561   2.31325s (threa
│ │ │ │ +00028ad0: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028af0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028b40: 0a7c 6f38 203d 202d 2d20 7261 7469 6f6e  .|o8 = -- ration
│ │ │ │ -00028b50: 616c 206d 6170 202d 2d20 2020 2020 2020  al map --       
│ │ │ │ +00028b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b40: 2020 2020 207c 0a7c 6f38 203d 202d 2d20       |.|o8 = -- 
│ │ │ │ +00028b50: 7261 7469 6f6e 616c 206d 6170 202d 2d20  rational map -- 
│ │ │ │  00028b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028b80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028b90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028b90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bb0: 2020 5a5a 2020 2020 2020 2020 2020 2020    ZZ            
│ │ │ │ +00028bb0: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │  00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028bd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028be0: 0a7c 2020 2020 2073 6f75 7263 653a 2073  .|     source: s
│ │ │ │ -00028bf0: 7562 7661 7269 6574 7920 6f66 2050 726f  ubvariety of Pro
│ │ │ │ -00028c00: 6a28 2d2d 5b78 202c 2078 202c 2078 202c  j(--[x , x , x ,
│ │ │ │ -00028c10: 2078 202c 2078 202c 2078 202c 2020 2020   x , x , x ,    
│ │ │ │ -00028c20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028c30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028be0: 2020 2020 207c 0a7c 2020 2020 2073 6f75       |.|     sou
│ │ │ │ +00028bf0: 7263 653a 2073 7562 7661 7269 6574 7920  rce: subvariety 
│ │ │ │ +00028c00: 6f66 2050 726f 6a28 2d2d 5b78 202c 2078  of Proj(--[x , x
│ │ │ │ +00028c10: 202c 2078 202c 2078 202c 2078 202c 2078   , x , x , x , x
│ │ │ │ +00028c20: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +00028c30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028c50: 2020 3937 2020 3020 2020 3120 2020 3220    97  0   1   2 
│ │ │ │ -00028c60: 2020 3320 2020 3420 2020 3520 2020 2020    3   4   5     
│ │ │ │ -00028c70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028c80: 0a7c 2020 2020 2020 2020 2020 2020 207b  .|             {
│ │ │ │ -00028c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028c50: 2020 2020 2020 2020 3937 2020 3020 2020          97  0   
│ │ │ │ +00028c60: 3120 2020 3220 2020 3320 2020 3420 2020  1   2   3   4   
│ │ │ │ +00028c70: 3520 2020 2020 2020 2020 2020 2020 2020  5               
│ │ │ │ +00028c80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028c90: 2020 2020 207b 2020 2020 2020 2020 2020       {          
│ │ │ │  00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028cd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028cf0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00028cf0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00028d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028d10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028d20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00028d30: 7820 7820 202d 2038 7820 7820 202b 2032  x x  - 8x x  + 2
│ │ │ │ -00028d40: 3578 2020 2d20 3235 7820 7820 202d 2032  5x  - 25x x  - 2
│ │ │ │ -00028d50: 3278 2078 2020 2b20 7820 7820 2020 2020  2x x  + x x     
│ │ │ │ -00028d60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028d70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00028d80: 2031 2033 2020 2020 2032 2033 2020 2020   1 3     2 3    
│ │ │ │ -00028d90: 2020 3320 2020 2020 2032 2034 2020 2020    3      2 4    
│ │ │ │ -00028da0: 2020 3320 3420 2020 2030 2035 2020 2020    3 4    0 5    
│ │ │ │ -00028db0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028dc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028d20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028d30: 2020 2020 2020 7820 7820 202d 2038 7820        x x  - 8x 
│ │ │ │ +00028d40: 7820 202b 2032 3578 2020 2d20 3235 7820  x  + 25x  - 25x 
│ │ │ │ +00028d50: 7820 202d 2032 3278 2078 2020 2b20 7820  x  - 22x x  + x 
│ │ │ │ +00028d60: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +00028d70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028d80: 2020 2020 2020 2031 2033 2020 2020 2032         1 3     2
│ │ │ │ +00028d90: 2033 2020 2020 2020 3320 2020 2020 2032   3      3      2
│ │ │ │ +00028da0: 2034 2020 2020 2020 3320 3420 2020 2030   4      3 4    0
│ │ │ │ +00028db0: 2035 2020 2020 2020 2020 2020 2020 2020   5              
│ │ │ │ +00028dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028e10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00028e20: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -00028e30: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00028e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028e20: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00028e30: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00028e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028e50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028e60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00028e70: 7820 202b 2031 3778 2078 2020 2d20 3134  x  + 17x x  - 14
│ │ │ │ -00028e80: 7820 202d 2031 3378 2078 2020 2b20 3334  x  - 13x x  + 34
│ │ │ │ -00028e90: 7820 7820 202b 2034 3478 2078 2020 2020  x x  + 44x x    
│ │ │ │ -00028ea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028eb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00028ec0: 2032 2020 2020 2020 3220 3320 2020 2020   2      2 3     
│ │ │ │ -00028ed0: 2033 2020 2020 2020 3220 3420 2020 2020   3      2 4     
│ │ │ │ -00028ee0: 2033 2034 2020 2020 2020 3020 3520 2020   3 4      0 5   
│ │ │ │ -00028ef0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028f00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028e60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028e70: 2020 2020 2020 7820 202b 2031 3778 2078        x  + 17x x
│ │ │ │ +00028e80: 2020 2d20 3134 7820 202d 2031 3378 2078    - 14x  - 13x x
│ │ │ │ +00028e90: 2020 2b20 3334 7820 7820 202b 2034 3478    + 34x x  + 44x
│ │ │ │ +00028ea0: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +00028eb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028ec0: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ +00028ed0: 3320 2020 2020 2033 2020 2020 2020 3220  3      3      2 
│ │ │ │ +00028ee0: 3420 2020 2020 2033 2034 2020 2020 2020  4      3 4      
│ │ │ │ +00028ef0: 3020 3520 2020 2020 2020 2020 2020 2020  0 5             
│ │ │ │ +00028f00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00028f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028f50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00028f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00028f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f70: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00028f70: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  00028f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00028f90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00028fb0: 7820 7820 202d 2034 3078 2078 2020 2b20  x x  - 40x x  + 
│ │ │ │ -00028fc0: 3238 7820 202d 2078 2078 2020 2b20 3578  28x  - x x  + 5x
│ │ │ │ -00028fd0: 2078 2020 2d20 3136 7820 7820 2020 2020   x  - 16x x     
│ │ │ │ -00028fe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00028ff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029000: 2031 2032 2020 2020 2020 3220 3320 2020   1 2      2 3   
│ │ │ │ -00029010: 2020 2033 2020 2020 3020 3420 2020 2020     3    0 4     
│ │ │ │ -00029020: 3220 3420 2020 2020 2033 2034 2020 2020  2 4      3 4    
│ │ │ │ -00029030: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029040: 0a7c 2020 2020 2020 2020 2020 2020 207d  .|             }
│ │ │ │ -00029050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00028fa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00028fb0: 2020 2020 2020 7820 7820 202d 2034 3078        x x  - 40x
│ │ │ │ +00028fc0: 2078 2020 2b20 3238 7820 202d 2078 2078   x  + 28x  - x x
│ │ │ │ +00028fd0: 2020 2b20 3578 2078 2020 2d20 3136 7820    + 5x x  - 16x 
│ │ │ │ +00028fe0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +00028ff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029000: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +00029010: 3220 3320 2020 2020 2033 2020 2020 3020  2 3      3    0 
│ │ │ │ +00029020: 3420 2020 2020 3220 3420 2020 2020 2033  4     2 4      3
│ │ │ │ +00029030: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +00029040: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029050: 2020 2020 207d 2020 2020 2020 2020 2020       }          
│ │ │ │  00029060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029080: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000290a0: 2020 2020 5a5a 2020 2020 2020 2020 2020      ZZ          
│ │ │ │ +00029080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000290a0: 2020 2020 2020 2020 2020 5a5a 2020 2020            ZZ    
│ │ │ │  000290b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000290c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000290d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000290e0: 0a7c 2020 2020 2074 6172 6765 743a 2050  .|     target: P
│ │ │ │ -000290f0: 726f 6a28 2d2d 5b74 202c 2074 202c 2074  roj(--[t , t , t
│ │ │ │ -00029100: 202c 2074 202c 2074 202c 2074 202c 2074   , t , t , t , t
│ │ │ │ -00029110: 202c 2074 202c 2074 205d 2920 2020 2020   , t , t ])     
│ │ │ │ -00029120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029130: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029140: 2020 2020 3937 2020 3020 2020 3120 2020      97  0   1   
│ │ │ │ -00029150: 3220 2020 3320 2020 3420 2020 3520 2020  2   3   4   5   
│ │ │ │ -00029160: 3620 2020 3720 2020 3820 2020 2020 2020  6   7   8       
│ │ │ │ -00029170: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029180: 0a7c 2020 2020 2064 6566 696e 696e 6720  .|     defining 
│ │ │ │ -00029190: 666f 726d 733a 207b 2020 2020 2020 2020  forms: {        
│ │ │ │ +000290d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000290e0: 2020 2020 207c 0a7c 2020 2020 2074 6172       |.|     tar
│ │ │ │ +000290f0: 6765 743a 2050 726f 6a28 2d2d 5b74 202c  get: Proj(--[t ,
│ │ │ │ +00029100: 2074 202c 2074 202c 2074 202c 2074 202c   t , t , t , t ,
│ │ │ │ +00029110: 2074 202c 2074 202c 2074 202c 2074 205d   t , t , t , t ]
│ │ │ │ +00029120: 2920 2020 2020 2020 2020 2020 2020 2020  )               
│ │ │ │ +00029130: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029140: 2020 2020 2020 2020 2020 3937 2020 3020            97  0 
│ │ │ │ +00029150: 2020 3120 2020 3220 2020 3320 2020 3420    1   2   3   4 
│ │ │ │ +00029160: 2020 3520 2020 3620 2020 3720 2020 3820    5   6   7   8 
│ │ │ │ +00029170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029180: 2020 2020 207c 0a7c 2020 2020 2064 6566       |.|     def
│ │ │ │ +00029190: 696e 696e 6720 666f 726d 733a 207b 2020  ining forms: {  
│ │ │ │  000291a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000291b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000291c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000291d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000291e0: 2020 2020 2020 2020 7820 7820 202b 2032          x x  + 2
│ │ │ │ -000291f0: 7820 7820 202b 2078 2078 2020 2b20 3278  x x  + x x  + 2x
│ │ │ │ -00029200: 2078 2020 2d20 3478 2078 2020 2d20 2020   x  - 4x x  -   
│ │ │ │ -00029210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029220: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029230: 2020 2020 2020 2020 2033 2035 2020 2020           3 5    
│ │ │ │ -00029240: 2035 2037 2020 2020 3120 3820 2020 2020   5 7    1 8     
│ │ │ │ -00029250: 3220 3820 2020 2020 3320 3820 2020 2020  2 8     3 8     
│ │ │ │ -00029260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029270: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000291c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000291d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000291e0: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +000291f0: 7820 202b 2032 7820 7820 202b 2078 2078  x  + 2x x  + x x
│ │ │ │ +00029200: 2020 2b20 3278 2078 2020 2d20 3478 2078    + 2x x  - 4x x
│ │ │ │ +00029210: 2020 2d20 2020 2020 2020 2020 2020 2020    -             
│ │ │ │ +00029220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029230: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00029240: 2035 2020 2020 2035 2037 2020 2020 3120   5     5 7    1 
│ │ │ │ +00029250: 3820 2020 2020 3220 3820 2020 2020 3320  8     2 8     3 
│ │ │ │ +00029260: 3820 2020 2020 2020 2020 2020 2020 2020  8               
│ │ │ │ +00029270: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000292a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000292b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000292c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000292d0: 2020 2020 2020 2020 7820 7820 202b 2036          x x  + 6
│ │ │ │ -000292e0: 7820 7820 202b 2032 7820 7820 202b 2035  x x  + 2x x  + 5
│ │ │ │ -000292f0: 7820 7820 202d 2031 3078 2078 2020 2020  x x  - 10x x    
│ │ │ │ -00029300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029310: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029320: 2020 2020 2020 2020 2032 2035 2020 2020           2 5    
│ │ │ │ -00029330: 2035 2037 2020 2020 2031 2038 2020 2020   5 7     1 8    
│ │ │ │ -00029340: 2032 2038 2020 2020 2020 3320 3820 2020   2 8      3 8   
│ │ │ │ -00029350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000292b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000292c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000292d0: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +000292e0: 7820 202b 2036 7820 7820 202b 2032 7820  x  + 6x x  + 2x 
│ │ │ │ +000292f0: 7820 202b 2035 7820 7820 202d 2031 3078  x  + 5x x  - 10x
│ │ │ │ +00029300: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +00029310: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029320: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00029330: 2035 2020 2020 2035 2037 2020 2020 2031   5     5 7     1
│ │ │ │ +00029340: 2038 2020 2020 2032 2038 2020 2020 2020   8     2 8      
│ │ │ │ +00029350: 3320 3820 2020 2020 2020 2020 2020 2020  3 8             
│ │ │ │ +00029360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000293a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000293b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000293c0: 2020 2020 2020 2020 7820 7820 202b 2038          x x  + 8
│ │ │ │ -000293d0: 7820 7820 202b 2078 2078 2020 2b20 3478  x x  + x x  + 4x
│ │ │ │ -000293e0: 2078 2020 2b20 3130 7820 7820 2020 2020   x  + 10x x     
│ │ │ │ -000293f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029400: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029410: 2020 2020 2020 2020 2030 2035 2020 2020           0 5    
│ │ │ │ -00029420: 2035 2037 2020 2020 3020 3820 2020 2020   5 7    0 8     
│ │ │ │ -00029430: 3120 3820 2020 2020 2032 2038 2020 2020  1 8      2 8    
│ │ │ │ -00029440: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000293a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000293b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000293c0: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +000293d0: 7820 202b 2038 7820 7820 202b 2078 2078  x  + 8x x  + x x
│ │ │ │ +000293e0: 2020 2b20 3478 2078 2020 2b20 3130 7820    + 4x x  + 10x 
│ │ │ │ +000293f0: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +00029400: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029410: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00029420: 2035 2020 2020 2035 2037 2020 2020 3020   5     5 7    0 
│ │ │ │ +00029430: 3820 2020 2020 3120 3820 2020 2020 2032  8     1 8      2
│ │ │ │ +00029440: 2038 2020 2020 2020 2020 2020 2020 2020   8              
│ │ │ │ +00029450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000294a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000294b0: 2020 2020 2020 2020 7820 7820 202d 2034          x x  - 4
│ │ │ │ -000294c0: 7820 7820 202b 2034 3678 2078 2020 2d20  x x  + 46x x  - 
│ │ │ │ -000294d0: 3338 7820 7820 202d 2034 3378 2020 2020  38x x  - 43x    
│ │ │ │ -000294e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000294f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029500: 2020 2020 2020 2020 2033 2034 2020 2020           3 4    
│ │ │ │ -00029510: 2032 2036 2020 2020 2020 3320 3620 2020   2 6      3 6   
│ │ │ │ -00029520: 2020 2033 2037 2020 2020 2020 3520 2020     3 7      5   
│ │ │ │ -00029530: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029540: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000294a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000294b0: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +000294c0: 7820 202d 2034 7820 7820 202b 2034 3678  x  - 4x x  + 46x
│ │ │ │ +000294d0: 2078 2020 2d20 3338 7820 7820 202d 2034   x  - 38x x  - 4
│ │ │ │ +000294e0: 3378 2020 2020 2020 2020 2020 2020 2020  3x              
│ │ │ │ +000294f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029500: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00029510: 2034 2020 2020 2032 2036 2020 2020 2020   4     2 6      
│ │ │ │ +00029520: 3320 3620 2020 2020 2033 2037 2020 2020  3 6      3 7    
│ │ │ │ +00029530: 2020 3520 2020 2020 2020 2020 2020 2020    5             
│ │ │ │ +00029540: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029590: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000295a0: 2020 2020 2020 2020 7820 7820 202b 2031          x x  + 1
│ │ │ │ -000295b0: 3778 2078 2020 2b20 3431 7820 7820 202d  7x x  + 41x x  -
│ │ │ │ -000295c0: 2078 2078 2020 2d20 3138 7820 7820 2020   x x  - 18x x   
│ │ │ │ -000295d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000295e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000295f0: 2020 2020 2020 2020 2032 2034 2020 2020           2 4    
│ │ │ │ -00029600: 2020 3220 3620 2020 2020 2033 2036 2020    2 6      3 6  
│ │ │ │ -00029610: 2020 3120 3720 2020 2020 2033 2020 2020    1 7      3    
│ │ │ │ -00029620: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029630: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000295a0: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +000295b0: 7820 202b 2031 3778 2078 2020 2b20 3431  x  + 17x x  + 41
│ │ │ │ +000295c0: 7820 7820 202d 2078 2078 2020 2d20 3138  x x  - x x  - 18
│ │ │ │ +000295d0: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +000295e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000295f0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00029600: 2034 2020 2020 2020 3220 3620 2020 2020   4      2 6     
│ │ │ │ +00029610: 2033 2036 2020 2020 3120 3720 2020 2020   3 6    1 7     
│ │ │ │ +00029620: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00029630: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029680: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029690: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00029670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029690: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  000296a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000296b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000296c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000296d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000296e0: 2020 2020 2020 2020 7820 202d 2033 3278          x  - 32x
│ │ │ │ -000296f0: 2078 2020 2b20 3432 7820 7820 202b 2032   x  + 42x x  + 2
│ │ │ │ -00029700: 3778 2078 2020 2d20 3432 7820 7820 2020  7x x  - 42x x   
│ │ │ │ -00029710: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029720: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029730: 2020 2020 2020 2020 2033 2020 2020 2020           3      
│ │ │ │ -00029740: 3020 3420 2020 2020 2030 2036 2020 2020  0 4      0 6    
│ │ │ │ -00029750: 2020 3220 3620 2020 2020 2033 2020 2020    2 6      3    
│ │ │ │ -00029760: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029770: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000296c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000296d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000296e0: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +000296f0: 202d 2033 3278 2078 2020 2b20 3432 7820   - 32x x  + 42x 
│ │ │ │ +00029700: 7820 202b 2032 3778 2078 2020 2d20 3432  x  + 27x x  - 42
│ │ │ │ +00029710: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +00029720: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029730: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +00029740: 2020 2020 2020 3020 3420 2020 2020 2030        0 4      0
│ │ │ │ +00029750: 2036 2020 2020 2020 3220 3620 2020 2020   6      2 6     
│ │ │ │ +00029760: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │ +00029770: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000297a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000297b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000297c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000297d0: 2020 2020 2020 2020 7820 7820 202b 2078          x x  + x
│ │ │ │ -000297e0: 2078 2020 2d20 3136 7820 7820 202d 2031   x  - 16x x  - 1
│ │ │ │ -000297f0: 3078 2078 2020 2b20 3278 2078 2020 2020  0x x  + 2x x    
│ │ │ │ -00029800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029810: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029820: 2020 2020 2020 2020 2032 2033 2020 2020           2 3    
│ │ │ │ -00029830: 3020 3420 2020 2020 2032 2036 2020 2020  0 4      2 6    
│ │ │ │ -00029840: 2020 3320 3620 2020 2020 3020 3720 2020    3 6     0 7   
│ │ │ │ -00029850: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000297b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000297c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000297d0: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +000297e0: 7820 202b 2078 2078 2020 2d20 3136 7820  x  + x x  - 16x 
│ │ │ │ +000297f0: 7820 202d 2031 3078 2078 2020 2b20 3278  x  - 10x x  + 2x
│ │ │ │ +00029800: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +00029810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029820: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00029830: 2033 2020 2020 3020 3420 2020 2020 2032   3    0 4      2
│ │ │ │ +00029840: 2036 2020 2020 2020 3320 3620 2020 2020   6      3 6     
│ │ │ │ +00029850: 3020 3720 2020 2020 2020 2020 2020 2020  0 7             
│ │ │ │ +00029860: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000298b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -000298c0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +000298a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000298b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000298c0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  000298d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000298e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000298f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029900: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029910: 2020 2020 2020 2020 7820 202b 2033 3678          x  + 36x
│ │ │ │ -00029920: 2078 2020 2b20 7820 7820 202b 2037 7820   x  + x x  + 7x 
│ │ │ │ -00029930: 7820 202d 2034 7820 7820 202d 2020 2020  x  - 4x x  -    
│ │ │ │ -00029940: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029960: 2020 2020 2020 2020 2031 2020 2020 2020           1      
│ │ │ │ -00029970: 3020 3420 2020 2034 2035 2020 2020 2030  0 4    4 5     0
│ │ │ │ -00029980: 2036 2020 2020 2031 2036 2020 2020 2020   6     1 6      
│ │ │ │ -00029990: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000299a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000298f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029900: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029910: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +00029920: 202b 2033 3678 2078 2020 2b20 7820 7820   + 36x x  + x x 
│ │ │ │ +00029930: 202b 2037 7820 7820 202d 2034 7820 7820   + 7x x  - 4x x 
│ │ │ │ +00029940: 202d 2020 2020 2020 2020 2020 2020 2020   -              
│ │ │ │ +00029950: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029960: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +00029970: 2020 2020 2020 3020 3420 2020 2034 2035        0 4    4 5
│ │ │ │ +00029980: 2020 2020 2030 2036 2020 2020 2031 2036       0 6     1 6
│ │ │ │ +00029990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000299a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000299b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000299c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000299d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000299e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000299f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029a00: 2020 2020 2020 2020 7820 7820 202b 2032          x x  + 2
│ │ │ │ -00029a10: 7820 7820 202d 2035 7820 7820 202d 2033  x x  - 5x x  - 3
│ │ │ │ -00029a20: 3478 2078 2020 2d20 3136 7820 7820 2020  4x x  - 16x x   
│ │ │ │ -00029a30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029a40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029a50: 2020 2020 2020 2020 2030 2031 2020 2020           0 1    
│ │ │ │ -00029a60: 2030 2032 2020 2020 2030 2033 2020 2020   0 2     0 3    
│ │ │ │ -00029a70: 2020 3020 3420 2020 2020 2030 2020 2020    0 4      0    
│ │ │ │ -00029a80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029a90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -00029aa0: 2020 2020 2020 207d 2020 2020 2020 2020         }        
│ │ │ │ +000299e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000299f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029a00: 2020 2020 2020 2020 2020 2020 2020 7820                x 
│ │ │ │ +00029a10: 7820 202b 2032 7820 7820 202d 2035 7820  x  + 2x x  - 5x 
│ │ │ │ +00029a20: 7820 202d 2033 3478 2078 2020 2d20 3136  x  - 34x x  - 16
│ │ │ │ +00029a30: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +00029a40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029a50: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ +00029a60: 2031 2020 2020 2030 2032 2020 2020 2030   1     0 2     0
│ │ │ │ +00029a70: 2033 2020 2020 2020 3020 3420 2020 2020   3      0 4     
│ │ │ │ +00029a80: 2030 2020 2020 2020 2020 2020 2020 2020   0              
│ │ │ │ +00029a90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00029aa0: 2020 2020 2020 2020 2020 2020 207d 2020               }  
│ │ │ │  00029ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ad0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029ae0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029b20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029b30: 0a7c 6f38 203a 2052 6174 696f 6e61 6c4d  .|o8 : RationalM
│ │ │ │ -00029b40: 6170 2028 7175 6164 7261 7469 6320 7261  ap (quadratic ra
│ │ │ │ -00029b50: 7469 6f6e 616c 206d 6170 2066 726f 6d20  tional map from 
│ │ │ │ -00029b60: 382d 6469 6d65 6e73 696f 6e61 6c20 2020  8-dimensional   
│ │ │ │ -00029b70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029b80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00029b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029b30: 2020 2020 207c 0a7c 6f38 203a 2052 6174       |.|o8 : Rat
│ │ │ │ +00029b40: 696f 6e61 6c4d 6170 2028 7175 6164 7261  ionalMap (quadra
│ │ │ │ +00029b50: 7469 6320 7261 7469 6f6e 616c 206d 6170  tic rational map
│ │ │ │ +00029b60: 2066 726f 6d20 382d 6469 6d65 6e73 696f   from 8-dimensio
│ │ │ │ +00029b70: 6e61 6c20 2020 2020 2020 2020 2020 2020  nal             
│ │ │ │ +00029b80: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00029b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00029bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00029bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00029bd0: 0a7c 7472 6970 6c65 2074 696d 6520 2020  .|triple time   
│ │ │ │ -00029be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00029bd0: 2d2d 2d2d 2d7c 0a7c 7472 6970 6c65 2074  -----|.|triple t
│ │ │ │ +00029be0: 696d 6520 2020 2020 2020 2020 2020 2020  ime             
│ │ │ │  00029bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029c20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029c20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029c60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029c70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029c70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029cc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029d10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029d10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029d50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029d60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029d60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029da0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029db0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029db0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029e00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029e00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029e50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029e50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029e90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029ea0: 0a7c 7820 2c20 7820 2c20 7820 2c20 7820  .|x , x , x , x 
│ │ │ │ -00029eb0: 2c20 7820 202c 2078 2020 5d29 2064 6566  , x  , x  ]) def
│ │ │ │ -00029ec0: 696e 6564 2062 7920 2020 2020 2020 2020  ined by         
│ │ │ │ +00029e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ea0: 2020 2020 207c 0a7c 7820 2c20 7820 2c20       |.|x , x , 
│ │ │ │ +00029eb0: 7820 2c20 7820 2c20 7820 202c 2078 2020  x , x , x  , x  
│ │ │ │ +00029ec0: 5d29 2064 6566 696e 6564 2062 7920 2020  ]) defined by   
│ │ │ │  00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029ef0: 0a7c 2036 2020 2037 2020 2038 2020 2039  .| 6   7   8   9
│ │ │ │ -00029f00: 2020 2031 3020 2020 3131 2020 2020 2020     10   11      
│ │ │ │ +00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029ef0: 2020 2020 207c 0a7c 2036 2020 2037 2020       |.| 6   7  
│ │ │ │ +00029f00: 2038 2020 2039 2020 2031 3020 2020 3131   8   9   10   11
│ │ │ │  00029f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029f40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029f80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029f90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029f90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00029fd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00029fe0: 0a7c 2b20 3133 7820 7820 202b 2034 3178  .|+ 13x x  + 41x
│ │ │ │ -00029ff0: 2078 2020 2d20 7820 7820 202b 2031 3278   x  - x x  + 12x
│ │ │ │ -0002a000: 2078 2020 2b20 3235 7820 7820 202b 2032   x  + 25x x  + 2
│ │ │ │ -0002a010: 3578 2078 2020 2b20 3233 7820 7820 202d  5x x  + 23x x  -
│ │ │ │ -0002a020: 2033 7820 7820 202b 2032 7820 7820 207c   3x x  + 2x x  |
│ │ │ │ -0002a030: 0a7c 2020 2020 2032 2035 2020 2020 2020  .|     2 5      
│ │ │ │ -0002a040: 3320 3520 2020 2030 2036 2020 2020 2020  3 5    0 6      
│ │ │ │ -0002a050: 3220 3620 2020 2020 2031 2037 2020 2020  2 6      1 7    
│ │ │ │ -0002a060: 2020 3320 3720 2020 2020 2035 2037 2020    3 7      5 7  
│ │ │ │ -0002a070: 2020 2036 2037 2020 2020 2030 2020 207c     6 7     0   |
│ │ │ │ -0002a080: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00029fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00029fe0: 2020 2020 207c 0a7c 2b20 3133 7820 7820       |.|+ 13x x 
│ │ │ │ +00029ff0: 202b 2034 3178 2078 2020 2d20 7820 7820   + 41x x  - x x 
│ │ │ │ +0002a000: 202b 2031 3278 2078 2020 2b20 3235 7820   + 12x x  + 25x 
│ │ │ │ +0002a010: 7820 202b 2032 3578 2078 2020 2b20 3233  x  + 25x x  + 23
│ │ │ │ +0002a020: 7820 7820 202d 2033 7820 7820 202b 2032  x x  - 3x x  + 2
│ │ │ │ +0002a030: 7820 7820 207c 0a7c 2020 2020 2032 2035  x x  |.|     2 5
│ │ │ │ +0002a040: 2020 2020 2020 3320 3520 2020 2030 2036        3 5    0 6
│ │ │ │ +0002a050: 2020 2020 2020 3220 3620 2020 2020 2031        2 6      1
│ │ │ │ +0002a060: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ +0002a070: 2035 2037 2020 2020 2036 2037 2020 2020   5 7     6 7    
│ │ │ │ +0002a080: 2030 2020 207c 0a7c 2020 2020 2020 2020   0   |.|        
│ │ │ │  0002a090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a0c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a0d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a120: 0a7c 202d 2033 3078 2078 2020 2b20 3237  .| - 30x x  + 27
│ │ │ │ -0002a130: 7820 7820 202b 2033 3178 2078 2020 2d20  x x  + 31x x  - 
│ │ │ │ -0002a140: 3336 7820 7820 202d 2078 2078 2020 2b20  36x x  - x x  + 
│ │ │ │ -0002a150: 3133 7820 7820 202b 2038 7820 7820 202b  13x x  + 8x x  +
│ │ │ │ -0002a160: 2039 7820 7820 202b 2034 3678 2020 207c   9x x  + 46x   |
│ │ │ │ -0002a170: 0a7c 2020 2020 2020 3220 3520 2020 2020  .|      2 5     
│ │ │ │ -0002a180: 2033 2035 2020 2020 2020 3220 3620 2020   3 5      2 6   
│ │ │ │ -0002a190: 2020 2033 2036 2020 2020 3020 3720 2020     3 6    0 7   
│ │ │ │ -0002a1a0: 2020 2031 2037 2020 2020 2033 2037 2020     1 7     3 7  
│ │ │ │ -0002a1b0: 2020 2035 2037 2020 2020 2020 3620 207c     5 7      6  |
│ │ │ │ -0002a1c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a120: 2020 2020 207c 0a7c 202d 2033 3078 2078       |.| - 30x x
│ │ │ │ +0002a130: 2020 2b20 3237 7820 7820 202b 2033 3178    + 27x x  + 31x
│ │ │ │ +0002a140: 2078 2020 2d20 3336 7820 7820 202d 2078   x  - 36x x  - x
│ │ │ │ +0002a150: 2078 2020 2b20 3133 7820 7820 202b 2038   x  + 13x x  + 8
│ │ │ │ +0002a160: 7820 7820 202b 2039 7820 7820 202b 2034  x x  + 9x x  + 4
│ │ │ │ +0002a170: 3678 2020 207c 0a7c 2020 2020 2020 3220  6x   |.|      2 
│ │ │ │ +0002a180: 3520 2020 2020 2033 2035 2020 2020 2020  5      3 5      
│ │ │ │ +0002a190: 3220 3620 2020 2020 2033 2036 2020 2020  2 6      3 6    
│ │ │ │ +0002a1a0: 3020 3720 2020 2020 2031 2037 2020 2020  0 7      1 7    
│ │ │ │ +0002a1b0: 2033 2037 2020 2020 2035 2037 2020 2020   3 7     5 7    
│ │ │ │ +0002a1c0: 2020 3620 207c 0a7c 2020 2020 2020 2020    6  |.|        
│ │ │ │  0002a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a200: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a210: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a260: 0a7c 2b20 3578 2078 2020 2d20 3336 7820  .|+ 5x x  - 36x 
│ │ │ │ -0002a270: 7820 202b 2033 3778 2078 2020 2b20 3438  x  + 37x x  + 48
│ │ │ │ -0002a280: 7820 7820 202d 2035 7820 7820 202d 2035  x x  - 5x x  - 5
│ │ │ │ -0002a290: 7820 7820 202b 2078 2078 2020 2b20 3230  x x  + x x  + 20
│ │ │ │ -0002a2a0: 7820 7820 202b 2031 3078 2078 2020 207c  x x  + 10x x   |
│ │ │ │ -0002a2b0: 0a7c 2020 2020 3020 3520 2020 2020 2032  .|    0 5      2
│ │ │ │ -0002a2c0: 2035 2020 2020 2020 3320 3520 2020 2020   5      3 5     
│ │ │ │ -0002a2d0: 2032 2036 2020 2020 2031 2037 2020 2020   2 6     1 7    
│ │ │ │ -0002a2e0: 2033 2037 2020 2020 3520 3720 2020 2020   3 7    5 7     
│ │ │ │ -0002a2f0: 2036 2037 2020 2020 2020 3020 3820 207c   6 7      0 8  |
│ │ │ │ -0002a300: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a260: 2020 2020 207c 0a7c 2b20 3578 2078 2020       |.|+ 5x x  
│ │ │ │ +0002a270: 2d20 3336 7820 7820 202b 2033 3778 2078  - 36x x  + 37x x
│ │ │ │ +0002a280: 2020 2b20 3438 7820 7820 202d 2035 7820    + 48x x  - 5x 
│ │ │ │ +0002a290: 7820 202d 2035 7820 7820 202b 2078 2078  x  - 5x x  + x x
│ │ │ │ +0002a2a0: 2020 2b20 3230 7820 7820 202b 2031 3078    + 20x x  + 10x
│ │ │ │ +0002a2b0: 2078 2020 207c 0a7c 2020 2020 3020 3520   x   |.|    0 5 
│ │ │ │ +0002a2c0: 2020 2020 2032 2035 2020 2020 2020 3320       2 5      3 
│ │ │ │ +0002a2d0: 3520 2020 2020 2032 2036 2020 2020 2031  5      2 6     1
│ │ │ │ +0002a2e0: 2037 2020 2020 2033 2037 2020 2020 3520   7     3 7    5 
│ │ │ │ +0002a2f0: 3720 2020 2020 2036 2037 2020 2020 2020  7      6 7      
│ │ │ │ +0002a300: 3020 3820 207c 0a7c 2020 2020 2020 2020  0 8  |.|        
│ │ │ │  0002a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a3a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a3e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a3f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a3f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a430: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a440: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a440: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a480: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a490: 0a7c 2032 7820 7820 202b 2078 2078 2020  .| 2x x  + x x  
│ │ │ │ -0002a4a0: 2d20 3678 2078 2020 2d20 3578 2078 2020  - 6x x  - 5x x  
│ │ │ │ -0002a4b0: 202d 2031 3078 2078 2020 202b 2032 3578   - 10x x   + 25x
│ │ │ │ -0002a4c0: 2078 2020 202b 2035 7820 7820 2020 2d20   x   + 5x x   - 
│ │ │ │ -0002a4d0: 3578 2078 2020 2c20 2020 2020 2020 207c  5x x  ,        |
│ │ │ │ -0002a4e0: 0a7c 2020 2034 2038 2020 2020 3520 3820  .|   4 8    5 8 
│ │ │ │ -0002a4f0: 2020 2020 3520 3920 2020 2020 3120 3130      5 9     1 10
│ │ │ │ -0002a500: 2020 2020 2020 3220 3130 2020 2020 2020        2 10      
│ │ │ │ -0002a510: 3320 3130 2020 2020 2034 2031 3020 2020  3 10     4 10   
│ │ │ │ -0002a520: 2020 3520 3130 2020 2020 2020 2020 207c    5 10         |
│ │ │ │ -0002a530: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a490: 2020 2020 207c 0a7c 2032 7820 7820 202b       |.| 2x x  +
│ │ │ │ +0002a4a0: 2078 2078 2020 2d20 3678 2078 2020 2d20   x x  - 6x x  - 
│ │ │ │ +0002a4b0: 3578 2078 2020 202d 2031 3078 2078 2020  5x x   - 10x x  
│ │ │ │ +0002a4c0: 202b 2032 3578 2078 2020 202b 2035 7820   + 25x x   + 5x 
│ │ │ │ +0002a4d0: 7820 2020 2d20 3578 2078 2020 2c20 2020  x   - 5x x  ,   
│ │ │ │ +0002a4e0: 2020 2020 207c 0a7c 2020 2034 2038 2020       |.|   4 8  
│ │ │ │ +0002a4f0: 2020 3520 3820 2020 2020 3520 3920 2020    5 8     5 9   
│ │ │ │ +0002a500: 2020 3120 3130 2020 2020 2020 3220 3130    1 10      2 10
│ │ │ │ +0002a510: 2020 2020 2020 3320 3130 2020 2020 2034        3 10     4
│ │ │ │ +0002a520: 2031 3020 2020 2020 3520 3130 2020 2020   10     5 10    
│ │ │ │ +0002a530: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a570: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a580: 0a7c 202d 2036 7820 7820 202d 2034 3678  .| - 6x x  - 46x
│ │ │ │ -0002a590: 2078 2020 2d20 3434 7820 7820 202d 2032   x  - 44x x  - 2
│ │ │ │ -0002a5a0: 3278 2078 2020 2b20 3336 7820 7820 2020  2x x  + 36x x   
│ │ │ │ -0002a5b0: 2d20 3235 7820 7820 2020 2b20 3134 7820  - 25x x   + 14x 
│ │ │ │ -0002a5c0: 7820 2020 2b20 3135 7820 7820 2020 207c  x   + 15x x    |
│ │ │ │ -0002a5d0: 0a7c 2020 2020 2034 2038 2020 2020 2020  .|     4 8      
│ │ │ │ -0002a5e0: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ -0002a5f0: 2020 3520 3920 2020 2020 2031 2031 3020    5 9      1 10 
│ │ │ │ -0002a600: 2020 2020 2032 2031 3020 2020 2020 2033       2 10      3
│ │ │ │ -0002a610: 2031 3020 2020 2020 2034 2031 3020 207c   10      4 10  |
│ │ │ │ -0002a620: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a580: 2020 2020 207c 0a7c 202d 2036 7820 7820       |.| - 6x x 
│ │ │ │ +0002a590: 202d 2034 3678 2078 2020 2d20 3434 7820   - 46x x  - 44x 
│ │ │ │ +0002a5a0: 7820 202d 2032 3278 2078 2020 2b20 3336  x  - 22x x  + 36
│ │ │ │ +0002a5b0: 7820 7820 2020 2d20 3235 7820 7820 2020  x x   - 25x x   
│ │ │ │ +0002a5c0: 2b20 3134 7820 7820 2020 2b20 3135 7820  + 14x x   + 15x 
│ │ │ │ +0002a5d0: 7820 2020 207c 0a7c 2020 2020 2034 2038  x    |.|     4 8
│ │ │ │ +0002a5e0: 2020 2020 2020 3520 3820 2020 2020 2036        5 8      6
│ │ │ │ +0002a5f0: 2038 2020 2020 2020 3520 3920 2020 2020   8      5 9     
│ │ │ │ +0002a600: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ +0002a610: 2020 2020 2033 2031 3020 2020 2020 2034       3 10      4
│ │ │ │ +0002a620: 2031 3020 207c 0a7c 2020 2020 2020 2020   10  |.|        
│ │ │ │  0002a630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a660: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a670: 0a7c 2d20 3235 7820 7820 202d 2038 7820  .|- 25x x  - 8x 
│ │ │ │ -0002a680: 7820 202b 2035 7820 7820 202d 2032 3578  x  + 5x x  - 25x
│ │ │ │ -0002a690: 2078 2020 2d20 3230 7820 7820 2020 2d20   x  - 20x x   - 
│ │ │ │ -0002a6a0: 3435 7820 7820 2020 2b20 3238 7820 7820  45x x   + 28x x 
│ │ │ │ -0002a6b0: 2020 2b20 3230 7820 7820 2020 2d20 207c    + 20x x   -  |
│ │ │ │ -0002a6c0: 0a7c 2020 2020 2033 2038 2020 2020 2034  .|     3 8     4
│ │ │ │ -0002a6d0: 2038 2020 2020 2035 2038 2020 2020 2020   8     5 8      
│ │ │ │ -0002a6e0: 3520 3920 2020 2020 2031 2031 3020 2020  5 9      1 10   
│ │ │ │ -0002a6f0: 2020 2032 2031 3020 2020 2020 2033 2031     2 10      3 1
│ │ │ │ -0002a700: 3020 2020 2020 2034 2031 3020 2020 207c  0      4 10    |
│ │ │ │ -0002a710: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a670: 2020 2020 207c 0a7c 2d20 3235 7820 7820       |.|- 25x x 
│ │ │ │ +0002a680: 202d 2038 7820 7820 202b 2035 7820 7820   - 8x x  + 5x x 
│ │ │ │ +0002a690: 202d 2032 3578 2078 2020 2d20 3230 7820   - 25x x  - 20x 
│ │ │ │ +0002a6a0: 7820 2020 2d20 3435 7820 7820 2020 2b20  x   - 45x x   + 
│ │ │ │ +0002a6b0: 3238 7820 7820 2020 2b20 3230 7820 7820  28x x   + 20x x 
│ │ │ │ +0002a6c0: 2020 2d20 207c 0a7c 2020 2020 2033 2038    -  |.|     3 8
│ │ │ │ +0002a6d0: 2020 2020 2034 2038 2020 2020 2035 2038       4 8     5 8
│ │ │ │ +0002a6e0: 2020 2020 2020 3520 3920 2020 2020 2031        5 9      1
│ │ │ │ +0002a6f0: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ +0002a700: 2020 2033 2031 3020 2020 2020 2034 2031     3 10      4 1
│ │ │ │ +0002a710: 3020 2020 207c 0a7c 2020 2020 2020 2020  0    |.|        
│ │ │ │  0002a720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a750: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a760: 0a7c 7820 202d 2033 3978 2078 2020 2d20  .|x  - 39x x  - 
│ │ │ │ -0002a770: 3431 7820 7820 202b 2031 3978 2078 2020  41x x  + 19x x  
│ │ │ │ -0002a780: 2b20 3430 7820 7820 202b 2034 3378 2078  + 40x x  + 43x x
│ │ │ │ -0002a790: 2020 2d20 3339 7820 7820 202d 2033 3778    - 39x x  - 37x
│ │ │ │ -0002a7a0: 2078 2020 2b20 3378 2078 2020 2b20 207c   x  + 3x x  +  |
│ │ │ │ -0002a7b0: 0a7c 2037 2020 2020 2020 3620 3720 2020  .| 7      6 7   
│ │ │ │ -0002a7c0: 2020 2031 2038 2020 2020 2020 3220 3820     1 8      2 8 
│ │ │ │ -0002a7d0: 2020 2020 2033 2038 2020 2020 2020 3420       3 8      4 
│ │ │ │ -0002a7e0: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ -0002a7f0: 3620 3820 2020 2020 3120 3920 2020 207c  6 8     1 9    |
│ │ │ │ -0002a800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a760: 2020 2020 207c 0a7c 7820 202d 2033 3978       |.|x  - 39x
│ │ │ │ +0002a770: 2078 2020 2d20 3431 7820 7820 202b 2031   x  - 41x x  + 1
│ │ │ │ +0002a780: 3978 2078 2020 2b20 3430 7820 7820 202b  9x x  + 40x x  +
│ │ │ │ +0002a790: 2034 3378 2078 2020 2d20 3339 7820 7820   43x x  - 39x x 
│ │ │ │ +0002a7a0: 202d 2033 3778 2078 2020 2b20 3378 2078   - 37x x  + 3x x
│ │ │ │ +0002a7b0: 2020 2b20 207c 0a7c 2037 2020 2020 2020    +  |.| 7      
│ │ │ │ +0002a7c0: 3620 3720 2020 2020 2031 2038 2020 2020  6 7      1 8    
│ │ │ │ +0002a7d0: 2020 3220 3820 2020 2020 2033 2038 2020    2 8      3 8  
│ │ │ │ +0002a7e0: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ +0002a7f0: 2020 2020 2020 3620 3820 2020 2020 3120        6 8     1 
│ │ │ │ +0002a800: 3920 2020 207c 0a7c 2020 2020 2020 2020  9    |.|        
│ │ │ │  0002a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a850: 0a7c 2020 2b20 3436 7820 7820 202d 2031  .|  + 46x x  - 1
│ │ │ │ -0002a860: 3678 2078 2020 2d20 3336 7820 7820 202b  6x x  - 36x x  +
│ │ │ │ -0002a870: 2039 7820 7820 202d 2032 3778 2078 2020   9x x  - 27x x  
│ │ │ │ -0002a880: 2d20 3436 7820 7820 202d 2034 3478 2078  - 46x x  - 44x x
│ │ │ │ -0002a890: 2020 2d20 3239 7820 7820 202d 2020 207c    - 29x x  -   |
│ │ │ │ -0002a8a0: 0a7c 3720 2020 2020 2035 2037 2020 2020  .|7      5 7    
│ │ │ │ -0002a8b0: 2020 3620 3720 2020 2020 2031 2038 2020    6 7      1 8  
│ │ │ │ -0002a8c0: 2020 2032 2038 2020 2020 2020 3320 3820     2 8      3 8 
│ │ │ │ -0002a8d0: 2020 2020 2034 2038 2020 2020 2020 3520       4 8      5 
│ │ │ │ -0002a8e0: 3820 2020 2020 2036 2038 2020 2020 207c  8      6 8     |
│ │ │ │ -0002a8f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a850: 2020 2020 207c 0a7c 2020 2b20 3436 7820       |.|  + 46x 
│ │ │ │ +0002a860: 7820 202d 2031 3678 2078 2020 2d20 3336  x  - 16x x  - 36
│ │ │ │ +0002a870: 7820 7820 202b 2039 7820 7820 202d 2032  x x  + 9x x  - 2
│ │ │ │ +0002a880: 3778 2078 2020 2d20 3436 7820 7820 202d  7x x  - 46x x  -
│ │ │ │ +0002a890: 2034 3478 2078 2020 2d20 3239 7820 7820   44x x  - 29x x 
│ │ │ │ +0002a8a0: 202d 2020 207c 0a7c 3720 2020 2020 2035   -   |.|7      5
│ │ │ │ +0002a8b0: 2037 2020 2020 2020 3620 3720 2020 2020   7      6 7     
│ │ │ │ +0002a8c0: 2031 2038 2020 2020 2032 2038 2020 2020   1 8     2 8    
│ │ │ │ +0002a8d0: 2020 3320 3820 2020 2020 2034 2038 2020    3 8      4 8  
│ │ │ │ +0002a8e0: 2020 2020 3520 3820 2020 2020 2036 2038      5 8      6 8
│ │ │ │ +0002a8f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a940: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a940: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002a970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002a980: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002a990: 0a7c 2020 2b20 3333 7820 7820 202b 2033  .|  + 33x x  + 3
│ │ │ │ -0002a9a0: 3378 2078 2020 2d20 3231 7820 7820 202d  3x x  - 21x x  -
│ │ │ │ -0002a9b0: 2032 3578 2078 2020 2b20 3435 7820 7820   25x x  + 45x x 
│ │ │ │ -0002a9c0: 202b 2032 3378 2078 2020 2b20 3130 7820   + 23x x  + 10x 
│ │ │ │ -0002a9d0: 7820 202d 2037 7820 7820 202b 2020 207c  x  - 7x x  +   |
│ │ │ │ -0002a9e0: 0a7c 3620 2020 2020 2030 2037 2020 2020  .|6      0 7    
│ │ │ │ -0002a9f0: 2020 3220 3720 2020 2020 2033 2037 2020    2 7      3 7  
│ │ │ │ -0002aa00: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ -0002aa10: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ -0002aa20: 2038 2020 2020 2032 2038 2020 2020 207c   8     2 8     |
│ │ │ │ -0002aa30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002a980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002a990: 2020 2020 207c 0a7c 2020 2b20 3333 7820       |.|  + 33x 
│ │ │ │ +0002a9a0: 7820 202b 2033 3378 2078 2020 2d20 3231  x  + 33x x  - 21
│ │ │ │ +0002a9b0: 7820 7820 202d 2032 3578 2078 2020 2b20  x x  - 25x x  + 
│ │ │ │ +0002a9c0: 3435 7820 7820 202b 2032 3378 2078 2020  45x x  + 23x x  
│ │ │ │ +0002a9d0: 2b20 3130 7820 7820 202d 2037 7820 7820  + 10x x  - 7x x 
│ │ │ │ +0002a9e0: 202b 2020 207c 0a7c 3620 2020 2020 2030   +   |.|6      0
│ │ │ │ +0002a9f0: 2037 2020 2020 2020 3220 3720 2020 2020   7      2 7     
│ │ │ │ +0002aa00: 2033 2037 2020 2020 2020 3520 3720 2020   3 7      5 7   
│ │ │ │ +0002aa10: 2020 2036 2037 2020 2020 2020 3020 3820     6 7      0 8 
│ │ │ │ +0002aa20: 2020 2020 2031 2038 2020 2020 2032 2038       1 8     2 8
│ │ │ │ +0002aa30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002aa40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aa70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002aa80: 0a7c 202b 2032 7820 7820 202b 2033 3478  .| + 2x x  + 34x
│ │ │ │ -0002aa90: 2078 2020 2b20 3232 7820 7820 202b 2033   x  + 22x x  + 3
│ │ │ │ -0002aaa0: 3878 2078 2020 2b20 7820 7820 202b 2033  8x x  + x x  + 3
│ │ │ │ -0002aab0: 3078 2078 2020 2d20 3231 7820 7820 202d  0x x  - 21x x  -
│ │ │ │ -0002aac0: 2033 3478 2078 2020 2d20 3231 7820 207c   34x x  - 21x  |
│ │ │ │ -0002aad0: 0a7c 2020 2020 2032 2037 2020 2020 2020  .|     2 7      
│ │ │ │ -0002aae0: 3320 3720 2020 2020 2035 2037 2020 2020  3 7      5 7    
│ │ │ │ -0002aaf0: 2020 3620 3720 2020 2030 2038 2020 2020    6 7    0 8    
│ │ │ │ -0002ab00: 2020 3120 3820 2020 2020 2032 2038 2020    1 8      2 8  
│ │ │ │ -0002ab10: 2020 2020 3320 3820 2020 2020 2020 207c      3 8        |
│ │ │ │ -0002ab20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002aa80: 2020 2020 207c 0a7c 202b 2032 7820 7820       |.| + 2x x 
│ │ │ │ +0002aa90: 202b 2033 3478 2078 2020 2b20 3232 7820   + 34x x  + 22x 
│ │ │ │ +0002aaa0: 7820 202b 2033 3878 2078 2020 2b20 7820  x  + 38x x  + x 
│ │ │ │ +0002aab0: 7820 202b 2033 3078 2078 2020 2d20 3231  x  + 30x x  - 21
│ │ │ │ +0002aac0: 7820 7820 202d 2033 3478 2078 2020 2d20  x x  - 34x x  - 
│ │ │ │ +0002aad0: 3231 7820 207c 0a7c 2020 2020 2032 2037  21x  |.|     2 7
│ │ │ │ +0002aae0: 2020 2020 2020 3320 3720 2020 2020 2035        3 7      5
│ │ │ │ +0002aaf0: 2037 2020 2020 2020 3620 3720 2020 2030   7      6 7    0
│ │ │ │ +0002ab00: 2038 2020 2020 2020 3120 3820 2020 2020   8      1 8     
│ │ │ │ +0002ab10: 2032 2038 2020 2020 2020 3320 3820 2020   2 8      3 8   
│ │ │ │ +0002ab20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ab30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ab60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ab70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ab60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ab70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ab80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ab90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002aba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002abb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002abc0: 0a7c 3231 7820 7820 202b 2033 3078 2078  .|21x x  + 30x x
│ │ │ │ -0002abd0: 2020 2d20 7820 7820 202d 2035 7820 7820    - x x  - 5x x 
│ │ │ │ -0002abe0: 202d 2032 3978 2078 2020 2b20 3278 2078   - 29x x  + 2x x
│ │ │ │ -0002abf0: 2020 2d20 3239 7820 7820 202d 2032 3778    - 29x x  - 27x
│ │ │ │ -0002ac00: 2078 2020 2d20 3578 2078 2020 2b20 207c   x  - 5x x  +  |
│ │ │ │ -0002ac10: 0a7c 2020 2032 2036 2020 2020 2020 3320  .|   2 6      3 
│ │ │ │ -0002ac20: 3620 2020 2034 2036 2020 2020 2035 2036  6    4 6     5 6
│ │ │ │ -0002ac30: 2020 2020 2020 3020 3720 2020 2020 3120        0 7     1 
│ │ │ │ -0002ac40: 3720 2020 2020 2032 2037 2020 2020 2020  7      2 7      
│ │ │ │ -0002ac50: 3320 3720 2020 2020 3520 3720 2020 207c  3 7     5 7    |
│ │ │ │ -0002ac60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002abb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002abc0: 2020 2020 207c 0a7c 3231 7820 7820 202b       |.|21x x  +
│ │ │ │ +0002abd0: 2033 3078 2078 2020 2d20 7820 7820 202d   30x x  - x x  -
│ │ │ │ +0002abe0: 2035 7820 7820 202d 2032 3978 2078 2020   5x x  - 29x x  
│ │ │ │ +0002abf0: 2b20 3278 2078 2020 2d20 3239 7820 7820  + 2x x  - 29x x 
│ │ │ │ +0002ac00: 202d 2032 3778 2078 2020 2d20 3578 2078   - 27x x  - 5x x
│ │ │ │ +0002ac10: 2020 2b20 207c 0a7c 2020 2032 2036 2020    +  |.|   2 6  
│ │ │ │ +0002ac20: 2020 2020 3320 3620 2020 2034 2036 2020      3 6    4 6  
│ │ │ │ +0002ac30: 2020 2035 2036 2020 2020 2020 3020 3720     5 6      0 7 
│ │ │ │ +0002ac40: 2020 2020 3120 3720 2020 2020 2032 2037      1 7      2 7
│ │ │ │ +0002ac50: 2020 2020 2020 3320 3720 2020 2020 3520        3 7     5 
│ │ │ │ +0002ac60: 3720 2020 207c 0a7c 2020 2020 2020 2020  7    |.|        
│ │ │ │  0002ac70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ac90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002aca0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002acb0: 0a7c 2020 2b20 3331 7820 7820 202b 2032  .|  + 31x x  + 2
│ │ │ │ -0002acc0: 3978 2078 2020 2b20 3130 7820 7820 202d  9x x  + 10x x  -
│ │ │ │ -0002acd0: 2032 3778 2078 2020 2d20 3331 7820 7820   27x x  - 31x x 
│ │ │ │ -0002ace0: 202d 2031 3778 2078 2020 2d20 3238 7820   - 17x x  - 28x 
│ │ │ │ -0002acf0: 7820 202b 2033 3978 2078 2020 2d20 207c  x  + 39x x  -  |
│ │ │ │ -0002ad00: 0a7c 3620 2020 2020 2030 2037 2020 2020  .|6      0 7    
│ │ │ │ -0002ad10: 2020 3220 3720 2020 2020 2033 2037 2020    2 7      3 7  
│ │ │ │ -0002ad20: 2020 2020 3520 3720 2020 2020 2036 2037      5 7      6 7
│ │ │ │ -0002ad30: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ -0002ad40: 2038 2020 2020 2020 3220 3820 2020 207c   8      2 8    |
│ │ │ │ -0002ad50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002aca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002acb0: 2020 2020 207c 0a7c 2020 2b20 3331 7820       |.|  + 31x 
│ │ │ │ +0002acc0: 7820 202b 2032 3978 2078 2020 2b20 3130  x  + 29x x  + 10
│ │ │ │ +0002acd0: 7820 7820 202d 2032 3778 2078 2020 2d20  x x  - 27x x  - 
│ │ │ │ +0002ace0: 3331 7820 7820 202d 2031 3778 2078 2020  31x x  - 17x x  
│ │ │ │ +0002acf0: 2d20 3238 7820 7820 202b 2033 3978 2078  - 28x x  + 39x x
│ │ │ │ +0002ad00: 2020 2d20 207c 0a7c 3620 2020 2020 2030    -  |.|6      0
│ │ │ │ +0002ad10: 2037 2020 2020 2020 3220 3720 2020 2020   7      2 7     
│ │ │ │ +0002ad20: 2033 2037 2020 2020 2020 3520 3720 2020   3 7      5 7   
│ │ │ │ +0002ad30: 2020 2036 2037 2020 2020 2020 3020 3820     6 7      0 8 
│ │ │ │ +0002ad40: 2020 2020 2031 2038 2020 2020 2020 3220       1 8      2 
│ │ │ │ +0002ad50: 3820 2020 207c 0a7c 2020 2020 2020 2020  8    |.|        
│ │ │ │  0002ad60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ad80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ad90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ada0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ad90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ada0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002adb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002adc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002add0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ade0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002adf0: 0a7c 2073 7562 7661 7269 6574 7920 6f66  .| subvariety of
│ │ │ │ -0002ae00: 2050 505e 3131 2074 6f20 5050 5e38 2920   PP^11 to PP^8) 
│ │ │ │ -0002ae10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ade0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002adf0: 2020 2020 207c 0a7c 2073 7562 7661 7269       |.| subvari
│ │ │ │ +0002ae00: 6574 7920 6f66 2050 505e 3131 2074 6f20  ety of PP^11 to 
│ │ │ │ +0002ae10: 5050 5e38 2920 2020 2020 2020 2020 2020  PP^8)           
│ │ │ │  0002ae20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ae30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ae40: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0002ae30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ae40: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  0002ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0002ae90: 0a7c 2020 2b20 3131 7820 7820 202d 2033  .|  + 11x x  - 3
│ │ │ │ -0002aea0: 3778 2078 2020 2d20 3233 7820 7820 202d  7x x  - 23x x  -
│ │ │ │ -0002aeb0: 2033 3378 2078 2020 2b20 3878 2078 2020   33x x  + 8x x  
│ │ │ │ -0002aec0: 2b20 3130 7820 7820 202d 2032 3578 2078  + 10x x  - 25x x
│ │ │ │ -0002aed0: 2020 2d20 3978 2078 2020 2b20 3378 207c    - 9x x  + 3x |
│ │ │ │ -0002aee0: 0a7c 3820 2020 2020 2031 2038 2020 2020  .|8      1 8    
│ │ │ │ -0002aef0: 2020 3320 3820 2020 2020 2034 2038 2020    3 8      4 8  
│ │ │ │ -0002af00: 2020 2020 3620 3820 2020 2020 3020 3920      6 8     0 9 
│ │ │ │ -0002af10: 2020 2020 2031 2039 2020 2020 2020 3220       1 9      2 
│ │ │ │ -0002af20: 3920 2020 2020 3320 3920 2020 2020 347c  9     3 9     4|
│ │ │ │ -0002af30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ae90: 2d2d 2d2d 2d7c 0a7c 2020 2b20 3131 7820  -----|.|  + 11x 
│ │ │ │ +0002aea0: 7820 202d 2033 3778 2078 2020 2d20 3233  x  - 37x x  - 23
│ │ │ │ +0002aeb0: 7820 7820 202d 2033 3378 2078 2020 2b20  x x  - 33x x  + 
│ │ │ │ +0002aec0: 3878 2078 2020 2b20 3130 7820 7820 202d  8x x  + 10x x  -
│ │ │ │ +0002aed0: 2032 3578 2078 2020 2d20 3978 2078 2020   25x x  - 9x x  
│ │ │ │ +0002aee0: 2b20 3378 207c 0a7c 3820 2020 2020 2031  + 3x |.|8      1
│ │ │ │ +0002aef0: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ +0002af00: 2034 2038 2020 2020 2020 3620 3820 2020   4 8      6 8   
│ │ │ │ +0002af10: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ +0002af20: 2020 2020 3220 3920 2020 2020 3320 3920      2 9     3 9 
│ │ │ │ +0002af30: 2020 2020 347c 0a7c 2020 2020 2020 2020      4|.|        
│ │ │ │  0002af40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002af60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002af70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002af80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002af70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002af80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002af90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002afc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002afd0: 0a7c 7820 202b 2034 3178 2078 2020 2d20  .|x  + 41x x  - 
│ │ │ │ -0002afe0: 3778 2078 2020 2d20 3334 7820 7820 202d  7x x  - 34x x  -
│ │ │ │ -0002aff0: 2039 7820 7820 202d 2034 3678 2078 2020   9x x  - 46x x  
│ │ │ │ -0002b000: 2d20 3137 7820 7820 202b 2033 3278 2078  - 17x x  + 32x x
│ │ │ │ -0002b010: 2020 2d20 3878 2078 2020 2d20 3335 787c    - 8x x  - 35x|
│ │ │ │ -0002b020: 0a7c 2037 2020 2020 2020 3020 3820 2020  .| 7      0 8   
│ │ │ │ -0002b030: 2020 3120 3820 2020 2020 2033 2038 2020    1 8      3 8  
│ │ │ │ -0002b040: 2020 2034 2038 2020 2020 2020 3620 3820     4 8      6 8 
│ │ │ │ -0002b050: 2020 2020 2030 2039 2020 2020 2020 3120       0 9      1 
│ │ │ │ -0002b060: 3920 2020 2020 3220 3920 2020 2020 207c  9     2 9      |
│ │ │ │ -0002b070: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002afd0: 2020 2020 207c 0a7c 7820 202b 2034 3178       |.|x  + 41x
│ │ │ │ +0002afe0: 2078 2020 2d20 3778 2078 2020 2d20 3334   x  - 7x x  - 34
│ │ │ │ +0002aff0: 7820 7820 202d 2039 7820 7820 202d 2034  x x  - 9x x  - 4
│ │ │ │ +0002b000: 3678 2078 2020 2d20 3137 7820 7820 202b  6x x  - 17x x  +
│ │ │ │ +0002b010: 2033 3278 2078 2020 2d20 3878 2078 2020   32x x  - 8x x  
│ │ │ │ +0002b020: 2d20 3335 787c 0a7c 2037 2020 2020 2020  - 35x|.| 7      
│ │ │ │ +0002b030: 3020 3820 2020 2020 3120 3820 2020 2020  0 8     1 8     
│ │ │ │ +0002b040: 2033 2038 2020 2020 2034 2038 2020 2020   3 8     4 8    
│ │ │ │ +0002b050: 2020 3620 3820 2020 2020 2030 2039 2020    6 8      0 9  
│ │ │ │ +0002b060: 2020 2020 3120 3920 2020 2020 3220 3920      1 9     2 9 
│ │ │ │ +0002b070: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b0b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b0c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b0c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b100: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b110: 0a7c 202b 2033 3478 2078 2020 2b20 3431  .| + 34x x  + 41
│ │ │ │ -0002b120: 7820 7820 202d 2078 2078 2020 2b20 7820  x x  - x x  + x 
│ │ │ │ -0002b130: 7820 202b 2034 3078 2078 2020 2d20 3332  x  + 40x x  - 32
│ │ │ │ -0002b140: 7820 7820 202b 2035 7820 7820 202d 2031  x x  + 5x x  - 1
│ │ │ │ -0002b150: 3178 2078 2020 2d20 3230 7820 7820 207c  1x x  - 20x x  |
│ │ │ │ -0002b160: 0a7c 2020 2020 2020 3120 3820 2020 2020  .|      1 8     
│ │ │ │ -0002b170: 2033 2038 2020 2020 3420 3820 2020 2036   3 8    4 8    6
│ │ │ │ -0002b180: 2038 2020 2020 2020 3020 3920 2020 2020   8      0 9     
│ │ │ │ -0002b190: 2031 2039 2020 2020 2032 2039 2020 2020   1 9     2 9    
│ │ │ │ -0002b1a0: 2020 3320 3920 2020 2020 2034 2039 207c    3 9      4 9 |
│ │ │ │ -0002b1b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b110: 2020 2020 207c 0a7c 202b 2033 3478 2078       |.| + 34x x
│ │ │ │ +0002b120: 2020 2b20 3431 7820 7820 202d 2078 2078    + 41x x  - x x
│ │ │ │ +0002b130: 2020 2b20 7820 7820 202b 2034 3078 2078    + x x  + 40x x
│ │ │ │ +0002b140: 2020 2d20 3332 7820 7820 202b 2035 7820    - 32x x  + 5x 
│ │ │ │ +0002b150: 7820 202d 2031 3178 2078 2020 2d20 3230  x  - 11x x  - 20
│ │ │ │ +0002b160: 7820 7820 207c 0a7c 2020 2020 2020 3120  x x  |.|      1 
│ │ │ │ +0002b170: 3820 2020 2020 2033 2038 2020 2020 3420  8      3 8    4 
│ │ │ │ +0002b180: 3820 2020 2036 2038 2020 2020 2020 3020  8    6 8      0 
│ │ │ │ +0002b190: 3920 2020 2020 2031 2039 2020 2020 2032  9      1 9     2
│ │ │ │ +0002b1a0: 2039 2020 2020 2020 3320 3920 2020 2020   9      3 9     
│ │ │ │ +0002b1b0: 2034 2039 207c 0a7c 2020 2020 2020 2020   4 9 |.|        
│ │ │ │  0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b1f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b200: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b250: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b2a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b2a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b2e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b2f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b2f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b330: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b340: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b340: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b380: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b390: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b3d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b3e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b3e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b430: 0a7c 202b 2033 3678 2078 2020 202b 2033  .| + 36x x   + 3
│ │ │ │ -0002b440: 3678 2078 2020 202d 2033 3678 2078 2020  6x x   - 36x x  
│ │ │ │ -0002b450: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0002b420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b430: 2020 2020 207c 0a7c 202b 2033 3678 2078       |.| + 36x x
│ │ │ │ +0002b440: 2020 202b 2033 3678 2078 2020 202d 2033     + 36x x   - 3
│ │ │ │ +0002b450: 3678 2078 2020 2c20 2020 2020 2020 2020  6x x  ,         
│ │ │ │  0002b460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b480: 0a7c 2020 2020 2020 3520 3130 2020 2020  .|      5 10    
│ │ │ │ -0002b490: 2020 3620 3130 2020 2020 2020 3520 3131    6 10      5 11
│ │ │ │ -0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b480: 2020 2020 207c 0a7c 2020 2020 2020 3520       |.|      5 
│ │ │ │ +0002b490: 3130 2020 2020 2020 3620 3130 2020 2020  10      6 10    
│ │ │ │ +0002b4a0: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │  0002b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b4c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b4d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b4d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b510: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b520: 0a7c 2032 3578 2078 2020 2c20 2020 2020  .| 25x x  ,     
│ │ │ │ -0002b530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b520: 2020 2020 207c 0a7c 2032 3578 2078 2020       |.| 25x x  
│ │ │ │ +0002b530: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │  0002b540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b560: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b570: 0a7c 2020 2020 3520 3130 2020 2020 2020  .|    5 10      
│ │ │ │ +0002b560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b570: 2020 2020 207c 0a7c 2020 2020 3520 3130       |.|    5 10
│ │ │ │  0002b580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b5c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b5c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b610: 0a7c 2033 3878 2078 2020 2d20 3278 2078  .| 38x x  - 2x x
│ │ │ │ -0002b620: 2020 2b20 3339 7820 7820 202d 2031 3178    + 39x x  - 11x
│ │ │ │ -0002b630: 2078 2020 2d20 3238 7820 7820 2020 2b20   x  - 28x x   + 
│ │ │ │ -0002b640: 3231 7820 7820 2020 2d20 3578 2078 2020  21x x   - 5x x  
│ │ │ │ -0002b650: 202d 2031 3078 2078 2020 202b 2078 207c   - 10x x   + x |
│ │ │ │ -0002b660: 0a7c 2020 2020 3220 3920 2020 2020 3320  .|    2 9     3 
│ │ │ │ -0002b670: 3920 2020 2020 2034 2039 2020 2020 2020  9      4 9      
│ │ │ │ -0002b680: 3520 3920 2020 2020 2031 2031 3020 2020  5 9      1 10   
│ │ │ │ -0002b690: 2020 2032 2031 3020 2020 2020 3320 3130     2 10     3 10
│ │ │ │ -0002b6a0: 2020 2020 2020 3420 3130 2020 2020 357c        4 10    5|
│ │ │ │ -0002b6b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b610: 2020 2020 207c 0a7c 2033 3878 2078 2020       |.| 38x x  
│ │ │ │ +0002b620: 2d20 3278 2078 2020 2b20 3339 7820 7820  - 2x x  + 39x x 
│ │ │ │ +0002b630: 202d 2031 3178 2078 2020 2d20 3238 7820   - 11x x  - 28x 
│ │ │ │ +0002b640: 7820 2020 2b20 3231 7820 7820 2020 2d20  x   + 21x x   - 
│ │ │ │ +0002b650: 3578 2078 2020 202d 2031 3078 2078 2020  5x x   - 10x x  
│ │ │ │ +0002b660: 202b 2078 207c 0a7c 2020 2020 3220 3920   + x |.|    2 9 
│ │ │ │ +0002b670: 2020 2020 3320 3920 2020 2020 2034 2039      3 9      4 9
│ │ │ │ +0002b680: 2020 2020 2020 3520 3920 2020 2020 2031        5 9      1
│ │ │ │ +0002b690: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ +0002b6a0: 2020 3320 3130 2020 2020 2020 3420 3130    3 10      4 10
│ │ │ │ +0002b6b0: 2020 2020 357c 0a7c 2020 2020 2020 2020      5|.|        
│ │ │ │  0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b6f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b700: 0a7c 3137 7820 7820 202b 2031 3878 2078  .|17x x  + 18x x
│ │ │ │ -0002b710: 2020 2b20 3435 7820 7820 202b 2031 3678    + 45x x  + 16x
│ │ │ │ -0002b720: 2078 2020 2b20 3578 2078 2020 2b20 3336   x  + 5x x  + 36
│ │ │ │ -0002b730: 7820 7820 202b 2033 3978 2078 2020 202d  x x  + 39x x   -
│ │ │ │ -0002b740: 2033 3678 2078 2020 202d 2033 3378 207c   36x x   - 33x |
│ │ │ │ -0002b750: 0a7c 2020 2031 2039 2020 2020 2020 3220  .|   1 9      2 
│ │ │ │ -0002b760: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ -0002b770: 3420 3920 2020 2020 3520 3920 2020 2020  4 9     5 9     
│ │ │ │ -0002b780: 2036 2039 2020 2020 2020 3120 3130 2020   6 9      1 10  
│ │ │ │ -0002b790: 2020 2020 3220 3130 2020 2020 2020 337c      2 10      3|
│ │ │ │ -0002b7a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b700: 2020 2020 207c 0a7c 3137 7820 7820 202b       |.|17x x  +
│ │ │ │ +0002b710: 2031 3878 2078 2020 2b20 3435 7820 7820   18x x  + 45x x 
│ │ │ │ +0002b720: 202b 2031 3678 2078 2020 2b20 3578 2078   + 16x x  + 5x x
│ │ │ │ +0002b730: 2020 2b20 3336 7820 7820 202b 2033 3978    + 36x x  + 39x
│ │ │ │ +0002b740: 2078 2020 202d 2033 3678 2078 2020 202d   x   - 36x x   -
│ │ │ │ +0002b750: 2033 3378 207c 0a7c 2020 2031 2039 2020   33x |.|   1 9  
│ │ │ │ +0002b760: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ +0002b770: 2020 2020 2020 3420 3920 2020 2020 3520        4 9     5 
│ │ │ │ +0002b780: 3920 2020 2020 2036 2039 2020 2020 2020  9      6 9      
│ │ │ │ +0002b790: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ +0002b7a0: 2020 2020 337c 0a7c 2020 2020 2020 2020      3|.|        
│ │ │ │  0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b7e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b7f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b7f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002b800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b840: 0a7c 3578 2078 2020 2d20 3778 2078 2020  .|5x x  - 7x x  
│ │ │ │ -0002b850: 2b20 3435 7820 7820 202b 2032 3578 2078  + 45x x  + 25x x
│ │ │ │ -0002b860: 2020 2d20 3437 7820 7820 202b 2034 7820    - 47x x  + 4x 
│ │ │ │ -0002b870: 7820 202d 2032 3978 2078 2020 2b20 3332  x  - 29x x  + 32
│ │ │ │ -0002b880: 7820 7820 202d 2031 3378 2078 2020 2d7c  x x  - 13x x  -|
│ │ │ │ -0002b890: 0a7c 2020 3320 3820 2020 2020 3420 3820  .|  3 8     4 8 
│ │ │ │ -0002b8a0: 2020 2020 2035 2038 2020 2020 2020 3620       5 8      6 
│ │ │ │ -0002b8b0: 3820 2020 2020 2030 2039 2020 2020 2031  8      0 9     1
│ │ │ │ -0002b8c0: 2039 2020 2020 2020 3220 3920 2020 2020   9      2 9     
│ │ │ │ -0002b8d0: 2033 2039 2020 2020 2020 3420 3920 207c   3 9      4 9  |
│ │ │ │ -0002b8e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b840: 2020 2020 207c 0a7c 3578 2078 2020 2d20       |.|5x x  - 
│ │ │ │ +0002b850: 3778 2078 2020 2b20 3435 7820 7820 202b  7x x  + 45x x  +
│ │ │ │ +0002b860: 2032 3578 2078 2020 2d20 3437 7820 7820   25x x  - 47x x 
│ │ │ │ +0002b870: 202b 2034 7820 7820 202d 2032 3978 2078   + 4x x  - 29x x
│ │ │ │ +0002b880: 2020 2b20 3332 7820 7820 202d 2031 3378    + 32x x  - 13x
│ │ │ │ +0002b890: 2078 2020 2d7c 0a7c 2020 3320 3820 2020   x  -|.|  3 8   
│ │ │ │ +0002b8a0: 2020 3420 3820 2020 2020 2035 2038 2020    4 8      5 8  
│ │ │ │ +0002b8b0: 2020 2020 3620 3820 2020 2020 2030 2039      6 8      0 9
│ │ │ │ +0002b8c0: 2020 2020 2031 2039 2020 2020 2020 3220       1 9      2 
│ │ │ │ +0002b8d0: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ +0002b8e0: 3420 3920 207c 0a7c 2020 2020 2020 2020  4 9  |.|        
│ │ │ │  0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002b920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002b930: 0a7c 2078 2020 2b20 3338 7820 7820 202b  .| x  + 38x x  +
│ │ │ │ -0002b940: 2034 3678 2078 2020 2d20 3678 2078 2020   46x x  - 6x x  
│ │ │ │ -0002b950: 2b20 3132 7820 7820 202d 2034 3878 2078  + 12x x  - 48x x
│ │ │ │ -0002b960: 2020 2b20 3137 7820 7820 202d 2033 3978    + 17x x  - 39x
│ │ │ │ -0002b970: 2078 2020 2d20 3434 7820 7820 202d 207c   x  - 44x x  - |
│ │ │ │ -0002b980: 0a7c 3420 3820 2020 2020 2035 2038 2020  .|4 8      5 8  
│ │ │ │ -0002b990: 2020 2020 3620 3820 2020 2020 3020 3920      6 8     0 9 
│ │ │ │ -0002b9a0: 2020 2020 2031 2039 2020 2020 2020 3220       1 9      2 
│ │ │ │ -0002b9b0: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ -0002b9c0: 3420 3920 2020 2020 2035 2039 2020 207c  4 9      5 9   |
│ │ │ │ -0002b9d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002b920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002b930: 2020 2020 207c 0a7c 2078 2020 2b20 3338       |.| x  + 38
│ │ │ │ +0002b940: 7820 7820 202b 2034 3678 2078 2020 2d20  x x  + 46x x  - 
│ │ │ │ +0002b950: 3678 2078 2020 2b20 3132 7820 7820 202d  6x x  + 12x x  -
│ │ │ │ +0002b960: 2034 3878 2078 2020 2b20 3137 7820 7820   48x x  + 17x x 
│ │ │ │ +0002b970: 202d 2033 3978 2078 2020 2d20 3434 7820   - 39x x  - 44x 
│ │ │ │ +0002b980: 7820 202d 207c 0a7c 3420 3820 2020 2020  x  - |.|4 8     
│ │ │ │ +0002b990: 2035 2038 2020 2020 2020 3620 3820 2020   5 8      6 8   
│ │ │ │ +0002b9a0: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ +0002b9b0: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ +0002b9c0: 2020 2020 2020 3420 3920 2020 2020 2035        4 9      5
│ │ │ │ +0002b9d0: 2039 2020 207c 0a7c 2020 2020 2020 2020   9   |.|        
│ │ │ │  0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ba20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ba10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ba50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ba60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ba70: 0a7c 2034 7820 7820 202b 2031 3778 2078  .| 4x x  + 17x x
│ │ │ │ -0002ba80: 2020 2d20 3434 7820 7820 202b 2031 3878    - 44x x  + 18x
│ │ │ │ -0002ba90: 2078 2020 2d20 3432 7820 7820 202b 2033   x  - 42x x  + 3
│ │ │ │ -0002baa0: 3978 2078 2020 2b20 3978 2078 2020 2d20  9x x  + 9x x  - 
│ │ │ │ -0002bab0: 3133 7820 7820 202b 2034 3878 2078 207c  13x x  + 48x x |
│ │ │ │ -0002bac0: 0a7c 2020 2036 2037 2020 2020 2020 3020  .|   6 7      0 
│ │ │ │ -0002bad0: 3820 2020 2020 2031 2038 2020 2020 2020  8      1 8      
│ │ │ │ -0002bae0: 3220 3820 2020 2020 2033 2038 2020 2020  2 8      3 8    
│ │ │ │ -0002baf0: 2020 3420 3820 2020 2020 3520 3820 2020    4 8     5 8   
│ │ │ │ -0002bb00: 2020 2036 2038 2020 2020 2020 3020 397c     6 8      0 9|
│ │ │ │ -0002bb10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ba60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ba70: 2020 2020 207c 0a7c 2034 7820 7820 202b       |.| 4x x  +
│ │ │ │ +0002ba80: 2031 3778 2078 2020 2d20 3434 7820 7820   17x x  - 44x x 
│ │ │ │ +0002ba90: 202b 2031 3878 2078 2020 2d20 3432 7820   + 18x x  - 42x 
│ │ │ │ +0002baa0: 7820 202b 2033 3978 2078 2020 2b20 3978  x  + 39x x  + 9x
│ │ │ │ +0002bab0: 2078 2020 2d20 3133 7820 7820 202b 2034   x  - 13x x  + 4
│ │ │ │ +0002bac0: 3878 2078 207c 0a7c 2020 2036 2037 2020  8x x |.|   6 7  
│ │ │ │ +0002bad0: 2020 2020 3020 3820 2020 2020 2031 2038      0 8      1 8
│ │ │ │ +0002bae0: 2020 2020 2020 3220 3820 2020 2020 2033        2 8      3
│ │ │ │ +0002baf0: 2038 2020 2020 2020 3420 3820 2020 2020   8      4 8     
│ │ │ │ +0002bb00: 3520 3820 2020 2020 2036 2038 2020 2020  5 8      6 8    
│ │ │ │ +0002bb10: 2020 3020 397c 0a7c 2020 2020 2020 2020    0 9|.|        
│ │ │ │  0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bb50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002bb60: 0a7c 2033 3278 2078 2020 2d20 3778 2078  .| 32x x  - 7x x
│ │ │ │ -0002bb70: 2020 2d20 3239 7820 7820 202d 2034 3778    - 29x x  - 47x
│ │ │ │ -0002bb80: 2078 2020 2b20 3335 7820 7820 202b 2033   x  + 35x x  + 3
│ │ │ │ -0002bb90: 3878 2078 2020 2d20 3433 7820 7820 202d  8x x  - 43x x  -
│ │ │ │ -0002bba0: 2032 7820 7820 202d 2033 3278 2078 207c   2x x  - 32x x |
│ │ │ │ -0002bbb0: 0a7c 2020 2020 3320 3820 2020 2020 3420  .|    3 8     4 
│ │ │ │ -0002bbc0: 3820 2020 2020 2035 2038 2020 2020 2020  8      5 8      
│ │ │ │ -0002bbd0: 3620 3820 2020 2020 2030 2039 2020 2020  6 8      0 9    
│ │ │ │ -0002bbe0: 2020 3120 3920 2020 2020 2032 2039 2020    1 9      2 9  
│ │ │ │ -0002bbf0: 2020 2033 2039 2020 2020 2020 3420 397c     3 9      4 9|
│ │ │ │ -0002bc00: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0002bb50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bb60: 2020 2020 207c 0a7c 2033 3278 2078 2020       |.| 32x x  
│ │ │ │ +0002bb70: 2d20 3778 2078 2020 2d20 3239 7820 7820  - 7x x  - 29x x 
│ │ │ │ +0002bb80: 202d 2034 3778 2078 2020 2b20 3335 7820   - 47x x  + 35x 
│ │ │ │ +0002bb90: 7820 202b 2033 3878 2078 2020 2d20 3433  x  + 38x x  - 43
│ │ │ │ +0002bba0: 7820 7820 202d 2032 7820 7820 202d 2033  x x  - 2x x  - 3
│ │ │ │ +0002bbb0: 3278 2078 207c 0a7c 2020 2020 3320 3820  2x x |.|    3 8 
│ │ │ │ +0002bbc0: 2020 2020 3420 3820 2020 2020 2035 2038      4 8      5 8
│ │ │ │ +0002bbd0: 2020 2020 2020 3620 3820 2020 2020 2030        6 8      0
│ │ │ │ +0002bbe0: 2039 2020 2020 2020 3120 3920 2020 2020   9      1 9     
│ │ │ │ +0002bbf0: 2032 2039 2020 2020 2033 2039 2020 2020   2 9     3 9    
│ │ │ │ +0002bc00: 2020 3420 397c 0a7c 2d2d 2d2d 2d2d 2d2d    4 9|.|--------
│ │ │ │  0002bc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0002bc50: 0a7c 7820 202b 2032 3478 2078 2020 2d20  .|x  + 24x x  - 
│ │ │ │ -0002bc60: 3237 7820 7820 202d 2035 7820 7820 2020  27x x  - 5x x   
│ │ │ │ -0002bc70: 2b20 3238 7820 7820 2020 2b20 3337 7820  + 28x x   + 37x 
│ │ │ │ -0002bc80: 7820 2020 2b20 3978 2078 2020 202b 2032  x   + 9x x   + 2
│ │ │ │ -0002bc90: 3778 2078 2020 202d 2032 3578 2078 207c  7x x   - 25x x |
│ │ │ │ -0002bca0: 0a7c 2039 2020 2020 2020 3520 3920 2020  .| 9      5 9   
│ │ │ │ -0002bcb0: 2020 2036 2039 2020 2020 2030 2031 3020     6 9     0 10 
│ │ │ │ -0002bcc0: 2020 2020 2031 2031 3020 2020 2020 2032       1 10      2
│ │ │ │ -0002bcd0: 2031 3020 2020 2020 3420 3130 2020 2020   10     4 10    
│ │ │ │ -0002bce0: 2020 3620 3130 2020 2020 2020 3020 207c    6 10      0  |
│ │ │ │ -0002bcf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002bc50: 2d2d 2d2d 2d7c 0a7c 7820 202b 2032 3478  -----|.|x  + 24x
│ │ │ │ +0002bc60: 2078 2020 2d20 3237 7820 7820 202d 2035   x  - 27x x  - 5
│ │ │ │ +0002bc70: 7820 7820 2020 2b20 3238 7820 7820 2020  x x   + 28x x   
│ │ │ │ +0002bc80: 2b20 3337 7820 7820 2020 2b20 3978 2078  + 37x x   + 9x x
│ │ │ │ +0002bc90: 2020 202b 2032 3778 2078 2020 202d 2032     + 27x x   - 2
│ │ │ │ +0002bca0: 3578 2078 207c 0a7c 2039 2020 2020 2020  5x x |.| 9      
│ │ │ │ +0002bcb0: 3520 3920 2020 2020 2036 2039 2020 2020  5 9      6 9    
│ │ │ │ +0002bcc0: 2030 2031 3020 2020 2020 2031 2031 3020   0 10      1 10 
│ │ │ │ +0002bcd0: 2020 2020 2032 2031 3020 2020 2020 3420       2 10     4 
│ │ │ │ +0002bce0: 3130 2020 2020 2020 3620 3130 2020 2020  10      6 10    
│ │ │ │ +0002bcf0: 2020 3020 207c 0a7c 2020 2020 2020 2020    0  |.|        
│ │ │ │  0002bd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002bd40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002bd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bd40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bd80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002bd90: 0a7c 2078 2020 2d20 3436 7820 7820 202b  .| x  - 46x x  +
│ │ │ │ -0002bda0: 2032 3678 2078 2020 2b20 3137 7820 7820   26x x  + 17x x 
│ │ │ │ -0002bdb0: 202b 2031 3578 2078 2020 202b 2033 3578   + 15x x   + 35x
│ │ │ │ -0002bdc0: 2078 2020 202b 2033 3478 2078 2020 202b   x   + 34x x   +
│ │ │ │ -0002bdd0: 2032 3078 2078 2020 202b 2031 3478 207c   20x x   + 14x |
│ │ │ │ -0002bde0: 0a7c 3320 3920 2020 2020 2034 2039 2020  .|3 9      4 9  
│ │ │ │ -0002bdf0: 2020 2020 3520 3920 2020 2020 2036 2039      5 9      6 9
│ │ │ │ -0002be00: 2020 2020 2020 3020 3130 2020 2020 2020        0 10      
│ │ │ │ -0002be10: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ -0002be20: 2020 2020 3420 3130 2020 2020 2020 207c      4 10       |
│ │ │ │ -0002be30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bd90: 2020 2020 207c 0a7c 2078 2020 2d20 3436       |.| x  - 46
│ │ │ │ +0002bda0: 7820 7820 202b 2032 3678 2078 2020 2b20  x x  + 26x x  + 
│ │ │ │ +0002bdb0: 3137 7820 7820 202b 2031 3578 2078 2020  17x x  + 15x x  
│ │ │ │ +0002bdc0: 202b 2033 3578 2078 2020 202b 2033 3478   + 35x x   + 34x
│ │ │ │ +0002bdd0: 2078 2020 202b 2032 3078 2078 2020 202b   x   + 20x x   +
│ │ │ │ +0002bde0: 2031 3478 207c 0a7c 3320 3920 2020 2020   14x |.|3 9     
│ │ │ │ +0002bdf0: 2034 2039 2020 2020 2020 3520 3920 2020   4 9      5 9   
│ │ │ │ +0002be00: 2020 2036 2039 2020 2020 2020 3020 3130     6 9      0 10
│ │ │ │ +0002be10: 2020 2020 2020 3120 3130 2020 2020 2020        1 10      
│ │ │ │ +0002be20: 3220 3130 2020 2020 2020 3420 3130 2020  2 10      4 10  
│ │ │ │ +0002be30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002be40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002be60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002be70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002be80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002be70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002be80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bec0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002bed0: 0a7c 2b20 3435 7820 7820 202d 2031 3478  .|+ 45x x  - 14x
│ │ │ │ -0002bee0: 2078 2020 2d20 3235 7820 7820 2020 2b20   x  - 25x x   + 
│ │ │ │ -0002bef0: 3435 7820 7820 2020 2d20 3431 7820 7820  45x x   - 41x x 
│ │ │ │ -0002bf00: 2020 2d20 3436 7820 7820 2020 2b20 3878    - 46x x   + 8x
│ │ │ │ -0002bf10: 2078 2020 202d 2032 3878 2078 2020 207c   x   - 28x x   |
│ │ │ │ -0002bf20: 0a7c 2020 2020 2035 2039 2020 2020 2020  .|     5 9      
│ │ │ │ -0002bf30: 3620 3920 2020 2020 2030 2031 3020 2020  6 9      0 10   
│ │ │ │ -0002bf40: 2020 2031 2031 3020 2020 2020 2032 2031     1 10      2 1
│ │ │ │ -0002bf50: 3020 2020 2020 2034 2031 3020 2020 2020  0      4 10     
│ │ │ │ -0002bf60: 3620 3130 2020 2020 2020 3020 3131 207c  6 10      0 11 |
│ │ │ │ -0002bf70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bed0: 2020 2020 207c 0a7c 2b20 3435 7820 7820       |.|+ 45x x 
│ │ │ │ +0002bee0: 202d 2031 3478 2078 2020 2d20 3235 7820   - 14x x  - 25x 
│ │ │ │ +0002bef0: 7820 2020 2b20 3435 7820 7820 2020 2d20  x   + 45x x   - 
│ │ │ │ +0002bf00: 3431 7820 7820 2020 2d20 3436 7820 7820  41x x   - 46x x 
│ │ │ │ +0002bf10: 2020 2b20 3878 2078 2020 202d 2032 3878    + 8x x   - 28x
│ │ │ │ +0002bf20: 2078 2020 207c 0a7c 2020 2020 2035 2039   x   |.|     5 9
│ │ │ │ +0002bf30: 2020 2020 2020 3620 3920 2020 2020 2030        6 9      0
│ │ │ │ +0002bf40: 2031 3020 2020 2020 2031 2031 3020 2020   10      1 10   
│ │ │ │ +0002bf50: 2020 2032 2031 3020 2020 2020 2034 2031     2 10      4 1
│ │ │ │ +0002bf60: 3020 2020 2020 3620 3130 2020 2020 2020  0     6 10      
│ │ │ │ +0002bf70: 3020 3131 207c 0a7c 2020 2020 2020 2020  0 11 |.|        
│ │ │ │  0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002bfb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002bfc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002bfc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c010: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c060: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c060: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c0b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c0b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c0f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c100: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c100: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c150: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c150: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c190: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c1a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c1a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c1e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c1f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c1f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c240: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c290: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c2d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c2e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c2e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c330: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c330: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c380: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c380: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c3c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c3d0: 0a7c 7820 2020 2d20 7820 7820 2020 2d20  .|x   - x x   - 
│ │ │ │ -0002c3e0: 3436 7820 7820 2020 2b20 3278 2078 2020  46x x   + 2x x  
│ │ │ │ -0002c3f0: 202b 2078 2078 2020 2c20 2020 2020 2020   + x x  ,       
│ │ │ │ +0002c3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c3d0: 2020 2020 207c 0a7c 7820 2020 2d20 7820       |.|x   - x 
│ │ │ │ +0002c3e0: 7820 2020 2d20 3436 7820 7820 2020 2b20  x   - 46x x   + 
│ │ │ │ +0002c3f0: 3278 2078 2020 202b 2078 2078 2020 2c20  2x x   + x x  , 
│ │ │ │  0002c400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c420: 0a7c 2031 3020 2020 2036 2031 3020 2020  .| 10    6 10   
│ │ │ │ -0002c430: 2020 2031 2031 3120 2020 2020 3220 3131     1 11     2 11
│ │ │ │ -0002c440: 2020 2020 3520 3131 2020 2020 2020 2020      5 11        
│ │ │ │ +0002c410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c420: 2020 2020 207c 0a7c 2031 3020 2020 2036       |.| 10    6
│ │ │ │ +0002c430: 2031 3020 2020 2020 2031 2031 3120 2020   10      1 11   
│ │ │ │ +0002c440: 2020 3220 3131 2020 2020 3520 3131 2020    2 11    5 11  
│ │ │ │  0002c450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c460: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c470: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c470: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c4b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c4c0: 0a7c 7820 2020 2d20 3239 7820 7820 2020  .|x   - 29x x   
│ │ │ │ -0002c4d0: 2b20 3236 7820 7820 2020 2d20 3236 7820  + 26x x   - 26x 
│ │ │ │ -0002c4e0: 7820 2020 2d20 3431 7820 7820 2020 2d20  x   - 41x x   - 
│ │ │ │ -0002c4f0: 3435 7820 7820 2020 2d20 3336 7820 7820  45x x   - 36x x 
│ │ │ │ -0002c500: 2020 2b20 3236 7820 7820 202c 2020 207c    + 26x x  ,   |
│ │ │ │ -0002c510: 0a7c 2031 3020 2020 2020 2034 2031 3020  .| 10      4 10 
│ │ │ │ -0002c520: 2020 2020 2035 2031 3020 2020 2020 2036       5 10      6
│ │ │ │ -0002c530: 2031 3020 2020 2020 2031 2031 3120 2020   10      1 11   
│ │ │ │ -0002c540: 2020 2032 2031 3120 2020 2020 2034 2031     2 11      4 1
│ │ │ │ -0002c550: 3120 2020 2020 2035 2031 3120 2020 207c  1      5 11    |
│ │ │ │ -0002c560: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c4c0: 2020 2020 207c 0a7c 7820 2020 2d20 3239       |.|x   - 29
│ │ │ │ +0002c4d0: 7820 7820 2020 2b20 3236 7820 7820 2020  x x   + 26x x   
│ │ │ │ +0002c4e0: 2d20 3236 7820 7820 2020 2d20 3431 7820  - 26x x   - 41x 
│ │ │ │ +0002c4f0: 7820 2020 2d20 3435 7820 7820 2020 2d20  x   - 45x x   - 
│ │ │ │ +0002c500: 3336 7820 7820 2020 2b20 3236 7820 7820  36x x   + 26x x 
│ │ │ │ +0002c510: 202c 2020 207c 0a7c 2031 3020 2020 2020   ,   |.| 10     
│ │ │ │ +0002c520: 2034 2031 3020 2020 2020 2035 2031 3020   4 10      5 10 
│ │ │ │ +0002c530: 2020 2020 2036 2031 3020 2020 2020 2031       6 10      1
│ │ │ │ +0002c540: 2031 3120 2020 2020 2032 2031 3120 2020   11      2 11   
│ │ │ │ +0002c550: 2020 2034 2031 3120 2020 2020 2035 2031     4 11      5 1
│ │ │ │ +0002c560: 3120 2020 207c 0a7c 2020 2020 2020 2020  1    |.|        
│ │ │ │  0002c570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c5b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c5b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c5f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c600: 0a7c 2034 3778 2078 2020 2d20 3432 7820  .| 47x x  - 42x 
│ │ │ │ -0002c610: 7820 202d 2031 3878 2078 2020 202d 2035  x  - 18x x   - 5
│ │ │ │ -0002c620: 7820 7820 2020 2b20 3434 7820 7820 2020  x x   + 44x x   
│ │ │ │ -0002c630: 2d20 3339 7820 7820 2020 2d20 3135 7820  - 39x x   - 15x 
│ │ │ │ -0002c640: 7820 2020 2d20 3331 7820 7820 2020 207c  x   - 31x x    |
│ │ │ │ -0002c650: 0a7c 2020 2020 3520 3920 2020 2020 2036  .|    5 9      6
│ │ │ │ -0002c660: 2039 2020 2020 2020 3020 3130 2020 2020   9      0 10    
│ │ │ │ -0002c670: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ -0002c680: 2020 2020 2033 2031 3020 2020 2020 2034       3 10      4
│ │ │ │ -0002c690: 2031 3020 2020 2020 2035 2031 3020 207c   10      5 10  |
│ │ │ │ -0002c6a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c600: 2020 2020 207c 0a7c 2034 3778 2078 2020       |.| 47x x  
│ │ │ │ +0002c610: 2d20 3432 7820 7820 202d 2031 3878 2078  - 42x x  - 18x x
│ │ │ │ +0002c620: 2020 202d 2035 7820 7820 2020 2b20 3434     - 5x x   + 44
│ │ │ │ +0002c630: 7820 7820 2020 2d20 3339 7820 7820 2020  x x   - 39x x   
│ │ │ │ +0002c640: 2d20 3135 7820 7820 2020 2d20 3331 7820  - 15x x   - 31x 
│ │ │ │ +0002c650: 7820 2020 207c 0a7c 2020 2020 3520 3920  x    |.|    5 9 
│ │ │ │ +0002c660: 2020 2020 2036 2039 2020 2020 2020 3020       6 9      0 
│ │ │ │ +0002c670: 3130 2020 2020 2031 2031 3020 2020 2020  10     1 10     
│ │ │ │ +0002c680: 2032 2031 3020 2020 2020 2033 2031 3020   2 10      3 10 
│ │ │ │ +0002c690: 2020 2020 2034 2031 3020 2020 2020 2035       4 10      5
│ │ │ │ +0002c6a0: 2031 3020 207c 0a7c 2020 2020 2020 2020   10  |.|        
│ │ │ │  0002c6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c6e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c6f0: 0a7c 3578 2078 2020 202d 2031 3578 2078  .|5x x   - 15x x
│ │ │ │ -0002c700: 2020 202d 2031 3378 2078 2020 202d 2032     - 13x x   - 2
│ │ │ │ -0002c710: 3078 2078 2020 202d 2034 3578 2078 2020  0x x   - 45x x  
│ │ │ │ -0002c720: 202b 2034 7820 7820 2020 2d20 3478 2078   + 4x x   - 4x x
│ │ │ │ -0002c730: 2020 202b 2031 3078 2078 2020 202b 207c     + 10x x   + |
│ │ │ │ -0002c740: 0a7c 2020 3020 3130 2020 2020 2020 3120  .|  0 10      1 
│ │ │ │ -0002c750: 3130 2020 2020 2020 3220 3130 2020 2020  10      2 10    
│ │ │ │ -0002c760: 2020 3320 3130 2020 2020 2020 3420 3130    3 10      4 10
│ │ │ │ -0002c770: 2020 2020 2035 2031 3020 2020 2020 3620       5 10     6 
│ │ │ │ -0002c780: 3130 2020 2020 2020 3120 3131 2020 207c  10      1 11   |
│ │ │ │ -0002c790: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c6f0: 2020 2020 207c 0a7c 3578 2078 2020 202d       |.|5x x   -
│ │ │ │ +0002c700: 2031 3578 2078 2020 202d 2031 3378 2078   15x x   - 13x x
│ │ │ │ +0002c710: 2020 202d 2032 3078 2078 2020 202d 2034     - 20x x   - 4
│ │ │ │ +0002c720: 3578 2078 2020 202b 2034 7820 7820 2020  5x x   + 4x x   
│ │ │ │ +0002c730: 2d20 3478 2078 2020 202b 2031 3078 2078  - 4x x   + 10x x
│ │ │ │ +0002c740: 2020 202b 207c 0a7c 2020 3020 3130 2020     + |.|  0 10  
│ │ │ │ +0002c750: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ +0002c760: 3130 2020 2020 2020 3320 3130 2020 2020  10      3 10    
│ │ │ │ +0002c770: 2020 3420 3130 2020 2020 2035 2031 3020    4 10     5 10 
│ │ │ │ +0002c780: 2020 2020 3620 3130 2020 2020 2020 3120      6 10      1 
│ │ │ │ +0002c790: 3131 2020 207c 0a7c 2020 2020 2020 2020  11   |.|        
│ │ │ │  0002c7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c7d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c7e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c7e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002c7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c830: 0a7c 202d 2033 3278 2078 2020 2d20 3337  .| - 32x x  - 37
│ │ │ │ -0002c840: 7820 7820 202d 2034 7820 7820 202d 2033  x x  - 4x x  - 3
│ │ │ │ -0002c850: 3878 2078 2020 2b20 3130 7820 7820 202b  8x x  + 10x x  +
│ │ │ │ -0002c860: 2031 3378 2078 2020 2b20 3132 7820 7820   13x x  + 12x x 
│ │ │ │ -0002c870: 2020 2b20 3335 7820 7820 2020 2b20 207c    + 35x x   +  |
│ │ │ │ -0002c880: 0a7c 2020 2020 2020 3120 3920 2020 2020  .|      1 9     
│ │ │ │ -0002c890: 2032 2039 2020 2020 2033 2039 2020 2020   2 9     3 9    
│ │ │ │ -0002c8a0: 2020 3420 3920 2020 2020 2035 2039 2020    4 9      5 9  
│ │ │ │ -0002c8b0: 2020 2020 3620 3920 2020 2020 2030 2031      6 9      0 1
│ │ │ │ -0002c8c0: 3020 2020 2020 2031 2031 3020 2020 207c  0      1 10    |
│ │ │ │ -0002c8d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002c820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c830: 2020 2020 207c 0a7c 202d 2033 3278 2078       |.| - 32x x
│ │ │ │ +0002c840: 2020 2d20 3337 7820 7820 202d 2034 7820    - 37x x  - 4x 
│ │ │ │ +0002c850: 7820 202d 2033 3878 2078 2020 2b20 3130  x  - 38x x  + 10
│ │ │ │ +0002c860: 7820 7820 202b 2031 3378 2078 2020 2b20  x x  + 13x x  + 
│ │ │ │ +0002c870: 3132 7820 7820 2020 2b20 3335 7820 7820  12x x   + 35x x 
│ │ │ │ +0002c880: 2020 2b20 207c 0a7c 2020 2020 2020 3120    +  |.|      1 
│ │ │ │ +0002c890: 3920 2020 2020 2032 2039 2020 2020 2033  9      2 9     3
│ │ │ │ +0002c8a0: 2039 2020 2020 2020 3420 3920 2020 2020   9      4 9     
│ │ │ │ +0002c8b0: 2035 2039 2020 2020 2020 3620 3920 2020   5 9      6 9   
│ │ │ │ +0002c8c0: 2020 2030 2031 3020 2020 2020 2031 2031     0 10      1 1
│ │ │ │ +0002c8d0: 3020 2020 207c 0a7c 2020 2020 2020 2020  0    |.|        
│ │ │ │  0002c8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002c900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002c910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002c920: 0a7c 202d 2034 3378 2078 2020 2b20 3132  .| - 43x x  + 12
│ │ │ │ -0002c930: 7820 7820 202d 2031 3278 2078 2020 202b  x x  - 12x x   +
│ │ │ │ -0002c940: 2031 3478 2078 2020 202d 2034 3778 2078   14x x   - 47x x
│ │ │ │ -0002c950: 2020 202d 2034 3678 2078 2020 202d 2031     - 46x x   - 1
│ │ │ │ -0002c960: 3478 2078 2020 202b 2034 3878 2078 207c  4x x   + 48x x |
│ │ │ │ -0002c970: 0a7c 2020 2020 2020 3520 3920 2020 2020  .|      5 9     
│ │ │ │ -0002c980: 2036 2039 2020 2020 2020 3020 3130 2020   6 9      0 10  
│ │ │ │ -0002c990: 2020 2020 3120 3130 2020 2020 2020 3220      1 10      2 
│ │ │ │ -0002c9a0: 3130 2020 2020 2020 3320 3130 2020 2020  10      3 10    
│ │ │ │ -0002c9b0: 2020 3420 3130 2020 2020 2020 3520 207c    4 10      5  |
│ │ │ │ -0002c9c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0002c910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002c920: 2020 2020 207c 0a7c 202d 2034 3378 2078       |.| - 43x x
│ │ │ │ +0002c930: 2020 2b20 3132 7820 7820 202d 2031 3278    + 12x x  - 12x
│ │ │ │ +0002c940: 2078 2020 202b 2031 3478 2078 2020 202d   x   + 14x x   -
│ │ │ │ +0002c950: 2034 3778 2078 2020 202d 2034 3678 2078   47x x   - 46x x
│ │ │ │ +0002c960: 2020 202d 2031 3478 2078 2020 202b 2034     - 14x x   + 4
│ │ │ │ +0002c970: 3878 2078 207c 0a7c 2020 2020 2020 3520  8x x |.|      5 
│ │ │ │ +0002c980: 3920 2020 2020 2036 2039 2020 2020 2020  9      6 9      
│ │ │ │ +0002c990: 3020 3130 2020 2020 2020 3120 3130 2020  0 10      1 10  
│ │ │ │ +0002c9a0: 2020 2020 3220 3130 2020 2020 2020 3320      2 10      3 
│ │ │ │ +0002c9b0: 3130 2020 2020 2020 3420 3130 2020 2020  10      4 10    
│ │ │ │ +0002c9c0: 2020 3520 207c 0a7c 2d2d 2d2d 2d2d 2d2d    5  |.|--------
│ │ │ │  0002c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0002ca10: 0a7c 2020 202b 2039 7820 7820 2020 2b20  .|   + 9x x   + 
│ │ │ │ -0002ca20: 3237 7820 7820 2020 2d20 3237 7820 7820  27x x   - 27x x 
│ │ │ │ -0002ca30: 202c 2020 2020 2020 2020 2020 2020 2020   ,              
│ │ │ │ +0002ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002ca10: 2d2d 2d2d 2d7c 0a7c 2020 202b 2039 7820  -----|.|   + 9x 
│ │ │ │ +0002ca20: 7820 2020 2b20 3237 7820 7820 2020 2d20  x   + 27x x   - 
│ │ │ │ +0002ca30: 3237 7820 7820 202c 2020 2020 2020 2020  27x x  ,        
│ │ │ │  0002ca40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ca50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ca60: 0a7c 3131 2020 2020 2032 2031 3120 2020  .|11     2 11   
│ │ │ │ -0002ca70: 2020 2034 2031 3120 2020 2020 2035 2031     4 11      5 1
│ │ │ │ -0002ca80: 3120 2020 2020 2020 2020 2020 2020 2020  1               
│ │ │ │ +0002ca50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ca60: 2020 2020 207c 0a7c 3131 2020 2020 2032       |.|11     2
│ │ │ │ +0002ca70: 2031 3120 2020 2020 2034 2031 3120 2020   11      4 11   
│ │ │ │ +0002ca80: 2020 2035 2031 3120 2020 2020 2020 2020     5 11         
│ │ │ │  0002ca90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002caa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cab0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002caf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cb00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002caf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cb00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cb50: 0a7c 2078 2020 202b 2033 3678 2078 2020  .| x   + 36x x  
│ │ │ │ -0002cb60: 202b 2033 3578 2078 2020 202d 2031 3778   + 35x x   - 17x
│ │ │ │ -0002cb70: 2078 2020 2c20 2020 2020 2020 2020 2020   x  ,           
│ │ │ │ +0002cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cb50: 2020 2020 207c 0a7c 2078 2020 202b 2033       |.| x   + 3
│ │ │ │ +0002cb60: 3678 2078 2020 202b 2033 3578 2078 2020  6x x   + 35x x  
│ │ │ │ +0002cb70: 202d 2031 3778 2078 2020 2c20 2020 2020   - 17x x  ,     
│ │ │ │  0002cb80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cb90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cba0: 0a7c 3020 3131 2020 2020 2020 3120 3131  .|0 11      1 11
│ │ │ │ -0002cbb0: 2020 2020 2020 3220 3131 2020 2020 2020        2 11      
│ │ │ │ -0002cbc0: 3420 3131 2020 2020 2020 2020 2020 2020  4 11            
│ │ │ │ +0002cb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cba0: 2020 2020 207c 0a7c 3020 3131 2020 2020       |.|0 11    
│ │ │ │ +0002cbb0: 2020 3120 3131 2020 2020 2020 3220 3131    1 11      2 11
│ │ │ │ +0002cbc0: 2020 2020 2020 3420 3131 2020 2020 2020        4 11      
│ │ │ │  0002cbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cbe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cbf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cbf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cc40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cc80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cc90: 0a7c 202b 2031 3178 2078 2020 202b 2031  .| + 11x x   + 1
│ │ │ │ -0002cca0: 3478 2078 2020 202d 2038 7820 7820 2020  4x x   - 8x x   
│ │ │ │ -0002ccb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cc90: 2020 2020 207c 0a7c 202b 2031 3178 2078       |.| + 11x x
│ │ │ │ +0002cca0: 2020 202b 2031 3478 2078 2020 202d 2038     + 14x x   - 8
│ │ │ │ +0002ccb0: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │  0002ccc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ccd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cce0: 0a7c 2020 2020 2020 3220 3131 2020 2020  .|      2 11    
│ │ │ │ -0002ccf0: 2020 3420 3131 2020 2020 2035 2031 3120    4 11     5 11 
│ │ │ │ -0002cd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ccd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cce0: 2020 2020 207c 0a7c 2020 2020 2020 3220       |.|      2 
│ │ │ │ +0002ccf0: 3131 2020 2020 2020 3420 3131 2020 2020  11      4 11    
│ │ │ │ +0002cd00: 2035 2031 3120 2020 2020 2020 2020 2020   5 11           
│ │ │ │  0002cd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cd30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cd30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cd70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cd80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cd80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cdc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cdd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cdd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cdf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ce20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ce10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ce20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ce30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ce60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ce70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ce60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ce70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ce80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ce90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ceb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ceb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ced0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cf10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cf10: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cf20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cf50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cf60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cf50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cf60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cf70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cf90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cfa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002cfb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cfa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002cfb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002cfc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002cfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002cff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d000: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002cff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d050: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d0a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d0a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d0e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d0f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d0f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d140: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d190: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d1d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d1e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d1e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d230: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d230: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d270: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d280: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d280: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d2c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d2d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d2d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d320: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d320: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d370: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d370: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d3b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d3c0: 0a7c 2b20 3135 7820 7820 2020 2d20 3137  .|+ 15x x   - 17
│ │ │ │ -0002d3d0: 7820 7820 2020 2d20 3239 7820 7820 2020  x x   - 29x x   
│ │ │ │ -0002d3e0: 2b20 3335 7820 7820 2020 2d20 3137 7820  + 35x x   - 17x 
│ │ │ │ -0002d3f0: 7820 2020 2d20 3331 7820 7820 202c 2020  x   - 31x x  ,  
│ │ │ │ -0002d400: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d410: 0a7c 2020 2020 2036 2031 3020 2020 2020  .|     6 10     
│ │ │ │ -0002d420: 2030 2031 3120 2020 2020 2031 2031 3120   0 11      1 11 
│ │ │ │ -0002d430: 2020 2020 2032 2031 3120 2020 2020 2033       2 11      3
│ │ │ │ -0002d440: 2031 3120 2020 2020 2035 2031 3120 2020   11      5 11   
│ │ │ │ -0002d450: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d460: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d3c0: 2020 2020 207c 0a7c 2b20 3135 7820 7820       |.|+ 15x x 
│ │ │ │ +0002d3d0: 2020 2d20 3137 7820 7820 2020 2d20 3239    - 17x x   - 29
│ │ │ │ +0002d3e0: 7820 7820 2020 2b20 3335 7820 7820 2020  x x   + 35x x   
│ │ │ │ +0002d3f0: 2d20 3137 7820 7820 2020 2d20 3331 7820  - 17x x   - 31x 
│ │ │ │ +0002d400: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
│ │ │ │ +0002d410: 2020 2020 207c 0a7c 2020 2020 2036 2031       |.|     6 1
│ │ │ │ +0002d420: 3020 2020 2020 2030 2031 3120 2020 2020  0      0 11     
│ │ │ │ +0002d430: 2031 2031 3120 2020 2020 2032 2031 3120   1 11      2 11 
│ │ │ │ +0002d440: 2020 2020 2033 2031 3120 2020 2020 2035       3 11      5
│ │ │ │ +0002d450: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │ +0002d460: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d4b0: 0a7c 2038 7820 7820 2020 2b20 3478 2078  .| 8x x   + 4x x
│ │ │ │ -0002d4c0: 2020 2c20 2020 2020 2020 2020 2020 2020    ,             
│ │ │ │ +0002d4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d4b0: 2020 2020 207c 0a7c 2038 7820 7820 2020       |.| 8x x   
│ │ │ │ +0002d4c0: 2b20 3478 2078 2020 2c20 2020 2020 2020  + 4x x  ,       
│ │ │ │  0002d4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d4f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d500: 0a7c 2020 2032 2031 3120 2020 2020 3520  .|   2 11     5 
│ │ │ │ -0002d510: 3131 2020 2020 2020 2020 2020 2020 2020  11              
│ │ │ │ +0002d4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d500: 2020 2020 207c 0a7c 2020 2032 2031 3120       |.|   2 11 
│ │ │ │ +0002d510: 2020 2020 3520 3131 2020 2020 2020 2020      5 11        
│ │ │ │  0002d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d550: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d5a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d5a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d5e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d5f0: 0a7c 3133 7820 7820 2020 2b20 3331 7820  .|13x x   + 31x 
│ │ │ │ -0002d600: 7820 2020 2b20 3330 7820 7820 2020 2d20  x   + 30x x   - 
│ │ │ │ -0002d610: 3435 7820 7820 2020 2d20 3135 7820 7820  45x x   - 15x x 
│ │ │ │ -0002d620: 2020 2b20 3337 7820 7820 2020 2b20 3434    + 37x x   + 44
│ │ │ │ -0002d630: 7820 7820 2020 2b20 3778 2078 2020 207c  x x   + 7x x   |
│ │ │ │ -0002d640: 0a7c 2020 2032 2031 3020 2020 2020 2033  .|   2 10      3
│ │ │ │ -0002d650: 2031 3020 2020 2020 2034 2031 3020 2020   10      4 10   
│ │ │ │ -0002d660: 2020 2035 2031 3020 2020 2020 2036 2031     5 10      6 1
│ │ │ │ -0002d670: 3020 2020 2020 2030 2031 3120 2020 2020  0      0 11     
│ │ │ │ -0002d680: 2031 2031 3120 2020 2020 3220 3131 207c   1 11     2 11 |
│ │ │ │ -0002d690: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d5f0: 2020 2020 207c 0a7c 3133 7820 7820 2020       |.|13x x   
│ │ │ │ +0002d600: 2b20 3331 7820 7820 2020 2b20 3330 7820  + 31x x   + 30x 
│ │ │ │ +0002d610: 7820 2020 2d20 3435 7820 7820 2020 2d20  x   - 45x x   - 
│ │ │ │ +0002d620: 3135 7820 7820 2020 2b20 3337 7820 7820  15x x   + 37x x 
│ │ │ │ +0002d630: 2020 2b20 3434 7820 7820 2020 2b20 3778    + 44x x   + 7x
│ │ │ │ +0002d640: 2078 2020 207c 0a7c 2020 2032 2031 3020   x   |.|   2 10 
│ │ │ │ +0002d650: 2020 2020 2033 2031 3020 2020 2020 2034       3 10      4
│ │ │ │ +0002d660: 2031 3020 2020 2020 2035 2031 3020 2020   10      5 10   
│ │ │ │ +0002d670: 2020 2036 2031 3020 2020 2020 2030 2031     6 10      0 1
│ │ │ │ +0002d680: 3120 2020 2020 2031 2031 3120 2020 2020  1      1 11     
│ │ │ │ +0002d690: 3220 3131 207c 0a7c 2020 2020 2020 2020  2 11 |.|        
│ │ │ │  0002d6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d6d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d6e0: 0a7c 2020 202b 2031 3278 2078 2020 202d  .|   + 12x x   -
│ │ │ │ -0002d6f0: 2031 3278 2078 2020 202b 2034 3878 2078   12x x   + 48x x
│ │ │ │ -0002d700: 2020 202d 2078 2078 2020 202d 2039 7820     - x x   - 9x 
│ │ │ │ -0002d710: 7820 2020 2b20 3438 7820 7820 2020 2020  x   + 48x x     
│ │ │ │ -0002d720: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d730: 0a7c 3130 2020 2020 2020 3620 3130 2020  .|10      6 10  
│ │ │ │ -0002d740: 2020 2020 3020 3131 2020 2020 2020 3120      0 11      1 
│ │ │ │ -0002d750: 3131 2020 2020 3220 3131 2020 2020 2033  11    2 11     3
│ │ │ │ -0002d760: 2031 3120 2020 2020 2035 2031 3120 2020   11      5 11   
│ │ │ │ -0002d770: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d780: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0002d6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d6e0: 2020 2020 207c 0a7c 2020 202b 2031 3278       |.|   + 12x
│ │ │ │ +0002d6f0: 2078 2020 202d 2031 3278 2078 2020 202b   x   - 12x x   +
│ │ │ │ +0002d700: 2034 3878 2078 2020 202d 2078 2078 2020   48x x   - x x  
│ │ │ │ +0002d710: 202d 2039 7820 7820 2020 2b20 3438 7820   - 9x x   + 48x 
│ │ │ │ +0002d720: 7820 2020 2020 2020 2020 2020 2020 2020  x               
│ │ │ │ +0002d730: 2020 2020 207c 0a7c 3130 2020 2020 2020       |.|10      
│ │ │ │ +0002d740: 3620 3130 2020 2020 2020 3020 3131 2020  6 10      0 11  
│ │ │ │ +0002d750: 2020 2020 3120 3131 2020 2020 3220 3131      1 11    2 11
│ │ │ │ +0002d760: 2020 2020 2033 2031 3120 2020 2020 2035       3 11      5
│ │ │ │ +0002d770: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │ +0002d780: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  0002d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0002d7d0: 0a7c 2b20 3978 2078 2020 202d 2032 3578  .|+ 9x x   - 25x
│ │ │ │ -0002d7e0: 2078 2020 202d 2034 3578 2078 2020 2c20   x   - 45x x  , 
│ │ │ │ -0002d7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d7d0: 2d2d 2d2d 2d7c 0a7c 2b20 3978 2078 2020  -----|.|+ 9x x  
│ │ │ │ +0002d7e0: 202d 2032 3578 2078 2020 202d 2034 3578   - 25x x   - 45x
│ │ │ │ +0002d7f0: 2078 2020 2c20 2020 2020 2020 2020 2020   x  ,           
│ │ │ │  0002d800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d810: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d820: 0a7c 2020 2020 3320 3131 2020 2020 2020  .|    3 11      
│ │ │ │ -0002d830: 3420 3131 2020 2020 2020 3520 3131 2020  4 11      5 11  
│ │ │ │ -0002d840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d820: 2020 2020 207c 0a7c 2020 2020 3320 3131       |.|    3 11
│ │ │ │ +0002d830: 2020 2020 2020 3420 3131 2020 2020 2020        4 11      
│ │ │ │ +0002d840: 3520 3131 2020 2020 2020 2020 2020 2020  5 11            
│ │ │ │  0002d850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d860: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d870: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002d860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d870: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0002d880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002d8c0: 0a7c 6939 203a 202d 2d20 6275 742e 2e2e  .|i9 : -- but...
│ │ │ │ -0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002d8c0: 2d2d 2d2d 2d2b 0a7c 6939 203a 202d 2d20  -----+.|i9 : -- 
│ │ │ │ +0002d8d0: 6275 742e 2e2e 2020 2020 2020 2020 2020  but...          
│ │ │ │  0002d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d900: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d910: 0a7c 2020 2020 2070 6869 202a 2070 7369  .|     phi * psi
│ │ │ │ -0002d920: 203d 3d20 3120 2020 2020 2020 2020 2020   == 1           
│ │ │ │ +0002d900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d910: 2020 2020 207c 0a7c 2020 2020 2070 6869       |.|     phi
│ │ │ │ +0002d920: 202a 2070 7369 203d 3d20 3120 2020 2020   * psi == 1     
│ │ │ │  0002d930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d950: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d960: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002d950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d960: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002d970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002d9b0: 0a7c 6f39 203d 2066 616c 7365 2020 2020  .|o9 = false    
│ │ │ │ -0002d9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002d9b0: 2020 2020 207c 0a7c 6f39 203d 2066 616c       |.|o9 = fal
│ │ │ │ +0002d9c0: 7365 2020 2020 2020 2020 2020 2020 2020  se              
│ │ │ │  0002d9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002d9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002d9f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002da00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +0002d9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002da00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  0002da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -0002da50: 0a7c 6931 3020 3a20 2d2d 2069 6e20 7468  .|i10 : -- in th
│ │ │ │ -0002da60: 6973 2063 6173 6520 7765 2063 616e 2072  is case we can r
│ │ │ │ -0002da70: 656d 6564 7920 656e 6162 6c69 6e67 2074  emedy enabling t
│ │ │ │ -0002da80: 6865 206f 7074 696f 6e20 4365 7274 6966  he option Certif
│ │ │ │ -0002da90: 7920 2020 2020 2020 2020 2020 2020 207c  y              |
│ │ │ │ -0002daa0: 0a7c 2020 2020 2020 7469 6d65 2070 7369  .|      time psi
│ │ │ │ -0002dab0: 203d 2061 7070 726f 7869 6d61 7465 496e   = approximateIn
│ │ │ │ -0002dac0: 7665 7273 654d 6170 2870 6869 2c43 6f64  verseMap(phi,Cod
│ │ │ │ -0002dad0: 696d 4273 496e 763d 3e34 2c43 6572 7469  imBsInv=>4,Certi
│ │ │ │ -0002dae0: 6679 3d3e 7472 7565 2920 2020 2020 207c  fy=>true)      |
│ │ │ │ -0002daf0: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ -0002db00: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ -0002db10: 2031 206f 6620 3320 2020 2020 2020 2020   1 of 3         
│ │ │ │ +0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002da50: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 2d2d  -----+.|i10 : --
│ │ │ │ +0002da60: 2069 6e20 7468 6973 2063 6173 6520 7765   in this case we
│ │ │ │ +0002da70: 2063 616e 2072 656d 6564 7920 656e 6162   can remedy enab
│ │ │ │ +0002da80: 6c69 6e67 2074 6865 206f 7074 696f 6e20  ling the option 
│ │ │ │ +0002da90: 4365 7274 6966 7920 2020 2020 2020 2020  Certify         
│ │ │ │ +0002daa0: 2020 2020 207c 0a7c 2020 2020 2020 7469       |.|      ti
│ │ │ │ +0002dab0: 6d65 2070 7369 203d 2061 7070 726f 7869  me psi = approxi
│ │ │ │ +0002dac0: 6d61 7465 496e 7665 7273 654d 6170 2870  mateInverseMap(p
│ │ │ │ +0002dad0: 6869 2c43 6f64 696d 4273 496e 763d 3e34  hi,CodimBsInv=>4
│ │ │ │ +0002dae0: 2c43 6572 7469 6679 3d3e 7472 7565 2920  ,Certify=>true) 
│ │ │ │ +0002daf0: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ +0002db00: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ +0002db10: 3a20 7374 6570 2031 206f 6620 3320 2020  : step 1 of 3   
│ │ │ │  0002db20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002db40: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ -0002db50: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ -0002db60: 2032 206f 6620 3320 2020 2020 2020 2020   2 of 3         
│ │ │ │ +0002db30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db40: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ +0002db50: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ +0002db60: 3a20 7374 6570 2032 206f 6620 3320 2020  : step 2 of 3   
│ │ │ │  0002db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002db80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002db90: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465  .|-- approximate
│ │ │ │ -0002dba0: 496e 7665 7273 654d 6170 3a20 7374 6570  InverseMap: step
│ │ │ │ -0002dbb0: 2033 206f 6620 3320 2020 2020 2020 2020   3 of 3         
│ │ │ │ +0002db80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002db90: 2020 2020 207c 0a7c 2d2d 2061 7070 726f       |.|-- appro
│ │ │ │ +0002dba0: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ +0002dbb0: 3a20 7374 6570 2033 206f 6620 3320 2020  : step 3 of 3   
│ │ │ │  0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dbd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dbe0: 0a7c 4365 7274 6966 793a 206f 7574 7075  .|Certify: outpu
│ │ │ │ -0002dbf0: 7420 6365 7274 6966 6965 6421 2020 2020  t certified!    
│ │ │ │ -0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dbe0: 2020 2020 207c 0a7c 4365 7274 6966 793a       |.|Certify:
│ │ │ │ +0002dbf0: 206f 7574 7075 7420 6365 7274 6966 6965   output certifie
│ │ │ │ +0002dc00: 6421 2020 2020 2020 2020 2020 2020 2020  d!              
│ │ │ │  0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dc30: 0a7c 202d 2d20 7573 6564 2033 2e38 3935  .| -- used 3.895
│ │ │ │ -0002dc40: 3535 7320 2863 7075 293b 2032 2e39 3232  55s (cpu); 2.922
│ │ │ │ -0002dc50: 3338 7320 2874 6872 6561 6429 3b20 3073  38s (thread); 0s
│ │ │ │ -0002dc60: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ -0002dc70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dc80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc30: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +0002dc40: 2033 2e35 3930 3635 7320 2863 7075 293b   3.59065s (cpu);
│ │ │ │ +0002dc50: 2032 2e39 3438 3239 7320 2874 6872 6561   2.94829s (threa
│ │ │ │ +0002dc60: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0002dc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dc80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dcd0: 0a7c 6f31 3020 3d20 2d2d 2072 6174 696f  .|o10 = -- ratio
│ │ │ │ -0002dce0: 6e61 6c20 6d61 7020 2d2d 2020 2020 2020  nal map --      
│ │ │ │ +0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dcd0: 2020 2020 207c 0a7c 6f31 3020 3d20 2d2d       |.|o10 = --
│ │ │ │ +0002dce0: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d   rational map --
│ │ │ │  0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dd20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002dd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd40: 2020 205a 5a20 2020 2020 2020 2020 2020     ZZ           
│ │ │ │ +0002dd40: 2020 2020 2020 2020 205a 5a20 2020 2020           ZZ     
│ │ │ │  0002dd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dd60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dd70: 0a7c 2020 2020 2020 736f 7572 6365 3a20  .|      source: 
│ │ │ │ -0002dd80: 7375 6276 6172 6965 7479 206f 6620 5072  subvariety of Pr
│ │ │ │ -0002dd90: 6f6a 282d 2d5b 7820 2c20 7820 2c20 7820  oj(--[x , x , x 
│ │ │ │ -0002dda0: 2c20 7820 2c20 7820 2c20 7820 2c20 7820  , x , x , x , x 
│ │ │ │ -0002ddb0: 2c20 7820 2c20 7820 2c20 7820 2c20 787c  , x , x , x , x|
│ │ │ │ -0002ddc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dd70: 2020 2020 207c 0a7c 2020 2020 2020 736f       |.|      so
│ │ │ │ +0002dd80: 7572 6365 3a20 7375 6276 6172 6965 7479  urce: subvariety
│ │ │ │ +0002dd90: 206f 6620 5072 6f6a 282d 2d5b 7820 2c20   of Proj(--[x , 
│ │ │ │ +0002dda0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +0002ddb0: 7820 2c20 7820 2c20 7820 2c20 7820 2c20  x , x , x , x , 
│ │ │ │ +0002ddc0: 7820 2c20 787c 0a7c 2020 2020 2020 2020  x , x|.|        
│ │ │ │  0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dde0: 2020 2039 3720 2030 2020 2031 2020 2032     97  0   1   2
│ │ │ │ -0002ddf0: 2020 2033 2020 2034 2020 2035 2020 2036     3   4   5   6
│ │ │ │ -0002de00: 2020 2037 2020 2038 2020 2039 2020 207c     7   8   9   |
│ │ │ │ -0002de10: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002de20: 7b20 2020 2020 2020 2020 2020 2020 2020  {               
│ │ │ │ +0002dde0: 2020 2020 2020 2020 2039 3720 2030 2020           97  0  
│ │ │ │ +0002ddf0: 2031 2020 2032 2020 2033 2020 2034 2020   1   2   3   4  
│ │ │ │ +0002de00: 2035 2020 2036 2020 2037 2020 2038 2020   5   6   7   8  
│ │ │ │ +0002de10: 2039 2020 207c 0a7c 2020 2020 2020 2020   9   |.|        
│ │ │ │ +0002de20: 2020 2020 2020 7b20 2020 2020 2020 2020        {         
│ │ │ │  0002de30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002de40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002de60: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002de50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002de60: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002de70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002de80: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0002de80: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │  0002de90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dea0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002deb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002dec0: 2078 2078 2020 2d20 3878 2078 2020 2b20   x x  - 8x x  + 
│ │ │ │ -0002ded0: 3235 7820 202d 2032 3578 2078 2020 2d20  25x  - 25x x  - 
│ │ │ │ -0002dee0: 3232 7820 7820 202b 2078 2078 2020 2b20  22x x  + x x  + 
│ │ │ │ -0002def0: 3133 7820 7820 202b 2034 3178 2078 207c  13x x  + 41x x |
│ │ │ │ -0002df00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002df10: 2020 3120 3320 2020 2020 3220 3320 2020    1 3     2 3   
│ │ │ │ -0002df20: 2020 2033 2020 2020 2020 3220 3420 2020     3      2 4   
│ │ │ │ -0002df30: 2020 2033 2034 2020 2020 3020 3520 2020     3 4    0 5   
│ │ │ │ -0002df40: 2020 2032 2035 2020 2020 2020 3320 357c     2 5      3 5|
│ │ │ │ -0002df50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002deb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002dec0: 2020 2020 2020 2078 2078 2020 2d20 3878         x x  - 8x
│ │ │ │ +0002ded0: 2078 2020 2b20 3235 7820 202d 2032 3578   x  + 25x  - 25x
│ │ │ │ +0002dee0: 2078 2020 2d20 3232 7820 7820 202b 2078   x  - 22x x  + x
│ │ │ │ +0002def0: 2078 2020 2b20 3133 7820 7820 202b 2034   x  + 13x x  + 4
│ │ │ │ +0002df00: 3178 2078 207c 0a7c 2020 2020 2020 2020  1x x |.|        
│ │ │ │ +0002df10: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ +0002df20: 3220 3320 2020 2020 2033 2020 2020 2020  2 3      3      
│ │ │ │ +0002df30: 3220 3420 2020 2020 2033 2034 2020 2020  2 4      3 4    
│ │ │ │ +0002df40: 3020 3520 2020 2020 2032 2035 2020 2020  0 5      2 5    
│ │ │ │ +0002df50: 2020 3320 357c 0a7c 2020 2020 2020 2020    3 5|.|        
│ │ │ │  0002df60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002df80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002df90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dfa0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002dfb0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0002dfc0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0002df90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dfa0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002dfb0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0002dfc0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002dfe0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002dff0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e000: 2078 2020 2b20 3137 7820 7820 202d 2031   x  + 17x x  - 1
│ │ │ │ -0002e010: 3478 2020 2d20 3133 7820 7820 202b 2033  4x  - 13x x  + 3
│ │ │ │ -0002e020: 3478 2078 2020 2b20 3434 7820 7820 202d  4x x  + 44x x  -
│ │ │ │ -0002e030: 2033 3078 2078 2020 2b20 3237 7820 787c   30x x  + 27x x|
│ │ │ │ -0002e040: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e050: 2020 3220 2020 2020 2032 2033 2020 2020    2      2 3    
│ │ │ │ -0002e060: 2020 3320 2020 2020 2032 2034 2020 2020    3      2 4    
│ │ │ │ -0002e070: 2020 3320 3420 2020 2020 2030 2035 2020    3 4      0 5  
│ │ │ │ -0002e080: 2020 2020 3220 3520 2020 2020 2033 207c      2 5      3 |
│ │ │ │ -0002e090: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002dff0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e000: 2020 2020 2020 2078 2020 2b20 3137 7820         x  + 17x 
│ │ │ │ +0002e010: 7820 202d 2031 3478 2020 2d20 3133 7820  x  - 14x  - 13x 
│ │ │ │ +0002e020: 7820 202b 2033 3478 2078 2020 2b20 3434  x  + 34x x  + 44
│ │ │ │ +0002e030: 7820 7820 202d 2033 3078 2078 2020 2b20  x x  - 30x x  + 
│ │ │ │ +0002e040: 3237 7820 787c 0a7c 2020 2020 2020 2020  27x x|.|        
│ │ │ │ +0002e050: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ +0002e060: 2033 2020 2020 2020 3320 2020 2020 2032   3      3      2
│ │ │ │ +0002e070: 2034 2020 2020 2020 3320 3420 2020 2020   4      3 4     
│ │ │ │ +0002e080: 2030 2035 2020 2020 2020 3220 3520 2020   0 5      2 5   
│ │ │ │ +0002e090: 2020 2033 207c 0a7c 2020 2020 2020 2020     3 |.|        
│ │ │ │  0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e0d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e0e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e0e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e100: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002e100: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0002e110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e120: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e130: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e140: 2078 2078 2020 2d20 3430 7820 7820 202b   x x  - 40x x  +
│ │ │ │ -0002e150: 2032 3878 2020 2d20 7820 7820 202b 2035   28x  - x x  + 5
│ │ │ │ -0002e160: 7820 7820 202d 2031 3678 2078 2020 2b20  x x  - 16x x  + 
│ │ │ │ -0002e170: 3578 2078 2020 2d20 3336 7820 7820 207c  5x x  - 36x x  |
│ │ │ │ -0002e180: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e190: 2020 3120 3220 2020 2020 2032 2033 2020    1 2      2 3  
│ │ │ │ -0002e1a0: 2020 2020 3320 2020 2030 2034 2020 2020      3    0 4    
│ │ │ │ -0002e1b0: 2032 2034 2020 2020 2020 3320 3420 2020   2 4      3 4   
│ │ │ │ -0002e1c0: 2020 3020 3520 2020 2020 2032 2035 207c    0 5      2 5 |
│ │ │ │ -0002e1d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e1e0: 7d20 2020 2020 2020 2020 2020 2020 2020  }               
│ │ │ │ +0002e120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e130: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e140: 2020 2020 2020 2078 2078 2020 2d20 3430         x x  - 40
│ │ │ │ +0002e150: 7820 7820 202b 2032 3878 2020 2d20 7820  x x  + 28x  - x 
│ │ │ │ +0002e160: 7820 202b 2035 7820 7820 202d 2031 3678  x  + 5x x  - 16x
│ │ │ │ +0002e170: 2078 2020 2b20 3578 2078 2020 2d20 3336   x  + 5x x  - 36
│ │ │ │ +0002e180: 7820 7820 207c 0a7c 2020 2020 2020 2020  x x  |.|        
│ │ │ │ +0002e190: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ +0002e1a0: 2032 2033 2020 2020 2020 3320 2020 2030   2 3      3    0
│ │ │ │ +0002e1b0: 2034 2020 2020 2032 2034 2020 2020 2020   4     2 4      
│ │ │ │ +0002e1c0: 3320 3420 2020 2020 3020 3520 2020 2020  3 4     0 5     
│ │ │ │ +0002e1d0: 2032 2035 207c 0a7c 2020 2020 2020 2020   2 5 |.|        
│ │ │ │ +0002e1e0: 2020 2020 2020 7d20 2020 2020 2020 2020        }         
│ │ │ │  0002e1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e210: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e220: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e230: 2020 2020 205a 5a20 2020 2020 2020 2020       ZZ         
│ │ │ │ +0002e210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e220: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e230: 2020 2020 2020 2020 2020 205a 5a20 2020             ZZ   
│ │ │ │  0002e240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e260: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e270: 0a7c 2020 2020 2020 7461 7267 6574 3a20  .|      target: 
│ │ │ │ -0002e280: 5072 6f6a 282d 2d5b 7420 2c20 7420 2c20  Proj(--[t , t , 
│ │ │ │ -0002e290: 7420 2c20 7420 2c20 7420 2c20 7420 2c20  t , t , t , t , 
│ │ │ │ -0002e2a0: 7420 2c20 7420 2c20 7420 5d29 2020 2020  t , t , t ])    
│ │ │ │ -0002e2b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e2c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e2d0: 2020 2020 2039 3720 2030 2020 2031 2020       97  0   1  
│ │ │ │ -0002e2e0: 2032 2020 2033 2020 2034 2020 2035 2020   2   3   4   5  
│ │ │ │ -0002e2f0: 2036 2020 2037 2020 2038 2020 2020 2020   6   7   8      
│ │ │ │ -0002e300: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e310: 0a7c 2020 2020 2020 6465 6669 6e69 6e67  .|      defining
│ │ │ │ -0002e320: 2066 6f72 6d73 3a20 7b20 2020 2020 2020   forms: {       
│ │ │ │ +0002e260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e270: 2020 2020 207c 0a7c 2020 2020 2020 7461       |.|      ta
│ │ │ │ +0002e280: 7267 6574 3a20 5072 6f6a 282d 2d5b 7420  rget: Proj(--[t 
│ │ │ │ +0002e290: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ +0002e2a0: 2c20 7420 2c20 7420 2c20 7420 2c20 7420  , t , t , t , t 
│ │ │ │ +0002e2b0: 5d29 2020 2020 2020 2020 2020 2020 2020  ])              
│ │ │ │ +0002e2c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e2d0: 2020 2020 2020 2020 2020 2039 3720 2030             97  0
│ │ │ │ +0002e2e0: 2020 2031 2020 2032 2020 2033 2020 2034     1   2   3   4
│ │ │ │ +0002e2f0: 2020 2035 2020 2036 2020 2037 2020 2038     5   6   7   8
│ │ │ │ +0002e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e310: 2020 2020 207c 0a7c 2020 2020 2020 6465       |.|      de
│ │ │ │ +0002e320: 6669 6e69 6e67 2066 6f72 6d73 3a20 7b20  fining forms: { 
│ │ │ │  0002e330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e350: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e360: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e360: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002e370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e380: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0002e380: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  0002e390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e3b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e3c0: 2020 2020 2020 2020 202d 2032 7820 7820           - 2x x 
│ │ │ │ -0002e3d0: 202b 2036 7820 202d 2032 3478 2078 2020   + 6x  - 24x x  
│ │ │ │ -0002e3e0: 2d20 3235 7820 7820 202b 2033 3978 2078  - 25x x  + 39x x
│ │ │ │ -0002e3f0: 2020 2b20 3378 2078 2020 2b20 3436 787c    + 3x x  + 46x|
│ │ │ │ -0002e400: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e410: 2020 2020 2020 2020 2020 2020 2032 2033               2 3
│ │ │ │ -0002e420: 2020 2020 2033 2020 2020 2020 3220 3420       3      2 4 
│ │ │ │ -0002e430: 2020 2020 2033 2034 2020 2020 2020 3020       3 4      0 
│ │ │ │ -0002e440: 3520 2020 2020 3220 3520 2020 2020 207c  5     2 5      |
│ │ │ │ -0002e450: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e3b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e3c0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +0002e3d0: 2032 7820 7820 202b 2036 7820 202d 2032   2x x  + 6x  - 2
│ │ │ │ +0002e3e0: 3478 2078 2020 2d20 3235 7820 7820 202b  4x x  - 25x x  +
│ │ │ │ +0002e3f0: 2033 3978 2078 2020 2b20 3378 2078 2020   39x x  + 3x x  
│ │ │ │ +0002e400: 2b20 3436 787c 0a7c 2020 2020 2020 2020  + 46x|.|        
│ │ │ │ +0002e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e420: 2020 2032 2033 2020 2020 2033 2020 2020     2 3     3    
│ │ │ │ +0002e430: 2020 3220 3420 2020 2020 2033 2034 2020    2 4      3 4  
│ │ │ │ +0002e440: 2020 2020 3020 3520 2020 2020 3220 3520      0 5     2 5 
│ │ │ │ +0002e450: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002e460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e490: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e4a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e4b0: 2020 2020 2020 2020 202d 2031 3978 2078           - 19x x
│ │ │ │ -0002e4c0: 2020 2d20 3430 7820 7820 202b 2031 3978    - 40x x  + 19x
│ │ │ │ -0002e4d0: 2078 2020 2b20 3139 7820 7820 202d 2033   x  + 19x x  - 3
│ │ │ │ -0002e4e0: 3878 2078 2020 2d20 3339 7820 7820 207c  8x x  - 39x x  |
│ │ │ │ -0002e4f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e500: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0002e510: 3520 2020 2020 2033 2035 2020 2020 2020  5      3 5      
│ │ │ │ -0002e520: 3120 3820 2020 2020 2032 2038 2020 2020  1 8      2 8    
│ │ │ │ -0002e530: 2020 3320 3820 2020 2020 2035 2038 207c    3 8      5 8 |
│ │ │ │ -0002e540: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e4a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e4b0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +0002e4c0: 2031 3978 2078 2020 2d20 3430 7820 7820   19x x  - 40x x 
│ │ │ │ +0002e4d0: 202b 2031 3978 2078 2020 2b20 3139 7820   + 19x x  + 19x 
│ │ │ │ +0002e4e0: 7820 202d 2033 3878 2078 2020 2d20 3339  x  - 38x x  - 39
│ │ │ │ +0002e4f0: 7820 7820 207c 0a7c 2020 2020 2020 2020  x x  |.|        
│ │ │ │ +0002e500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e510: 2020 2020 3220 3520 2020 2020 2033 2035      2 5      3 5
│ │ │ │ +0002e520: 2020 2020 2020 3120 3820 2020 2020 2032        1 8      2
│ │ │ │ +0002e530: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ +0002e540: 2035 2038 207c 0a7c 2020 2020 2020 2020   5 8 |.|        
│ │ │ │  0002e550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e580: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e590: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e5a0: 2020 2020 2020 2020 202d 2033 3978 2078           - 39x x
│ │ │ │ -0002e5b0: 2020 2d20 3339 7820 7820 202d 2033 3878    - 39x x  - 38x
│ │ │ │ -0002e5c0: 2078 2020 2d20 3339 7820 7820 202b 2031   x  - 39x x  + 1
│ │ │ │ -0002e5d0: 3978 2078 2020 2b20 3278 2078 2020 2b7c  9x x  + 2x x  +|
│ │ │ │ -0002e5e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e5f0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0002e600: 3320 2020 2020 2030 2034 2020 2020 2020  3      0 4      
│ │ │ │ -0002e610: 3320 3420 2020 2020 2030 2035 2020 2020  3 4      0 5    
│ │ │ │ -0002e620: 2020 3220 3520 2020 2020 3320 3520 207c    2 5     3 5  |
│ │ │ │ -0002e630: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e590: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e5a0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +0002e5b0: 2033 3978 2078 2020 2d20 3339 7820 7820   39x x  - 39x x 
│ │ │ │ +0002e5c0: 202d 2033 3878 2078 2020 2d20 3339 7820   - 38x x  - 39x 
│ │ │ │ +0002e5d0: 7820 202b 2031 3978 2078 2020 2b20 3278  x  + 19x x  + 2x
│ │ │ │ +0002e5e0: 2078 2020 2b7c 0a7c 2020 2020 2020 2020   x  +|.|        
│ │ │ │ +0002e5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e600: 2020 2020 3220 3320 2020 2020 2030 2034      2 3      0 4
│ │ │ │ +0002e610: 2020 2020 2020 3320 3420 2020 2020 2030        3 4      0
│ │ │ │ +0002e620: 2035 2020 2020 2020 3220 3520 2020 2020   5      2 5     
│ │ │ │ +0002e630: 3320 3520 207c 0a7c 2020 2020 2020 2020  3 5  |.|        
│ │ │ │  0002e640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e680: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e690: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0002e6a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0002e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e6c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e6d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e6e0: 2020 2020 2020 2020 202d 2033 3978 2020           - 39x  
│ │ │ │ -0002e6f0: 2d20 3878 2078 2020 2b20 3235 7820 202b  - 8x x  + 25x  +
│ │ │ │ -0002e700: 2031 3978 2078 2020 2b20 3436 7820 7820   19x x  + 46x x 
│ │ │ │ -0002e710: 202d 2032 3278 2078 2020 2b20 7820 787c   - 22x x  + x x|
│ │ │ │ -0002e720: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e730: 2020 2020 2020 2020 2020 2020 2020 3120                1 
│ │ │ │ -0002e740: 2020 2020 3220 3320 2020 2020 2033 2020      2 3      3  
│ │ │ │ -0002e750: 2020 2020 3020 3420 2020 2020 2032 2034      0 4      2 4
│ │ │ │ -0002e760: 2020 2020 2020 3320 3420 2020 2030 207c        3 4    0 |
│ │ │ │ -0002e770: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e6a0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0002e6b0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0002e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e6d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e6e0: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +0002e6f0: 2033 3978 2020 2d20 3878 2078 2020 2b20   39x  - 8x x  + 
│ │ │ │ +0002e700: 3235 7820 202b 2031 3978 2078 2020 2b20  25x  + 19x x  + 
│ │ │ │ +0002e710: 3436 7820 7820 202d 2032 3278 2078 2020  46x x  - 22x x  
│ │ │ │ +0002e720: 2b20 7820 787c 0a7c 2020 2020 2020 2020  + x x|.|        
│ │ │ │ +0002e730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e740: 2020 2020 3120 2020 2020 3220 3320 2020      1     2 3   
│ │ │ │ +0002e750: 2020 2033 2020 2020 2020 3020 3420 2020     3      0 4   
│ │ │ │ +0002e760: 2020 2032 2034 2020 2020 2020 3320 3420     2 4      3 4 
│ │ │ │ +0002e770: 2020 2030 207c 0a7c 2020 2020 2020 2020     0 |.|        
│ │ │ │  0002e780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e7a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e7c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e7c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0002e800: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e810: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e820: 2020 2020 2020 2020 202d 2033 3978 2078           - 39x x
│ │ │ │ -0002e830: 2020 2b20 3139 7820 7820 202b 2078 2078    + 19x x  + x x
│ │ │ │ -0002e840: 2020 2b20 3878 2078 2020 2d20 3235 7820    + 8x x  - 25x 
│ │ │ │ -0002e850: 202d 2032 3378 2078 2020 2d20 3330 787c   - 23x x  - 30x|
│ │ │ │ -0002e860: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e870: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -0002e880: 3120 2020 2020 2030 2032 2020 2020 3020  1      0 2    0 
│ │ │ │ -0002e890: 3320 2020 2020 3220 3320 2020 2020 2033  3     2 3      3
│ │ │ │ -0002e8a0: 2020 2020 2020 3220 3420 2020 2020 207c        2 4      |
│ │ │ │ -0002e8b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e800: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0002e810: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e820: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +0002e830: 2033 3978 2078 2020 2b20 3139 7820 7820   39x x  + 19x x 
│ │ │ │ +0002e840: 202b 2078 2078 2020 2b20 3878 2078 2020   + x x  + 8x x  
│ │ │ │ +0002e850: 2d20 3235 7820 202d 2032 3378 2078 2020  - 25x  - 23x x  
│ │ │ │ +0002e860: 2d20 3330 787c 0a7c 2020 2020 2020 2020  - 30x|.|        
│ │ │ │ +0002e870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e880: 2020 2020 3020 3120 2020 2020 2030 2032      0 1      0 2
│ │ │ │ +0002e890: 2020 2020 3020 3320 2020 2020 3220 3320      0 3     2 3 
│ │ │ │ +0002e8a0: 2020 2020 2033 2020 2020 2020 3220 3420       3      2 4 
│ │ │ │ +0002e8b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002e8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e8f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e900: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e910: 2020 2020 2020 2020 202d 2032 3478 2078           - 24x x
│ │ │ │ -0002e920: 2020 2d20 3235 7820 7820 202b 2032 3278    - 25x x  + 22x
│ │ │ │ -0002e930: 2078 2020 2b20 3438 7820 7820 202d 2031   x  + 48x x  - 1
│ │ │ │ -0002e940: 3778 2078 2020 2b20 3234 7820 7820 207c  7x x  + 24x x  |
│ │ │ │ -0002e950: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002e960: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0002e970: 3420 2020 2020 2033 2034 2020 2020 2020  4      3 4      
│ │ │ │ -0002e980: 3220 3520 2020 2020 2033 2035 2020 2020  2 5      3 5    
│ │ │ │ -0002e990: 2020 3220 3620 2020 2020 2031 2037 207c    2 6      1 7 |
│ │ │ │ -0002e9a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e900: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002e910: 2020 2020 2020 2020 2020 2020 2020 202d                 -
│ │ │ │ +0002e920: 2032 3478 2078 2020 2d20 3235 7820 7820   24x x  - 25x x 
│ │ │ │ +0002e930: 202b 2032 3278 2078 2020 2b20 3438 7820   + 22x x  + 48x 
│ │ │ │ +0002e940: 7820 202d 2031 3778 2078 2020 2b20 3234  x  - 17x x  + 24
│ │ │ │ +0002e950: 7820 7820 207c 0a7c 2020 2020 2020 2020  x x  |.|        
│ │ │ │ +0002e960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e970: 2020 2020 3220 3420 2020 2020 2033 2034      2 4      3 4
│ │ │ │ +0002e980: 2020 2020 2020 3220 3520 2020 2020 2033        2 5      3
│ │ │ │ +0002e990: 2035 2020 2020 2020 3220 3620 2020 2020   5      2 6     
│ │ │ │ +0002e9a0: 2031 2037 207c 0a7c 2020 2020 2020 2020   1 7 |.|        
│ │ │ │  0002e9b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002e9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002e9e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002e9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002ea00: 2020 2020 2020 2020 2031 3978 2078 2020           19x x  
│ │ │ │ -0002ea10: 2b20 7820 7820 202b 2031 3978 2078 2020  + x x  + 19x x  
│ │ │ │ -0002ea20: 2b20 3339 7820 7820 202d 2031 3978 2078  + 39x x  - 19x x
│ │ │ │ -0002ea30: 2020 2b20 3337 7820 7820 202d 2033 367c    + 37x x  - 36|
│ │ │ │ -0002ea40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002ea50: 2020 2020 2020 2020 2020 2020 3220 3520              2 5 
│ │ │ │ -0002ea60: 2020 2033 2035 2020 2020 2020 3520 3720     3 5      5 7 
│ │ │ │ -0002ea70: 2020 2020 2031 2038 2020 2020 2020 3420       1 8      4 
│ │ │ │ -0002ea80: 3820 2020 2020 2036 2038 2020 2020 207c  8      6 8     |
│ │ │ │ -0002ea90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002e9f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ea00: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ +0002ea10: 3978 2078 2020 2b20 7820 7820 202b 2031  9x x  + x x  + 1
│ │ │ │ +0002ea20: 3978 2078 2020 2b20 3339 7820 7820 202d  9x x  + 39x x  -
│ │ │ │ +0002ea30: 2031 3978 2078 2020 2b20 3337 7820 7820   19x x  + 37x x 
│ │ │ │ +0002ea40: 202d 2033 367c 0a7c 2020 2020 2020 2020   - 36|.|        
│ │ │ │ +0002ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ea60: 2020 3220 3520 2020 2033 2035 2020 2020    2 5    3 5    
│ │ │ │ +0002ea70: 2020 3520 3720 2020 2020 2031 2038 2020    5 7      1 8  
│ │ │ │ +0002ea80: 2020 2020 3420 3820 2020 2020 2036 2038      4 8      6 8
│ │ │ │ +0002ea90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002eaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ead0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002eae0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002eaf0: 2020 2020 2020 2020 2033 3978 2078 2020           39x x  
│ │ │ │ -0002eb00: 2d20 3139 7820 7820 202d 2032 7820 7820  - 19x x  - 2x x 
│ │ │ │ -0002eb10: 202b 2033 3978 2078 2020 2b20 3139 7820   + 39x x  + 19x 
│ │ │ │ -0002eb20: 7820 202d 2078 2078 2020 2d20 3337 787c  x  - x x  - 37x|
│ │ │ │ -0002eb30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002eb40: 2020 2020 2020 2020 2020 2020 3020 3520              0 5 
│ │ │ │ -0002eb50: 2020 2020 2032 2035 2020 2020 2033 2035       2 5     3 5
│ │ │ │ -0002eb60: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ -0002eb70: 2038 2020 2020 3320 3820 2020 2020 207c   8    3 8      |
│ │ │ │ -0002eb80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ead0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eae0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2033                 3
│ │ │ │ +0002eb00: 3978 2078 2020 2d20 3139 7820 7820 202d  9x x  - 19x x  -
│ │ │ │ +0002eb10: 2032 7820 7820 202b 2033 3978 2078 2020   2x x  + 39x x  
│ │ │ │ +0002eb20: 2b20 3139 7820 7820 202d 2078 2078 2020  + 19x x  - x x  
│ │ │ │ +0002eb30: 2d20 3337 787c 0a7c 2020 2020 2020 2020  - 37x|.|        
│ │ │ │ +0002eb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eb50: 2020 3020 3520 2020 2020 2032 2035 2020    0 5      2 5  
│ │ │ │ +0002eb60: 2020 2033 2035 2020 2020 2020 3020 3820     3 5      0 8 
│ │ │ │ +0002eb70: 2020 2020 2031 2038 2020 2020 3320 3820       1 8    3 8 
│ │ │ │ +0002eb80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ebc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ebd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002ebe0: 2020 2020 2020 2020 2034 3378 2078 2020           43x x  
│ │ │ │ -0002ebf0: 2d20 3330 7820 7820 202d 2031 3978 2078  - 30x x  - 19x x
│ │ │ │ -0002ec00: 2020 2d20 3430 7820 7820 202b 2034 3478    - 40x x  + 44x
│ │ │ │ -0002ec10: 2078 2020 2d20 3232 7820 7820 202d 207c   x  - 22x x  - |
│ │ │ │ -0002ec20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002ec30: 2020 2020 2020 2020 2020 2020 3220 3420              2 4 
│ │ │ │ -0002ec40: 2020 2020 2033 2034 2020 2020 2020 3020       3 4      0 
│ │ │ │ -0002ec50: 3520 2020 2020 2032 2035 2020 2020 2020  5      2 5      
│ │ │ │ -0002ec60: 3320 3520 2020 2020 2032 2036 2020 207c  3 5      2 6   |
│ │ │ │ -0002ec70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ -0002ec80: 2020 2020 2020 2020 7d20 2020 2020 2020          }       
│ │ │ │ +0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ebd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2034                 4
│ │ │ │ +0002ebf0: 3378 2078 2020 2d20 3330 7820 7820 202d  3x x  - 30x x  -
│ │ │ │ +0002ec00: 2031 3978 2078 2020 2d20 3430 7820 7820   19x x  - 40x x 
│ │ │ │ +0002ec10: 202b 2034 3478 2078 2020 2d20 3232 7820   + 44x x  - 22x 
│ │ │ │ +0002ec20: 7820 202d 207c 0a7c 2020 2020 2020 2020  x  - |.|        
│ │ │ │ +0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ec40: 2020 3220 3420 2020 2020 2033 2034 2020    2 4      3 4  
│ │ │ │ +0002ec50: 2020 2020 3020 3520 2020 2020 2032 2035      0 5      2 5
│ │ │ │ +0002ec60: 2020 2020 2020 3320 3520 2020 2020 2032        3 5      2
│ │ │ │ +0002ec70: 2036 2020 207c 0a7c 2020 2020 2020 2020   6   |.|        
│ │ │ │ +0002ec80: 2020 2020 2020 2020 2020 2020 2020 7d20                } 
│ │ │ │  0002ec90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ecb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ecc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ecb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ecc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ecd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ed00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ed10: 0a7c 6f31 3020 3a20 5261 7469 6f6e 616c  .|o10 : Rational
│ │ │ │ -0002ed20: 4d61 7020 2871 7561 6472 6174 6963 2062  Map (quadratic b
│ │ │ │ -0002ed30: 6972 6174 696f 6e61 6c20 6d61 7020 6672  irational map fr
│ │ │ │ -0002ed40: 6f6d 2038 2d64 696d 656e 7369 6f6e 616c  om 8-dimensional
│ │ │ │ -0002ed50: 2073 7562 7661 7269 6574 7920 6f66 207c   subvariety of |
│ │ │ │ -0002ed60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0002ed00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ed10: 2020 2020 207c 0a7c 6f31 3020 3a20 5261       |.|o10 : Ra
│ │ │ │ +0002ed20: 7469 6f6e 616c 4d61 7020 2871 7561 6472  tionalMap (quadr
│ │ │ │ +0002ed30: 6174 6963 2062 6972 6174 696f 6e61 6c20  atic birational 
│ │ │ │ +0002ed40: 6d61 7020 6672 6f6d 2038 2d64 696d 656e  map from 8-dimen
│ │ │ │ +0002ed50: 7369 6f6e 616c 2073 7562 7661 7269 6574  sional subvariet
│ │ │ │ +0002ed60: 7920 6f66 207c 0a7c 2d2d 2d2d 2d2d 2d2d  y of |.|--------
│ │ │ │  0002ed70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ed80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002ed90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0002edb0: 0a7c 2020 2c20 7820 205d 2920 6465 6669  .|  , x  ]) defi
│ │ │ │ -0002edc0: 6e65 6420 6279 2020 2020 2020 2020 2020  ned by          
│ │ │ │ +0002eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002edb0: 2d2d 2d2d 2d7c 0a7c 2020 2c20 7820 205d  -----|.|  , x  ]
│ │ │ │ +0002edc0: 2920 6465 6669 6e65 6420 6279 2020 2020  ) defined by    
│ │ │ │  0002edd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ede0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002edf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ee00: 0a7c 3130 2020 2031 3120 2020 2020 2020  .|10   11       
│ │ │ │ +0002edf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ee00: 2020 2020 207c 0a7c 3130 2020 2031 3120       |.|10   11 
│ │ │ │  0002ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ee50: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ee50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ee60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ee90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002eea0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eea0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002eeb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002eed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002eee0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002eef0: 0a7c 202d 2078 2078 2020 2b20 3132 7820  .| - x x  + 12x 
│ │ │ │ -0002ef00: 7820 202b 2032 3578 2078 2020 2b20 3235  x  + 25x x  + 25
│ │ │ │ -0002ef10: 7820 7820 202b 2032 3378 2078 2020 2d20  x x  + 23x x  - 
│ │ │ │ -0002ef20: 3378 2078 2020 2b20 3278 2078 2020 2b20  3x x  + 2x x  + 
│ │ │ │ -0002ef30: 3131 7820 7820 202d 2033 3778 2020 207c  11x x  - 37x   |
│ │ │ │ -0002ef40: 0a7c 2020 2020 3020 3620 2020 2020 2032  .|    0 6      2
│ │ │ │ -0002ef50: 2036 2020 2020 2020 3120 3720 2020 2020   6      1 7     
│ │ │ │ -0002ef60: 2033 2037 2020 2020 2020 3520 3720 2020   3 7      5 7   
│ │ │ │ -0002ef70: 2020 3620 3720 2020 2020 3020 3820 2020    6 7     0 8   
│ │ │ │ -0002ef80: 2020 2031 2038 2020 2020 2020 3320 207c     1 8      3  |
│ │ │ │ -0002ef90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002eee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002eef0: 2020 2020 207c 0a7c 202d 2078 2078 2020       |.| - x x  
│ │ │ │ +0002ef00: 2b20 3132 7820 7820 202b 2032 3578 2078  + 12x x  + 25x x
│ │ │ │ +0002ef10: 2020 2b20 3235 7820 7820 202b 2032 3378    + 25x x  + 23x
│ │ │ │ +0002ef20: 2078 2020 2d20 3378 2078 2020 2b20 3278   x  - 3x x  + 2x
│ │ │ │ +0002ef30: 2078 2020 2b20 3131 7820 7820 202d 2033   x  + 11x x  - 3
│ │ │ │ +0002ef40: 3778 2020 207c 0a7c 2020 2020 3020 3620  7x   |.|    0 6 
│ │ │ │ +0002ef50: 2020 2020 2032 2036 2020 2020 2020 3120       2 6      1 
│ │ │ │ +0002ef60: 3720 2020 2020 2033 2037 2020 2020 2020  7      3 7      
│ │ │ │ +0002ef70: 3520 3720 2020 2020 3620 3720 2020 2020  5 7     6 7     
│ │ │ │ +0002ef80: 3020 3820 2020 2020 2031 2038 2020 2020  0 8      1 8    
│ │ │ │ +0002ef90: 2020 3320 207c 0a7c 2020 2020 2020 2020    3  |.|        
│ │ │ │  0002efa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002efb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002efc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002efd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002efe0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002efd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002efe0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002eff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f020: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f030: 0a7c 2020 2b20 3331 7820 7820 202d 2033  .|  + 31x x  - 3
│ │ │ │ -0002f040: 3678 2078 2020 2d20 7820 7820 202b 2031  6x x  - x x  + 1
│ │ │ │ -0002f050: 3378 2078 2020 2b20 3878 2078 2020 2b20  3x x  + 8x x  + 
│ │ │ │ -0002f060: 3978 2078 2020 2b20 3436 7820 7820 202b  9x x  + 46x x  +
│ │ │ │ -0002f070: 2034 3178 2078 2020 2d20 3778 2020 207c   41x x  - 7x   |
│ │ │ │ -0002f080: 0a7c 3520 2020 2020 2032 2036 2020 2020  .|5      2 6    
│ │ │ │ -0002f090: 2020 3320 3620 2020 2030 2037 2020 2020    3 6    0 7    
│ │ │ │ -0002f0a0: 2020 3120 3720 2020 2020 3320 3720 2020    1 7     3 7   
│ │ │ │ -0002f0b0: 2020 3520 3720 2020 2020 2036 2037 2020    5 7      6 7  
│ │ │ │ -0002f0c0: 2020 2020 3020 3820 2020 2020 3120 207c      0 8     1  |
│ │ │ │ -0002f0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f030: 2020 2020 207c 0a7c 2020 2b20 3331 7820       |.|  + 31x 
│ │ │ │ +0002f040: 7820 202d 2033 3678 2078 2020 2d20 7820  x  - 36x x  - x 
│ │ │ │ +0002f050: 7820 202b 2031 3378 2078 2020 2b20 3878  x  + 13x x  + 8x
│ │ │ │ +0002f060: 2078 2020 2b20 3978 2078 2020 2b20 3436   x  + 9x x  + 46
│ │ │ │ +0002f070: 7820 7820 202b 2034 3178 2078 2020 2d20  x x  + 41x x  - 
│ │ │ │ +0002f080: 3778 2020 207c 0a7c 3520 2020 2020 2032  7x   |.|5      2
│ │ │ │ +0002f090: 2036 2020 2020 2020 3320 3620 2020 2030   6      3 6    0
│ │ │ │ +0002f0a0: 2037 2020 2020 2020 3120 3720 2020 2020   7      1 7     
│ │ │ │ +0002f0b0: 3320 3720 2020 2020 3520 3720 2020 2020  3 7     5 7     
│ │ │ │ +0002f0c0: 2036 2037 2020 2020 2020 3020 3820 2020   6 7      0 8   
│ │ │ │ +0002f0d0: 2020 3120 207c 0a7c 2020 2020 2020 2020    1  |.|        
│ │ │ │  0002f0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f0f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f110: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f120: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f120: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f160: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f170: 0a7c 2b20 3337 7820 7820 202b 2034 3878  .|+ 37x x  + 48x
│ │ │ │ -0002f180: 2078 2020 2d20 3578 2078 2020 2d20 3578   x  - 5x x  - 5x
│ │ │ │ -0002f190: 2078 2020 2b20 7820 7820 202b 2032 3078   x  + x x  + 20x
│ │ │ │ -0002f1a0: 2078 2020 2b20 3130 7820 7820 202b 2033   x  + 10x x  + 3
│ │ │ │ -0002f1b0: 3478 2078 2020 2b20 3431 7820 7820 207c  4x x  + 41x x  |
│ │ │ │ -0002f1c0: 0a7c 2020 2020 2033 2035 2020 2020 2020  .|     3 5      
│ │ │ │ -0002f1d0: 3220 3620 2020 2020 3120 3720 2020 2020  2 6     1 7     
│ │ │ │ -0002f1e0: 3320 3720 2020 2035 2037 2020 2020 2020  3 7    5 7      
│ │ │ │ -0002f1f0: 3620 3720 2020 2020 2030 2038 2020 2020  6 7      0 8    
│ │ │ │ -0002f200: 2020 3120 3820 2020 2020 2033 2020 207c    1 8      3   |
│ │ │ │ -0002f210: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f170: 2020 2020 207c 0a7c 2b20 3337 7820 7820       |.|+ 37x x 
│ │ │ │ +0002f180: 202b 2034 3878 2078 2020 2d20 3578 2078   + 48x x  - 5x x
│ │ │ │ +0002f190: 2020 2d20 3578 2078 2020 2b20 7820 7820    - 5x x  + x x 
│ │ │ │ +0002f1a0: 202b 2032 3078 2078 2020 2b20 3130 7820   + 20x x  + 10x 
│ │ │ │ +0002f1b0: 7820 202b 2033 3478 2078 2020 2b20 3431  x  + 34x x  + 41
│ │ │ │ +0002f1c0: 7820 7820 207c 0a7c 2020 2020 2033 2035  x x  |.|     3 5
│ │ │ │ +0002f1d0: 2020 2020 2020 3220 3620 2020 2020 3120        2 6     1 
│ │ │ │ +0002f1e0: 3720 2020 2020 3320 3720 2020 2035 2037  7     3 7    5 7
│ │ │ │ +0002f1f0: 2020 2020 2020 3620 3720 2020 2020 2030        6 7      0
│ │ │ │ +0002f200: 2038 2020 2020 2020 3120 3820 2020 2020   8      1 8     
│ │ │ │ +0002f210: 2033 2020 207c 0a7c 2020 2020 2020 2020   3   |.|        
│ │ │ │  0002f220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f250: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f260: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f260: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f2b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f2b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f2f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f300: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f300: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f340: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f350: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f350: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f390: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f3e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f3f0: 0a7c 2078 2020 2d20 3339 7820 7820 202d  .| x  - 39x x  -
│ │ │ │ -0002f400: 2031 3778 2078 2020 2d20 3338 7820 7820   17x x  - 38x x 
│ │ │ │ -0002f410: 202b 2032 3478 2078 2020 2b20 3234 7820   + 24x x  + 24x 
│ │ │ │ -0002f420: 7820 202b 2035 7820 7820 202b 2078 2078  x  + 5x x  + x x
│ │ │ │ -0002f430: 2020 2d20 3139 7820 7820 202d 2020 207c    - 19x x  -   |
│ │ │ │ -0002f440: 0a7c 3320 3520 2020 2020 2030 2036 2020  .|3 5      0 6  
│ │ │ │ -0002f450: 2020 2020 3220 3620 2020 2020 2033 2036      2 6      3 6
│ │ │ │ -0002f460: 2020 2020 2020 3120 3720 2020 2020 2033        1 7      3
│ │ │ │ -0002f470: 2037 2020 2020 2035 2037 2020 2020 3620   7     5 7    6 
│ │ │ │ -0002f480: 3720 2020 2020 2030 2038 2020 2020 207c  7      0 8     |
│ │ │ │ -0002f490: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f3f0: 2020 2020 207c 0a7c 2078 2020 2d20 3339       |.| x  - 39
│ │ │ │ +0002f400: 7820 7820 202d 2031 3778 2078 2020 2d20  x x  - 17x x  - 
│ │ │ │ +0002f410: 3338 7820 7820 202b 2032 3478 2078 2020  38x x  + 24x x  
│ │ │ │ +0002f420: 2b20 3234 7820 7820 202b 2035 7820 7820  + 24x x  + 5x x 
│ │ │ │ +0002f430: 202b 2078 2078 2020 2d20 3139 7820 7820   + x x  - 19x x 
│ │ │ │ +0002f440: 202d 2020 207c 0a7c 3320 3520 2020 2020   -   |.|3 5     
│ │ │ │ +0002f450: 2030 2036 2020 2020 2020 3220 3620 2020   0 6      2 6   
│ │ │ │ +0002f460: 2020 2033 2036 2020 2020 2020 3120 3720     3 6      1 7 
│ │ │ │ +0002f470: 2020 2020 2033 2037 2020 2020 2035 2037       3 7     5 7
│ │ │ │ +0002f480: 2020 2020 3620 3720 2020 2020 2030 2038      6 7      0 8
│ │ │ │ +0002f490: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f4d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f4e0: 0a7c 2d20 3337 7820 7820 202d 2032 3178  .|- 37x x  - 21x
│ │ │ │ -0002f4f0: 2078 2020 2b20 7820 7820 2020 2b20 3278   x  + x x   + 2x
│ │ │ │ -0002f500: 2078 2020 202d 2035 7820 7820 2020 2b20   x   - 5x x   + 
│ │ │ │ -0002f510: 7820 7820 2020 2d20 3578 2078 2020 202b  x x   - 5x x   +
│ │ │ │ -0002f520: 2035 7820 7820 202c 2020 2020 2020 207c   5x x  ,       |
│ │ │ │ -0002f530: 0a7c 2020 2020 2036 2038 2020 2020 2020  .|     6 8      
│ │ │ │ -0002f540: 3520 3920 2020 2031 2031 3020 2020 2020  5 9    1 10     
│ │ │ │ -0002f550: 3220 3130 2020 2020 2033 2031 3020 2020  2 10     3 10   
│ │ │ │ -0002f560: 2035 2031 3020 2020 2020 3620 3130 2020   5 10     6 10  
│ │ │ │ -0002f570: 2020 2035 2031 3120 2020 2020 2020 207c     5 11        |
│ │ │ │ -0002f580: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f4e0: 2020 2020 207c 0a7c 2d20 3337 7820 7820       |.|- 37x x 
│ │ │ │ +0002f4f0: 202d 2032 3178 2078 2020 2b20 7820 7820   - 21x x  + x x 
│ │ │ │ +0002f500: 2020 2b20 3278 2078 2020 202d 2035 7820    + 2x x   - 5x 
│ │ │ │ +0002f510: 7820 2020 2b20 7820 7820 2020 2d20 3578  x   + x x   - 5x
│ │ │ │ +0002f520: 2078 2020 202b 2035 7820 7820 202c 2020   x   + 5x x  ,  
│ │ │ │ +0002f530: 2020 2020 207c 0a7c 2020 2020 2036 2038       |.|     6 8
│ │ │ │ +0002f540: 2020 2020 2020 3520 3920 2020 2031 2031        5 9    1 1
│ │ │ │ +0002f550: 3020 2020 2020 3220 3130 2020 2020 2033  0     2 10     3
│ │ │ │ +0002f560: 2031 3020 2020 2035 2031 3020 2020 2020   10    5 10     
│ │ │ │ +0002f570: 3620 3130 2020 2020 2035 2031 3120 2020  6 10     5 11   
│ │ │ │ +0002f580: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f5c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f5d0: 0a7c 2031 3978 2078 2020 2b20 3139 7820  .| 19x x  + 19x 
│ │ │ │ -0002f5e0: 7820 202b 2032 3178 2078 2020 2b20 3139  x  + 21x x  + 19
│ │ │ │ -0002f5f0: 7820 7820 202d 2031 3978 2078 2020 2b20  x x  - 19x x  + 
│ │ │ │ -0002f600: 7820 7820 202d 2033 3978 2078 2020 2b20  x x  - 39x x  + 
│ │ │ │ -0002f610: 3337 7820 7820 202b 2034 3078 2020 207c  37x x  + 40x   |
│ │ │ │ -0002f620: 0a7c 2020 2020 3020 3720 2020 2020 2032  .|    0 7      2
│ │ │ │ -0002f630: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ -0002f640: 2030 2038 2020 2020 2020 3120 3820 2020   0 8      1 8   
│ │ │ │ -0002f650: 2033 2038 2020 2020 2020 3420 3820 2020   3 8      4 8   
│ │ │ │ -0002f660: 2020 2036 2038 2020 2020 2020 3020 207c     6 8      0  |
│ │ │ │ -0002f670: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f5d0: 2020 2020 207c 0a7c 2031 3978 2078 2020       |.| 19x x  
│ │ │ │ +0002f5e0: 2b20 3139 7820 7820 202b 2032 3178 2078  + 19x x  + 21x x
│ │ │ │ +0002f5f0: 2020 2b20 3139 7820 7820 202d 2031 3978    + 19x x  - 19x
│ │ │ │ +0002f600: 2078 2020 2b20 7820 7820 202d 2033 3978   x  + x x  - 39x
│ │ │ │ +0002f610: 2078 2020 2b20 3337 7820 7820 202b 2034   x  + 37x x  + 4
│ │ │ │ +0002f620: 3078 2020 207c 0a7c 2020 2020 3020 3720  0x   |.|    0 7 
│ │ │ │ +0002f630: 2020 2020 2032 2037 2020 2020 2020 3320       2 7      3 
│ │ │ │ +0002f640: 3720 2020 2020 2030 2038 2020 2020 2020  7      0 8      
│ │ │ │ +0002f650: 3120 3820 2020 2033 2038 2020 2020 2020  1 8    3 8      
│ │ │ │ +0002f660: 3420 3820 2020 2020 2036 2038 2020 2020  4 8      6 8    
│ │ │ │ +0002f670: 2020 3020 207c 0a7c 2020 2020 2020 2020    0  |.|        
│ │ │ │  0002f680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f6b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f6c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f6c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f700: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f710: 0a7c 2020 2d20 3878 2078 2020 2b20 3332  .|  - 8x x  + 32
│ │ │ │ -0002f720: 7820 7820 202d 2033 3978 2078 2020 2b20  x x  - 39x x  + 
│ │ │ │ -0002f730: 7820 7820 202d 2033 3878 2078 2020 2d20  x x  - 38x x  - 
│ │ │ │ -0002f740: 3330 7820 7820 202d 2035 7820 7820 202b  30x x  - 5x x  +
│ │ │ │ -0002f750: 2033 3978 2078 2020 2b20 7820 7820 207c   39x x  + x x  |
│ │ │ │ -0002f760: 0a7c 3520 2020 2020 3220 3520 2020 2020  .|5     2 5     
│ │ │ │ -0002f770: 2033 2035 2020 2020 2020 3420 3520 2020   3 5      4 5   
│ │ │ │ -0002f780: 2030 2036 2020 2020 2020 3120 3620 2020   0 6      1 6   
│ │ │ │ -0002f790: 2020 2032 2036 2020 2020 2033 2036 2020     2 6     3 6  
│ │ │ │ -0002f7a0: 2020 2020 3420 3620 2020 2035 2020 207c      4 6    5   |
│ │ │ │ -0002f7b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f710: 2020 2020 207c 0a7c 2020 2d20 3878 2078       |.|  - 8x x
│ │ │ │ +0002f720: 2020 2b20 3332 7820 7820 202d 2033 3978    + 32x x  - 39x
│ │ │ │ +0002f730: 2078 2020 2b20 7820 7820 202d 2033 3878   x  + x x  - 38x
│ │ │ │ +0002f740: 2078 2020 2d20 3330 7820 7820 202d 2035   x  - 30x x  - 5
│ │ │ │ +0002f750: 7820 7820 202b 2033 3978 2078 2020 2b20  x x  + 39x x  + 
│ │ │ │ +0002f760: 7820 7820 207c 0a7c 3520 2020 2020 3220  x x  |.|5     2 
│ │ │ │ +0002f770: 3520 2020 2020 2033 2035 2020 2020 2020  5      3 5      
│ │ │ │ +0002f780: 3420 3520 2020 2030 2036 2020 2020 2020  4 5    0 6      
│ │ │ │ +0002f790: 3120 3620 2020 2020 2032 2036 2020 2020  1 6      2 6    
│ │ │ │ +0002f7a0: 2033 2036 2020 2020 2020 3420 3620 2020   3 6      4 6   
│ │ │ │ +0002f7b0: 2035 2020 207c 0a7c 2020 2020 2020 2020   5   |.|        
│ │ │ │  0002f7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f7f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f800: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f800: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f840: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f850: 0a7c 2078 2020 2d20 3231 7820 7820 202d  .| x  - 21x x  -
│ │ │ │ -0002f860: 2031 3078 2078 2020 2d20 3334 7820 7820   10x x  - 34x x 
│ │ │ │ -0002f870: 202d 2033 3878 2078 2020 2d20 3778 2078   - 38x x  - 7x x
│ │ │ │ -0002f880: 2020 2b20 3578 2078 2020 2b20 3139 7820    + 5x x  + 19x 
│ │ │ │ -0002f890: 7820 202b 2032 3378 2078 2020 2b20 207c  x  + 23x x  +  |
│ │ │ │ -0002f8a0: 0a7c 3320 3420 2020 2020 2030 2035 2020  .|3 4      0 5  
│ │ │ │ -0002f8b0: 2020 2020 3220 3520 2020 2020 2033 2035      2 5      3 5
│ │ │ │ -0002f8c0: 2020 2020 2020 3020 3620 2020 2020 3220        0 6     2 
│ │ │ │ -0002f8d0: 3620 2020 2020 3320 3620 2020 2020 2030  6     3 6      0
│ │ │ │ -0002f8e0: 2037 2020 2020 2020 3120 3720 2020 207c   7      1 7    |
│ │ │ │ -0002f8f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f850: 2020 2020 207c 0a7c 2078 2020 2d20 3231       |.| x  - 21
│ │ │ │ +0002f860: 7820 7820 202d 2031 3078 2078 2020 2d20  x x  - 10x x  - 
│ │ │ │ +0002f870: 3334 7820 7820 202d 2033 3878 2078 2020  34x x  - 38x x  
│ │ │ │ +0002f880: 2d20 3778 2078 2020 2b20 3578 2078 2020  - 7x x  + 5x x  
│ │ │ │ +0002f890: 2b20 3139 7820 7820 202b 2032 3378 2078  + 19x x  + 23x x
│ │ │ │ +0002f8a0: 2020 2b20 207c 0a7c 3320 3420 2020 2020    +  |.|3 4     
│ │ │ │ +0002f8b0: 2030 2035 2020 2020 2020 3220 3520 2020   0 5      2 5   
│ │ │ │ +0002f8c0: 2020 2033 2035 2020 2020 2020 3020 3620     3 5      0 6 
│ │ │ │ +0002f8d0: 2020 2020 3220 3620 2020 2020 3320 3620      2 6     3 6 
│ │ │ │ +0002f8e0: 2020 2020 2030 2037 2020 2020 2020 3120       0 7      1 
│ │ │ │ +0002f8f0: 3720 2020 207c 0a7c 2020 2020 2020 2020  7    |.|        
│ │ │ │  0002f900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002f920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002f930: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002f940: 0a7c 2b20 3234 7820 7820 202b 2035 7820  .|+ 24x x  + 5x 
│ │ │ │ -0002f950: 7820 202b 2078 2078 2020 2b20 3431 7820  x  + x x  + 41x 
│ │ │ │ -0002f960: 7820 202b 2031 3278 2078 2020 2d20 3578  x  + 12x x  - 5x
│ │ │ │ -0002f970: 2078 2020 2d20 3236 7820 7820 202b 2034   x  - 26x x  + 4
│ │ │ │ -0002f980: 3278 2078 2020 2d20 3234 7820 7820 207c  2x x  - 24x x  |
│ │ │ │ -0002f990: 0a7c 2020 2020 2033 2037 2020 2020 2035  .|     3 7     5
│ │ │ │ -0002f9a0: 2037 2020 2020 3620 3720 2020 2020 2031   7    6 7      1
│ │ │ │ -0002f9b0: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ -0002f9c0: 3420 3820 2020 2020 2036 2038 2020 2020  4 8      6 8    
│ │ │ │ -0002f9d0: 2020 3120 3920 2020 2020 2032 2020 207c    1 9      2   |
│ │ │ │ -0002f9e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002f930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002f940: 2020 2020 207c 0a7c 2b20 3234 7820 7820       |.|+ 24x x 
│ │ │ │ +0002f950: 202b 2035 7820 7820 202b 2078 2078 2020   + 5x x  + x x  
│ │ │ │ +0002f960: 2b20 3431 7820 7820 202b 2031 3278 2078  + 41x x  + 12x x
│ │ │ │ +0002f970: 2020 2d20 3578 2078 2020 2d20 3236 7820    - 5x x  - 26x 
│ │ │ │ +0002f980: 7820 202b 2034 3278 2078 2020 2d20 3234  x  + 42x x  - 24
│ │ │ │ +0002f990: 7820 7820 207c 0a7c 2020 2020 2033 2037  x x  |.|     3 7
│ │ │ │ +0002f9a0: 2020 2020 2035 2037 2020 2020 3620 3720       5 7    6 7 
│ │ │ │ +0002f9b0: 2020 2020 2031 2038 2020 2020 2020 3320       1 8      3 
│ │ │ │ +0002f9c0: 3820 2020 2020 3420 3820 2020 2020 2036  8     4 8      6
│ │ │ │ +0002f9d0: 2038 2020 2020 2020 3120 3920 2020 2020   8      1 9     
│ │ │ │ +0002f9e0: 2032 2020 207c 0a7c 2020 2020 2020 2020   2   |.|        
│ │ │ │  0002f9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fa00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fa10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fa20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fa30: 0a7c 7820 7820 202d 2078 2078 2020 202b  .|x x  - x x   +
│ │ │ │ -0002fa40: 2035 7820 7820 2020 2d20 3578 2078 2020   5x x   - 5x x  
│ │ │ │ -0002fa50: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ +0002fa20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fa30: 2020 2020 207c 0a7c 7820 7820 202d 2078       |.|x x  - x
│ │ │ │ +0002fa40: 2078 2020 202b 2035 7820 7820 2020 2d20   x   + 5x x   - 
│ │ │ │ +0002fa50: 3578 2078 2020 2c20 2020 2020 2020 2020  5x x  ,         
│ │ │ │  0002fa60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fa70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fa80: 0a7c 2035 2039 2020 2020 3420 3130 2020  .| 5 9    4 10  
│ │ │ │ -0002fa90: 2020 2036 2031 3020 2020 2020 3520 3131     6 10     5 11
│ │ │ │ -0002faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fa70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fa80: 2020 2020 207c 0a7c 2035 2039 2020 2020       |.| 5 9    
│ │ │ │ +0002fa90: 3420 3130 2020 2020 2036 2031 3020 2020  4 10     6 10   
│ │ │ │ +0002faa0: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │  0002fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fac0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fad0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fad0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002fae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002faf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fb10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fb20: 0a7c 2078 2020 2b20 3337 7820 7820 202b  .| x  + 37x x  +
│ │ │ │ -0002fb30: 2078 2078 2020 202b 2078 2078 2020 202d   x x   + x x   -
│ │ │ │ -0002fb40: 2035 7820 7820 2020 2b20 3578 2078 2020   5x x   + 5x x  
│ │ │ │ -0002fb50: 2c20 2020 2020 2020 2020 2020 2020 2020  ,               
│ │ │ │ -0002fb60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fb70: 0a7c 3620 3820 2020 2020 2035 2039 2020  .|6 8      5 9  
│ │ │ │ -0002fb80: 2020 3120 3130 2020 2020 3220 3130 2020    1 10    2 10  
│ │ │ │ -0002fb90: 2020 2036 2031 3020 2020 2020 3520 3131     6 10     5 11
│ │ │ │ -0002fba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fbb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002fb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fb20: 2020 2020 207c 0a7c 2078 2020 2b20 3337       |.| x  + 37
│ │ │ │ +0002fb30: 7820 7820 202b 2078 2078 2020 202b 2078  x x  + x x   + x
│ │ │ │ +0002fb40: 2078 2020 202d 2035 7820 7820 2020 2b20   x   - 5x x   + 
│ │ │ │ +0002fb50: 3578 2078 2020 2c20 2020 2020 2020 2020  5x x  ,         
│ │ │ │ +0002fb60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fb70: 2020 2020 207c 0a7c 3620 3820 2020 2020       |.|6 8     
│ │ │ │ +0002fb80: 2035 2039 2020 2020 3120 3130 2020 2020   5 9    1 10    
│ │ │ │ +0002fb90: 3220 3130 2020 2020 2036 2031 3020 2020  2 10     6 10   
│ │ │ │ +0002fba0: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │ +0002fbb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fbc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fc00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fc10: 0a7c 3578 2078 2020 2d20 3433 7820 7820  .|5x x  - 43x x 
│ │ │ │ -0002fc20: 202d 2032 3278 2078 2020 2d20 3433 7820   - 22x x  - 43x 
│ │ │ │ -0002fc30: 7820 202d 2033 7820 7820 202d 2031 3978  x  - 3x x  - 19x
│ │ │ │ -0002fc40: 2078 2020 2d20 3432 7820 7820 202d 2031   x  - 42x x  - 1
│ │ │ │ -0002fc50: 3378 2078 2020 2b20 3433 7820 7820 207c  3x x  + 43x x  |
│ │ │ │ -0002fc60: 0a7c 2020 3320 3620 2020 2020 2031 2037  .|  3 6      1 7
│ │ │ │ -0002fc70: 2020 2020 2020 3320 3720 2020 2020 2035        3 7      5
│ │ │ │ -0002fc80: 2037 2020 2020 2036 2037 2020 2020 2020   7     6 7      
│ │ │ │ -0002fc90: 3020 3820 2020 2020 2031 2038 2020 2020  0 8      1 8    
│ │ │ │ -0002fca0: 2020 3320 3820 2020 2020 2034 2020 207c    3 8      4   |
│ │ │ │ -0002fcb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fc10: 2020 2020 207c 0a7c 3578 2078 2020 2d20       |.|5x x  - 
│ │ │ │ +0002fc20: 3433 7820 7820 202d 2032 3278 2078 2020  43x x  - 22x x  
│ │ │ │ +0002fc30: 2d20 3433 7820 7820 202d 2033 7820 7820  - 43x x  - 3x x 
│ │ │ │ +0002fc40: 202d 2031 3978 2078 2020 2d20 3432 7820   - 19x x  - 42x 
│ │ │ │ +0002fc50: 7820 202d 2031 3378 2078 2020 2b20 3433  x  - 13x x  + 43
│ │ │ │ +0002fc60: 7820 7820 207c 0a7c 2020 3320 3620 2020  x x  |.|  3 6   
│ │ │ │ +0002fc70: 2020 2031 2037 2020 2020 2020 3320 3720     1 7      3 7 
│ │ │ │ +0002fc80: 2020 2020 2035 2037 2020 2020 2036 2037       5 7     6 7
│ │ │ │ +0002fc90: 2020 2020 2020 3020 3820 2020 2020 2031        0 8      1
│ │ │ │ +0002fca0: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ +0002fcb0: 2034 2020 207c 0a7c 2020 2020 2020 2020   4   |.|        
│ │ │ │  0002fcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fcd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fcf0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fd00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002fcf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fd00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002fd10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fd40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fd50: 0a7c 5050 5e31 3120 746f 2050 505e 3829  .|PP^11 to PP^8)
│ │ │ │ -0002fd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fd50: 2020 2020 207c 0a7c 5050 5e31 3120 746f       |.|PP^11 to
│ │ │ │ +0002fd60: 2050 505e 3829 2020 2020 2020 2020 2020   PP^8)          
│ │ │ │  0002fd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fd90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fda0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +0002fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fda0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  0002fdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0002fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0002fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -0002fdf0: 0a7c 7820 202d 2032 3378 2078 2020 2d20  .|x  - 23x x  - 
│ │ │ │ -0002fe00: 3333 7820 7820 202b 2038 7820 7820 202b  33x x  + 8x x  +
│ │ │ │ -0002fe10: 2031 3078 2078 2020 2d20 3235 7820 7820   10x x  - 25x x 
│ │ │ │ -0002fe20: 202d 2039 7820 7820 202b 2033 7820 7820   - 9x x  + 3x x 
│ │ │ │ -0002fe30: 202b 2032 3478 2078 2020 2020 2020 207c   + 24x x       |
│ │ │ │ -0002fe40: 0a7c 2038 2020 2020 2020 3420 3820 2020  .| 8      4 8   
│ │ │ │ -0002fe50: 2020 2036 2038 2020 2020 2030 2039 2020     6 8     0 9  
│ │ │ │ -0002fe60: 2020 2020 3120 3920 2020 2020 2032 2039      1 9      2 9
│ │ │ │ -0002fe70: 2020 2020 2033 2039 2020 2020 2034 2039       3 9     4 9
│ │ │ │ -0002fe80: 2020 2020 2020 3520 2020 2020 2020 207c        5        |
│ │ │ │ -0002fe90: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0002fdf0: 2d2d 2d2d 2d7c 0a7c 7820 202d 2032 3378  -----|.|x  - 23x
│ │ │ │ +0002fe00: 2078 2020 2d20 3333 7820 7820 202b 2038   x  - 33x x  + 8
│ │ │ │ +0002fe10: 7820 7820 202b 2031 3078 2078 2020 2d20  x x  + 10x x  - 
│ │ │ │ +0002fe20: 3235 7820 7820 202d 2039 7820 7820 202b  25x x  - 9x x  +
│ │ │ │ +0002fe30: 2033 7820 7820 202b 2032 3478 2078 2020   3x x  + 24x x  
│ │ │ │ +0002fe40: 2020 2020 207c 0a7c 2038 2020 2020 2020       |.| 8      
│ │ │ │ +0002fe50: 3420 3820 2020 2020 2036 2038 2020 2020  4 8      6 8    
│ │ │ │ +0002fe60: 2030 2039 2020 2020 2020 3120 3920 2020   0 9      1 9   
│ │ │ │ +0002fe70: 2020 2032 2039 2020 2020 2033 2039 2020     2 9     3 9  
│ │ │ │ +0002fe80: 2020 2034 2039 2020 2020 2020 3520 2020     4 9      5   
│ │ │ │ +0002fe90: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002fea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002feb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002fed0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002fee0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002fed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002fee0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002fef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ff00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002ff10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0002ff20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -0002ff30: 0a7c 7820 202d 2033 3478 2078 2020 2d20  .|x  - 34x x  - 
│ │ │ │ -0002ff40: 3978 2078 2020 2d20 3436 7820 7820 202d  9x x  - 46x x  -
│ │ │ │ -0002ff50: 2031 3778 2078 2020 2b20 3332 7820 7820   17x x  + 32x x 
│ │ │ │ -0002ff60: 202d 2038 7820 7820 202d 2033 3578 2078   - 8x x  - 35x x
│ │ │ │ -0002ff70: 2020 2d20 3436 7820 2020 2020 2020 207c    - 46x        |
│ │ │ │ -0002ff80: 0a7c 2038 2020 2020 2020 3320 3820 2020  .| 8      3 8   
│ │ │ │ -0002ff90: 2020 3420 3820 2020 2020 2036 2038 2020    4 8      6 8  
│ │ │ │ -0002ffa0: 2020 2020 3020 3920 2020 2020 2031 2039      0 9      1 9
│ │ │ │ -0002ffb0: 2020 2020 2032 2039 2020 2020 2020 3320       2 9      3 
│ │ │ │ -0002ffc0: 3920 2020 2020 2034 2020 2020 2020 207c  9      4       |
│ │ │ │ -0002ffd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +0002ff20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0002ff30: 2020 2020 207c 0a7c 7820 202d 2033 3478       |.|x  - 34x
│ │ │ │ +0002ff40: 2078 2020 2d20 3978 2078 2020 2d20 3436   x  - 9x x  - 46
│ │ │ │ +0002ff50: 7820 7820 202d 2031 3778 2078 2020 2b20  x x  - 17x x  + 
│ │ │ │ +0002ff60: 3332 7820 7820 202d 2038 7820 7820 202d  32x x  - 8x x  -
│ │ │ │ +0002ff70: 2033 3578 2078 2020 2d20 3436 7820 2020   35x x  - 46x   
│ │ │ │ +0002ff80: 2020 2020 207c 0a7c 2038 2020 2020 2020       |.| 8      
│ │ │ │ +0002ff90: 3320 3820 2020 2020 3420 3820 2020 2020  3 8     4 8     
│ │ │ │ +0002ffa0: 2036 2038 2020 2020 2020 3020 3920 2020   6 8      0 9   
│ │ │ │ +0002ffb0: 2020 2031 2039 2020 2020 2032 2039 2020     1 9     2 9  
│ │ │ │ +0002ffc0: 2020 2020 3320 3920 2020 2020 2034 2020      3 9      4  
│ │ │ │ +0002ffd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  0002ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0002fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030010: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030020: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030020: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030060: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030070: 0a7c 2020 2d20 7820 7820 202b 2078 2078  .|  - x x  + x x
│ │ │ │ -00030080: 2020 2b20 3430 7820 7820 202d 2033 3278    + 40x x  - 32x
│ │ │ │ -00030090: 2078 2020 2b20 3578 2078 2020 2d20 3131   x  + 5x x  - 11
│ │ │ │ -000300a0: 7820 7820 202d 2032 3078 2078 2020 2b20  x x  - 20x x  + 
│ │ │ │ -000300b0: 3435 7820 7820 202d 2020 2020 2020 207c  45x x  -       |
│ │ │ │ -000300c0: 0a7c 3820 2020 2034 2038 2020 2020 3620  .|8    4 8    6 
│ │ │ │ -000300d0: 3820 2020 2020 2030 2039 2020 2020 2020  8      0 9      
│ │ │ │ -000300e0: 3120 3920 2020 2020 3220 3920 2020 2020  1 9     2 9     
│ │ │ │ -000300f0: 2033 2039 2020 2020 2020 3420 3920 2020   3 9      4 9   
│ │ │ │ -00030100: 2020 2035 2039 2020 2020 2020 2020 207c     5 9         |
│ │ │ │ -00030110: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030070: 2020 2020 207c 0a7c 2020 2d20 7820 7820       |.|  - x x 
│ │ │ │ +00030080: 202b 2078 2078 2020 2b20 3430 7820 7820   + x x  + 40x x 
│ │ │ │ +00030090: 202d 2033 3278 2078 2020 2b20 3578 2078   - 32x x  + 5x x
│ │ │ │ +000300a0: 2020 2d20 3131 7820 7820 202d 2032 3078    - 11x x  - 20x
│ │ │ │ +000300b0: 2078 2020 2b20 3435 7820 7820 202d 2020   x  + 45x x  -  
│ │ │ │ +000300c0: 2020 2020 207c 0a7c 3820 2020 2034 2038       |.|8    4 8
│ │ │ │ +000300d0: 2020 2020 3620 3820 2020 2020 2030 2039      6 8      0 9
│ │ │ │ +000300e0: 2020 2020 2020 3120 3920 2020 2020 3220        1 9     2 
│ │ │ │ +000300f0: 3920 2020 2020 2033 2039 2020 2020 2020  9      3 9      
│ │ │ │ +00030100: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ +00030110: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030150: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030160: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030160: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000301a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000301b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000301a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000301b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000301c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000301d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000301e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000301f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030200: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000301f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030200: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030240: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030250: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030250: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030290: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000302a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000302a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000302b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000302c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000302d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000302e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000302f0: 0a7c 3337 7820 7820 202b 2031 3278 2078  .|37x x  + 12x x
│ │ │ │ -00030300: 2020 2d20 3578 2078 2020 2d20 3578 2078    - 5x x  - 5x x
│ │ │ │ -00030310: 2020 2b20 3231 7820 7820 202b 2034 3278    + 21x x  + 42x
│ │ │ │ -00030320: 2078 2020 2d20 3578 2078 2020 2b20 7820   x  - 5x x  + x 
│ │ │ │ -00030330: 7820 202d 2078 2078 2020 2020 2020 207c  x  - x x       |
│ │ │ │ -00030340: 0a7c 2020 2031 2038 2020 2020 2020 3320  .|   1 8      3 
│ │ │ │ -00030350: 3820 2020 2020 3420 3820 2020 2020 3620  8     4 8     6 
│ │ │ │ -00030360: 3820 2020 2020 2030 2039 2020 2020 2020  8      0 9      
│ │ │ │ -00030370: 3120 3920 2020 2020 3220 3920 2020 2033  1 9     2 9    3
│ │ │ │ -00030380: 2039 2020 2020 3420 2020 2020 2020 207c   9    4        |
│ │ │ │ -00030390: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000302e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000302f0: 2020 2020 207c 0a7c 3337 7820 7820 202b       |.|37x x  +
│ │ │ │ +00030300: 2031 3278 2078 2020 2d20 3578 2078 2020   12x x  - 5x x  
│ │ │ │ +00030310: 2d20 3578 2078 2020 2b20 3231 7820 7820  - 5x x  + 21x x 
│ │ │ │ +00030320: 202b 2034 3278 2078 2020 2d20 3578 2078   + 42x x  - 5x x
│ │ │ │ +00030330: 2020 2b20 7820 7820 202d 2078 2078 2020    + x x  - x x  
│ │ │ │ +00030340: 2020 2020 207c 0a7c 2020 2031 2038 2020       |.|   1 8  
│ │ │ │ +00030350: 2020 2020 3320 3820 2020 2020 3420 3820      3 8     4 8 
│ │ │ │ +00030360: 2020 2020 3620 3820 2020 2020 2030 2039      6 8      0 9
│ │ │ │ +00030370: 2020 2020 2020 3120 3920 2020 2020 3220        1 9     2 
│ │ │ │ +00030380: 3920 2020 2033 2039 2020 2020 3420 2020  9    3 9    4   
│ │ │ │ +00030390: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000303a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000303c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000303d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000303e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000303d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000303e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000303f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030420: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030430: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030430: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030470: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030480: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030480: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000304b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000304c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000304d0: 0a7c 7820 202b 2034 3078 2078 2020 2d20  .|x  + 40x x  - 
│ │ │ │ -000304e0: 3578 2078 2020 2b20 3339 7820 7820 202d  5x x  + 39x x  -
│ │ │ │ -000304f0: 2033 3778 2078 2020 2b20 7820 7820 2020   37x x  + x x   
│ │ │ │ -00030500: 2d20 7820 7820 2020 2d20 7820 7820 2020  - x x   - x x   
│ │ │ │ -00030510: 2b20 7820 7820 2020 2020 2020 2020 207c  + x x          |
│ │ │ │ -00030520: 0a7c 2039 2020 2020 2020 3220 3920 2020  .| 9      2 9   
│ │ │ │ -00030530: 2020 3320 3920 2020 2020 2034 2039 2020    3 9      4 9  
│ │ │ │ -00030540: 2020 2020 3520 3920 2020 2030 2031 3020      5 9    0 10 
│ │ │ │ -00030550: 2020 2031 2031 3020 2020 2032 2031 3020     1 10    2 10 
│ │ │ │ -00030560: 2020 2034 2031 3020 2020 2020 2020 207c     4 10        |
│ │ │ │ -00030570: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000304c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000304d0: 2020 2020 207c 0a7c 7820 202b 2034 3078       |.|x  + 40x
│ │ │ │ +000304e0: 2078 2020 2d20 3578 2078 2020 2b20 3339   x  - 5x x  + 39
│ │ │ │ +000304f0: 7820 7820 202d 2033 3778 2078 2020 2b20  x x  - 37x x  + 
│ │ │ │ +00030500: 7820 7820 2020 2d20 7820 7820 2020 2d20  x x   - x x   - 
│ │ │ │ +00030510: 7820 7820 2020 2b20 7820 7820 2020 2020  x x   + x x     
│ │ │ │ +00030520: 2020 2020 207c 0a7c 2039 2020 2020 2020       |.| 9      
│ │ │ │ +00030530: 3220 3920 2020 2020 3320 3920 2020 2020  2 9     3 9     
│ │ │ │ +00030540: 2034 2039 2020 2020 2020 3520 3920 2020   4 9      5 9   
│ │ │ │ +00030550: 2030 2031 3020 2020 2031 2031 3020 2020   0 10    1 10   
│ │ │ │ +00030560: 2032 2031 3020 2020 2034 2031 3020 2020   2 10    4 10   
│ │ │ │ +00030570: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000305b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000305c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000305b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000305c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000305d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000305f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030600: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030610: 0a7c 2020 2d20 3237 7820 7820 202d 2032  .|  - 27x x  - 2
│ │ │ │ -00030620: 3778 2078 2020 2d20 3432 7820 7820 202b  7x x  - 42x x  +
│ │ │ │ -00030630: 2031 3178 2078 2020 2b20 3278 2078 2020   11x x  + 2x x  
│ │ │ │ -00030640: 2b20 3232 7820 7820 202b 2033 3478 2078  + 22x x  + 34x x
│ │ │ │ -00030650: 2020 2b20 3432 7820 2020 2020 2020 207c    + 42x        |
│ │ │ │ -00030660: 0a7c 3620 2020 2020 2031 2037 2020 2020  .|6      1 7    
│ │ │ │ -00030670: 2020 3320 3720 2020 2020 2035 2037 2020    3 7      5 7  
│ │ │ │ -00030680: 2020 2020 3620 3720 2020 2020 3020 3820      6 7     0 8 
│ │ │ │ -00030690: 2020 2020 2031 2038 2020 2020 2020 3320       1 8      3 
│ │ │ │ -000306a0: 3820 2020 2020 2034 2020 2020 2020 207c  8      4       |
│ │ │ │ -000306b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030610: 2020 2020 207c 0a7c 2020 2d20 3237 7820       |.|  - 27x 
│ │ │ │ +00030620: 7820 202d 2032 3778 2078 2020 2d20 3432  x  - 27x x  - 42
│ │ │ │ +00030630: 7820 7820 202b 2031 3178 2078 2020 2b20  x x  + 11x x  + 
│ │ │ │ +00030640: 3278 2078 2020 2b20 3232 7820 7820 202b  2x x  + 22x x  +
│ │ │ │ +00030650: 2033 3478 2078 2020 2b20 3432 7820 2020   34x x  + 42x   
│ │ │ │ +00030660: 2020 2020 207c 0a7c 3620 2020 2020 2031       |.|6      1
│ │ │ │ +00030670: 2037 2020 2020 2020 3320 3720 2020 2020   7      3 7     
│ │ │ │ +00030680: 2035 2037 2020 2020 2020 3620 3720 2020   5 7      6 7   
│ │ │ │ +00030690: 2020 3020 3820 2020 2020 2031 2038 2020    0 8      1 8  
│ │ │ │ +000306a0: 2020 2020 3320 3820 2020 2020 2034 2020      3 8      4  
│ │ │ │ +000306b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000306c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000306e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000306f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030700: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000306f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030700: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030740: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030750: 0a7c 2032 3178 2078 2020 2b20 3434 7820  .| 21x x  + 44x 
│ │ │ │ -00030760: 7820 202d 2031 3478 2078 2020 2d20 3232  x  - 14x x  - 22
│ │ │ │ -00030770: 7820 7820 202d 2035 7820 7820 202d 2033  x x  - 5x x  - 3
│ │ │ │ -00030780: 3478 2078 2020 2d20 3434 7820 7820 202b  4x x  - 44x x  +
│ │ │ │ -00030790: 2031 3078 2078 2020 2020 2020 2020 207c   10x x         |
│ │ │ │ -000307a0: 0a7c 2020 2020 3320 3720 2020 2020 2035  .|    3 7      5
│ │ │ │ -000307b0: 2037 2020 2020 2020 3620 3720 2020 2020   7      6 7     
│ │ │ │ -000307c0: 2030 2038 2020 2020 2031 2038 2020 2020   0 8     1 8    
│ │ │ │ -000307d0: 2020 3320 3820 2020 2020 2034 2038 2020    3 8      4 8  
│ │ │ │ -000307e0: 2020 2020 3620 3820 2020 2020 2020 207c      6 8        |
│ │ │ │ -000307f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030750: 2020 2020 207c 0a7c 2032 3178 2078 2020       |.| 21x x  
│ │ │ │ +00030760: 2b20 3434 7820 7820 202d 2031 3478 2078  + 44x x  - 14x x
│ │ │ │ +00030770: 2020 2d20 3232 7820 7820 202d 2035 7820    - 22x x  - 5x 
│ │ │ │ +00030780: 7820 202d 2033 3478 2078 2020 2d20 3434  x  - 34x x  - 44
│ │ │ │ +00030790: 7820 7820 202b 2031 3078 2078 2020 2020  x x  + 10x x    
│ │ │ │ +000307a0: 2020 2020 207c 0a7c 2020 2020 3320 3720       |.|    3 7 
│ │ │ │ +000307b0: 2020 2020 2035 2037 2020 2020 2020 3620       5 7      6 
│ │ │ │ +000307c0: 3720 2020 2020 2030 2038 2020 2020 2031  7      0 8     1
│ │ │ │ +000307d0: 2038 2020 2020 2020 3320 3820 2020 2020   8      3 8     
│ │ │ │ +000307e0: 2034 2038 2020 2020 2020 3620 3820 2020   4 8      6 8   
│ │ │ │ +000307f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030830: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030840: 0a7c 2020 2b20 3337 7820 7820 202d 2078  .|  + 37x x  - x
│ │ │ │ -00030850: 2078 2020 2d20 3335 7820 7820 202b 2039   x  - 35x x  + 9
│ │ │ │ -00030860: 7820 7820 202b 2032 3578 2078 2020 202d  x x  + 25x x   -
│ │ │ │ -00030870: 2031 3278 2078 2020 202d 2033 3678 2078   12x x   - 36x x
│ │ │ │ -00030880: 2020 202d 2031 3478 2020 2020 2020 207c     - 14x       |
│ │ │ │ -00030890: 0a7c 3920 2020 2020 2033 2039 2020 2020  .|9      3 9    
│ │ │ │ -000308a0: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ -000308b0: 2036 2039 2020 2020 2020 3120 3130 2020   6 9      1 10  
│ │ │ │ -000308c0: 2020 2020 3220 3130 2020 2020 2020 3420      2 10      4 
│ │ │ │ -000308d0: 3130 2020 2020 2020 2020 2020 2020 207c  10             |
│ │ │ │ -000308e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030840: 2020 2020 207c 0a7c 2020 2b20 3337 7820       |.|  + 37x 
│ │ │ │ +00030850: 7820 202d 2078 2078 2020 2d20 3335 7820  x  - x x  - 35x 
│ │ │ │ +00030860: 7820 202b 2039 7820 7820 202b 2032 3578  x  + 9x x  + 25x
│ │ │ │ +00030870: 2078 2020 202d 2031 3278 2078 2020 202d   x   - 12x x   -
│ │ │ │ +00030880: 2033 3678 2078 2020 202d 2031 3478 2020   36x x   - 14x  
│ │ │ │ +00030890: 2020 2020 207c 0a7c 3920 2020 2020 2033       |.|9      3
│ │ │ │ +000308a0: 2039 2020 2020 3420 3920 2020 2020 2035   9    4 9      5
│ │ │ │ +000308b0: 2039 2020 2020 2036 2039 2020 2020 2020   9     6 9      
│ │ │ │ +000308c0: 3120 3130 2020 2020 2020 3220 3130 2020  1 10      2 10  
│ │ │ │ +000308d0: 2020 2020 3420 3130 2020 2020 2020 2020      4 10        
│ │ │ │ +000308e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000308f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030920: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030930: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030930: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030970: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030980: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030980: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000309c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000309d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000309c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000309d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000309e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000309f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030a20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030a20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030a60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030a70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030a70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ab0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030ac0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030ac0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030b00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030b10: 0a7c 2020 2d20 3236 7820 7820 202d 2034  .|  - 26x x  - 4
│ │ │ │ -00030b20: 3578 2078 2020 2b20 3232 7820 7820 202d  5x x  + 22x x  -
│ │ │ │ -00030b30: 2034 3278 2078 2020 2b20 3378 2078 2020   42x x  + 3x x  
│ │ │ │ -00030b40: 2d20 3133 7820 7820 202d 2034 7820 7820  - 13x x  - 4x x 
│ │ │ │ -00030b50: 202d 2032 3478 2078 2020 2020 2020 207c   - 24x x       |
│ │ │ │ -00030b60: 0a7c 3820 2020 2020 2036 2038 2020 2020  .|8      6 8    
│ │ │ │ -00030b70: 2020 3120 3920 2020 2020 2032 2039 2020    1 9      2 9  
│ │ │ │ -00030b80: 2020 2020 3320 3920 2020 2020 3420 3920      3 9     4 9 
│ │ │ │ -00030b90: 2020 2020 2035 2039 2020 2020 2036 2039       5 9     6 9
│ │ │ │ -00030ba0: 2020 2020 2020 3120 2020 2020 2020 207c        1        |
│ │ │ │ -00030bb0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00030b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030b10: 2020 2020 207c 0a7c 2020 2d20 3236 7820       |.|  - 26x 
│ │ │ │ +00030b20: 7820 202d 2034 3578 2078 2020 2b20 3232  x  - 45x x  + 22
│ │ │ │ +00030b30: 7820 7820 202d 2034 3278 2078 2020 2b20  x x  - 42x x  + 
│ │ │ │ +00030b40: 3378 2078 2020 2d20 3133 7820 7820 202d  3x x  - 13x x  -
│ │ │ │ +00030b50: 2034 7820 7820 202d 2032 3478 2078 2020   4x x  - 24x x  
│ │ │ │ +00030b60: 2020 2020 207c 0a7c 3820 2020 2020 2036       |.|8      6
│ │ │ │ +00030b70: 2038 2020 2020 2020 3120 3920 2020 2020   8      1 9     
│ │ │ │ +00030b80: 2032 2039 2020 2020 2020 3320 3920 2020   2 9      3 9   
│ │ │ │ +00030b90: 2020 3420 3920 2020 2020 2035 2039 2020    4 9      5 9  
│ │ │ │ +00030ba0: 2020 2036 2039 2020 2020 2020 3120 2020     6 9      1   
│ │ │ │ +00030bb0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00030bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00030be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00030c00: 0a7c 2020 2d20 3237 7820 7820 202d 2035  .|  - 27x x  - 5
│ │ │ │ -00030c10: 7820 7820 2020 2b20 3238 7820 7820 2020  x x   + 28x x   
│ │ │ │ -00030c20: 2b20 3337 7820 7820 2020 2b20 3978 2078  + 37x x   + 9x x
│ │ │ │ -00030c30: 2020 202b 2032 3778 2078 2020 202d 2032     + 27x x   - 2
│ │ │ │ -00030c40: 3578 2078 2020 202b 2039 7820 7820 207c  5x x   + 9x x  |
│ │ │ │ -00030c50: 0a7c 3920 2020 2020 2036 2039 2020 2020  .|9      6 9    
│ │ │ │ -00030c60: 2030 2031 3020 2020 2020 2031 2031 3020   0 10      1 10 
│ │ │ │ -00030c70: 2020 2020 2032 2031 3020 2020 2020 3420       2 10     4 
│ │ │ │ -00030c80: 3130 2020 2020 2020 3620 3130 2020 2020  10      6 10    
│ │ │ │ -00030c90: 2020 3020 3131 2020 2020 2032 2031 317c    0 11     2 11|
│ │ │ │ -00030ca0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00030c00: 2d2d 2d2d 2d7c 0a7c 2020 2d20 3237 7820  -----|.|  - 27x 
│ │ │ │ +00030c10: 7820 202d 2035 7820 7820 2020 2b20 3238  x  - 5x x   + 28
│ │ │ │ +00030c20: 7820 7820 2020 2b20 3337 7820 7820 2020  x x   + 37x x   
│ │ │ │ +00030c30: 2b20 3978 2078 2020 202b 2032 3778 2078  + 9x x   + 27x x
│ │ │ │ +00030c40: 2020 202d 2032 3578 2078 2020 202b 2039     - 25x x   + 9
│ │ │ │ +00030c50: 7820 7820 207c 0a7c 3920 2020 2020 2036  x x  |.|9      6
│ │ │ │ +00030c60: 2039 2020 2020 2030 2031 3020 2020 2020   9     0 10     
│ │ │ │ +00030c70: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ +00030c80: 2020 2020 3420 3130 2020 2020 2020 3620      4 10      6 
│ │ │ │ +00030c90: 3130 2020 2020 2020 3020 3131 2020 2020  10      0 11    
│ │ │ │ +00030ca0: 2032 2031 317c 0a7c 2020 2020 2020 2020   2 11|.|        
│ │ │ │  00030cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030ce0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030cf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030d30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030d40: 0a7c 7820 202b 2032 3678 2078 2020 2b20  .|x  + 26x x  + 
│ │ │ │ -00030d50: 3137 7820 7820 202b 2031 3578 2078 2020  17x x  + 15x x  
│ │ │ │ -00030d60: 202b 2033 3578 2078 2020 202b 2033 3478   + 35x x   + 34x
│ │ │ │ -00030d70: 2078 2020 202b 2032 3078 2078 2020 202b   x   + 20x x   +
│ │ │ │ -00030d80: 2031 3478 2078 2020 202b 2033 3678 207c   14x x   + 36x |
│ │ │ │ -00030d90: 0a7c 2039 2020 2020 2020 3520 3920 2020  .| 9      5 9   
│ │ │ │ -00030da0: 2020 2036 2039 2020 2020 2020 3020 3130     6 9      0 10
│ │ │ │ -00030db0: 2020 2020 2020 3120 3130 2020 2020 2020        1 10      
│ │ │ │ -00030dc0: 3220 3130 2020 2020 2020 3420 3130 2020  2 10      4 10  
│ │ │ │ -00030dd0: 2020 2020 3020 3131 2020 2020 2020 317c      0 11      1|
│ │ │ │ -00030de0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030d40: 2020 2020 207c 0a7c 7820 202b 2032 3678       |.|x  + 26x
│ │ │ │ +00030d50: 2078 2020 2b20 3137 7820 7820 202b 2031   x  + 17x x  + 1
│ │ │ │ +00030d60: 3578 2078 2020 202b 2033 3578 2078 2020  5x x   + 35x x  
│ │ │ │ +00030d70: 202b 2033 3478 2078 2020 202b 2032 3078   + 34x x   + 20x
│ │ │ │ +00030d80: 2078 2020 202b 2031 3478 2078 2020 202b   x   + 14x x   +
│ │ │ │ +00030d90: 2033 3678 207c 0a7c 2039 2020 2020 2020   36x |.| 9      
│ │ │ │ +00030da0: 3520 3920 2020 2020 2036 2039 2020 2020  5 9      6 9    
│ │ │ │ +00030db0: 2020 3020 3130 2020 2020 2020 3120 3130    0 10      1 10
│ │ │ │ +00030dc0: 2020 2020 2020 3220 3130 2020 2020 2020        2 10      
│ │ │ │ +00030dd0: 3420 3130 2020 2020 2020 3020 3131 2020  4 10      0 11  
│ │ │ │ +00030de0: 2020 2020 317c 0a7c 2020 2020 2020 2020      1|.|        
│ │ │ │  00030df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030e30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030e30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030e70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030e80: 0a7c 2031 3478 2078 2020 2d20 3235 7820  .| 14x x  - 25x 
│ │ │ │ -00030e90: 7820 2020 2b20 3435 7820 7820 2020 2d20  x   + 45x x   - 
│ │ │ │ -00030ea0: 3431 7820 7820 2020 2d20 3436 7820 7820  41x x   - 46x x 
│ │ │ │ -00030eb0: 2020 2b20 3878 2078 2020 202d 2032 3878    + 8x x   - 28x
│ │ │ │ -00030ec0: 2078 2020 202b 2031 3178 2078 2020 207c   x   + 11x x   |
│ │ │ │ -00030ed0: 0a7c 2020 2020 3620 3920 2020 2020 2030  .|    6 9      0
│ │ │ │ -00030ee0: 2031 3020 2020 2020 2031 2031 3020 2020   10      1 10   
│ │ │ │ -00030ef0: 2020 2032 2031 3020 2020 2020 2034 2031     2 10      4 1
│ │ │ │ -00030f00: 3020 2020 2020 3620 3130 2020 2020 2020  0     6 10      
│ │ │ │ -00030f10: 3020 3131 2020 2020 2020 3220 3131 207c  0 11      2 11 |
│ │ │ │ -00030f20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030e80: 2020 2020 207c 0a7c 2031 3478 2078 2020       |.| 14x x  
│ │ │ │ +00030e90: 2d20 3235 7820 7820 2020 2b20 3435 7820  - 25x x   + 45x 
│ │ │ │ +00030ea0: 7820 2020 2d20 3431 7820 7820 2020 2d20  x   - 41x x   - 
│ │ │ │ +00030eb0: 3436 7820 7820 2020 2b20 3878 2078 2020  46x x   + 8x x  
│ │ │ │ +00030ec0: 202d 2032 3878 2078 2020 202b 2031 3178   - 28x x   + 11x
│ │ │ │ +00030ed0: 2078 2020 207c 0a7c 2020 2020 3620 3920   x   |.|    6 9 
│ │ │ │ +00030ee0: 2020 2020 2030 2031 3020 2020 2020 2031       0 10      1
│ │ │ │ +00030ef0: 2031 3020 2020 2020 2032 2031 3020 2020   10      2 10   
│ │ │ │ +00030f00: 2020 2034 2031 3020 2020 2020 3620 3130     4 10     6 10
│ │ │ │ +00030f10: 2020 2020 2020 3020 3131 2020 2020 2020        0 11      
│ │ │ │ +00030f20: 3220 3131 207c 0a7c 2020 2020 2020 2020  2 11 |.|        
│ │ │ │  00030f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030f60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030f70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030f70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00030fb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00030fc0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00030fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00030fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00030fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00030ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031000: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031010: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031010: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031050: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031060: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031060: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000310a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000310b0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000310a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000310b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000310c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000310d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000310e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000310f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031100: 0a7c 2020 2b20 3278 2078 2020 2b20 3438  .|  + 2x x  + 48
│ │ │ │ -00031110: 7820 7820 202d 2078 2078 2020 202b 2032  x x  - x x   + 2
│ │ │ │ -00031120: 3678 2078 2020 202d 2031 3278 2078 2020  6x x   - 12x x  
│ │ │ │ -00031130: 202d 2033 3678 2078 2020 202d 2031 3878   - 36x x   - 18x
│ │ │ │ -00031140: 2078 2020 202d 2035 7820 7820 2020 2d7c   x   - 5x x   -|
│ │ │ │ -00031150: 0a7c 3920 2020 2020 3520 3920 2020 2020  .|9     5 9     
│ │ │ │ -00031160: 2036 2039 2020 2020 3020 3130 2020 2020   6 9    0 10    
│ │ │ │ -00031170: 2020 3120 3130 2020 2020 2020 3220 3130    1 10      2 10
│ │ │ │ -00031180: 2020 2020 2020 3420 3130 2020 2020 2020        4 10      
│ │ │ │ -00031190: 3620 3130 2020 2020 2030 2031 3120 207c  6 10     0 11  |
│ │ │ │ -000311a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000310f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031100: 2020 2020 207c 0a7c 2020 2b20 3278 2078       |.|  + 2x x
│ │ │ │ +00031110: 2020 2b20 3438 7820 7820 202d 2078 2078    + 48x x  - x x
│ │ │ │ +00031120: 2020 202b 2032 3678 2078 2020 202d 2031     + 26x x   - 1
│ │ │ │ +00031130: 3278 2078 2020 202d 2033 3678 2078 2020  2x x   - 36x x  
│ │ │ │ +00031140: 202d 2031 3878 2078 2020 202d 2035 7820   - 18x x   - 5x 
│ │ │ │ +00031150: 7820 2020 2d7c 0a7c 3920 2020 2020 3520  x   -|.|9     5 
│ │ │ │ +00031160: 3920 2020 2020 2036 2039 2020 2020 3020  9      6 9    0 
│ │ │ │ +00031170: 3130 2020 2020 2020 3120 3130 2020 2020  10      1 10    
│ │ │ │ +00031180: 2020 3220 3130 2020 2020 2020 3420 3130    2 10      4 10
│ │ │ │ +00031190: 2020 2020 2020 3620 3130 2020 2020 2030        6 10     0
│ │ │ │ +000311a0: 2031 3120 207c 0a7c 2020 2020 2020 2020   11  |.|        
│ │ │ │  000311b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000311c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000311d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000311e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000311f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000311e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000311f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031230: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031240: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031240: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031280: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031290: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031290: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000312a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000312b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000312c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000312d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000312e0: 0a7c 2b20 3578 2078 2020 202d 2035 7820  .|+ 5x x   - 5x 
│ │ │ │ -000312f0: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
│ │ │ │ +000312d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000312e0: 2020 2020 207c 0a7c 2b20 3578 2078 2020       |.|+ 5x x  
│ │ │ │ +000312f0: 202d 2035 7820 7820 202c 2020 2020 2020   - 5x x  ,      
│ │ │ │  00031300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031320: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031330: 0a7c 2020 2020 3620 3130 2020 2020 2035  .|    6 10     5
│ │ │ │ -00031340: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │ +00031320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031330: 2020 2020 207c 0a7c 2020 2020 3620 3130       |.|    6 10
│ │ │ │ +00031340: 2020 2020 2035 2031 3120 2020 2020 2020       5 11       
│ │ │ │  00031350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031370: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031380: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031380: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000313c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000313d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000313c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000313d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000313e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000313f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031410: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031420: 0a7c 7820 202b 2031 3478 2078 2020 2b20  .|x  + 14x x  + 
│ │ │ │ -00031430: 3878 2078 2020 2b20 3136 7820 7820 202b  8x x  + 16x x  +
│ │ │ │ -00031440: 2032 3778 2078 2020 2b20 3234 7820 7820   27x x  + 24x x 
│ │ │ │ -00031450: 202d 2031 3178 2078 2020 2d20 3678 2078   - 11x x  - 6x x
│ │ │ │ -00031460: 2020 2b20 3278 2078 2020 2d20 3578 207c    + 2x x  - 5x |
│ │ │ │ -00031470: 0a7c 2038 2020 2020 2020 3620 3820 2020  .| 8      6 8   
│ │ │ │ -00031480: 2020 3020 3920 2020 2020 2031 2039 2020    0 9      1 9  
│ │ │ │ -00031490: 2020 2020 3220 3920 2020 2020 2033 2039      2 9      3 9
│ │ │ │ -000314a0: 2020 2020 2020 3420 3920 2020 2020 3520        4 9     5 
│ │ │ │ -000314b0: 3920 2020 2020 3620 3920 2020 2020 307c  9     6 9     0|
│ │ │ │ -000314c0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031420: 2020 2020 207c 0a7c 7820 202b 2031 3478       |.|x  + 14x
│ │ │ │ +00031430: 2078 2020 2b20 3878 2078 2020 2b20 3136   x  + 8x x  + 16
│ │ │ │ +00031440: 7820 7820 202b 2032 3778 2078 2020 2b20  x x  + 27x x  + 
│ │ │ │ +00031450: 3234 7820 7820 202d 2031 3178 2078 2020  24x x  - 11x x  
│ │ │ │ +00031460: 2d20 3678 2078 2020 2b20 3278 2078 2020  - 6x x  + 2x x  
│ │ │ │ +00031470: 2d20 3578 207c 0a7c 2038 2020 2020 2020  - 5x |.| 8      
│ │ │ │ +00031480: 3620 3820 2020 2020 3020 3920 2020 2020  6 8     0 9     
│ │ │ │ +00031490: 2031 2039 2020 2020 2020 3220 3920 2020   1 9      2 9   
│ │ │ │ +000314a0: 2020 2033 2039 2020 2020 2020 3420 3920     3 9      4 9 
│ │ │ │ +000314b0: 2020 2020 3520 3920 2020 2020 3620 3920      5 9     6 9 
│ │ │ │ +000314c0: 2020 2020 307c 0a7c 2020 2020 2020 2020      0|.|        
│ │ │ │  000314d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000314f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031510: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031510: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031550: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031560: 0a7c 2d20 3434 7820 7820 202d 2032 3178  .|- 44x x  - 21x
│ │ │ │ -00031570: 2078 2020 2d20 3231 7820 7820 202d 2039   x  - 21x x  - 9
│ │ │ │ -00031580: 7820 7820 202b 2031 3478 2078 2020 2d20  x x  + 14x x  - 
│ │ │ │ -00031590: 3232 7820 7820 202b 2034 3578 2078 2020  22x x  + 45x x  
│ │ │ │ -000315a0: 2b20 3578 2078 2020 202b 2032 3078 207c  + 5x x   + 20x |
│ │ │ │ -000315b0: 0a7c 2020 2020 2030 2039 2020 2020 2020  .|     0 9      
│ │ │ │ -000315c0: 3120 3920 2020 2020 2032 2039 2020 2020  1 9      2 9    
│ │ │ │ -000315d0: 2033 2039 2020 2020 2020 3420 3920 2020   3 9      4 9   
│ │ │ │ -000315e0: 2020 2035 2039 2020 2020 2020 3620 3920     5 9      6 9 
│ │ │ │ -000315f0: 2020 2020 3020 3130 2020 2020 2020 317c      0 10      1|
│ │ │ │ -00031600: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031560: 2020 2020 207c 0a7c 2d20 3434 7820 7820       |.|- 44x x 
│ │ │ │ +00031570: 202d 2032 3178 2078 2020 2d20 3231 7820   - 21x x  - 21x 
│ │ │ │ +00031580: 7820 202d 2039 7820 7820 202b 2031 3478  x  - 9x x  + 14x
│ │ │ │ +00031590: 2078 2020 2d20 3232 7820 7820 202b 2034   x  - 22x x  + 4
│ │ │ │ +000315a0: 3578 2078 2020 2b20 3578 2078 2020 202b  5x x  + 5x x   +
│ │ │ │ +000315b0: 2032 3078 207c 0a7c 2020 2020 2030 2039   20x |.|     0 9
│ │ │ │ +000315c0: 2020 2020 2020 3120 3920 2020 2020 2032        1 9      2
│ │ │ │ +000315d0: 2039 2020 2020 2033 2039 2020 2020 2020   9     3 9      
│ │ │ │ +000315e0: 3420 3920 2020 2020 2035 2039 2020 2020  4 9      5 9    
│ │ │ │ +000315f0: 2020 3620 3920 2020 2020 3020 3130 2020    6 9     0 10  
│ │ │ │ +00031600: 2020 2020 317c 0a7c 2020 2020 2020 2020      1|.|        
│ │ │ │  00031610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031640: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031650: 0a7c 2078 2020 202d 2033 3778 2078 2020  .| x   - 37x x  
│ │ │ │ -00031660: 202d 2039 7820 7820 2020 2b20 3134 7820   - 9x x   + 14x 
│ │ │ │ -00031670: 7820 202c 2020 2020 2020 2020 2020 2020  x  ,            
│ │ │ │ +00031640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031650: 2020 2020 207c 0a7c 2078 2020 202d 2033       |.| x   - 3
│ │ │ │ +00031660: 3778 2078 2020 202d 2039 7820 7820 2020  7x x   - 9x x   
│ │ │ │ +00031670: 2b20 3134 7820 7820 202c 2020 2020 2020  + 14x x  ,      
│ │ │ │  00031680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031690: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000316a0: 0a7c 3620 3130 2020 2020 2020 3220 3131  .|6 10      2 11
│ │ │ │ -000316b0: 2020 2020 2034 2031 3120 2020 2020 2035       4 11      5
│ │ │ │ -000316c0: 2031 3120 2020 2020 2020 2020 2020 2020   11             
│ │ │ │ +00031690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000316a0: 2020 2020 207c 0a7c 3620 3130 2020 2020       |.|6 10    
│ │ │ │ +000316b0: 2020 3220 3131 2020 2020 2034 2031 3120    2 11     4 11 
│ │ │ │ +000316c0: 2020 2020 2035 2031 3120 2020 2020 2020       5 11       
│ │ │ │  000316d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000316e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000316f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000316e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000316f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031730: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031740: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031740: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031780: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031790: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031790: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000317a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000317b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000317c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000317d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000317e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000317d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000317e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000317f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031820: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031830: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031830: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031870: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031880: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031880: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000318c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000318d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000318c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000318d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000318e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000318f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031910: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031920: 0a7c 2020 202b 2031 3378 2078 2020 202b  .|   + 13x x   +
│ │ │ │ -00031930: 2033 3978 2078 2020 202d 2036 7820 7820   39x x   - 6x x 
│ │ │ │ -00031940: 2020 2b20 3578 2078 2020 202b 2034 3278    + 5x x   + 42x
│ │ │ │ -00031950: 2078 2020 202b 2034 7820 7820 2020 2b20   x   + 4x x   + 
│ │ │ │ -00031960: 3678 2078 2020 2020 2020 2020 2020 207c  6x x           |
│ │ │ │ -00031970: 0a7c 3130 2020 2020 2020 3220 3130 2020  .|10      2 10  
│ │ │ │ -00031980: 2020 2020 3420 3130 2020 2020 2036 2031      4 10     6 1
│ │ │ │ -00031990: 3020 2020 2020 3120 3131 2020 2020 2020  0     1 11      
│ │ │ │ -000319a0: 3220 3131 2020 2020 2034 2031 3120 2020  2 11     4 11   
│ │ │ │ -000319b0: 2020 3520 3131 2020 2020 2020 2020 207c    5 11         |
│ │ │ │ -000319c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00031910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031920: 2020 2020 207c 0a7c 2020 202b 2031 3378       |.|   + 13x
│ │ │ │ +00031930: 2078 2020 202b 2033 3978 2078 2020 202d   x   + 39x x   -
│ │ │ │ +00031940: 2036 7820 7820 2020 2b20 3578 2078 2020   6x x   + 5x x  
│ │ │ │ +00031950: 202b 2034 3278 2078 2020 202b 2034 7820   + 42x x   + 4x 
│ │ │ │ +00031960: 7820 2020 2b20 3678 2078 2020 2020 2020  x   + 6x x      
│ │ │ │ +00031970: 2020 2020 207c 0a7c 3130 2020 2020 2020       |.|10      
│ │ │ │ +00031980: 3220 3130 2020 2020 2020 3420 3130 2020  2 10      4 10  
│ │ │ │ +00031990: 2020 2036 2031 3020 2020 2020 3120 3131     6 10     1 11
│ │ │ │ +000319a0: 2020 2020 2020 3220 3131 2020 2020 2034        2 11     4
│ │ │ │ +000319b0: 2031 3120 2020 2020 3520 3131 2020 2020   11     5 11    
│ │ │ │ +000319c0: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  000319d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000319e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000319f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00031a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00031a10: 0a7c 202b 2032 3778 2078 2020 202d 2032  .| + 27x x   - 2
│ │ │ │ -00031a20: 3778 2078 2020 2c20 2020 2020 2020 2020  7x x  ,         
│ │ │ │ +00031a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00031a10: 2d2d 2d2d 2d7c 0a7c 202b 2032 3778 2078  -----|.| + 27x x
│ │ │ │ +00031a20: 2020 202d 2032 3778 2078 2020 2c20 2020     - 27x x  ,   
│ │ │ │  00031a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031a50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031a60: 0a7c 2020 2020 2020 3420 3131 2020 2020  .|      4 11    
│ │ │ │ -00031a70: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │ +00031a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031a60: 2020 2020 207c 0a7c 2020 2020 2020 3420       |.|      4 
│ │ │ │ +00031a70: 3131 2020 2020 2020 3520 3131 2020 2020  11      5 11    
│ │ │ │  00031a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031aa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031ab0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031ab0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031af0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031b00: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031b00: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b40: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031b50: 0a7c 7820 2020 2b20 3335 7820 7820 2020  .|x   + 35x x   
│ │ │ │ -00031b60: 2d20 3137 7820 7820 202c 2020 2020 2020  - 17x x  ,      
│ │ │ │ +00031b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031b50: 2020 2020 207c 0a7c 7820 2020 2b20 3335       |.|x   + 35
│ │ │ │ +00031b60: 7820 7820 2020 2d20 3137 7820 7820 202c  x x   - 17x x  ,
│ │ │ │  00031b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031b90: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031ba0: 0a7c 2031 3120 2020 2020 2032 2031 3120  .| 11      2 11 
│ │ │ │ -00031bb0: 2020 2020 2034 2031 3120 2020 2020 2020       4 11       
│ │ │ │ +00031b90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031ba0: 2020 2020 207c 0a7c 2031 3120 2020 2020       |.| 11     
│ │ │ │ +00031bb0: 2032 2031 3120 2020 2020 2034 2031 3120   2 11      4 11 
│ │ │ │  00031bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031be0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031bf0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031bf0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c30: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031c40: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031c40: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031c80: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031c90: 0a7c 2b20 3134 7820 7820 2020 2d20 3878  .|+ 14x x   - 8x
│ │ │ │ -00031ca0: 2078 2020 2020 2020 2020 2020 2020 2020   x              
│ │ │ │ +00031c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031c90: 2020 2020 207c 0a7c 2b20 3134 7820 7820       |.|+ 14x x 
│ │ │ │ +00031ca0: 2020 2d20 3878 2078 2020 2020 2020 2020    - 8x x        
│ │ │ │  00031cb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031cd0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031ce0: 0a7c 2020 2020 2034 2031 3120 2020 2020  .|     4 11     
│ │ │ │ -00031cf0: 3520 3131 2020 2020 2020 2020 2020 2020  5 11            
│ │ │ │ +00031cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031ce0: 2020 2020 207c 0a7c 2020 2020 2034 2031       |.|     4 1
│ │ │ │ +00031cf0: 3120 2020 2020 3520 3131 2020 2020 2020  1     5 11      
│ │ │ │  00031d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031d30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031d30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031d70: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031d80: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031d80: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031dc0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031dd0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031de0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e10: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031e20: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031e20: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031e60: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031e70: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031e70: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031eb0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031ec0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031ed0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031f00: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031f10: 0a7c 2033 3778 2078 2020 202d 2035 7820  .| 37x x   - 5x 
│ │ │ │ -00031f20: 7820 2020 2d20 3978 2078 2020 202b 2031  x   - 9x x   + 1
│ │ │ │ -00031f30: 3978 2078 2020 2c20 2020 2020 2020 2020  9x x  ,         
│ │ │ │ +00031f00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031f10: 2020 2020 207c 0a7c 2033 3778 2078 2020       |.| 37x x  
│ │ │ │ +00031f20: 202d 2035 7820 7820 2020 2d20 3978 2078   - 5x x   - 9x x
│ │ │ │ +00031f30: 2020 202b 2031 3978 2078 2020 2c20 2020     + 19x x  ,   
│ │ │ │  00031f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031f50: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031f60: 0a7c 2020 2020 3220 3131 2020 2020 2033  .|    2 11     3
│ │ │ │ -00031f70: 2031 3120 2020 2020 3420 3131 2020 2020   11     4 11    
│ │ │ │ -00031f80: 2020 3520 3131 2020 2020 2020 2020 2020    5 11          
│ │ │ │ +00031f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031f60: 2020 2020 207c 0a7c 2020 2020 3220 3131       |.|    2 11
│ │ │ │ +00031f70: 2020 2020 2033 2031 3120 2020 2020 3420       3 11     4 
│ │ │ │ +00031f80: 3131 2020 2020 2020 3520 3131 2020 2020  11      5 11    
│ │ │ │  00031f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031fa0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00031fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00031fb0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00031fc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031fd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00031fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00031ff0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032000: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00031ff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032000: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00032010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032040: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032050: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00032040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032050: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00032060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032090: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000320a0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00032090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000320a0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000320b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000320c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000320d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000320e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000320f0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000320e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000320f0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00032100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032130: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032140: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00032130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032140: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00032150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032180: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032190: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00032180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032190: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000321a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000321c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000321d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000321e0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000321d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000321e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000321f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032220: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032230: 0a7c 7820 2020 2d20 3130 7820 7820 2020  .|x   - 10x x   
│ │ │ │ -00032240: 2d20 3334 7820 7820 2020 2d20 3878 2078  - 34x x   - 8x x
│ │ │ │ -00032250: 2020 202b 2031 3578 2078 2020 202d 2032     + 15x x   - 2
│ │ │ │ -00032260: 3578 2078 2020 202d 2032 3478 2078 2020  5x x   - 24x x  
│ │ │ │ -00032270: 202d 2032 7820 7820 2020 2d20 2020 207c   - 2x x   -    |
│ │ │ │ -00032280: 0a7c 2031 3020 2020 2020 2031 2031 3020  .| 10      1 10 
│ │ │ │ -00032290: 2020 2020 2032 2031 3020 2020 2020 3420       2 10     4 
│ │ │ │ -000322a0: 3130 2020 2020 2020 3620 3130 2020 2020  10      6 10    
│ │ │ │ -000322b0: 2020 3020 3131 2020 2020 2020 3220 3131    0 11      2 11
│ │ │ │ -000322c0: 2020 2020 2034 2031 3120 2020 2020 207c       4 11      |
│ │ │ │ -000322d0: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00032220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032230: 2020 2020 207c 0a7c 7820 2020 2d20 3130       |.|x   - 10
│ │ │ │ +00032240: 7820 7820 2020 2d20 3334 7820 7820 2020  x x   - 34x x   
│ │ │ │ +00032250: 2d20 3878 2078 2020 202b 2031 3578 2078  - 8x x   + 15x x
│ │ │ │ +00032260: 2020 202d 2032 3578 2078 2020 202d 2032     - 25x x   - 2
│ │ │ │ +00032270: 3478 2078 2020 202d 2032 7820 7820 2020  4x x   - 2x x   
│ │ │ │ +00032280: 2d20 2020 207c 0a7c 2031 3020 2020 2020  -    |.| 10     
│ │ │ │ +00032290: 2031 2031 3020 2020 2020 2032 2031 3020   1 10      2 10 
│ │ │ │ +000322a0: 2020 2020 3420 3130 2020 2020 2020 3620      4 10      6 
│ │ │ │ +000322b0: 3130 2020 2020 2020 3020 3131 2020 2020  10      0 11    
│ │ │ │ +000322c0: 2020 3220 3131 2020 2020 2034 2031 3120    2 11     4 11 
│ │ │ │ +000322d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  000322e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000322f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032310: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032320: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00032310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032320: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00032330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032360: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032370: 0a7c 7820 2020 2b20 3334 7820 7820 2020  .|x   + 34x x   
│ │ │ │ -00032380: 2b20 3132 7820 7820 2020 2d20 3235 7820  + 12x x   - 25x 
│ │ │ │ -00032390: 7820 2020 2b20 3230 7820 7820 2020 2d20  x   + 20x x   - 
│ │ │ │ -000323a0: 3578 2078 2020 202b 2039 7820 7820 2020  5x x   + 9x x   
│ │ │ │ -000323b0: 2d20 3435 7820 7820 2020 2b20 2020 207c  - 45x x   +    |
│ │ │ │ -000323c0: 0a7c 2031 3020 2020 2020 2032 2031 3020  .| 10      2 10 
│ │ │ │ -000323d0: 2020 2020 2034 2031 3020 2020 2020 2036       4 10      6
│ │ │ │ -000323e0: 2031 3020 2020 2020 2030 2031 3120 2020   10      0 11   
│ │ │ │ -000323f0: 2020 3120 3131 2020 2020 2032 2031 3120    1 11     2 11 
│ │ │ │ -00032400: 2020 2020 2034 2031 3120 2020 2020 207c       4 11      |
│ │ │ │ -00032410: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .|--------------
│ │ │ │ +00032360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032370: 2020 2020 207c 0a7c 7820 2020 2b20 3334       |.|x   + 34
│ │ │ │ +00032380: 7820 7820 2020 2b20 3132 7820 7820 2020  x x   + 12x x   
│ │ │ │ +00032390: 2d20 3235 7820 7820 2020 2b20 3230 7820  - 25x x   + 20x 
│ │ │ │ +000323a0: 7820 2020 2d20 3578 2078 2020 202b 2039  x   - 5x x   + 9
│ │ │ │ +000323b0: 7820 7820 2020 2d20 3435 7820 7820 2020  x x   - 45x x   
│ │ │ │ +000323c0: 2b20 2020 207c 0a7c 2031 3020 2020 2020  +    |.| 10     
│ │ │ │ +000323d0: 2032 2031 3020 2020 2020 2034 2031 3020   2 10      4 10 
│ │ │ │ +000323e0: 2020 2020 2036 2031 3020 2020 2020 2030       6 10      0
│ │ │ │ +000323f0: 2031 3120 2020 2020 3120 3131 2020 2020   11     1 11    
│ │ │ │ +00032400: 2032 2031 3120 2020 2020 2034 2031 3120   2 11      4 11 
│ │ │ │ +00032410: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d       |.|--------
│ │ │ │  00032420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c  ---------------|
│ │ │ │ -00032460: 0a7c 3135 7820 7820 202c 2020 2020 2020  .|15x x  ,      
│ │ │ │ +00032450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032460: 2d2d 2d2d 2d7c 0a7c 3135 7820 7820 202c  -----|.|15x x  ,
│ │ │ │  00032470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000324a0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000324b0: 0a7c 2020 2035 2031 3120 2020 2020 2020  .|   5 11       
│ │ │ │ +000324a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000324b0: 2020 2020 207c 0a7c 2020 2035 2031 3120       |.|   5 11 
│ │ │ │  000324c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000324e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000324f0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032500: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +000324f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032500: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00032510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032540: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032550: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00032540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032550: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00032560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032590: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000325a0: 0a7c 3235 7820 7820 202c 2020 2020 2020  .|25x x  ,      
│ │ │ │ +00032590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000325a0: 2020 2020 207c 0a7c 3235 7820 7820 202c       |.|25x x  ,
│ │ │ │  000325b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000325c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000325d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000325e0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000325f0: 0a7c 2020 2035 2031 3120 2020 2020 2020  .|   5 11       
│ │ │ │ +000325e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000325f0: 2020 2020 207c 0a7c 2020 2035 2031 3120       |.|   5 11 
│ │ │ │  00032600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00032620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00032630: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00032640: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +00032630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  00032650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00032690: 0a7c 6931 3120 3a20 6173 7365 7274 2870  .|i11 : assert(p
│ │ │ │ -000326a0: 6869 202a 2070 7369 203d 3d20 3129 2020  hi * psi == 1)  
│ │ │ │ -000326b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00032680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032690: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 6173  -----+.|i11 : as
│ │ │ │ +000326a0: 7365 7274 2870 6869 202a 2070 7369 203d  sert(phi * psi =
│ │ │ │ +000326b0: 3d20 3129 2020 2020 2020 2020 2020 2020  = 1)            
│ │ │ │  000326c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000326d0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000326e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │ +000326d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000326e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │  000326f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ -00032730: 0a0a 5468 6520 6d65 7468 6f64 2061 6c73  ..The method als
│ │ │ │ -00032740: 6f20 6163 6365 7074 7320 6173 2069 6e70  o accepts as inp
│ │ │ │ -00032750: 7574 2061 202a 6e6f 7465 2072 696e 6720  ut a *note ring 
│ │ │ │ -00032760: 6d61 703a 2028 4d61 6361 756c 6179 3244  map: (Macaulay2D
│ │ │ │ -00032770: 6f63 2952 696e 674d 6170 2c0a 7265 7072  oc)RingMap,.repr
│ │ │ │ -00032780: 6573 656e 7469 6e67 2061 2072 6174 696f  esenting a ratio
│ │ │ │ -00032790: 6e61 6c20 6d61 7020 6265 7477 6565 6e20  nal map between 
│ │ │ │ -000327a0: 7072 6f6a 6563 7469 7665 2076 6172 6965  projective varie
│ │ │ │ -000327b0: 7469 6573 2e20 496e 2074 6869 7320 6361  ties. In this ca
│ │ │ │ -000327c0: 7365 2c20 6120 2a6e 6f74 650a 7269 6e67  se, a *note.ring
│ │ │ │ -000327d0: 206d 6170 3a20 284d 6163 6175 6c61 7932   map: (Macaulay2
│ │ │ │ -000327e0: 446f 6329 5269 6e67 4d61 702c 2069 7320  Doc)RingMap, is 
│ │ │ │ -000327f0: 7265 7475 726e 6564 2061 7320 7765 6c6c  returned as well
│ │ │ │ -00032800: 2e0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d  ...Caveat.======
│ │ │ │ -00032810: 0a0a 466f 7220 7468 6520 7075 7270 6f73  ..For the purpos
│ │ │ │ -00032820: 6520 6f66 2074 6869 7320 6d65 7468 6f64  e of this method
│ │ │ │ -00032830: 2c20 7468 6520 6f70 7469 6f6e 202a 6e6f  , the option *no
│ │ │ │ -00032840: 7465 2043 6572 7469 6679 3a20 4365 7274  te Certify: Cert
│ │ │ │ -00032850: 6966 792c 3d3e 7472 7565 2069 7320 746f  ify,=>true is to
│ │ │ │ -00032860: 6f0a 7269 6769 642c 2065 7370 6563 6961  o.rigid, especia
│ │ │ │ -00032870: 6c6c 7920 7768 656e 2074 6865 2073 6f75  lly when the sou
│ │ │ │ -00032880: 7263 6520 6f66 2074 6865 2070 6173 7365  rce of the passe
│ │ │ │ -00032890: 6420 6d61 7020 6973 206e 6f74 2061 2070  d map is not a p
│ │ │ │ -000328a0: 726f 6a65 6374 6976 6520 7370 6163 652e  rojective space.
│ │ │ │ -000328b0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d  ..See also.=====
│ │ │ │ -000328c0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2069  ===..  * *note i
│ │ │ │ -000328d0: 6e76 6572 7365 4d61 703a 2069 6e76 6572  nverseMap: inver
│ │ │ │ -000328e0: 7365 4d61 702c 202d 2d20 696e 7665 7273  seMap, -- invers
│ │ │ │ -000328f0: 6520 6f66 2061 2062 6972 6174 696f 6e61  e of a birationa
│ │ │ │ -00032900: 6c20 6d61 700a 2020 2a20 2a6e 6f74 6520  l map.  * *note 
│ │ │ │ -00032910: 696e 7665 7273 6528 5261 7469 6f6e 616c  inverse(Rational
│ │ │ │ -00032920: 4d61 7029 3a20 696e 7665 7273 655f 6c70  Map): inverse_lp
│ │ │ │ -00032930: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ -00032940: 2d2d 2069 6e76 6572 7365 206f 6620 610a  -- inverse of a.
│ │ │ │ -00032950: 2020 2020 6269 7261 7469 6f6e 616c 206d      birational m
│ │ │ │ -00032960: 6170 0a0a 5761 7973 2074 6f20 7573 6520  ap..Ways to use 
│ │ │ │ -00032970: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ -00032980: 7365 4d61 703a 0a3d 3d3d 3d3d 3d3d 3d3d  seMap:.=========
│ │ │ │ +00032720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00032730: 2d2d 2d2d 2d2b 0a0a 5468 6520 6d65 7468  -----+..The meth
│ │ │ │ +00032740: 6f64 2061 6c73 6f20 6163 6365 7074 7320  od also accepts 
│ │ │ │ +00032750: 6173 2069 6e70 7574 2061 202a 6e6f 7465  as input a *note
│ │ │ │ +00032760: 2072 696e 6720 6d61 703a 2028 4d61 6361   ring map: (Maca
│ │ │ │ +00032770: 756c 6179 3244 6f63 2952 696e 674d 6170  ulay2Doc)RingMap
│ │ │ │ +00032780: 2c0a 7265 7072 6573 656e 7469 6e67 2061  ,.representing a
│ │ │ │ +00032790: 2072 6174 696f 6e61 6c20 6d61 7020 6265   rational map be
│ │ │ │ +000327a0: 7477 6565 6e20 7072 6f6a 6563 7469 7665  tween projective
│ │ │ │ +000327b0: 2076 6172 6965 7469 6573 2e20 496e 2074   varieties. In t
│ │ │ │ +000327c0: 6869 7320 6361 7365 2c20 6120 2a6e 6f74  his case, a *not
│ │ │ │ +000327d0: 650a 7269 6e67 206d 6170 3a20 284d 6163  e.ring map: (Mac
│ │ │ │ +000327e0: 6175 6c61 7932 446f 6329 5269 6e67 4d61  aulay2Doc)RingMa
│ │ │ │ +000327f0: 702c 2069 7320 7265 7475 726e 6564 2061  p, is returned a
│ │ │ │ +00032800: 7320 7765 6c6c 2e0a 0a43 6176 6561 740a  s well...Caveat.
│ │ │ │ +00032810: 3d3d 3d3d 3d3d 0a0a 466f 7220 7468 6520  ======..For the 
│ │ │ │ +00032820: 7075 7270 6f73 6520 6f66 2074 6869 7320  purpose of this 
│ │ │ │ +00032830: 6d65 7468 6f64 2c20 7468 6520 6f70 7469  method, the opti
│ │ │ │ +00032840: 6f6e 202a 6e6f 7465 2043 6572 7469 6679  on *note Certify
│ │ │ │ +00032850: 3a20 4365 7274 6966 792c 3d3e 7472 7565  : Certify,=>true
│ │ │ │ +00032860: 2069 7320 746f 6f0a 7269 6769 642c 2065   is too.rigid, e
│ │ │ │ +00032870: 7370 6563 6961 6c6c 7920 7768 656e 2074  specially when t
│ │ │ │ +00032880: 6865 2073 6f75 7263 6520 6f66 2074 6865  he source of the
│ │ │ │ +00032890: 2070 6173 7365 6420 6d61 7020 6973 206e   passed map is n
│ │ │ │ +000328a0: 6f74 2061 2070 726f 6a65 6374 6976 6520  ot a projective 
│ │ │ │ +000328b0: 7370 6163 652e 0a0a 5365 6520 616c 736f  space...See also
│ │ │ │ +000328c0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ +000328d0: 6e6f 7465 2069 6e76 6572 7365 4d61 703a  note inverseMap:
│ │ │ │ +000328e0: 2069 6e76 6572 7365 4d61 702c 202d 2d20   inverseMap, -- 
│ │ │ │ +000328f0: 696e 7665 7273 6520 6f66 2061 2062 6972  inverse of a bir
│ │ │ │ +00032900: 6174 696f 6e61 6c20 6d61 700a 2020 2a20  ational map.  * 
│ │ │ │ +00032910: 2a6e 6f74 6520 696e 7665 7273 6528 5261  *note inverse(Ra
│ │ │ │ +00032920: 7469 6f6e 616c 4d61 7029 3a20 696e 7665  tionalMap): inve
│ │ │ │ +00032930: 7273 655f 6c70 5261 7469 6f6e 616c 4d61  rse_lpRationalMa
│ │ │ │ +00032940: 705f 7270 2c20 2d2d 2069 6e76 6572 7365  p_rp, -- inverse
│ │ │ │ +00032950: 206f 6620 610a 2020 2020 6269 7261 7469   of a.    birati
│ │ │ │ +00032960: 6f6e 616c 206d 6170 0a0a 5761 7973 2074  onal map..Ways t
│ │ │ │ +00032970: 6f20 7573 6520 6170 7072 6f78 696d 6174  o use approximat
│ │ │ │ +00032980: 6549 6e76 6572 7365 4d61 703a 0a3d 3d3d  eInverseMap:.===
│ │ │ │  00032990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -000329a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ -000329b0: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ -000329c0: 7365 4d61 7028 5261 7469 6f6e 616c 4d61  seMap(RationalMa
│ │ │ │ -000329d0: 7029 220a 2020 2a20 2261 7070 726f 7869  p)".  * "approxi
│ │ │ │ -000329e0: 6d61 7465 496e 7665 7273 654d 6170 2852  mateInverseMap(R
│ │ │ │ -000329f0: 6174 696f 6e61 6c4d 6170 2c5a 5a29 220a  ationalMap,ZZ)".
│ │ │ │ -00032a00: 2020 2a20 2261 7070 726f 7869 6d61 7465    * "approximate
│ │ │ │ -00032a10: 496e 7665 7273 654d 6170 2852 696e 674d  InverseMap(RingM
│ │ │ │ -00032a20: 6170 2922 0a20 202a 2022 6170 7072 6f78  ap)".  * "approx
│ │ │ │ -00032a30: 696d 6174 6549 6e76 6572 7365 4d61 7028  imateInverseMap(
│ │ │ │ -00032a40: 5269 6e67 4d61 702c 5a5a 2922 0a0a 466f  RingMap,ZZ)"..Fo
│ │ │ │ -00032a50: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00032a60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00032a70: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00032a80: 2a6e 6f74 6520 6170 7072 6f78 696d 6174  *note approximat
│ │ │ │ -00032a90: 6549 6e76 6572 7365 4d61 703a 2061 7070  eInverseMap: app
│ │ │ │ -00032aa0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ -00032ab0: 6170 2c20 6973 2061 202a 6e6f 7465 0a6d  ap, is a *note.m
│ │ │ │ -00032ac0: 6574 686f 6420 6675 6e63 7469 6f6e 2077  ethod function w
│ │ │ │ -00032ad0: 6974 6820 6f70 7469 6f6e 733a 2028 4d61  ith options: (Ma
│ │ │ │ -00032ae0: 6361 756c 6179 3244 6f63 294d 6574 686f  caulay2Doc)Metho
│ │ │ │ -00032af0: 6446 756e 6374 696f 6e57 6974 684f 7074  dFunctionWithOpt
│ │ │ │ -00032b00: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  ions,...--------
│ │ │ │ +000329a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ===============.
│ │ │ │ +000329b0: 0a20 202a 2022 6170 7072 6f78 696d 6174  .  * "approximat
│ │ │ │ +000329c0: 6549 6e76 6572 7365 4d61 7028 5261 7469  eInverseMap(Rati
│ │ │ │ +000329d0: 6f6e 616c 4d61 7029 220a 2020 2a20 2261  onalMap)".  * "a
│ │ │ │ +000329e0: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ +000329f0: 654d 6170 2852 6174 696f 6e61 6c4d 6170  eMap(RationalMap
│ │ │ │ +00032a00: 2c5a 5a29 220a 2020 2a20 2261 7070 726f  ,ZZ)".  * "appro
│ │ │ │ +00032a10: 7869 6d61 7465 496e 7665 7273 654d 6170  ximateInverseMap
│ │ │ │ +00032a20: 2852 696e 674d 6170 2922 0a20 202a 2022  (RingMap)".  * "
│ │ │ │ +00032a30: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +00032a40: 7365 4d61 7028 5269 6e67 4d61 702c 5a5a  seMap(RingMap,ZZ
│ │ │ │ +00032a50: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00032a60: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00032a70: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00032a80: 626a 6563 7420 2a6e 6f74 6520 6170 7072  bject *note appr
│ │ │ │ +00032a90: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +00032aa0: 703a 2061 7070 726f 7869 6d61 7465 496e  p: approximateIn
│ │ │ │ +00032ab0: 7665 7273 654d 6170 2c20 6973 2061 202a  verseMap, is a *
│ │ │ │ +00032ac0: 6e6f 7465 0a6d 6574 686f 6420 6675 6e63  note.method func
│ │ │ │ +00032ad0: 7469 6f6e 2077 6974 6820 6f70 7469 6f6e  tion with option
│ │ │ │ +00032ae0: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +00032af0: 294d 6574 686f 6446 756e 6374 696f 6e57  )MethodFunctionW
│ │ │ │ +00032b00: 6974 684f 7074 696f 6e73 2c2e 0a0a 2d2d  ithOptions,...--
│ │ │ │  00032b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00032b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00032b50: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -00032b60: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -00032b70: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -00032b80: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -00032b90: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00032ba0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ -00032bb0: 756c 6179 322f 7061 636b 6167 6573 2f43  ulay2/packages/C
│ │ │ │ -00032bc0: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74  remona/.document
│ │ │ │ -00032bd0: 6174 696f 6e2e 6d32 3a32 3237 3a30 2e0a  ation.m2:227:0..
│ │ │ │ -00032be0: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e  ..File: Cremona.
│ │ │ │ -00032bf0: 696e 666f 2c20 4e6f 6465 3a20 426c 6f77  info, Node: Blow
│ │ │ │ -00032c00: 5570 5374 7261 7465 6779 2c20 4e65 7874  UpStrategy, Next
│ │ │ │ -00032c10: 3a20 4365 7274 6966 792c 2050 7265 763a  : Certify, Prev:
│ │ │ │ -00032c20: 2061 7070 726f 7869 6d61 7465 496e 7665   approximateInve
│ │ │ │ -00032c30: 7273 654d 6170 2c20 5570 3a20 546f 700a  rseMap, Up: Top.
│ │ │ │ -00032c40: 0a42 6c6f 7755 7053 7472 6174 6567 790a  .BlowUpStrategy.
│ │ │ │ -00032c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a  **************..
│ │ │ │ -00032c60: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -00032c70: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6973  =======..This is
│ │ │ │ -00032c80: 2061 6e20 6f70 7469 6f6e 616c 2061 7267   an optional arg
│ │ │ │ -00032c90: 756d 656e 7420 666f 7220 2a6e 6f74 6520  ument for *note 
│ │ │ │ -00032ca0: 6772 6170 683a 2067 7261 7068 2c2c 2061  graph: graph,, a
│ │ │ │ -00032cb0: 6e64 2066 6f72 2074 6865 206d 6574 686f  nd for the metho
│ │ │ │ -00032cc0: 6473 2074 6861 740a 6576 656e 7475 616c  ds that.eventual
│ │ │ │ -00032cd0: 6c79 2063 616c 6c20 6974 2e20 4375 7272  ly call it. Curr
│ │ │ │ -00032ce0: 656e 746c 792c 2074 6865 2070 6f73 7369  ently, the possi
│ │ │ │ -00032cf0: 626c 6520 7661 6c75 6573 2061 7265 2022  ble values are "
│ │ │ │ -00032d00: 456c 696d 696e 6174 6522 2061 6e64 0a22  Eliminate" and."
│ │ │ │ -00032d10: 5361 7475 7261 7465 222c 2077 6869 6368  Saturate", which
│ │ │ │ -00032d20: 2069 6e64 6963 6174 6520 7477 6f20 6469   indicate two di
│ │ │ │ -00032d30: 6666 6572 656e 7420 7761 7973 206f 6620  fferent ways of 
│ │ │ │ -00032d40: 636f 6d70 7574 696e 6720 7468 6520 2863  computing the (c
│ │ │ │ -00032d50: 6c6f 7375 7265 206f 6620 7468 6529 0a67  losure of the).g
│ │ │ │ -00032d60: 7261 7068 206f 6620 6120 7261 7469 6f6e  raph of a ration
│ │ │ │ -00032d70: 616c 206d 6170 2e20 5468 6520 6465 6661  al map. The defa
│ │ │ │ -00032d80: 756c 7420 6368 6f69 6365 2069 7320 2245  ult choice is "E
│ │ │ │ -00032d90: 6c69 6d69 6e61 7465 2220 616e 6420 7468  liminate" and th
│ │ │ │ -00032da0: 6973 2069 730a 6765 6e65 7261 6c6c 7920  is is.generally 
│ │ │ │ -00032db0: 7072 6566 6572 6162 6c65 2e0a 0a46 756e  preferable...Fun
│ │ │ │ -00032dc0: 6374 696f 6e73 2077 6974 6820 6f70 7469  ctions with opti
│ │ │ │ -00032dd0: 6f6e 616c 2061 7267 756d 656e 7420 6e61  onal argument na
│ │ │ │ -00032de0: 6d65 6420 426c 6f77 5570 5374 7261 7465  med BlowUpStrate
│ │ │ │ -00032df0: 6779 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  gy:.============
│ │ │ │ +00032b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +00032b60: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +00032b70: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +00032b80: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +00032b90: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +00032ba0: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ +00032bb0: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +00032bc0: 6167 6573 2f43 7265 6d6f 6e61 2f0a 646f  ages/Cremona/.do
│ │ │ │ +00032bd0: 6375 6d65 6e74 6174 696f 6e2e 6d32 3a32  cumentation.m2:2
│ │ │ │ +00032be0: 3237 3a30 2e0a 1f0a 4669 6c65 3a20 4372  27:0....File: Cr
│ │ │ │ +00032bf0: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465  emona.info, Node
│ │ │ │ +00032c00: 3a20 426c 6f77 5570 5374 7261 7465 6779  : BlowUpStrategy
│ │ │ │ +00032c10: 2c20 4e65 7874 3a20 4365 7274 6966 792c  , Next: Certify,
│ │ │ │ +00032c20: 2050 7265 763a 2061 7070 726f 7869 6d61   Prev: approxima
│ │ │ │ +00032c30: 7465 496e 7665 7273 654d 6170 2c20 5570  teInverseMap, Up
│ │ │ │ +00032c40: 3a20 546f 700a 0a42 6c6f 7755 7053 7472  : Top..BlowUpStr
│ │ │ │ +00032c50: 6174 6567 790a 2a2a 2a2a 2a2a 2a2a 2a2a  ategy.**********
│ │ │ │ +00032c60: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f  ****..Descriptio
│ │ │ │ +00032c70: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54  n.===========..T
│ │ │ │ +00032c80: 6869 7320 6973 2061 6e20 6f70 7469 6f6e  his is an option
│ │ │ │ +00032c90: 616c 2061 7267 756d 656e 7420 666f 7220  al argument for 
│ │ │ │ +00032ca0: 2a6e 6f74 6520 6772 6170 683a 2067 7261  *note graph: gra
│ │ │ │ +00032cb0: 7068 2c2c 2061 6e64 2066 6f72 2074 6865  ph,, and for the
│ │ │ │ +00032cc0: 206d 6574 686f 6473 2074 6861 740a 6576   methods that.ev
│ │ │ │ +00032cd0: 656e 7475 616c 6c79 2063 616c 6c20 6974  entually call it
│ │ │ │ +00032ce0: 2e20 4375 7272 656e 746c 792c 2074 6865  . Currently, the
│ │ │ │ +00032cf0: 2070 6f73 7369 626c 6520 7661 6c75 6573   possible values
│ │ │ │ +00032d00: 2061 7265 2022 456c 696d 696e 6174 6522   are "Eliminate"
│ │ │ │ +00032d10: 2061 6e64 0a22 5361 7475 7261 7465 222c   and."Saturate",
│ │ │ │ +00032d20: 2077 6869 6368 2069 6e64 6963 6174 6520   which indicate 
│ │ │ │ +00032d30: 7477 6f20 6469 6666 6572 656e 7420 7761  two different wa
│ │ │ │ +00032d40: 7973 206f 6620 636f 6d70 7574 696e 6720  ys of computing 
│ │ │ │ +00032d50: 7468 6520 2863 6c6f 7375 7265 206f 6620  the (closure of 
│ │ │ │ +00032d60: 7468 6529 0a67 7261 7068 206f 6620 6120  the).graph of a 
│ │ │ │ +00032d70: 7261 7469 6f6e 616c 206d 6170 2e20 5468  rational map. Th
│ │ │ │ +00032d80: 6520 6465 6661 756c 7420 6368 6f69 6365  e default choice
│ │ │ │ +00032d90: 2069 7320 2245 6c69 6d69 6e61 7465 2220   is "Eliminate" 
│ │ │ │ +00032da0: 616e 6420 7468 6973 2069 730a 6765 6e65  and this is.gene
│ │ │ │ +00032db0: 7261 6c6c 7920 7072 6566 6572 6162 6c65  rally preferable
│ │ │ │ +00032dc0: 2e0a 0a46 756e 6374 696f 6e73 2077 6974  ...Functions wit
│ │ │ │ +00032dd0: 6820 6f70 7469 6f6e 616c 2061 7267 756d  h optional argum
│ │ │ │ +00032de0: 656e 7420 6e61 6d65 6420 426c 6f77 5570  ent named BlowUp
│ │ │ │ +00032df0: 5374 7261 7465 6779 3a0a 3d3d 3d3d 3d3d  Strategy:.======
│ │ │ │  00032e00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00032e10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00032e20: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00032e30: 2243 6865 726e 5363 6877 6172 747a 4d61  "ChernSchwartzMa
│ │ │ │ -00032e40: 6350 6865 7273 6f6e 282e 2e2e 2c42 6c6f  cPherson(...,Blo
│ │ │ │ -00032e50: 7755 7053 7472 6174 6567 793d 3e2e 2e2e  wUpStrategy=>...
│ │ │ │ -00032e60: 2922 0a20 202a 2022 6465 6772 6565 4d61  )".  * "degreeMa
│ │ │ │ -00032e70: 7028 2e2e 2e2c 426c 6f77 5570 5374 7261  p(...,BlowUpStra
│ │ │ │ -00032e80: 7465 6779 3d3e 2e2e 2e29 220a 2020 2a20  tegy=>...)".  * 
│ │ │ │ -00032e90: 2245 756c 6572 4368 6172 6163 7465 7269  "EulerCharacteri
│ │ │ │ -00032ea0: 7374 6963 282e 2e2e 2c42 6c6f 7755 7053  stic(...,BlowUpS
│ │ │ │ -00032eb0: 7472 6174 6567 793d 3e2e 2e2e 2922 0a20  trategy=>...)". 
│ │ │ │ -00032ec0: 202a 2022 6772 6170 6828 2e2e 2e2c 426c   * "graph(...,Bl
│ │ │ │ -00032ed0: 6f77 5570 5374 7261 7465 6779 3d3e 2e2e  owUpStrategy=>..
│ │ │ │ -00032ee0: 2e29 220a 2020 2a20 2269 6e76 6572 7365  .)".  * "inverse
│ │ │ │ -00032ef0: 4d61 7028 2e2e 2e2c 426c 6f77 5570 5374  Map(...,BlowUpSt
│ │ │ │ -00032f00: 7261 7465 6779 3d3e 2e2e 2e29 220a 2020  rategy=>...)".  
│ │ │ │ -00032f10: 2a20 2269 7342 6972 6174 696f 6e61 6c28  * "isBirational(
│ │ │ │ -00032f20: 2e2e 2e2c 426c 6f77 5570 5374 7261 7465  ...,BlowUpStrate
│ │ │ │ -00032f30: 6779 3d3e 2e2e 2e29 220a 2020 2a20 2270  gy=>...)".  * "p
│ │ │ │ -00032f40: 726f 6a65 6374 6976 6544 6567 7265 6573  rojectiveDegrees
│ │ │ │ -00032f50: 282e 2e2e 2c42 6c6f 7755 7053 7472 6174  (...,BlowUpStrat
│ │ │ │ -00032f60: 6567 793d 3e2e 2e2e 2922 0a20 202a 2022  egy=>...)".  * "
│ │ │ │ -00032f70: 5365 6772 6543 6c61 7373 282e 2e2e 2c42  SegreClass(...,B
│ │ │ │ -00032f80: 6c6f 7755 7053 7472 6174 6567 793d 3e2e  lowUpStrategy=>.
│ │ │ │ -00032f90: 2e2e 2922 0a0a 466f 7220 7468 6520 7072  ..)"..For the pr
│ │ │ │ -00032fa0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d  ogrammer.=======
│ │ │ │ -00032fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865  ===========..The
│ │ │ │ -00032fc0: 206f 626a 6563 7420 2a6e 6f74 6520 426c   object *note Bl
│ │ │ │ -00032fd0: 6f77 5570 5374 7261 7465 6779 3a20 426c  owUpStrategy: Bl
│ │ │ │ -00032fe0: 6f77 5570 5374 7261 7465 6779 2c20 6973  owUpStrategy, is
│ │ │ │ -00032ff0: 2061 202a 6e6f 7465 2073 796d 626f 6c3a   a *note symbol:
│ │ │ │ -00033000: 0a28 4d61 6361 756c 6179 3244 6f63 2953  .(Macaulay2Doc)S
│ │ │ │ -00033010: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d  ymbol,...-------
│ │ │ │ +00032e20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00032e30: 0a0a 2020 2a20 2243 6865 726e 5363 6877  ..  * "ChernSchw
│ │ │ │ +00032e40: 6172 747a 4d61 6350 6865 7273 6f6e 282e  artzMacPherson(.
│ │ │ │ +00032e50: 2e2e 2c42 6c6f 7755 7053 7472 6174 6567  ..,BlowUpStrateg
│ │ │ │ +00032e60: 793d 3e2e 2e2e 2922 0a20 202a 2022 6465  y=>...)".  * "de
│ │ │ │ +00032e70: 6772 6565 4d61 7028 2e2e 2e2c 426c 6f77  greeMap(...,Blow
│ │ │ │ +00032e80: 5570 5374 7261 7465 6779 3d3e 2e2e 2e29  UpStrategy=>...)
│ │ │ │ +00032e90: 220a 2020 2a20 2245 756c 6572 4368 6172  ".  * "EulerChar
│ │ │ │ +00032ea0: 6163 7465 7269 7374 6963 282e 2e2e 2c42  acteristic(...,B
│ │ │ │ +00032eb0: 6c6f 7755 7053 7472 6174 6567 793d 3e2e  lowUpStrategy=>.
│ │ │ │ +00032ec0: 2e2e 2922 0a20 202a 2022 6772 6170 6828  ..)".  * "graph(
│ │ │ │ +00032ed0: 2e2e 2e2c 426c 6f77 5570 5374 7261 7465  ...,BlowUpStrate
│ │ │ │ +00032ee0: 6779 3d3e 2e2e 2e29 220a 2020 2a20 2269  gy=>...)".  * "i
│ │ │ │ +00032ef0: 6e76 6572 7365 4d61 7028 2e2e 2e2c 426c  nverseMap(...,Bl
│ │ │ │ +00032f00: 6f77 5570 5374 7261 7465 6779 3d3e 2e2e  owUpStrategy=>..
│ │ │ │ +00032f10: 2e29 220a 2020 2a20 2269 7342 6972 6174  .)".  * "isBirat
│ │ │ │ +00032f20: 696f 6e61 6c28 2e2e 2e2c 426c 6f77 5570  ional(...,BlowUp
│ │ │ │ +00032f30: 5374 7261 7465 6779 3d3e 2e2e 2e29 220a  Strategy=>...)".
│ │ │ │ +00032f40: 2020 2a20 2270 726f 6a65 6374 6976 6544    * "projectiveD
│ │ │ │ +00032f50: 6567 7265 6573 282e 2e2e 2c42 6c6f 7755  egrees(...,BlowU
│ │ │ │ +00032f60: 7053 7472 6174 6567 793d 3e2e 2e2e 2922  pStrategy=>...)"
│ │ │ │ +00032f70: 0a20 202a 2022 5365 6772 6543 6c61 7373  .  * "SegreClass
│ │ │ │ +00032f80: 282e 2e2e 2c42 6c6f 7755 7053 7472 6174  (...,BlowUpStrat
│ │ │ │ +00032f90: 6567 793d 3e2e 2e2e 2922 0a0a 466f 7220  egy=>...)"..For 
│ │ │ │ +00032fa0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d  the programmer.=
│ │ │ │ +00032fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00032fc0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e  =..The object *n
│ │ │ │ +00032fd0: 6f74 6520 426c 6f77 5570 5374 7261 7465  ote BlowUpStrate
│ │ │ │ +00032fe0: 6779 3a20 426c 6f77 5570 5374 7261 7465  gy: BlowUpStrate
│ │ │ │ +00032ff0: 6779 2c20 6973 2061 202a 6e6f 7465 2073  gy, is a *note s
│ │ │ │ +00033000: 796d 626f 6c3a 0a28 4d61 6361 756c 6179  ymbol:.(Macaulay
│ │ │ │ +00033010: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d  2Doc)Symbol,...-
│ │ │ │  00033020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033060: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -00033070: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -00033080: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -00033090: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -000330a0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -000330b0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ -000330c0: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -000330d0: 4372 656d 6f6e 612f 0a64 6f63 756d 656e  Cremona/.documen
│ │ │ │ -000330e0: 7461 7469 6f6e 2e6d 323a 3936 333a 302e  tation.m2:963:0.
│ │ │ │ -000330f0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ -00033100: 2e69 6e66 6f2c 204e 6f64 653a 2043 6572  .info, Node: Cer
│ │ │ │ -00033110: 7469 6679 2c20 4e65 7874 3a20 4368 6572  tify, Next: Cher
│ │ │ │ -00033120: 6e53 6368 7761 7274 7a4d 6163 5068 6572  nSchwartzMacPher
│ │ │ │ -00033130: 736f 6e2c 2050 7265 763a 2042 6c6f 7755  son, Prev: BlowU
│ │ │ │ -00033140: 7053 7472 6174 6567 792c 2055 703a 2054  pStrategy, Up: T
│ │ │ │ -00033150: 6f70 0a0a 4365 7274 6966 7920 2d2d 2077  op..Certify -- w
│ │ │ │ -00033160: 6865 7468 6572 2074 6f20 656e 7375 7265  hether to ensure
│ │ │ │ -00033170: 2063 6f72 7265 6374 6e65 7373 206f 6620   correctness of 
│ │ │ │ -00033180: 6f75 7470 7574 0a2a 2a2a 2a2a 2a2a 2a2a  output.*********
│ │ │ │ +00033060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +00033070: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +00033080: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +00033090: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +000330a0: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +000330b0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ +000330c0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +000330d0: 6b61 6765 732f 4372 656d 6f6e 612f 0a64  kages/Cremona/.d
│ │ │ │ +000330e0: 6f63 756d 656e 7461 7469 6f6e 2e6d 323a  ocumentation.m2:
│ │ │ │ +000330f0: 3936 333a 302e 0a1f 0a46 696c 653a 2043  963:0....File: C
│ │ │ │ +00033100: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ +00033110: 653a 2043 6572 7469 6679 2c20 4e65 7874  e: Certify, Next
│ │ │ │ +00033120: 3a20 4368 6572 6e53 6368 7761 7274 7a4d  : ChernSchwartzM
│ │ │ │ +00033130: 6163 5068 6572 736f 6e2c 2050 7265 763a  acPherson, Prev:
│ │ │ │ +00033140: 2042 6c6f 7755 7053 7472 6174 6567 792c   BlowUpStrategy,
│ │ │ │ +00033150: 2055 703a 2054 6f70 0a0a 4365 7274 6966   Up: Top..Certif
│ │ │ │ +00033160: 7920 2d2d 2077 6865 7468 6572 2074 6f20  y -- whether to 
│ │ │ │ +00033170: 656e 7375 7265 2063 6f72 7265 6374 6e65  ensure correctne
│ │ │ │ +00033180: 7373 206f 6620 6f75 7470 7574 0a2a 2a2a  ss of output.***
│ │ │ │  00033190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  000331a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -000331b0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372  *********..Descr
│ │ │ │ -000331c0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -000331d0: 3d3d 0a0a 5468 6973 206f 7074 696f 6e20  ==..This option 
│ │ │ │ -000331e0: 6163 6365 7074 7320 6120 2a6e 6f74 6520  accepts a *note 
│ │ │ │ -000331f0: 426f 6f6c 6561 6e3a 2028 4d61 6361 756c  Boolean: (Macaul
│ │ │ │ -00033200: 6179 3244 6f63 2942 6f6f 6c65 616e 2c20  ay2Doc)Boolean, 
│ │ │ │ -00033210: 7661 6c75 652c 2064 6566 6175 6c74 0a76  value, default.v
│ │ │ │ -00033220: 616c 7565 2066 616c 7365 2e0a 0a49 6620  alue false...If 
│ │ │ │ -00033230: 7475 726e 6564 206f 6e20 696e 2074 6865  turned on in the
│ │ │ │ -00033240: 206d 6574 686f 6473 202a 6e6f 7465 2069   methods *note i
│ │ │ │ -00033250: 6e76 6572 7365 4d61 703a 2069 6e76 6572  nverseMap: inver
│ │ │ │ -00033260: 7365 4d61 702c 2061 6e64 202a 6e6f 7465  seMap, and *note
│ │ │ │ -00033270: 0a61 7070 726f 7869 6d61 7465 496e 7665  .approximateInve
│ │ │ │ -00033280: 7273 654d 6170 3a20 6170 7072 6f78 696d  rseMap: approxim
│ │ │ │ -00033290: 6174 6549 6e76 6572 7365 4d61 702c 2c20  ateInverseMap,, 
│ │ │ │ -000332a0: 7468 656e 2069 7420 7769 6c6c 2062 6520  then it will be 
│ │ │ │ -000332b0: 6368 6563 6b65 6420 7768 6574 6865 720a  checked whether.
│ │ │ │ -000332c0: 7468 6520 6d61 7073 2069 6e20 696e 7075  the maps in inpu
│ │ │ │ -000332d0: 7420 616e 6420 6f75 7470 7574 2061 7265  t and output are
│ │ │ │ -000332e0: 206f 6e65 2074 6865 2069 6e76 6572 7365   one the inverse
│ │ │ │ -000332f0: 206f 6620 7468 6520 6f74 6865 722c 2074   of the other, t
│ │ │ │ -00033300: 6872 6f77 696e 6720 616e 0a65 7272 6f72  hrowing an.error
│ │ │ │ -00033310: 2069 6620 7468 6579 2061 7265 206e 6f74   if they are not
│ │ │ │ -00033320: 2e20 4163 7475 616c 6c79 2c20 2a6e 6f74  . Actually, *not
│ │ │ │ -00033330: 6520 6170 7072 6f78 696d 6174 6549 6e76  e approximateInv
│ │ │ │ -00033340: 6572 7365 4d61 703a 0a61 7070 726f 7869  erseMap:.approxi
│ │ │ │ -00033350: 6d61 7465 496e 7665 7273 654d 6170 2c20  mateInverseMap, 
│ │ │ │ -00033360: 7769 6c6c 2066 6972 7374 2074 7279 2074  will first try t
│ │ │ │ -00033370: 6f20 6669 7820 7468 6520 6572 726f 7220  o fix the error 
│ │ │ │ -00033380: 6f66 2074 6865 2061 7070 726f 7869 6d61  of the approxima
│ │ │ │ -00033390: 7469 6f6e 2e0a 5768 656e 2074 7572 6e65  tion..When turne
│ │ │ │ -000333a0: 6420 6f6e 2069 6e20 7468 6520 6d65 7468  d on in the meth
│ │ │ │ -000333b0: 6f64 7320 2a6e 6f74 6520 7072 6f6a 6563  ods *note projec
│ │ │ │ -000333c0: 7469 7665 4465 6772 6565 733a 2070 726f  tiveDegrees: pro
│ │ │ │ -000333d0: 6a65 6374 6976 6544 6567 7265 6573 2c2c  jectiveDegrees,,
│ │ │ │ -000333e0: 0a2a 6e6f 7465 2064 6567 7265 654d 6170  .*note degreeMap
│ │ │ │ -000333f0: 3a20 6465 6772 6565 4d61 702c 2c20 2a6e  : degreeMap,, *n
│ │ │ │ -00033400: 6f74 6520 6973 4269 7261 7469 6f6e 616c  ote isBirational
│ │ │ │ -00033410: 3a20 6973 4269 7261 7469 6f6e 616c 2c2c  : isBirational,,
│ │ │ │ -00033420: 202a 6e6f 7465 0a69 7344 6f6d 696e 616e   *note.isDominan
│ │ │ │ -00033430: 743a 2069 7344 6f6d 696e 616e 742c 2c20  t: isDominant,, 
│ │ │ │ -00033440: 2a6e 6f74 6520 5365 6772 6543 6c61 7373  *note SegreClass
│ │ │ │ -00033450: 3a20 5365 6772 6543 6c61 7373 2c2c 202a  : SegreClass,, *
│ │ │ │ -00033460: 6e6f 7465 0a45 756c 6572 4368 6172 6163  note.EulerCharac
│ │ │ │ -00033470: 7465 7269 7374 6963 3a20 4575 6c65 7243  teristic: EulerC
│ │ │ │ -00033480: 6861 7261 6374 6572 6973 7469 632c 2061  haracteristic, a
│ │ │ │ -00033490: 6e64 202a 6e6f 7465 2043 6865 726e 5363  nd *note ChernSc
│ │ │ │ -000334a0: 6877 6172 747a 4d61 6350 6865 7273 6f6e  hwartzMacPherson
│ │ │ │ -000334b0: 3a0a 4368 6572 6e53 6368 7761 7274 7a4d  :.ChernSchwartzM
│ │ │ │ -000334c0: 6163 5068 6572 736f 6e2c 2c20 6974 206d  acPherson,, it m
│ │ │ │ -000334d0: 6561 6e73 2077 6865 7468 6572 2074 6f20  eans whether to 
│ │ │ │ -000334e0: 7573 6520 6120 6e6f 6e2d 7072 6f62 6162  use a non-probab
│ │ │ │ -000334f0: 696c 6973 7469 630a 616c 676f 7269 7468  ilistic.algorith
│ │ │ │ -00033500: 6d2e 0a0a 4675 6e63 7469 6f6e 7320 7769  m...Functions wi
│ │ │ │ -00033510: 7468 206f 7074 696f 6e61 6c20 6172 6775  th optional argu
│ │ │ │ -00033520: 6d65 6e74 206e 616d 6564 2043 6572 7469  ment named Certi
│ │ │ │ -00033530: 6679 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  fy:.============
│ │ │ │ +000331b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a  ***************.
│ │ │ │ +000331c0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +000331d0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 206f  ========..This o
│ │ │ │ +000331e0: 7074 696f 6e20 6163 6365 7074 7320 6120  ption accepts a 
│ │ │ │ +000331f0: 2a6e 6f74 6520 426f 6f6c 6561 6e3a 2028  *note Boolean: (
│ │ │ │ +00033200: 4d61 6361 756c 6179 3244 6f63 2942 6f6f  Macaulay2Doc)Boo
│ │ │ │ +00033210: 6c65 616e 2c20 7661 6c75 652c 2064 6566  lean, value, def
│ │ │ │ +00033220: 6175 6c74 0a76 616c 7565 2066 616c 7365  ault.value false
│ │ │ │ +00033230: 2e0a 0a49 6620 7475 726e 6564 206f 6e20  ...If turned on 
│ │ │ │ +00033240: 696e 2074 6865 206d 6574 686f 6473 202a  in the methods *
│ │ │ │ +00033250: 6e6f 7465 2069 6e76 6572 7365 4d61 703a  note inverseMap:
│ │ │ │ +00033260: 2069 6e76 6572 7365 4d61 702c 2061 6e64   inverseMap, and
│ │ │ │ +00033270: 202a 6e6f 7465 0a61 7070 726f 7869 6d61   *note.approxima
│ │ │ │ +00033280: 7465 496e 7665 7273 654d 6170 3a20 6170  teInverseMap: ap
│ │ │ │ +00033290: 7072 6f78 696d 6174 6549 6e76 6572 7365  proximateInverse
│ │ │ │ +000332a0: 4d61 702c 2c20 7468 656e 2069 7420 7769  Map,, then it wi
│ │ │ │ +000332b0: 6c6c 2062 6520 6368 6563 6b65 6420 7768  ll be checked wh
│ │ │ │ +000332c0: 6574 6865 720a 7468 6520 6d61 7073 2069  ether.the maps i
│ │ │ │ +000332d0: 6e20 696e 7075 7420 616e 6420 6f75 7470  n input and outp
│ │ │ │ +000332e0: 7574 2061 7265 206f 6e65 2074 6865 2069  ut are one the i
│ │ │ │ +000332f0: 6e76 6572 7365 206f 6620 7468 6520 6f74  nverse of the ot
│ │ │ │ +00033300: 6865 722c 2074 6872 6f77 696e 6720 616e  her, throwing an
│ │ │ │ +00033310: 0a65 7272 6f72 2069 6620 7468 6579 2061  .error if they a
│ │ │ │ +00033320: 7265 206e 6f74 2e20 4163 7475 616c 6c79  re not. Actually
│ │ │ │ +00033330: 2c20 2a6e 6f74 6520 6170 7072 6f78 696d  , *note approxim
│ │ │ │ +00033340: 6174 6549 6e76 6572 7365 4d61 703a 0a61  ateInverseMap:.a
│ │ │ │ +00033350: 7070 726f 7869 6d61 7465 496e 7665 7273  pproximateInvers
│ │ │ │ +00033360: 654d 6170 2c20 7769 6c6c 2066 6972 7374  eMap, will first
│ │ │ │ +00033370: 2074 7279 2074 6f20 6669 7820 7468 6520   try to fix the 
│ │ │ │ +00033380: 6572 726f 7220 6f66 2074 6865 2061 7070  error of the app
│ │ │ │ +00033390: 726f 7869 6d61 7469 6f6e 2e0a 5768 656e  roximation..When
│ │ │ │ +000333a0: 2074 7572 6e65 6420 6f6e 2069 6e20 7468   turned on in th
│ │ │ │ +000333b0: 6520 6d65 7468 6f64 7320 2a6e 6f74 6520  e methods *note 
│ │ │ │ +000333c0: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ +000333d0: 733a 2070 726f 6a65 6374 6976 6544 6567  s: projectiveDeg
│ │ │ │ +000333e0: 7265 6573 2c2c 0a2a 6e6f 7465 2064 6567  rees,,.*note deg
│ │ │ │ +000333f0: 7265 654d 6170 3a20 6465 6772 6565 4d61  reeMap: degreeMa
│ │ │ │ +00033400: 702c 2c20 2a6e 6f74 6520 6973 4269 7261  p,, *note isBira
│ │ │ │ +00033410: 7469 6f6e 616c 3a20 6973 4269 7261 7469  tional: isBirati
│ │ │ │ +00033420: 6f6e 616c 2c2c 202a 6e6f 7465 0a69 7344  onal,, *note.isD
│ │ │ │ +00033430: 6f6d 696e 616e 743a 2069 7344 6f6d 696e  ominant: isDomin
│ │ │ │ +00033440: 616e 742c 2c20 2a6e 6f74 6520 5365 6772  ant,, *note Segr
│ │ │ │ +00033450: 6543 6c61 7373 3a20 5365 6772 6543 6c61  eClass: SegreCla
│ │ │ │ +00033460: 7373 2c2c 202a 6e6f 7465 0a45 756c 6572  ss,, *note.Euler
│ │ │ │ +00033470: 4368 6172 6163 7465 7269 7374 6963 3a20  Characteristic: 
│ │ │ │ +00033480: 4575 6c65 7243 6861 7261 6374 6572 6973  EulerCharacteris
│ │ │ │ +00033490: 7469 632c 2061 6e64 202a 6e6f 7465 2043  tic, and *note C
│ │ │ │ +000334a0: 6865 726e 5363 6877 6172 747a 4d61 6350  hernSchwartzMacP
│ │ │ │ +000334b0: 6865 7273 6f6e 3a0a 4368 6572 6e53 6368  herson:.ChernSch
│ │ │ │ +000334c0: 7761 7274 7a4d 6163 5068 6572 736f 6e2c  wartzMacPherson,
│ │ │ │ +000334d0: 2c20 6974 206d 6561 6e73 2077 6865 7468  , it means wheth
│ │ │ │ +000334e0: 6572 2074 6f20 7573 6520 6120 6e6f 6e2d  er to use a non-
│ │ │ │ +000334f0: 7072 6f62 6162 696c 6973 7469 630a 616c  probabilistic.al
│ │ │ │ +00033500: 676f 7269 7468 6d2e 0a0a 4675 6e63 7469  gorithm...Functi
│ │ │ │ +00033510: 6f6e 7320 7769 7468 206f 7074 696f 6e61  ons with optiona
│ │ │ │ +00033520: 6c20 6172 6775 6d65 6e74 206e 616d 6564  l argument named
│ │ │ │ +00033530: 2043 6572 7469 6679 3a0a 3d3d 3d3d 3d3d   Certify:.======
│ │ │ │  00033540: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00033550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00033560: 3d3d 3d0a 0a20 202a 2022 6170 7072 6f78  ===..  * "approx
│ │ │ │ -00033570: 696d 6174 6549 6e76 6572 7365 4d61 7028  imateInverseMap(
│ │ │ │ -00033580: 2e2e 2e2c 4365 7274 6966 793d 3e2e 2e2e  ...,Certify=>...
│ │ │ │ -00033590: 2922 0a20 202a 2022 4368 6572 6e53 6368  )".  * "ChernSch
│ │ │ │ -000335a0: 7761 7274 7a4d 6163 5068 6572 736f 6e28  wartzMacPherson(
│ │ │ │ -000335b0: 2e2e 2e2c 4365 7274 6966 793d 3e2e 2e2e  ...,Certify=>...
│ │ │ │ -000335c0: 2922 0a20 202a 2022 6465 6772 6565 4d61  )".  * "degreeMa
│ │ │ │ -000335d0: 7028 2e2e 2e2c 4365 7274 6966 793d 3e2e  p(...,Certify=>.
│ │ │ │ -000335e0: 2e2e 2922 0a20 202a 2022 4575 6c65 7243  ..)".  * "EulerC
│ │ │ │ -000335f0: 6861 7261 6374 6572 6973 7469 6328 2e2e  haracteristic(..
│ │ │ │ -00033600: 2e2c 4365 7274 6966 793d 3e2e 2e2e 2922  .,Certify=>...)"
│ │ │ │ -00033610: 0a20 202a 2022 6578 6365 7074 696f 6e61  .  * "exceptiona
│ │ │ │ -00033620: 6c4c 6f63 7573 282e 2e2e 2c43 6572 7469  lLocus(...,Certi
│ │ │ │ -00033630: 6679 3d3e 2e2e 2e29 220a 2020 2a20 2269  fy=>...)".  * "i
│ │ │ │ -00033640: 6e76 6572 7365 4d61 7028 2e2e 2e2c 4365  nverseMap(...,Ce
│ │ │ │ -00033650: 7274 6966 793d 3e2e 2e2e 2922 0a20 202a  rtify=>...)".  *
│ │ │ │ -00033660: 2022 6973 4269 7261 7469 6f6e 616c 282e   "isBirational(.
│ │ │ │ -00033670: 2e2e 2c43 6572 7469 6679 3d3e 2e2e 2e29  ..,Certify=>...)
│ │ │ │ -00033680: 220a 2020 2a20 2269 7344 6f6d 696e 616e  ".  * "isDominan
│ │ │ │ -00033690: 7428 2e2e 2e2c 4365 7274 6966 793d 3e2e  t(...,Certify=>.
│ │ │ │ -000336a0: 2e2e 2922 0a20 202a 2022 7072 6f6a 6563  ..)".  * "projec
│ │ │ │ -000336b0: 7469 7665 4465 6772 6565 7328 2e2e 2e2c  tiveDegrees(...,
│ │ │ │ -000336c0: 4365 7274 6966 793d 3e2e 2e2e 2922 0a20  Certify=>...)". 
│ │ │ │ -000336d0: 202a 2022 5365 6772 6543 6c61 7373 282e   * "SegreClass(.
│ │ │ │ -000336e0: 2e2e 2c43 6572 7469 6679 3d3e 2e2e 2e29  ..,Certify=>...)
│ │ │ │ -000336f0: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00033700: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00033710: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -00033720: 6a65 6374 202a 6e6f 7465 2043 6572 7469  ject *note Certi
│ │ │ │ -00033730: 6679 3a20 4365 7274 6966 792c 2069 7320  fy: Certify, is 
│ │ │ │ -00033740: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20  a *note symbol: 
│ │ │ │ -00033750: 284d 6163 6175 6c61 7932 446f 6329 5379  (Macaulay2Doc)Sy
│ │ │ │ -00033760: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d  mbol,...--------
│ │ │ │ +00033560: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022  =========..  * "
│ │ │ │ +00033570: 6170 7072 6f78 696d 6174 6549 6e76 6572  approximateInver
│ │ │ │ +00033580: 7365 4d61 7028 2e2e 2e2c 4365 7274 6966  seMap(...,Certif
│ │ │ │ +00033590: 793d 3e2e 2e2e 2922 0a20 202a 2022 4368  y=>...)".  * "Ch
│ │ │ │ +000335a0: 6572 6e53 6368 7761 7274 7a4d 6163 5068  ernSchwartzMacPh
│ │ │ │ +000335b0: 6572 736f 6e28 2e2e 2e2c 4365 7274 6966  erson(...,Certif
│ │ │ │ +000335c0: 793d 3e2e 2e2e 2922 0a20 202a 2022 6465  y=>...)".  * "de
│ │ │ │ +000335d0: 6772 6565 4d61 7028 2e2e 2e2c 4365 7274  greeMap(...,Cert
│ │ │ │ +000335e0: 6966 793d 3e2e 2e2e 2922 0a20 202a 2022  ify=>...)".  * "
│ │ │ │ +000335f0: 4575 6c65 7243 6861 7261 6374 6572 6973  EulerCharacteris
│ │ │ │ +00033600: 7469 6328 2e2e 2e2c 4365 7274 6966 793d  tic(...,Certify=
│ │ │ │ +00033610: 3e2e 2e2e 2922 0a20 202a 2022 6578 6365  >...)".  * "exce
│ │ │ │ +00033620: 7074 696f 6e61 6c4c 6f63 7573 282e 2e2e  ptionalLocus(...
│ │ │ │ +00033630: 2c43 6572 7469 6679 3d3e 2e2e 2e29 220a  ,Certify=>...)".
│ │ │ │ +00033640: 2020 2a20 2269 6e76 6572 7365 4d61 7028    * "inverseMap(
│ │ │ │ +00033650: 2e2e 2e2c 4365 7274 6966 793d 3e2e 2e2e  ...,Certify=>...
│ │ │ │ +00033660: 2922 0a20 202a 2022 6973 4269 7261 7469  )".  * "isBirati
│ │ │ │ +00033670: 6f6e 616c 282e 2e2e 2c43 6572 7469 6679  onal(...,Certify
│ │ │ │ +00033680: 3d3e 2e2e 2e29 220a 2020 2a20 2269 7344  =>...)".  * "isD
│ │ │ │ +00033690: 6f6d 696e 616e 7428 2e2e 2e2c 4365 7274  ominant(...,Cert
│ │ │ │ +000336a0: 6966 793d 3e2e 2e2e 2922 0a20 202a 2022  ify=>...)".  * "
│ │ │ │ +000336b0: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ +000336c0: 7328 2e2e 2e2c 4365 7274 6966 793d 3e2e  s(...,Certify=>.
│ │ │ │ +000336d0: 2e2e 2922 0a20 202a 2022 5365 6772 6543  ..)".  * "SegreC
│ │ │ │ +000336e0: 6c61 7373 282e 2e2e 2c43 6572 7469 6679  lass(...,Certify
│ │ │ │ +000336f0: 3d3e 2e2e 2e29 220a 0a46 6f72 2074 6865  =>...)"..For the
│ │ │ │ +00033700: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +00033710: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00033720: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00033730: 2043 6572 7469 6679 3a20 4365 7274 6966   Certify: Certif
│ │ │ │ +00033740: 792c 2069 7320 6120 2a6e 6f74 6520 7379  y, is a *note sy
│ │ │ │ +00033750: 6d62 6f6c 3a20 284d 6163 6175 6c61 7932  mbol: (Macaulay2
│ │ │ │ +00033760: 446f 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d  Doc)Symbol,...--
│ │ │ │  00033770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000337a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000337b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75  -------..The sou
│ │ │ │ -000337c0: 7263 6520 6f66 2074 6869 7320 646f 6375  rce of this docu
│ │ │ │ -000337d0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c  ment is in./buil
│ │ │ │ -000337e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70  d/reproducible-p
│ │ │ │ -000337f0: 6174 682f 6d61 6361 756c 6179 322d 312e  ath/macaulay2-1.
│ │ │ │ -00033800: 3236 2e30 362b 6473 2f4d 322f 4d61 6361  26.06+ds/M2/Maca
│ │ │ │ -00033810: 756c 6179 322f 7061 636b 6167 6573 2f43  ulay2/packages/C
│ │ │ │ -00033820: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74  remona/.document
│ │ │ │ -00033830: 6174 696f 6e2e 6d32 3a38 333a 302e 0a1f  ation.m2:83:0...
│ │ │ │ -00033840: 0a46 696c 653a 2043 7265 6d6f 6e61 2e69  .File: Cremona.i
│ │ │ │ -00033850: 6e66 6f2c 204e 6f64 653a 2043 6865 726e  nfo, Node: Chern
│ │ │ │ -00033860: 5363 6877 6172 747a 4d61 6350 6865 7273  SchwartzMacPhers
│ │ │ │ -00033870: 6f6e 2c20 4e65 7874 3a20 436f 6469 6d42  on, Next: CodimB
│ │ │ │ -00033880: 7349 6e76 2c20 5072 6576 3a20 4365 7274  sInv, Prev: Cert
│ │ │ │ -00033890: 6966 792c 2055 703a 2054 6f70 0a0a 4368  ify, Up: Top..Ch
│ │ │ │ -000338a0: 6572 6e53 6368 7761 7274 7a4d 6163 5068  ernSchwartzMacPh
│ │ │ │ -000338b0: 6572 736f 6e20 2d2d 2043 6865 726e 2d53  erson -- Chern-S
│ │ │ │ -000338c0: 6368 7761 7274 7a2d 4d61 6350 6865 7273  chwartz-MacPhers
│ │ │ │ -000338d0: 6f6e 2063 6c61 7373 206f 6620 6120 7072  on class of a pr
│ │ │ │ -000338e0: 6f6a 6563 7469 7665 2073 6368 656d 650a  ojective scheme.
│ │ │ │ -000338f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +000337b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54  -------------..T
│ │ │ │ +000337c0: 6865 2073 6f75 7263 6520 6f66 2074 6869  he source of thi
│ │ │ │ +000337d0: 7320 646f 6375 6d65 6e74 2069 7320 696e  s document is in
│ │ │ │ +000337e0: 0a2f 6275 696c 642f 7265 7072 6f64 7563  ./build/reproduc
│ │ │ │ +000337f0: 6962 6c65 2d70 6174 682f 6d61 6361 756c  ible-path/macaul
│ │ │ │ +00033800: 6179 322d 312e 3236 2e30 362b 6473 2f4d  ay2-1.26.06+ds/M
│ │ │ │ +00033810: 322f 4d61 6361 756c 6179 322f 7061 636b  2/Macaulay2/pack
│ │ │ │ +00033820: 6167 6573 2f43 7265 6d6f 6e61 2f0a 646f  ages/Cremona/.do
│ │ │ │ +00033830: 6375 6d65 6e74 6174 696f 6e2e 6d32 3a38  cumentation.m2:8
│ │ │ │ +00033840: 333a 302e 0a1f 0a46 696c 653a 2043 7265  3:0....File: Cre
│ │ │ │ +00033850: 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 653a  mona.info, Node:
│ │ │ │ +00033860: 2043 6865 726e 5363 6877 6172 747a 4d61   ChernSchwartzMa
│ │ │ │ +00033870: 6350 6865 7273 6f6e 2c20 4e65 7874 3a20  cPherson, Next: 
│ │ │ │ +00033880: 436f 6469 6d42 7349 6e76 2c20 5072 6576  CodimBsInv, Prev
│ │ │ │ +00033890: 3a20 4365 7274 6966 792c 2055 703a 2054  : Certify, Up: T
│ │ │ │ +000338a0: 6f70 0a0a 4368 6572 6e53 6368 7761 7274  op..ChernSchwart
│ │ │ │ +000338b0: 7a4d 6163 5068 6572 736f 6e20 2d2d 2043  zMacPherson -- C
│ │ │ │ +000338c0: 6865 726e 2d53 6368 7761 7274 7a2d 4d61  hern-Schwartz-Ma
│ │ │ │ +000338d0: 6350 6865 7273 6f6e 2063 6c61 7373 206f  cPherson class o
│ │ │ │ +000338e0: 6620 6120 7072 6f6a 6563 7469 7665 2073  f a projective s
│ │ │ │ +000338f0: 6368 656d 650a 2a2a 2a2a 2a2a 2a2a 2a2a  cheme.**********
│ │ │ │  00033900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00033910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00033920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00033930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00033940: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20  *..  * Usage: . 
│ │ │ │ -00033950: 2020 2020 2020 2043 6865 726e 5363 6877         ChernSchw
│ │ │ │ -00033960: 6172 747a 4d61 6350 6865 7273 6f6e 2049  artzMacPherson I
│ │ │ │ -00033970: 0a20 202a 2049 6e70 7574 733a 0a20 2020  .  * Inputs:.   
│ │ │ │ -00033980: 2020 202a 2049 2c20 616e 202a 6e6f 7465     * I, an *note
│ │ │ │ -00033990: 2069 6465 616c 3a20 284d 6163 6175 6c61   ideal: (Macaula
│ │ │ │ -000339a0: 7932 446f 6329 4964 6561 6c2c 2c20 6120  y2Doc)Ideal,, a 
│ │ │ │ -000339b0: 686f 6d6f 6765 6e65 6f75 7320 6964 6561  homogeneous idea
│ │ │ │ -000339c0: 6c20 6465 6669 6e69 6e67 2061 0a20 2020  l defining a.   
│ │ │ │ -000339d0: 2020 2020 2063 6c6f 7365 6420 7375 6273       closed subs
│ │ │ │ -000339e0: 6368 656d 6520 2458 5c73 7562 7365 745c  cheme $X\subset\
│ │ │ │ -000339f0: 6d61 7468 6262 7b50 7d5e 6e24 0a20 202a  mathbb{P}^n$.  *
│ │ │ │ -00033a00: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ -00033a10: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ -00033a20: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ -00033a30: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ -00033a40: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ -00033a50: 2020 202a 202a 6e6f 7465 2042 6c6f 7755     * *note BlowU
│ │ │ │ -00033a60: 7053 7472 6174 6567 793a 2042 6c6f 7755  pStrategy: BlowU
│ │ │ │ -00033a70: 7053 7472 6174 6567 792c 203d 3e20 2e2e  pStrategy, => ..
│ │ │ │ -00033a80: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ -00033a90: 0a20 2020 2020 2020 2022 456c 696d 696e  .        "Elimin
│ │ │ │ -00033aa0: 6174 6522 2c0a 2020 2020 2020 2a20 2a6e  ate",.      * *n
│ │ │ │ -00033ab0: 6f74 6520 4365 7274 6966 793a 2043 6572  ote Certify: Cer
│ │ │ │ -00033ac0: 7469 6679 2c20 3d3e 202e 2e2e 2c20 6465  tify, => ..., de
│ │ │ │ -00033ad0: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ -00033ae0: 652c 2077 6865 7468 6572 2074 6f20 656e  e, whether to en
│ │ │ │ -00033af0: 7375 7265 0a20 2020 2020 2020 2063 6f72  sure.        cor
│ │ │ │ -00033b00: 7265 6374 6e65 7373 206f 6620 6f75 7470  rectness of outp
│ │ │ │ -00033b10: 7574 0a20 2020 2020 202a 202a 6e6f 7465  ut.      * *note
│ │ │ │ -00033b20: 2056 6572 626f 7365 3a20 696e 7665 7273   Verbose: invers
│ │ │ │ -00033b30: 654d 6170 5f6c 705f 7064 5f70 645f 7064  eMap_lp_pd_pd_pd
│ │ │ │ -00033b40: 5f63 6d56 6572 626f 7365 3d3e 5f70 645f  _cmVerbose=>_pd_
│ │ │ │ -00033b50: 7064 5f70 645f 7270 2c20 3d3e 202e 2e2e  pd_pd_rp, => ...
│ │ │ │ -00033b60: 2c0a 2020 2020 2020 2020 6465 6661 756c  ,.        defaul
│ │ │ │ -00033b70: 7420 7661 6c75 6520 7472 7565 2c0a 2020  t value true,.  
│ │ │ │ -00033b80: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -00033b90: 202a 2061 202a 6e6f 7465 2072 696e 6720   * a *note ring 
│ │ │ │ -00033ba0: 656c 656d 656e 743a 2028 4d61 6361 756c  element: (Macaul
│ │ │ │ -00033bb0: 6179 3244 6f63 2952 696e 6745 6c65 6d65  ay2Doc)RingEleme
│ │ │ │ -00033bc0: 6e74 2c2c 2074 6865 2070 7573 682d 666f  nt,, the push-fo
│ │ │ │ -00033bd0: 7277 6172 6420 746f 0a20 2020 2020 2020  rward to.       
│ │ │ │ -00033be0: 2074 6865 2043 686f 7720 7269 6e67 206f   the Chow ring o
│ │ │ │ -00033bf0: 6620 245c 6d61 7468 6262 7b50 7d5e 6e24  f $\mathbb{P}^n$
│ │ │ │ -00033c00: 206f 6620 7468 6520 4368 6572 6e2d 5363   of the Chern-Sc
│ │ │ │ -00033c10: 6877 6172 747a 2d4d 6163 5068 6572 736f  hwartz-MacPherso
│ │ │ │ -00033c20: 6e20 636c 6173 730a 2020 2020 2020 2020  n class.        
│ │ │ │ -00033c30: 2463 5f7b 534d 7d28 5829 2420 6f66 2024  $c_{SM}(X)$ of $
│ │ │ │ -00033c40: 5824 2e20 496e 2070 6172 7469 6375 6c61  X$. In particula
│ │ │ │ -00033c50: 722c 2074 6865 2063 6f65 6666 6963 6965  r, the coefficie
│ │ │ │ -00033c60: 6e74 206f 6620 2448 5e6e 2420 6769 7665  nt of $H^n$ give
│ │ │ │ -00033c70: 7320 7468 650a 2020 2020 2020 2020 4575  s the.        Eu
│ │ │ │ -00033c80: 6c65 7220 6368 6172 6163 7465 7269 7374  ler characterist
│ │ │ │ -00033c90: 6963 206f 6620 7468 6520 7375 7070 6f72  ic of the suppor
│ │ │ │ -00033ca0: 7420 6f66 2024 5824 2c20 7768 6572 6520  t of $X$, where 
│ │ │ │ -00033cb0: 2448 2420 6465 6e6f 7465 7320 7468 650a  $H$ denotes the.
│ │ │ │ -00033cc0: 2020 2020 2020 2020 6879 7065 7270 6c61          hyperpla
│ │ │ │ -00033cd0: 6e65 2063 6c61 7373 2e0a 0a44 6573 6372  ne class...Descr
│ │ │ │ -00033ce0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d  iption.=========
│ │ │ │ -00033cf0: 3d3d 0a0a 5468 6973 2069 7320 616e 2065  ==..This is an e
│ │ │ │ -00033d00: 7861 6d70 6c65 206f 6620 6170 706c 6963  xample of applic
│ │ │ │ -00033d10: 6174 696f 6e20 6f66 2074 6865 206d 6574  ation of the met
│ │ │ │ -00033d20: 686f 6420 2a6e 6f74 6520 7072 6f6a 6563  hod *note projec
│ │ │ │ -00033d30: 7469 7665 4465 6772 6565 733a 0a70 726f  tiveDegrees:.pro
│ │ │ │ -00033d40: 6a65 6374 6976 6544 6567 7265 6573 2c2c  jectiveDegrees,,
│ │ │ │ -00033d50: 2064 7565 2074 6f20 7265 7375 6c74 7320   due to results 
│ │ │ │ -00033d60: 7368 6f77 6e20 696e 2043 6f6d 7075 7469  shown in Computi
│ │ │ │ -00033d70: 6e67 2063 6861 7261 6374 6572 6973 7469  ng characteristi
│ │ │ │ -00033d80: 6320 636c 6173 7365 7320 6f66 0a70 726f  c classes of.pro
│ │ │ │ -00033d90: 6a65 6374 6976 6520 7363 6865 6d65 7320  jective schemes 
│ │ │ │ -00033da0: 2873 6565 2068 7474 703a 2f2f 7777 772e  (see http://www.
│ │ │ │ -00033db0: 7363 6965 6e63 6564 6972 6563 742e 636f  sciencedirect.co
│ │ │ │ -00033dc0: 6d2f 7363 6965 6e63 652f 6172 7469 636c  m/science/articl
│ │ │ │ -00033dd0: 652f 7069 692f 0a53 3037 3437 3731 3731  e/pii/.S07477171
│ │ │ │ -00033de0: 3032 3030 3038 3935 2029 2c20 6279 2050  02000895 ), by P
│ │ │ │ -00033df0: 2e20 416c 7566 6669 2e20 5365 6520 616c  . Aluffi. See al
│ │ │ │ -00033e00: 736f 2074 6865 2063 6f72 7265 7370 6f6e  so the correspon
│ │ │ │ -00033e10: 6469 6e67 206d 6574 686f 6473 2069 6e20  ding methods in 
│ │ │ │ -00033e20: 7468 650a 7061 636b 6167 6573 2043 534d  the.packages CSM
│ │ │ │ -00033e30: 2d41 2028 7365 6520 6874 7470 3a2f 2f77  -A (see http://w
│ │ │ │ -00033e40: 7777 2e6d 6174 682e 6673 752e 6564 752f  ww.math.fsu.edu/
│ │ │ │ -00033e50: 7e61 6c75 6666 692f 4353 4d2f 4353 4d2e  ~aluffi/CSM/CSM.
│ │ │ │ -00033e60: 6874 6d6c 2029 2c20 6279 2050 2e0a 416c  html ), by P..Al
│ │ │ │ -00033e70: 7566 6669 2c20 616e 6420 2a6e 6f74 6520  uffi, and *note 
│ │ │ │ -00033e80: 4368 6172 6163 7465 7269 7374 6963 436c  CharacteristicCl
│ │ │ │ -00033e90: 6173 7365 733a 2028 4368 6172 6163 7465  asses: (Characte
│ │ │ │ -00033ea0: 7269 7374 6963 436c 6173 7365 7329 546f  risticClasses)To
│ │ │ │ -00033eb0: 702c 2c20 6279 204d 2e0a 4865 6c6d 6572  p,, by M..Helmer
│ │ │ │ -00033ec0: 2061 6e64 2043 2e20 4a6f 7374 2e0a 0a49   and C. Jost...I
│ │ │ │ -00033ed0: 6e20 7468 6520 6578 616d 706c 6520 6265  n the example be
│ │ │ │ -00033ee0: 6c6f 772c 2077 6520 636f 6d70 7574 6520  low, we compute 
│ │ │ │ -00033ef0: 7468 6520 7075 7368 2d66 6f72 7761 7264  the push-forward
│ │ │ │ -00033f00: 2074 6f20 7468 6520 4368 6f77 2072 696e   to the Chow rin
│ │ │ │ -00033f10: 6720 6f66 0a24 5c6d 6174 6862 627b 507d  g of.$\mathbb{P}
│ │ │ │ -00033f20: 5e34 2420 6f66 2074 6865 2043 6865 726e  ^4$ of the Chern
│ │ │ │ -00033f30: 2d53 6368 7761 7274 7a2d 4d61 6350 6865  -Schwartz-MacPhe
│ │ │ │ -00033f40: 7273 6f6e 2063 6c61 7373 206f 6620 7468  rson class of th
│ │ │ │ -00033f50: 6520 636f 6e65 206f 7665 7220 7468 650a  e cone over the.
│ │ │ │ -00033f60: 7477 6973 7465 6420 6375 6269 6320 6375  twisted cubic cu
│ │ │ │ -00033f70: 7276 652c 2075 7369 6e67 2062 6f74 6820  rve, using both 
│ │ │ │ -00033f80: 6120 7072 6f62 6162 696c 6973 7469 6320  a probabilistic 
│ │ │ │ -00033f90: 616e 6420 6120 6e6f 6e2d 7072 6f62 6162  and a non-probab
│ │ │ │ -00033fa0: 696c 6973 7469 630a 6170 7072 6f61 6368  ilistic.approach
│ │ │ │ -00033fb0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │ +00033940: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361  *******..  * Usa
│ │ │ │ +00033950: 6765 3a20 0a20 2020 2020 2020 2043 6865  ge: .        Che
│ │ │ │ +00033960: 726e 5363 6877 6172 747a 4d61 6350 6865  rnSchwartzMacPhe
│ │ │ │ +00033970: 7273 6f6e 2049 0a20 202a 2049 6e70 7574  rson I.  * Input
│ │ │ │ +00033980: 733a 0a20 2020 2020 202a 2049 2c20 616e  s:.      * I, an
│ │ │ │ +00033990: 202a 6e6f 7465 2069 6465 616c 3a20 284d   *note ideal: (M
│ │ │ │ +000339a0: 6163 6175 6c61 7932 446f 6329 4964 6561  acaulay2Doc)Idea
│ │ │ │ +000339b0: 6c2c 2c20 6120 686f 6d6f 6765 6e65 6f75  l,, a homogeneou
│ │ │ │ +000339c0: 7320 6964 6561 6c20 6465 6669 6e69 6e67  s ideal defining
│ │ │ │ +000339d0: 2061 0a20 2020 2020 2020 2063 6c6f 7365   a.        close
│ │ │ │ +000339e0: 6420 7375 6273 6368 656d 6520 2458 5c73  d subscheme $X\s
│ │ │ │ +000339f0: 7562 7365 745c 6d61 7468 6262 7b50 7d5e  ubset\mathbb{P}^
│ │ │ │ +00033a00: 6e24 0a20 202a 202a 6e6f 7465 204f 7074  n$.  * *note Opt
│ │ │ │ +00033a10: 696f 6e61 6c20 696e 7075 7473 3a20 284d  ional inputs: (M
│ │ │ │ +00033a20: 6163 6175 6c61 7932 446f 6329 7573 696e  acaulay2Doc)usin
│ │ │ │ +00033a30: 6720 6675 6e63 7469 6f6e 7320 7769 7468  g functions with
│ │ │ │ +00033a40: 206f 7074 696f 6e61 6c20 696e 7075 7473   optional inputs
│ │ │ │ +00033a50: 2c3a 0a20 2020 2020 202a 202a 6e6f 7465  ,:.      * *note
│ │ │ │ +00033a60: 2042 6c6f 7755 7053 7472 6174 6567 793a   BlowUpStrategy:
│ │ │ │ +00033a70: 2042 6c6f 7755 7053 7472 6174 6567 792c   BlowUpStrategy,
│ │ │ │ +00033a80: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74   => ..., default
│ │ │ │ +00033a90: 2076 616c 7565 0a20 2020 2020 2020 2022   value.        "
│ │ │ │ +00033aa0: 456c 696d 696e 6174 6522 2c0a 2020 2020  Eliminate",.    
│ │ │ │ +00033ab0: 2020 2a20 2a6e 6f74 6520 4365 7274 6966    * *note Certif
│ │ │ │ +00033ac0: 793a 2043 6572 7469 6679 2c20 3d3e 202e  y: Certify, => .
│ │ │ │ +00033ad0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75  .., default valu
│ │ │ │ +00033ae0: 6520 6661 6c73 652c 2077 6865 7468 6572  e false, whether
│ │ │ │ +00033af0: 2074 6f20 656e 7375 7265 0a20 2020 2020   to ensure.     
│ │ │ │ +00033b00: 2020 2063 6f72 7265 6374 6e65 7373 206f     correctness o
│ │ │ │ +00033b10: 6620 6f75 7470 7574 0a20 2020 2020 202a  f output.      *
│ │ │ │ +00033b20: 202a 6e6f 7465 2056 6572 626f 7365 3a20   *note Verbose: 
│ │ │ │ +00033b30: 696e 7665 7273 654d 6170 5f6c 705f 7064  inverseMap_lp_pd
│ │ │ │ +00033b40: 5f70 645f 7064 5f63 6d56 6572 626f 7365  _pd_pd_cmVerbose
│ │ │ │ +00033b50: 3d3e 5f70 645f 7064 5f70 645f 7270 2c20  =>_pd_pd_pd_rp, 
│ │ │ │ +00033b60: 3d3e 202e 2e2e 2c0a 2020 2020 2020 2020  => ...,.        
│ │ │ │ +00033b70: 6465 6661 756c 7420 7661 6c75 6520 7472  default value tr
│ │ │ │ +00033b80: 7565 2c0a 2020 2a20 4f75 7470 7574 733a  ue,.  * Outputs:
│ │ │ │ +00033b90: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ +00033ba0: 2072 696e 6720 656c 656d 656e 743a 2028   ring element: (
│ │ │ │ +00033bb0: 4d61 6361 756c 6179 3244 6f63 2952 696e  Macaulay2Doc)Rin
│ │ │ │ +00033bc0: 6745 6c65 6d65 6e74 2c2c 2074 6865 2070  gElement,, the p
│ │ │ │ +00033bd0: 7573 682d 666f 7277 6172 6420 746f 0a20  ush-forward to. 
│ │ │ │ +00033be0: 2020 2020 2020 2074 6865 2043 686f 7720         the Chow 
│ │ │ │ +00033bf0: 7269 6e67 206f 6620 245c 6d61 7468 6262  ring of $\mathbb
│ │ │ │ +00033c00: 7b50 7d5e 6e24 206f 6620 7468 6520 4368  {P}^n$ of the Ch
│ │ │ │ +00033c10: 6572 6e2d 5363 6877 6172 747a 2d4d 6163  ern-Schwartz-Mac
│ │ │ │ +00033c20: 5068 6572 736f 6e20 636c 6173 730a 2020  Pherson class.  
│ │ │ │ +00033c30: 2020 2020 2020 2463 5f7b 534d 7d28 5829        $c_{SM}(X)
│ │ │ │ +00033c40: 2420 6f66 2024 5824 2e20 496e 2070 6172  $ of $X$. In par
│ │ │ │ +00033c50: 7469 6375 6c61 722c 2074 6865 2063 6f65  ticular, the coe
│ │ │ │ +00033c60: 6666 6963 6965 6e74 206f 6620 2448 5e6e  fficient of $H^n
│ │ │ │ +00033c70: 2420 6769 7665 7320 7468 650a 2020 2020  $ gives the.    
│ │ │ │ +00033c80: 2020 2020 4575 6c65 7220 6368 6172 6163      Euler charac
│ │ │ │ +00033c90: 7465 7269 7374 6963 206f 6620 7468 6520  teristic of the 
│ │ │ │ +00033ca0: 7375 7070 6f72 7420 6f66 2024 5824 2c20  support of $X$, 
│ │ │ │ +00033cb0: 7768 6572 6520 2448 2420 6465 6e6f 7465  where $H$ denote
│ │ │ │ +00033cc0: 7320 7468 650a 2020 2020 2020 2020 6879  s the.        hy
│ │ │ │ +00033cd0: 7065 7270 6c61 6e65 2063 6c61 7373 2e0a  perplane class..
│ │ │ │ +00033ce0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d  .Description.===
│ │ │ │ +00033cf0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2069  ========..This i
│ │ │ │ +00033d00: 7320 616e 2065 7861 6d70 6c65 206f 6620  s an example of 
│ │ │ │ +00033d10: 6170 706c 6963 6174 696f 6e20 6f66 2074  application of t
│ │ │ │ +00033d20: 6865 206d 6574 686f 6420 2a6e 6f74 6520  he method *note 
│ │ │ │ +00033d30: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ +00033d40: 733a 0a70 726f 6a65 6374 6976 6544 6567  s:.projectiveDeg
│ │ │ │ +00033d50: 7265 6573 2c2c 2064 7565 2074 6f20 7265  rees,, due to re
│ │ │ │ +00033d60: 7375 6c74 7320 7368 6f77 6e20 696e 2043  sults shown in C
│ │ │ │ +00033d70: 6f6d 7075 7469 6e67 2063 6861 7261 6374  omputing charact
│ │ │ │ +00033d80: 6572 6973 7469 6320 636c 6173 7365 7320  eristic classes 
│ │ │ │ +00033d90: 6f66 0a70 726f 6a65 6374 6976 6520 7363  of.projective sc
│ │ │ │ +00033da0: 6865 6d65 7320 2873 6565 2068 7474 703a  hemes (see http:
│ │ │ │ +00033db0: 2f2f 7777 772e 7363 6965 6e63 6564 6972  //www.sciencedir
│ │ │ │ +00033dc0: 6563 742e 636f 6d2f 7363 6965 6e63 652f  ect.com/science/
│ │ │ │ +00033dd0: 6172 7469 636c 652f 7069 692f 0a53 3037  article/pii/.S07
│ │ │ │ +00033de0: 3437 3731 3731 3032 3030 3038 3935 2029  47717102000895 )
│ │ │ │ +00033df0: 2c20 6279 2050 2e20 416c 7566 6669 2e20  , by P. Aluffi. 
│ │ │ │ +00033e00: 5365 6520 616c 736f 2074 6865 2063 6f72  See also the cor
│ │ │ │ +00033e10: 7265 7370 6f6e 6469 6e67 206d 6574 686f  responding metho
│ │ │ │ +00033e20: 6473 2069 6e20 7468 650a 7061 636b 6167  ds in the.packag
│ │ │ │ +00033e30: 6573 2043 534d 2d41 2028 7365 6520 6874  es CSM-A (see ht
│ │ │ │ +00033e40: 7470 3a2f 2f77 7777 2e6d 6174 682e 6673  tp://www.math.fs
│ │ │ │ +00033e50: 752e 6564 752f 7e61 6c75 6666 692f 4353  u.edu/~aluffi/CS
│ │ │ │ +00033e60: 4d2f 4353 4d2e 6874 6d6c 2029 2c20 6279  M/CSM.html ), by
│ │ │ │ +00033e70: 2050 2e0a 416c 7566 6669 2c20 616e 6420   P..Aluffi, and 
│ │ │ │ +00033e80: 2a6e 6f74 6520 4368 6172 6163 7465 7269  *note Characteri
│ │ │ │ +00033e90: 7374 6963 436c 6173 7365 733a 2028 4368  sticClasses: (Ch
│ │ │ │ +00033ea0: 6172 6163 7465 7269 7374 6963 436c 6173  aracteristicClas
│ │ │ │ +00033eb0: 7365 7329 546f 702c 2c20 6279 204d 2e0a  ses)Top,, by M..
│ │ │ │ +00033ec0: 4865 6c6d 6572 2061 6e64 2043 2e20 4a6f  Helmer and C. Jo
│ │ │ │ +00033ed0: 7374 2e0a 0a49 6e20 7468 6520 6578 616d  st...In the exam
│ │ │ │ +00033ee0: 706c 6520 6265 6c6f 772c 2077 6520 636f  ple below, we co
│ │ │ │ +00033ef0: 6d70 7574 6520 7468 6520 7075 7368 2d66  mpute the push-f
│ │ │ │ +00033f00: 6f72 7761 7264 2074 6f20 7468 6520 4368  orward to the Ch
│ │ │ │ +00033f10: 6f77 2072 696e 6720 6f66 0a24 5c6d 6174  ow ring of.$\mat
│ │ │ │ +00033f20: 6862 627b 507d 5e34 2420 6f66 2074 6865  hbb{P}^4$ of the
│ │ │ │ +00033f30: 2043 6865 726e 2d53 6368 7761 7274 7a2d   Chern-Schwartz-
│ │ │ │ +00033f40: 4d61 6350 6865 7273 6f6e 2063 6c61 7373  MacPherson class
│ │ │ │ +00033f50: 206f 6620 7468 6520 636f 6e65 206f 7665   of the cone ove
│ │ │ │ +00033f60: 7220 7468 650a 7477 6973 7465 6420 6375  r the.twisted cu
│ │ │ │ +00033f70: 6269 6320 6375 7276 652c 2075 7369 6e67  bic curve, using
│ │ │ │ +00033f80: 2062 6f74 6820 6120 7072 6f62 6162 696c   both a probabil
│ │ │ │ +00033f90: 6973 7469 6320 616e 6420 6120 6e6f 6e2d  istic and a non-
│ │ │ │ +00033fa0: 7072 6f62 6162 696c 6973 7469 630a 6170  probabilistic.ap
│ │ │ │ +00033fb0: 7072 6f61 6368 2e0a 0a2b 2d2d 2d2d 2d2d  proach...+------
│ │ │ │  00033fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00033fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00033fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120  ----------+.|i1 
│ │ │ │ -00033ff0: 3a20 4746 2835 5e37 295b 785f 302e 2e78  : GF(5^7)[x_0..x
│ │ │ │ -00034000: 5f34 5d20 2020 2020 2020 2020 2020 2020  _4]             
│ │ │ │ +00033fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00033ff0: 2b0a 7c69 3120 3a20 4746 2835 5e37 295b  +.|i1 : GF(5^7)[
│ │ │ │ +00034000: 785f 302e 2e78 5f34 5d20 2020 2020 2020  x_0..x_4]       
│ │ │ │  00034010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034020: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00034020: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00034030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034050: 2020 2020 2020 2020 2020 2020 7c0a 7c6f              |.|o
│ │ │ │ -00034060: 3120 3d20 4746 2037 3831 3235 5b78 202e  1 = GF 78125[x .
│ │ │ │ -00034070: 2e78 205d 2020 2020 2020 2020 2020 2020  .x ]            
│ │ │ │ +00034050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034060: 2020 7c0a 7c6f 3120 3d20 4746 2037 3831    |.|o1 = GF 781
│ │ │ │ +00034070: 3235 5b78 202e 2e78 205d 2020 2020 2020  25[x ..x ]      
│ │ │ │  00034080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034090: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000340a0: 2020 2020 2020 2030 2020 2034 2020 2020         0   4    
│ │ │ │ -000340b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000340c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000340d0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034090: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000340a0: 2020 2020 2020 2020 2020 2020 2030 2020               0  
│ │ │ │ +000340b0: 2034 2020 2020 2020 2020 2020 2020 2020   4              
│ │ │ │ +000340c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000340d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020      |.|         
│ │ │ │  000340e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000340f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034100: 2020 2020 2020 207c 0a7c 6f31 203a 2050         |.|o1 : P
│ │ │ │ -00034110: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020  olynomialRing   
│ │ │ │ -00034120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034100: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034110: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52  o1 : PolynomialR
│ │ │ │ +00034120: 696e 6720 2020 2020 2020 2020 2020 2020  ing             
│ │ │ │  00034130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034140: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00034140: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00034150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034170: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a  ---------+.|i2 :
│ │ │ │ -00034180: 2043 203d 206d 696e 6f72 7328 322c 6d61   C = minors(2,ma
│ │ │ │ -00034190: 7472 6978 7b7b 785f 302c 785f 312c 785f  trix{{x_0,x_1,x_
│ │ │ │ -000341a0: 327d 2c7b 785f 312c 785f 322c 785f 337d  2},{x_1,x_2,x_3}
│ │ │ │ -000341b0: 7d29 7c0a 7c20 2020 2020 2020 2020 2020  })|.|           
│ │ │ │ +00034170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +00034180: 0a7c 6932 203a 2043 203d 206d 696e 6f72  .|i2 : C = minor
│ │ │ │ +00034190: 7328 322c 6d61 7472 6978 7b7b 785f 302c  s(2,matrix{{x_0,
│ │ │ │ +000341a0: 785f 312c 785f 327d 2c7b 785f 312c 785f  x_1,x_2},{x_1,x_
│ │ │ │ +000341b0: 322c 785f 337d 7d29 7c0a 7c20 2020 2020  2,x_3}})|.|     
│ │ │ │  000341c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000341d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000341e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000341f0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00034200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034210: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ -00034220: 2020 2020 7c0a 7c6f 3220 3d20 6964 6561      |.|o2 = idea
│ │ │ │ -00034230: 6c20 282d 2078 2020 2b20 7820 7820 2c20  l (- x  + x x , 
│ │ │ │ -00034240: 2d20 7820 7820 202b 2078 2078 202c 202d  - x x  + x x , -
│ │ │ │ -00034250: 2078 2020 2b20 7820 7820 2920 207c 0a7c   x  + x x )  |.|
│ │ │ │ -00034260: 2020 2020 2020 2020 2020 2020 2020 2031                 1
│ │ │ │ -00034270: 2020 2020 3020 3220 2020 2020 3120 3220      0 2     1 2 
│ │ │ │ -00034280: 2020 2030 2033 2020 2020 2032 2020 2020     0 3     2    
│ │ │ │ -00034290: 3120 3320 2020 7c0a 7c20 2020 2020 2020  1 3   |.|       
│ │ │ │ +000341e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000341f0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00034200: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00034210: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00034220: 2020 2020 2020 2020 2020 7c0a 7c6f 3220            |.|o2 
│ │ │ │ +00034230: 3d20 6964 6561 6c20 282d 2078 2020 2b20  = ideal (- x  + 
│ │ │ │ +00034240: 7820 7820 2c20 2d20 7820 7820 202b 2078  x x , - x x  + x
│ │ │ │ +00034250: 2078 202c 202d 2078 2020 2b20 7820 7820   x , - x  + x x 
│ │ │ │ +00034260: 2920 207c 0a7c 2020 2020 2020 2020 2020  )  |.|          
│ │ │ │ +00034270: 2020 2020 2031 2020 2020 3020 3220 2020       1    0 2   
│ │ │ │ +00034280: 2020 3120 3220 2020 2030 2033 2020 2020    1 2    0 3    
│ │ │ │ +00034290: 2032 2020 2020 3120 3320 2020 7c0a 7c20   2    1 3   |.| 
│ │ │ │  000342a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000342b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000342c0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -000342d0: 0a7c 6f32 203a 2049 6465 616c 206f 6620  .|o2 : Ideal of 
│ │ │ │ -000342e0: 4746 2037 3831 3235 5b78 202e 2e78 205d  GF 78125[x ..x ]
│ │ │ │ -000342f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034300: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ -00034310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034320: 2020 2030 2020 2034 2020 2020 2020 2020     0   4        
│ │ │ │ +000342c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000342d0: 2020 2020 207c 0a7c 6f32 203a 2049 6465       |.|o2 : Ide
│ │ │ │ +000342e0: 616c 206f 6620 4746 2037 3831 3235 5b78  al of GF 78125[x
│ │ │ │ +000342f0: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │ +00034300: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ +00034310: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │ +00034320: 2020 2020 2020 2020 2030 2020 2034 2020           0   4  
│ │ │ │  00034330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034340: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │ +00034340: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d         |.+------
│ │ │ │  00034350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320  ----------+.|i3 
│ │ │ │ -00034380: 3a20 7469 6d65 2043 6865 726e 5363 6877  : time ChernSchw
│ │ │ │ -00034390: 6172 747a 4d61 6350 6865 7273 6f6e 2043  artzMacPherson C
│ │ │ │ -000343a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000343b0: 2020 207c 0a7c 202d 2d20 7573 6564 2031     |.| -- used 1
│ │ │ │ -000343c0: 2e32 3639 3536 7320 2863 7075 293b 2030  .26956s (cpu); 0
│ │ │ │ -000343d0: 2e38 3934 3033 3173 2028 7468 7265 6164  .894031s (thread
│ │ │ │ -000343e0: 293b 2030 7320 2867 6329 2020 7c0a 7c20  ); 0s (gc)  |.| 
│ │ │ │ -000343f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034380: 2b0a 7c69 3320 3a20 7469 6d65 2043 6865  +.|i3 : time Che
│ │ │ │ +00034390: 726e 5363 6877 6172 747a 4d61 6350 6865  rnSchwartzMacPhe
│ │ │ │ +000343a0: 7273 6f6e 2043 2020 2020 2020 2020 2020  rson C          
│ │ │ │ +000343b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20           |.| -- 
│ │ │ │ +000343c0: 7573 6564 2031 2e35 3532 3634 7320 2863  used 1.55264s (c
│ │ │ │ +000343d0: 7075 293b 2031 2e32 3330 3631 7320 2874  pu); 1.23061s (t
│ │ │ │ +000343e0: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ +000343f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │  00034400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034420: 2020 2020 207c 0a7c 2020 2020 2020 2034       |.|       4
│ │ │ │ -00034430: 2020 2020 2033 2020 2020 2032 2020 2020       3     2    
│ │ │ │ -00034440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034450: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00034460: 7c6f 3320 3d20 3348 2020 2b20 3548 2020  |o3 = 3H  + 5H  
│ │ │ │ -00034470: 2b20 3348 2020 2020 2020 2020 2020 2020  + 3H            
│ │ │ │ +00034420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00034430: 2020 2020 2034 2020 2020 2033 2020 2020       4     3    
│ │ │ │ +00034440: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00034450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034460: 2020 2020 7c0a 7c6f 3320 3d20 3348 2020      |.|o3 = 3H  
│ │ │ │ +00034470: 2b20 3548 2020 2b20 3348 2020 2020 2020  + 5H  + 3H      
│ │ │ │  00034480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034490: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +00034490: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │  000344a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000344b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000344c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000344d0: 7c0a 7c20 2020 2020 5a5a 5b48 5d20 2020  |.|     ZZ[H]   
│ │ │ │ -000344e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000344d0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a        |.|     ZZ
│ │ │ │ +000344e0: 5b48 5d20 2020 2020 2020 2020 2020 2020  [H]             
│ │ │ │  000344f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034500: 2020 2020 2020 2020 207c 0a7c 6f33 203a           |.|o3 :
│ │ │ │ -00034510: 202d 2d2d 2d2d 2020 2020 2020 2020 2020   -----          
│ │ │ │ +00034500: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +00034510: 0a7c 6f33 203a 202d 2d2d 2d2d 2020 2020  .|o3 : -----    
│ │ │ │  00034520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034540: 2020 7c0a 7c20 2020 2020 2020 2035 2020    |.|        5  
│ │ │ │ -00034550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034540: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │ +00034550: 2020 2035 2020 2020 2020 2020 2020 2020     5            
│ │ │ │  00034560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034570: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00034580: 2020 2020 2048 2020 2020 2020 2020 2020       H          
│ │ │ │ +00034570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034580: 207c 0a7c 2020 2020 2020 2048 2020 2020   |.|       H    
│ │ │ │  00034590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000345a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000345b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +000345b0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  000345c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000345d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000345e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -000345f0: 6934 203a 2074 696d 6520 4368 6572 6e53  i4 : time ChernS
│ │ │ │ -00034600: 6368 7761 7274 7a4d 6163 5068 6572 736f  chwartzMacPherso
│ │ │ │ -00034610: 6e28 432c 4365 7274 6966 793d 3e74 7275  n(C,Certify=>tru
│ │ │ │ -00034620: 6529 2020 2020 7c0a 7c43 6572 7469 6679  e)    |.|Certify
│ │ │ │ -00034630: 3a20 6f75 7470 7574 2063 6572 7469 6669  : output certifi
│ │ │ │ -00034640: 6564 2120 2020 2020 2020 2020 2020 2020  ed!             
│ │ │ │ -00034650: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00034660: 0a7c 202d 2d20 7573 6564 2031 2e34 3639  .| -- used 1.469
│ │ │ │ -00034670: 3032 7320 2863 7075 293b 2031 2e30 3033  02s (cpu); 1.003
│ │ │ │ -00034680: 3039 7320 2874 6872 6561 6429 3b20 3073  09s (thread); 0s
│ │ │ │ -00034690: 2028 6763 2920 2020 7c0a 7c20 2020 2020   (gc)   |.|     
│ │ │ │ -000346a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000345e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000345f0: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520  ---+.|i4 : time 
│ │ │ │ +00034600: 4368 6572 6e53 6368 7761 7274 7a4d 6163  ChernSchwartzMac
│ │ │ │ +00034610: 5068 6572 736f 6e28 432c 4365 7274 6966  Pherson(C,Certif
│ │ │ │ +00034620: 793d 3e74 7275 6529 2020 2020 7c0a 7c43  y=>true)    |.|C
│ │ │ │ +00034630: 6572 7469 6679 3a20 6f75 7470 7574 2063  ertify: output c
│ │ │ │ +00034640: 6572 7469 6669 6564 2120 2020 2020 2020  ertified!       
│ │ │ │ +00034650: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034660: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ +00034670: 2031 2e36 3534 3834 7320 2863 7075 293b   1.65484s (cpu);
│ │ │ │ +00034680: 2031 2e31 3635 3831 7320 2874 6872 6561   1.16581s (threa
│ │ │ │ +00034690: 6429 3b20 3073 2028 6763 2920 2020 7c0a  d); 0s (gc)   |.
│ │ │ │ +000346a0: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  000346b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000346c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000346d0: 207c 0a7c 2020 2020 2020 2034 2020 2020   |.|       4    
│ │ │ │ -000346e0: 2033 2020 2020 2032 2020 2020 2020 2020   3     2        
│ │ │ │ +000346d0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ +000346e0: 2034 2020 2020 2033 2020 2020 2032 2020   4     3     2  
│ │ │ │  000346f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034700: 2020 2020 2020 2020 2020 7c0a 7c6f 3420            |.|o4 
│ │ │ │ -00034710: 3d20 3348 2020 2b20 3548 2020 2b20 3348  = 3H  + 5H  + 3H
│ │ │ │ -00034720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034710: 7c0a 7c6f 3420 3d20 3348 2020 2b20 3548  |.|o4 = 3H  + 5H
│ │ │ │ +00034720: 2020 2b20 3348 2020 2020 2020 2020 2020    + 3H          
│ │ │ │  00034730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034740: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00034740: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  00034750: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034770: 2020 2020 2020 2020 2020 2020 7c0a 7c20              |.| 
│ │ │ │ -00034780: 2020 2020 5a5a 5b48 5d20 2020 2020 2020      ZZ[H]       
│ │ │ │ +00034770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034780: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20    |.|     ZZ[H] 
│ │ │ │  00034790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000347a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000347b0: 2020 2020 207c 0a7c 6f34 203a 202d 2d2d       |.|o4 : ---
│ │ │ │ -000347c0: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +000347b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ +000347c0: 203a 202d 2d2d 2d2d 2020 2020 2020 2020   : -----        
│ │ │ │  000347d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000347e0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000347f0: 7c20 2020 2020 2020 2035 2020 2020 2020  |        5      
│ │ │ │ +000347e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000347f0: 2020 2020 7c0a 7c20 2020 2020 2020 2035      |.|        5
│ │ │ │  00034800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034820: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -00034830: 2048 2020 2020 2020 2020 2020 2020 2020   H              
│ │ │ │ +00034820: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +00034830: 2020 2020 2020 2048 2020 2020 2020 2020         H        
│ │ │ │  00034840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034860: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  |.+-------------
│ │ │ │ +00034860: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d        |.+-------
│ │ │ │  00034870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034890: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a  ---------+.|i5 :
│ │ │ │ -000348a0: 206f 6f20 3d3d 206f 6f6f 2020 2020 2020   oo == ooo      
│ │ │ │ +00034890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b  ---------------+
│ │ │ │ +000348a0: 0a7c 6935 203a 206f 6f20 3d3d 206f 6f6f  .|i5 : oo == ooo
│ │ │ │  000348b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000348c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000348d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020    |.|           
│ │ │ │ +000348d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020          |.|     
│ │ │ │  000348e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000348f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034900: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -00034910: 203d 2074 7275 6520 2020 2020 2020 2020   = true         
│ │ │ │ +00034900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034910: 207c 0a7c 6f35 203d 2074 7275 6520 2020   |.|o5 = true   
│ │ │ │  00034920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d      |.+---------
│ │ │ │ +00034940: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d            |.+---
│ │ │ │  00034950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00034980: 496e 2074 6865 2063 6173 6520 7768 656e  In the case when
│ │ │ │ -00034990: 2074 6865 2069 6e70 7574 2069 6465 616c   the input ideal
│ │ │ │ -000349a0: 2049 2064 6566 696e 6573 2061 2073 6d6f   I defines a smo
│ │ │ │ -000349b0: 6f74 6820 7072 6f6a 6563 7469 7665 2076  oth projective v
│ │ │ │ -000349c0: 6172 6965 7479 2024 5824 2c20 7468 650a  ariety $X$, the.
│ │ │ │ -000349d0: 7075 7368 2d66 6f72 7761 7264 206f 6620  push-forward of 
│ │ │ │ -000349e0: 2463 5f7b 534d 7d28 5829 2420 6361 6e20  $c_{SM}(X)$ can 
│ │ │ │ -000349f0: 6265 2063 6f6d 7075 7465 6420 6d75 6368  be computed much
│ │ │ │ -00034a00: 206d 6f72 6520 6566 6669 6369 656e 746c   more efficientl
│ │ │ │ -00034a10: 7920 7573 696e 6720 2a6e 6f74 650a 5365  y using *note.Se
│ │ │ │ -00034a20: 6772 6543 6c61 7373 3a20 5365 6772 6543  greClass: SegreC
│ │ │ │ -00034a30: 6c61 7373 2c2e 2049 6e64 6565 642c 2069  lass,. Indeed, i
│ │ │ │ -00034a40: 6e20 7468 6973 2063 6173 652c 2024 635f  n this case, $c_
│ │ │ │ -00034a50: 7b53 4d7d 2858 2924 2063 6f69 6e63 6964  {SM}(X)$ coincid
│ │ │ │ -00034a60: 6573 2077 6974 6820 7468 650a 2874 6f74  es with the.(tot
│ │ │ │ -00034a70: 616c 2920 4368 6572 6e20 636c 6173 7320  al) Chern class 
│ │ │ │ -00034a80: 6f66 2074 6865 2074 616e 6765 6e74 2062  of the tangent b
│ │ │ │ -00034a90: 756e 646c 6520 6f66 2024 5824 2061 6e64  undle of $X$ and
│ │ │ │ -00034aa0: 2063 616e 2062 6520 6f62 7461 696e 6564   can be obtained
│ │ │ │ -00034ab0: 2061 7320 666f 6c6c 6f77 730a 2869 6e20   as follows.(in 
│ │ │ │ -00034ac0: 6765 6e65 7261 6c20 7468 6520 6d65 7468  general the meth
│ │ │ │ -00034ad0: 6f64 2062 656c 6f77 2067 6976 6573 2074  od below gives t
│ │ │ │ -00034ae0: 6865 2070 7573 682d 666f 7277 6172 6420  he push-forward 
│ │ │ │ -00034af0: 6f66 2074 6865 2073 6f2d 6361 6c6c 6564  of the so-called
│ │ │ │ -00034b00: 0a43 6865 726e 2d46 756c 746f 6e20 636c  .Chern-Fulton cl
│ │ │ │ -00034b10: 6173 7329 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d  ass)...+--------
│ │ │ │ +00034970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034980: 2d2d 2d2b 0a0a 496e 2074 6865 2063 6173  ---+..In the cas
│ │ │ │ +00034990: 6520 7768 656e 2074 6865 2069 6e70 7574  e when the input
│ │ │ │ +000349a0: 2069 6465 616c 2049 2064 6566 696e 6573   ideal I defines
│ │ │ │ +000349b0: 2061 2073 6d6f 6f74 6820 7072 6f6a 6563   a smooth projec
│ │ │ │ +000349c0: 7469 7665 2076 6172 6965 7479 2024 5824  tive variety $X$
│ │ │ │ +000349d0: 2c20 7468 650a 7075 7368 2d66 6f72 7761  , the.push-forwa
│ │ │ │ +000349e0: 7264 206f 6620 2463 5f7b 534d 7d28 5829  rd of $c_{SM}(X)
│ │ │ │ +000349f0: 2420 6361 6e20 6265 2063 6f6d 7075 7465  $ can be compute
│ │ │ │ +00034a00: 6420 6d75 6368 206d 6f72 6520 6566 6669  d much more effi
│ │ │ │ +00034a10: 6369 656e 746c 7920 7573 696e 6720 2a6e  ciently using *n
│ │ │ │ +00034a20: 6f74 650a 5365 6772 6543 6c61 7373 3a20  ote.SegreClass: 
│ │ │ │ +00034a30: 5365 6772 6543 6c61 7373 2c2e 2049 6e64  SegreClass,. Ind
│ │ │ │ +00034a40: 6565 642c 2069 6e20 7468 6973 2063 6173  eed, in this cas
│ │ │ │ +00034a50: 652c 2024 635f 7b53 4d7d 2858 2924 2063  e, $c_{SM}(X)$ c
│ │ │ │ +00034a60: 6f69 6e63 6964 6573 2077 6974 6820 7468  oincides with th
│ │ │ │ +00034a70: 650a 2874 6f74 616c 2920 4368 6572 6e20  e.(total) Chern 
│ │ │ │ +00034a80: 636c 6173 7320 6f66 2074 6865 2074 616e  class of the tan
│ │ │ │ +00034a90: 6765 6e74 2062 756e 646c 6520 6f66 2024  gent bundle of $
│ │ │ │ +00034aa0: 5824 2061 6e64 2063 616e 2062 6520 6f62  X$ and can be ob
│ │ │ │ +00034ab0: 7461 696e 6564 2061 7320 666f 6c6c 6f77  tained as follow
│ │ │ │ +00034ac0: 730a 2869 6e20 6765 6e65 7261 6c20 7468  s.(in general th
│ │ │ │ +00034ad0: 6520 6d65 7468 6f64 2062 656c 6f77 2067  e method below g
│ │ │ │ +00034ae0: 6976 6573 2074 6865 2070 7573 682d 666f  ives the push-fo
│ │ │ │ +00034af0: 7277 6172 6420 6f66 2074 6865 2073 6f2d  rward of the so-
│ │ │ │ +00034b00: 6361 6c6c 6564 0a43 6865 726e 2d46 756c  called.Chern-Ful
│ │ │ │ +00034b10: 746f 6e20 636c 6173 7329 2e0a 0a2b 2d2d  ton class)...+--
│ │ │ │  00034b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034b60: 2d2d 2d2d 2d2b 0a7c 6936 203a 2043 6865  -----+.|i6 : Che
│ │ │ │ -00034b70: 726e 436c 6173 7320 3d20 6d65 7468 6f64  rnClass = method
│ │ │ │ -00034b80: 284f 7074 696f 6e73 3d3e 7b43 6572 7469  (Options=>{Certi
│ │ │ │ -00034b90: 6679 3d3e 6661 6c73 657d 293b 2020 2020  fy=>false});    
│ │ │ │ -00034ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034bb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00034b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ +00034b70: 203a 2043 6865 726e 436c 6173 7320 3d20   : ChernClass = 
│ │ │ │ +00034b80: 6d65 7468 6f64 284f 7074 696f 6e73 3d3e  method(Options=>
│ │ │ │ +00034b90: 7b43 6572 7469 6679 3d3e 6661 6c73 657d  {Certify=>false}
│ │ │ │ +00034ba0: 293b 2020 2020 2020 2020 2020 2020 2020  );              
│ │ │ │ +00034bb0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00034bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034c00: 2d2d 2d2d 2d2b 0a7c 6937 203a 2043 6865  -----+.|i7 : Che
│ │ │ │ -00034c10: 726e 436c 6173 7320 2849 6465 616c 2920  rnClass (Ideal) 
│ │ │ │ -00034c20: 3a3d 206f 202d 3e20 2849 2920 2d3e 2028  := o -> (I) -> (
│ │ │ │ -00034c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937  -----------+.|i7
│ │ │ │ +00034c10: 203a 2043 6865 726e 436c 6173 7320 2849   : ChernClass (I
│ │ │ │ +00034c20: 6465 616c 2920 3a3d 206f 202d 3e20 2849  deal) := o -> (I
│ │ │ │ +00034c30: 2920 2d3e 2028 2020 2020 2020 2020 2020  ) -> (          
│ │ │ │  00034c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034c50: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00034c60: 7320 3a3d 2053 6567 7265 436c 6173 7328  s := SegreClass(
│ │ │ │ -00034c70: 492c 4365 7274 6966 793d 3e6f 2e43 6572  I,Certify=>o.Cer
│ │ │ │ -00034c80: 7469 6679 293b 2020 2020 2020 2020 2020  tify);          
│ │ │ │ +00034c50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00034c60: 2020 2020 2020 7320 3a3d 2053 6567 7265        s := Segre
│ │ │ │ +00034c70: 436c 6173 7328 492c 4365 7274 6966 793d  Class(I,Certify=
│ │ │ │ +00034c80: 3e6f 2e43 6572 7469 6679 293b 2020 2020  >o.Certify);    
│ │ │ │  00034c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034ca0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00034cb0: 732a 2831 2b66 6972 7374 2067 656e 7320  s*(1+first gens 
│ │ │ │ -00034cc0: 7269 6e67 2073 295e 286e 756d 6765 6e73  ring s)^(numgens
│ │ │ │ -00034cd0: 2072 696e 6720 4929 293b 2020 2020 2020   ring I));      
│ │ │ │ +00034ca0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00034cb0: 2020 2020 2020 732a 2831 2b66 6972 7374        s*(1+first
│ │ │ │ +00034cc0: 2067 656e 7320 7269 6e67 2073 295e 286e   gens ring s)^(n
│ │ │ │ +00034cd0: 756d 6765 6e73 2072 696e 6720 4929 293b  umgens ring I));
│ │ │ │  00034ce0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034cf0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00034cf0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00034d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034d40: 2d2d 2d2d 2d2b 0a7c 6938 203a 202d 2d20  -----+.|i8 : -- 
│ │ │ │ -00034d50: 6578 616d 706c 653a 2043 6865 726e 2063  example: Chern c
│ │ │ │ -00034d60: 6c61 7373 206f 6620 4728 312c 3429 2020  lass of G(1,4)  
│ │ │ │ -00034d70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00034d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938  -----------+.|i8
│ │ │ │ +00034d50: 203a 202d 2d20 6578 616d 706c 653a 2043   : -- example: C
│ │ │ │ +00034d60: 6865 726e 2063 6c61 7373 206f 6620 4728  hern class of G(
│ │ │ │ +00034d70: 312c 3429 2020 2020 2020 2020 2020 2020  1,4)            
│ │ │ │  00034d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034d90: 2020 2020 207c 0a7c 2020 2020 2047 203d       |.|     G =
│ │ │ │ -00034da0: 2047 7261 7373 6d61 6e6e 6961 6e28 312c   Grassmannian(1,
│ │ │ │ -00034db0: 342c 436f 6566 6669 6369 656e 7452 696e  4,CoefficientRin
│ │ │ │ -00034dc0: 673d 3e5a 5a2f 3139 3031 3831 2920 2020  g=>ZZ/190181)   
│ │ │ │ -00034dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034de0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00034d90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00034da0: 2020 2047 203d 2047 7261 7373 6d61 6e6e     G = Grassmann
│ │ │ │ +00034db0: 6961 6e28 312c 342c 436f 6566 6669 6369  ian(1,4,Coeffici
│ │ │ │ +00034dc0: 656e 7452 696e 673d 3e5a 5a2f 3139 3031  entRing=>ZZ/1901
│ │ │ │ +00034dd0: 3831 2920 2020 2020 2020 2020 2020 2020  81)             
│ │ │ │ +00034de0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00034df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00034e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00034e30: 2020 2020 207c 0a7c 6f38 203d 2069 6465       |.|o8 = ide
│ │ │ │ -00034e40: 616c 2028 7020 2020 7020 2020 202d 2070  al (p   p    - p
│ │ │ │ -00034e50: 2020 2070 2020 2020 2b20 7020 2020 7020     p    + p   p 
│ │ │ │ -00034e60: 2020 2c20 7020 2020 7020 2020 202d 2070    , p   p    - p
│ │ │ │ -00034e70: 2020 2070 2020 2020 2b20 7020 2020 7020     p    + p   p 
│ │ │ │ -00034e80: 2020 2c20 207c 0a7c 2020 2020 2020 2020    ,  |.|        
│ │ │ │ -00034e90: 2020 2020 2032 2c33 2031 2c34 2020 2020       2,3 1,4    
│ │ │ │ -00034ea0: 312c 3320 322c 3420 2020 2031 2c32 2033  1,3 2,4    1,2 3
│ │ │ │ -00034eb0: 2c34 2020 2032 2c33 2030 2c34 2020 2020  ,4   2,3 0,4    
│ │ │ │ -00034ec0: 302c 3320 322c 3420 2020 2030 2c32 2033  0,3 2,4    0,2 3
│ │ │ │ -00034ed0: 2c34 2020 207c 0a7c 2020 2020 202d 2d2d  ,4   |.|     ---
│ │ │ │ -00034ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034e30: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +00034e40: 203d 2069 6465 616c 2028 7020 2020 7020   = ideal (p   p 
│ │ │ │ +00034e50: 2020 202d 2070 2020 2070 2020 2020 2b20     - p   p    + 
│ │ │ │ +00034e60: 7020 2020 7020 2020 2c20 7020 2020 7020  p   p   , p   p 
│ │ │ │ +00034e70: 2020 202d 2070 2020 2070 2020 2020 2b20     - p   p    + 
│ │ │ │ +00034e80: 7020 2020 7020 2020 2c20 207c 0a7c 2020  p   p   ,  |.|  
│ │ │ │ +00034e90: 2020 2020 2020 2020 2020 2032 2c33 2031             2,3 1
│ │ │ │ +00034ea0: 2c34 2020 2020 312c 3320 322c 3420 2020  ,4    1,3 2,4   
│ │ │ │ +00034eb0: 2031 2c32 2033 2c34 2020 2032 2c33 2030   1,2 3,4   2,3 0
│ │ │ │ +00034ec0: 2c34 2020 2020 302c 3320 322c 3420 2020  ,4    0,3 2,4   
│ │ │ │ +00034ed0: 2030 2c32 2033 2c34 2020 207c 0a7c 2020   0,2 3,4   |.|  
│ │ │ │ +00034ee0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  00034ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00034f20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2070 2020  -----|.|     p  
│ │ │ │ -00034f30: 2070 2020 2020 2d20 7020 2020 7020 2020   p    - p   p   
│ │ │ │ -00034f40: 202b 2070 2020 2070 2020 202c 2070 2020   + p   p   , p  
│ │ │ │ -00034f50: 2070 2020 2020 2d20 7020 2020 7020 2020   p    - p   p   
│ │ │ │ -00034f60: 202b 2070 2020 2070 2020 202c 2070 2020   + p   p   , p  
│ │ │ │ -00034f70: 2070 2020 207c 0a7c 2020 2020 2020 312c   p   |.|      1,
│ │ │ │ -00034f80: 3320 302c 3420 2020 2030 2c33 2031 2c34  3 0,4    0,3 1,4
│ │ │ │ -00034f90: 2020 2020 302c 3120 332c 3420 2020 312c      0,1 3,4   1,
│ │ │ │ -00034fa0: 3220 302c 3420 2020 2030 2c32 2031 2c34  2 0,4    0,2 1,4
│ │ │ │ -00034fb0: 2020 2020 302c 3120 322c 3420 2020 312c      0,1 2,4   1,
│ │ │ │ -00034fc0: 3220 302c 337c 0a7c 2020 2020 202d 2d2d  2 0,3|.|     ---
│ │ │ │ -00034fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00034f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00034f30: 2020 2070 2020 2070 2020 2020 2d20 7020     p   p    - p 
│ │ │ │ +00034f40: 2020 7020 2020 202b 2070 2020 2070 2020    p    + p   p  
│ │ │ │ +00034f50: 202c 2070 2020 2070 2020 2020 2d20 7020   , p   p    - p 
│ │ │ │ +00034f60: 2020 7020 2020 202b 2070 2020 2070 2020    p    + p   p  
│ │ │ │ +00034f70: 202c 2070 2020 2070 2020 207c 0a7c 2020   , p   p   |.|  
│ │ │ │ +00034f80: 2020 2020 312c 3320 302c 3420 2020 2030      1,3 0,4    0
│ │ │ │ +00034f90: 2c33 2031 2c34 2020 2020 302c 3120 332c  ,3 1,4    0,1 3,
│ │ │ │ +00034fa0: 3420 2020 312c 3220 302c 3420 2020 2030  4   1,2 0,4    0
│ │ │ │ +00034fb0: 2c32 2031 2c34 2020 2020 302c 3120 322c  ,2 1,4    0,1 2,
│ │ │ │ +00034fc0: 3420 2020 312c 3220 302c 337c 0a7c 2020  4   1,2 0,3|.|  
│ │ │ │ +00034fd0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d     -------------
│ │ │ │  00034fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00034ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035010: 2d2d 2d2d 2d7c 0a7c 2020 2020 202d 2070  -----|.|     - p
│ │ │ │ -00035020: 2020 2070 2020 2020 2b20 7020 2020 7020     p    + p   p 
│ │ │ │ -00035030: 2020 2920 2020 2020 2020 2020 2020 2020    )             
│ │ │ │ +00035010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00035020: 2020 202d 2070 2020 2070 2020 2020 2b20     - p   p    + 
│ │ │ │ +00035030: 7020 2020 7020 2020 2920 2020 2020 2020  p   p   )       
│ │ │ │  00035040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035060: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00035070: 302c 3220 312c 3320 2020 2030 2c31 2032  0,2 1,3    0,1 2
│ │ │ │ -00035080: 2c33 2020 2020 2020 2020 2020 2020 2020  ,3              
│ │ │ │ +00035060: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00035070: 2020 2020 2020 302c 3220 312c 3320 2020        0,2 1,3   
│ │ │ │ +00035080: 2030 2c31 2032 2c33 2020 2020 2020 2020   0,1 2,3        
│ │ │ │  00035090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000350b0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000350b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000350c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000350f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035100: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00035110: 2020 2020 2020 2020 5a5a 2020 2020 2020          ZZ      
│ │ │ │ +00035100: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00035110: 2020 2020 2020 2020 2020 2020 2020 5a5a                ZZ
│ │ │ │  00035120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035150: 2020 2020 207c 0a7c 6f38 203a 2049 6465       |.|o8 : Ide
│ │ │ │ -00035160: 616c 206f 6620 2d2d 2d2d 2d2d 5b70 2020  al of ------[p  
│ │ │ │ -00035170: 202e 2e70 2020 202c 2070 2020 202c 2070   ..p   , p   , p
│ │ │ │ +00035150: 2020 2020 2020 2020 2020 207c 0a7c 6f38             |.|o8
│ │ │ │ +00035160: 203a 2049 6465 616c 206f 6620 2d2d 2d2d   : Ideal of ----
│ │ │ │ +00035170: 2d2d 5b70 2020 202e 2e70 2020 202c 2070  --[p   ..p   , p
│ │ │ │  00035180: 2020 202c 2070 2020 202c 2070 2020 202c     , p   , p   ,
│ │ │ │  00035190: 2070 2020 202c 2070 2020 202c 2070 2020   p   , p   , p  
│ │ │ │ -000351a0: 202c 2070 207c 0a7c 2020 2020 2020 2020   , p |.|        
│ │ │ │ -000351b0: 2020 2020 2020 3139 3031 3831 2020 302c        190181  0,
│ │ │ │ -000351c0: 3120 2020 302c 3220 2020 312c 3220 2020  1   0,2   1,2   
│ │ │ │ -000351d0: 302c 3320 2020 312c 3320 2020 322c 3320  0,3   1,3   2,3 
│ │ │ │ -000351e0: 2020 302c 3420 2020 312c 3420 2020 322c    0,4   1,4   2,
│ │ │ │ -000351f0: 3420 2020 337c 0a7c 2d2d 2d2d 2d2d 2d2d  4   3|.|--------
│ │ │ │ +000351a0: 202c 2070 2020 202c 2070 207c 0a7c 2020   , p   , p |.|  
│ │ │ │ +000351b0: 2020 2020 2020 2020 2020 2020 3139 3031              1901
│ │ │ │ +000351c0: 3831 2020 302c 3120 2020 302c 3220 2020  81  0,1   0,2   
│ │ │ │ +000351d0: 312c 3220 2020 302c 3320 2020 312c 3320  1,2   0,3   1,3 
│ │ │ │ +000351e0: 2020 322c 3320 2020 302c 3420 2020 312c    2,3   0,4   1,
│ │ │ │ +000351f0: 3420 2020 322c 3420 2020 337c 0a7c 2d2d  4   2,4   3|.|--
│ │ │ │  00035200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035240: 2d2d 2d2d 2d7c 0a7c 2020 5d20 2020 2020  -----|.|  ]     
│ │ │ │ -00035250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ +00035250: 5d20 2020 2020 2020 2020 2020 2020 2020  ]               
│ │ │ │  00035260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035290: 2020 2020 207c 0a7c 2c34 2020 2020 2020       |.|,4      
│ │ │ │ +00035290: 2020 2020 2020 2020 2020 207c 0a7c 2c34             |.|,4
│ │ │ │  000352a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000352b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000352c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000352d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000352e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +000352e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  000352f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035330: 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 696d  -----+.|i9 : tim
│ │ │ │ -00035340: 6520 4368 6572 6e43 6c61 7373 2047 2020  e ChernClass G  
│ │ │ │ -00035350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939  -----------+.|i9
│ │ │ │ +00035340: 203a 2074 696d 6520 4368 6572 6e43 6c61   : time ChernCla
│ │ │ │ +00035350: 7373 2047 2020 2020 2020 2020 2020 2020  ss G            
│ │ │ │  00035360: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035380: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00035390: 2030 2e31 3130 3239 3873 2028 6370 7529   0.110298s (cpu)
│ │ │ │ -000353a0: 3b20 302e 3131 3033 3035 7320 2874 6872  ; 0.110305s (thr
│ │ │ │ -000353b0: 6561 6429 3b20 3073 2028 6763 2920 2020  ead); 0s (gc)   
│ │ │ │ -000353c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000353d0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00035380: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00035390: 2d20 7573 6564 2030 2e31 3532 3230 3773  - used 0.152207s
│ │ │ │ +000353a0: 2028 6370 7529 3b20 302e 3135 3231 3737   (cpu); 0.152177
│ │ │ │ +000353b0: 7320 2874 6872 6561 6429 3b20 3073 2028  s (thread); 0s (
│ │ │ │ +000353c0: 6763 2920 2020 2020 2020 2020 2020 2020  gc)             
│ │ │ │ +000353d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000353e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000353f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035420: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00035430: 3920 2020 2020 2038 2020 2020 2020 3720  9      8      7 
│ │ │ │ -00035440: 2020 2020 2036 2020 2020 2020 3520 2020       6      5   
│ │ │ │ -00035450: 2020 2034 2020 2020 2033 2020 2020 2020     4     3      
│ │ │ │ +00035420: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00035430: 2020 2020 2020 3920 2020 2020 2038 2020        9      8  
│ │ │ │ +00035440: 2020 2020 3720 2020 2020 2036 2020 2020      7      6    
│ │ │ │ +00035450: 2020 3520 2020 2020 2034 2020 2020 2033    5      4     3
│ │ │ │  00035460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035470: 2020 2020 207c 0a7c 6f39 203d 2031 3048       |.|o9 = 10H
│ │ │ │ -00035480: 2020 2b20 3330 4820 202b 2036 3048 2020    + 30H  + 60H  
│ │ │ │ -00035490: 2b20 3735 4820 202b 2035 3748 2020 2b20  + 75H  + 57H  + 
│ │ │ │ -000354a0: 3235 4820 202b 2035 4820 2020 2020 2020  25H  + 5H       
│ │ │ │ +00035470: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ +00035480: 203d 2031 3048 2020 2b20 3330 4820 202b   = 10H  + 30H  +
│ │ │ │ +00035490: 2036 3048 2020 2b20 3735 4820 202b 2035   60H  + 75H  + 5
│ │ │ │ +000354a0: 3748 2020 2b20 3235 4820 202b 2035 4820  7H  + 25H  + 5H 
│ │ │ │  000354b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000354c0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +000354c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000354d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000354f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035510: 2020 2020 207c 0a7c 2020 2020 205a 5a5b       |.|     ZZ[
│ │ │ │ -00035520: 485d 2020 2020 2020 2020 2020 2020 2020  H]              
│ │ │ │ +00035510: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00035520: 2020 205a 5a5b 485d 2020 2020 2020 2020     ZZ[H]        
│ │ │ │  00035530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035560: 2020 2020 207c 0a7c 6f39 203a 202d 2d2d       |.|o9 : ---
│ │ │ │ -00035570: 2d2d 2020 2020 2020 2020 2020 2020 2020  --              
│ │ │ │ +00035560: 2020 2020 2020 2020 2020 207c 0a7c 6f39             |.|o9
│ │ │ │ +00035570: 203a 202d 2d2d 2d2d 2020 2020 2020 2020   : -----        
│ │ │ │  00035580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000355b0: 2020 2020 207c 0a7c 2020 2020 2020 2031       |.|       1
│ │ │ │ -000355c0: 3020 2020 2020 2020 2020 2020 2020 2020  0               
│ │ │ │ +000355b0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000355c0: 2020 2020 2031 3020 2020 2020 2020 2020       10         
│ │ │ │  000355d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000355f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035600: 2020 2020 207c 0a7c 2020 2020 2020 4820       |.|      H 
│ │ │ │ -00035610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00035600: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00035610: 2020 2020 4820 2020 2020 2020 2020 2020      H           
│ │ │ │  00035620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035650: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00035650: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00035660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000356a0: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 7469  -----+.|i10 : ti
│ │ │ │ -000356b0: 6d65 2043 6865 726e 436c 6173 7328 472c  me ChernClass(G,
│ │ │ │ -000356c0: 4365 7274 6966 793d 3e74 7275 6529 2020  Certify=>true)  
│ │ │ │ -000356d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000356a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ +000356b0: 3020 3a20 7469 6d65 2043 6865 726e 436c  0 : time ChernCl
│ │ │ │ +000356c0: 6173 7328 472c 4365 7274 6966 793d 3e74  ass(G,Certify=>t
│ │ │ │ +000356d0: 7275 6529 2020 2020 2020 2020 2020 2020  rue)            
│ │ │ │  000356e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000356f0: 2020 2020 207c 0a7c 4365 7274 6966 793a       |.|Certify:
│ │ │ │ -00035700: 206f 7574 7075 7420 6365 7274 6966 6965   output certifie
│ │ │ │ -00035710: 6421 2020 2020 2020 2020 2020 2020 2020  d!              
│ │ │ │ +000356f0: 2020 2020 2020 2020 2020 207c 0a7c 4365             |.|Ce
│ │ │ │ +00035700: 7274 6966 793a 206f 7574 7075 7420 6365  rtify: output ce
│ │ │ │ +00035710: 7274 6966 6965 6421 2020 2020 2020 2020  rtified!        
│ │ │ │  00035720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035740: 2020 2020 207c 0a7c 202d 2d20 7573 6564       |.| -- used
│ │ │ │ -00035750: 2030 2e30 3131 3333 3539 7320 2863 7075   0.0113359s (cpu
│ │ │ │ -00035760: 293b 2030 2e30 3130 3732 3335 7320 2874  ); 0.0107235s (t
│ │ │ │ -00035770: 6872 6561 6429 3b20 3073 2028 6763 2920  hread); 0s (gc) 
│ │ │ │ -00035780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035790: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00035740: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ +00035750: 2d20 7573 6564 2030 2e30 3432 3131 3138  - used 0.0421118
│ │ │ │ +00035760: 7320 2863 7075 293b 2030 2e30 3138 3135  s (cpu); 0.01815
│ │ │ │ +00035770: 3533 7320 2874 6872 6561 6429 3b20 3073  53s (thread); 0s
│ │ │ │ +00035780: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ +00035790: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  000357a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000357b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000357c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000357d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000357e0: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -000357f0: 2039 2020 2020 2020 3820 2020 2020 2037   9      8      7
│ │ │ │ -00035800: 2020 2020 2020 3620 2020 2020 2035 2020        6      5  
│ │ │ │ -00035810: 2020 2020 3420 2020 2020 3320 2020 2020      4     3     
│ │ │ │ -00035820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035830: 2020 2020 207c 0a7c 6f31 3020 3d20 3130       |.|o10 = 10
│ │ │ │ -00035840: 4820 202b 2033 3048 2020 2b20 3630 4820  H  + 30H  + 60H 
│ │ │ │ -00035850: 202b 2037 3548 2020 2b20 3537 4820 202b   + 75H  + 57H  +
│ │ │ │ -00035860: 2032 3548 2020 2b20 3548 2020 2020 2020   25H  + 5H      
│ │ │ │ +000357e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000357f0: 2020 2020 2020 2039 2020 2020 2020 3820         9      8 
│ │ │ │ +00035800: 2020 2020 2037 2020 2020 2020 3620 2020       7      6   
│ │ │ │ +00035810: 2020 2035 2020 2020 2020 3420 2020 2020     5      4     
│ │ │ │ +00035820: 3320 2020 2020 2020 2020 2020 2020 2020  3               
│ │ │ │ +00035830: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00035840: 3020 3d20 3130 4820 202b 2033 3048 2020  0 = 10H  + 30H  
│ │ │ │ +00035850: 2b20 3630 4820 202b 2037 3548 2020 2b20  + 60H  + 75H  + 
│ │ │ │ +00035860: 3537 4820 202b 2032 3548 2020 2b20 3548  57H  + 25H  + 5H
│ │ │ │  00035870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035880: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ +00035880: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │  00035890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000358a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000358b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000358c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000358d0: 2020 2020 207c 0a7c 2020 2020 2020 5a5a       |.|      ZZ
│ │ │ │ -000358e0: 5b48 5d20 2020 2020 2020 2020 2020 2020  [H]             
│ │ │ │ +000358d0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000358e0: 2020 2020 5a5a 5b48 5d20 2020 2020 2020      ZZ[H]       
│ │ │ │  000358f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035920: 2020 2020 207c 0a7c 6f31 3020 3a20 2d2d       |.|o10 : --
│ │ │ │ -00035930: 2d2d 2d20 2020 2020 2020 2020 2020 2020  ---             
│ │ │ │ +00035920: 2020 2020 2020 2020 2020 207c 0a7c 6f31             |.|o1
│ │ │ │ +00035930: 3020 3a20 2d2d 2d2d 2d20 2020 2020 2020  0 : -----       
│ │ │ │  00035940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035970: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │ -00035980: 3130 2020 2020 2020 2020 2020 2020 2020  10              
│ │ │ │ +00035970: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +00035980: 2020 2020 2020 3130 2020 2020 2020 2020        10        
│ │ │ │  00035990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000359a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000359b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000359c0: 2020 2020 207c 0a7c 2020 2020 2020 2048       |.|       H
│ │ │ │ -000359d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000359c0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ +000359d0: 2020 2020 2048 2020 2020 2020 2020 2020       H          
│ │ │ │  000359e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000359f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00035a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00035a10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00035a10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00035a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035a60: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f  -----+..See also
│ │ │ │ -00035a70: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a  .========..  * *
│ │ │ │ -00035a80: 6e6f 7465 2053 6567 7265 436c 6173 733a  note SegreClass:
│ │ │ │ -00035a90: 2053 6567 7265 436c 6173 732c 202d 2d20   SegreClass, -- 
│ │ │ │ -00035aa0: 5365 6772 6520 636c 6173 7320 6f66 2061  Segre class of a
│ │ │ │ -00035ab0: 2063 6c6f 7365 6420 7375 6273 6368 656d   closed subschem
│ │ │ │ -00035ac0: 6520 6f66 2061 0a20 2020 2070 726f 6a65  e of a.    proje
│ │ │ │ -00035ad0: 6374 6976 6520 7661 7269 6574 790a 2020  ctive variety.  
│ │ │ │ -00035ae0: 2a20 2a6e 6f74 6520 4575 6c65 7243 6861  * *note EulerCha
│ │ │ │ -00035af0: 7261 6374 6572 6973 7469 633a 2045 756c  racteristic: Eul
│ │ │ │ -00035b00: 6572 4368 6172 6163 7465 7269 7374 6963  erCharacteristic
│ │ │ │ -00035b10: 2c20 2d2d 2074 6f70 6f6c 6f67 6963 616c  , -- topological
│ │ │ │ -00035b20: 2045 756c 6572 0a20 2020 2063 6861 7261   Euler.    chara
│ │ │ │ -00035b30: 6374 6572 6973 7469 6320 6f66 2061 2028  cteristic of a (
│ │ │ │ -00035b40: 736d 6f6f 7468 2920 7072 6f6a 6563 7469  smooth) projecti
│ │ │ │ -00035b50: 7665 2076 6172 6965 7479 0a20 202a 202a  ve variety.  * *
│ │ │ │ -00035b60: 6e6f 7465 2065 756c 6572 2850 726f 6a65  note euler(Proje
│ │ │ │ -00035b70: 6374 6976 6556 6172 6965 7479 293a 2028  ctiveVariety): (
│ │ │ │ -00035b80: 5661 7269 6574 6965 7329 6575 6c65 725f  Varieties)euler_
│ │ │ │ -00035b90: 6c70 5072 6f6a 6563 7469 7665 5661 7269  lpProjectiveVari
│ │ │ │ -00035ba0: 6574 795f 7270 2c20 2d2d 0a20 2020 2074  ety_rp, --.    t
│ │ │ │ -00035bb0: 6f70 6f6c 6f67 6963 616c 2045 756c 6572  opological Euler
│ │ │ │ -00035bc0: 2063 6861 7261 6374 6572 6973 7469 6320   characteristic 
│ │ │ │ -00035bd0: 6f66 2061 2028 736d 6f6f 7468 2920 7072  of a (smooth) pr
│ │ │ │ -00035be0: 6f6a 6563 7469 7665 2076 6172 6965 7479  ojective variety
│ │ │ │ -00035bf0: 0a0a 5761 7973 2074 6f20 7573 6520 4368  ..Ways to use Ch
│ │ │ │ -00035c00: 6572 6e53 6368 7761 7274 7a4d 6163 5068  ernSchwartzMacPh
│ │ │ │ -00035c10: 6572 736f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d  erson:.=========
│ │ │ │ +00035a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365  -----------+..Se
│ │ │ │ +00035a70: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a  e also.========.
│ │ │ │ +00035a80: 0a20 202a 202a 6e6f 7465 2053 6567 7265  .  * *note Segre
│ │ │ │ +00035a90: 436c 6173 733a 2053 6567 7265 436c 6173  Class: SegreClas
│ │ │ │ +00035aa0: 732c 202d 2d20 5365 6772 6520 636c 6173  s, -- Segre clas
│ │ │ │ +00035ab0: 7320 6f66 2061 2063 6c6f 7365 6420 7375  s of a closed su
│ │ │ │ +00035ac0: 6273 6368 656d 6520 6f66 2061 0a20 2020  bscheme of a.   
│ │ │ │ +00035ad0: 2070 726f 6a65 6374 6976 6520 7661 7269   projective vari
│ │ │ │ +00035ae0: 6574 790a 2020 2a20 2a6e 6f74 6520 4575  ety.  * *note Eu
│ │ │ │ +00035af0: 6c65 7243 6861 7261 6374 6572 6973 7469  lerCharacteristi
│ │ │ │ +00035b00: 633a 2045 756c 6572 4368 6172 6163 7465  c: EulerCharacte
│ │ │ │ +00035b10: 7269 7374 6963 2c20 2d2d 2074 6f70 6f6c  ristic, -- topol
│ │ │ │ +00035b20: 6f67 6963 616c 2045 756c 6572 0a20 2020  ogical Euler.   
│ │ │ │ +00035b30: 2063 6861 7261 6374 6572 6973 7469 6320   characteristic 
│ │ │ │ +00035b40: 6f66 2061 2028 736d 6f6f 7468 2920 7072  of a (smooth) pr
│ │ │ │ +00035b50: 6f6a 6563 7469 7665 2076 6172 6965 7479  ojective variety
│ │ │ │ +00035b60: 0a20 202a 202a 6e6f 7465 2065 756c 6572  .  * *note euler
│ │ │ │ +00035b70: 2850 726f 6a65 6374 6976 6556 6172 6965  (ProjectiveVarie
│ │ │ │ +00035b80: 7479 293a 2028 5661 7269 6574 6965 7329  ty): (Varieties)
│ │ │ │ +00035b90: 6575 6c65 725f 6c70 5072 6f6a 6563 7469  euler_lpProjecti
│ │ │ │ +00035ba0: 7665 5661 7269 6574 795f 7270 2c20 2d2d  veVariety_rp, --
│ │ │ │ +00035bb0: 0a20 2020 2074 6f70 6f6c 6f67 6963 616c  .    topological
│ │ │ │ +00035bc0: 2045 756c 6572 2063 6861 7261 6374 6572   Euler character
│ │ │ │ +00035bd0: 6973 7469 6320 6f66 2061 2028 736d 6f6f  istic of a (smoo
│ │ │ │ +00035be0: 7468 2920 7072 6f6a 6563 7469 7665 2076  th) projective v
│ │ │ │ +00035bf0: 6172 6965 7479 0a0a 5761 7973 2074 6f20  ariety..Ways to 
│ │ │ │ +00035c00: 7573 6520 4368 6572 6e53 6368 7761 7274  use ChernSchwart
│ │ │ │ +00035c10: 7a4d 6163 5068 6572 736f 6e3a 0a3d 3d3d  zMacPherson:.===
│ │ │ │  00035c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00035c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ -00035c40: 2022 4368 6572 6e53 6368 7761 7274 7a4d   "ChernSchwartzM
│ │ │ │ -00035c50: 6163 5068 6572 736f 6e28 4964 6561 6c29  acPherson(Ideal)
│ │ │ │ -00035c60: 220a 0a46 6f72 2074 6865 2070 726f 6772  "..For the progr
│ │ │ │ -00035c70: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d  ammer.==========
│ │ │ │ -00035c80: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62  ========..The ob
│ │ │ │ -00035c90: 6a65 6374 202a 6e6f 7465 2043 6865 726e  ject *note Chern
│ │ │ │ -00035ca0: 5363 6877 6172 747a 4d61 6350 6865 7273  SchwartzMacPhers
│ │ │ │ -00035cb0: 6f6e 3a20 4368 6572 6e53 6368 7761 7274  on: ChernSchwart
│ │ │ │ -00035cc0: 7a4d 6163 5068 6572 736f 6e2c 2069 7320  zMacPherson, is 
│ │ │ │ -00035cd0: 6120 2a6e 6f74 650a 6d65 7468 6f64 2066  a *note.method f
│ │ │ │ -00035ce0: 756e 6374 696f 6e20 7769 7468 206f 7074  unction with opt
│ │ │ │ -00035cf0: 696f 6e73 3a20 284d 6163 6175 6c61 7932  ions: (Macaulay2
│ │ │ │ -00035d00: 446f 6329 4d65 7468 6f64 4675 6e63 7469  Doc)MethodFuncti
│ │ │ │ -00035d10: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a  onWithOptions,..
│ │ │ │ -00035d20: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │ +00035c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00035c40: 3d0a 0a20 202a 2022 4368 6572 6e53 6368  =..  * "ChernSch
│ │ │ │ +00035c50: 7761 7274 7a4d 6163 5068 6572 736f 6e28  wartzMacPherson(
│ │ │ │ +00035c60: 4964 6561 6c29 220a 0a46 6f72 2074 6865  Ideal)"..For the
│ │ │ │ +00035c70: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d   programmer.====
│ │ │ │ +00035c80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a  ==============..
│ │ │ │ +00035c90: 5468 6520 6f62 6a65 6374 202a 6e6f 7465  The object *note
│ │ │ │ +00035ca0: 2043 6865 726e 5363 6877 6172 747a 4d61   ChernSchwartzMa
│ │ │ │ +00035cb0: 6350 6865 7273 6f6e 3a20 4368 6572 6e53  cPherson: ChernS
│ │ │ │ +00035cc0: 6368 7761 7274 7a4d 6163 5068 6572 736f  chwartzMacPherso
│ │ │ │ +00035cd0: 6e2c 2069 7320 6120 2a6e 6f74 650a 6d65  n, is a *note.me
│ │ │ │ +00035ce0: 7468 6f64 2066 756e 6374 696f 6e20 7769  thod function wi
│ │ │ │ +00035cf0: 7468 206f 7074 696f 6e73 3a20 284d 6163  th options: (Mac
│ │ │ │ +00035d00: 6175 6c61 7932 446f 6329 4d65 7468 6f64  aulay2Doc)Method
│ │ │ │ +00035d10: 4675 6e63 7469 6f6e 5769 7468 4f70 7469  FunctionWithOpti
│ │ │ │ +00035d20: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d  ons,...---------
│ │ │ │  00035d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00035d70: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ -00035d80: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ -00035d90: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ -00035da0: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ -00035db0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64  aulay2-1.26.06+d
│ │ │ │ -00035dc0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ -00035dd0: 6163 6b61 6765 732f 4372 656d 6f6e 612f  ackages/Cremona/
│ │ │ │ -00035de0: 0a64 6f63 756d 656e 7461 7469 6f6e 2e6d  .documentation.m
│ │ │ │ -00035df0: 323a 3332 363a 302e 0a1f 0a46 696c 653a  2:326:0....File:
│ │ │ │ -00035e00: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ -00035e10: 6f64 653a 2043 6f64 696d 4273 496e 762c  ode: CodimBsInv,
│ │ │ │ -00035e20: 204e 6578 743a 2063 6f65 6666 6963 6965   Next: coefficie
│ │ │ │ -00035e30: 6e74 5269 6e67 5f6c 7052 6174 696f 6e61  ntRing_lpRationa
│ │ │ │ -00035e40: 6c4d 6170 5f72 702c 2050 7265 763a 2043  lMap_rp, Prev: C
│ │ │ │ -00035e50: 6865 726e 5363 6877 6172 747a 4d61 6350  hernSchwartzMacP
│ │ │ │ -00035e60: 6865 7273 6f6e 2c20 5570 3a20 546f 700a  herson, Up: Top.
│ │ │ │ -00035e70: 0a43 6f64 696d 4273 496e 760a 2a2a 2a2a  .CodimBsInv.****
│ │ │ │ -00035e80: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074  ******..Descript
│ │ │ │ -00035e90: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a  ion.===========.
│ │ │ │ -00035ea0: 0a54 6869 7320 6973 2061 2074 6563 686e  .This is a techn
│ │ │ │ -00035eb0: 6963 616c 206f 7074 696f 6e20 666f 7220  ical option for 
│ │ │ │ -00035ec0: 2a6e 6f74 6520 6170 7072 6f78 696d 6174  *note approximat
│ │ │ │ -00035ed0: 6549 6e76 6572 7365 4d61 703a 0a61 7070  eInverseMap:.app
│ │ │ │ -00035ee0: 726f 7869 6d61 7465 496e 7665 7273 654d  roximateInverseM
│ │ │ │ -00035ef0: 6170 2c2e 2049 7420 6163 6365 7074 7320  ap,. It accepts 
│ │ │ │ -00035f00: 616e 2069 6e74 6567 6572 2077 6869 6368  an integer which
│ │ │ │ -00035f10: 2073 686f 756c 6420 6265 2061 206c 6f77   should be a low
│ │ │ │ -00035f20: 6572 2062 6f75 6e64 2066 6f72 0a74 6865  er bound for.the
│ │ │ │ -00035f30: 2063 6f64 696d 656e 7369 6f6e 206f 6620   codimension of 
│ │ │ │ -00035f40: 7468 6520 6261 7365 206c 6f63 7573 206f  the base locus o
│ │ │ │ -00035f50: 6620 7468 6520 696e 7665 7273 6520 6d61  f the inverse ma
│ │ │ │ -00035f60: 702e 2049 6e20 6d6f 7374 2063 6173 6573  p. In most cases
│ │ │ │ -00035f70: 2c20 6f6e 6520 6361 6e0a 6f62 7461 696e  , one can.obtain
│ │ │ │ -00035f80: 2074 6865 206f 7074 696d 616c 2076 616c   the optimal val
│ │ │ │ -00035f90: 7565 2074 6f20 6265 2070 6173 7365 6420  ue to be passed 
│ │ │ │ -00035fa0: 6173 2069 6e20 7468 6520 666f 6c6c 6f77  as in the follow
│ │ │ │ -00035fb0: 696e 6720 6578 616d 706c 652e 0a0a 0a2b  ing example....+
│ │ │ │ -00035fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00035d70: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572  ------..The sour
│ │ │ │ +00035d80: 6365 206f 6620 7468 6973 2064 6f63 756d  ce of this docum
│ │ │ │ +00035d90: 656e 7420 6973 2069 6e0a 2f62 7569 6c64  ent is in./build
│ │ │ │ +00035da0: 2f72 6570 726f 6475 6369 626c 652d 7061  /reproducible-pa
│ │ │ │ +00035db0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32  th/macaulay2-1.2
│ │ │ │ +00035dc0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175  6.06+ds/M2/Macau
│ │ │ │ +00035dd0: 6c61 7932 2f70 6163 6b61 6765 732f 4372  lay2/packages/Cr
│ │ │ │ +00035de0: 656d 6f6e 612f 0a64 6f63 756d 656e 7461  emona/.documenta
│ │ │ │ +00035df0: 7469 6f6e 2e6d 323a 3332 363a 302e 0a1f  tion.m2:326:0...
│ │ │ │ +00035e00: 0a46 696c 653a 2043 7265 6d6f 6e61 2e69  .File: Cremona.i
│ │ │ │ +00035e10: 6e66 6f2c 204e 6f64 653a 2043 6f64 696d  nfo, Node: Codim
│ │ │ │ +00035e20: 4273 496e 762c 204e 6578 743a 2063 6f65  BsInv, Next: coe
│ │ │ │ +00035e30: 6666 6963 6965 6e74 5269 6e67 5f6c 7052  fficientRing_lpR
│ │ │ │ +00035e40: 6174 696f 6e61 6c4d 6170 5f72 702c 2050  ationalMap_rp, P
│ │ │ │ +00035e50: 7265 763a 2043 6865 726e 5363 6877 6172  rev: ChernSchwar
│ │ │ │ +00035e60: 747a 4d61 6350 6865 7273 6f6e 2c20 5570  tzMacPherson, Up
│ │ │ │ +00035e70: 3a20 546f 700a 0a43 6f64 696d 4273 496e  : Top..CodimBsIn
│ │ │ │ +00035e80: 760a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465  v.**********..De
│ │ │ │ +00035e90: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ +00035ea0: 3d3d 3d3d 3d0a 0a54 6869 7320 6973 2061  =====..This is a
│ │ │ │ +00035eb0: 2074 6563 686e 6963 616c 206f 7074 696f   technical optio
│ │ │ │ +00035ec0: 6e20 666f 7220 2a6e 6f74 6520 6170 7072  n for *note appr
│ │ │ │ +00035ed0: 6f78 696d 6174 6549 6e76 6572 7365 4d61  oximateInverseMa
│ │ │ │ +00035ee0: 703a 0a61 7070 726f 7869 6d61 7465 496e  p:.approximateIn
│ │ │ │ +00035ef0: 7665 7273 654d 6170 2c2e 2049 7420 6163  verseMap,. It ac
│ │ │ │ +00035f00: 6365 7074 7320 616e 2069 6e74 6567 6572  cepts an integer
│ │ │ │ +00035f10: 2077 6869 6368 2073 686f 756c 6420 6265   which should be
│ │ │ │ +00035f20: 2061 206c 6f77 6572 2062 6f75 6e64 2066   a lower bound f
│ │ │ │ +00035f30: 6f72 0a74 6865 2063 6f64 696d 656e 7369  or.the codimensi
│ │ │ │ +00035f40: 6f6e 206f 6620 7468 6520 6261 7365 206c  on of the base l
│ │ │ │ +00035f50: 6f63 7573 206f 6620 7468 6520 696e 7665  ocus of the inve
│ │ │ │ +00035f60: 7273 6520 6d61 702e 2049 6e20 6d6f 7374  rse map. In most
│ │ │ │ +00035f70: 2063 6173 6573 2c20 6f6e 6520 6361 6e0a   cases, one can.
│ │ │ │ +00035f80: 6f62 7461 696e 2074 6865 206f 7074 696d  obtain the optim
│ │ │ │ +00035f90: 616c 2076 616c 7565 2074 6f20 6265 2070  al value to be p
│ │ │ │ +00035fa0: 6173 7365 6420 6173 2069 6e20 7468 6520  assed as in the 
│ │ │ │ +00035fb0: 666f 6c6c 6f77 696e 6720 6578 616d 706c  following exampl
│ │ │ │ +00035fc0: 652e 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d  e....+----------
│ │ │ │  00035fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00035ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00036010: 6931 203a 2063 6f64 696d 4273 496e 7620  i1 : codimBsInv 
│ │ │ │ -00036020: 3d20 286d 2920 2d3e 2028 2020 2020 2020  = (m) -> (      
│ │ │ │ +00036000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036010: 2d2d 2d2b 0a7c 6931 203a 2063 6f64 696d  ---+.|i1 : codim
│ │ │ │ +00036020: 4273 496e 7620 3d20 286d 2920 2d3e 2028  BsInv = (m) -> (
│ │ │ │  00036030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036050: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00036060: 2020 2020 2020 2020 2d2d 2069 6e70 7574          -- input
│ │ │ │ -00036070: 3a20 6d2c 2074 6865 206c 6973 7420 6f66  : m, the list of
│ │ │ │ -00036080: 2070 726f 6a65 6374 6976 6520 6465 6772   projective degr
│ │ │ │ -00036090: 6565 7320 6f66 2061 2062 6972 6174 696f  ees of a biratio
│ │ │ │ -000360a0: 6e61 6c20 6d61 7020 2020 2020 207c 0a7c  nal map      |.|
│ │ │ │ -000360b0: 2020 2020 2020 2020 2d2d 206f 7574 7075          -- outpu
│ │ │ │ -000360c0: 743a 2074 6865 2063 6f64 696d 656e 7369  t: the codimensi
│ │ │ │ -000360d0: 6f6e 206f 6620 7468 6520 6261 7365 206c  on of the base l
│ │ │ │ -000360e0: 6f63 7573 206f 6620 7468 6520 696e 7665  ocus of the inve
│ │ │ │ -000360f0: 7273 6520 6d61 7020 2020 2020 207c 0a7c  rse map      |.|
│ │ │ │ -00036100: 2020 2020 2020 2020 6b3a 3d23 6d20 2d31          k:=#m -1
│ │ │ │ -00036110: 3b20 7a3a 3d6d 5f6b 3b20 643a 3d66 6c6f  ; z:=m_k; d:=flo
│ │ │ │ -00036120: 6f72 286d 5f28 6b2d 3129 2f7a 293b 2020  or(m_(k-1)/z);  
│ │ │ │ -00036130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036140: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00036150: 2020 2020 2020 2020 666f 7220 6920 6672          for i fr
│ │ │ │ -00036160: 6f6d 2032 2074 6f20 6b20 646f 2069 6620  om 2 to k do if 
│ │ │ │ -00036170: 7a2a 645e 6920 2d20 6d5f 286b 2d69 2920  z*d^i - m_(k-i) 
│ │ │ │ -00036180: 3e20 3020 7468 656e 2072 6574 7572 6e20  > 0 then return 
│ │ │ │ -00036190: 693b 2020 2020 2020 2020 2020 207c 0a7c  i;           |.|
│ │ │ │ -000361a0: 2020 2020 2029 3b20 2020 2020 2020 2020       );         
│ │ │ │ +00036050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036060: 2020 207c 0a7c 2020 2020 2020 2020 2d2d     |.|        --
│ │ │ │ +00036070: 2069 6e70 7574 3a20 6d2c 2074 6865 206c   input: m, the l
│ │ │ │ +00036080: 6973 7420 6f66 2070 726f 6a65 6374 6976  ist of projectiv
│ │ │ │ +00036090: 6520 6465 6772 6565 7320 6f66 2061 2062  e degrees of a b
│ │ │ │ +000360a0: 6972 6174 696f 6e61 6c20 6d61 7020 2020  irational map   
│ │ │ │ +000360b0: 2020 207c 0a7c 2020 2020 2020 2020 2d2d     |.|        --
│ │ │ │ +000360c0: 206f 7574 7075 743a 2074 6865 2063 6f64   output: the cod
│ │ │ │ +000360d0: 696d 656e 7369 6f6e 206f 6620 7468 6520  imension of the 
│ │ │ │ +000360e0: 6261 7365 206c 6f63 7573 206f 6620 7468  base locus of th
│ │ │ │ +000360f0: 6520 696e 7665 7273 6520 6d61 7020 2020  e inverse map   
│ │ │ │ +00036100: 2020 207c 0a7c 2020 2020 2020 2020 6b3a     |.|        k:
│ │ │ │ +00036110: 3d23 6d20 2d31 3b20 7a3a 3d6d 5f6b 3b20  =#m -1; z:=m_k; 
│ │ │ │ +00036120: 643a 3d66 6c6f 6f72 286d 5f28 6b2d 3129  d:=floor(m_(k-1)
│ │ │ │ +00036130: 2f7a 293b 2020 2020 2020 2020 2020 2020  /z);            
│ │ │ │ +00036140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036150: 2020 207c 0a7c 2020 2020 2020 2020 666f     |.|        fo
│ │ │ │ +00036160: 7220 6920 6672 6f6d 2032 2074 6f20 6b20  r i from 2 to k 
│ │ │ │ +00036170: 646f 2069 6620 7a2a 645e 6920 2d20 6d5f  do if z*d^i - m_
│ │ │ │ +00036180: 286b 2d69 2920 3e20 3020 7468 656e 2072  (k-i) > 0 then r
│ │ │ │ +00036190: 6574 7572 6e20 693b 2020 2020 2020 2020  eturn i;        
│ │ │ │ +000361a0: 2020 207c 0a7c 2020 2020 2029 3b20 2020     |.|     );   
│ │ │ │  000361b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000361c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000361d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000361e0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -000361f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000361e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000361f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00036200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00036240: 6932 203a 2070 6869 203d 2074 6f4d 6170  i2 : phi = toMap
│ │ │ │ -00036250: 2074 7269 6d20 6d69 6e6f 7273 2832 2c67   trim minors(2,g
│ │ │ │ -00036260: 656e 6572 6963 5379 6d6d 6574 7269 634d  enericSymmetricM
│ │ │ │ -00036270: 6174 7269 7828 5151 5b78 5f30 2e2e 785f  atrix(QQ[x_0..x_
│ │ │ │ -00036280: 355d 2c33 2929 2020 2020 2020 207c 0a7c  5],3))       |.|
│ │ │ │ -00036290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036240: 2d2d 2d2b 0a7c 6932 203a 2070 6869 203d  ---+.|i2 : phi =
│ │ │ │ +00036250: 2074 6f4d 6170 2074 7269 6d20 6d69 6e6f   toMap trim mino
│ │ │ │ +00036260: 7273 2832 2c67 656e 6572 6963 5379 6d6d  rs(2,genericSymm
│ │ │ │ +00036270: 6574 7269 634d 6174 7269 7828 5151 5b78  etricMatrix(QQ[x
│ │ │ │ +00036280: 5f30 2e2e 785f 355d 2c33 2929 2020 2020  _0..x_5],3))    
│ │ │ │ +00036290: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000362a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000362b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000362c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000362d0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000362e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000362d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000362e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000362f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036300: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00036300: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  00036310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036320: 2020 2020 2020 2020 2032 2020 207c 0a7c           2   |.|
│ │ │ │ -00036330: 6f32 203d 206d 6170 2028 5151 5b78 202e  o2 = map (QQ[x .
│ │ │ │ -00036340: 2e78 205d 2c20 5151 5b78 202e 2e78 205d  .x ], QQ[x ..x ]
│ │ │ │ -00036350: 2c20 7b78 2020 2d20 7820 7820 2c20 7820  , {x  - x x , x 
│ │ │ │ -00036360: 7820 202d 2078 2078 202c 2078 2078 2020  x  - x x , x x  
│ │ │ │ -00036370: 2d20 7820 7820 2c20 7820 202d 207c 0a7c  - x x , x  - |.|
│ │ │ │ -00036380: 2020 2020 2020 2020 2020 2020 2020 3020                0 
│ │ │ │ -00036390: 2020 3520 2020 2020 2020 3020 2020 3520    5       0   5 
│ │ │ │ -000363a0: 2020 2020 3420 2020 2033 2035 2020 2032      4    3 5   2
│ │ │ │ -000363b0: 2034 2020 2020 3120 3520 2020 3220 3320   4    1 5   2 3 
│ │ │ │ -000363c0: 2020 2031 2034 2020 2032 2020 207c 0a7c     1 4   2   |.|
│ │ │ │ -000363d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036320: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00036330: 2020 207c 0a7c 6f32 203d 206d 6170 2028     |.|o2 = map (
│ │ │ │ +00036340: 5151 5b78 202e 2e78 205d 2c20 5151 5b78  QQ[x ..x ], QQ[x
│ │ │ │ +00036350: 202e 2e78 205d 2c20 7b78 2020 2d20 7820   ..x ], {x  - x 
│ │ │ │ +00036360: 7820 2c20 7820 7820 202d 2078 2078 202c  x , x x  - x x ,
│ │ │ │ +00036370: 2078 2078 2020 2d20 7820 7820 2c20 7820   x x  - x x , x 
│ │ │ │ +00036380: 202d 207c 0a7c 2020 2020 2020 2020 2020   - |.|          
│ │ │ │ +00036390: 2020 2020 3020 2020 3520 2020 2020 2020      0   5       
│ │ │ │ +000363a0: 3020 2020 3520 2020 2020 3420 2020 2033  0   5     4    3
│ │ │ │ +000363b0: 2035 2020 2032 2034 2020 2020 3120 3520   5   2 4    1 5 
│ │ │ │ +000363c0: 2020 3220 3320 2020 2031 2034 2020 2032    2 3    1 4   2
│ │ │ │ +000363d0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000363e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000363f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036410: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00036420: 6f32 203a 2052 696e 674d 6170 2051 515b  o2 : RingMap QQ[
│ │ │ │ -00036430: 7820 2e2e 7820 5d20 3c2d 2d20 5151 5b78  x ..x ] <-- QQ[x
│ │ │ │ -00036440: 202e 2e78 205d 2020 2020 2020 2020 2020   ..x ]          
│ │ │ │ +00036410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036420: 2020 207c 0a7c 6f32 203a 2052 696e 674d     |.|o2 : RingM
│ │ │ │ +00036430: 6170 2051 515b 7820 2e2e 7820 5d20 3c2d  ap QQ[x ..x ] <-
│ │ │ │ +00036440: 2d20 5151 5b78 202e 2e78 205d 2020 2020  - QQ[x ..x ]    
│ │ │ │  00036450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036460: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00036470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036480: 2030 2020 2035 2020 2020 2020 2020 2020   0   5          
│ │ │ │ -00036490: 3020 2020 3520 2020 2020 2020 2020 2020  0   5           
│ │ │ │ +00036460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036470: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +00036480: 2020 2020 2020 2030 2020 2035 2020 2020         0   5    
│ │ │ │ +00036490: 2020 2020 2020 3020 2020 3520 2020 2020        0   5     
│ │ │ │  000364a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000364b0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000364c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000364b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000364c0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d     |.|----------
│ │ │ │  000364d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000364e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000364f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c  -------------|.|
│ │ │ │ -00036510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036520: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00036500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036510: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020  ---|.|          
│ │ │ │ +00036520: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  00036530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036550: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -00036560: 7820 7820 2c20 7820 7820 202d 2078 2078  x x , x x  - x x
│ │ │ │ -00036570: 202c 2078 2020 2d20 7820 7820 7d29 2020   , x  - x x })  
│ │ │ │ -00036580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036560: 2020 207c 0a7c 7820 7820 2c20 7820 7820     |.|x x , x x 
│ │ │ │ +00036570: 202d 2078 2078 202c 2078 2020 2d20 7820   - x x , x  - x 
│ │ │ │ +00036580: 7820 7d29 2020 2020 2020 2020 2020 2020  x })            
│ │ │ │  00036590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000365b0: 2030 2035 2020 2031 2032 2020 2020 3020   0 5   1 2    0 
│ │ │ │ -000365c0: 3420 2020 3120 2020 2030 2033 2020 2020  4   1    0 3    
│ │ │ │ -000365d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000365a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000365b0: 2020 207c 0a7c 2030 2035 2020 2031 2032     |.| 0 5   1 2
│ │ │ │ +000365c0: 2020 2020 3020 3420 2020 3120 2020 2030      0 4   1    0
│ │ │ │ +000365d0: 2033 2020 2020 2020 2020 2020 2020 2020   3              
│ │ │ │  000365e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000365f0: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00036600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000365f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036600: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00036610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c  -------------+.|
│ │ │ │ -00036650: 6933 203a 2063 6f64 696d 4273 496e 7620  i3 : codimBsInv 
│ │ │ │ -00036660: 7072 6f6a 6563 7469 7665 4465 6772 6565  projectiveDegree
│ │ │ │ -00036670: 7320 7068 6920 2020 2020 2020 2020 2020  s phi           
│ │ │ │ +00036640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036650: 2d2d 2d2b 0a7c 6933 203a 2063 6f64 696d  ---+.|i3 : codim
│ │ │ │ +00036660: 4273 496e 7620 7072 6f6a 6563 7469 7665  BsInv projective
│ │ │ │ +00036670: 4465 6772 6565 7320 7068 6920 2020 2020  Degrees phi     
│ │ │ │  00036680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036690: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000366a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000366a0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │  000366b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000366c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000366d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000366e0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ -000366f0: 6f33 203d 2033 2020 2020 2020 2020 2020  o3 = 3          
│ │ │ │ +000366e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000366f0: 2020 207c 0a7c 6f33 203d 2033 2020 2020     |.|o3 = 3    
│ │ │ │  00036700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00036720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00036730: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00036740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00036740: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00036750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a  -------------+..
│ │ │ │ -00036790: 486f 7765 7665 722c 2073 6f6d 6574 696d  However, sometim
│ │ │ │ -000367a0: 6573 206c 6172 6765 7220 7661 6c75 6573  es larger values
│ │ │ │ -000367b0: 206d 6179 2062 6520 7072 6566 6572 6162   may be preferab
│ │ │ │ -000367c0: 6c65 2e0a 0a46 756e 6374 696f 6e73 2077  le...Functions w
│ │ │ │ -000367d0: 6974 6820 6f70 7469 6f6e 616c 2061 7267  ith optional arg
│ │ │ │ -000367e0: 756d 656e 7420 6e61 6d65 6420 436f 6469  ument named Codi
│ │ │ │ -000367f0: 6d42 7349 6e76 3a0a 3d3d 3d3d 3d3d 3d3d  mBsInv:.========
│ │ │ │ +00036780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036790: 2d2d 2d2b 0a0a 486f 7765 7665 722c 2073  ---+..However, s
│ │ │ │ +000367a0: 6f6d 6574 696d 6573 206c 6172 6765 7220  ometimes larger 
│ │ │ │ +000367b0: 7661 6c75 6573 206d 6179 2062 6520 7072  values may be pr
│ │ │ │ +000367c0: 6566 6572 6162 6c65 2e0a 0a46 756e 6374  eferable...Funct
│ │ │ │ +000367d0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e  ions with option
│ │ │ │ +000367e0: 616c 2061 7267 756d 656e 7420 6e61 6d65  al argument name
│ │ │ │ +000367f0: 6420 436f 6469 6d42 7349 6e76 3a0a 3d3d  d CodimBsInv:.==
│ │ │ │  00036800: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │  00036810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00036820: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ -00036830: 2261 7070 726f 7869 6d61 7465 496e 7665  "approximateInve
│ │ │ │ -00036840: 7273 654d 6170 282e 2e2e 2c43 6f64 696d  rseMap(...,Codim
│ │ │ │ -00036850: 4273 496e 763d 3e2e 2e2e 2922 0a0a 466f  BsInv=>...)"..Fo
│ │ │ │ -00036860: 7220 7468 6520 7072 6f67 7261 6d6d 6572  r the programmer
│ │ │ │ -00036870: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  .===============
│ │ │ │ -00036880: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420  ===..The object 
│ │ │ │ -00036890: 2a6e 6f74 6520 436f 6469 6d42 7349 6e76  *note CodimBsInv
│ │ │ │ -000368a0: 3a20 436f 6469 6d42 7349 6e76 2c20 6973  : CodimBsInv, is
│ │ │ │ -000368b0: 2061 202a 6e6f 7465 2073 796d 626f 6c3a   a *note symbol:
│ │ │ │ -000368c0: 0a28 4d61 6361 756c 6179 3244 6f63 2953  .(Macaulay2Doc)S
│ │ │ │ -000368d0: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d  ymbol,...-------
│ │ │ │ +00036820: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00036830: 0a0a 2020 2a20 2261 7070 726f 7869 6d61  ..  * "approxima
│ │ │ │ +00036840: 7465 496e 7665 7273 654d 6170 282e 2e2e  teInverseMap(...
│ │ │ │ +00036850: 2c43 6f64 696d 4273 496e 763d 3e2e 2e2e  ,CodimBsInv=>...
│ │ │ │ +00036860: 2922 0a0a 466f 7220 7468 6520 7072 6f67  )"..For the prog
│ │ │ │ +00036870: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d  rammer.=========
│ │ │ │ +00036880: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f  =========..The o
│ │ │ │ +00036890: 626a 6563 7420 2a6e 6f74 6520 436f 6469  bject *note Codi
│ │ │ │ +000368a0: 6d42 7349 6e76 3a20 436f 6469 6d42 7349  mBsInv: CodimBsI
│ │ │ │ +000368b0: 6e76 2c20 6973 2061 202a 6e6f 7465 2073  nv, is a *note s
│ │ │ │ +000368c0: 796d 626f 6c3a 0a28 4d61 6361 756c 6179  ymbol:.(Macaulay
│ │ │ │ +000368d0: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d  2Doc)Symbol,...-
│ │ │ │  000368e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000368f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036920: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f  --------..The so
│ │ │ │ -00036930: 7572 6365 206f 6620 7468 6973 2064 6f63  urce of this doc
│ │ │ │ -00036940: 756d 656e 7420 6973 2069 6e0a 2f62 7569  ument is in./bui
│ │ │ │ -00036950: 6c64 2f72 6570 726f 6475 6369 626c 652d  ld/reproducible-
│ │ │ │ -00036960: 7061 7468 2f6d 6163 6175 6c61 7932 2d31  path/macaulay2-1
│ │ │ │ -00036970: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163  .26.06+ds/M2/Mac
│ │ │ │ -00036980: 6175 6c61 7932 2f70 6163 6b61 6765 732f  aulay2/packages/
│ │ │ │ -00036990: 4372 656d 6f6e 612f 0a64 6f63 756d 656e  Cremona/.documen
│ │ │ │ -000369a0: 7461 7469 6f6e 2e6d 323a 3130 373a 302e  tation.m2:107:0.
│ │ │ │ -000369b0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61  ...File: Cremona
│ │ │ │ -000369c0: 2e69 6e66 6f2c 204e 6f64 653a 2063 6f65  .info, Node: coe
│ │ │ │ -000369d0: 6666 6963 6965 6e74 5269 6e67 5f6c 7052  fficientRing_lpR
│ │ │ │ -000369e0: 6174 696f 6e61 6c4d 6170 5f72 702c 204e  ationalMap_rp, N
│ │ │ │ -000369f0: 6578 743a 2063 6f65 6666 6963 6965 6e74  ext: coefficient
│ │ │ │ -00036a00: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ -00036a10: 7270 2c20 5072 6576 3a20 436f 6469 6d42  rp, Prev: CodimB
│ │ │ │ -00036a20: 7349 6e76 2c20 5570 3a20 546f 700a 0a63  sInv, Up: Top..c
│ │ │ │ -00036a30: 6f65 6666 6963 6965 6e74 5269 6e67 2852  oefficientRing(R
│ │ │ │ -00036a40: 6174 696f 6e61 6c4d 6170 2920 2d2d 2063  ationalMap) -- c
│ │ │ │ -00036a50: 6f65 6666 6963 6965 6e74 2072 696e 6720  oefficient ring 
│ │ │ │ -00036a60: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ -00036a70: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  p.**************
│ │ │ │ +00036920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a  --------------..
│ │ │ │ +00036930: 5468 6520 736f 7572 6365 206f 6620 7468  The source of th
│ │ │ │ +00036940: 6973 2064 6f63 756d 656e 7420 6973 2069  is document is i
│ │ │ │ +00036950: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475  n./build/reprodu
│ │ │ │ +00036960: 6369 626c 652d 7061 7468 2f6d 6163 6175  cible-path/macau
│ │ │ │ +00036970: 6c61 7932 2d31 2e32 362e 3036 2b64 732f  lay2-1.26.06+ds/
│ │ │ │ +00036980: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163  M2/Macaulay2/pac
│ │ │ │ +00036990: 6b61 6765 732f 4372 656d 6f6e 612f 0a64  kages/Cremona/.d
│ │ │ │ +000369a0: 6f63 756d 656e 7461 7469 6f6e 2e6d 323a  ocumentation.m2:
│ │ │ │ +000369b0: 3130 373a 302e 0a1f 0a46 696c 653a 2043  107:0....File: C
│ │ │ │ +000369c0: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64  remona.info, Nod
│ │ │ │ +000369d0: 653a 2063 6f65 6666 6963 6965 6e74 5269  e: coefficientRi
│ │ │ │ +000369e0: 6e67 5f6c 7052 6174 696f 6e61 6c4d 6170  ng_lpRationalMap
│ │ │ │ +000369f0: 5f72 702c 204e 6578 743a 2063 6f65 6666  _rp, Next: coeff
│ │ │ │ +00036a00: 6963 6965 6e74 735f 6c70 5261 7469 6f6e  icients_lpRation
│ │ │ │ +00036a10: 616c 4d61 705f 7270 2c20 5072 6576 3a20  alMap_rp, Prev: 
│ │ │ │ +00036a20: 436f 6469 6d42 7349 6e76 2c20 5570 3a20  CodimBsInv, Up: 
│ │ │ │ +00036a30: 546f 700a 0a63 6f65 6666 6963 6965 6e74  Top..coefficient
│ │ │ │ +00036a40: 5269 6e67 2852 6174 696f 6e61 6c4d 6170  Ring(RationalMap
│ │ │ │ +00036a50: 2920 2d2d 2063 6f65 6666 6963 6965 6e74  ) -- coefficient
│ │ │ │ +00036a60: 2072 696e 6720 6f66 2061 2072 6174 696f   ring of a ratio
│ │ │ │ +00036a70: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a  nal map.********
│ │ │ │  00036a80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00036ab0: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469  ****..  * Functi
│ │ │ │ -00036ac0: 6f6e 3a20 2a6e 6f74 6520 636f 6566 6669  on: *note coeffi
│ │ │ │ -00036ad0: 6369 656e 7452 696e 673a 2028 4d61 6361  cientRing: (Maca
│ │ │ │ -00036ae0: 756c 6179 3244 6f63 2963 6f65 6666 6963  ulay2Doc)coeffic
│ │ │ │ -00036af0: 6965 6e74 5269 6e67 2c0a 2020 2a20 5573  ientRing,.  * Us
│ │ │ │ -00036b00: 6167 653a 200a 2020 2020 2020 2020 636f  age: .        co
│ │ │ │ -00036b10: 6566 6669 6369 656e 7452 696e 6720 7068  efficientRing ph
│ │ │ │ -00036b20: 690a 2020 2a20 496e 7075 7473 3a0a 2020  i.  * Inputs:.  
│ │ │ │ -00036b30: 2020 2020 2a20 7068 692c 2061 202a 6e6f      * phi, a *no
│ │ │ │ -00036b40: 7465 2072 6174 696f 6e61 6c20 6d61 703a  te rational map:
│ │ │ │ -00036b50: 2052 6174 696f 6e61 6c4d 6170 2c0a 2020   RationalMap,.  
│ │ │ │ -00036b60: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ -00036b70: 202a 2061 202a 6e6f 7465 2072 696e 673a   * a *note ring:
│ │ │ │ -00036b80: 2028 4d61 6361 756c 6179 3244 6f63 2952   (Macaulay2Doc)R
│ │ │ │ -00036b90: 696e 672c 2c20 7468 6520 636f 6566 6669  ing,, the coeffi
│ │ │ │ -00036ba0: 6369 656e 7420 7269 6e67 206f 6620 7068  cient ring of ph
│ │ │ │ -00036bb0: 690a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d  i..See also.====
│ │ │ │ -00036bc0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -00036bd0: 5261 7469 6f6e 616c 4d61 7020 2a2a 2052  RationalMap ** R
│ │ │ │ -00036be0: 696e 673a 2052 6174 696f 6e61 6c4d 6170  ing: RationalMap
│ │ │ │ -00036bf0: 205f 7374 5f73 7420 5269 6e67 2c20 2d2d   _st_st Ring, --
│ │ │ │ -00036c00: 2063 6861 6e67 6520 7468 650a 2020 2020   change the.    
│ │ │ │ -00036c10: 636f 6566 6669 6369 656e 7420 7269 6e67  coefficient ring
│ │ │ │ -00036c20: 206f 6620 6120 7261 7469 6f6e 616c 206d   of a rational m
│ │ │ │ -00036c30: 6170 0a0a 5761 7973 2074 6f20 7573 6520  ap..Ways to use 
│ │ │ │ -00036c40: 7468 6973 206d 6574 686f 643a 0a3d 3d3d  this method:.===
│ │ │ │ -00036c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00036c60: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465  =====..  * *note
│ │ │ │ -00036c70: 2063 6f65 6666 6963 6965 6e74 5269 6e67   coefficientRing
│ │ │ │ -00036c80: 2852 6174 696f 6e61 6c4d 6170 293a 2063  (RationalMap): c
│ │ │ │ -00036c90: 6f65 6666 6963 6965 6e74 5269 6e67 5f6c  oefficientRing_l
│ │ │ │ -00036ca0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c  pRationalMap_rp,
│ │ │ │ -00036cb0: 202d 2d0a 2020 2020 636f 6566 6669 6369   --.    coeffici
│ │ │ │ -00036cc0: 656e 7420 7269 6e67 206f 6620 6120 7261  ent ring of a ra
│ │ │ │ -00036cd0: 7469 6f6e 616c 206d 6170 0a2d 2d2d 2d2d  tional map.-----
│ │ │ │ -00036ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036ab0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00036ac0: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520  Function: *note 
│ │ │ │ +00036ad0: 636f 6566 6669 6369 656e 7452 696e 673a  coefficientRing:
│ │ │ │ +00036ae0: 2028 4d61 6361 756c 6179 3244 6f63 2963   (Macaulay2Doc)c
│ │ │ │ +00036af0: 6f65 6666 6963 6965 6e74 5269 6e67 2c0a  oefficientRing,.
│ │ │ │ +00036b00: 2020 2a20 5573 6167 653a 200a 2020 2020    * Usage: .    
│ │ │ │ +00036b10: 2020 2020 636f 6566 6669 6369 656e 7452      coefficientR
│ │ │ │ +00036b20: 696e 6720 7068 690a 2020 2a20 496e 7075  ing phi.  * Inpu
│ │ │ │ +00036b30: 7473 3a0a 2020 2020 2020 2a20 7068 692c  ts:.      * phi,
│ │ │ │ +00036b40: 2061 202a 6e6f 7465 2072 6174 696f 6e61   a *note rationa
│ │ │ │ +00036b50: 6c20 6d61 703a 2052 6174 696f 6e61 6c4d  l map: RationalM
│ │ │ │ +00036b60: 6170 2c0a 2020 2a20 4f75 7470 7574 733a  ap,.  * Outputs:
│ │ │ │ +00036b70: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ +00036b80: 2072 696e 673a 2028 4d61 6361 756c 6179   ring: (Macaulay
│ │ │ │ +00036b90: 3244 6f63 2952 696e 672c 2c20 7468 6520  2Doc)Ring,, the 
│ │ │ │ +00036ba0: 636f 6566 6669 6369 656e 7420 7269 6e67  coefficient ring
│ │ │ │ +00036bb0: 206f 6620 7068 690a 0a53 6565 2061 6c73   of phi..See als
│ │ │ │ +00036bc0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  o.========..  * 
│ │ │ │ +00036bd0: 2a6e 6f74 6520 5261 7469 6f6e 616c 4d61  *note RationalMa
│ │ │ │ +00036be0: 7020 2a2a 2052 696e 673a 2052 6174 696f  p ** Ring: Ratio
│ │ │ │ +00036bf0: 6e61 6c4d 6170 205f 7374 5f73 7420 5269  nalMap _st_st Ri
│ │ │ │ +00036c00: 6e67 2c20 2d2d 2063 6861 6e67 6520 7468  ng, -- change th
│ │ │ │ +00036c10: 650a 2020 2020 636f 6566 6669 6369 656e  e.    coefficien
│ │ │ │ +00036c20: 7420 7269 6e67 206f 6620 6120 7261 7469  t ring of a rati
│ │ │ │ +00036c30: 6f6e 616c 206d 6170 0a0a 5761 7973 2074  onal map..Ways t
│ │ │ │ +00036c40: 6f20 7573 6520 7468 6973 206d 6574 686f  o use this metho
│ │ │ │ +00036c50: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  d:.=============
│ │ │ │ +00036c60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a  ===========..  *
│ │ │ │ +00036c70: 202a 6e6f 7465 2063 6f65 6666 6963 6965   *note coefficie
│ │ │ │ +00036c80: 6e74 5269 6e67 2852 6174 696f 6e61 6c4d  ntRing(RationalM
│ │ │ │ +00036c90: 6170 293a 2063 6f65 6666 6963 6965 6e74  ap): coefficient
│ │ │ │ +00036ca0: 5269 6e67 5f6c 7052 6174 696f 6e61 6c4d  Ring_lpRationalM
│ │ │ │ +00036cb0: 6170 5f72 702c 202d 2d0a 2020 2020 636f  ap_rp, --.    co
│ │ │ │ +00036cc0: 6566 6669 6369 656e 7420 7269 6e67 206f  efficient ring o
│ │ │ │ +00036cd0: 6620 6120 7261 7469 6f6e 616c 206d 6170  f a rational map
│ │ │ │ +00036ce0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .---------------
│ │ │ │  00036cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00036d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00036d20: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520  ----------..The 
│ │ │ │ -00036d30: 736f 7572 6365 206f 6620 7468 6973 2064  source of this d
│ │ │ │ -00036d40: 6f63 756d 656e 7420 6973 2069 6e0a 2f62  ocument is in./b
│ │ │ │ -00036d50: 7569 6c64 2f72 6570 726f 6475 6369 626c  uild/reproducibl
│ │ │ │ -00036d60: 652d 7061 7468 2f6d 6163 6175 6c61 7932  e-path/macaulay2
│ │ │ │ -00036d70: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d  -1.26.06+ds/M2/M
│ │ │ │ -00036d80: 6163 6175 6c61 7932 2f70 6163 6b61 6765  acaulay2/package
│ │ │ │ -00036d90: 732f 4372 656d 6f6e 612f 0a64 6f63 756d  s/Cremona/.docum
│ │ │ │ -00036da0: 656e 7461 7469 6f6e 2e6d 323a 3438 323a  entation.m2:482:
│ │ │ │ -00036db0: 302e 0a1f 0a46 696c 653a 2043 7265 6d6f  0....File: Cremo
│ │ │ │ -00036dc0: 6e61 2e69 6e66 6f2c 204e 6f64 653a 2063  na.info, Node: c
│ │ │ │ -00036dd0: 6f65 6666 6963 6965 6e74 735f 6c70 5261  oefficients_lpRa
│ │ │ │ -00036de0: 7469 6f6e 616c 4d61 705f 7270 2c20 4e65  tionalMap_rp, Ne
│ │ │ │ -00036df0: 7874 3a20 6465 6772 6565 5f6c 7052 6174  xt: degree_lpRat
│ │ │ │ -00036e00: 696f 6e61 6c4d 6170 5f72 702c 2050 7265  ionalMap_rp, Pre
│ │ │ │ -00036e10: 763a 2063 6f65 6666 6963 6965 6e74 5269  v: coefficientRi
│ │ │ │ -00036e20: 6e67 5f6c 7052 6174 696f 6e61 6c4d 6170  ng_lpRationalMap
│ │ │ │ -00036e30: 5f72 702c 2055 703a 2054 6f70 0a0a 636f  _rp, Up: Top..co
│ │ │ │ -00036e40: 6566 6669 6369 656e 7473 2852 6174 696f  efficients(Ratio
│ │ │ │ -00036e50: 6e61 6c4d 6170 2920 2d2d 2063 6f65 6666  nalMap) -- coeff
│ │ │ │ -00036e60: 6963 6965 6e74 206d 6174 7269 7820 6f66  icient matrix of
│ │ │ │ -00036e70: 2061 2072 6174 696f 6e61 6c20 6d61 700a   a rational map.
│ │ │ │ -00036e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ +00036d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00036d30: 0a0a 5468 6520 736f 7572 6365 206f 6620  ..The source of 
│ │ │ │ +00036d40: 7468 6973 2064 6f63 756d 656e 7420 6973  this document is
│ │ │ │ +00036d50: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f   in./build/repro
│ │ │ │ +00036d60: 6475 6369 626c 652d 7061 7468 2f6d 6163  ducible-path/mac
│ │ │ │ +00036d70: 6175 6c61 7932 2d31 2e32 362e 3036 2b64  aulay2-1.26.06+d
│ │ │ │ +00036d80: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70  s/M2/Macaulay2/p
│ │ │ │ +00036d90: 6163 6b61 6765 732f 4372 656d 6f6e 612f  ackages/Cremona/
│ │ │ │ +00036da0: 0a64 6f63 756d 656e 7461 7469 6f6e 2e6d  .documentation.m
│ │ │ │ +00036db0: 323a 3438 323a 302e 0a1f 0a46 696c 653a  2:482:0....File:
│ │ │ │ +00036dc0: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e   Cremona.info, N
│ │ │ │ +00036dd0: 6f64 653a 2063 6f65 6666 6963 6965 6e74  ode: coefficient
│ │ │ │ +00036de0: 735f 6c70 5261 7469 6f6e 616c 4d61 705f  s_lpRationalMap_
│ │ │ │ +00036df0: 7270 2c20 4e65 7874 3a20 6465 6772 6565  rp, Next: degree
│ │ │ │ +00036e00: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ +00036e10: 702c 2050 7265 763a 2063 6f65 6666 6963  p, Prev: coeffic
│ │ │ │ +00036e20: 6965 6e74 5269 6e67 5f6c 7052 6174 696f  ientRing_lpRatio
│ │ │ │ +00036e30: 6e61 6c4d 6170 5f72 702c 2055 703a 2054  nalMap_rp, Up: T
│ │ │ │ +00036e40: 6f70 0a0a 636f 6566 6669 6369 656e 7473  op..coefficients
│ │ │ │ +00036e50: 2852 6174 696f 6e61 6c4d 6170 2920 2d2d  (RationalMap) --
│ │ │ │ +00036e60: 2063 6f65 6666 6963 6965 6e74 206d 6174   coefficient mat
│ │ │ │ +00036e70: 7269 7820 6f66 2061 2072 6174 696f 6e61  rix of a rationa
│ │ │ │ +00036e80: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a  l map.**********
│ │ │ │  00036e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036ea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00036eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00036ec0: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a  *..  * Function:
│ │ │ │ -00036ed0: 202a 6e6f 7465 2063 6f65 6666 6963 6965   *note coefficie
│ │ │ │ -00036ee0: 6e74 733a 2028 4d61 6361 756c 6179 3244  nts: (Macaulay2D
│ │ │ │ -00036ef0: 6f63 2963 6f65 6666 6963 6965 6e74 732c  oc)coefficients,
│ │ │ │ -00036f00: 0a20 202a 2055 7361 6765 3a20 0a20 2020  .  * Usage: .   
│ │ │ │ -00036f10: 2020 2020 2063 6f65 6666 6963 6965 6e74       coefficient
│ │ │ │ -00036f20: 7320 7068 690a 2020 2a20 496e 7075 7473  s phi.  * Inputs
│ │ │ │ -00036f30: 3a0a 2020 2020 2020 2a20 7068 692c 2061  :.      * phi, a
│ │ │ │ -00036f40: 202a 6e6f 7465 2072 6174 696f 6e61 6c20   *note rational 
│ │ │ │ -00036f50: 6d61 703a 2052 6174 696f 6e61 6c4d 6170  map: RationalMap
│ │ │ │ -00036f60: 2c0a 2020 2a20 2a6e 6f74 6520 4f70 7469  ,.  * *note Opti
│ │ │ │ -00036f70: 6f6e 616c 2069 6e70 7574 733a 2028 4d61  onal inputs: (Ma
│ │ │ │ -00036f80: 6361 756c 6179 3244 6f63 2975 7369 6e67  caulay2Doc)using
│ │ │ │ -00036f90: 2066 756e 6374 696f 6e73 2077 6974 6820   functions with 
│ │ │ │ -00036fa0: 6f70 7469 6f6e 616c 2069 6e70 7574 732c  optional inputs,
│ │ │ │ -00036fb0: 3a0a 2020 2020 2020 2a20 2a6e 6f74 6520  :.      * *note 
│ │ │ │ -00036fc0: 4d6f 6e6f 6d69 616c 733a 2028 4d61 6361  Monomials: (Maca
│ │ │ │ -00036fd0: 756c 6179 3244 6f63 2963 6f65 6666 6963  ulay2Doc)coeffic
│ │ │ │ -00036fe0: 6965 6e74 732c 203d 3e20 2e2e 2e2c 2064  ients, => ..., d
│ │ │ │ -00036ff0: 6566 6175 6c74 2076 616c 7565 0a20 2020  efault value.   
│ │ │ │ -00037000: 2020 2020 206e 756c 6c2c 0a20 2020 2020       null,.     
│ │ │ │ -00037010: 202a 202a 6e6f 7465 2056 6172 6961 626c   * *note Variabl
│ │ │ │ -00037020: 6573 3a20 284d 6163 6175 6c61 7932 446f  es: (Macaulay2Do
│ │ │ │ -00037030: 6329 636f 6566 6669 6369 656e 7473 2c20  c)coefficients, 
│ │ │ │ -00037040: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -00037050: 7661 6c75 650a 2020 2020 2020 2020 6e75  value.        nu
│ │ │ │ -00037060: 6c6c 2c0a 2020 2a20 4f75 7470 7574 733a  ll,.  * Outputs:
│ │ │ │ -00037070: 0a20 2020 2020 202a 2061 202a 6e6f 7465  .      * a *note
│ │ │ │ -00037080: 206d 6174 7269 783a 2028 4d61 6361 756c   matrix: (Macaul
│ │ │ │ -00037090: 6179 3244 6f63 294d 6174 7269 782c 2c20  ay2Doc)Matrix,, 
│ │ │ │ -000370a0: 7468 6520 636f 6566 6669 6369 656e 7420  the coefficient 
│ │ │ │ -000370b0: 6d61 7472 6978 206f 6620 7468 650a 2020  matrix of the.  
│ │ │ │ -000370c0: 2020 2020 2020 706f 6c79 6e6f 6d69 616c        polynomial
│ │ │ │ -000370d0: 7320 6465 6669 6e69 6e67 2070 6869 0a0a  s defining phi..
│ │ │ │ -000370e0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d  Description.====
│ │ │ │ -000370f0: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d  =======..+------
│ │ │ │ +00036ec0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e  *******..  * Fun
│ │ │ │ +00036ed0: 6374 696f 6e3a 202a 6e6f 7465 2063 6f65  ction: *note coe
│ │ │ │ +00036ee0: 6666 6963 6965 6e74 733a 2028 4d61 6361  fficients: (Maca
│ │ │ │ +00036ef0: 756c 6179 3244 6f63 2963 6f65 6666 6963  ulay2Doc)coeffic
│ │ │ │ +00036f00: 6965 6e74 732c 0a20 202a 2055 7361 6765  ients,.  * Usage
│ │ │ │ +00036f10: 3a20 0a20 2020 2020 2020 2063 6f65 6666  : .        coeff
│ │ │ │ +00036f20: 6963 6965 6e74 7320 7068 690a 2020 2a20  icients phi.  * 
│ │ │ │ +00036f30: 496e 7075 7473 3a0a 2020 2020 2020 2a20  Inputs:.      * 
│ │ │ │ +00036f40: 7068 692c 2061 202a 6e6f 7465 2072 6174  phi, a *note rat
│ │ │ │ +00036f50: 696f 6e61 6c20 6d61 703a 2052 6174 696f  ional map: Ratio
│ │ │ │ +00036f60: 6e61 6c4d 6170 2c0a 2020 2a20 2a6e 6f74  nalMap,.  * *not
│ │ │ │ +00036f70: 6520 4f70 7469 6f6e 616c 2069 6e70 7574  e Optional input
│ │ │ │ +00036f80: 733a 2028 4d61 6361 756c 6179 3244 6f63  s: (Macaulay2Doc
│ │ │ │ +00036f90: 2975 7369 6e67 2066 756e 6374 696f 6e73  )using functions
│ │ │ │ +00036fa0: 2077 6974 6820 6f70 7469 6f6e 616c 2069   with optional i
│ │ │ │ +00036fb0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20  nputs,:.      * 
│ │ │ │ +00036fc0: 2a6e 6f74 6520 4d6f 6e6f 6d69 616c 733a  *note Monomials:
│ │ │ │ +00036fd0: 2028 4d61 6361 756c 6179 3244 6f63 2963   (Macaulay2Doc)c
│ │ │ │ +00036fe0: 6f65 6666 6963 6965 6e74 732c 203d 3e20  oefficients, => 
│ │ │ │ +00036ff0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c  ..., default val
│ │ │ │ +00037000: 7565 0a20 2020 2020 2020 206e 756c 6c2c  ue.        null,
│ │ │ │ +00037010: 0a20 2020 2020 202a 202a 6e6f 7465 2056  .      * *note V
│ │ │ │ +00037020: 6172 6961 626c 6573 3a20 284d 6163 6175  ariables: (Macau
│ │ │ │ +00037030: 6c61 7932 446f 6329 636f 6566 6669 6369  lay2Doc)coeffici
│ │ │ │ +00037040: 656e 7473 2c20 3d3e 202e 2e2e 2c20 6465  ents, => ..., de
│ │ │ │ +00037050: 6661 756c 7420 7661 6c75 650a 2020 2020  fault value.    
│ │ │ │ +00037060: 2020 2020 6e75 6c6c 2c0a 2020 2a20 4f75      null,.  * Ou
│ │ │ │ +00037070: 7470 7574 733a 0a20 2020 2020 202a 2061  tputs:.      * a
│ │ │ │ +00037080: 202a 6e6f 7465 206d 6174 7269 783a 2028   *note matrix: (
│ │ │ │ +00037090: 4d61 6361 756c 6179 3244 6f63 294d 6174  Macaulay2Doc)Mat
│ │ │ │ +000370a0: 7269 782c 2c20 7468 6520 636f 6566 6669  rix,, the coeffi
│ │ │ │ +000370b0: 6369 656e 7420 6d61 7472 6978 206f 6620  cient matrix of 
│ │ │ │ +000370c0: 7468 650a 2020 2020 2020 2020 706f 6c79  the.        poly
│ │ │ │ +000370d0: 6e6f 6d69 616c 7320 6465 6669 6e69 6e67  nomials defining
│ │ │ │ +000370e0: 2070 6869 0a0a 4465 7363 7269 7074 696f   phi..Descriptio
│ │ │ │ +000370f0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b  n.===========..+
│ │ │ │  00037100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037130: 2d2d 2b0a 7c69 3120 3a20 4b20 3d20 5151  --+.|i1 : K = QQ
│ │ │ │ -00037140: 3b20 7269 6e67 5039 203d 204b 5b78 5f30  ; ringP9 = K[x_0
│ │ │ │ -00037150: 2e2e 785f 395d 3b20 2020 2020 2020 2020  ..x_9];         
│ │ │ │ -00037160: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00037170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00037130: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20  --------+.|i1 : 
│ │ │ │ +00037140: 4b20 3d20 5151 3b20 7269 6e67 5039 203d  K = QQ; ringP9 =
│ │ │ │ +00037150: 204b 5b78 5f30 2e2e 785f 395d 3b20 2020   K[x_0..x_9];   
│ │ │ │ +00037160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037170: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00037180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000371a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20  --------+.|i3 : 
│ │ │ │ -000371b0: 4d20 3d20 7261 6e64 6f6d 284b 5e31 302c  M = random(K^10,
│ │ │ │ -000371c0: 4b5e 3130 2920 2020 2020 2020 2020 2020  K^10)           
│ │ │ │ +000371a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +000371b0: 7c69 3320 3a20 4d20 3d20 7261 6e64 6f6d  |i3 : M = random
│ │ │ │ +000371c0: 284b 5e31 302c 4b5e 3130 2920 2020 2020  (K^10,K^10)     
│ │ │ │  000371d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000371e0: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ +000371e0: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │  000371f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037210: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00037220: 7c6f 3320 3d20 7c20 392f 3220 2036 2f37  |o3 = | 9/2  6/7
│ │ │ │ -00037230: 2020 372f 3820 352f 3320 2035 2f34 2020    7/8 5/3  5/4  
│ │ │ │ -00037240: 392f 3130 2031 3020 2020 342f 3920 2035  9/10 10   4/9  5
│ │ │ │ -00037250: 2f36 2020 3320 2020 7c7c 0a7c 2020 2020  /6  3   ||.|    
│ │ │ │ -00037260: 207c 2039 2f34 2020 3620 2020 2035 2f36   | 9/4  6    5/6
│ │ │ │ -00037270: 2031 2f31 3020 322f 3920 2035 2f34 2020   1/10 2/9  5/4  
│ │ │ │ -00037280: 3130 2f39 2039 2f31 3020 372f 3130 2037  10/9 9/10 7/10 7
│ │ │ │ -00037290: 2f35 207c 7c0a 7c20 2020 2020 7c20 332f  /5 ||.|     | 3/
│ │ │ │ -000372a0: 3420 2035 2f34 2020 3520 2020 342f 3320  4  5/4  5   4/3 
│ │ │ │ -000372b0: 2038 2f35 2020 312f 3720 2038 2f33 2020   8/5  1/7  8/3  
│ │ │ │ -000372c0: 332f 3220 2031 2f38 2020 372f 3320 7c7c  3/2  1/8  7/3 ||
│ │ │ │ -000372d0: 0a7c 2020 2020 207c 2037 2f34 2020 322f  .|     | 7/4  2/
│ │ │ │ -000372e0: 3920 2032 2f35 2033 2f37 2020 392f 3420  9  2/5 3/7  9/4 
│ │ │ │ -000372f0: 2037 2f35 2020 332f 3420 2034 2f33 2020   7/5  3/4  4/3  
│ │ │ │ -00037300: 382f 3720 2032 2f37 207c 7c0a 7c20 2020  8/7  2/7 ||.|   
│ │ │ │ -00037310: 2020 7c20 372f 3920 2033 2f31 3020 352f    | 7/9  3/10 5/
│ │ │ │ -00037320: 3320 392f 3130 2032 2f39 2020 312f 3520  3 9/10 2/9  1/5 
│ │ │ │ -00037330: 2039 2f35 2020 312f 3820 2037 2f33 2020   9/5  1/8  7/3  
│ │ │ │ -00037340: 312f 3520 7c7c 0a7c 2020 2020 207c 2037  1/5 ||.|     | 7
│ │ │ │ -00037350: 2f31 3020 332f 3720 2037 2f32 2034 2f37  /10 3/7  7/2 4/7
│ │ │ │ -00037360: 2020 392f 3820 2035 2f37 2020 332f 3520    9/8  5/7  3/5 
│ │ │ │ -00037370: 2037 2f38 2020 3130 2f39 2035 2f37 207c   7/8  10/9 5/7 |
│ │ │ │ -00037380: 7c0a 7c20 2020 2020 7c20 372f 3130 2035  |.|     | 7/10 5
│ │ │ │ -00037390: 2020 2020 322f 3520 352f 3920 2031 2f38      2/5 5/9  1/8
│ │ │ │ -000373a0: 2020 332f 3820 2034 2020 2020 3130 2f39    3/8  4    10/9
│ │ │ │ -000373b0: 2038 2020 2020 372f 3320 7c7c 0a7c 2020   8    7/3 ||.|  
│ │ │ │ -000373c0: 2020 207c 2037 2f33 2020 3130 2f39 2036     | 7/3  10/9 6
│ │ │ │ -000373d0: 2f35 2035 2f39 2020 3130 2f33 2035 2f32  /5 5/9  10/3 5/2
│ │ │ │ -000373e0: 2020 372f 3520 2036 2f37 2020 3130 2f39    7/5  6/7  10/9
│ │ │ │ -000373f0: 2032 2020 207c 7c0a 7c20 2020 2020 7c20   2   ||.|     | 
│ │ │ │ -00037400: 3720 2020 2031 3020 2020 352f 3720 362f  7    10   5/7 6/
│ │ │ │ -00037410: 3720 2034 2020 2020 312f 3620 2035 2020  7  4    1/6  5  
│ │ │ │ -00037420: 2020 372f 3620 2038 2020 2020 3420 2020    7/6  8    4   
│ │ │ │ -00037430: 7c7c 0a7c 2020 2020 207c 2033 2f37 2020  ||.|     | 3/7  
│ │ │ │ -00037440: 332f 3220 2035 2f39 2036 2020 2020 312f  3/2  5/9 6    1/
│ │ │ │ -00037450: 3320 2038 2f35 2020 372f 3130 2035 2f36  3  8/5  7/10 5/6
│ │ │ │ -00037460: 2020 3220 2020 2038 2f33 207c 7c0a 7c20    2    8/3 ||.| 
│ │ │ │ -00037470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037220: 2020 2020 7c0a 7c6f 3320 3d20 7c20 392f      |.|o3 = | 9/
│ │ │ │ +00037230: 3220 2036 2f37 2020 372f 3820 352f 3320  2  6/7  7/8 5/3 
│ │ │ │ +00037240: 2035 2f34 2020 392f 3130 2031 3020 2020   5/4  9/10 10   
│ │ │ │ +00037250: 342f 3920 2035 2f36 2020 3320 2020 7c7c  4/9  5/6  3   ||
│ │ │ │ +00037260: 0a7c 2020 2020 207c 2039 2f34 2020 3620  .|     | 9/4  6 
│ │ │ │ +00037270: 2020 2035 2f36 2031 2f31 3020 322f 3920     5/6 1/10 2/9 
│ │ │ │ +00037280: 2035 2f34 2020 3130 2f39 2039 2f31 3020   5/4  10/9 9/10 
│ │ │ │ +00037290: 372f 3130 2037 2f35 207c 7c0a 7c20 2020  7/10 7/5 ||.|   
│ │ │ │ +000372a0: 2020 7c20 332f 3420 2035 2f34 2020 3520    | 3/4  5/4  5 
│ │ │ │ +000372b0: 2020 342f 3320 2038 2f35 2020 312f 3720    4/3  8/5  1/7 
│ │ │ │ +000372c0: 2038 2f33 2020 332f 3220 2031 2f38 2020   8/3  3/2  1/8  
│ │ │ │ +000372d0: 372f 3320 7c7c 0a7c 2020 2020 207c 2037  7/3 ||.|     | 7
│ │ │ │ +000372e0: 2f34 2020 322f 3920 2032 2f35 2033 2f37  /4  2/9  2/5 3/7
│ │ │ │ +000372f0: 2020 392f 3420 2037 2f35 2020 332f 3420    9/4  7/5  3/4 
│ │ │ │ +00037300: 2034 2f33 2020 382f 3720 2032 2f37 207c   4/3  8/7  2/7 |
│ │ │ │ +00037310: 7c0a 7c20 2020 2020 7c20 372f 3920 2033  |.|     | 7/9  3
│ │ │ │ +00037320: 2f31 3020 352f 3320 392f 3130 2032 2f39  /10 5/3 9/10 2/9
│ │ │ │ +00037330: 2020 312f 3520 2039 2f35 2020 312f 3820    1/5  9/5  1/8 
│ │ │ │ +00037340: 2037 2f33 2020 312f 3520 7c7c 0a7c 2020   7/3  1/5 ||.|  
│ │ │ │ +00037350: 2020 207c 2037 2f31 3020 332f 3720 2037     | 7/10 3/7  7
│ │ │ │ +00037360: 2f32 2034 2f37 2020 392f 3820 2035 2f37  /2 4/7  9/8  5/7
│ │ │ │ +00037370: 2020 332f 3520 2037 2f38 2020 3130 2f39    3/5  7/8  10/9
│ │ │ │ +00037380: 2035 2f37 207c 7c0a 7c20 2020 2020 7c20   5/7 ||.|     | 
│ │ │ │ +00037390: 372f 3130 2035 2020 2020 322f 3520 352f  7/10 5    2/5 5/
│ │ │ │ +000373a0: 3920 2031 2f38 2020 332f 3820 2034 2020  9  1/8  3/8  4  
│ │ │ │ +000373b0: 2020 3130 2f39 2038 2020 2020 372f 3320    10/9 8    7/3 
│ │ │ │ +000373c0: 7c7c 0a7c 2020 2020 207c 2037 2f33 2020  ||.|     | 7/3  
│ │ │ │ +000373d0: 3130 2f39 2036 2f35 2035 2f39 2020 3130  10/9 6/5 5/9  10
│ │ │ │ +000373e0: 2f33 2035 2f32 2020 372f 3520 2036 2f37  /3 5/2  7/5  6/7
│ │ │ │ +000373f0: 2020 3130 2f39 2032 2020 207c 7c0a 7c20    10/9 2   ||.| 
│ │ │ │ +00037400: 2020 2020 7c20 3720 2020 2031 3020 2020      | 7    10   
│ │ │ │ +00037410: 352f 3720 362f 3720 2034 2020 2020 312f  5/7 6/7  4    1/
│ │ │ │ +00037420: 3620 2035 2020 2020 372f 3620 2038 2020  6  5    7/6  8  
│ │ │ │ +00037430: 2020 3420 2020 7c7c 0a7c 2020 2020 207c    4   ||.|     |
│ │ │ │ +00037440: 2033 2f37 2020 332f 3220 2035 2f39 2036   3/7  3/2  5/9 6
│ │ │ │ +00037450: 2020 2020 312f 3320 2038 2f35 2020 372f      1/3  8/5  7/
│ │ │ │ +00037460: 3130 2035 2f36 2020 3220 2020 2038 2f33  10 5/6  2    8/3
│ │ │ │ +00037470: 207c 7c0a 7c20 2020 2020 2020 2020 2020   ||.|           
│ │ │ │  00037480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374a0: 2020 2020 2020 207c 0a7c 2020 2020 2020         |.|      
│ │ │ │ -000374b0: 2020 2020 2020 2020 3130 2020 2020 2020          10      
│ │ │ │ -000374c0: 2031 3020 2020 2020 2020 2020 2020 2020   10             
│ │ │ │ +000374a0: 2020 2020 2020 2020 2020 2020 207c 0a7c               |.|
│ │ │ │ +000374b0: 2020 2020 2020 2020 2020 2020 2020 3130                10
│ │ │ │ +000374c0: 2020 2020 2020 2031 3020 2020 2020 2020         10       
│ │ │ │  000374d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000374e0: 2020 7c0a 7c6f 3320 3a20 4d61 7472 6978    |.|o3 : Matrix
│ │ │ │ -000374f0: 2051 5120 2020 3c2d 2d20 5151 2020 2020   QQ   <-- QQ    
│ │ │ │ -00037500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037510: 2020 2020 2020 2020 2020 2020 207c 0a2b               |.+
│ │ │ │ -00037520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000374e0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20          |.|o3 : 
│ │ │ │ +000374f0: 4d61 7472 6978 2051 5120 2020 3c2d 2d20  Matrix QQ   <-- 
│ │ │ │ +00037500: 5151 2020 2020 2020 2020 2020 2020 2020  QQ              
│ │ │ │ +00037510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037520: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d     |.+----------
│ │ │ │  00037530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037550: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20  --------+.|i4 : 
│ │ │ │ -00037560: 7068 6920 3d20 7261 7469 6f6e 616c 4d61  phi = rationalMa
│ │ │ │ -00037570: 7020 2828 7661 7273 2072 696e 6750 3929  p ((vars ringP9)
│ │ │ │ -00037580: 202a 2028 7472 616e 7370 6f73 6520 4d29   * (transpose M)
│ │ │ │ -00037590: 293b 207c 0a7c 2020 2020 2020 2020 2020  ); |.|          
│ │ │ │ +00037550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a  --------------+.
│ │ │ │ +00037560: 7c69 3420 3a20 7068 6920 3d20 7261 7469  |i4 : phi = rati
│ │ │ │ +00037570: 6f6e 616c 4d61 7020 2828 7661 7273 2072  onalMap ((vars r
│ │ │ │ +00037580: 696e 6750 3929 202a 2028 7472 616e 7370  ingP9) * (transp
│ │ │ │ +00037590: 6f73 6520 4d29 293b 207c 0a7c 2020 2020  ose M)); |.|    
│ │ │ │  000375a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000375b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000375c0: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -000375d0: 7c6f 3420 3a20 5261 7469 6f6e 616c 4d61  |o4 : RationalMa
│ │ │ │ -000375e0: 7020 286c 696e 6561 7220 7261 7469 6f6e  p (linear ration
│ │ │ │ -000375f0: 616c 206d 6170 2066 726f 6d20 5050 5e39  al map from PP^9
│ │ │ │ -00037600: 2074 6f20 5050 5e39 297c 0a2b 2d2d 2d2d   to PP^9)|.+----
│ │ │ │ -00037610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000375c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000375d0: 2020 2020 7c0a 7c6f 3420 3a20 5261 7469      |.|o4 : Rati
│ │ │ │ +000375e0: 6f6e 616c 4d61 7020 286c 696e 6561 7220  onalMap (linear 
│ │ │ │ +000375f0: 7261 7469 6f6e 616c 206d 6170 2066 726f  rational map fro
│ │ │ │ +00037600: 6d20 5050 5e39 2074 6f20 5050 5e39 297c  m PP^9 to PP^9)|
│ │ │ │ +00037610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  00037620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037640: 2d2d 2d2d 2b0a 7c69 3520 3a20 4d27 203d  ----+.|i5 : M' =
│ │ │ │ -00037650: 2063 6f65 6666 6963 6965 6e74 7320 7068   coefficients ph
│ │ │ │ -00037660: 6920 2020 2020 2020 2020 2020 2020 2020  i               
│ │ │ │ -00037670: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00037680: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00037640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520  ----------+.|i5 
│ │ │ │ +00037650: 3a20 4d27 203d 2063 6f65 6666 6963 6965  : M' = coefficie
│ │ │ │ +00037660: 6e74 7320 7068 6920 2020 2020 2020 2020  nts phi         
│ │ │ │ +00037670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037680: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00037690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000376a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000376b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3520            |.|o5 
│ │ │ │ -000376c0: 3d20 7c20 392f 3220 2036 2f37 2020 372f  = | 9/2  6/7  7/
│ │ │ │ -000376d0: 3820 352f 3320 2035 2f34 2020 392f 3130  8 5/3  5/4  9/10
│ │ │ │ -000376e0: 2031 3020 2020 342f 3920 2035 2f36 2020   10   4/9  5/6  
│ │ │ │ -000376f0: 3320 2020 7c7c 0a7c 2020 2020 207c 2039  3   ||.|     | 9
│ │ │ │ -00037700: 2f34 2020 3620 2020 2035 2f36 2031 2f31  /4  6    5/6 1/1
│ │ │ │ -00037710: 3020 322f 3920 2035 2f34 2020 3130 2f39  0 2/9  5/4  10/9
│ │ │ │ -00037720: 2039 2f31 3020 372f 3130 2037 2f35 207c   9/10 7/10 7/5 |
│ │ │ │ -00037730: 7c0a 7c20 2020 2020 7c20 332f 3420 2035  |.|     | 3/4  5
│ │ │ │ -00037740: 2f34 2020 3520 2020 342f 3320 2038 2f35  /4  5   4/3  8/5
│ │ │ │ -00037750: 2020 312f 3720 2038 2f33 2020 332f 3220    1/7  8/3  3/2 
│ │ │ │ -00037760: 2031 2f38 2020 372f 3320 7c7c 0a7c 2020   1/8  7/3 ||.|  
│ │ │ │ -00037770: 2020 207c 2037 2f34 2020 322f 3920 2032     | 7/4  2/9  2
│ │ │ │ -00037780: 2f35 2033 2f37 2020 392f 3420 2037 2f35  /5 3/7  9/4  7/5
│ │ │ │ -00037790: 2020 332f 3420 2034 2f33 2020 382f 3720    3/4  4/3  8/7 
│ │ │ │ -000377a0: 2032 2f37 207c 7c0a 7c20 2020 2020 7c20   2/7 ||.|     | 
│ │ │ │ -000377b0: 372f 3920 2033 2f31 3020 352f 3320 392f  7/9  3/10 5/3 9/
│ │ │ │ -000377c0: 3130 2032 2f39 2020 312f 3520 2039 2f35  10 2/9  1/5  9/5
│ │ │ │ -000377d0: 2020 312f 3820 2037 2f33 2020 312f 3520    1/8  7/3  1/5 
│ │ │ │ -000377e0: 7c7c 0a7c 2020 2020 207c 2037 2f31 3020  ||.|     | 7/10 
│ │ │ │ -000377f0: 332f 3720 2037 2f32 2034 2f37 2020 392f  3/7  7/2 4/7  9/
│ │ │ │ -00037800: 3820 2035 2f37 2020 332f 3520 2037 2f38  8  5/7  3/5  7/8
│ │ │ │ -00037810: 2020 3130 2f39 2035 2f37 207c 7c0a 7c20    10/9 5/7 ||.| 
│ │ │ │ -00037820: 2020 2020 7c20 372f 3130 2035 2020 2020      | 7/10 5    
│ │ │ │ -00037830: 322f 3520 352f 3920 2031 2f38 2020 332f  2/5 5/9  1/8  3/
│ │ │ │ -00037840: 3820 2034 2020 2020 3130 2f39 2038 2020  8  4    10/9 8  
│ │ │ │ -00037850: 2020 372f 3320 7c7c 0a7c 2020 2020 207c    7/3 ||.|     |
│ │ │ │ -00037860: 2037 2f33 2020 3130 2f39 2036 2f35 2035   7/3  10/9 6/5 5
│ │ │ │ -00037870: 2f39 2020 3130 2f33 2035 2f32 2020 372f  /9  10/3 5/2  7/
│ │ │ │ -00037880: 3520 2036 2f37 2020 3130 2f39 2032 2020  5  6/7  10/9 2  
│ │ │ │ -00037890: 207c 7c0a 7c20 2020 2020 7c20 3720 2020   ||.|     | 7   
│ │ │ │ -000378a0: 2031 3020 2020 352f 3720 362f 3720 2034   10   5/7 6/7  4
│ │ │ │ -000378b0: 2020 2020 312f 3620 2035 2020 2020 372f      1/6  5    7/
│ │ │ │ -000378c0: 3620 2038 2020 2020 3420 2020 7c7c 0a7c  6  8    4   ||.|
│ │ │ │ -000378d0: 2020 2020 207c 2033 2f37 2020 332f 3220       | 3/7  3/2 
│ │ │ │ -000378e0: 2035 2f39 2036 2020 2020 312f 3320 2038   5/9 6    1/3  8
│ │ │ │ -000378f0: 2f35 2020 372f 3130 2035 2f36 2020 3220  /5  7/10 5/6  2 
│ │ │ │ -00037900: 2020 2038 2f33 207c 7c0a 7c20 2020 2020     8/3 ||.|     
│ │ │ │ -00037910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000376b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000376c0: 7c0a 7c6f 3520 3d20 7c20 392f 3220 2036  |.|o5 = | 9/2  6
│ │ │ │ +000376d0: 2f37 2020 372f 3820 352f 3320 2035 2f34  /7  7/8 5/3  5/4
│ │ │ │ +000376e0: 2020 392f 3130 2031 3020 2020 342f 3920    9/10 10   4/9 
│ │ │ │ +000376f0: 2035 2f36 2020 3320 2020 7c7c 0a7c 2020   5/6  3   ||.|  
│ │ │ │ +00037700: 2020 207c 2039 2f34 2020 3620 2020 2035     | 9/4  6    5
│ │ │ │ +00037710: 2f36 2031 2f31 3020 322f 3920 2035 2f34  /6 1/10 2/9  5/4
│ │ │ │ +00037720: 2020 3130 2f39 2039 2f31 3020 372f 3130    10/9 9/10 7/10
│ │ │ │ +00037730: 2037 2f35 207c 7c0a 7c20 2020 2020 7c20   7/5 ||.|     | 
│ │ │ │ +00037740: 332f 3420 2035 2f34 2020 3520 2020 342f  3/4  5/4  5   4/
│ │ │ │ +00037750: 3320 2038 2f35 2020 312f 3720 2038 2f33  3  8/5  1/7  8/3
│ │ │ │ +00037760: 2020 332f 3220 2031 2f38 2020 372f 3320    3/2  1/8  7/3 
│ │ │ │ +00037770: 7c7c 0a7c 2020 2020 207c 2037 2f34 2020  ||.|     | 7/4  
│ │ │ │ +00037780: 322f 3920 2032 2f35 2033 2f37 2020 392f  2/9  2/5 3/7  9/
│ │ │ │ +00037790: 3420 2037 2f35 2020 332f 3420 2034 2f33  4  7/5  3/4  4/3
│ │ │ │ +000377a0: 2020 382f 3720 2032 2f37 207c 7c0a 7c20    8/7  2/7 ||.| 
│ │ │ │ +000377b0: 2020 2020 7c20 372f 3920 2033 2f31 3020      | 7/9  3/10 
│ │ │ │ +000377c0: 352f 3320 392f 3130 2032 2f39 2020 312f  5/3 9/10 2/9  1/
│ │ │ │ +000377d0: 3520 2039 2f35 2020 312f 3820 2037 2f33  5  9/5  1/8  7/3
│ │ │ │ +000377e0: 2020 312f 3520 7c7c 0a7c 2020 2020 207c    1/5 ||.|     |
│ │ │ │ +000377f0: 2037 2f31 3020 332f 3720 2037 2f32 2034   7/10 3/7  7/2 4
│ │ │ │ +00037800: 2f37 2020 392f 3820 2035 2f37 2020 332f  /7  9/8  5/7  3/
│ │ │ │ +00037810: 3520 2037 2f38 2020 3130 2f39 2035 2f37  5  7/8  10/9 5/7
│ │ │ │ +00037820: 207c 7c0a 7c20 2020 2020 7c20 372f 3130   ||.|     | 7/10
│ │ │ │ +00037830: 2035 2020 2020 322f 3520 352f 3920 2031   5    2/5 5/9  1
│ │ │ │ +00037840: 2f38 2020 332f 3820 2034 2020 2020 3130  /8  3/8  4    10
│ │ │ │ +00037850: 2f39 2038 2020 2020 372f 3320 7c7c 0a7c  /9 8    7/3 ||.|
│ │ │ │ +00037860: 2020 2020 207c 2037 2f33 2020 3130 2f39       | 7/3  10/9
│ │ │ │ +00037870: 2036 2f35 2035 2f39 2020 3130 2f33 2035   6/5 5/9  10/3 5
│ │ │ │ +00037880: 2f32 2020 372f 3520 2036 2f37 2020 3130  /2  7/5  6/7  10
│ │ │ │ +00037890: 2f39 2032 2020 207c 7c0a 7c20 2020 2020  /9 2   ||.|     
│ │ │ │ +000378a0: 7c20 3720 2020 2031 3020 2020 352f 3720  | 7    10   5/7 
│ │ │ │ +000378b0: 362f 3720 2034 2020 2020 312f 3620 2035  6/7  4    1/6  5
│ │ │ │ +000378c0: 2020 2020 372f 3620 2038 2020 2020 3420      7/6  8    4 
│ │ │ │ +000378d0: 2020 7c7c 0a7c 2020 2020 207c 2033 2f37    ||.|     | 3/7
│ │ │ │ +000378e0: 2020 332f 3220 2035 2f39 2036 2020 2020    3/2  5/9 6    
│ │ │ │ +000378f0: 312f 3320 2038 2f35 2020 372f 3130 2035  1/3  8/5  7/10 5
│ │ │ │ +00037900: 2f36 2020 3220 2020 2038 2f33 207c 7c0a  /6  2    8/3 ||.
│ │ │ │ +00037910: 7c20 2020 2020 2020 2020 2020 2020 2020  |               
│ │ │ │  00037920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037940: 2020 207c 0a7c 2020 2020 2020 2020 2020     |.|          
│ │ │ │ -00037950: 2020 2020 3130 2020 2020 2020 2031 3020      10       10 
│ │ │ │ -00037960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037970: 2020 2020 2020 2020 2020 2020 2020 7c0a                |.
│ │ │ │ -00037980: 7c6f 3520 3a20 4d61 7472 6978 2051 5120  |o5 : Matrix QQ 
│ │ │ │ -00037990: 2020 3c2d 2d20 5151 2020 2020 2020 2020    <-- QQ        
│ │ │ │ +00037940: 2020 2020 2020 2020 207c 0a7c 2020 2020           |.|    
│ │ │ │ +00037950: 2020 2020 2020 2020 2020 3130 2020 2020            10    
│ │ │ │ +00037960: 2020 2031 3020 2020 2020 2020 2020 2020     10           
│ │ │ │ +00037970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037980: 2020 2020 7c0a 7c6f 3520 3a20 4d61 7472      |.|o5 : Matr
│ │ │ │ +00037990: 6978 2051 5120 2020 3c2d 2d20 5151 2020  ix QQ   <-- QQ  
│ │ │ │  000379a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000379b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d           |.+----
│ │ │ │ -000379c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000379b0: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ +000379c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  .+--------------
│ │ │ │  000379d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000379e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000379f0: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20 3d3d  ----+.|i6 : M ==
│ │ │ │ -00037a00: 204d 2720 2020 2020 2020 2020 2020 2020   M'             
│ │ │ │ +000379f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620  ----------+.|i6 
│ │ │ │ +00037a00: 3a20 4d20 3d3d 204d 2720 2020 2020 2020  : M == M'       
│ │ │ │  00037a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a20: 2020 2020 2020 2020 2020 2020 2020 207c                 |
│ │ │ │ -00037a30: 0a7c 2020 2020 2020 2020 2020 2020 2020  .|              
│ │ │ │ +00037a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037a30: 2020 2020 207c 0a7c 2020 2020 2020 2020       |.|        
│ │ │ │  00037a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037a60: 2020 2020 2020 2020 2020 7c0a 7c6f 3620            |.|o6 
│ │ │ │ -00037a70: 3d20 7472 7565 2020 2020 2020 2020 2020  = true          
│ │ │ │ +00037a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00037a70: 7c0a 7c6f 3620 3d20 7472 7565 2020 2020  |.|o6 = true    
│ │ │ │  00037a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00037a90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00037aa0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d       |.+--------
│ │ │ │ +00037aa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │  00037ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037ae0: 2b0a 0a57 6179 7320 746f 2075 7365 2074  +..Ways to use t
│ │ │ │ -00037af0: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d  his method:.====
│ │ │ │ -00037b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ -00037b10: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520  ====..  * *note 
│ │ │ │ -00037b20: 636f 6566 6669 6369 656e 7473 2852 6174  coefficients(Rat
│ │ │ │ -00037b30: 696f 6e61 6c4d 6170 293a 2063 6f65 6666  ionalMap): coeff
│ │ │ │ -00037b40: 6963 6965 6e74 735f 6c70 5261 7469 6f6e  icients_lpRation
│ │ │ │ -00037b50: 616c 4d61 705f 7270 2c20 2d2d 0a20 2020  alMap_rp, --.   
│ │ │ │ -00037b60: 2063 6f65 6666 6963 6965 6e74 206d 6174   coefficient mat
│ │ │ │ -00037b70: 7269 7820 6f66 2061 2072 6174 696f 6e61  rix of a rationa
│ │ │ │ -00037b80: 6c20 6d61 700a 2d2d 2d2d 2d2d 2d2d 2d2d  l map.----------
│ │ │ │ +00037ae0: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f  ------+..Ways to
│ │ │ │ +00037af0: 2075 7365 2074 6869 7320 6d65 7468 6f64   use this method
│ │ │ │ +00037b00: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  :.==============
│ │ │ │ +00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20  ==========..  * 
│ │ │ │ +00037b20: 2a6e 6f74 6520 636f 6566 6669 6369 656e  *note coefficien
│ │ │ │ +00037b30: 7473 2852 6174 696f 6e61 6c4d 6170 293a  ts(RationalMap):
│ │ │ │ +00037b40: 2063 6f65 6666 6963 6965 6e74 735f 6c70   coefficients_lp
│ │ │ │ +00037b50: 5261 7469 6f6e 616c 4d61 705f 7270 2c20  RationalMap_rp, 
│ │ │ │ +00037b60: 2d2d 0a20 2020 2063 6f65 6666 6963 6965  --.    coefficie
│ │ │ │ +00037b70: 6e74 206d 6174 7269 7820 6f66 2061 2072  nt matrix of a r
│ │ │ │ +00037b80: 6174 696f 6e61 6c20 6d61 700a 2d2d 2d2d  ational map.----
│ │ │ │  00037b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037bd0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -00037be0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -00037bf0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -00037c00: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -00037c10: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ -00037c20: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ -00037c30: 6179 322f 7061 636b 6167 6573 2f43 7265  ay2/packages/Cre
│ │ │ │ -00037c40: 6d6f 6e61 2f0a 646f 6375 6d65 6e74 6174  mona/.documentat
│ │ │ │ -00037c50: 696f 6e2e 6d32 3a35 3338 3a30 2e0a 1f0a  ion.m2:538:0....
│ │ │ │ -00037c60: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ -00037c70: 666f 2c20 4e6f 6465 3a20 6465 6772 6565  fo, Node: degree
│ │ │ │ -00037c80: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72  _lpRationalMap_r
│ │ │ │ -00037c90: 702c 204e 6578 743a 2064 6567 7265 654d  p, Next: degreeM
│ │ │ │ -00037ca0: 6170 2c20 5072 6576 3a20 636f 6566 6669  ap, Prev: coeffi
│ │ │ │ -00037cb0: 6369 656e 7473 5f6c 7052 6174 696f 6e61  cients_lpRationa
│ │ │ │ -00037cc0: 6c4d 6170 5f72 702c 2055 703a 2054 6f70  lMap_rp, Up: Top
│ │ │ │ -00037cd0: 0a0a 6465 6772 6565 2852 6174 696f 6e61  ..degree(Rationa
│ │ │ │ -00037ce0: 6c4d 6170 2920 2d2d 2064 6567 7265 6520  lMap) -- degree 
│ │ │ │ -00037cf0: 6f66 2061 2072 6174 696f 6e61 6c20 6d61  of a rational ma
│ │ │ │ -00037d00: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  p.**************
│ │ │ │ +00037bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00037be0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00037bf0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00037c00: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00037c10: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00037c20: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ +00037c30: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00037c40: 6573 2f43 7265 6d6f 6e61 2f0a 646f 6375  es/Cremona/.docu
│ │ │ │ +00037c50: 6d65 6e74 6174 696f 6e2e 6d32 3a35 3338  mentation.m2:538
│ │ │ │ +00037c60: 3a30 2e0a 1f0a 4669 6c65 3a20 4372 656d  :0....File: Crem
│ │ │ │ +00037c70: 6f6e 612e 696e 666f 2c20 4e6f 6465 3a20  ona.info, Node: 
│ │ │ │ +00037c80: 6465 6772 6565 5f6c 7052 6174 696f 6e61  degree_lpRationa
│ │ │ │ +00037c90: 6c4d 6170 5f72 702c 204e 6578 743a 2064  lMap_rp, Next: d
│ │ │ │ +00037ca0: 6567 7265 654d 6170 2c20 5072 6576 3a20  egreeMap, Prev: 
│ │ │ │ +00037cb0: 636f 6566 6669 6369 656e 7473 5f6c 7052  coefficients_lpR
│ │ │ │ +00037cc0: 6174 696f 6e61 6c4d 6170 5f72 702c 2055  ationalMap_rp, U
│ │ │ │ +00037cd0: 703a 2054 6f70 0a0a 6465 6772 6565 2852  p: Top..degree(R
│ │ │ │ +00037ce0: 6174 696f 6e61 6c4d 6170 2920 2d2d 2064  ationalMap) -- d
│ │ │ │ +00037cf0: 6567 7265 6520 6f66 2061 2072 6174 696f  egree of a ratio
│ │ │ │ +00037d00: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a  nal map.********
│ │ │ │  00037d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00037d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00037d30: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a  *..  * Function:
│ │ │ │ -00037d40: 202a 6e6f 7465 2064 6567 7265 653a 2028   *note degree: (
│ │ │ │ -00037d50: 4d61 6361 756c 6179 3244 6f63 2964 6567  Macaulay2Doc)deg
│ │ │ │ -00037d60: 7265 652c 0a20 202a 2055 7361 6765 3a20  ree,.  * Usage: 
│ │ │ │ -00037d70: 0a20 2020 2020 2020 2064 6567 7265 6520  .        degree 
│ │ │ │ -00037d80: 7068 690a 2020 2a20 496e 7075 7473 3a0a  phi.  * Inputs:.
│ │ │ │ -00037d90: 2020 2020 2020 2a20 7068 692c 2061 202a        * phi, a *
│ │ │ │ -00037da0: 6e6f 7465 2072 6174 696f 6e61 6c20 6d61  note rational ma
│ │ │ │ -00037db0: 703a 2052 6174 696f 6e61 6c4d 6170 2c0a  p: RationalMap,.
│ │ │ │ -00037dc0: 2020 2a20 4f75 7470 7574 733a 0a20 2020    * Outputs:.   
│ │ │ │ -00037dd0: 2020 202a 2061 6e20 2a6e 6f74 6520 696e     * an *note in
│ │ │ │ -00037de0: 7465 6765 723a 2028 4d61 6361 756c 6179  teger: (Macaulay
│ │ │ │ -00037df0: 3244 6f63 295a 5a2c 2c20 7468 6520 6465  2Doc)ZZ,, the de
│ │ │ │ -00037e00: 6772 6565 206f 6620 7068 692e 2053 6f20  gree of phi. So 
│ │ │ │ -00037e10: 7468 6973 2076 616c 7565 0a20 2020 2020  this value.     
│ │ │ │ -00037e20: 2020 2069 7320 3120 6966 2061 6e64 206f     is 1 if and o
│ │ │ │ -00037e30: 6e6c 7920 6966 2074 6865 206d 6170 2069  nly if the map i
│ │ │ │ -00037e40: 7320 6269 7261 7469 6f6e 616c 206f 6e74  s birational ont
│ │ │ │ -00037e50: 6f20 6974 7320 696d 6167 652e 0a0a 4465  o its image...De
│ │ │ │ -00037e60: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d  scription.======
│ │ │ │ -00037e70: 3d3d 3d3d 3d0a 0a54 6869 7320 6973 2061  =====..This is a
│ │ │ │ -00037e80: 2073 686f 7274 6375 7420 666f 7220 6465   shortcut for de
│ │ │ │ -00037e90: 6772 6565 4d61 7028 7068 692c 4365 7274  greeMap(phi,Cert
│ │ │ │ -00037ea0: 6966 793d 3e74 7275 652c 5665 7262 6f73  ify=>true,Verbos
│ │ │ │ -00037eb0: 653d 3e66 616c 7365 292c 2073 6565 202a  e=>false), see *
│ │ │ │ -00037ec0: 6e6f 7465 0a64 6567 7265 654d 6170 2852  note.degreeMap(R
│ │ │ │ -00037ed0: 6174 696f 6e61 6c4d 6170 293a 2064 6567  ationalMap): deg
│ │ │ │ -00037ee0: 7265 654d 6170 5f6c 7052 6174 696f 6e61  reeMap_lpRationa
│ │ │ │ -00037ef0: 6c4d 6170 5f72 702c 2e0a 0a57 6179 7320  lMap_rp,...Ways 
│ │ │ │ -00037f00: 746f 2075 7365 2074 6869 7320 6d65 7468  to use this meth
│ │ │ │ -00037f10: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  od:.============
│ │ │ │ -00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020  ============..  
│ │ │ │ -00037f30: 2a20 2a6e 6f74 6520 6465 6772 6565 2852  * *note degree(R
│ │ │ │ -00037f40: 6174 696f 6e61 6c4d 6170 293a 2064 6567  ationalMap): deg
│ │ │ │ -00037f50: 7265 655f 6c70 5261 7469 6f6e 616c 4d61  ree_lpRationalMa
│ │ │ │ -00037f60: 705f 7270 2c20 2d2d 2064 6567 7265 6520  p_rp, -- degree 
│ │ │ │ -00037f70: 6f66 2061 2072 6174 696f 6e61 6c0a 2020  of a rational.  
│ │ │ │ -00037f80: 2020 6d61 700a 2d2d 2d2d 2d2d 2d2d 2d2d    map.----------
│ │ │ │ +00037d30: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e  *******..  * Fun
│ │ │ │ +00037d40: 6374 696f 6e3a 202a 6e6f 7465 2064 6567  ction: *note deg
│ │ │ │ +00037d50: 7265 653a 2028 4d61 6361 756c 6179 3244  ree: (Macaulay2D
│ │ │ │ +00037d60: 6f63 2964 6567 7265 652c 0a20 202a 2055  oc)degree,.  * U
│ │ │ │ +00037d70: 7361 6765 3a20 0a20 2020 2020 2020 2064  sage: .        d
│ │ │ │ +00037d80: 6567 7265 6520 7068 690a 2020 2a20 496e  egree phi.  * In
│ │ │ │ +00037d90: 7075 7473 3a0a 2020 2020 2020 2a20 7068  puts:.      * ph
│ │ │ │ +00037da0: 692c 2061 202a 6e6f 7465 2072 6174 696f  i, a *note ratio
│ │ │ │ +00037db0: 6e61 6c20 6d61 703a 2052 6174 696f 6e61  nal map: Rationa
│ │ │ │ +00037dc0: 6c4d 6170 2c0a 2020 2a20 4f75 7470 7574  lMap,.  * Output
│ │ │ │ +00037dd0: 733a 0a20 2020 2020 202a 2061 6e20 2a6e  s:.      * an *n
│ │ │ │ +00037de0: 6f74 6520 696e 7465 6765 723a 2028 4d61  ote integer: (Ma
│ │ │ │ +00037df0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20  caulay2Doc)ZZ,, 
│ │ │ │ +00037e00: 7468 6520 6465 6772 6565 206f 6620 7068  the degree of ph
│ │ │ │ +00037e10: 692e 2053 6f20 7468 6973 2076 616c 7565  i. So this value
│ │ │ │ +00037e20: 0a20 2020 2020 2020 2069 7320 3120 6966  .        is 1 if
│ │ │ │ +00037e30: 2061 6e64 206f 6e6c 7920 6966 2074 6865   and only if the
│ │ │ │ +00037e40: 206d 6170 2069 7320 6269 7261 7469 6f6e   map is biration
│ │ │ │ +00037e50: 616c 206f 6e74 6f20 6974 7320 696d 6167  al onto its imag
│ │ │ │ +00037e60: 652e 0a0a 4465 7363 7269 7074 696f 6e0a  e...Description.
│ │ │ │ +00037e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869  ===========..Thi
│ │ │ │ +00037e80: 7320 6973 2061 2073 686f 7274 6375 7420  s is a shortcut 
│ │ │ │ +00037e90: 666f 7220 6465 6772 6565 4d61 7028 7068  for degreeMap(ph
│ │ │ │ +00037ea0: 692c 4365 7274 6966 793d 3e74 7275 652c  i,Certify=>true,
│ │ │ │ +00037eb0: 5665 7262 6f73 653d 3e66 616c 7365 292c  Verbose=>false),
│ │ │ │ +00037ec0: 2073 6565 202a 6e6f 7465 0a64 6567 7265   see *note.degre
│ │ │ │ +00037ed0: 654d 6170 2852 6174 696f 6e61 6c4d 6170  eMap(RationalMap
│ │ │ │ +00037ee0: 293a 2064 6567 7265 654d 6170 5f6c 7052  ): degreeMap_lpR
│ │ │ │ +00037ef0: 6174 696f 6e61 6c4d 6170 5f72 702c 2e0a  ationalMap_rp,..
│ │ │ │ +00037f00: 0a57 6179 7320 746f 2075 7365 2074 6869  .Ways to use thi
│ │ │ │ +00037f10: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d  s method:.======
│ │ │ │ +00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d  ================
│ │ │ │ +00037f30: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6465  ==..  * *note de
│ │ │ │ +00037f40: 6772 6565 2852 6174 696f 6e61 6c4d 6170  gree(RationalMap
│ │ │ │ +00037f50: 293a 2064 6567 7265 655f 6c70 5261 7469  ): degree_lpRati
│ │ │ │ +00037f60: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2064  onalMap_rp, -- d
│ │ │ │ +00037f70: 6567 7265 6520 6f66 2061 2072 6174 696f  egree of a ratio
│ │ │ │ +00037f80: 6e61 6c0a 2020 2020 6d61 700a 2d2d 2d2d  nal.    map.----
│ │ │ │  00037f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00037fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00037fd0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263  -----..The sourc
│ │ │ │ -00037fe0: 6520 6f66 2074 6869 7320 646f 6375 6d65  e of this docume
│ │ │ │ -00037ff0: 6e74 2069 7320 696e 0a2f 6275 696c 642f  nt is in./build/
│ │ │ │ -00038000: 7265 7072 6f64 7563 6962 6c65 2d70 6174  reproducible-pat
│ │ │ │ -00038010: 682f 6d61 6361 756c 6179 322d 312e 3236  h/macaulay2-1.26
│ │ │ │ -00038020: 2e30 362b 6473 2f4d 322f 4d61 6361 756c  .06+ds/M2/Macaul
│ │ │ │ -00038030: 6179 322f 7061 636b 6167 6573 2f43 7265  ay2/packages/Cre
│ │ │ │ -00038040: 6d6f 6e61 2f0a 646f 6375 6d65 6e74 6174  mona/.documentat
│ │ │ │ -00038050: 696f 6e2e 6d32 3a34 3932 3a30 2e0a 1f0a  ion.m2:492:0....
│ │ │ │ -00038060: 4669 6c65 3a20 4372 656d 6f6e 612e 696e  File: Cremona.in
│ │ │ │ -00038070: 666f 2c20 4e6f 6465 3a20 6465 6772 6565  fo, Node: degree
│ │ │ │ -00038080: 4d61 702c 204e 6578 743a 2064 6567 7265  Map, Next: degre
│ │ │ │ -00038090: 654d 6170 5f6c 7052 6174 696f 6e61 6c4d  eMap_lpRationalM
│ │ │ │ -000380a0: 6170 5f72 702c 2050 7265 763a 2064 6567  ap_rp, Prev: deg
│ │ │ │ -000380b0: 7265 655f 6c70 5261 7469 6f6e 616c 4d61  ree_lpRationalMa
│ │ │ │ -000380c0: 705f 7270 2c20 5570 3a20 546f 700a 0a64  p_rp, Up: Top..d
│ │ │ │ -000380d0: 6567 7265 654d 6170 202d 2d20 6465 6772  egreeMap -- degr
│ │ │ │ -000380e0: 6565 206f 6620 6120 7261 7469 6f6e 616c  ee of a rational
│ │ │ │ -000380f0: 206d 6170 2062 6574 7765 656e 2070 726f   map between pro
│ │ │ │ -00038100: 6a65 6374 6976 6520 7661 7269 6574 6965  jective varietie
│ │ │ │ -00038110: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  s.**************
│ │ │ │ +00037fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865  -----------..The
│ │ │ │ +00037fe0: 2073 6f75 7263 6520 6f66 2074 6869 7320   source of this 
│ │ │ │ +00037ff0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f  document is in./
│ │ │ │ +00038000: 6275 696c 642f 7265 7072 6f64 7563 6962  build/reproducib
│ │ │ │ +00038010: 6c65 2d70 6174 682f 6d61 6361 756c 6179  le-path/macaulay
│ │ │ │ +00038020: 322d 312e 3236 2e30 362b 6473 2f4d 322f  2-1.26.06+ds/M2/
│ │ │ │ +00038030: 4d61 6361 756c 6179 322f 7061 636b 6167  Macaulay2/packag
│ │ │ │ +00038040: 6573 2f43 7265 6d6f 6e61 2f0a 646f 6375  es/Cremona/.docu
│ │ │ │ +00038050: 6d65 6e74 6174 696f 6e2e 6d32 3a34 3932  mentation.m2:492
│ │ │ │ +00038060: 3a30 2e0a 1f0a 4669 6c65 3a20 4372 656d  :0....File: Crem
│ │ │ │ +00038070: 6f6e 612e 696e 666f 2c20 4e6f 6465 3a20  ona.info, Node: 
│ │ │ │ +00038080: 6465 6772 6565 4d61 702c 204e 6578 743a  degreeMap, Next:
│ │ │ │ +00038090: 2064 6567 7265 654d 6170 5f6c 7052 6174   degreeMap_lpRat
│ │ │ │ +000380a0: 696f 6e61 6c4d 6170 5f72 702c 2050 7265  ionalMap_rp, Pre
│ │ │ │ +000380b0: 763a 2064 6567 7265 655f 6c70 5261 7469  v: degree_lpRati
│ │ │ │ +000380c0: 6f6e 616c 4d61 705f 7270 2c20 5570 3a20  onalMap_rp, Up: 
│ │ │ │ +000380d0: 546f 700a 0a64 6567 7265 654d 6170 202d  Top..degreeMap -
│ │ │ │ +000380e0: 2d20 6465 6772 6565 206f 6620 6120 7261  - degree of a ra
│ │ │ │ +000380f0: 7469 6f6e 616c 206d 6170 2062 6574 7765  tional map betwe
│ │ │ │ +00038100: 656e 2070 726f 6a65 6374 6976 6520 7661  en projective va
│ │ │ │ +00038110: 7269 6574 6965 730a 2a2a 2a2a 2a2a 2a2a  rieties.********
│ │ │ │  00038120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00038130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │  00038140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a  ****************
│ │ │ │ -00038150: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a  ****..  * Usage:
│ │ │ │ -00038160: 200a 2020 2020 2020 2020 6465 6772 6565   .        degree
│ │ │ │ -00038170: 4d61 7020 7068 690a 2020 2a20 496e 7075  Map phi.  * Inpu
│ │ │ │ -00038180: 7473 3a0a 2020 2020 2020 2a20 7068 692c  ts:.      * phi,
│ │ │ │ -00038190: 2061 202a 6e6f 7465 2072 696e 6720 6d61   a *note ring ma
│ │ │ │ -000381a0: 703a 2028 4d61 6361 756c 6179 3244 6f63  p: (Macaulay2Doc
│ │ │ │ -000381b0: 2952 696e 674d 6170 2c2c 2077 6869 6368  )RingMap,, which
│ │ │ │ -000381c0: 2072 6570 7265 7365 6e74 7320 610a 2020   represents a.  
│ │ │ │ -000381d0: 2020 2020 2020 7261 7469 6f6e 616c 206d        rational m
│ │ │ │ -000381e0: 6170 2024 5c50 6869 2420 6265 7477 6565  ap $\Phi$ betwee
│ │ │ │ -000381f0: 6e20 7072 6f6a 6563 7469 7665 2076 6172  n projective var
│ │ │ │ -00038200: 6965 7469 6573 0a20 202a 202a 6e6f 7465  ieties.  * *note
│ │ │ │ -00038210: 204f 7074 696f 6e61 6c20 696e 7075 7473   Optional inputs
│ │ │ │ -00038220: 3a20 284d 6163 6175 6c61 7932 446f 6329  : (Macaulay2Doc)
│ │ │ │ -00038230: 7573 696e 6720 6675 6e63 7469 6f6e 7320  using functions 
│ │ │ │ -00038240: 7769 7468 206f 7074 696f 6e61 6c20 696e  with optional in
│ │ │ │ -00038250: 7075 7473 2c3a 0a20 2020 2020 202a 202a  puts,:.      * *
│ │ │ │ -00038260: 6e6f 7465 2042 6c6f 7755 7053 7472 6174  note BlowUpStrat
│ │ │ │ -00038270: 6567 793a 2042 6c6f 7755 7053 7472 6174  egy: BlowUpStrat
│ │ │ │ -00038280: 6567 792c 203d 3e20 2e2e 2e2c 2064 6566  egy, => ..., def
│ │ │ │ -00038290: 6175 6c74 2076 616c 7565 0a20 2020 2020  ault value.     
│ │ │ │ -000382a0: 2020 2022 456c 696d 696e 6174 6522 2c0a     "Eliminate",.
│ │ │ │ -000382b0: 2020 2020 2020 2a20 2a6e 6f74 6520 4365        * *note Ce
│ │ │ │ -000382c0: 7274 6966 793a 2043 6572 7469 6679 2c20  rtify: Certify, 
│ │ │ │ -000382d0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420  => ..., default 
│ │ │ │ -000382e0: 7661 6c75 6520 6661 6c73 652c 2077 6865  value false, whe
│ │ │ │ -000382f0: 7468 6572 2074 6f20 656e 7375 7265 0a20  ther to ensure. 
│ │ │ │ -00038300: 2020 2020 2020 2063 6f72 7265 6374 6e65         correctne
│ │ │ │ -00038310: 7373 206f 6620 6f75 7470 7574 0a20 2020  ss of output.   
│ │ │ │ -00038320: 2020 202a 202a 6e6f 7465 2056 6572 626f     * *note Verbo
│ │ │ │ -00038330: 7365 3a20 696e 7665 7273 654d 6170 5f6c  se: inverseMap_l
│ │ │ │ -00038340: 705f 7064 5f70 645f 7064 5f63 6d56 6572  p_pd_pd_pd_cmVer
│ │ │ │ -00038350: 626f 7365 3d3e 5f70 645f 7064 5f70 645f  bose=>_pd_pd_pd_
│ │ │ │ -00038360: 7270 2c20 3d3e 202e 2e2e 2c0a 2020 2020  rp, => ...,.    
│ │ │ │ -00038370: 2020 2020 6465 6661 756c 7420 7661 6c75      default valu
│ │ │ │ -00038380: 6520 7472 7565 2c0a 2020 2a20 4f75 7470  e true,.  * Outp
│ │ │ │ -00038390: 7574 733a 0a20 2020 2020 202a 2061 6e20  uts:.      * an 
│ │ │ │ -000383a0: 2a6e 6f74 6520 696e 7465 6765 723a 2028  *note integer: (
│ │ │ │ -000383b0: 4d61 6361 756c 6179 3244 6f63 295a 5a2c  Macaulay2Doc)ZZ,
│ │ │ │ -000383c0: 2c20 7468 6520 6465 6772 6565 206f 6620  , the degree of 
│ │ │ │ -000383d0: 245c 5068 6924 2e20 536f 2074 6869 730a  $\Phi$. So this.
│ │ │ │ -000383e0: 2020 2020 2020 2020 7661 6c75 6520 6973          value is
│ │ │ │ -000383f0: 2031 2069 6620 616e 6420 6f6e 6c79 2069   1 if and only i
│ │ │ │ -00038400: 6620 7468 6520 6d61 7020 6973 2062 6972  f the map is bir
│ │ │ │ -00038410: 6174 696f 6e61 6c20 6f6e 746f 2069 7473  ational onto its
│ │ │ │ -00038420: 2069 6d61 6765 2e0a 0a44 6573 6372 6970   image...Descrip
│ │ │ │ -00038430: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d  tion.===========
│ │ │ │ -00038440: 0a0a 4f6e 6520 696d 706f 7274 616e 7420  ..One important 
│ │ │ │ -00038450: 6361 7365 2069 7320 7768 656e 2024 5c50  case is when $\P
│ │ │ │ -00038460: 6869 3a5c 6d61 7468 6262 7b50 7d5e 6e3d  hi:\mathbb{P}^n=
│ │ │ │ -00038470: 5072 6f6a 284b 5b78 5f30 2c5c 6c64 6f74  Proj(K[x_0,\ldot
│ │ │ │ -00038480: 732c 785f 6e5d 290a 5c64 6173 6872 6967  s,x_n]).\dashrig
│ │ │ │ -00038490: 6874 6172 726f 7720 5c6d 6174 6862 627b  htarrow \mathbb{
│ │ │ │ -000384a0: 507d 5e6d 3d50 726f 6a28 4b5b 795f 302c  P}^m=Proj(K[y_0,
│ │ │ │ -000384b0: 5c6c 646f 7473 2c79 5f6d 5d29 2420 6973  \ldots,y_m])$ is
│ │ │ │ -000384c0: 2061 2072 6174 696f 6e61 6c20 6d61 7020   a rational map 
│ │ │ │ -000384d0: 6265 7477 6565 6e0a 7072 6f6a 6563 7469  between.projecti
│ │ │ │ -000384e0: 7665 2073 7061 6365 732c 2063 6f72 7265  ve spaces, corre
│ │ │ │ -000384f0: 7370 6f6e 6469 6e67 2074 6f20 6120 7269  sponding to a ri
│ │ │ │ -00038500: 6e67 206d 6170 2024 5c70 6869 242e 2049  ng map $\phi$. I
│ │ │ │ -00038510: 6620 2470 2420 6973 2061 2067 656e 6572  f $p$ is a gener
│ │ │ │ -00038520: 616c 0a70 6f69 6e74 206f 6620 245c 6d61  al.point of $\ma
│ │ │ │ -00038530: 7468 6262 7b50 7d5e 6e24 2c20 6465 6e6f  thbb{P}^n$, deno
│ │ │ │ -00038540: 7465 2062 7920 2446 5f70 285c 5068 6929  te by $F_p(\Phi)
│ │ │ │ -00038550: 2420 7468 6520 636c 6f73 7572 6520 6f66  $ the closure of
│ │ │ │ -00038560: 0a24 5c50 6869 5e7b 2d31 7d28 5c50 6869  .$\Phi^{-1}(\Phi
│ │ │ │ -00038570: 2870 2929 5c73 7562 7365 7465 7120 5c6d  (p))\subseteq \m
│ │ │ │ -00038580: 6174 6862 627b 507d 5e6e 242e 2054 6865  athbb{P}^n$. The
│ │ │ │ -00038590: 2064 6567 7265 6520 6f66 2024 5c50 6869   degree of $\Phi
│ │ │ │ -000385a0: 2420 6973 2064 6566 696e 6564 2061 730a  $ is defined as.
│ │ │ │ -000385b0: 7468 6520 6465 6772 6565 206f 6620 2446  the degree of $F
│ │ │ │ -000385c0: 5f70 285c 5068 6929 2420 6966 2024 6469  _p(\Phi)$ if $di
│ │ │ │ -000385d0: 6d20 465f 7028 5c50 6869 2920 3d20 3024  m F_p(\Phi) = 0$
│ │ │ │ -000385e0: 2061 6e64 2024 3024 206f 7468 6572 7769   and $0$ otherwi
│ │ │ │ -000385f0: 7365 2e20 4966 2024 5c50 6869 240a 6973  se. If $\Phi$.is
│ │ │ │ -00038600: 2064 6566 696e 6564 2062 7920 666f 726d   defined by form
│ │ │ │ -00038610: 7320 2446 5f30 2878 5f30 2c5c 6c64 6f74  s $F_0(x_0,\ldot
│ │ │ │ -00038620: 732c 785f 6e29 2c5c 6c64 6f74 732c 465f  s,x_n),\ldots,F_
│ │ │ │ -00038630: 6d28 785f 302c 5c6c 646f 7473 2c78 5f6e  m(x_0,\ldots,x_n
│ │ │ │ -00038640: 2924 2061 6e64 2024 495f 7024 0a69 7320  )$ and $I_p$.is 
│ │ │ │ -00038650: 7468 6520 6964 6561 6c20 6f66 2074 6865  the ideal of the
│ │ │ │ -00038660: 2070 6f69 6e74 2024 7024 2c20 7468 656e   point $p$, then
│ │ │ │ -00038670: 2074 6865 2069 6465 616c 206f 6620 2446   the ideal of $F
│ │ │ │ -00038680: 5f70 285c 5068 6929 2420 6973 206e 6f74  _p(\Phi)$ is not
│ │ │ │ -00038690: 6869 6e67 2062 7574 2074 6865 0a73 6174  hing but the.sat
│ │ │ │ -000386a0: 7572 6174 696f 6e20 247b 285c 7068 6928  uration ${(\phi(
│ │ │ │ -000386b0: 5c70 6869 5e7b 2d31 7d28 495f 7029 2929  \phi^{-1}(I_p)))
│ │ │ │ -000386c0: 3a28 465f 302c 2e2e 2e2e 2c46 5f6d 297d  :(F_0,....,F_m)}
│ │ │ │ -000386d0: 5e7b 5c69 6e66 7479 7d24 2e0a 0a2b 2d2d  ^{\infty}$...+--
│ │ │ │ -000386e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038150: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20  **********..  * 
│ │ │ │ +00038160: 5573 6167 653a 200a 2020 2020 2020 2020  Usage: .        
│ │ │ │ +00038170: 6465 6772 6565 4d61 7020 7068 690a 2020  degreeMap phi.  
│ │ │ │ +00038180: 2a20 496e 7075 7473 3a0a 2020 2020 2020  * Inputs:.      
│ │ │ │ +00038190: 2a20 7068 692c 2061 202a 6e6f 7465 2072  * phi, a *note r
│ │ │ │ +000381a0: 696e 6720 6d61 703a 2028 4d61 6361 756c  ing map: (Macaul
│ │ │ │ +000381b0: 6179 3244 6f63 2952 696e 674d 6170 2c2c  ay2Doc)RingMap,,
│ │ │ │ +000381c0: 2077 6869 6368 2072 6570 7265 7365 6e74   which represent
│ │ │ │ +000381d0: 7320 610a 2020 2020 2020 2020 7261 7469  s a.        rati
│ │ │ │ +000381e0: 6f6e 616c 206d 6170 2024 5c50 6869 2420  onal map $\Phi$ 
│ │ │ │ +000381f0: 6265 7477 6565 6e20 7072 6f6a 6563 7469  between projecti
│ │ │ │ +00038200: 7665 2076 6172 6965 7469 6573 0a20 202a  ve varieties.  *
│ │ │ │ +00038210: 202a 6e6f 7465 204f 7074 696f 6e61 6c20   *note Optional 
│ │ │ │ +00038220: 696e 7075 7473 3a20 284d 6163 6175 6c61  inputs: (Macaula
│ │ │ │ +00038230: 7932 446f 6329 7573 696e 6720 6675 6e63  y2Doc)using func
│ │ │ │ +00038240: 7469 6f6e 7320 7769 7468 206f 7074 696f  tions with optio
│ │ │ │ +00038250: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020  nal inputs,:.   
│ │ │ │ +00038260: 2020 202a 202a 6e6f 7465 2042 6c6f 7755     * *note BlowU
│ │ │ │ +00038270: 7053 7472 6174 6567 793a 2042 6c6f 7755  pStrategy: BlowU
│ │ │ │ +00038280: 7053 7472 6174 6567 792c 203d 3e20 2e2e  pStrategy, => ..
│ │ │ │ +00038290: 2e2c 2064 6566 6175 6c74 2076 616c 7565  ., default value
│ │ │ │ +000382a0: 0a20 2020 2020 2020 2022 456c 696d 696e  .        "Elimin
│ │ │ │ +000382b0: 6174 6522 2c0a 2020 2020 2020 2a20 2a6e  ate",.      * *n
│ │ │ │ +000382c0: 6f74 6520 4365 7274 6966 793a 2043 6572  ote Certify: Cer
│ │ │ │ +000382d0: 7469 6679 2c20 3d3e 202e 2e2e 2c20 6465  tify, => ..., de
│ │ │ │ +000382e0: 6661 756c 7420 7661 6c75 6520 6661 6c73  fault value fals
│ │ │ │ +000382f0: 652c 2077 6865 7468 6572 2074 6f20 656e  e, whether to en
│ │ │ │ +00038300: 7375 7265 0a20 2020 2020 2020 2063 6f72  sure.        cor
│ │ │ │ +00038310: 7265 6374 6e65 7373 206f 6620 6f75 7470  rectness of outp
│ │ │ │ +00038320: 7574 0a20 2020 2020 202a 202a 6e6f 7465  ut.      * *note
│ │ │ │ +00038330: 2056 6572 626f 7365 3a20 696e 7665 7273   Verbose: invers
│ │ │ │ +00038340: 654d 6170 5f6c 705f 7064 5f70 645f 7064  eMap_lp_pd_pd_pd
│ │ │ │ +00038350: 5f63 6d56 6572 626f 7365 3d3e 5f70 645f  _cmVerbose=>_pd_
│ │ │ │ +00038360: 7064 5f70 645f 7270 2c20 3d3e 202e 2e2e  pd_pd_rp, => ...
│ │ │ │ +00038370: 2c0a 2020 2020 2020 2020 6465 6661 756c  ,.        defaul
│ │ │ │ +00038380: 7420 7661 6c75 6520 7472 7565 2c0a 2020  t value true,.  
│ │ │ │ +00038390: 2a20 4f75 7470 7574 733a 0a20 2020 2020  * Outputs:.     
│ │ │ │ +000383a0: 202a 2061 6e20 2a6e 6f74 6520 696e 7465   * an *note inte
│ │ │ │ +000383b0: 6765 723a 2028 4d61 6361 756c 6179 3244  ger: (Macaulay2D
│ │ │ │ +000383c0: 6f63 295a 5a2c 2c20 7468 6520 6465 6772  oc)ZZ,, the degr
│ │ │ │ +000383d0: 6565 206f 6620 245c 5068 6924 2e20 536f  ee of $\Phi$. So
│ │ │ │ +000383e0: 2074 6869 730a 2020 2020 2020 2020 7661   this.        va
│ │ │ │ +000383f0: 6c75 6520 6973 2031 2069 6620 616e 6420  lue is 1 if and 
│ │ │ │ +00038400: 6f6e 6c79 2069 6620 7468 6520 6d61 7020  only if the map 
│ │ │ │ +00038410: 6973 2062 6972 6174 696f 6e61 6c20 6f6e  is birational on
│ │ │ │ +00038420: 746f 2069 7473 2069 6d61 6765 2e0a 0a44  to its image...D
│ │ │ │ +00038430: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d  escription.=====
│ │ │ │ +00038440: 3d3d 3d3d 3d3d 0a0a 4f6e 6520 696d 706f  ======..One impo
│ │ │ │ +00038450: 7274 616e 7420 6361 7365 2069 7320 7768  rtant case is wh
│ │ │ │ +00038460: 656e 2024 5c50 6869 3a5c 6d61 7468 6262  en $\Phi:\mathbb
│ │ │ │ +00038470: 7b50 7d5e 6e3d 5072 6f6a 284b 5b78 5f30  {P}^n=Proj(K[x_0
│ │ │ │ +00038480: 2c5c 6c64 6f74 732c 785f 6e5d 290a 5c64  ,\ldots,x_n]).\d
│ │ │ │ +00038490: 6173 6872 6967 6874 6172 726f 7720 5c6d  ashrightarrow \m
│ │ │ │ +000384a0: 6174 6862 627b 507d 5e6d 3d50 726f 6a28  athbb{P}^m=Proj(
│ │ │ │ +000384b0: 4b5b 795f 302c 5c6c 646f 7473 2c79 5f6d  K[y_0,\ldots,y_m
│ │ │ │ +000384c0: 5d29 2420 6973 2061 2072 6174 696f 6e61  ])$ is a rationa
│ │ │ │ +000384d0: 6c20 6d61 7020 6265 7477 6565 6e0a 7072  l map between.pr
│ │ │ │ +000384e0: 6f6a 6563 7469 7665 2073 7061 6365 732c  ojective spaces,
│ │ │ │ +000384f0: 2063 6f72 7265 7370 6f6e 6469 6e67 2074   corresponding t
│ │ │ │ +00038500: 6f20 6120 7269 6e67 206d 6170 2024 5c70  o a ring map $\p
│ │ │ │ +00038510: 6869 242e 2049 6620 2470 2420 6973 2061  hi$. If $p$ is a
│ │ │ │ +00038520: 2067 656e 6572 616c 0a70 6f69 6e74 206f   general.point o
│ │ │ │ +00038530: 6620 245c 6d61 7468 6262 7b50 7d5e 6e24  f $\mathbb{P}^n$
│ │ │ │ +00038540: 2c20 6465 6e6f 7465 2062 7920 2446 5f70  , denote by $F_p
│ │ │ │ +00038550: 285c 5068 6929 2420 7468 6520 636c 6f73  (\Phi)$ the clos
│ │ │ │ +00038560: 7572 6520 6f66 0a24 5c50 6869 5e7b 2d31  ure of.$\Phi^{-1
│ │ │ │ +00038570: 7d28 5c50 6869 2870 2929 5c73 7562 7365  }(\Phi(p))\subse
│ │ │ │ +00038580: 7465 7120 5c6d 6174 6862 627b 507d 5e6e  teq \mathbb{P}^n
│ │ │ │ +00038590: 242e 2054 6865 2064 6567 7265 6520 6f66  $. The degree of
│ │ │ │ +000385a0: 2024 5c50 6869 2420 6973 2064 6566 696e   $\Phi$ is defin
│ │ │ │ +000385b0: 6564 2061 730a 7468 6520 6465 6772 6565  ed as.the degree
│ │ │ │ +000385c0: 206f 6620 2446 5f70 285c 5068 6929 2420   of $F_p(\Phi)$ 
│ │ │ │ +000385d0: 6966 2024 6469 6d20 465f 7028 5c50 6869  if $dim F_p(\Phi
│ │ │ │ +000385e0: 2920 3d20 3024 2061 6e64 2024 3024 206f  ) = 0$ and $0$ o
│ │ │ │ +000385f0: 7468 6572 7769 7365 2e20 4966 2024 5c50  therwise. If $\P
│ │ │ │ +00038600: 6869 240a 6973 2064 6566 696e 6564 2062  hi$.is defined b
│ │ │ │ +00038610: 7920 666f 726d 7320 2446 5f30 2878 5f30  y forms $F_0(x_0
│ │ │ │ +00038620: 2c5c 6c64 6f74 732c 785f 6e29 2c5c 6c64  ,\ldots,x_n),\ld
│ │ │ │ +00038630: 6f74 732c 465f 6d28 785f 302c 5c6c 646f  ots,F_m(x_0,\ldo
│ │ │ │ +00038640: 7473 2c78 5f6e 2924 2061 6e64 2024 495f  ts,x_n)$ and $I_
│ │ │ │ +00038650: 7024 0a69 7320 7468 6520 6964 6561 6c20  p$.is the ideal 
│ │ │ │ +00038660: 6f66 2074 6865 2070 6f69 6e74 2024 7024  of the point $p$
│ │ │ │ +00038670: 2c20 7468 656e 2074 6865 2069 6465 616c  , then the ideal
│ │ │ │ +00038680: 206f 6620 2446 5f70 285c 5068 6929 2420   of $F_p(\Phi)$ 
│ │ │ │ +00038690: 6973 206e 6f74 6869 6e67 2062 7574 2074  is nothing but t
│ │ │ │ +000386a0: 6865 0a73 6174 7572 6174 696f 6e20 247b  he.saturation ${
│ │ │ │ +000386b0: 285c 7068 6928 5c70 6869 5e7b 2d31 7d28  (\phi(\phi^{-1}(
│ │ │ │ +000386c0: 495f 7029 2929 3a28 465f 302c 2e2e 2e2e  I_p))):(F_0,....
│ │ │ │ +000386d0: 2c46 5f6d 297d 5e7b 5c69 6e66 7479 7d24  ,F_m)}^{\infty}$
│ │ │ │ +000386e0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ...+------------
│ │ │ │  000386f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931  -----------+.|i1
│ │ │ │ -00038730: 203a 202d 2d20 5461 6b65 2061 2072 6174   : -- Take a rat
│ │ │ │ -00038740: 696f 6e61 6c20 6d61 7020 7068 693a 505e  ional map phi:P^
│ │ │ │ -00038750: 382d 2d2d 3e47 2831 2c35 2920 7375 6273  8--->G(1,5) subs
│ │ │ │ -00038760: 6574 2050 5e31 3420 6465 6669 6e65 6420  et P^14 defined 
│ │ │ │ -00038770: 6279 2020 2020 2020 2020 207c 0a7c 2020  by         |.|  
│ │ │ │ -00038780: 2020 202d 2d20 6f66 2061 2067 656e 6572     -- of a gener
│ │ │ │ -00038790: 6963 2032 2078 2036 206d 6174 7269 7820  ic 2 x 6 matrix 
│ │ │ │ -000387a0: 6f66 206c 696e 6561 7220 666f 726d 7320  of linear forms 
│ │ │ │ -000387b0: 6f6e 2050 5e38 2028 7468 7573 2070 6869  on P^8 (thus phi
│ │ │ │ -000387c0: 2069 7320 2020 2020 2020 207c 0a7c 2020   is        |.|  
│ │ │ │ -000387d0: 2020 204b 3d5a 5a2f 3333 3331 3b20 7269     K=ZZ/3331; ri
│ │ │ │ -000387e0: 6e67 5038 3d4b 5b78 5f30 2e2e 785f 385d  ngP8=K[x_0..x_8]
│ │ │ │ -000387f0: 3b20 7269 6e67 5031 343d 4b5b 745f 302e  ; ringP14=K[t_0.
│ │ │ │ -00038800: 2e74 5f31 345d 3b20 2020 2020 2020 2020  .t_14];         
│ │ │ │ -00038810: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00038820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038730: 2d2b 0a7c 6931 203a 202d 2d20 5461 6b65  -+.|i1 : -- Take
│ │ │ │ +00038740: 2061 2072 6174 696f 6e61 6c20 6d61 7020   a rational map 
│ │ │ │ +00038750: 7068 693a 505e 382d 2d2d 3e47 2831 2c35  phi:P^8--->G(1,5
│ │ │ │ +00038760: 2920 7375 6273 6574 2050 5e31 3420 6465  ) subset P^14 de
│ │ │ │ +00038770: 6669 6e65 6420 6279 2020 2020 2020 2020  fined by        
│ │ │ │ +00038780: 207c 0a7c 2020 2020 202d 2d20 6f66 2061   |.|     -- of a
│ │ │ │ +00038790: 2067 656e 6572 6963 2032 2078 2036 206d   generic 2 x 6 m
│ │ │ │ +000387a0: 6174 7269 7820 6f66 206c 696e 6561 7220  atrix of linear 
│ │ │ │ +000387b0: 666f 726d 7320 6f6e 2050 5e38 2028 7468  forms on P^8 (th
│ │ │ │ +000387c0: 7573 2070 6869 2069 7320 2020 2020 2020  us phi is       
│ │ │ │ +000387d0: 207c 0a7c 2020 2020 204b 3d5a 5a2f 3333   |.|     K=ZZ/33
│ │ │ │ +000387e0: 3331 3b20 7269 6e67 5038 3d4b 5b78 5f30  31; ringP8=K[x_0
│ │ │ │ +000387f0: 2e2e 785f 385d 3b20 7269 6e67 5031 343d  ..x_8]; ringP14=
│ │ │ │ +00038800: 4b5b 745f 302e 2e74 5f31 345d 3b20 2020  K[t_0..t_14];   
│ │ │ │ +00038810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038820: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00038830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7468  -----------|.|th
│ │ │ │ -00038870: 6520 6d61 7869 6d61 6c20 6d69 6e6f 7273  e maximal minors
│ │ │ │ -00038880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038870: 2d7c 0a7c 7468 6520 6d61 7869 6d61 6c20  -|.|the maximal 
│ │ │ │ +00038880: 6d69 6e6f 7273 2020 2020 2020 2020 2020  minors          
│ │ │ │  00038890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000388b0: 2020 2020 2020 2020 2020 207c 0a7c 2062             |.| b
│ │ │ │ -000388c0: 6972 6174 696f 6e61 6c20 6f6e 746f 2069  irational onto i
│ │ │ │ -000388d0: 7473 2069 6d61 6765 2920 2020 2020 2020  ts image)       
│ │ │ │ +000388b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000388c0: 207c 0a7c 2062 6972 6174 696f 6e61 6c20   |.| birational 
│ │ │ │ +000388d0: 6f6e 746f 2069 7473 2069 6d61 6765 2920  onto its image) 
│ │ │ │  000388e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000388f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038900: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -00038910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038910: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  00038920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934  -----------+.|i4
│ │ │ │ -00038960: 203a 2070 6869 3d6d 6170 2872 696e 6750   : phi=map(ringP
│ │ │ │ -00038970: 382c 7269 6e67 5031 342c 6765 6e73 206d  8,ringP14,gens m
│ │ │ │ -00038980: 696e 6f72 7328 322c 6d61 7472 6978 2070  inors(2,matrix p
│ │ │ │ -00038990: 6163 6b28 362c 666f 7220 6920 746f 2020  ack(6,for i to  
│ │ │ │ -000389a0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000389b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038960: 2d2b 0a7c 6934 203a 2070 6869 3d6d 6170  -+.|i4 : phi=map
│ │ │ │ +00038970: 2872 696e 6750 382c 7269 6e67 5031 342c  (ringP8,ringP14,
│ │ │ │ +00038980: 6765 6e73 206d 696e 6f72 7328 322c 6d61  gens minors(2,ma
│ │ │ │ +00038990: 7472 6978 2070 6163 6b28 362c 666f 7220  trix pack(6,for 
│ │ │ │ +000389a0: 6920 746f 2020 2020 2020 2020 2020 2020  i to            
│ │ │ │ +000389b0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000389c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000389d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000389e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000389f0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00038a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038a10: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00038a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038a30: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00038a40: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -00038a50: 203d 206d 6170 2028 7269 6e67 5038 2c20   = map (ringP8, 
│ │ │ │ -00038a60: 7269 6e67 5031 342c 207b 2d20 3935 7820  ringP14, {- 95x 
│ │ │ │ -00038a70: 202b 2031 3831 7820 7820 202b 2031 3032   + 181x x  + 102
│ │ │ │ -00038a80: 3878 2020 2d20 3133 3834 7820 7820 2020  8x  - 1384x x   
│ │ │ │ -00038a90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00038aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038ab0: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00038ac0: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ -00038ad0: 2020 3120 2020 2020 2020 2030 2032 2020    1        0 2  
│ │ │ │ -00038ae0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00038af0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000389f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a00: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a20: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00038a30: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00038a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038a50: 207c 0a7c 6f34 203d 206d 6170 2028 7269   |.|o4 = map (ri
│ │ │ │ +00038a60: 6e67 5038 2c20 7269 6e67 5031 342c 207b  ngP8, ringP14, {
│ │ │ │ +00038a70: 2d20 3935 7820 202b 2031 3831 7820 7820  - 95x  + 181x x 
│ │ │ │ +00038a80: 202b 2031 3032 3878 2020 2d20 3133 3834   + 1028x  - 1384
│ │ │ │ +00038a90: 7820 7820 2020 2020 2020 2020 2020 2020  x x             
│ │ │ │ +00038aa0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ac0: 2020 2020 2030 2020 2020 2020 2030 2031       0       0 1
│ │ │ │ +00038ad0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +00038ae0: 2030 2032 2020 2020 2020 2020 2020 2020   0 2            
│ │ │ │ +00038af0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00038b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038b30: 2020 2020 2020 2020 2020 207c 0a7c 6f34             |.|o4
│ │ │ │ -00038b40: 203a 2052 696e 674d 6170 2072 696e 6750   : RingMap ringP
│ │ │ │ -00038b50: 3820 3c2d 2d20 7269 6e67 5031 3420 2020  8 <-- ringP14   
│ │ │ │ -00038b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038b40: 207c 0a7c 6f34 203a 2052 696e 674d 6170   |.|o4 : RingMap
│ │ │ │ +00038b50: 2072 696e 6750 3820 3c2d 2d20 7269 6e67   ringP8 <-- ring
│ │ │ │ +00038b60: 5031 3420 2020 2020 2020 2020 2020 2020  P14             
│ │ │ │  00038b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038b80: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00038b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038b90: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00038ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3131  -----------|.|11
│ │ │ │ -00038be0: 206c 6973 7420 7261 6e64 6f6d 2831 2c72   list random(1,r
│ │ │ │ -00038bf0: 696e 6750 3829 2929 2920 2020 2020 2020  ingP8))))       
│ │ │ │ +00038bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038be0: 2d7c 0a7c 3131 206c 6973 7420 7261 6e64  -|.|11 list rand
│ │ │ │ +00038bf0: 6f6d 2831 2c72 696e 6750 3829 2929 2920  om(1,ringP8)))) 
│ │ │ │  00038c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c20: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00038c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038c30: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00038c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038c70: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00038c80: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00038c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038c80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00038c90: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00038ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038cb0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00038cc0: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -00038cd0: 3134 3535 7820 7820 202b 2035 3539 7820  1455x x  + 559x 
│ │ │ │ -00038ce0: 202d 2035 3032 7820 7820 202b 2031 3236   - 502x x  + 126
│ │ │ │ -00038cf0: 3478 2078 2020 2d20 3136 3278 2078 2020  4x x  - 162x x  
│ │ │ │ -00038d00: 2b20 3132 3039 7820 202d 2031 3830 7820  + 1209x  - 180x 
│ │ │ │ -00038d10: 7820 202d 2020 2020 2020 207c 0a7c 2020  x  -       |.|  
│ │ │ │ -00038d20: 2020 2020 2031 2032 2020 2020 2020 2032       1 2       2
│ │ │ │ -00038d30: 2020 2020 2020 2030 2033 2020 2020 2020         0 3      
│ │ │ │ -00038d40: 2020 3120 3320 2020 2020 2020 3220 3320    1 3       2 3 
│ │ │ │ -00038d50: 2020 2020 2020 2033 2020 2020 2020 2030         3       0
│ │ │ │ -00038d60: 2034 2020 2020 2020 2020 207c 0a7c 2d2d   4         |.|--
│ │ │ │ -00038d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038cb0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00038cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038cd0: 207c 0a7c 2d20 3134 3535 7820 7820 202b   |.|- 1455x x  +
│ │ │ │ +00038ce0: 2035 3539 7820 202d 2035 3032 7820 7820   559x  - 502x x 
│ │ │ │ +00038cf0: 202b 2031 3236 3478 2078 2020 2d20 3136   + 1264x x  - 16
│ │ │ │ +00038d00: 3278 2078 2020 2b20 3132 3039 7820 202d  2x x  + 1209x  -
│ │ │ │ +00038d10: 2031 3830 7820 7820 202d 2020 2020 2020   180x x  -      
│ │ │ │ +00038d20: 207c 0a7c 2020 2020 2020 2031 2032 2020   |.|       1 2  
│ │ │ │ +00038d30: 2020 2020 2032 2020 2020 2020 2030 2033       2       0 3
│ │ │ │ +00038d40: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ +00038d50: 2020 3220 3320 2020 2020 2020 2033 2020    2 3        3  
│ │ │ │ +00038d60: 2020 2020 2030 2034 2020 2020 2020 2020       0 4        
│ │ │ │ +00038d70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00038d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00038dc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038dc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00038dd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038de0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00038de0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  00038df0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038e00: 2020 2020 2020 2020 2020 207c 0a7c 3530             |.|50
│ │ │ │ -00038e10: 3478 2078 2020 2d20 3131 3638 7820 7820  4x x  - 1168x x 
│ │ │ │ -00038e20: 202d 2036 3736 7820 7820 202b 2035 3031   - 676x x  + 501
│ │ │ │ -00038e30: 7820 202b 2037 3378 2078 2020 2b20 3132  x  + 73x x  + 12
│ │ │ │ -00038e40: 3633 7820 7820 202b 2031 3033 3578 2078  63x x  + 1035x x
│ │ │ │ -00038e50: 2020 2b20 3834 3478 2078 207c 0a7c 2020    + 844x x |.|  
│ │ │ │ -00038e60: 2020 3120 3420 2020 2020 2020 2032 2034    1 4        2 4
│ │ │ │ -00038e70: 2020 2020 2020 2033 2034 2020 2020 2020         3 4      
│ │ │ │ -00038e80: 2034 2020 2020 2020 3020 3520 2020 2020   4      0 5     
│ │ │ │ -00038e90: 2020 2031 2035 2020 2020 2020 2020 3220     1 5        2 
│ │ │ │ -00038ea0: 3520 2020 2020 2020 3320 357c 0a7c 2d2d  5       3 5|.|--
│ │ │ │ -00038eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038e10: 207c 0a7c 3530 3478 2078 2020 2d20 3131   |.|504x x  - 11
│ │ │ │ +00038e20: 3638 7820 7820 202d 2036 3736 7820 7820  68x x  - 676x x 
│ │ │ │ +00038e30: 202b 2035 3031 7820 202b 2037 3378 2078   + 501x  + 73x x
│ │ │ │ +00038e40: 2020 2b20 3132 3633 7820 7820 202b 2031    + 1263x x  + 1
│ │ │ │ +00038e50: 3033 3578 2078 2020 2b20 3834 3478 2078  035x x  + 844x x
│ │ │ │ +00038e60: 207c 0a7c 2020 2020 3120 3420 2020 2020   |.|    1 4     
│ │ │ │ +00038e70: 2020 2032 2034 2020 2020 2020 2033 2034     2 4       3 4
│ │ │ │ +00038e80: 2020 2020 2020 2034 2020 2020 2020 3020         4      0 
│ │ │ │ +00038e90: 3520 2020 2020 2020 2031 2035 2020 2020  5        1 5    
│ │ │ │ +00038ea0: 2020 2020 3220 3520 2020 2020 2020 3320      2 5       3 
│ │ │ │ +00038eb0: 357c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  5|.|------------
│ │ │ │  00038ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00038ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00038ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00038f00: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00038f10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038f00: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00038f10: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00038f20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00038f30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00038f40: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -00038f50: 3135 3933 7820 7820 202b 2037 3835 7820  1593x x  + 785x 
│ │ │ │ -00038f60: 202b 2039 3832 7820 7820 202d 2034 3132   + 982x x  - 412
│ │ │ │ -00038f70: 7820 7820 202b 2031 3333 3578 2078 2020  x x  + 1335x x  
│ │ │ │ -00038f80: 2b20 3131 3336 7820 7820 202b 2038 3236  + 1136x x  + 826
│ │ │ │ -00038f90: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -00038fa0: 2020 2020 2034 2035 2020 2020 2020 2035       4 5       5
│ │ │ │ -00038fb0: 2020 2020 2020 2030 2036 2020 2020 2020         0 6      
│ │ │ │ -00038fc0: 2031 2036 2020 2020 2020 2020 3220 3620   1 6        2 6 
│ │ │ │ -00038fd0: 2020 2020 2020 2033 2036 2020 2020 2020         3 6      
│ │ │ │ -00038fe0: 2034 2036 2020 2020 2020 207c 0a7c 2d2d   4 6       |.|--
│ │ │ │ -00038ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00038f40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00038f50: 207c 0a7c 2b20 3135 3933 7820 7820 202b   |.|+ 1593x x  +
│ │ │ │ +00038f60: 2037 3835 7820 202b 2039 3832 7820 7820   785x  + 982x x 
│ │ │ │ +00038f70: 202d 2034 3132 7820 7820 202b 2031 3333   - 412x x  + 133
│ │ │ │ +00038f80: 3578 2078 2020 2b20 3131 3336 7820 7820  5x x  + 1136x x 
│ │ │ │ +00038f90: 202b 2038 3236 7820 7820 202b 2020 2020   + 826x x  +    
│ │ │ │ +00038fa0: 207c 0a7c 2020 2020 2020 2034 2035 2020   |.|       4 5  
│ │ │ │ +00038fb0: 2020 2020 2035 2020 2020 2020 2030 2036       5       0 6
│ │ │ │ +00038fc0: 2020 2020 2020 2031 2036 2020 2020 2020         1 6      
│ │ │ │ +00038fd0: 2020 3220 3620 2020 2020 2020 2033 2036    2 6        3 6
│ │ │ │ +00038fe0: 2020 2020 2020 2034 2036 2020 2020 2020         4 6      
│ │ │ │ +00038ff0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039040: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00039050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039040: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00039050: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  00039060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039080: 2020 2020 2020 2020 2020 207c 0a7c 3130             |.|10
│ │ │ │ -00039090: 3738 7820 7820 202b 2031 3135 3878 2020  78x x  + 1158x  
│ │ │ │ -000390a0: 2b20 3333 3578 2078 2020 2d20 3938 3278  + 335x x  - 982x
│ │ │ │ -000390b0: 2078 2020 2d20 3134 3739 7820 7820 202d   x  - 1479x x  -
│ │ │ │ -000390c0: 2031 3578 2078 2020 2b20 3133 3633 7820   15x x  + 1363x 
│ │ │ │ -000390d0: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -000390e0: 2020 2035 2036 2020 2020 2020 2020 3620     5 6        6 
│ │ │ │ -000390f0: 2020 2020 2020 3020 3720 2020 2020 2020        0 7       
│ │ │ │ -00039100: 3120 3720 2020 2020 2020 2032 2037 2020  1 7        2 7  
│ │ │ │ -00039110: 2020 2020 3320 3720 2020 2020 2020 2034      3 7        4
│ │ │ │ -00039120: 2037 2020 2020 2020 2020 207c 0a7c 2d2d   7         |.|--
│ │ │ │ -00039130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039090: 207c 0a7c 3130 3738 7820 7820 202b 2031   |.|1078x x  + 1
│ │ │ │ +000390a0: 3135 3878 2020 2b20 3333 3578 2078 2020  158x  + 335x x  
│ │ │ │ +000390b0: 2d20 3938 3278 2078 2020 2d20 3134 3739  - 982x x  - 1479
│ │ │ │ +000390c0: 7820 7820 202d 2031 3578 2078 2020 2b20  x x  - 15x x  + 
│ │ │ │ +000390d0: 3133 3633 7820 7820 202b 2020 2020 2020  1363x x  +      
│ │ │ │ +000390e0: 207c 0a7c 2020 2020 2035 2036 2020 2020   |.|     5 6    
│ │ │ │ +000390f0: 2020 2020 3620 2020 2020 2020 3020 3720      6       0 7 
│ │ │ │ +00039100: 2020 2020 2020 3120 3720 2020 2020 2020        1 7       
│ │ │ │ +00039110: 2032 2037 2020 2020 2020 3320 3720 2020   2 7      3 7   
│ │ │ │ +00039120: 2020 2020 2034 2037 2020 2020 2020 2020       4 7        
│ │ │ │ +00039130: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039190: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00039170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039180: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00039190: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  000391a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000391b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000391c0: 2020 2020 2020 2020 2020 207c 0a7c 3133             |.|13
│ │ │ │ -000391d0: 3937 7820 7820 202d 2035 3735 7820 7820  97x x  - 575x x 
│ │ │ │ -000391e0: 202d 2037 3178 2020 2b20 3132 3535 7820   - 71x  + 1255x 
│ │ │ │ -000391f0: 7820 202d 2031 3133 3878 2078 2020 2d20  x  - 1138x x  - 
│ │ │ │ -00039200: 3135 3930 7820 7820 202b 2036 3034 7820  1590x x  + 604x 
│ │ │ │ -00039210: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -00039220: 2020 2035 2037 2020 2020 2020 2036 2037     5 7       6 7
│ │ │ │ -00039230: 2020 2020 2020 3720 2020 2020 2020 2030        7        0
│ │ │ │ -00039240: 2038 2020 2020 2020 2020 3120 3820 2020   8        1 8   
│ │ │ │ -00039250: 2020 2020 2032 2038 2020 2020 2020 2033       2 8       3
│ │ │ │ -00039260: 2038 2020 2020 2020 2020 207c 0a7c 2d2d   8         |.|--
│ │ │ │ -00039270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000391c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000391d0: 207c 0a7c 3133 3937 7820 7820 202d 2035   |.|1397x x  - 5
│ │ │ │ +000391e0: 3735 7820 7820 202d 2037 3178 2020 2b20  75x x  - 71x  + 
│ │ │ │ +000391f0: 3132 3535 7820 7820 202d 2031 3133 3878  1255x x  - 1138x
│ │ │ │ +00039200: 2078 2020 2d20 3135 3930 7820 7820 202b   x  - 1590x x  +
│ │ │ │ +00039210: 2036 3034 7820 7820 202b 2020 2020 2020   604x x  +      
│ │ │ │ +00039220: 207c 0a7c 2020 2020 2035 2037 2020 2020   |.|     5 7    
│ │ │ │ +00039230: 2020 2036 2037 2020 2020 2020 3720 2020     6 7      7   
│ │ │ │ +00039240: 2020 2020 2030 2038 2020 2020 2020 2020       0 8        
│ │ │ │ +00039250: 3120 3820 2020 2020 2020 2032 2038 2020  1 8        2 8  
│ │ │ │ +00039260: 2020 2020 2033 2038 2020 2020 2020 2020       3 8        
│ │ │ │ +00039270: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000392a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000392b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000392c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000392b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000392c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  000392d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000392e0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000392f0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00039300: 2020 2020 2020 2020 3220 207c 0a7c 3131          2  |.|11
│ │ │ │ -00039310: 3832 7820 7820 202d 2036 3378 2078 2020  82x x  - 63x x  
│ │ │ │ -00039320: 2d20 3133 3832 7820 7820 202d 2031 3235  - 1382x x  - 125
│ │ │ │ -00039330: 3578 2078 2020 2d20 3631 3378 202c 202d  5x x  - 613x , -
│ │ │ │ -00039340: 2031 3434 3478 2020 2b20 3537 3578 2078   1444x  + 575x x
│ │ │ │ -00039350: 2020 2b20 3736 3778 2020 2d7c 0a7c 2020    + 767x  -|.|  
│ │ │ │ -00039360: 2020 2034 2038 2020 2020 2020 3520 3820     4 8      5 8 
│ │ │ │ -00039370: 2020 2020 2020 2036 2038 2020 2020 2020         6 8      
│ │ │ │ -00039380: 2020 3720 3820 2020 2020 2020 3820 2020    7 8       8   
│ │ │ │ -00039390: 2020 2020 2020 3020 2020 2020 2020 3020        0       0 
│ │ │ │ -000393a0: 3120 2020 2020 2020 3120 207c 0a7c 2d2d  1       1  |.|--
│ │ │ │ -000393b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000392e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000392f0: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ +00039300: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00039310: 207c 0a7c 3131 3832 7820 7820 202d 2036   |.|1182x x  - 6
│ │ │ │ +00039320: 3378 2078 2020 2d20 3133 3832 7820 7820  3x x  - 1382x x 
│ │ │ │ +00039330: 202d 2031 3235 3578 2078 2020 2d20 3631   - 1255x x  - 61
│ │ │ │ +00039340: 3378 202c 202d 2031 3434 3478 2020 2b20  3x , - 1444x  + 
│ │ │ │ +00039350: 3537 3578 2078 2020 2b20 3736 3778 2020  575x x  + 767x  
│ │ │ │ +00039360: 2d7c 0a7c 2020 2020 2034 2038 2020 2020  -|.|     4 8    
│ │ │ │ +00039370: 2020 3520 3820 2020 2020 2020 2036 2038    5 8        6 8
│ │ │ │ +00039380: 2020 2020 2020 2020 3720 3820 2020 2020          7 8     
│ │ │ │ +00039390: 2020 3820 2020 2020 2020 2020 3020 2020    8         0   
│ │ │ │ +000393a0: 2020 2020 3020 3120 2020 2020 2020 3120      0 1       1 
│ │ │ │ +000393b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000393c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000393d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000393e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000393f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039410: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000393f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039400: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00039410: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00039420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039440: 3220 2020 2020 2020 2020 207c 0a7c 3134  2          |.|14
│ │ │ │ -00039450: 3935 7820 7820 202b 2031 3633 3178 2078  95x x  + 1631x x
│ │ │ │ -00039460: 2020 2d20 3231 3778 2020 2d20 3239 3478    - 217x  - 294x
│ │ │ │ -00039470: 2078 2020 2d20 3135 3131 7820 7820 202d   x  - 1511x x  -
│ │ │ │ -00039480: 2035 3034 7820 7820 202d 2031 3238 3478   504x x  - 1284x
│ │ │ │ -00039490: 2020 2d20 2020 2020 2020 207c 0a7c 2020    -        |.|  
│ │ │ │ -000394a0: 2020 2030 2032 2020 2020 2020 2020 3120     0 2        1 
│ │ │ │ -000394b0: 3220 2020 2020 2020 3220 2020 2020 2020  2       2       
│ │ │ │ -000394c0: 3020 3320 2020 2020 2020 2031 2033 2020  0 3        1 3  
│ │ │ │ -000394d0: 2020 2020 2032 2033 2020 2020 2020 2020       2 3        
│ │ │ │ -000394e0: 3320 2020 2020 2020 2020 207c 0a7c 2d2d  3          |.|--
│ │ │ │ -000394f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039440: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00039450: 207c 0a7c 3134 3935 7820 7820 202b 2031   |.|1495x x  + 1
│ │ │ │ +00039460: 3633 3178 2078 2020 2d20 3231 3778 2020  631x x  - 217x  
│ │ │ │ +00039470: 2d20 3239 3478 2078 2020 2d20 3135 3131  - 294x x  - 1511
│ │ │ │ +00039480: 7820 7820 202d 2035 3034 7820 7820 202d  x x  - 504x x  -
│ │ │ │ +00039490: 2031 3238 3478 2020 2d20 2020 2020 2020   1284x  -       
│ │ │ │ +000394a0: 207c 0a7c 2020 2020 2030 2032 2020 2020   |.|     0 2    
│ │ │ │ +000394b0: 2020 2020 3120 3220 2020 2020 2020 3220      1 2       2 
│ │ │ │ +000394c0: 2020 2020 2020 3020 3320 2020 2020 2020        0 3       
│ │ │ │ +000394d0: 2031 2033 2020 2020 2020 2032 2033 2020   1 3       2 3  
│ │ │ │ +000394e0: 2020 2020 2020 3320 2020 2020 2020 2020        3         
│ │ │ │ +000394f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039540: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00039550: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039560: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00039560: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00039570: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039580: 2020 2020 2020 2020 2020 207c 0a7c 3134             |.|14
│ │ │ │ -00039590: 3539 7820 7820 202b 2031 3532 7820 7820  59x x  + 152x x 
│ │ │ │ -000395a0: 202b 2031 3431 7820 7820 202d 2031 3078   + 141x x  - 10x
│ │ │ │ -000395b0: 2078 2020 2d20 3935 7820 202b 2031 3035   x  - 95x  + 105
│ │ │ │ -000395c0: 3678 2078 2020 2b20 3635 3478 2078 2020  6x x  + 654x x  
│ │ │ │ -000395d0: 2b20 3133 3937 7820 7820 207c 0a7c 2020  + 1397x x  |.|  
│ │ │ │ -000395e0: 2020 2030 2034 2020 2020 2020 2031 2034     0 4       1 4
│ │ │ │ -000395f0: 2020 2020 2020 2032 2034 2020 2020 2020         2 4      
│ │ │ │ -00039600: 3320 3420 2020 2020 2034 2020 2020 2020  3 4      4      
│ │ │ │ -00039610: 2020 3020 3520 2020 2020 2020 3120 3520    0 5       1 5 
│ │ │ │ -00039620: 2020 2020 2020 2032 2035 207c 0a7c 2d2d         2 5 |.|--
│ │ │ │ -00039630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039590: 207c 0a7c 3134 3539 7820 7820 202b 2031   |.|1459x x  + 1
│ │ │ │ +000395a0: 3532 7820 7820 202b 2031 3431 7820 7820  52x x  + 141x x 
│ │ │ │ +000395b0: 202d 2031 3078 2078 2020 2d20 3935 7820   - 10x x  - 95x 
│ │ │ │ +000395c0: 202b 2031 3035 3678 2078 2020 2b20 3635   + 1056x x  + 65
│ │ │ │ +000395d0: 3478 2078 2020 2b20 3133 3937 7820 7820  4x x  + 1397x x 
│ │ │ │ +000395e0: 207c 0a7c 2020 2020 2030 2034 2020 2020   |.|     0 4    
│ │ │ │ +000395f0: 2020 2031 2034 2020 2020 2020 2032 2034     1 4       2 4
│ │ │ │ +00039600: 2020 2020 2020 3320 3420 2020 2020 2034        3 4      4
│ │ │ │ +00039610: 2020 2020 2020 2020 3020 3520 2020 2020          0 5     
│ │ │ │ +00039620: 2020 3120 3520 2020 2020 2020 2032 2035    1 5        2 5
│ │ │ │ +00039630: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039680: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039690: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00039670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039680: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00039690: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  000396a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000396b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000396c0: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -000396d0: 3933 3078 2078 2020 2b20 3537 3878 2078  930x x  + 578x x
│ │ │ │ -000396e0: 2020 2d20 3639 3678 2020 2b20 3735 3978    - 696x  + 759x
│ │ │ │ -000396f0: 2078 2020 2b20 3733 3378 2078 2020 2b20   x  + 733x x  + 
│ │ │ │ -00039700: 3530 3578 2078 2020 2d20 3630 3978 2078  505x x  - 609x x
│ │ │ │ -00039710: 2020 2b20 3532 3678 2078 207c 0a7c 2020    + 526x x |.|  
│ │ │ │ -00039720: 2020 2020 3320 3520 2020 2020 2020 3420      3 5       4 
│ │ │ │ -00039730: 3520 2020 2020 2020 3520 2020 2020 2020  5       5       
│ │ │ │ -00039740: 3020 3620 2020 2020 2020 3120 3620 2020  0 6       1 6   
│ │ │ │ -00039750: 2020 2020 3220 3620 2020 2020 2020 3320      2 6       3 
│ │ │ │ -00039760: 3620 2020 2020 2020 3420 367c 0a7c 2d2d  6       4 6|.|--
│ │ │ │ -00039770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000396c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000396d0: 207c 0a7c 2d20 3933 3078 2078 2020 2b20   |.|- 930x x  + 
│ │ │ │ +000396e0: 3537 3878 2078 2020 2d20 3639 3678 2020  578x x  - 696x  
│ │ │ │ +000396f0: 2b20 3735 3978 2078 2020 2b20 3733 3378  + 759x x  + 733x
│ │ │ │ +00039700: 2078 2020 2b20 3530 3578 2078 2020 2d20   x  + 505x x  - 
│ │ │ │ +00039710: 3630 3978 2078 2020 2b20 3532 3678 2078  609x x  + 526x x
│ │ │ │ +00039720: 207c 0a7c 2020 2020 2020 3320 3520 2020   |.|      3 5   
│ │ │ │ +00039730: 2020 2020 3420 3520 2020 2020 2020 3520      4 5       5 
│ │ │ │ +00039740: 2020 2020 2020 3020 3620 2020 2020 2020        0 6       
│ │ │ │ +00039750: 3120 3620 2020 2020 2020 3220 3620 2020  1 6       2 6   
│ │ │ │ +00039760: 2020 2020 3320 3620 2020 2020 2020 3420      3 6       4 
│ │ │ │ +00039770: 367c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  6|.|------------
│ │ │ │  00039780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000397a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000397b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000397c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000397d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000397b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000397c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +000397d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000397e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000397f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039800: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -00039810: 3635 3978 2078 2020 2b20 3834 3678 2020  659x x  + 846x  
│ │ │ │ -00039820: 2b20 3132 3533 7820 7820 202d 2031 3531  + 1253x x  - 151
│ │ │ │ -00039830: 3978 2078 2020 2b20 3633 3578 2078 2020  9x x  + 635x x  
│ │ │ │ -00039840: 2b20 3537 3678 2078 2020 2b20 3534 7820  + 576x x  + 54x 
│ │ │ │ -00039850: 7820 202d 2020 2020 2020 207c 0a7c 2020  x  -       |.|  
│ │ │ │ -00039860: 2020 2020 3520 3620 2020 2020 2020 3620      5 6       6 
│ │ │ │ -00039870: 2020 2020 2020 2030 2037 2020 2020 2020         0 7      
│ │ │ │ -00039880: 2020 3120 3720 2020 2020 2020 3220 3720    1 7       2 7 
│ │ │ │ -00039890: 2020 2020 2020 3320 3720 2020 2020 2034        3 7      4
│ │ │ │ -000398a0: 2037 2020 2020 2020 2020 207c 0a7c 2d2d   7         |.|--
│ │ │ │ -000398b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039810: 207c 0a7c 2d20 3635 3978 2078 2020 2b20   |.|- 659x x  + 
│ │ │ │ +00039820: 3834 3678 2020 2b20 3132 3533 7820 7820  846x  + 1253x x 
│ │ │ │ +00039830: 202d 2031 3531 3978 2078 2020 2b20 3633   - 1519x x  + 63
│ │ │ │ +00039840: 3578 2078 2020 2b20 3537 3678 2078 2020  5x x  + 576x x  
│ │ │ │ +00039850: 2b20 3534 7820 7820 202d 2020 2020 2020  + 54x x  -      
│ │ │ │ +00039860: 207c 0a7c 2020 2020 2020 3520 3620 2020   |.|      5 6   
│ │ │ │ +00039870: 2020 2020 3620 2020 2020 2020 2030 2037      6        0 7
│ │ │ │ +00039880: 2020 2020 2020 2020 3120 3720 2020 2020          1 7     
│ │ │ │ +00039890: 2020 3220 3720 2020 2020 2020 3320 3720    2 7       3 7 
│ │ │ │ +000398a0: 2020 2020 2034 2037 2020 2020 2020 2020       4 7        
│ │ │ │ +000398b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000398c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000398d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000398e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000398f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039910: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000398f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039900: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00039910: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  00039920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039940: 2020 2020 2020 2020 2020 207c 0a7c 3132             |.|12
│ │ │ │ -00039950: 3631 7820 7820 202d 2038 3232 7820 7820  61x x  - 822x x 
│ │ │ │ -00039960: 202d 2032 3537 7820 202d 2039 3836 7820   - 257x  - 986x 
│ │ │ │ -00039970: 7820 202b 2033 3536 7820 7820 202d 2031  x  + 356x x  - 1
│ │ │ │ -00039980: 3438 3878 2078 2020 2d20 3135 3631 7820  488x x  - 1561x 
│ │ │ │ -00039990: 7820 202d 2020 2020 2020 207c 0a7c 2020  x  -       |.|  
│ │ │ │ -000399a0: 2020 2035 2037 2020 2020 2020 2036 2037     5 7       6 7
│ │ │ │ -000399b0: 2020 2020 2020 2037 2020 2020 2020 2030         7       0
│ │ │ │ -000399c0: 2038 2020 2020 2020 2031 2038 2020 2020   8       1 8    
│ │ │ │ -000399d0: 2020 2020 3220 3820 2020 2020 2020 2033      2 8        3
│ │ │ │ -000399e0: 2038 2020 2020 2020 2020 207c 0a7c 2d2d   8         |.|--
│ │ │ │ -000399f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039950: 207c 0a7c 3132 3631 7820 7820 202d 2038   |.|1261x x  - 8
│ │ │ │ +00039960: 3232 7820 7820 202d 2032 3537 7820 202d  22x x  - 257x  -
│ │ │ │ +00039970: 2039 3836 7820 7820 202b 2033 3536 7820   986x x  + 356x 
│ │ │ │ +00039980: 7820 202d 2031 3438 3878 2078 2020 2d20  x  - 1488x x  - 
│ │ │ │ +00039990: 3135 3631 7820 7820 202d 2020 2020 2020  1561x x  -      
│ │ │ │ +000399a0: 207c 0a7c 2020 2020 2035 2037 2020 2020   |.|     5 7    
│ │ │ │ +000399b0: 2020 2036 2037 2020 2020 2020 2037 2020     6 7       7  
│ │ │ │ +000399c0: 2020 2020 2030 2038 2020 2020 2020 2031       0 8       1
│ │ │ │ +000399d0: 2038 2020 2020 2020 2020 3220 3820 2020   8        2 8   
│ │ │ │ +000399e0: 2020 2020 2033 2038 2020 2020 2020 2020       3 8        
│ │ │ │ +000399f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039a40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039a40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00039a50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039a60: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00039a70: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -00039a80: 2020 2020 2020 2020 3220 207c 0a7c 3835          2  |.|85
│ │ │ │ -00039a90: 3078 2078 2020 2d20 3835 7820 7820 202d  0x x  - 85x x  -
│ │ │ │ -00039aa0: 2031 3335 3078 2078 2020 2d20 3738 3378   1350x x  - 783x
│ │ │ │ -00039ab0: 2078 2020 2d20 3133 3335 7820 2c20 2d20   x  - 1335x , - 
│ │ │ │ -00039ac0: 3837 3178 2020 2b20 3130 3036 7820 7820  871x  + 1006x x 
│ │ │ │ -00039ad0: 202d 2031 3339 3978 2020 2d7c 0a7c 2020   - 1399x  -|.|  
│ │ │ │ -00039ae0: 2020 3420 3820 2020 2020 2035 2038 2020    4 8      5 8  
│ │ │ │ -00039af0: 2020 2020 2020 3620 3820 2020 2020 2020        6 8       
│ │ │ │ -00039b00: 3720 3820 2020 2020 2020 2038 2020 2020  7 8        8    
│ │ │ │ -00039b10: 2020 2020 3020 2020 2020 2020 2030 2031      0        0 1
│ │ │ │ -00039b20: 2020 2020 2020 2020 3120 207c 0a7c 2d2d          1  |.|--
│ │ │ │ -00039b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039a60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039a70: 2032 2020 2020 2020 2020 3220 2020 2020   2        2     
│ │ │ │ +00039a80: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00039a90: 207c 0a7c 3835 3078 2078 2020 2d20 3835   |.|850x x  - 85
│ │ │ │ +00039aa0: 7820 7820 202d 2031 3335 3078 2078 2020  x x  - 1350x x  
│ │ │ │ +00039ab0: 2d20 3738 3378 2078 2020 2d20 3133 3335  - 783x x  - 1335
│ │ │ │ +00039ac0: 7820 2c20 2d20 3837 3178 2020 2b20 3130  x , - 871x  + 10
│ │ │ │ +00039ad0: 3036 7820 7820 202d 2031 3339 3978 2020  06x x  - 1399x  
│ │ │ │ +00039ae0: 2d7c 0a7c 2020 2020 3420 3820 2020 2020  -|.|    4 8     
│ │ │ │ +00039af0: 2035 2038 2020 2020 2020 2020 3620 3820   5 8        6 8 
│ │ │ │ +00039b00: 2020 2020 2020 3720 3820 2020 2020 2020        7 8       
│ │ │ │ +00039b10: 2038 2020 2020 2020 2020 3020 2020 2020   8        0     
│ │ │ │ +00039b20: 2020 2030 2031 2020 2020 2020 2020 3120     0 1        1 
│ │ │ │ +00039b30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039b80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039b90: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00039b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039b80: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00039b90: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  00039ba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039bb0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00039bc0: 2020 2020 2020 2020 2020 207c 0a7c 3136             |.|16
│ │ │ │ -00039bd0: 3336 7820 7820 202d 2036 3939 7820 7820  36x x  - 699x x 
│ │ │ │ -00039be0: 202d 2037 3639 7820 202d 2033 3037 7820   - 769x  - 307x 
│ │ │ │ -00039bf0: 7820 202d 2031 3634 3578 2078 2020 2d20  x  - 1645x x  - 
│ │ │ │ -00039c00: 3530 3278 2078 2020 2d20 3731 3978 2020  502x x  - 719x  
│ │ │ │ -00039c10: 2b20 3134 3035 7820 7820 207c 0a7c 2020  + 1405x x  |.|  
│ │ │ │ -00039c20: 2020 2030 2032 2020 2020 2020 2031 2032     0 2       1 2
│ │ │ │ -00039c30: 2020 2020 2020 2032 2020 2020 2020 2030         2       0
│ │ │ │ -00039c40: 2033 2020 2020 2020 2020 3120 3320 2020   3        1 3   
│ │ │ │ -00039c50: 2020 2020 3220 3320 2020 2020 2020 3320      2 3       3 
│ │ │ │ -00039c60: 2020 2020 2020 2030 2034 207c 0a7c 2d2d         0 4 |.|--
│ │ │ │ -00039c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039bb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039bc0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00039bd0: 207c 0a7c 3136 3336 7820 7820 202d 2036   |.|1636x x  - 6
│ │ │ │ +00039be0: 3939 7820 7820 202d 2037 3639 7820 202d  99x x  - 769x  -
│ │ │ │ +00039bf0: 2033 3037 7820 7820 202d 2031 3634 3578   307x x  - 1645x
│ │ │ │ +00039c00: 2078 2020 2d20 3530 3278 2078 2020 2d20   x  - 502x x  - 
│ │ │ │ +00039c10: 3731 3978 2020 2b20 3134 3035 7820 7820  719x  + 1405x x 
│ │ │ │ +00039c20: 207c 0a7c 2020 2020 2030 2032 2020 2020   |.|     0 2    
│ │ │ │ +00039c30: 2020 2031 2032 2020 2020 2020 2032 2020     1 2       2  
│ │ │ │ +00039c40: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ +00039c50: 3120 3320 2020 2020 2020 3220 3320 2020  1 3       2 3   
│ │ │ │ +00039c60: 2020 2020 3320 2020 2020 2020 2030 2034      3        0 4
│ │ │ │ +00039c70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039cc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039cc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00039cd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039ce0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00039ce0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  00039cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039d00: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -00039d10: 3837 3078 2078 2020 2d20 3131 3333 7820  870x x  - 1133x 
│ │ │ │ -00039d20: 7820 202b 2034 3235 7820 7820 202d 2031  x  + 425x x  - 1
│ │ │ │ -00039d30: 3230 3378 2020 2d20 3136 3031 7820 7820  203x  - 1601x x 
│ │ │ │ -00039d40: 202b 2031 3137 7820 7820 202d 2033 3832   + 117x x  - 382
│ │ │ │ -00039d50: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -00039d60: 2020 2020 3120 3420 2020 2020 2020 2032      1 4        2
│ │ │ │ -00039d70: 2034 2020 2020 2020 2033 2034 2020 2020   4       3 4    
│ │ │ │ -00039d80: 2020 2020 3420 2020 2020 2020 2030 2035      4        0 5
│ │ │ │ -00039d90: 2020 2020 2020 2031 2035 2020 2020 2020         1 5      
│ │ │ │ -00039da0: 2032 2035 2020 2020 2020 207c 0a7c 2d2d   2 5       |.|--
│ │ │ │ -00039db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039d10: 207c 0a7c 2b20 3837 3078 2078 2020 2d20   |.|+ 870x x  - 
│ │ │ │ +00039d20: 3131 3333 7820 7820 202b 2034 3235 7820  1133x x  + 425x 
│ │ │ │ +00039d30: 7820 202d 2031 3230 3378 2020 2d20 3136  x  - 1203x  - 16
│ │ │ │ +00039d40: 3031 7820 7820 202b 2031 3137 7820 7820  01x x  + 117x x 
│ │ │ │ +00039d50: 202d 2033 3832 7820 7820 202b 2020 2020   - 382x x  +    
│ │ │ │ +00039d60: 207c 0a7c 2020 2020 2020 3120 3420 2020   |.|      1 4   
│ │ │ │ +00039d70: 2020 2020 2032 2034 2020 2020 2020 2033       2 4       3
│ │ │ │ +00039d80: 2034 2020 2020 2020 2020 3420 2020 2020   4        4     
│ │ │ │ +00039d90: 2020 2030 2035 2020 2020 2020 2031 2035     0 5       1 5
│ │ │ │ +00039da0: 2020 2020 2020 2032 2035 2020 2020 2020         2 5      
│ │ │ │ +00039db0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00039dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039e00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039e10: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00039df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039e00: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00039e10: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  00039e20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039e40: 2020 2020 2020 2020 2020 207c 0a7c 3331             |.|31
│ │ │ │ -00039e50: 3878 2078 2020 2d20 3131 3778 2078 2020  8x x  - 117x x  
│ │ │ │ -00039e60: 2d20 3536 3078 2020 2b20 3131 3335 7820  - 560x  + 1135x 
│ │ │ │ -00039e70: 7820 202b 2031 3436 3878 2078 2020 2b20  x  + 1468x x  + 
│ │ │ │ -00039e80: 3836 3978 2078 2020 2d20 3934 3378 2078  869x x  - 943x x
│ │ │ │ -00039e90: 2020 2d20 3333 3578 2078 207c 0a7c 2020    - 335x x |.|  
│ │ │ │ -00039ea0: 2020 3320 3520 2020 2020 2020 3420 3520    3 5       4 5 
│ │ │ │ -00039eb0: 2020 2020 2020 3520 2020 2020 2020 2030        5        0
│ │ │ │ -00039ec0: 2036 2020 2020 2020 2020 3120 3620 2020   6        1 6   
│ │ │ │ -00039ed0: 2020 2020 3220 3620 2020 2020 2020 3320      2 6       3 
│ │ │ │ -00039ee0: 3620 2020 2020 2020 3420 367c 0a7c 2d2d  6       4 6|.|--
│ │ │ │ -00039ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039e50: 207c 0a7c 3331 3878 2078 2020 2d20 3131   |.|318x x  - 11
│ │ │ │ +00039e60: 3778 2078 2020 2d20 3536 3078 2020 2b20  7x x  - 560x  + 
│ │ │ │ +00039e70: 3131 3335 7820 7820 202b 2031 3436 3878  1135x x  + 1468x
│ │ │ │ +00039e80: 2078 2020 2b20 3836 3978 2078 2020 2d20   x  + 869x x  - 
│ │ │ │ +00039e90: 3934 3378 2078 2020 2d20 3333 3578 2078  943x x  - 335x x
│ │ │ │ +00039ea0: 207c 0a7c 2020 2020 3320 3520 2020 2020   |.|    3 5     
│ │ │ │ +00039eb0: 2020 3420 3520 2020 2020 2020 3520 2020    4 5       5   
│ │ │ │ +00039ec0: 2020 2020 2030 2036 2020 2020 2020 2020       0 6        
│ │ │ │ +00039ed0: 3120 3620 2020 2020 2020 3220 3620 2020  1 6       2 6   
│ │ │ │ +00039ee0: 2020 2020 3320 3620 2020 2020 2020 3420      3 6       4 
│ │ │ │ +00039ef0: 367c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  6|.|------------
│ │ │ │  00039f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00039f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00039f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00039f40: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00039f50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039f40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00039f50: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00039f60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00039f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00039f80: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -00039f90: 3132 3138 7820 7820 202b 2032 3031 7820  1218x x  + 201x 
│ │ │ │ -00039fa0: 202d 2031 3178 2078 2020 2b20 3534 3078   - 11x x  + 540x
│ │ │ │ -00039fb0: 2078 2020 2d20 3731 3078 2078 2020 2d20   x  - 710x x  - 
│ │ │ │ -00039fc0: 3438 3978 2078 2020 2b20 3136 3035 7820  489x x  + 1605x 
│ │ │ │ -00039fd0: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -00039fe0: 2020 2020 2035 2036 2020 2020 2020 2036       5 6       6
│ │ │ │ -00039ff0: 2020 2020 2020 3020 3720 2020 2020 2020        0 7       
│ │ │ │ -0003a000: 3120 3720 2020 2020 2020 3220 3720 2020  1 7       2 7   
│ │ │ │ -0003a010: 2020 2020 3320 3720 2020 2020 2020 2034      3 7        4
│ │ │ │ -0003a020: 2037 2020 2020 2020 2020 207c 0a7c 2d2d   7         |.|--
│ │ │ │ -0003a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00039f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00039f90: 207c 0a7c 2d20 3132 3138 7820 7820 202b   |.|- 1218x x  +
│ │ │ │ +00039fa0: 2032 3031 7820 202d 2031 3178 2078 2020   201x  - 11x x  
│ │ │ │ +00039fb0: 2b20 3534 3078 2078 2020 2d20 3731 3078  + 540x x  - 710x
│ │ │ │ +00039fc0: 2078 2020 2d20 3438 3978 2078 2020 2b20   x  - 489x x  + 
│ │ │ │ +00039fd0: 3136 3035 7820 7820 202b 2020 2020 2020  1605x x  +      
│ │ │ │ +00039fe0: 207c 0a7c 2020 2020 2020 2035 2036 2020   |.|       5 6  
│ │ │ │ +00039ff0: 2020 2020 2036 2020 2020 2020 3020 3720       6      0 7 
│ │ │ │ +0003a000: 2020 2020 2020 3120 3720 2020 2020 2020        1 7       
│ │ │ │ +0003a010: 3220 3720 2020 2020 2020 3320 3720 2020  2 7       3 7   
│ │ │ │ +0003a020: 2020 2020 2034 2037 2020 2020 2020 2020       4 7        
│ │ │ │ +0003a030: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003a080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a090: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a080: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003a090: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0003a0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a0c0: 2020 2020 2020 2020 2020 207c 0a7c 3136             |.|16
│ │ │ │ -0003a0d0: 3633 7820 7820 202d 2034 3233 7820 7820  63x x  - 423x x 
│ │ │ │ -0003a0e0: 202b 2031 3234 3678 2020 2b20 3937 7820   + 1246x  + 97x 
│ │ │ │ -0003a0f0: 7820 202d 2036 3434 7820 7820 202b 2031  x  - 644x x  + 1
│ │ │ │ -0003a100: 3635 3578 2078 2020 2b20 3132 3139 7820  655x x  + 1219x 
│ │ │ │ -0003a110: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -0003a120: 2020 2035 2037 2020 2020 2020 2036 2037     5 7       6 7
│ │ │ │ -0003a130: 2020 2020 2020 2020 3720 2020 2020 2030          7      0
│ │ │ │ -0003a140: 2038 2020 2020 2020 2031 2038 2020 2020   8       1 8    
│ │ │ │ -0003a150: 2020 2020 3220 3820 2020 2020 2020 2033      2 8        3
│ │ │ │ -0003a160: 2038 2020 2020 2020 2020 207c 0a7c 2d2d   8         |.|--
│ │ │ │ -0003a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a0d0: 207c 0a7c 3136 3633 7820 7820 202d 2034   |.|1663x x  - 4
│ │ │ │ +0003a0e0: 3233 7820 7820 202b 2031 3234 3678 2020  23x x  + 1246x  
│ │ │ │ +0003a0f0: 2b20 3937 7820 7820 202d 2036 3434 7820  + 97x x  - 644x 
│ │ │ │ +0003a100: 7820 202b 2031 3635 3578 2078 2020 2b20  x  + 1655x x  + 
│ │ │ │ +0003a110: 3132 3139 7820 7820 202b 2020 2020 2020  1219x x  +      
│ │ │ │ +0003a120: 207c 0a7c 2020 2020 2035 2037 2020 2020   |.|     5 7    
│ │ │ │ +0003a130: 2020 2036 2037 2020 2020 2020 2020 3720     6 7        7 
│ │ │ │ +0003a140: 2020 2020 2030 2038 2020 2020 2020 2031       0 8       1
│ │ │ │ +0003a150: 2038 2020 2020 2020 2020 3220 3820 2020   8        2 8   
│ │ │ │ +0003a160: 2020 2020 2033 2038 2020 2020 2020 2020       3 8        
│ │ │ │ +0003a170: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003a1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003a1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a1e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0003a1f0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0003a200: 2020 2020 2020 2020 2020 207c 0a7c 3134             |.|14
│ │ │ │ -0003a210: 3736 7820 7820 202b 2031 3335 3578 2078  76x x  + 1355x x
│ │ │ │ -0003a220: 2020 2b20 3135 3934 7820 7820 202b 2038    + 1594x x  + 8
│ │ │ │ -0003a230: 3933 7820 7820 202b 2031 3135 3078 202c  93x x  + 1150x ,
│ │ │ │ -0003a240: 202d 2031 3433 7820 202b 2031 3234 3078   - 143x  + 1240x
│ │ │ │ -0003a250: 2078 2020 2d20 2020 2020 207c 0a7c 2020   x  -      |.|  
│ │ │ │ -0003a260: 2020 2034 2038 2020 2020 2020 2020 3520     4 8        5 
│ │ │ │ -0003a270: 3820 2020 2020 2020 2036 2038 2020 2020  8        6 8    
│ │ │ │ -0003a280: 2020 2037 2038 2020 2020 2020 2020 3820     7 8        8 
│ │ │ │ -0003a290: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ -0003a2a0: 3020 3120 2020 2020 2020 207c 0a7c 2d2d  0 1        |.|--
│ │ │ │ -0003a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a1f0: 2020 2020 3220 2020 2020 2020 2032 2020      2        2  
│ │ │ │ +0003a200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a210: 207c 0a7c 3134 3736 7820 7820 202b 2031   |.|1476x x  + 1
│ │ │ │ +0003a220: 3335 3578 2078 2020 2b20 3135 3934 7820  355x x  + 1594x 
│ │ │ │ +0003a230: 7820 202b 2038 3933 7820 7820 202b 2031  x  + 893x x  + 1
│ │ │ │ +0003a240: 3135 3078 202c 202d 2031 3433 7820 202b  150x , - 143x  +
│ │ │ │ +0003a250: 2031 3234 3078 2078 2020 2d20 2020 2020   1240x x  -     
│ │ │ │ +0003a260: 207c 0a7c 2020 2020 2034 2038 2020 2020   |.|     4 8    
│ │ │ │ +0003a270: 2020 2020 3520 3820 2020 2020 2020 2036      5 8        6
│ │ │ │ +0003a280: 2038 2020 2020 2020 2037 2038 2020 2020   8       7 8    
│ │ │ │ +0003a290: 2020 2020 3820 2020 2020 2020 2030 2020      8        0  
│ │ │ │ +0003a2a0: 2020 2020 2020 3020 3120 2020 2020 2020        0 1       
│ │ │ │ +0003a2b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003a300: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0003a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a300: 2d7c 0a7c 2020 2020 2032 2020 2020 2020  -|.|     2      
│ │ │ │  0003a310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a320: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0003a320: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0003a330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a340: 2020 2020 2020 2020 3220 207c 0a7c 3130          2  |.|10
│ │ │ │ -0003a350: 3432 7820 202b 2031 3634 3978 2078 2020  42x  + 1649x x  
│ │ │ │ -0003a360: 2b20 3130 3234 7820 7820 202b 2037 3934  + 1024x x  + 794
│ │ │ │ -0003a370: 7820 202b 2031 3434 3278 2078 2020 2d20  x  + 1442x x  - 
│ │ │ │ -0003a380: 3132 3633 7820 7820 202b 2035 3337 7820  1263x x  + 537x 
│ │ │ │ -0003a390: 7820 202d 2038 3278 2020 2d7c 0a7c 2020  x  - 82x  -|.|  
│ │ │ │ -0003a3a0: 2020 2031 2020 2020 2020 2020 3020 3220     1        0 2 
│ │ │ │ -0003a3b0: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ -0003a3c0: 2032 2020 2020 2020 2020 3020 3320 2020   2        0 3   
│ │ │ │ -0003a3d0: 2020 2020 2031 2033 2020 2020 2020 2032       1 3       2
│ │ │ │ -0003a3e0: 2033 2020 2020 2020 3320 207c 0a7c 2d2d   3      3  |.|--
│ │ │ │ -0003a3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a340: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0003a350: 207c 0a7c 3130 3432 7820 202b 2031 3634   |.|1042x  + 164
│ │ │ │ +0003a360: 3978 2078 2020 2b20 3130 3234 7820 7820  9x x  + 1024x x 
│ │ │ │ +0003a370: 202b 2037 3934 7820 202b 2031 3434 3278   + 794x  + 1442x
│ │ │ │ +0003a380: 2078 2020 2d20 3132 3633 7820 7820 202b   x  - 1263x x  +
│ │ │ │ +0003a390: 2035 3337 7820 7820 202d 2038 3278 2020   537x x  - 82x  
│ │ │ │ +0003a3a0: 2d7c 0a7c 2020 2020 2031 2020 2020 2020  -|.|     1      
│ │ │ │ +0003a3b0: 2020 3020 3220 2020 2020 2020 2031 2032    0 2        1 2
│ │ │ │ +0003a3c0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003a3d0: 3020 3320 2020 2020 2020 2031 2033 2020  0 3        1 3  
│ │ │ │ +0003a3e0: 2020 2020 2032 2033 2020 2020 2020 3320       2 3      3 
│ │ │ │ +0003a3f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003a400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003a440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a440: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003a450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a460: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0003a470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a480: 2020 2020 2020 2020 2020 207c 0a7c 3733             |.|73
│ │ │ │ -0003a490: 3478 2078 2020 2d20 3135 3639 7820 7820  4x x  - 1569x x 
│ │ │ │ -0003a4a0: 202d 2037 3938 7820 7820 202d 2033 3636   - 798x x  - 366
│ │ │ │ -0003a4b0: 7820 7820 202b 2031 3238 3978 2020 2d20  x x  + 1289x  - 
│ │ │ │ -0003a4c0: 3536 3978 2078 2020 2d20 3235 3478 2078  569x x  - 254x x
│ │ │ │ -0003a4d0: 2020 2b20 3233 3778 2078 207c 0a7c 2020    + 237x x |.|  
│ │ │ │ -0003a4e0: 2020 3020 3420 2020 2020 2020 2031 2034    0 4        1 4
│ │ │ │ -0003a4f0: 2020 2020 2020 2032 2034 2020 2020 2020         2 4      
│ │ │ │ -0003a500: 2033 2034 2020 2020 2020 2020 3420 2020   3 4        4   
│ │ │ │ -0003a510: 2020 2020 3020 3520 2020 2020 2020 3120      0 5       1 
│ │ │ │ -0003a520: 3520 2020 2020 2020 3220 357c 0a7c 2d2d  5       2 5|.|--
│ │ │ │ -0003a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a470: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003a480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a490: 207c 0a7c 3733 3478 2078 2020 2d20 3135   |.|734x x  - 15
│ │ │ │ +0003a4a0: 3639 7820 7820 202d 2037 3938 7820 7820  69x x  - 798x x 
│ │ │ │ +0003a4b0: 202d 2033 3636 7820 7820 202b 2031 3238   - 366x x  + 128
│ │ │ │ +0003a4c0: 3978 2020 2d20 3536 3978 2078 2020 2d20  9x  - 569x x  - 
│ │ │ │ +0003a4d0: 3235 3478 2078 2020 2b20 3233 3778 2078  254x x  + 237x x
│ │ │ │ +0003a4e0: 207c 0a7c 2020 2020 3020 3420 2020 2020   |.|    0 4     
│ │ │ │ +0003a4f0: 2020 2031 2034 2020 2020 2020 2032 2034     1 4       2 4
│ │ │ │ +0003a500: 2020 2020 2020 2033 2034 2020 2020 2020         3 4      
│ │ │ │ +0003a510: 2020 3420 2020 2020 2020 3020 3520 2020    4       0 5   
│ │ │ │ +0003a520: 2020 2020 3120 3520 2020 2020 2020 3220      1 5       2 
│ │ │ │ +0003a530: 357c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  5|.|------------
│ │ │ │  0003a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003a580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a590: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0003a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a580: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003a590: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  0003a5a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a5c0: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -0003a5d0: 3132 3334 7820 7820 202d 2038 3037 7820  1234x x  - 807x 
│ │ │ │ -0003a5e0: 7820 202b 2032 3634 7820 202d 2032 3032  x  + 264x  - 202
│ │ │ │ -0003a5f0: 7820 7820 202d 2036 3136 7820 7820 202b  x x  - 616x x  +
│ │ │ │ -0003a600: 2034 3478 2078 2020 2b20 3134 3635 7820   44x x  + 1465x 
│ │ │ │ -0003a610: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -0003a620: 2020 2020 2033 2035 2020 2020 2020 2034       3 5       4
│ │ │ │ -0003a630: 2035 2020 2020 2020 2035 2020 2020 2020   5       5      
│ │ │ │ -0003a640: 2030 2036 2020 2020 2020 2031 2036 2020   0 6       1 6  
│ │ │ │ -0003a650: 2020 2020 3220 3620 2020 2020 2020 2033      2 6        3
│ │ │ │ -0003a660: 2036 2020 2020 2020 2020 207c 0a7c 2d2d   6         |.|--
│ │ │ │ -0003a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a5d0: 207c 0a7c 2d20 3132 3334 7820 7820 202d   |.|- 1234x x  -
│ │ │ │ +0003a5e0: 2038 3037 7820 7820 202b 2032 3634 7820   807x x  + 264x 
│ │ │ │ +0003a5f0: 202d 2032 3032 7820 7820 202d 2036 3136   - 202x x  - 616
│ │ │ │ +0003a600: 7820 7820 202b 2034 3478 2078 2020 2b20  x x  + 44x x  + 
│ │ │ │ +0003a610: 3134 3635 7820 7820 202b 2020 2020 2020  1465x x  +      
│ │ │ │ +0003a620: 207c 0a7c 2020 2020 2020 2033 2035 2020   |.|       3 5  
│ │ │ │ +0003a630: 2020 2020 2034 2035 2020 2020 2020 2035       4 5       5
│ │ │ │ +0003a640: 2020 2020 2020 2030 2036 2020 2020 2020         0 6      
│ │ │ │ +0003a650: 2031 2036 2020 2020 2020 3220 3620 2020   1 6      2 6   
│ │ │ │ +0003a660: 2020 2020 2033 2036 2020 2020 2020 2020       3 6        
│ │ │ │ +0003a670: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003a6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a6d0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003a6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a6c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003a6d0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0003a6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003a6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a700: 2020 2020 2020 2020 2020 207c 0a7c 3638             |.|68
│ │ │ │ -0003a710: 3578 2078 2020 2b20 3136 3330 7820 7820  5x x  + 1630x x 
│ │ │ │ -0003a720: 202d 2034 3036 7820 202d 2031 3233 7820   - 406x  - 123x 
│ │ │ │ -0003a730: 7820 202d 2034 7820 7820 202b 2031 3538  x  - 4x x  + 158
│ │ │ │ -0003a740: 3378 2078 2020 2b20 3132 3335 7820 7820  3x x  + 1235x x 
│ │ │ │ -0003a750: 202b 2031 3632 7820 7820 207c 0a7c 2020   + 162x x  |.|  
│ │ │ │ -0003a760: 2020 3420 3620 2020 2020 2020 2035 2036    4 6        5 6
│ │ │ │ -0003a770: 2020 2020 2020 2036 2020 2020 2020 2030         6       0
│ │ │ │ -0003a780: 2037 2020 2020 2031 2037 2020 2020 2020   7     1 7      
│ │ │ │ -0003a790: 2020 3220 3720 2020 2020 2020 2033 2037    2 7        3 7
│ │ │ │ -0003a7a0: 2020 2020 2020 2034 2037 207c 0a7c 2d2d         4 7 |.|--
│ │ │ │ -0003a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a710: 207c 0a7c 3638 3578 2078 2020 2b20 3136   |.|685x x  + 16
│ │ │ │ +0003a720: 3330 7820 7820 202d 2034 3036 7820 202d  30x x  - 406x  -
│ │ │ │ +0003a730: 2031 3233 7820 7820 202d 2034 7820 7820   123x x  - 4x x 
│ │ │ │ +0003a740: 202b 2031 3538 3378 2078 2020 2b20 3132   + 1583x x  + 12
│ │ │ │ +0003a750: 3335 7820 7820 202b 2031 3632 7820 7820  35x x  + 162x x 
│ │ │ │ +0003a760: 207c 0a7c 2020 2020 3420 3620 2020 2020   |.|    4 6     
│ │ │ │ +0003a770: 2020 2035 2036 2020 2020 2020 2036 2020     5 6       6  
│ │ │ │ +0003a780: 2020 2020 2030 2037 2020 2020 2031 2037       0 7     1 7
│ │ │ │ +0003a790: 2020 2020 2020 2020 3220 3720 2020 2020          2 7     
│ │ │ │ +0003a7a0: 2020 2033 2037 2020 2020 2020 2034 2037     3 7       4 7
│ │ │ │ +0003a7b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003a800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a810: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0003a820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a800: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003a810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a820: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0003a830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a840: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -0003a850: 3130 3334 7820 7820 202d 2031 3033 3578  1034x x  - 1035x
│ │ │ │ -0003a860: 2078 2020 2b20 3733 3778 2020 2b20 3636   x  + 737x  + 66
│ │ │ │ -0003a870: 3078 2078 2020 2b20 3134 3539 7820 7820  0x x  + 1459x x 
│ │ │ │ -0003a880: 202d 2033 3539 7820 7820 202d 2031 3239   - 359x x  - 129
│ │ │ │ -0003a890: 3178 2078 2020 2b20 2020 207c 0a7c 2020  1x x  +    |.|  
│ │ │ │ -0003a8a0: 2020 2020 2035 2037 2020 2020 2020 2020       5 7        
│ │ │ │ -0003a8b0: 3620 3720 2020 2020 2020 3720 2020 2020  6 7       7     
│ │ │ │ -0003a8c0: 2020 3020 3820 2020 2020 2020 2031 2038    0 8        1 8
│ │ │ │ -0003a8d0: 2020 2020 2020 2032 2038 2020 2020 2020         2 8      
│ │ │ │ -0003a8e0: 2020 3320 3820 2020 2020 207c 0a7c 2d2d    3 8      |.|--
│ │ │ │ -0003a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a850: 207c 0a7c 2b20 3130 3334 7820 7820 202d   |.|+ 1034x x  -
│ │ │ │ +0003a860: 2031 3033 3578 2078 2020 2b20 3733 3778   1035x x  + 737x
│ │ │ │ +0003a870: 2020 2b20 3636 3078 2078 2020 2b20 3134    + 660x x  + 14
│ │ │ │ +0003a880: 3539 7820 7820 202d 2033 3539 7820 7820  59x x  - 359x x 
│ │ │ │ +0003a890: 202d 2031 3239 3178 2078 2020 2b20 2020   - 1291x x  +   
│ │ │ │ +0003a8a0: 207c 0a7c 2020 2020 2020 2035 2037 2020   |.|       5 7  
│ │ │ │ +0003a8b0: 2020 2020 2020 3620 3720 2020 2020 2020        6 7       
│ │ │ │ +0003a8c0: 3720 2020 2020 2020 3020 3820 2020 2020  7       0 8     
│ │ │ │ +0003a8d0: 2020 2031 2038 2020 2020 2020 2032 2038     1 8       2 8
│ │ │ │ +0003a8e0: 2020 2020 2020 2020 3320 3820 2020 2020          3 8     
│ │ │ │ +0003a8f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003a940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a940: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003a950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003a960: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0003a970: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0003a980: 2020 2020 2020 3220 2020 207c 0a7c 3136        2    |.|16
│ │ │ │ -0003a990: 3338 7820 7820 202d 2033 3235 7820 7820  38x x  - 325x x 
│ │ │ │ -0003a9a0: 202d 2036 3331 7820 7820 202b 2037 3378   - 631x x  + 73x
│ │ │ │ -0003a9b0: 2078 2020 2d20 3134 3731 7820 2c20 2d20   x  - 1471x , - 
│ │ │ │ -0003a9c0: 3133 3430 7820 202b 2033 3178 2078 2020  1340x  + 31x x  
│ │ │ │ -0003a9d0: 2d20 3939 3478 2020 2d20 207c 0a7c 2020  - 994x  -  |.|  
│ │ │ │ -0003a9e0: 2020 2034 2038 2020 2020 2020 2035 2038     4 8       5 8
│ │ │ │ -0003a9f0: 2020 2020 2020 2036 2038 2020 2020 2020         6 8      
│ │ │ │ -0003aa00: 3720 3820 2020 2020 2020 2038 2020 2020  7 8        8    
│ │ │ │ -0003aa10: 2020 2020 2030 2020 2020 2020 3020 3120       0      0 1 
│ │ │ │ -0003aa20: 2020 2020 2020 3120 2020 207c 0a7c 2d2d        1    |.|--
│ │ │ │ -0003aa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003a960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003a970: 2032 2020 2020 2020 2020 2032 2020 2020   2         2    
│ │ │ │ +0003a980: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0003a990: 207c 0a7c 3136 3338 7820 7820 202d 2033   |.|1638x x  - 3
│ │ │ │ +0003a9a0: 3235 7820 7820 202d 2036 3331 7820 7820  25x x  - 631x x 
│ │ │ │ +0003a9b0: 202b 2037 3378 2078 2020 2d20 3134 3731   + 73x x  - 1471
│ │ │ │ +0003a9c0: 7820 2c20 2d20 3133 3430 7820 202b 2033  x , - 1340x  + 3
│ │ │ │ +0003a9d0: 3178 2078 2020 2d20 3939 3478 2020 2d20  1x x  - 994x  - 
│ │ │ │ +0003a9e0: 207c 0a7c 2020 2020 2034 2038 2020 2020   |.|     4 8    
│ │ │ │ +0003a9f0: 2020 2035 2038 2020 2020 2020 2036 2038     5 8       6 8
│ │ │ │ +0003aa00: 2020 2020 2020 3720 3820 2020 2020 2020        7 8       
│ │ │ │ +0003aa10: 2038 2020 2020 2020 2020 2030 2020 2020   8         0    
│ │ │ │ +0003aa20: 2020 3020 3120 2020 2020 2020 3120 2020    0 1       1   
│ │ │ │ +0003aa30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003aa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003aa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003aa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003aa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aa90: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0003aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003aa80: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003aa90: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │  0003aaa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003aab0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0003aac0: 2020 2020 2020 2020 2020 207c 0a7c 3838             |.|88
│ │ │ │ -0003aad0: 3078 2078 2020 2d20 3839 7820 7820 202b  0x x  - 89x x  +
│ │ │ │ -0003aae0: 2035 3734 7820 202b 2037 3630 7820 7820   574x  + 760x x 
│ │ │ │ -0003aaf0: 202d 2031 3035 3478 2078 2020 2b20 3737   - 1054x x  + 77
│ │ │ │ -0003ab00: 3278 2078 2020 2d20 3233 3978 2020 2d20  2x x  - 239x  - 
│ │ │ │ -0003ab10: 3434 3378 2078 2020 2b20 207c 0a7c 2020  443x x  +  |.|  
│ │ │ │ -0003ab20: 2020 3020 3220 2020 2020 2031 2032 2020    0 2      1 2  
│ │ │ │ -0003ab30: 2020 2020 2032 2020 2020 2020 2030 2033       2       0 3
│ │ │ │ -0003ab40: 2020 2020 2020 2020 3120 3320 2020 2020          1 3     
│ │ │ │ -0003ab50: 2020 3220 3320 2020 2020 2020 3320 2020    2 3       3   
│ │ │ │ -0003ab60: 2020 2020 3020 3420 2020 207c 0a7c 2d2d      0 4    |.|--
│ │ │ │ -0003ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003aab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003aac0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003aad0: 207c 0a7c 3838 3078 2078 2020 2d20 3839   |.|880x x  - 89
│ │ │ │ +0003aae0: 7820 7820 202b 2035 3734 7820 202b 2037  x x  + 574x  + 7
│ │ │ │ +0003aaf0: 3630 7820 7820 202d 2031 3035 3478 2078  60x x  - 1054x x
│ │ │ │ +0003ab00: 2020 2b20 3737 3278 2078 2020 2d20 3233    + 772x x  - 23
│ │ │ │ +0003ab10: 3978 2020 2d20 3434 3378 2078 2020 2b20  9x  - 443x x  + 
│ │ │ │ +0003ab20: 207c 0a7c 2020 2020 3020 3220 2020 2020   |.|    0 2     
│ │ │ │ +0003ab30: 2031 2032 2020 2020 2020 2032 2020 2020   1 2       2    
│ │ │ │ +0003ab40: 2020 2030 2033 2020 2020 2020 2020 3120     0 3        1 
│ │ │ │ +0003ab50: 3320 2020 2020 2020 3220 3320 2020 2020  3       2 3     
│ │ │ │ +0003ab60: 2020 3320 2020 2020 2020 3020 3420 2020    3       0 4   
│ │ │ │ +0003ab70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003abc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003abc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003abd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003abe0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003abe0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0003abf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ac00: 2020 2020 2020 2020 2020 207c 0a7c 3132             |.|12
│ │ │ │ -0003ac10: 3430 7820 7820 202b 2036 3337 7820 7820  40x x  + 637x x 
│ │ │ │ -0003ac20: 202d 2031 3432 3378 2078 2020 2b20 3332   - 1423x x  + 32
│ │ │ │ -0003ac30: 3078 2020 2d20 3133 3633 7820 7820 202d  0x  - 1363x x  -
│ │ │ │ -0003ac40: 2031 3133 3978 2078 2020 2d20 3135 3878   1139x x  - 158x
│ │ │ │ -0003ac50: 2078 2020 2d20 2020 2020 207c 0a7c 2020   x  -      |.|  
│ │ │ │ -0003ac60: 2020 2031 2034 2020 2020 2020 2032 2034     1 4       2 4
│ │ │ │ -0003ac70: 2020 2020 2020 2020 3320 3420 2020 2020          3 4     
│ │ │ │ -0003ac80: 2020 3420 2020 2020 2020 2030 2035 2020    4        0 5  
│ │ │ │ -0003ac90: 2020 2020 2020 3120 3520 2020 2020 2020        1 5       
│ │ │ │ -0003aca0: 3220 3520 2020 2020 2020 207c 0a7c 2d2d  2 5        |.|--
│ │ │ │ -0003acb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ac00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ac10: 207c 0a7c 3132 3430 7820 7820 202b 2036   |.|1240x x  + 6
│ │ │ │ +0003ac20: 3337 7820 7820 202d 2031 3432 3378 2078  37x x  - 1423x x
│ │ │ │ +0003ac30: 2020 2b20 3332 3078 2020 2d20 3133 3633    + 320x  - 1363
│ │ │ │ +0003ac40: 7820 7820 202d 2031 3133 3978 2078 2020  x x  - 1139x x  
│ │ │ │ +0003ac50: 2d20 3135 3878 2078 2020 2d20 2020 2020  - 158x x  -     
│ │ │ │ +0003ac60: 207c 0a7c 2020 2020 2031 2034 2020 2020   |.|     1 4    
│ │ │ │ +0003ac70: 2020 2032 2034 2020 2020 2020 2020 3320     2 4        3 
│ │ │ │ +0003ac80: 3420 2020 2020 2020 3420 2020 2020 2020  4       4       
│ │ │ │ +0003ac90: 2030 2035 2020 2020 2020 2020 3120 3520   0 5        1 5 
│ │ │ │ +0003aca0: 2020 2020 2020 3220 3520 2020 2020 2020        2 5       
│ │ │ │ +0003acb0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003acc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003acd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ace0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003ad00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad10: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0003acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ad00: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003ad10: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │  0003ad20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ad30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ad40: 2020 2020 2020 2020 2020 207c 0a7c 3332             |.|32
│ │ │ │ -0003ad50: 3578 2078 2020 2d20 3135 3738 7820 7820  5x x  - 1578x x 
│ │ │ │ -0003ad60: 202b 2033 3278 2020 2b20 3639 3578 2078   + 32x  + 695x x
│ │ │ │ -0003ad70: 2020 2b20 3330 3578 2078 2020 2b20 3130    + 305x x  + 10
│ │ │ │ -0003ad80: 3132 7820 7820 202b 2031 3439 3278 2078  12x x  + 1492x x
│ │ │ │ -0003ad90: 2020 2b20 2020 2020 2020 207c 0a7c 2020    +        |.|  
│ │ │ │ -0003ada0: 2020 3320 3520 2020 2020 2020 2034 2035    3 5        4 5
│ │ │ │ -0003adb0: 2020 2020 2020 3520 2020 2020 2020 3020        5       0 
│ │ │ │ -0003adc0: 3620 2020 2020 2020 3120 3620 2020 2020  6       1 6     
│ │ │ │ -0003add0: 2020 2032 2036 2020 2020 2020 2020 3320     2 6        3 
│ │ │ │ -0003ade0: 3620 2020 2020 2020 2020 207c 0a7c 2d2d  6          |.|--
│ │ │ │ -0003adf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ad40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ad50: 207c 0a7c 3332 3578 2078 2020 2d20 3135   |.|325x x  - 15
│ │ │ │ +0003ad60: 3738 7820 7820 202b 2033 3278 2020 2b20  78x x  + 32x  + 
│ │ │ │ +0003ad70: 3639 3578 2078 2020 2b20 3330 3578 2078  695x x  + 305x x
│ │ │ │ +0003ad80: 2020 2b20 3130 3132 7820 7820 202b 2031    + 1012x x  + 1
│ │ │ │ +0003ad90: 3439 3278 2078 2020 2b20 2020 2020 2020  492x x  +       
│ │ │ │ +0003ada0: 207c 0a7c 2020 2020 3320 3520 2020 2020   |.|    3 5     
│ │ │ │ +0003adb0: 2020 2034 2035 2020 2020 2020 3520 2020     4 5      5   
│ │ │ │ +0003adc0: 2020 2020 3020 3620 2020 2020 2020 3120      0 6       1 
│ │ │ │ +0003add0: 3620 2020 2020 2020 2032 2036 2020 2020  6        2 6    
│ │ │ │ +0003ade0: 2020 2020 3320 3620 2020 2020 2020 2020      3 6         
│ │ │ │ +0003adf0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003ae00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ae10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ae20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ae30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003ae40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae50: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003ae30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003ae50: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0003ae60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ae70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ae80: 2020 2020 2020 2020 2020 207c 0a7c 3132             |.|12
│ │ │ │ -0003ae90: 3930 7820 7820 202b 2031 3537 3978 2078  90x x  + 1579x x
│ │ │ │ -0003aea0: 2020 2d20 3334 3278 2020 2d20 3833 7820    - 342x  - 83x 
│ │ │ │ -0003aeb0: 7820 202d 2031 3034 7820 7820 202b 2039  x  - 104x x  + 9
│ │ │ │ -0003aec0: 3938 7820 7820 202d 2039 3278 2078 2020  98x x  - 92x x  
│ │ │ │ -0003aed0: 2b20 3135 3534 7820 7820 207c 0a7c 2020  + 1554x x  |.|  
│ │ │ │ -0003aee0: 2020 2034 2036 2020 2020 2020 2020 3520     4 6        5 
│ │ │ │ -0003aef0: 3620 2020 2020 2020 3620 2020 2020 2030  6       6      0
│ │ │ │ -0003af00: 2037 2020 2020 2020 2031 2037 2020 2020   7       1 7    
│ │ │ │ -0003af10: 2020 2032 2037 2020 2020 2020 3320 3720     2 7      3 7 
│ │ │ │ -0003af20: 2020 2020 2020 2034 2037 207c 0a7c 2d2d         4 7 |.|--
│ │ │ │ -0003af30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ae80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ae90: 207c 0a7c 3132 3930 7820 7820 202b 2031   |.|1290x x  + 1
│ │ │ │ +0003aea0: 3537 3978 2078 2020 2d20 3334 3278 2020  579x x  - 342x  
│ │ │ │ +0003aeb0: 2d20 3833 7820 7820 202d 2031 3034 7820  - 83x x  - 104x 
│ │ │ │ +0003aec0: 7820 202b 2039 3938 7820 7820 202d 2039  x  + 998x x  - 9
│ │ │ │ +0003aed0: 3278 2078 2020 2b20 3135 3534 7820 7820  2x x  + 1554x x 
│ │ │ │ +0003aee0: 207c 0a7c 2020 2020 2034 2036 2020 2020   |.|     4 6    
│ │ │ │ +0003aef0: 2020 2020 3520 3620 2020 2020 2020 3620      5 6       6 
│ │ │ │ +0003af00: 2020 2020 2030 2037 2020 2020 2020 2031       0 7       1
│ │ │ │ +0003af10: 2037 2020 2020 2020 2032 2037 2020 2020   7       2 7    
│ │ │ │ +0003af20: 2020 3320 3720 2020 2020 2020 2034 2037    3 7        4 7
│ │ │ │ +0003af30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003af40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003af80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003af90: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003af80: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003af90: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0003afa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003afb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003afc0: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -0003afd0: 3230 3178 2078 2020 2d20 3233 3778 2078  201x x  - 237x x
│ │ │ │ -0003afe0: 2020 2b20 3136 3078 2020 2d20 3232 3878    + 160x  - 228x
│ │ │ │ -0003aff0: 2078 2020 2d20 3534 3378 2078 2020 2d20   x  - 543x x  - 
│ │ │ │ -0003b000: 3131 3437 7820 7820 202d 2033 3736 7820  1147x x  - 376x 
│ │ │ │ -0003b010: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -0003b020: 2020 2020 3520 3720 2020 2020 2020 3620      5 7       6 
│ │ │ │ -0003b030: 3720 2020 2020 2020 3720 2020 2020 2020  7       7       
│ │ │ │ -0003b040: 3020 3820 2020 2020 2020 3120 3820 2020  0 8       1 8   
│ │ │ │ -0003b050: 2020 2020 2032 2038 2020 2020 2020 2033       2 8       3
│ │ │ │ -0003b060: 2038 2020 2020 2020 2020 207c 0a7c 2d2d   8         |.|--
│ │ │ │ -0003b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003afc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003afd0: 207c 0a7c 2b20 3230 3178 2078 2020 2d20   |.|+ 201x x  - 
│ │ │ │ +0003afe0: 3233 3778 2078 2020 2b20 3136 3078 2020  237x x  + 160x  
│ │ │ │ +0003aff0: 2d20 3232 3878 2078 2020 2d20 3534 3378  - 228x x  - 543x
│ │ │ │ +0003b000: 2078 2020 2d20 3131 3437 7820 7820 202d   x  - 1147x x  -
│ │ │ │ +0003b010: 2033 3736 7820 7820 202b 2020 2020 2020   376x x  +      
│ │ │ │ +0003b020: 207c 0a7c 2020 2020 2020 3520 3720 2020   |.|      5 7   
│ │ │ │ +0003b030: 2020 2020 3620 3720 2020 2020 2020 3720      6 7       7 
│ │ │ │ +0003b040: 2020 2020 2020 3020 3820 2020 2020 2020        0 8       
│ │ │ │ +0003b050: 3120 3820 2020 2020 2020 2032 2038 2020  1 8        2 8  
│ │ │ │ +0003b060: 2020 2020 2033 2038 2020 2020 2020 2020       3 8        
│ │ │ │ +0003b070: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003b0c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b0c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003b0d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b0e0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0003b0f0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0003b100: 2020 2020 2020 2020 3220 207c 0a7c 3133          2  |.|13
│ │ │ │ -0003b110: 3133 7820 7820 202b 2036 3033 7820 7820  13x x  + 603x x 
│ │ │ │ -0003b120: 202b 2031 3036 7820 7820 202d 2031 3336   + 106x x  - 136
│ │ │ │ -0003b130: 3178 2078 2020 2b20 3639 3978 202c 202d  1x x  + 699x , -
│ │ │ │ -0003b140: 2032 3238 7820 202d 2031 3531 3078 2078   228x  - 1510x x
│ │ │ │ -0003b150: 2020 2b20 3237 3778 2020 2d7c 0a7c 2020    + 277x  -|.|  
│ │ │ │ -0003b160: 2020 2034 2038 2020 2020 2020 2035 2038     4 8       5 8
│ │ │ │ -0003b170: 2020 2020 2020 2036 2038 2020 2020 2020         6 8      
│ │ │ │ -0003b180: 2020 3720 3820 2020 2020 2020 3820 2020    7 8       8   
│ │ │ │ -0003b190: 2020 2020 2030 2020 2020 2020 2020 3020       0        0 
│ │ │ │ -0003b1a0: 3120 2020 2020 2020 3120 207c 0a7c 2d2d  1       1  |.|--
│ │ │ │ -0003b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b0e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b0f0: 2020 3220 2020 2020 2020 2032 2020 2020    2        2    
│ │ │ │ +0003b100: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0003b110: 207c 0a7c 3133 3133 7820 7820 202b 2036   |.|1313x x  + 6
│ │ │ │ +0003b120: 3033 7820 7820 202b 2031 3036 7820 7820  03x x  + 106x x 
│ │ │ │ +0003b130: 202d 2031 3336 3178 2078 2020 2b20 3639   - 1361x x  + 69
│ │ │ │ +0003b140: 3978 202c 202d 2032 3238 7820 202d 2031  9x , - 228x  - 1
│ │ │ │ +0003b150: 3531 3078 2078 2020 2b20 3237 3778 2020  510x x  + 277x  
│ │ │ │ +0003b160: 2d7c 0a7c 2020 2020 2034 2038 2020 2020  -|.|     4 8    
│ │ │ │ +0003b170: 2020 2035 2038 2020 2020 2020 2036 2038     5 8       6 8
│ │ │ │ +0003b180: 2020 2020 2020 2020 3720 3820 2020 2020          7 8     
│ │ │ │ +0003b190: 2020 3820 2020 2020 2020 2030 2020 2020    8        0    
│ │ │ │ +0003b1a0: 2020 2020 3020 3120 2020 2020 2020 3120      0 1       1 
│ │ │ │ +0003b1b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003b200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b210: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0003b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b200: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003b210: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │  0003b220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b230: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0003b240: 2020 2020 2020 2020 2020 207c 0a7c 3478             |.|4x
│ │ │ │ -0003b250: 2078 2020 2d20 3232 7820 7820 202d 2031   x  - 22x x  - 1
│ │ │ │ -0003b260: 3532 3678 2020 2b20 3233 3478 2078 2020  526x  + 234x x  
│ │ │ │ -0003b270: 2b20 3936 3978 2078 2020 2b20 3132 3533  + 969x x  + 1253
│ │ │ │ -0003b280: 7820 7820 202d 2031 3432 3678 2020 2d20  x x  - 1426x  - 
│ │ │ │ -0003b290: 3134 3734 7820 7820 202b 207c 0a7c 2020  1474x x  + |.|  
│ │ │ │ -0003b2a0: 3020 3220 2020 2020 2031 2032 2020 2020  0 2      1 2    
│ │ │ │ -0003b2b0: 2020 2020 3220 2020 2020 2020 3020 3320      2       0 3 
│ │ │ │ -0003b2c0: 2020 2020 2020 3120 3320 2020 2020 2020        1 3       
│ │ │ │ -0003b2d0: 2032 2033 2020 2020 2020 2020 3320 2020   2 3        3   
│ │ │ │ -0003b2e0: 2020 2020 2030 2034 2020 207c 0a7c 2d2d       0 4   |.|--
│ │ │ │ -0003b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b240: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003b250: 207c 0a7c 3478 2078 2020 2d20 3232 7820   |.|4x x  - 22x 
│ │ │ │ +0003b260: 7820 202d 2031 3532 3678 2020 2b20 3233  x  - 1526x  + 23
│ │ │ │ +0003b270: 3478 2078 2020 2b20 3936 3978 2078 2020  4x x  + 969x x  
│ │ │ │ +0003b280: 2b20 3132 3533 7820 7820 202d 2031 3432  + 1253x x  - 142
│ │ │ │ +0003b290: 3678 2020 2d20 3134 3734 7820 7820 202b  6x  - 1474x x  +
│ │ │ │ +0003b2a0: 207c 0a7c 2020 3020 3220 2020 2020 2031   |.|  0 2      1
│ │ │ │ +0003b2b0: 2032 2020 2020 2020 2020 3220 2020 2020   2        2     
│ │ │ │ +0003b2c0: 2020 3020 3320 2020 2020 2020 3120 3320    0 3       1 3 
│ │ │ │ +0003b2d0: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ +0003b2e0: 2020 3320 2020 2020 2020 2030 2034 2020    3        0 4  
│ │ │ │ +0003b2f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003b300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003b340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b340: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003b350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b360: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003b360: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  0003b370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b380: 2020 2020 2020 2020 2020 207c 0a7c 3934             |.|94
│ │ │ │ -0003b390: 3778 2078 2020 2b20 3139 3478 2078 2020  7x x  + 194x x  
│ │ │ │ -0003b3a0: 2d20 3331 3678 2078 2020 2d20 3938 3878  - 316x x  - 988x
│ │ │ │ -0003b3b0: 2020 2d20 3132 3131 7820 7820 202b 2031    - 1211x x  + 1
│ │ │ │ -0003b3c0: 3038 3778 2078 2020 2b20 3533 3678 2078  087x x  + 536x x
│ │ │ │ -0003b3d0: 2020 2d20 3439 3178 2078 207c 0a7c 2020    - 491x x |.|  
│ │ │ │ -0003b3e0: 2020 3120 3420 2020 2020 2020 3220 3420    1 4       2 4 
│ │ │ │ -0003b3f0: 2020 2020 2020 3320 3420 2020 2020 2020        3 4       
│ │ │ │ -0003b400: 3420 2020 2020 2020 2030 2035 2020 2020  4        0 5    
│ │ │ │ -0003b410: 2020 2020 3120 3520 2020 2020 2020 3220      1 5       2 
│ │ │ │ -0003b420: 3520 2020 2020 2020 3320 357c 0a7c 2d2d  5       3 5|.|--
│ │ │ │ -0003b430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b390: 207c 0a7c 3934 3778 2078 2020 2b20 3139   |.|947x x  + 19
│ │ │ │ +0003b3a0: 3478 2078 2020 2d20 3331 3678 2078 2020  4x x  - 316x x  
│ │ │ │ +0003b3b0: 2d20 3938 3878 2020 2d20 3132 3131 7820  - 988x  - 1211x 
│ │ │ │ +0003b3c0: 7820 202b 2031 3038 3778 2078 2020 2b20  x  + 1087x x  + 
│ │ │ │ +0003b3d0: 3533 3678 2078 2020 2d20 3439 3178 2078  536x x  - 491x x
│ │ │ │ +0003b3e0: 207c 0a7c 2020 2020 3120 3420 2020 2020   |.|    1 4     
│ │ │ │ +0003b3f0: 2020 3220 3420 2020 2020 2020 3320 3420    2 4       3 4 
│ │ │ │ +0003b400: 2020 2020 2020 3420 2020 2020 2020 2030        4        0
│ │ │ │ +0003b410: 2035 2020 2020 2020 2020 3120 3520 2020   5        1 5   
│ │ │ │ +0003b420: 2020 2020 3220 3520 2020 2020 2020 3320      2 5       3 
│ │ │ │ +0003b430: 357c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  5|.|------------
│ │ │ │  0003b440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003b480: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0003b490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b480: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003b490: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0003b4a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b4b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b4c0: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -0003b4d0: 3837 3078 2078 2020 2d20 3635 3978 2020  870x x  - 659x  
│ │ │ │ -0003b4e0: 2b20 3134 3930 7820 7820 202d 2034 3639  + 1490x x  - 469
│ │ │ │ -0003b4f0: 7820 7820 202b 2031 3139 3078 2078 2020  x x  + 1190x x  
│ │ │ │ -0003b500: 2b20 3830 3778 2078 2020 2b20 3635 3078  + 807x x  + 650x
│ │ │ │ -0003b510: 2078 2020 2b20 2020 2020 207c 0a7c 2020   x  +      |.|  
│ │ │ │ -0003b520: 2020 2020 3420 3520 2020 2020 2020 3520      4 5       5 
│ │ │ │ -0003b530: 2020 2020 2020 2030 2036 2020 2020 2020         0 6      
│ │ │ │ -0003b540: 2031 2036 2020 2020 2020 2020 3220 3620   1 6        2 6 
│ │ │ │ -0003b550: 2020 2020 2020 3320 3620 2020 2020 2020        3 6       
│ │ │ │ -0003b560: 3420 3620 2020 2020 2020 207c 0a7c 2d2d  4 6        |.|--
│ │ │ │ -0003b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b4d0: 207c 0a7c 2b20 3837 3078 2078 2020 2d20   |.|+ 870x x  - 
│ │ │ │ +0003b4e0: 3635 3978 2020 2b20 3134 3930 7820 7820  659x  + 1490x x 
│ │ │ │ +0003b4f0: 202d 2034 3639 7820 7820 202b 2031 3139   - 469x x  + 119
│ │ │ │ +0003b500: 3078 2078 2020 2b20 3830 3778 2078 2020  0x x  + 807x x  
│ │ │ │ +0003b510: 2b20 3635 3078 2078 2020 2b20 2020 2020  + 650x x  +     
│ │ │ │ +0003b520: 207c 0a7c 2020 2020 2020 3420 3520 2020   |.|      4 5   
│ │ │ │ +0003b530: 2020 2020 3520 2020 2020 2020 2030 2036      5        0 6
│ │ │ │ +0003b540: 2020 2020 2020 2031 2036 2020 2020 2020         1 6      
│ │ │ │ +0003b550: 2020 3220 3620 2020 2020 2020 3320 3620    2 6       3 6 
│ │ │ │ +0003b560: 2020 2020 2020 3420 3620 2020 2020 2020        4 6       
│ │ │ │ +0003b570: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003b5c0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003b5d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b5c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003b5d0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0003b5e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b5f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b600: 2020 2020 2020 2020 2020 207c 0a7c 3434             |.|44
│ │ │ │ -0003b610: 3878 2078 2020 2d20 3133 3533 7820 202d  8x x  - 1353x  -
│ │ │ │ -0003b620: 2032 3138 7820 7820 202b 2037 3539 7820   218x x  + 759x 
│ │ │ │ -0003b630: 7820 202d 2032 3533 7820 7820 202b 2038  x  - 253x x  + 8
│ │ │ │ -0003b640: 3330 7820 7820 202d 2031 3038 3078 2078  30x x  - 1080x x
│ │ │ │ -0003b650: 2020 2d20 3134 3378 2078 207c 0a7c 2020    - 143x x |.|  
│ │ │ │ -0003b660: 2020 3520 3620 2020 2020 2020 2036 2020    5 6        6  
│ │ │ │ -0003b670: 2020 2020 2030 2037 2020 2020 2020 2031       0 7       1
│ │ │ │ -0003b680: 2037 2020 2020 2020 2032 2037 2020 2020   7       2 7    
│ │ │ │ -0003b690: 2020 2033 2037 2020 2020 2020 2020 3420     3 7        4 
│ │ │ │ -0003b6a0: 3720 2020 2020 2020 3520 377c 0a7c 2d2d  7       5 7|.|--
│ │ │ │ -0003b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b610: 207c 0a7c 3434 3878 2078 2020 2d20 3133   |.|448x x  - 13
│ │ │ │ +0003b620: 3533 7820 202d 2032 3138 7820 7820 202b  53x  - 218x x  +
│ │ │ │ +0003b630: 2037 3539 7820 7820 202d 2032 3533 7820   759x x  - 253x 
│ │ │ │ +0003b640: 7820 202b 2038 3330 7820 7820 202d 2031  x  + 830x x  - 1
│ │ │ │ +0003b650: 3038 3078 2078 2020 2d20 3134 3378 2078  080x x  - 143x x
│ │ │ │ +0003b660: 207c 0a7c 2020 2020 3520 3620 2020 2020   |.|    5 6     
│ │ │ │ +0003b670: 2020 2036 2020 2020 2020 2030 2037 2020     6       0 7  
│ │ │ │ +0003b680: 2020 2020 2031 2037 2020 2020 2020 2032       1 7       2
│ │ │ │ +0003b690: 2037 2020 2020 2020 2033 2037 2020 2020   7       3 7    
│ │ │ │ +0003b6a0: 2020 2020 3420 3720 2020 2020 2020 3520      4 7       5 
│ │ │ │ +0003b6b0: 377c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  7|.|------------
│ │ │ │  0003b6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003b700: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0003b710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b700: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003b710: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  0003b720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003b730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b740: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -0003b750: 3133 3133 7820 7820 202d 2033 3734 7820  1313x x  - 374x 
│ │ │ │ -0003b760: 202d 2031 3830 7820 7820 202b 2037 3431   - 180x x  + 741
│ │ │ │ -0003b770: 7820 7820 202b 2037 3432 7820 7820 202d  x x  + 742x x  -
│ │ │ │ -0003b780: 2031 3235 3478 2078 2020 2b20 3435 3878   1254x x  + 458x
│ │ │ │ -0003b790: 2078 2020 2d20 2020 2020 207c 0a7c 2020   x  -      |.|  
│ │ │ │ -0003b7a0: 2020 2020 2036 2037 2020 2020 2020 2037       6 7       7
│ │ │ │ -0003b7b0: 2020 2020 2020 2030 2038 2020 2020 2020         0 8      
│ │ │ │ -0003b7c0: 2031 2038 2020 2020 2020 2032 2038 2020   1 8       2 8  
│ │ │ │ -0003b7d0: 2020 2020 2020 3320 3820 2020 2020 2020        3 8       
│ │ │ │ -0003b7e0: 3420 3820 2020 2020 2020 207c 0a7c 2d2d  4 8        |.|--
│ │ │ │ -0003b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b750: 207c 0a7c 2d20 3133 3133 7820 7820 202d   |.|- 1313x x  -
│ │ │ │ +0003b760: 2033 3734 7820 202d 2031 3830 7820 7820   374x  - 180x x 
│ │ │ │ +0003b770: 202b 2037 3431 7820 7820 202b 2037 3432   + 741x x  + 742
│ │ │ │ +0003b780: 7820 7820 202d 2031 3235 3478 2078 2020  x x  - 1254x x  
│ │ │ │ +0003b790: 2b20 3435 3878 2078 2020 2d20 2020 2020  + 458x x  -     
│ │ │ │ +0003b7a0: 207c 0a7c 2020 2020 2020 2036 2037 2020   |.|       6 7  
│ │ │ │ +0003b7b0: 2020 2020 2037 2020 2020 2020 2030 2038       7       0 8
│ │ │ │ +0003b7c0: 2020 2020 2020 2031 2038 2020 2020 2020         1 8      
│ │ │ │ +0003b7d0: 2032 2038 2020 2020 2020 2020 3320 3820   2 8        3 8 
│ │ │ │ +0003b7e0: 2020 2020 2020 3420 3820 2020 2020 2020        4 8       
│ │ │ │ +0003b7f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003b840: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b840: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003b850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b860: 3220 2020 2020 2020 3220 2020 2020 2020  2       2       
│ │ │ │ -0003b870: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0003b880: 2020 2020 2020 2020 2020 207c 0a7c 3334             |.|34
│ │ │ │ -0003b890: 3578 2078 2020 2b20 3539 3778 2078 2020  5x x  + 597x x  
│ │ │ │ -0003b8a0: 2b20 3135 3637 7820 7820 202d 2033 3178  + 1567x x  - 31x
│ │ │ │ -0003b8b0: 202c 2031 3132 3078 2020 2b20 3730 3978   , 1120x  + 709x
│ │ │ │ -0003b8c0: 2078 2020 2d20 3135 3338 7820 202d 2031   x  - 1538x  - 1
│ │ │ │ -0003b8d0: 3034 3878 2078 2020 2d20 207c 0a7c 2020  048x x  -  |.|  
│ │ │ │ -0003b8e0: 2020 3520 3820 2020 2020 2020 3620 3820    5 8       6 8 
│ │ │ │ -0003b8f0: 2020 2020 2020 2037 2038 2020 2020 2020         7 8      
│ │ │ │ -0003b900: 3820 2020 2020 2020 3020 2020 2020 2020  8       0       
│ │ │ │ -0003b910: 3020 3120 2020 2020 2020 2031 2020 2020  0 1        1    
│ │ │ │ -0003b920: 2020 2020 3020 3220 2020 207c 0a7c 2d2d      0 2    |.|--
│ │ │ │ -0003b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b860: 2020 2020 2020 3220 2020 2020 2020 3220        2       2 
│ │ │ │ +0003b870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b880: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0003b890: 207c 0a7c 3334 3578 2078 2020 2b20 3539   |.|345x x  + 59
│ │ │ │ +0003b8a0: 3778 2078 2020 2b20 3135 3637 7820 7820  7x x  + 1567x x 
│ │ │ │ +0003b8b0: 202d 2033 3178 202c 2031 3132 3078 2020   - 31x , 1120x  
│ │ │ │ +0003b8c0: 2b20 3730 3978 2078 2020 2d20 3135 3338  + 709x x  - 1538
│ │ │ │ +0003b8d0: 7820 202d 2031 3034 3878 2078 2020 2d20  x  - 1048x x  - 
│ │ │ │ +0003b8e0: 207c 0a7c 2020 2020 3520 3820 2020 2020   |.|    5 8     
│ │ │ │ +0003b8f0: 2020 3620 3820 2020 2020 2020 2037 2038    6 8        7 8
│ │ │ │ +0003b900: 2020 2020 2020 3820 2020 2020 2020 3020        8       0 
│ │ │ │ +0003b910: 2020 2020 2020 3020 3120 2020 2020 2020        0 1       
│ │ │ │ +0003b920: 2031 2020 2020 2020 2020 3020 3220 2020   1        0 2   
│ │ │ │ +0003b930: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003b980: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003b990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b980: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003b990: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0003b9a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003b9b0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0003b9c0: 2020 2020 2020 2020 2020 207c 0a7c 3136             |.|16
│ │ │ │ -0003b9d0: 3278 2078 2020 2d20 3135 3138 7820 202d  2x x  - 1518x  -
│ │ │ │ -0003b9e0: 2037 3378 2078 2020 2b20 3338 3078 2078   73x x  + 380x x
│ │ │ │ -0003b9f0: 2020 2b20 3533 3378 2078 2020 2d20 3238    + 533x x  - 28
│ │ │ │ -0003ba00: 3678 2020 2b20 3133 3734 7820 7820 202d  6x  + 1374x x  -
│ │ │ │ -0003ba10: 2037 3478 2078 2020 2d20 207c 0a7c 2020   74x x  -  |.|  
│ │ │ │ -0003ba20: 2020 3120 3220 2020 2020 2020 2032 2020    1 2        2  
│ │ │ │ -0003ba30: 2020 2020 3020 3320 2020 2020 2020 3120      0 3       1 
│ │ │ │ -0003ba40: 3320 2020 2020 2020 3220 3320 2020 2020  3       2 3     
│ │ │ │ -0003ba50: 2020 3320 2020 2020 2020 2030 2034 2020    3        0 4  
│ │ │ │ -0003ba60: 2020 2020 3120 3420 2020 207c 0a7c 2d2d      1 4    |.|--
│ │ │ │ -0003ba70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003b9b0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003b9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003b9d0: 207c 0a7c 3136 3278 2078 2020 2d20 3135   |.|162x x  - 15
│ │ │ │ +0003b9e0: 3138 7820 202d 2037 3378 2078 2020 2b20  18x  - 73x x  + 
│ │ │ │ +0003b9f0: 3338 3078 2078 2020 2b20 3533 3378 2078  380x x  + 533x x
│ │ │ │ +0003ba00: 2020 2d20 3238 3678 2020 2b20 3133 3734    - 286x  + 1374
│ │ │ │ +0003ba10: 7820 7820 202d 2037 3478 2078 2020 2d20  x x  - 74x x  - 
│ │ │ │ +0003ba20: 207c 0a7c 2020 2020 3120 3220 2020 2020   |.|    1 2     
│ │ │ │ +0003ba30: 2020 2032 2020 2020 2020 3020 3320 2020     2      0 3   
│ │ │ │ +0003ba40: 2020 2020 3120 3320 2020 2020 2020 3220      1 3       2 
│ │ │ │ +0003ba50: 3320 2020 2020 2020 3320 2020 2020 2020  3       3       
│ │ │ │ +0003ba60: 2030 2034 2020 2020 2020 3120 3420 2020   0 4      1 4   
│ │ │ │ +0003ba70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003bac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bad0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bac0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003bad0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0003bae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003baf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bb00: 2020 2020 2020 2020 2020 207c 0a7c 3232             |.|22
│ │ │ │ -0003bb10: 7820 7820 202b 2031 3533 3578 2078 2020  x x  + 1535x x  
│ │ │ │ -0003bb20: 2d20 3130 3731 7820 202d 2038 3339 7820  - 1071x  - 839x 
│ │ │ │ -0003bb30: 7820 202d 2035 3630 7820 7820 202b 2039  x  - 560x x  + 9
│ │ │ │ -0003bb40: 3238 7820 7820 202b 2033 3335 7820 7820  28x x  + 335x x 
│ │ │ │ -0003bb50: 202d 2031 3030 3878 2078 207c 0a7c 2020   - 1008x x |.|  
│ │ │ │ -0003bb60: 2032 2034 2020 2020 2020 2020 3320 3420   2 4        3 4 
│ │ │ │ -0003bb70: 2020 2020 2020 2034 2020 2020 2020 2030         4       0
│ │ │ │ -0003bb80: 2035 2020 2020 2020 2031 2035 2020 2020   5       1 5    
│ │ │ │ -0003bb90: 2020 2032 2035 2020 2020 2020 2033 2035     2 5       3 5
│ │ │ │ -0003bba0: 2020 2020 2020 2020 3420 357c 0a7c 2d2d          4 5|.|--
│ │ │ │ -0003bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bb00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bb10: 207c 0a7c 3232 7820 7820 202b 2031 3533   |.|22x x  + 153
│ │ │ │ +0003bb20: 3578 2078 2020 2d20 3130 3731 7820 202d  5x x  - 1071x  -
│ │ │ │ +0003bb30: 2038 3339 7820 7820 202d 2035 3630 7820   839x x  - 560x 
│ │ │ │ +0003bb40: 7820 202b 2039 3238 7820 7820 202b 2033  x  + 928x x  + 3
│ │ │ │ +0003bb50: 3335 7820 7820 202d 2031 3030 3878 2078  35x x  - 1008x x
│ │ │ │ +0003bb60: 207c 0a7c 2020 2032 2034 2020 2020 2020   |.|   2 4      
│ │ │ │ +0003bb70: 2020 3320 3420 2020 2020 2020 2034 2020    3 4        4  
│ │ │ │ +0003bb80: 2020 2020 2030 2035 2020 2020 2020 2031       0 5       1
│ │ │ │ +0003bb90: 2035 2020 2020 2020 2032 2035 2020 2020   5       2 5    
│ │ │ │ +0003bba0: 2020 2033 2035 2020 2020 2020 2020 3420     3 5        4 
│ │ │ │ +0003bbb0: 357c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  5|.|------------
│ │ │ │  0003bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003bc00: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0003bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bc00: 2d7c 0a7c 2020 2020 2020 3220 2020 2020  -|.|      2     
│ │ │ │  0003bc10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bc40: 2020 2020 2020 2020 2032 207c 0a7c 2b20           2 |.|+ 
│ │ │ │ -0003bc50: 3831 3078 2020 2d20 3434 3878 2078 2020  810x  - 448x x  
│ │ │ │ -0003bc60: 2d20 3335 3778 2078 2020 2d20 3130 3778  - 357x x  - 107x
│ │ │ │ -0003bc70: 2078 2020 2b20 3430 7820 7820 202b 2037   x  + 40x x  + 7
│ │ │ │ -0003bc80: 3834 7820 7820 202d 2031 3432 3378 2078  84x x  - 1423x x
│ │ │ │ -0003bc90: 2020 2b20 3132 3736 7820 207c 0a7c 2020    + 1276x  |.|  
│ │ │ │ -0003bca0: 2020 2020 3520 2020 2020 2020 3020 3620      5       0 6 
│ │ │ │ -0003bcb0: 2020 2020 2020 3120 3620 2020 2020 2020        1 6       
│ │ │ │ -0003bcc0: 3220 3620 2020 2020 2033 2036 2020 2020  2 6      3 6    
│ │ │ │ -0003bcd0: 2020 2034 2036 2020 2020 2020 2020 3520     4 6        5 
│ │ │ │ -0003bce0: 3620 2020 2020 2020 2036 207c 0a7c 2d2d  6        6 |.|--
│ │ │ │ -0003bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bc40: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0003bc50: 207c 0a7c 2b20 3831 3078 2020 2d20 3434   |.|+ 810x  - 44
│ │ │ │ +0003bc60: 3878 2078 2020 2d20 3335 3778 2078 2020  8x x  - 357x x  
│ │ │ │ +0003bc70: 2d20 3130 3778 2078 2020 2b20 3430 7820  - 107x x  + 40x 
│ │ │ │ +0003bc80: 7820 202b 2037 3834 7820 7820 202d 2031  x  + 784x x  - 1
│ │ │ │ +0003bc90: 3432 3378 2078 2020 2b20 3132 3736 7820  423x x  + 1276x 
│ │ │ │ +0003bca0: 207c 0a7c 2020 2020 2020 3520 2020 2020   |.|      5     
│ │ │ │ +0003bcb0: 2020 3020 3620 2020 2020 2020 3120 3620    0 6       1 6 
│ │ │ │ +0003bcc0: 2020 2020 2020 3220 3620 2020 2020 2033        2 6      3
│ │ │ │ +0003bcd0: 2036 2020 2020 2020 2034 2036 2020 2020   6       4 6    
│ │ │ │ +0003bce0: 2020 2020 3520 3620 2020 2020 2020 2036      5 6        6
│ │ │ │ +0003bcf0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003bd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bd40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003bd50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bd70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bd80: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -0003bd90: 3134 3778 2078 2020 2b20 3434 3378 2078  147x x  + 443x x
│ │ │ │ -0003bda0: 2020 2d20 3539 3878 2078 2020 2d20 3130    - 598x x  - 10
│ │ │ │ -0003bdb0: 3737 7820 7820 202d 2031 3231 3478 2078  77x x  - 1214x x
│ │ │ │ -0003bdc0: 2020 2b20 3332 3278 2078 2020 2d20 3134    + 322x x  - 14
│ │ │ │ -0003bdd0: 3038 7820 7820 202b 2020 207c 0a7c 2020  08x x  +   |.|  
│ │ │ │ -0003bde0: 2020 2020 3020 3720 2020 2020 2020 3120      0 7       1 
│ │ │ │ -0003bdf0: 3720 2020 2020 2020 3220 3720 2020 2020  7       2 7     
│ │ │ │ -0003be00: 2020 2033 2037 2020 2020 2020 2020 3420     3 7        4 
│ │ │ │ -0003be10: 3720 2020 2020 2020 3520 3720 2020 2020  7       5 7     
│ │ │ │ -0003be20: 2020 2036 2037 2020 2020 207c 0a7c 2d2d     6 7     |.|--
│ │ │ │ -0003be30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bd80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bd90: 207c 0a7c 2b20 3134 3778 2078 2020 2b20   |.|+ 147x x  + 
│ │ │ │ +0003bda0: 3434 3378 2078 2020 2d20 3539 3878 2078  443x x  - 598x x
│ │ │ │ +0003bdb0: 2020 2d20 3130 3737 7820 7820 202d 2031    - 1077x x  - 1
│ │ │ │ +0003bdc0: 3231 3478 2078 2020 2b20 3332 3278 2078  214x x  + 322x x
│ │ │ │ +0003bdd0: 2020 2d20 3134 3038 7820 7820 202b 2020    - 1408x x  +  
│ │ │ │ +0003bde0: 207c 0a7c 2020 2020 2020 3020 3720 2020   |.|      0 7   
│ │ │ │ +0003bdf0: 2020 2020 3120 3720 2020 2020 2020 3220      1 7       2 
│ │ │ │ +0003be00: 3720 2020 2020 2020 2033 2037 2020 2020  7        3 7    
│ │ │ │ +0003be10: 2020 2020 3420 3720 2020 2020 2020 3520      4 7       5 
│ │ │ │ +0003be20: 3720 2020 2020 2020 2036 2037 2020 2020  7        6 7    
│ │ │ │ +0003be30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003be70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003be80: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0003be70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003be80: 2d7c 0a7c 2020 2032 2020 2020 2020 2020  -|.|   2        
│ │ │ │  0003be90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003bea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003beb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003bec0: 2020 2020 2020 2020 2020 207c 0a7c 3732             |.|72
│ │ │ │ -0003bed0: 7820 202d 2036 3378 2078 2020 2d20 3135  x  - 63x x  - 15
│ │ │ │ -0003bee0: 3133 7820 7820 202d 2037 3931 7820 7820  13x x  - 791x x 
│ │ │ │ -0003bef0: 202b 2031 3178 2078 2020 2b20 3737 7820   + 11x x  + 77x 
│ │ │ │ -0003bf00: 7820 202b 2038 3336 7820 7820 202d 2031  x  + 836x x  - 1
│ │ │ │ -0003bf10: 3130 3078 2078 2020 2b20 207c 0a7c 2020  100x x  +  |.|  
│ │ │ │ -0003bf20: 2037 2020 2020 2020 3020 3820 2020 2020   7      0 8     
│ │ │ │ -0003bf30: 2020 2031 2038 2020 2020 2020 2032 2038     1 8       2 8
│ │ │ │ -0003bf40: 2020 2020 2020 3320 3820 2020 2020 2034        3 8      4
│ │ │ │ -0003bf50: 2038 2020 2020 2020 2035 2038 2020 2020   8       5 8    
│ │ │ │ -0003bf60: 2020 2020 3620 3820 2020 207c 0a7c 2d2d      6 8    |.|--
│ │ │ │ -0003bf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003bed0: 207c 0a7c 3732 7820 202d 2036 3378 2078   |.|72x  - 63x x
│ │ │ │ +0003bee0: 2020 2d20 3135 3133 7820 7820 202d 2037    - 1513x x  - 7
│ │ │ │ +0003bef0: 3931 7820 7820 202b 2031 3178 2078 2020  91x x  + 11x x  
│ │ │ │ +0003bf00: 2b20 3737 7820 7820 202b 2038 3336 7820  + 77x x  + 836x 
│ │ │ │ +0003bf10: 7820 202d 2031 3130 3078 2078 2020 2b20  x  - 1100x x  + 
│ │ │ │ +0003bf20: 207c 0a7c 2020 2037 2020 2020 2020 3020   |.|   7      0 
│ │ │ │ +0003bf30: 3820 2020 2020 2020 2031 2038 2020 2020  8        1 8    
│ │ │ │ +0003bf40: 2020 2032 2038 2020 2020 2020 3320 3820     2 8      3 8 
│ │ │ │ +0003bf50: 2020 2020 2034 2038 2020 2020 2020 2035       4 8       5
│ │ │ │ +0003bf60: 2038 2020 2020 2020 2020 3620 3820 2020   8        6 8   
│ │ │ │ +0003bf70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003bf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003bfc0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003bfd0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0003bfe0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003bfc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003bfd0: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │ +0003bfe0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0003bff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c000: 2020 2032 2020 2020 2020 207c 0a7c 3136     2       |.|16
│ │ │ │ -0003c010: 3337 7820 7820 202d 2037 3838 7820 2c20  37x x  - 788x , 
│ │ │ │ -0003c020: 3133 3331 7820 202b 2033 3138 7820 7820  1331x  + 318x x 
│ │ │ │ -0003c030: 202d 2037 3034 7820 202b 2035 3178 2078   - 704x  + 51x x
│ │ │ │ -0003c040: 2020 2b20 3237 3578 2078 2020 2b20 3131    + 275x x  + 11
│ │ │ │ -0003c050: 3439 7820 202b 2020 2020 207c 0a7c 2020  49x  +     |.|  
│ │ │ │ -0003c060: 2020 2037 2038 2020 2020 2020 2038 2020     7 8       8  
│ │ │ │ -0003c070: 2020 2020 2030 2020 2020 2020 2030 2031       0       0 1
│ │ │ │ -0003c080: 2020 2020 2020 2031 2020 2020 2020 3020         1      0 
│ │ │ │ -0003c090: 3220 2020 2020 2020 3120 3220 2020 2020  2       1 2     
│ │ │ │ -0003c0a0: 2020 2032 2020 2020 2020 207c 0a7c 2d2d     2       |.|--
│ │ │ │ -0003c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c000: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0003c010: 207c 0a7c 3136 3337 7820 7820 202d 2037   |.|1637x x  - 7
│ │ │ │ +0003c020: 3838 7820 2c20 3133 3331 7820 202b 2033  88x , 1331x  + 3
│ │ │ │ +0003c030: 3138 7820 7820 202d 2037 3034 7820 202b  18x x  - 704x  +
│ │ │ │ +0003c040: 2035 3178 2078 2020 2b20 3237 3578 2078   51x x  + 275x x
│ │ │ │ +0003c050: 2020 2b20 3131 3439 7820 202b 2020 2020    + 1149x  +    
│ │ │ │ +0003c060: 207c 0a7c 2020 2020 2037 2038 2020 2020   |.|     7 8    
│ │ │ │ +0003c070: 2020 2038 2020 2020 2020 2030 2020 2020     8       0    
│ │ │ │ +0003c080: 2020 2030 2031 2020 2020 2020 2031 2020     0 1       1  
│ │ │ │ +0003c090: 2020 2020 3020 3220 2020 2020 2020 3120      0 2       1 
│ │ │ │ +0003c0a0: 3220 2020 2020 2020 2032 2020 2020 2020  2        2      
│ │ │ │ +0003c0b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003c100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c100: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003c110: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c120: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0003c120: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0003c130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c140: 2020 2020 2020 2020 2020 207c 0a7c 3135             |.|15
│ │ │ │ -0003c150: 3236 7820 7820 202b 2037 3638 7820 7820  26x x  + 768x x 
│ │ │ │ -0003c160: 202b 2034 3134 7820 7820 202d 2037 3832   + 414x x  - 782
│ │ │ │ -0003c170: 7820 202d 2032 3632 7820 7820 202b 2036  x  - 262x x  + 6
│ │ │ │ -0003c180: 3836 7820 7820 202d 2033 3830 7820 7820  86x x  - 380x x 
│ │ │ │ -0003c190: 202b 2031 3337 3778 2078 207c 0a7c 2020   + 1377x x |.|  
│ │ │ │ -0003c1a0: 2020 2030 2033 2020 2020 2020 2031 2033     0 3       1 3
│ │ │ │ -0003c1b0: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ -0003c1c0: 2033 2020 2020 2020 2030 2034 2020 2020   3       0 4    
│ │ │ │ -0003c1d0: 2020 2031 2034 2020 2020 2020 2032 2034     1 4       2 4
│ │ │ │ -0003c1e0: 2020 2020 2020 2020 3320 347c 0a7c 2d2d          3 4|.|--
│ │ │ │ -0003c1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c150: 207c 0a7c 3135 3236 7820 7820 202b 2037   |.|1526x x  + 7
│ │ │ │ +0003c160: 3638 7820 7820 202b 2034 3134 7820 7820  68x x  + 414x x 
│ │ │ │ +0003c170: 202d 2037 3832 7820 202d 2032 3632 7820   - 782x  - 262x 
│ │ │ │ +0003c180: 7820 202b 2036 3836 7820 7820 202d 2033  x  + 686x x  - 3
│ │ │ │ +0003c190: 3830 7820 7820 202b 2031 3337 3778 2078  80x x  + 1377x x
│ │ │ │ +0003c1a0: 207c 0a7c 2020 2020 2030 2033 2020 2020   |.|     0 3    
│ │ │ │ +0003c1b0: 2020 2031 2033 2020 2020 2020 2032 2033     1 3       2 3
│ │ │ │ +0003c1c0: 2020 2020 2020 2033 2020 2020 2020 2030         3       0
│ │ │ │ +0003c1d0: 2034 2020 2020 2020 2031 2034 2020 2020   4       1 4    
│ │ │ │ +0003c1e0: 2020 2032 2034 2020 2020 2020 2020 3320     2 4        3 
│ │ │ │ +0003c1f0: 347c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  4|.|------------
│ │ │ │  0003c200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003c240: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0003c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c240: 2d7c 0a7c 2020 2020 2020 2032 2020 2020  -|.|       2    
│ │ │ │  0003c250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c280: 2020 3220 2020 2020 2020 207c 0a7c 2b20    2        |.|+ 
│ │ │ │ -0003c290: 3130 3737 7820 202b 2031 3635 3078 2078  1077x  + 1650x x
│ │ │ │ -0003c2a0: 2020 2d20 3131 3239 7820 7820 202d 2035    - 1129x x  - 5
│ │ │ │ -0003c2b0: 3038 7820 7820 202b 2038 3436 7820 7820  08x x  + 846x x 
│ │ │ │ -0003c2c0: 202b 2031 3531 3378 2078 2020 2b20 3436   + 1513x x  + 46
│ │ │ │ -0003c2d0: 3078 2020 2d20 2020 2020 207c 0a7c 2020  0x  -      |.|  
│ │ │ │ -0003c2e0: 2020 2020 2034 2020 2020 2020 2020 3020       4        0 
│ │ │ │ -0003c2f0: 3520 2020 2020 2020 2031 2035 2020 2020  5        1 5    
│ │ │ │ -0003c300: 2020 2032 2035 2020 2020 2020 2033 2035     2 5       3 5
│ │ │ │ -0003c310: 2020 2020 2020 2020 3420 3520 2020 2020          4 5     
│ │ │ │ -0003c320: 2020 3520 2020 2020 2020 207c 0a7c 2d2d    5        |.|--
│ │ │ │ -0003c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c280: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003c290: 207c 0a7c 2b20 3130 3737 7820 202b 2031   |.|+ 1077x  + 1
│ │ │ │ +0003c2a0: 3635 3078 2078 2020 2d20 3131 3239 7820  650x x  - 1129x 
│ │ │ │ +0003c2b0: 7820 202d 2035 3038 7820 7820 202b 2038  x  - 508x x  + 8
│ │ │ │ +0003c2c0: 3436 7820 7820 202b 2031 3531 3378 2078  46x x  + 1513x x
│ │ │ │ +0003c2d0: 2020 2b20 3436 3078 2020 2d20 2020 2020    + 460x  -     
│ │ │ │ +0003c2e0: 207c 0a7c 2020 2020 2020 2034 2020 2020   |.|       4    
│ │ │ │ +0003c2f0: 2020 2020 3020 3520 2020 2020 2020 2031      0 5        1
│ │ │ │ +0003c300: 2035 2020 2020 2020 2032 2035 2020 2020   5       2 5    
│ │ │ │ +0003c310: 2020 2033 2035 2020 2020 2020 2020 3420     3 5        4 
│ │ │ │ +0003c320: 3520 2020 2020 2020 3520 2020 2020 2020  5       5       
│ │ │ │ +0003c330: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003c380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c380: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003c390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c3a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c3b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c3c0: 2020 3220 2020 2020 2020 207c 0a7c 3136    2        |.|16
│ │ │ │ -0003c3d0: 3236 7820 7820 202d 2031 3032 3478 2078  26x x  - 1024x x
│ │ │ │ -0003c3e0: 2020 2b20 3836 3278 2078 2020 2b20 3133    + 862x x  + 13
│ │ │ │ -0003c3f0: 3532 7820 7820 202d 2031 3838 7820 7820  52x x  - 188x x 
│ │ │ │ -0003c400: 202d 2031 3338 3278 2078 2020 2d20 3635   - 1382x x  - 65
│ │ │ │ -0003c410: 3078 2020 2b20 2020 2020 207c 0a7c 2020  0x  +      |.|  
│ │ │ │ -0003c420: 2020 2030 2036 2020 2020 2020 2020 3120     0 6        1 
│ │ │ │ -0003c430: 3620 2020 2020 2020 3220 3620 2020 2020  6       2 6     
│ │ │ │ -0003c440: 2020 2033 2036 2020 2020 2020 2034 2036     3 6       4 6
│ │ │ │ -0003c450: 2020 2020 2020 2020 3520 3620 2020 2020          5 6     
│ │ │ │ -0003c460: 2020 3620 2020 2020 2020 207c 0a7c 2d2d    6        |.|--
│ │ │ │ -0003c470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c3c0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003c3d0: 207c 0a7c 3136 3236 7820 7820 202d 2031   |.|1626x x  - 1
│ │ │ │ +0003c3e0: 3032 3478 2078 2020 2b20 3836 3278 2078  024x x  + 862x x
│ │ │ │ +0003c3f0: 2020 2b20 3133 3532 7820 7820 202d 2031    + 1352x x  - 1
│ │ │ │ +0003c400: 3838 7820 7820 202d 2031 3338 3278 2078  88x x  - 1382x x
│ │ │ │ +0003c410: 2020 2d20 3635 3078 2020 2b20 2020 2020    - 650x  +     
│ │ │ │ +0003c420: 207c 0a7c 2020 2020 2030 2036 2020 2020   |.|     0 6    
│ │ │ │ +0003c430: 2020 2020 3120 3620 2020 2020 2020 3220      1 6       2 
│ │ │ │ +0003c440: 3620 2020 2020 2020 2033 2036 2020 2020  6        3 6    
│ │ │ │ +0003c450: 2020 2034 2036 2020 2020 2020 2020 3520     4 6        5 
│ │ │ │ +0003c460: 3620 2020 2020 2020 3620 2020 2020 2020  6       6       
│ │ │ │ +0003c470: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003c480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003c4c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c4c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003c4d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c4e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c4f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c500: 2020 2020 2020 2020 2020 207c 0a7c 3535             |.|55
│ │ │ │ -0003c510: 7820 7820 202d 2033 3236 7820 7820 202b  x x  - 326x x  +
│ │ │ │ -0003c520: 2031 3033 3778 2078 2020 2b20 3730 3578   1037x x  + 705x
│ │ │ │ -0003c530: 2078 2020 2d20 3636 3778 2078 2020 2b20   x  - 667x x  + 
│ │ │ │ -0003c540: 3134 3833 7820 7820 202b 2031 3636 3178  1483x x  + 1661x
│ │ │ │ -0003c550: 2078 2020 2d20 2020 2020 207c 0a7c 2020   x  -      |.|  
│ │ │ │ -0003c560: 2030 2037 2020 2020 2020 2031 2037 2020   0 7       1 7  
│ │ │ │ -0003c570: 2020 2020 2020 3220 3720 2020 2020 2020        2 7       
│ │ │ │ -0003c580: 3320 3720 2020 2020 2020 3420 3720 2020  3 7       4 7   
│ │ │ │ -0003c590: 2020 2020 2035 2037 2020 2020 2020 2020       5 7        
│ │ │ │ -0003c5a0: 3620 3720 2020 2020 2020 207c 0a7c 2d2d  6 7        |.|--
│ │ │ │ -0003c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c510: 207c 0a7c 3535 7820 7820 202d 2033 3236   |.|55x x  - 326
│ │ │ │ +0003c520: 7820 7820 202b 2031 3033 3778 2078 2020  x x  + 1037x x  
│ │ │ │ +0003c530: 2b20 3730 3578 2078 2020 2d20 3636 3778  + 705x x  - 667x
│ │ │ │ +0003c540: 2078 2020 2b20 3134 3833 7820 7820 202b   x  + 1483x x  +
│ │ │ │ +0003c550: 2031 3636 3178 2078 2020 2d20 2020 2020   1661x x  -     
│ │ │ │ +0003c560: 207c 0a7c 2020 2030 2037 2020 2020 2020   |.|   0 7      
│ │ │ │ +0003c570: 2031 2037 2020 2020 2020 2020 3220 3720   1 7        2 7 
│ │ │ │ +0003c580: 2020 2020 2020 3320 3720 2020 2020 2020        3 7       
│ │ │ │ +0003c590: 3420 3720 2020 2020 2020 2035 2037 2020  4 7        5 7  
│ │ │ │ +0003c5a0: 2020 2020 2020 3620 3720 2020 2020 2020        6 7       
│ │ │ │ +0003c5b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003c600: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0003c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c600: 2d7c 0a7c 2020 2020 2032 2020 2020 2020  -|.|     2      
│ │ │ │  0003c610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003c630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c640: 2020 2020 2020 2020 2020 207c 0a7c 3136             |.|16
│ │ │ │ -0003c650: 3532 7820 202d 2031 3035 3278 2078 2020  52x  - 1052x x  
│ │ │ │ -0003c660: 2d20 3639 3278 2078 2020 2d20 3534 3278  - 692x x  - 542x
│ │ │ │ -0003c670: 2078 2020 2b20 3136 3278 2078 2020 2b20   x  + 162x x  + 
│ │ │ │ -0003c680: 3538 3278 2078 2020 2d20 3133 3639 7820  582x x  - 1369x 
│ │ │ │ -0003c690: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -0003c6a0: 2020 2037 2020 2020 2020 2020 3020 3820     7        0 8 
│ │ │ │ -0003c6b0: 2020 2020 2020 3120 3820 2020 2020 2020        1 8       
│ │ │ │ -0003c6c0: 3220 3820 2020 2020 2020 3320 3820 2020  2 8       3 8   
│ │ │ │ -0003c6d0: 2020 2020 3420 3820 2020 2020 2020 2035      4 8        5
│ │ │ │ -0003c6e0: 2038 2020 2020 2020 2020 207c 0a7c 2d2d   8         |.|--
│ │ │ │ -0003c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c640: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c650: 207c 0a7c 3136 3532 7820 202d 2031 3035   |.|1652x  - 105
│ │ │ │ +0003c660: 3278 2078 2020 2d20 3639 3278 2078 2020  2x x  - 692x x  
│ │ │ │ +0003c670: 2d20 3534 3278 2078 2020 2b20 3136 3278  - 542x x  + 162x
│ │ │ │ +0003c680: 2078 2020 2b20 3538 3278 2078 2020 2d20   x  + 582x x  - 
│ │ │ │ +0003c690: 3133 3639 7820 7820 202b 2020 2020 2020  1369x x  +      
│ │ │ │ +0003c6a0: 207c 0a7c 2020 2020 2037 2020 2020 2020   |.|     7      
│ │ │ │ +0003c6b0: 2020 3020 3820 2020 2020 2020 3120 3820    0 8       1 8 
│ │ │ │ +0003c6c0: 2020 2020 2020 3220 3820 2020 2020 2020        2 8       
│ │ │ │ +0003c6d0: 3320 3820 2020 2020 2020 3420 3820 2020  3 8       4 8   
│ │ │ │ +0003c6e0: 2020 2020 2035 2038 2020 2020 2020 2020       5 8        
│ │ │ │ +0003c6f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003c740: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c750: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -0003c760: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ -0003c770: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0003c780: 2020 2020 2020 2020 2020 207c 0a7c 3933             |.|93
│ │ │ │ -0003c790: 3478 2078 2020 2b20 3133 3932 7820 7820  4x x  + 1392x x 
│ │ │ │ -0003c7a0: 202b 2031 3232 3778 202c 202d 2033 3436   + 1227x , - 346
│ │ │ │ -0003c7b0: 7820 202b 2031 3430 3878 2078 2020 2d20  x  + 1408x x  - 
│ │ │ │ -0003c7c0: 3132 3235 7820 202d 2031 3533 3678 2078  1225x  - 1536x x
│ │ │ │ -0003c7d0: 2020 2d20 2020 2020 2020 207c 0a7c 2020    -        |.|  
│ │ │ │ -0003c7e0: 2020 3620 3820 2020 2020 2020 2037 2038    6 8        7 8
│ │ │ │ -0003c7f0: 2020 2020 2020 2020 3820 2020 2020 2020          8       
│ │ │ │ -0003c800: 2030 2020 2020 2020 2020 3020 3120 2020   0        0 1   
│ │ │ │ -0003c810: 2020 2020 2031 2020 2020 2020 2020 3020       1        0 
│ │ │ │ -0003c820: 3220 2020 2020 2020 2020 207c 0a7c 2d2d  2          |.|--
│ │ │ │ -0003c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c740: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003c750: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0003c760: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003c770: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0003c780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c790: 207c 0a7c 3933 3478 2078 2020 2b20 3133   |.|934x x  + 13
│ │ │ │ +0003c7a0: 3932 7820 7820 202b 2031 3232 3778 202c  92x x  + 1227x ,
│ │ │ │ +0003c7b0: 202d 2033 3436 7820 202b 2031 3430 3878   - 346x  + 1408x
│ │ │ │ +0003c7c0: 2078 2020 2d20 3132 3235 7820 202d 2031   x  - 1225x  - 1
│ │ │ │ +0003c7d0: 3533 3678 2078 2020 2d20 2020 2020 2020  536x x  -       
│ │ │ │ +0003c7e0: 207c 0a7c 2020 2020 3620 3820 2020 2020   |.|    6 8     
│ │ │ │ +0003c7f0: 2020 2037 2038 2020 2020 2020 2020 3820     7 8        8 
│ │ │ │ +0003c800: 2020 2020 2020 2030 2020 2020 2020 2020         0        
│ │ │ │ +0003c810: 3020 3120 2020 2020 2020 2031 2020 2020  0 1        1    
│ │ │ │ +0003c820: 2020 2020 3020 3220 2020 2020 2020 2020      0 2         
│ │ │ │ +0003c830: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003c880: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003c890: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c880: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003c890: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0003c8a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c8b0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0003c8c0: 2020 2020 2020 2020 2020 207c 0a7c 3130             |.|10
│ │ │ │ -0003c8d0: 3238 7820 7820 202d 2039 3835 7820 202d  28x x  - 985x  -
│ │ │ │ -0003c8e0: 2032 3130 7820 7820 202d 2031 3331 3278   210x x  - 1312x
│ │ │ │ -0003c8f0: 2078 2020 2b20 3931 3578 2078 2020 2b20   x  + 915x x  + 
│ │ │ │ -0003c900: 3136 3333 7820 202d 2032 3032 7820 7820  1633x  - 202x x 
│ │ │ │ -0003c910: 202d 2031 3633 3678 2078 207c 0a7c 2020   - 1636x x |.|  
│ │ │ │ -0003c920: 2020 2031 2032 2020 2020 2020 2032 2020     1 2       2  
│ │ │ │ -0003c930: 2020 2020 2030 2033 2020 2020 2020 2020       0 3        
│ │ │ │ -0003c940: 3120 3320 2020 2020 2020 3220 3320 2020  1 3       2 3   
│ │ │ │ -0003c950: 2020 2020 2033 2020 2020 2020 2030 2034       3       0 4
│ │ │ │ -0003c960: 2020 2020 2020 2020 3120 347c 0a7c 2d2d          1 4|.|--
│ │ │ │ -0003c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c8b0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +0003c8c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c8d0: 207c 0a7c 3130 3238 7820 7820 202d 2039   |.|1028x x  - 9
│ │ │ │ +0003c8e0: 3835 7820 202d 2032 3130 7820 7820 202d  85x  - 210x x  -
│ │ │ │ +0003c8f0: 2031 3331 3278 2078 2020 2b20 3931 3578   1312x x  + 915x
│ │ │ │ +0003c900: 2078 2020 2b20 3136 3333 7820 202d 2032   x  + 1633x  - 2
│ │ │ │ +0003c910: 3032 7820 7820 202d 2031 3633 3678 2078  02x x  - 1636x x
│ │ │ │ +0003c920: 207c 0a7c 2020 2020 2031 2032 2020 2020   |.|     1 2    
│ │ │ │ +0003c930: 2020 2032 2020 2020 2020 2030 2033 2020     2       0 3  
│ │ │ │ +0003c940: 2020 2020 2020 3120 3320 2020 2020 2020        1 3       
│ │ │ │ +0003c950: 3220 3320 2020 2020 2020 2033 2020 2020  2 3        3    
│ │ │ │ +0003c960: 2020 2030 2034 2020 2020 2020 2020 3120     0 4        1 
│ │ │ │ +0003c970: 347c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  4|.|------------
│ │ │ │  0003c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003c9c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003c9d0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0003c9e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003c9c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003c9d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003c9e0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │  0003c9f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ca00: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -0003ca10: 3136 3533 7820 7820 202d 2034 3830 7820  1653x x  - 480x 
│ │ │ │ -0003ca20: 7820 202d 2031 3236 3078 2020 2d20 3831  x  - 1260x  - 81
│ │ │ │ -0003ca30: 3378 2078 2020 2d20 3136 3233 7820 7820  3x x  - 1623x x 
│ │ │ │ -0003ca40: 202d 2031 3432 3978 2078 2020 2b20 3130   - 1429x x  + 10
│ │ │ │ -0003ca50: 3934 7820 7820 202d 2020 207c 0a7c 2020  94x x  -   |.|  
│ │ │ │ -0003ca60: 2020 2020 2032 2034 2020 2020 2020 2033       2 4       3
│ │ │ │ -0003ca70: 2034 2020 2020 2020 2020 3420 2020 2020   4        4     
│ │ │ │ -0003ca80: 2020 3020 3520 2020 2020 2020 2031 2035    0 5        1 5
│ │ │ │ -0003ca90: 2020 2020 2020 2020 3220 3520 2020 2020          2 5     
│ │ │ │ -0003caa0: 2020 2033 2035 2020 2020 207c 0a7c 2d2d     3 5     |.|--
│ │ │ │ -0003cab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ca00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ca10: 207c 0a7c 2d20 3136 3533 7820 7820 202d   |.|- 1653x x  -
│ │ │ │ +0003ca20: 2034 3830 7820 7820 202d 2031 3236 3078   480x x  - 1260x
│ │ │ │ +0003ca30: 2020 2d20 3831 3378 2078 2020 2d20 3136    - 813x x  - 16
│ │ │ │ +0003ca40: 3233 7820 7820 202d 2031 3432 3978 2078  23x x  - 1429x x
│ │ │ │ +0003ca50: 2020 2b20 3130 3934 7820 7820 202d 2020    + 1094x x  -  
│ │ │ │ +0003ca60: 207c 0a7c 2020 2020 2020 2032 2034 2020   |.|       2 4  
│ │ │ │ +0003ca70: 2020 2020 2033 2034 2020 2020 2020 2020       3 4        
│ │ │ │ +0003ca80: 3420 2020 2020 2020 3020 3520 2020 2020  4       0 5     
│ │ │ │ +0003ca90: 2020 2031 2035 2020 2020 2020 2020 3220     1 5        2 
│ │ │ │ +0003caa0: 3520 2020 2020 2020 2033 2035 2020 2020  5        3 5    
│ │ │ │ +0003cab0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003cac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003cb00: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0003cb10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003cb00: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003cb10: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │  0003cb20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cb30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cb40: 2020 2020 2020 2020 2020 207c 0a7c 3734             |.|74
│ │ │ │ -0003cb50: 3778 2078 2020 2b20 3935 3578 2020 2b20  7x x  + 955x  + 
│ │ │ │ -0003cb60: 3839 3878 2078 2020 2d20 3739 3578 2078  898x x  - 795x x
│ │ │ │ -0003cb70: 2020 2d20 3335 7820 7820 202d 2035 3636    - 35x x  - 566
│ │ │ │ -0003cb80: 7820 7820 202b 2031 3633 3178 2078 2020  x x  + 1631x x  
│ │ │ │ -0003cb90: 2d20 3332 3478 2078 2020 2b7c 0a7c 2020  - 324x x  +|.|  
│ │ │ │ -0003cba0: 2020 3420 3520 2020 2020 2020 3520 2020    4 5       5   
│ │ │ │ -0003cbb0: 2020 2020 3020 3620 2020 2020 2020 3120      0 6       1 
│ │ │ │ -0003cbc0: 3620 2020 2020 2032 2036 2020 2020 2020  6      2 6      
│ │ │ │ -0003cbd0: 2033 2036 2020 2020 2020 2020 3420 3620   3 6        4 6 
│ │ │ │ -0003cbe0: 2020 2020 2020 3520 3620 207c 0a7c 2d2d        5 6  |.|--
│ │ │ │ -0003cbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003cb40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cb50: 207c 0a7c 3734 3778 2078 2020 2b20 3935   |.|747x x  + 95
│ │ │ │ +0003cb60: 3578 2020 2b20 3839 3878 2078 2020 2d20  5x  + 898x x  - 
│ │ │ │ +0003cb70: 3739 3578 2078 2020 2d20 3335 7820 7820  795x x  - 35x x 
│ │ │ │ +0003cb80: 202d 2035 3636 7820 7820 202b 2031 3633   - 566x x  + 163
│ │ │ │ +0003cb90: 3178 2078 2020 2d20 3332 3478 2078 2020  1x x  - 324x x  
│ │ │ │ +0003cba0: 2b7c 0a7c 2020 2020 3420 3520 2020 2020  +|.|    4 5     
│ │ │ │ +0003cbb0: 2020 3520 2020 2020 2020 3020 3620 2020    5       0 6   
│ │ │ │ +0003cbc0: 2020 2020 3120 3620 2020 2020 2032 2036      1 6      2 6
│ │ │ │ +0003cbd0: 2020 2020 2020 2033 2036 2020 2020 2020         3 6      
│ │ │ │ +0003cbe0: 2020 3420 3620 2020 2020 2020 3520 3620    4 6       5 6 
│ │ │ │ +0003cbf0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003cc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003cc40: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003cc40: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  0003cc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cc80: 2020 2020 2020 2020 2020 207c 0a7c 3932             |.|92
│ │ │ │ -0003cc90: 3678 2020 2d20 3133 3278 2078 2020 2d20  6x  - 132x x  - 
│ │ │ │ -0003cca0: 3978 2078 2020 2d20 3132 3930 7820 7820  9x x  - 1290x x 
│ │ │ │ -0003ccb0: 202d 2035 3433 7820 7820 202b 2039 3032   - 543x x  + 902
│ │ │ │ -0003ccc0: 7820 7820 202b 2037 3335 7820 7820 202d  x x  + 735x x  -
│ │ │ │ -0003ccd0: 2033 3432 7820 7820 202d 207c 0a7c 2020   342x x  - |.|  
│ │ │ │ -0003cce0: 2020 3620 2020 2020 2020 3020 3720 2020    6       0 7   
│ │ │ │ -0003ccf0: 2020 3120 3720 2020 2020 2020 2032 2037    1 7        2 7
│ │ │ │ -0003cd00: 2020 2020 2020 2033 2037 2020 2020 2020         3 7      
│ │ │ │ -0003cd10: 2034 2037 2020 2020 2020 2035 2037 2020   4 7       5 7  
│ │ │ │ -0003cd20: 2020 2020 2036 2037 2020 207c 0a7c 2d2d       6 7   |.|--
│ │ │ │ -0003cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003cc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cc90: 207c 0a7c 3932 3678 2020 2d20 3133 3278   |.|926x  - 132x
│ │ │ │ +0003cca0: 2078 2020 2d20 3978 2078 2020 2d20 3132   x  - 9x x  - 12
│ │ │ │ +0003ccb0: 3930 7820 7820 202d 2035 3433 7820 7820  90x x  - 543x x 
│ │ │ │ +0003ccc0: 202b 2039 3032 7820 7820 202b 2037 3335   + 902x x  + 735
│ │ │ │ +0003ccd0: 7820 7820 202d 2033 3432 7820 7820 202d  x x  - 342x x  -
│ │ │ │ +0003cce0: 207c 0a7c 2020 2020 3620 2020 2020 2020   |.|    6       
│ │ │ │ +0003ccf0: 3020 3720 2020 2020 3120 3720 2020 2020  0 7     1 7     
│ │ │ │ +0003cd00: 2020 2032 2037 2020 2020 2020 2033 2037     2 7       3 7
│ │ │ │ +0003cd10: 2020 2020 2020 2034 2037 2020 2020 2020         4 7      
│ │ │ │ +0003cd20: 2035 2037 2020 2020 2020 2036 2037 2020   5 7       6 7  
│ │ │ │ +0003cd30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003cd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003cd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003cd80: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003cd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003cd80: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  0003cd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003cdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cdc0: 2020 2020 2020 2020 2020 207c 0a7c 3430             |.|40
│ │ │ │ -0003cdd0: 3078 2020 2b20 3930 3078 2078 2020 2d20  0x  + 900x x  - 
│ │ │ │ -0003cde0: 3436 3378 2078 2020 2b20 3639 3478 2078  463x x  + 694x x
│ │ │ │ -0003cdf0: 2020 2d20 3132 3632 7820 7820 202d 2031    - 1262x x  - 1
│ │ │ │ -0003ce00: 3434 3978 2078 2020 2d20 3434 3878 2078  449x x  - 448x x
│ │ │ │ -0003ce10: 2020 2d20 2020 2020 2020 207c 0a7c 2020    -        |.|  
│ │ │ │ -0003ce20: 2020 3720 2020 2020 2020 3020 3820 2020    7       0 8   
│ │ │ │ -0003ce30: 2020 2020 3120 3820 2020 2020 2020 3220      1 8       2 
│ │ │ │ -0003ce40: 3820 2020 2020 2020 2033 2038 2020 2020  8        3 8    
│ │ │ │ -0003ce50: 2020 2020 3420 3820 2020 2020 2020 3520      4 8       5 
│ │ │ │ -0003ce60: 3820 2020 2020 2020 2020 207c 0a7c 2d2d  8          |.|--
│ │ │ │ -0003ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003cdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cdd0: 207c 0a7c 3430 3078 2020 2b20 3930 3078   |.|400x  + 900x
│ │ │ │ +0003cde0: 2078 2020 2d20 3436 3378 2078 2020 2b20   x  - 463x x  + 
│ │ │ │ +0003cdf0: 3639 3478 2078 2020 2d20 3132 3632 7820  694x x  - 1262x 
│ │ │ │ +0003ce00: 7820 202d 2031 3434 3978 2078 2020 2d20  x  - 1449x x  - 
│ │ │ │ +0003ce10: 3434 3878 2078 2020 2d20 2020 2020 2020  448x x  -       
│ │ │ │ +0003ce20: 207c 0a7c 2020 2020 3720 2020 2020 2020   |.|    7       
│ │ │ │ +0003ce30: 3020 3820 2020 2020 2020 3120 3820 2020  0 8       1 8   
│ │ │ │ +0003ce40: 2020 2020 3220 3820 2020 2020 2020 2033      2 8        3
│ │ │ │ +0003ce50: 2038 2020 2020 2020 2020 3420 3820 2020   8        4 8   
│ │ │ │ +0003ce60: 2020 2020 3520 3820 2020 2020 2020 2020      5 8         
│ │ │ │ +0003ce70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003cec0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ced0: 2020 2020 2020 2032 2020 2020 2020 3220         2      2 
│ │ │ │ -0003cee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003cef0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ -0003cf00: 2020 2020 2020 2020 2020 207c 0a7c 3134             |.|14
│ │ │ │ -0003cf10: 3032 7820 7820 202d 2037 3331 7820 7820  02x x  - 731x x 
│ │ │ │ -0003cf20: 202d 2039 3936 7820 2c20 3330 3178 2020   - 996x , 301x  
│ │ │ │ -0003cf30: 2b20 3136 3678 2078 2020 2d20 3935 3578  + 166x x  - 955x
│ │ │ │ -0003cf40: 2020 2d20 3733 3978 2078 2020 2d20 3131    - 739x x  - 11
│ │ │ │ -0003cf50: 3939 7820 7820 202d 2020 207c 0a7c 2020  99x x  -   |.|  
│ │ │ │ -0003cf60: 2020 2036 2038 2020 2020 2020 2037 2038     6 8       7 8
│ │ │ │ -0003cf70: 2020 2020 2020 2038 2020 2020 2020 3020         8      0 
│ │ │ │ -0003cf80: 2020 2020 2020 3020 3120 2020 2020 2020        0 1       
│ │ │ │ -0003cf90: 3120 2020 2020 2020 3020 3220 2020 2020  1       0 2     
│ │ │ │ -0003cfa0: 2020 2031 2032 2020 2020 207c 0a7c 2d2d     1 2     |.|--
│ │ │ │ -0003cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003cec0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003ced0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0003cee0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +0003cef0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0003cf00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003cf10: 207c 0a7c 3134 3032 7820 7820 202d 2037   |.|1402x x  - 7
│ │ │ │ +0003cf20: 3331 7820 7820 202d 2039 3936 7820 2c20  31x x  - 996x , 
│ │ │ │ +0003cf30: 3330 3178 2020 2b20 3136 3678 2078 2020  301x  + 166x x  
│ │ │ │ +0003cf40: 2d20 3935 3578 2020 2d20 3733 3978 2078  - 955x  - 739x x
│ │ │ │ +0003cf50: 2020 2d20 3131 3939 7820 7820 202d 2020    - 1199x x  -  
│ │ │ │ +0003cf60: 207c 0a7c 2020 2020 2036 2038 2020 2020   |.|     6 8    
│ │ │ │ +0003cf70: 2020 2037 2038 2020 2020 2020 2038 2020     7 8       8  
│ │ │ │ +0003cf80: 2020 2020 3020 2020 2020 2020 3020 3120      0       0 1 
│ │ │ │ +0003cf90: 2020 2020 2020 3120 2020 2020 2020 3020        1       0 
│ │ │ │ +0003cfa0: 3220 2020 2020 2020 2031 2032 2020 2020  2        1 2    
│ │ │ │ +0003cfb0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003d000: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d000: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  0003d010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d020: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0003d030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d040: 2020 2020 2020 2020 2020 207c 0a7c 3331             |.|31
│ │ │ │ -0003d050: 3978 2020 2b20 3130 3437 7820 7820 202d  9x  + 1047x x  -
│ │ │ │ -0003d060: 2035 3332 7820 7820 202b 2039 3032 7820   532x x  + 902x 
│ │ │ │ -0003d070: 7820 202b 2031 3139 3578 2020 2d20 3636  x  + 1195x  - 66
│ │ │ │ -0003d080: 3378 2078 2020 2b20 3132 3135 7820 7820  3x x  + 1215x x 
│ │ │ │ -0003d090: 202d 2035 3334 7820 7820 207c 0a7c 2020   - 534x x  |.|  
│ │ │ │ -0003d0a0: 2020 3220 2020 2020 2020 2030 2033 2020    2        0 3  
│ │ │ │ -0003d0b0: 2020 2020 2031 2033 2020 2020 2020 2032       1 3       2
│ │ │ │ -0003d0c0: 2033 2020 2020 2020 2020 3320 2020 2020   3        3     
│ │ │ │ -0003d0d0: 2020 3020 3420 2020 2020 2020 2031 2034    0 4        1 4
│ │ │ │ -0003d0e0: 2020 2020 2020 2032 2034 207c 0a7c 2d2d         2 4 |.|--
│ │ │ │ -0003d0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d030: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003d040: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d050: 207c 0a7c 3331 3978 2020 2b20 3130 3437   |.|319x  + 1047
│ │ │ │ +0003d060: 7820 7820 202d 2035 3332 7820 7820 202b  x x  - 532x x  +
│ │ │ │ +0003d070: 2039 3032 7820 7820 202b 2031 3139 3578   902x x  + 1195x
│ │ │ │ +0003d080: 2020 2d20 3636 3378 2078 2020 2b20 3132    - 663x x  + 12
│ │ │ │ +0003d090: 3135 7820 7820 202d 2035 3334 7820 7820  15x x  - 534x x 
│ │ │ │ +0003d0a0: 207c 0a7c 2020 2020 3220 2020 2020 2020   |.|    2       
│ │ │ │ +0003d0b0: 2030 2033 2020 2020 2020 2031 2033 2020   0 3       1 3  
│ │ │ │ +0003d0c0: 2020 2020 2032 2033 2020 2020 2020 2020       2 3        
│ │ │ │ +0003d0d0: 3320 2020 2020 2020 3020 3420 2020 2020  3       0 4     
│ │ │ │ +0003d0e0: 2020 2031 2034 2020 2020 2020 2032 2034     1 4       2 4
│ │ │ │ +0003d0f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003d100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003d140: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0003d150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d140: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003d150: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0003d160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d170: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d180: 2020 2020 2020 2020 3220 207c 0a7c 2d20          2  |.|- 
│ │ │ │ -0003d190: 3333 3278 2078 2020 2d20 3937 3378 2020  332x x  - 973x  
│ │ │ │ -0003d1a0: 2b20 3737 3278 2078 2020 2d20 3330 3878  + 772x x  - 308x
│ │ │ │ -0003d1b0: 2078 2020 2b20 3331 3578 2078 2020 2d20   x  + 315x x  - 
│ │ │ │ -0003d1c0: 3435 3478 2078 2020 2d20 3438 3378 2078  454x x  - 483x x
│ │ │ │ -0003d1d0: 2020 2d20 3233 3978 2020 2d7c 0a7c 2020    - 239x  -|.|  
│ │ │ │ -0003d1e0: 2020 2020 3320 3420 2020 2020 2020 3420      3 4       4 
│ │ │ │ -0003d1f0: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ -0003d200: 3120 3520 2020 2020 2020 3220 3520 2020  1 5       2 5   
│ │ │ │ -0003d210: 2020 2020 3320 3520 2020 2020 2020 3420      3 5       4 
│ │ │ │ -0003d220: 3520 2020 2020 2020 3520 207c 0a7c 2d2d  5       5  |.|--
│ │ │ │ -0003d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d180: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0003d190: 207c 0a7c 2d20 3333 3278 2078 2020 2d20   |.|- 332x x  - 
│ │ │ │ +0003d1a0: 3937 3378 2020 2b20 3737 3278 2078 2020  973x  + 772x x  
│ │ │ │ +0003d1b0: 2d20 3330 3878 2078 2020 2b20 3331 3578  - 308x x  + 315x
│ │ │ │ +0003d1c0: 2078 2020 2d20 3435 3478 2078 2020 2d20   x  - 454x x  - 
│ │ │ │ +0003d1d0: 3438 3378 2078 2020 2d20 3233 3978 2020  483x x  - 239x  
│ │ │ │ +0003d1e0: 2d7c 0a7c 2020 2020 2020 3320 3420 2020  -|.|      3 4   
│ │ │ │ +0003d1f0: 2020 2020 3420 2020 2020 2020 3020 3520      4       0 5 
│ │ │ │ +0003d200: 2020 2020 2020 3120 3520 2020 2020 2020        1 5       
│ │ │ │ +0003d210: 3220 3520 2020 2020 2020 3320 3520 2020  2 5       3 5   
│ │ │ │ +0003d220: 2020 2020 3420 3520 2020 2020 2020 3520      4 5       5 
│ │ │ │ +0003d230: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003d280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d280: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003d290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d2a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d2b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d2c0: 2020 2032 2020 2020 2020 207c 0a7c 3133     2       |.|13
│ │ │ │ -0003d2d0: 3133 7820 7820 202d 2034 3139 7820 7820  13x x  - 419x x 
│ │ │ │ -0003d2e0: 202d 2031 3334 3078 2078 2020 2d20 3133   - 1340x x  - 13
│ │ │ │ -0003d2f0: 3838 7820 7820 202d 2031 3334 3078 2078  88x x  - 1340x x
│ │ │ │ -0003d300: 2020 2d20 3136 3635 7820 7820 202d 2033    - 1665x x  - 3
│ │ │ │ -0003d310: 3333 7820 202d 2020 2020 207c 0a7c 2020  33x  -     |.|  
│ │ │ │ -0003d320: 2020 2030 2036 2020 2020 2020 2031 2036     0 6       1 6
│ │ │ │ -0003d330: 2020 2020 2020 2020 3220 3620 2020 2020          2 6     
│ │ │ │ -0003d340: 2020 2033 2036 2020 2020 2020 2020 3420     3 6        4 
│ │ │ │ -0003d350: 3620 2020 2020 2020 2035 2036 2020 2020  6        5 6    
│ │ │ │ -0003d360: 2020 2036 2020 2020 2020 207c 0a7c 2d2d     6       |.|--
│ │ │ │ -0003d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d2c0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +0003d2d0: 207c 0a7c 3133 3133 7820 7820 202d 2034   |.|1313x x  - 4
│ │ │ │ +0003d2e0: 3139 7820 7820 202d 2031 3334 3078 2078  19x x  - 1340x x
│ │ │ │ +0003d2f0: 2020 2d20 3133 3838 7820 7820 202d 2031    - 1388x x  - 1
│ │ │ │ +0003d300: 3334 3078 2078 2020 2d20 3136 3635 7820  340x x  - 1665x 
│ │ │ │ +0003d310: 7820 202d 2033 3333 7820 202d 2020 2020  x  - 333x  -    
│ │ │ │ +0003d320: 207c 0a7c 2020 2020 2030 2036 2020 2020   |.|     0 6    
│ │ │ │ +0003d330: 2020 2031 2036 2020 2020 2020 2020 3220     1 6        2 
│ │ │ │ +0003d340: 3620 2020 2020 2020 2033 2036 2020 2020  6        3 6    
│ │ │ │ +0003d350: 2020 2020 3420 3620 2020 2020 2020 2035      4 6        5
│ │ │ │ +0003d360: 2036 2020 2020 2020 2036 2020 2020 2020   6       6      
│ │ │ │ +0003d370: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003d3c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d3c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003d3d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d3e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d3f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d400: 2020 2020 2020 2020 2020 327c 0a7c 3436            2|.|46
│ │ │ │ -0003d410: 3578 2078 2020 2d20 3130 3834 7820 7820  5x x  - 1084x x 
│ │ │ │ -0003d420: 202b 2036 3736 7820 7820 202d 2031 3631   + 676x x  - 161
│ │ │ │ -0003d430: 3278 2078 2020 2d20 3238 3878 2078 2020  2x x  - 288x x  
│ │ │ │ -0003d440: 2b20 3131 7820 7820 202d 2031 3137 3078  + 11x x  - 1170x
│ │ │ │ -0003d450: 2078 2020 2d20 3138 3978 207c 0a7c 2020   x  - 189x |.|  
│ │ │ │ -0003d460: 2020 3020 3720 2020 2020 2020 2031 2037    0 7        1 7
│ │ │ │ -0003d470: 2020 2020 2020 2032 2037 2020 2020 2020         2 7      
│ │ │ │ -0003d480: 2020 3320 3720 2020 2020 2020 3420 3720    3 7       4 7 
│ │ │ │ -0003d490: 2020 2020 2035 2037 2020 2020 2020 2020       5 7        
│ │ │ │ -0003d4a0: 3620 3720 2020 2020 2020 377c 0a7c 2d2d  6 7       7|.|--
│ │ │ │ -0003d4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d400: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d410: 327c 0a7c 3436 3578 2078 2020 2d20 3130  2|.|465x x  - 10
│ │ │ │ +0003d420: 3834 7820 7820 202b 2036 3736 7820 7820  84x x  + 676x x 
│ │ │ │ +0003d430: 202d 2031 3631 3278 2078 2020 2d20 3238   - 1612x x  - 28
│ │ │ │ +0003d440: 3878 2078 2020 2b20 3131 7820 7820 202d  8x x  + 11x x  -
│ │ │ │ +0003d450: 2031 3137 3078 2078 2020 2d20 3138 3978   1170x x  - 189x
│ │ │ │ +0003d460: 207c 0a7c 2020 2020 3020 3720 2020 2020   |.|    0 7     
│ │ │ │ +0003d470: 2020 2031 2037 2020 2020 2020 2032 2037     1 7       2 7
│ │ │ │ +0003d480: 2020 2020 2020 2020 3320 3720 2020 2020          3 7     
│ │ │ │ +0003d490: 2020 3420 3720 2020 2020 2035 2037 2020    4 7      5 7  
│ │ │ │ +0003d4a0: 2020 2020 2020 3620 3720 2020 2020 2020        6 7       
│ │ │ │ +0003d4b0: 377c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  7|.|------------
│ │ │ │  0003d4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003d500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d500: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003d510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d540: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -0003d550: 3439 3878 2078 2020 2d20 3838 3978 2078  498x x  - 889x x
│ │ │ │ -0003d560: 2020 2b20 3639 3378 2078 2020 2b20 3134    + 693x x  + 14
│ │ │ │ -0003d570: 3630 7820 7820 202d 2034 3733 7820 7820  60x x  - 473x x 
│ │ │ │ -0003d580: 202d 2034 3134 7820 7820 202d 2031 3232   - 414x x  - 122
│ │ │ │ -0003d590: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -0003d5a0: 2020 2020 3020 3820 2020 2020 2020 3120      0 8       1 
│ │ │ │ -0003d5b0: 3820 2020 2020 2020 3220 3820 2020 2020  8       2 8     
│ │ │ │ -0003d5c0: 2020 2033 2038 2020 2020 2020 2034 2038     3 8       4 8
│ │ │ │ -0003d5d0: 2020 2020 2020 2035 2038 2020 2020 2020         5 8      
│ │ │ │ -0003d5e0: 2036 2038 2020 2020 2020 207c 0a7c 2d2d   6 8       |.|--
│ │ │ │ -0003d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d550: 207c 0a7c 2b20 3439 3878 2078 2020 2d20   |.|+ 498x x  - 
│ │ │ │ +0003d560: 3838 3978 2078 2020 2b20 3639 3378 2078  889x x  + 693x x
│ │ │ │ +0003d570: 2020 2b20 3134 3630 7820 7820 202d 2034    + 1460x x  - 4
│ │ │ │ +0003d580: 3733 7820 7820 202d 2034 3134 7820 7820  73x x  - 414x x 
│ │ │ │ +0003d590: 202d 2031 3232 7820 7820 202d 2020 2020   - 122x x  -    
│ │ │ │ +0003d5a0: 207c 0a7c 2020 2020 2020 3020 3820 2020   |.|      0 8   
│ │ │ │ +0003d5b0: 2020 2020 3120 3820 2020 2020 2020 3220      1 8       2 
│ │ │ │ +0003d5c0: 3820 2020 2020 2020 2033 2038 2020 2020  8        3 8    
│ │ │ │ +0003d5d0: 2020 2034 2038 2020 2020 2020 2035 2038     4 8       5 8
│ │ │ │ +0003d5e0: 2020 2020 2020 2036 2038 2020 2020 2020         6 8      
│ │ │ │ +0003d5f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003d640: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0003d650: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ -0003d660: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d640: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003d650: 2020 2020 3220 2020 2020 3220 2020 2020      2     2     
│ │ │ │ +0003d660: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0003d670: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d680: 2020 2020 2020 3220 2020 207c 0a7c 3136        2    |.|16
│ │ │ │ -0003d690: 3539 7820 7820 202d 2031 3432 3178 202c  59x x  - 1421x ,
│ │ │ │ -0003d6a0: 2031 3478 2020 2d20 3130 3439 7820 7820   14x  - 1049x x 
│ │ │ │ -0003d6b0: 202b 2031 3530 3678 2020 2b20 3132 3335   + 1506x  + 1235
│ │ │ │ -0003d6c0: 7820 7820 202b 2036 3432 7820 7820 202d  x x  + 642x x  -
│ │ │ │ -0003d6d0: 2031 3033 3478 2020 2b20 207c 0a7c 2020   1034x  +  |.|  
│ │ │ │ -0003d6e0: 2020 2037 2038 2020 2020 2020 2020 3820     7 8        8 
│ │ │ │ -0003d6f0: 2020 2020 3020 2020 2020 2020 2030 2031      0        0 1
│ │ │ │ -0003d700: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -0003d710: 2030 2032 2020 2020 2020 2031 2032 2020   0 2       1 2  
│ │ │ │ -0003d720: 2020 2020 2020 3220 2020 207c 0a7c 2d2d        2    |.|--
│ │ │ │ -0003d730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d680: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0003d690: 207c 0a7c 3136 3539 7820 7820 202d 2031   |.|1659x x  - 1
│ │ │ │ +0003d6a0: 3432 3178 202c 2031 3478 2020 2d20 3130  421x , 14x  - 10
│ │ │ │ +0003d6b0: 3439 7820 7820 202b 2031 3530 3678 2020  49x x  + 1506x  
│ │ │ │ +0003d6c0: 2b20 3132 3335 7820 7820 202b 2036 3432  + 1235x x  + 642
│ │ │ │ +0003d6d0: 7820 7820 202d 2031 3033 3478 2020 2b20  x x  - 1034x  + 
│ │ │ │ +0003d6e0: 207c 0a7c 2020 2020 2037 2038 2020 2020   |.|     7 8    
│ │ │ │ +0003d6f0: 2020 2020 3820 2020 2020 3020 2020 2020      8     0     
│ │ │ │ +0003d700: 2020 2030 2031 2020 2020 2020 2020 3120     0 1        1 
│ │ │ │ +0003d710: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ +0003d720: 2031 2032 2020 2020 2020 2020 3220 2020   1 2        2   
│ │ │ │ +0003d730: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003d740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003d780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d780: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003d790: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d7a0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +0003d7a0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  0003d7b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d7c0: 2020 2020 2020 2020 2020 207c 0a7c 3436             |.|46
│ │ │ │ -0003d7d0: 3078 2078 2020 2b20 3135 3078 2078 2020  0x x  + 150x x  
│ │ │ │ -0003d7e0: 2b20 3736 3078 2078 2020 2d20 3132 3436  + 760x x  - 1246
│ │ │ │ -0003d7f0: 7820 202d 2031 3430 3778 2078 2020 2b20  x  - 1407x x  + 
│ │ │ │ -0003d800: 3135 3730 7820 7820 202b 2031 3430 3378  1570x x  + 1403x
│ │ │ │ -0003d810: 2078 2020 2d20 2020 2020 207c 0a7c 2020   x  -      |.|  
│ │ │ │ -0003d820: 2020 3020 3320 2020 2020 2020 3120 3320    0 3       1 3 
│ │ │ │ -0003d830: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ -0003d840: 2033 2020 2020 2020 2020 3020 3420 2020   3        0 4   
│ │ │ │ -0003d850: 2020 2020 2031 2034 2020 2020 2020 2020       1 4        
│ │ │ │ -0003d860: 3220 3420 2020 2020 2020 207c 0a7c 2d2d  2 4        |.|--
│ │ │ │ -0003d870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d7c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d7d0: 207c 0a7c 3436 3078 2078 2020 2b20 3135   |.|460x x  + 15
│ │ │ │ +0003d7e0: 3078 2078 2020 2b20 3736 3078 2078 2020  0x x  + 760x x  
│ │ │ │ +0003d7f0: 2d20 3132 3436 7820 202d 2031 3430 3778  - 1246x  - 1407x
│ │ │ │ +0003d800: 2078 2020 2b20 3135 3730 7820 7820 202b   x  + 1570x x  +
│ │ │ │ +0003d810: 2031 3430 3378 2078 2020 2d20 2020 2020   1403x x  -     
│ │ │ │ +0003d820: 207c 0a7c 2020 2020 3020 3320 2020 2020   |.|    0 3     
│ │ │ │ +0003d830: 2020 3120 3320 2020 2020 2020 3220 3320    1 3       2 3 
│ │ │ │ +0003d840: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +0003d850: 3020 3420 2020 2020 2020 2031 2034 2020  0 4        1 4  
│ │ │ │ +0003d860: 2020 2020 2020 3220 3420 2020 2020 2020        2 4       
│ │ │ │ +0003d870: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003d880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003d8c0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003d8d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d8c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003d8d0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003d8f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003d900: 2020 2020 2020 2020 3220 207c 0a7c 3136          2  |.|16
│ │ │ │ -0003d910: 3130 7820 7820 202d 2034 3331 7820 202b  10x x  - 431x  +
│ │ │ │ -0003d920: 2035 3734 7820 7820 202b 2038 3933 7820   574x x  + 893x 
│ │ │ │ -0003d930: 7820 202d 2036 3537 7820 7820 202b 2034  x  - 657x x  + 4
│ │ │ │ -0003d940: 3137 7820 7820 202b 2031 3336 3278 2078  17x x  + 1362x x
│ │ │ │ -0003d950: 2020 2b20 3232 3478 2020 2b7c 0a7c 2020    + 224x  +|.|  
│ │ │ │ -0003d960: 2020 2033 2034 2020 2020 2020 2034 2020     3 4       4  
│ │ │ │ -0003d970: 2020 2020 2030 2035 2020 2020 2020 2031       0 5       1
│ │ │ │ -0003d980: 2035 2020 2020 2020 2032 2035 2020 2020   5       2 5    
│ │ │ │ -0003d990: 2020 2033 2035 2020 2020 2020 2020 3420     3 5        4 
│ │ │ │ -0003d9a0: 3520 2020 2020 2020 3520 207c 0a7c 2d2d  5       5  |.|--
│ │ │ │ -0003d9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003d900: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0003d910: 207c 0a7c 3136 3130 7820 7820 202d 2034   |.|1610x x  - 4
│ │ │ │ +0003d920: 3331 7820 202b 2035 3734 7820 7820 202b  31x  + 574x x  +
│ │ │ │ +0003d930: 2038 3933 7820 7820 202d 2036 3537 7820   893x x  - 657x 
│ │ │ │ +0003d940: 7820 202b 2034 3137 7820 7820 202b 2031  x  + 417x x  + 1
│ │ │ │ +0003d950: 3336 3278 2078 2020 2b20 3232 3478 2020  362x x  + 224x  
│ │ │ │ +0003d960: 2b7c 0a7c 2020 2020 2033 2034 2020 2020  +|.|     3 4    
│ │ │ │ +0003d970: 2020 2034 2020 2020 2020 2030 2035 2020     4       0 5  
│ │ │ │ +0003d980: 2020 2020 2031 2035 2020 2020 2020 2032       1 5       2
│ │ │ │ +0003d990: 2035 2020 2020 2020 2033 2035 2020 2020   5       3 5    
│ │ │ │ +0003d9a0: 2020 2020 3420 3520 2020 2020 2020 3520      4 5       5 
│ │ │ │ +0003d9b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003d9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003d9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003da00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003da00: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003da10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003da20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003da30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003da40: 3220 2020 2020 2020 2020 207c 0a7c 3236  2          |.|26
│ │ │ │ -0003da50: 3878 2078 2020 2b20 3130 3937 7820 7820  8x x  + 1097x x 
│ │ │ │ -0003da60: 202b 2031 3133 3278 2078 2020 2b20 3134   + 1132x x  + 14
│ │ │ │ -0003da70: 3878 2078 2020 2b20 3133 3331 7820 7820  8x x  + 1331x x 
│ │ │ │ -0003da80: 202d 2037 3778 2078 2020 2d20 3735 3678   - 77x x  - 756x
│ │ │ │ -0003da90: 2020 2b20 3232 3878 2078 207c 0a7c 2020    + 228x x |.|  
│ │ │ │ -0003daa0: 2020 3020 3620 2020 2020 2020 2031 2036    0 6        1 6
│ │ │ │ -0003dab0: 2020 2020 2020 2020 3220 3620 2020 2020          2 6     
│ │ │ │ -0003dac0: 2020 3320 3620 2020 2020 2020 2034 2036    3 6        4 6
│ │ │ │ -0003dad0: 2020 2020 2020 3520 3620 2020 2020 2020        5 6       
│ │ │ │ -0003dae0: 3620 2020 2020 2020 3020 377c 0a7c 2d2d  6       0 7|.|--
│ │ │ │ -0003daf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003da40: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +0003da50: 207c 0a7c 3236 3878 2078 2020 2b20 3130   |.|268x x  + 10
│ │ │ │ +0003da60: 3937 7820 7820 202b 2031 3133 3278 2078  97x x  + 1132x x
│ │ │ │ +0003da70: 2020 2b20 3134 3878 2078 2020 2b20 3133    + 148x x  + 13
│ │ │ │ +0003da80: 3331 7820 7820 202d 2037 3778 2078 2020  31x x  - 77x x  
│ │ │ │ +0003da90: 2d20 3735 3678 2020 2b20 3232 3878 2078  - 756x  + 228x x
│ │ │ │ +0003daa0: 207c 0a7c 2020 2020 3020 3620 2020 2020   |.|    0 6     
│ │ │ │ +0003dab0: 2020 2031 2036 2020 2020 2020 2020 3220     1 6        2 
│ │ │ │ +0003dac0: 3620 2020 2020 2020 3320 3620 2020 2020  6       3 6     
│ │ │ │ +0003dad0: 2020 2034 2036 2020 2020 2020 3520 3620     4 6      5 6 
│ │ │ │ +0003dae0: 2020 2020 2020 3620 2020 2020 2020 3020        6       0 
│ │ │ │ +0003daf0: 377c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  7|.|------------
│ │ │ │  0003db00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003db40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003db40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003db50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003db60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003db70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003db80: 2020 3220 2020 2020 2020 207c 0a7c 2b20    2        |.|+ 
│ │ │ │ -0003db90: 3133 3678 2078 2020 2d20 3134 3834 7820  136x x  - 1484x 
│ │ │ │ -0003dba0: 7820 202d 2031 3437 3878 2078 2020 2d20  x  - 1478x x  - 
│ │ │ │ -0003dbb0: 3133 7820 7820 202b 2031 3632 3078 2078  13x x  + 1620x x
│ │ │ │ -0003dbc0: 2020 2d20 3730 3178 2078 2020 2d20 3736    - 701x x  - 76
│ │ │ │ -0003dbd0: 3978 2020 2d20 2020 2020 207c 0a7c 2020  9x  -      |.|  
│ │ │ │ -0003dbe0: 2020 2020 3120 3720 2020 2020 2020 2032      1 7        2
│ │ │ │ -0003dbf0: 2037 2020 2020 2020 2020 3320 3720 2020   7        3 7   
│ │ │ │ -0003dc00: 2020 2034 2037 2020 2020 2020 2020 3520     4 7        5 
│ │ │ │ -0003dc10: 3720 2020 2020 2020 3620 3720 2020 2020  7       6 7     
│ │ │ │ -0003dc20: 2020 3720 2020 2020 2020 207c 0a7c 2d2d    7        |.|--
│ │ │ │ -0003dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003db80: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003db90: 207c 0a7c 2b20 3133 3678 2078 2020 2d20   |.|+ 136x x  - 
│ │ │ │ +0003dba0: 3134 3834 7820 7820 202d 2031 3437 3878  1484x x  - 1478x
│ │ │ │ +0003dbb0: 2078 2020 2d20 3133 7820 7820 202b 2031   x  - 13x x  + 1
│ │ │ │ +0003dbc0: 3632 3078 2078 2020 2d20 3730 3178 2078  620x x  - 701x x
│ │ │ │ +0003dbd0: 2020 2d20 3736 3978 2020 2d20 2020 2020    - 769x  -     
│ │ │ │ +0003dbe0: 207c 0a7c 2020 2020 2020 3120 3720 2020   |.|      1 7   
│ │ │ │ +0003dbf0: 2020 2020 2032 2037 2020 2020 2020 2020       2 7        
│ │ │ │ +0003dc00: 3320 3720 2020 2020 2034 2037 2020 2020  3 7      4 7    
│ │ │ │ +0003dc10: 2020 2020 3520 3720 2020 2020 2020 3620      5 7       6 
│ │ │ │ +0003dc20: 3720 2020 2020 2020 3720 2020 2020 2020  7       7       
│ │ │ │ +0003dc30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003dc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003dc80: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003dc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003dcb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003dcc0: 2020 2020 2020 2020 2020 207c 0a7c 3736             |.|76
│ │ │ │ -0003dcd0: 3078 2078 2020 2d20 3439 3278 2078 2020  0x x  - 492x x  
│ │ │ │ -0003dce0: 2d20 3130 3737 7820 7820 202d 2031 3234  - 1077x x  - 124
│ │ │ │ -0003dcf0: 3978 2078 2020 2d20 3833 3478 2078 2020  9x x  - 834x x  
│ │ │ │ -0003dd00: 2d20 3339 3578 2078 2020 2d20 3133 3538  - 395x x  - 1358
│ │ │ │ -0003dd10: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -0003dd20: 2020 3020 3820 2020 2020 2020 3120 3820    0 8       1 8 
│ │ │ │ -0003dd30: 2020 2020 2020 2032 2038 2020 2020 2020         2 8      
│ │ │ │ -0003dd40: 2020 3320 3820 2020 2020 2020 3420 3820    3 8       4 8 
│ │ │ │ -0003dd50: 2020 2020 2020 3520 3820 2020 2020 2020        5 8       
│ │ │ │ -0003dd60: 2036 2038 2020 2020 2020 207c 0a7c 2d2d   6 8       |.|--
│ │ │ │ -0003dd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003dcc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003dcd0: 207c 0a7c 3736 3078 2078 2020 2d20 3439   |.|760x x  - 49
│ │ │ │ +0003dce0: 3278 2078 2020 2d20 3130 3737 7820 7820  2x x  - 1077x x 
│ │ │ │ +0003dcf0: 202d 2031 3234 3978 2078 2020 2d20 3833   - 1249x x  - 83
│ │ │ │ +0003dd00: 3478 2078 2020 2d20 3339 3578 2078 2020  4x x  - 395x x  
│ │ │ │ +0003dd10: 2d20 3133 3538 7820 7820 202d 2020 2020  - 1358x x  -    
│ │ │ │ +0003dd20: 207c 0a7c 2020 2020 3020 3820 2020 2020   |.|    0 8     
│ │ │ │ +0003dd30: 2020 3120 3820 2020 2020 2020 2032 2038    1 8        2 8
│ │ │ │ +0003dd40: 2020 2020 2020 2020 3320 3820 2020 2020          3 8     
│ │ │ │ +0003dd50: 2020 3420 3820 2020 2020 2020 3520 3820    4 8       5 8 
│ │ │ │ +0003dd60: 2020 2020 2020 2036 2038 2020 2020 2020         6 8      
│ │ │ │ +0003dd70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003dd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003dd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003dda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003ddc0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -0003ddd0: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -0003dde0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003ddb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ddc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003ddd0: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ +0003dde0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │  0003ddf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003de00: 2020 2020 3220 2020 2020 207c 0a7c 3938      2      |.|98
│ │ │ │ -0003de10: 3878 2078 2020 2b20 3131 3378 202c 202d  8x x  + 113x , -
│ │ │ │ -0003de20: 2031 3633 3478 2020 2d20 3133 7820 7820   1634x  - 13x x 
│ │ │ │ -0003de30: 202b 2038 3035 7820 202d 2032 3178 2078   + 805x  - 21x x
│ │ │ │ -0003de40: 2020 2d20 3136 3535 7820 7820 202b 2031    - 1655x x  + 1
│ │ │ │ -0003de50: 3437 3978 2020 2d20 2020 207c 0a7c 2020  479x  -    |.|  
│ │ │ │ -0003de60: 2020 3720 3820 2020 2020 2020 3820 2020    7 8       8   
│ │ │ │ -0003de70: 2020 2020 2020 3020 2020 2020 2030 2031        0      0 1
│ │ │ │ -0003de80: 2020 2020 2020 2031 2020 2020 2020 3020         1      0 
│ │ │ │ -0003de90: 3220 2020 2020 2020 2031 2032 2020 2020  2        1 2    
│ │ │ │ -0003dea0: 2020 2020 3220 2020 2020 207c 0a7c 2d2d      2      |.|--
│ │ │ │ -0003deb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003de00: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0003de10: 207c 0a7c 3938 3878 2078 2020 2b20 3131   |.|988x x  + 11
│ │ │ │ +0003de20: 3378 202c 202d 2031 3633 3478 2020 2d20  3x , - 1634x  - 
│ │ │ │ +0003de30: 3133 7820 7820 202b 2038 3035 7820 202d  13x x  + 805x  -
│ │ │ │ +0003de40: 2032 3178 2078 2020 2d20 3136 3535 7820   21x x  - 1655x 
│ │ │ │ +0003de50: 7820 202b 2031 3437 3978 2020 2d20 2020  x  + 1479x  -   
│ │ │ │ +0003de60: 207c 0a7c 2020 2020 3720 3820 2020 2020   |.|    7 8     
│ │ │ │ +0003de70: 2020 3820 2020 2020 2020 2020 3020 2020    8         0   
│ │ │ │ +0003de80: 2020 2030 2031 2020 2020 2020 2031 2020     0 1       1  
│ │ │ │ +0003de90: 2020 2020 3020 3220 2020 2020 2020 2031      0 2        1
│ │ │ │ +0003dea0: 2032 2020 2020 2020 2020 3220 2020 2020   2        2     
│ │ │ │ +0003deb0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003dec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003df00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003df00: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003df10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df20: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003df20: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0003df30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003df40: 2020 2020 2020 2020 2020 207c 0a7c 3135             |.|15
│ │ │ │ -0003df50: 3130 7820 7820 202d 2036 3436 7820 7820  10x x  - 646x x 
│ │ │ │ -0003df60: 202b 2032 3235 7820 7820 202d 2031 3431   + 225x x  - 141
│ │ │ │ -0003df70: 3178 2020 2b20 3132 3237 7820 7820 202d  1x  + 1227x x  -
│ │ │ │ -0003df80: 2031 3130 3878 2078 2020 2b20 3132 3931   1108x x  + 1291
│ │ │ │ -0003df90: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -0003dfa0: 2020 2030 2033 2020 2020 2020 2031 2033     0 3       1 3
│ │ │ │ -0003dfb0: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ -0003dfc0: 2020 3320 2020 2020 2020 2030 2034 2020    3        0 4  
│ │ │ │ -0003dfd0: 2020 2020 2020 3120 3420 2020 2020 2020        1 4       
│ │ │ │ -0003dfe0: 2032 2034 2020 2020 2020 207c 0a7c 2d2d   2 4       |.|--
│ │ │ │ -0003dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003df40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003df50: 207c 0a7c 3135 3130 7820 7820 202d 2036   |.|1510x x  - 6
│ │ │ │ +0003df60: 3436 7820 7820 202b 2032 3235 7820 7820  46x x  + 225x x 
│ │ │ │ +0003df70: 202d 2031 3431 3178 2020 2b20 3132 3237   - 1411x  + 1227
│ │ │ │ +0003df80: 7820 7820 202d 2031 3130 3878 2078 2020  x x  - 1108x x  
│ │ │ │ +0003df90: 2b20 3132 3931 7820 7820 202d 2020 2020  + 1291x x  -    
│ │ │ │ +0003dfa0: 207c 0a7c 2020 2020 2030 2033 2020 2020   |.|     0 3    
│ │ │ │ +0003dfb0: 2020 2031 2033 2020 2020 2020 2032 2033     1 3       2 3
│ │ │ │ +0003dfc0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ +0003dfd0: 2030 2034 2020 2020 2020 2020 3120 3420   0 4        1 4 
│ │ │ │ +0003dfe0: 2020 2020 2020 2032 2034 2020 2020 2020         2 4      
│ │ │ │ +0003dff0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003e040: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0003e050: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e040: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003e050: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0003e060: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e080: 2020 2020 2020 2032 2020 207c 0a7c 3539         2   |.|59
│ │ │ │ -0003e090: 7820 7820 202d 2031 3432 7820 202b 2035  x x  - 142x  + 5
│ │ │ │ -0003e0a0: 3836 7820 7820 202d 2036 3736 7820 7820  86x x  - 676x x 
│ │ │ │ -0003e0b0: 202b 2036 3535 7820 7820 202d 2031 3437   + 655x x  - 147
│ │ │ │ -0003e0c0: 3678 2078 2020 2b20 3435 3378 2078 2020  6x x  + 453x x  
│ │ │ │ -0003e0d0: 2d20 3130 3736 7820 202d 207c 0a7c 2020  - 1076x  - |.|  
│ │ │ │ -0003e0e0: 2033 2034 2020 2020 2020 2034 2020 2020   3 4       4    
│ │ │ │ -0003e0f0: 2020 2030 2035 2020 2020 2020 2031 2035     0 5       1 5
│ │ │ │ -0003e100: 2020 2020 2020 2032 2035 2020 2020 2020         2 5      
│ │ │ │ -0003e110: 2020 3320 3520 2020 2020 2020 3420 3520    3 5       4 5 
│ │ │ │ -0003e120: 2020 2020 2020 2035 2020 207c 0a7c 2d2d         5   |.|--
│ │ │ │ -0003e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e080: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0003e090: 207c 0a7c 3539 7820 7820 202d 2031 3432   |.|59x x  - 142
│ │ │ │ +0003e0a0: 7820 202b 2035 3836 7820 7820 202d 2036  x  + 586x x  - 6
│ │ │ │ +0003e0b0: 3736 7820 7820 202b 2036 3535 7820 7820  76x x  + 655x x 
│ │ │ │ +0003e0c0: 202d 2031 3437 3678 2078 2020 2b20 3435   - 1476x x  + 45
│ │ │ │ +0003e0d0: 3378 2078 2020 2d20 3130 3736 7820 202d  3x x  - 1076x  -
│ │ │ │ +0003e0e0: 207c 0a7c 2020 2033 2034 2020 2020 2020   |.|   3 4      
│ │ │ │ +0003e0f0: 2034 2020 2020 2020 2030 2035 2020 2020   4       0 5    
│ │ │ │ +0003e100: 2020 2031 2035 2020 2020 2020 2032 2035     1 5       2 5
│ │ │ │ +0003e110: 2020 2020 2020 2020 3320 3520 2020 2020          3 5     
│ │ │ │ +0003e120: 2020 3420 3520 2020 2020 2020 2035 2020    4 5        5  
│ │ │ │ +0003e130: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003e180: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e180: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003e190: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e1a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e1b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e1c0: 2032 2020 2020 2020 2020 207c 0a7c 3131   2         |.|11
│ │ │ │ -0003e1d0: 3532 7820 7820 202b 2031 3337 3378 2078  52x x  + 1373x x
│ │ │ │ -0003e1e0: 2020 2d20 3131 3931 7820 7820 202d 2034    - 1191x x  - 4
│ │ │ │ -0003e1f0: 3136 7820 7820 202b 2036 3939 7820 7820  16x x  + 699x x 
│ │ │ │ -0003e200: 202b 2033 3137 7820 7820 202b 2038 3235   + 317x x  + 825
│ │ │ │ -0003e210: 7820 202d 2020 2020 2020 207c 0a7c 2020  x  -       |.|  
│ │ │ │ -0003e220: 2020 2030 2036 2020 2020 2020 2020 3120     0 6        1 
│ │ │ │ -0003e230: 3620 2020 2020 2020 2032 2036 2020 2020  6        2 6    
│ │ │ │ -0003e240: 2020 2033 2036 2020 2020 2020 2034 2036     3 6       4 6
│ │ │ │ -0003e250: 2020 2020 2020 2035 2036 2020 2020 2020         5 6      
│ │ │ │ -0003e260: 2036 2020 2020 2020 2020 207c 0a7c 2d2d   6         |.|--
│ │ │ │ -0003e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e1c0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003e1d0: 207c 0a7c 3131 3532 7820 7820 202b 2031   |.|1152x x  + 1
│ │ │ │ +0003e1e0: 3337 3378 2078 2020 2d20 3131 3931 7820  373x x  - 1191x 
│ │ │ │ +0003e1f0: 7820 202d 2034 3136 7820 7820 202b 2036  x  - 416x x  + 6
│ │ │ │ +0003e200: 3939 7820 7820 202b 2033 3137 7820 7820  99x x  + 317x x 
│ │ │ │ +0003e210: 202b 2038 3235 7820 202d 2020 2020 2020   + 825x  -      
│ │ │ │ +0003e220: 207c 0a7c 2020 2020 2030 2036 2020 2020   |.|     0 6    
│ │ │ │ +0003e230: 2020 2020 3120 3620 2020 2020 2020 2032      1 6        2
│ │ │ │ +0003e240: 2036 2020 2020 2020 2033 2036 2020 2020   6       3 6    
│ │ │ │ +0003e250: 2020 2034 2036 2020 2020 2020 2035 2036     4 6       5 6
│ │ │ │ +0003e260: 2020 2020 2020 2036 2020 2020 2020 2020         6        
│ │ │ │ +0003e270: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003e2c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e2c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003e2d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e2e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e2f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e300: 2020 2020 2020 2020 2020 207c 0a7c 3135             |.|15
│ │ │ │ -0003e310: 3630 7820 7820 202d 2034 3838 7820 7820  60x x  - 488x x 
│ │ │ │ -0003e320: 202d 2031 3033 3578 2078 2020 2d20 3135   - 1035x x  - 15
│ │ │ │ -0003e330: 3631 7820 7820 202d 2036 3434 7820 7820  61x x  - 644x x 
│ │ │ │ -0003e340: 202d 2031 3137 3878 2078 2020 2d20 3133   - 1178x x  - 13
│ │ │ │ -0003e350: 3230 7820 7820 202b 2020 207c 0a7c 2020  20x x  +   |.|  
│ │ │ │ -0003e360: 2020 2030 2037 2020 2020 2020 2031 2037     0 7       1 7
│ │ │ │ -0003e370: 2020 2020 2020 2020 3220 3720 2020 2020          2 7     
│ │ │ │ -0003e380: 2020 2033 2037 2020 2020 2020 2034 2037     3 7       4 7
│ │ │ │ -0003e390: 2020 2020 2020 2020 3520 3720 2020 2020          5 7     
│ │ │ │ -0003e3a0: 2020 2036 2037 2020 2020 207c 0a7c 2d2d     6 7     |.|--
│ │ │ │ -0003e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e310: 207c 0a7c 3135 3630 7820 7820 202d 2034   |.|1560x x  - 4
│ │ │ │ +0003e320: 3838 7820 7820 202d 2031 3033 3578 2078  88x x  - 1035x x
│ │ │ │ +0003e330: 2020 2d20 3135 3631 7820 7820 202d 2036    - 1561x x  - 6
│ │ │ │ +0003e340: 3434 7820 7820 202d 2031 3137 3878 2078  44x x  - 1178x x
│ │ │ │ +0003e350: 2020 2d20 3133 3230 7820 7820 202b 2020    - 1320x x  +  
│ │ │ │ +0003e360: 207c 0a7c 2020 2020 2030 2037 2020 2020   |.|     0 7    
│ │ │ │ +0003e370: 2020 2031 2037 2020 2020 2020 2020 3220     1 7        2 
│ │ │ │ +0003e380: 3720 2020 2020 2020 2033 2037 2020 2020  7        3 7    
│ │ │ │ +0003e390: 2020 2034 2037 2020 2020 2020 2020 3520     4 7        5 
│ │ │ │ +0003e3a0: 3720 2020 2020 2020 2036 2037 2020 2020  7        6 7    
│ │ │ │ +0003e3b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003e400: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e400: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  0003e410: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e420: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e430: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e440: 2020 2020 2020 2020 2020 207c 0a7c 3135             |.|15
│ │ │ │ -0003e450: 3878 2020 2b20 3838 3978 2078 2020 2b20  8x  + 889x x  + 
│ │ │ │ -0003e460: 3134 3434 7820 7820 202d 2031 3438 3678  1444x x  - 1486x
│ │ │ │ -0003e470: 2078 2020 2d20 3132 3131 7820 7820 202b   x  - 1211x x  +
│ │ │ │ -0003e480: 2031 3236 3978 2078 2020 2d20 3132 3238   1269x x  - 1228
│ │ │ │ -0003e490: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -0003e4a0: 2020 3720 2020 2020 2020 3020 3820 2020    7       0 8   
│ │ │ │ -0003e4b0: 2020 2020 2031 2038 2020 2020 2020 2020       1 8        
│ │ │ │ -0003e4c0: 3220 3820 2020 2020 2020 2033 2038 2020  2 8        3 8  
│ │ │ │ -0003e4d0: 2020 2020 2020 3420 3820 2020 2020 2020        4 8       
│ │ │ │ -0003e4e0: 2035 2038 2020 2020 2020 207c 0a7c 2d2d   5 8       |.|--
│ │ │ │ -0003e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e450: 207c 0a7c 3135 3878 2020 2b20 3838 3978   |.|158x  + 889x
│ │ │ │ +0003e460: 2078 2020 2b20 3134 3434 7820 7820 202d   x  + 1444x x  -
│ │ │ │ +0003e470: 2031 3438 3678 2078 2020 2d20 3132 3131   1486x x  - 1211
│ │ │ │ +0003e480: 7820 7820 202b 2031 3236 3978 2078 2020  x x  + 1269x x  
│ │ │ │ +0003e490: 2d20 3132 3238 7820 7820 202b 2020 2020  - 1228x x  +    
│ │ │ │ +0003e4a0: 207c 0a7c 2020 2020 3720 2020 2020 2020   |.|    7       
│ │ │ │ +0003e4b0: 3020 3820 2020 2020 2020 2031 2038 2020  0 8        1 8  
│ │ │ │ +0003e4c0: 2020 2020 2020 3220 3820 2020 2020 2020        2 8       
│ │ │ │ +0003e4d0: 2033 2038 2020 2020 2020 2020 3420 3820   3 8        4 8 
│ │ │ │ +0003e4e0: 2020 2020 2020 2035 2038 2020 2020 2020         5 8      
│ │ │ │ +0003e4f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003e540: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e550: 2020 2020 2020 2020 3220 2020 2020 2032          2      2
│ │ │ │ -0003e560: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e570: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -0003e580: 2020 2020 2020 2020 2020 207c 0a7c 3536             |.|56
│ │ │ │ -0003e590: 3878 2078 2020 2b20 3135 3931 7820 7820  8x x  + 1591x x 
│ │ │ │ -0003e5a0: 202b 2031 3230 3778 202c 2031 3035 7820   + 1207x , 105x 
│ │ │ │ -0003e5b0: 202d 2035 3338 7820 7820 202d 2031 3232   - 538x x  - 122
│ │ │ │ -0003e5c0: 3278 2020 2d20 3237 3778 2078 2020 2b20  2x  - 277x x  + 
│ │ │ │ -0003e5d0: 3731 3678 2078 2020 2d20 207c 0a7c 2020  716x x  -  |.|  
│ │ │ │ -0003e5e0: 2020 3620 3820 2020 2020 2020 2037 2038    6 8        7 8
│ │ │ │ -0003e5f0: 2020 2020 2020 2020 3820 2020 2020 2030          8      0
│ │ │ │ -0003e600: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ -0003e610: 2020 3120 2020 2020 2020 3020 3220 2020    1       0 2   
│ │ │ │ -0003e620: 2020 2020 3120 3220 2020 207c 0a7c 2d2d      1 2    |.|--
│ │ │ │ -0003e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e540: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003e550: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +0003e560: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +0003e570: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003e580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e590: 207c 0a7c 3536 3878 2078 2020 2b20 3135   |.|568x x  + 15
│ │ │ │ +0003e5a0: 3931 7820 7820 202b 2031 3230 3778 202c  91x x  + 1207x ,
│ │ │ │ +0003e5b0: 2031 3035 7820 202d 2035 3338 7820 7820   105x  - 538x x 
│ │ │ │ +0003e5c0: 202d 2031 3232 3278 2020 2d20 3237 3778   - 1222x  - 277x
│ │ │ │ +0003e5d0: 2078 2020 2b20 3731 3678 2078 2020 2d20   x  + 716x x  - 
│ │ │ │ +0003e5e0: 207c 0a7c 2020 2020 3620 3820 2020 2020   |.|    6 8     
│ │ │ │ +0003e5f0: 2020 2037 2038 2020 2020 2020 2020 3820     7 8        8 
│ │ │ │ +0003e600: 2020 2020 2030 2020 2020 2020 2030 2031       0       0 1
│ │ │ │ +0003e610: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +0003e620: 3020 3220 2020 2020 2020 3120 3220 2020  0 2       1 2   
│ │ │ │ +0003e630: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003e680: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +0003e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e680: 2d7c 0a7c 2020 2020 2032 2020 2020 2020  -|.|     2      
│ │ │ │  0003e690: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e6a0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003e6a0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0003e6b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e6c0: 2020 2020 2020 2020 2020 207c 0a7c 3130             |.|10
│ │ │ │ -0003e6d0: 3637 7820 202d 2034 3238 7820 7820 202b  67x  - 428x x  +
│ │ │ │ -0003e6e0: 2031 3534 7820 7820 202d 2034 3639 7820   154x x  - 469x 
│ │ │ │ -0003e6f0: 7820 202b 2037 3778 2020 2b20 3533 3878  x  + 77x  + 538x
│ │ │ │ -0003e700: 2078 2020 2d20 3137 3978 2078 2020 2b20   x  - 179x x  + 
│ │ │ │ -0003e710: 3932 3178 2078 2020 2d20 207c 0a7c 2020  921x x  -  |.|  
│ │ │ │ -0003e720: 2020 2032 2020 2020 2020 2030 2033 2020     2       0 3  
│ │ │ │ -0003e730: 2020 2020 2031 2033 2020 2020 2020 2032       1 3       2
│ │ │ │ -0003e740: 2033 2020 2020 2020 3320 2020 2020 2020   3      3       
│ │ │ │ -0003e750: 3020 3420 2020 2020 2020 3120 3420 2020  0 4       1 4   
│ │ │ │ -0003e760: 2020 2020 3220 3420 2020 207c 0a7c 2d2d      2 4    |.|--
│ │ │ │ -0003e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e6c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e6d0: 207c 0a7c 3130 3637 7820 202d 2034 3238   |.|1067x  - 428
│ │ │ │ +0003e6e0: 7820 7820 202b 2031 3534 7820 7820 202d  x x  + 154x x  -
│ │ │ │ +0003e6f0: 2034 3639 7820 7820 202b 2037 3778 2020   469x x  + 77x  
│ │ │ │ +0003e700: 2b20 3533 3878 2078 2020 2d20 3137 3978  + 538x x  - 179x
│ │ │ │ +0003e710: 2078 2020 2b20 3932 3178 2078 2020 2d20   x  + 921x x  - 
│ │ │ │ +0003e720: 207c 0a7c 2020 2020 2032 2020 2020 2020   |.|     2      
│ │ │ │ +0003e730: 2030 2033 2020 2020 2020 2031 2033 2020   0 3       1 3  
│ │ │ │ +0003e740: 2020 2020 2032 2033 2020 2020 2020 3320       2 3      3 
│ │ │ │ +0003e750: 2020 2020 2020 3020 3420 2020 2020 2020        0 4       
│ │ │ │ +0003e760: 3120 3420 2020 2020 2020 3220 3420 2020  1 4       2 4   
│ │ │ │ +0003e770: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003e7c0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e7c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003e7d0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0003e7e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e7f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e800: 2020 2020 2020 2020 2020 327c 0a7c 3232            2|.|22
│ │ │ │ -0003e810: 3378 2078 2020 2b20 3130 3933 7820 202d  3x x  + 1093x  -
│ │ │ │ -0003e820: 2032 3632 7820 7820 202b 2031 3239 3978   262x x  + 1299x
│ │ │ │ -0003e830: 2078 2020 2b20 3633 3178 2078 2020 2b20   x  + 631x x  + 
│ │ │ │ -0003e840: 3134 3836 7820 7820 202d 2031 3238 3078  1486x x  - 1280x
│ │ │ │ -0003e850: 2078 2020 2d20 3132 3178 207c 0a7c 2020   x  - 121x |.|  
│ │ │ │ -0003e860: 2020 3320 3420 2020 2020 2020 2034 2020    3 4        4  
│ │ │ │ -0003e870: 2020 2020 2030 2035 2020 2020 2020 2020       0 5        
│ │ │ │ -0003e880: 3120 3520 2020 2020 2020 3220 3520 2020  1 5       2 5   
│ │ │ │ -0003e890: 2020 2020 2033 2035 2020 2020 2020 2020       3 5        
│ │ │ │ -0003e8a0: 3420 3520 2020 2020 2020 357c 0a7c 2d2d  4 5       5|.|--
│ │ │ │ -0003e8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e810: 327c 0a7c 3232 3378 2078 2020 2b20 3130  2|.|223x x  + 10
│ │ │ │ +0003e820: 3933 7820 202d 2032 3632 7820 7820 202b  93x  - 262x x  +
│ │ │ │ +0003e830: 2031 3239 3978 2078 2020 2b20 3633 3178   1299x x  + 631x
│ │ │ │ +0003e840: 2078 2020 2b20 3134 3836 7820 7820 202d   x  + 1486x x  -
│ │ │ │ +0003e850: 2031 3238 3078 2078 2020 2d20 3132 3178   1280x x  - 121x
│ │ │ │ +0003e860: 207c 0a7c 2020 2020 3320 3420 2020 2020   |.|    3 4     
│ │ │ │ +0003e870: 2020 2034 2020 2020 2020 2030 2035 2020     4       0 5  
│ │ │ │ +0003e880: 2020 2020 2020 3120 3520 2020 2020 2020        1 5       
│ │ │ │ +0003e890: 3220 3520 2020 2020 2020 2033 2035 2020  2 5        3 5  
│ │ │ │ +0003e8a0: 2020 2020 2020 3420 3520 2020 2020 2020        4 5       
│ │ │ │ +0003e8b0: 357c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  5|.|------------
│ │ │ │  0003e8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003e900: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e900: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003e910: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e920: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003e930: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003e940: 2032 2020 2020 2020 2020 207c 0a7c 2d20   2         |.|- 
│ │ │ │ -0003e950: 3530 7820 7820 202d 2039 3738 7820 7820  50x x  - 978x x 
│ │ │ │ -0003e960: 202d 2036 3934 7820 7820 202d 2035 3331   - 694x x  - 531
│ │ │ │ -0003e970: 7820 7820 202b 2035 3035 7820 7820 202b  x x  + 505x x  +
│ │ │ │ -0003e980: 2031 3431 3278 2078 2020 2d20 3130 3631   1412x x  - 1061
│ │ │ │ -0003e990: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -0003e9a0: 2020 2030 2036 2020 2020 2020 2031 2036     0 6       1 6
│ │ │ │ -0003e9b0: 2020 2020 2020 2032 2036 2020 2020 2020         2 6      
│ │ │ │ -0003e9c0: 2033 2036 2020 2020 2020 2034 2036 2020   3 6       4 6  
│ │ │ │ -0003e9d0: 2020 2020 2020 3520 3620 2020 2020 2020        5 6       
│ │ │ │ -0003e9e0: 2036 2020 2020 2020 2020 207c 0a7c 2d2d   6         |.|--
│ │ │ │ -0003e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003e940: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +0003e950: 207c 0a7c 2d20 3530 7820 7820 202d 2039   |.|- 50x x  - 9
│ │ │ │ +0003e960: 3738 7820 7820 202d 2036 3934 7820 7820  78x x  - 694x x 
│ │ │ │ +0003e970: 202d 2035 3331 7820 7820 202b 2035 3035   - 531x x  + 505
│ │ │ │ +0003e980: 7820 7820 202b 2031 3431 3278 2078 2020  x x  + 1412x x  
│ │ │ │ +0003e990: 2d20 3130 3631 7820 202b 2020 2020 2020  - 1061x  +      
│ │ │ │ +0003e9a0: 207c 0a7c 2020 2020 2030 2036 2020 2020   |.|     0 6    
│ │ │ │ +0003e9b0: 2020 2031 2036 2020 2020 2020 2032 2036     1 6       2 6
│ │ │ │ +0003e9c0: 2020 2020 2020 2033 2036 2020 2020 2020         3 6      
│ │ │ │ +0003e9d0: 2034 2036 2020 2020 2020 2020 3520 3620   4 6        5 6 
│ │ │ │ +0003e9e0: 2020 2020 2020 2036 2020 2020 2020 2020         6        
│ │ │ │ +0003e9f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003ea40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ea40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003ea50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ea60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ea70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ea80: 2020 2020 2020 2020 2020 207c 0a7c 3132             |.|12
│ │ │ │ -0003ea90: 3032 7820 7820 202b 2034 3438 7820 7820  02x x  + 448x x 
│ │ │ │ -0003eaa0: 202d 2031 3837 7820 7820 202b 2031 3237   - 187x x  + 127
│ │ │ │ -0003eab0: 3678 2078 2020 2d20 3132 3178 2078 2020  6x x  - 121x x  
│ │ │ │ -0003eac0: 2b20 3133 3631 7820 7820 202b 2036 3937  + 1361x x  + 697
│ │ │ │ -0003ead0: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -0003eae0: 2020 2030 2037 2020 2020 2020 2031 2037     0 7       1 7
│ │ │ │ -0003eaf0: 2020 2020 2020 2032 2037 2020 2020 2020         2 7      
│ │ │ │ -0003eb00: 2020 3320 3720 2020 2020 2020 3420 3720    3 7       4 7 
│ │ │ │ -0003eb10: 2020 2020 2020 2035 2037 2020 2020 2020         5 7      
│ │ │ │ -0003eb20: 2036 2037 2020 2020 2020 207c 0a7c 2d2d   6 7       |.|--
│ │ │ │ -0003eb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ea80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ea90: 207c 0a7c 3132 3032 7820 7820 202b 2034   |.|1202x x  + 4
│ │ │ │ +0003eaa0: 3438 7820 7820 202d 2031 3837 7820 7820  48x x  - 187x x 
│ │ │ │ +0003eab0: 202b 2031 3237 3678 2078 2020 2d20 3132   + 1276x x  - 12
│ │ │ │ +0003eac0: 3178 2078 2020 2b20 3133 3631 7820 7820  1x x  + 1361x x 
│ │ │ │ +0003ead0: 202b 2036 3937 7820 7820 202b 2020 2020   + 697x x  +    
│ │ │ │ +0003eae0: 207c 0a7c 2020 2020 2030 2037 2020 2020   |.|     0 7    
│ │ │ │ +0003eaf0: 2020 2031 2037 2020 2020 2020 2032 2037     1 7       2 7
│ │ │ │ +0003eb00: 2020 2020 2020 2020 3320 3720 2020 2020          3 7     
│ │ │ │ +0003eb10: 2020 3420 3720 2020 2020 2020 2035 2037    4 7        5 7
│ │ │ │ +0003eb20: 2020 2020 2020 2036 2037 2020 2020 2020         6 7      
│ │ │ │ +0003eb30: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003eb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003eb80: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003eb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003eb80: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  0003eb90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003eba0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ebb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ebc0: 2020 2020 2020 2020 2020 207c 0a7c 3638             |.|68
│ │ │ │ -0003ebd0: 3278 2020 2b20 3135 3932 7820 7820 202b  2x  + 1592x x  +
│ │ │ │ -0003ebe0: 2037 3035 7820 7820 202d 2032 3237 7820   705x x  - 227x 
│ │ │ │ -0003ebf0: 7820 202d 2037 7820 7820 202d 2031 3432  x  - 7x x  - 142
│ │ │ │ -0003ec00: 3378 2078 2020 2d20 3134 3436 7820 7820  3x x  - 1446x x 
│ │ │ │ -0003ec10: 202d 2031 3537 3878 2078 207c 0a7c 2020   - 1578x x |.|  
│ │ │ │ -0003ec20: 2020 3720 2020 2020 2020 2030 2038 2020    7        0 8  
│ │ │ │ -0003ec30: 2020 2020 2031 2038 2020 2020 2020 2032       1 8       2
│ │ │ │ -0003ec40: 2038 2020 2020 2033 2038 2020 2020 2020   8     3 8      
│ │ │ │ -0003ec50: 2020 3420 3820 2020 2020 2020 2035 2038    4 8        5 8
│ │ │ │ -0003ec60: 2020 2020 2020 2020 3620 387c 0a7c 2d2d          6 8|.|--
│ │ │ │ -0003ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ebc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ebd0: 207c 0a7c 3638 3278 2020 2b20 3135 3932   |.|682x  + 1592
│ │ │ │ +0003ebe0: 7820 7820 202b 2037 3035 7820 7820 202d  x x  + 705x x  -
│ │ │ │ +0003ebf0: 2032 3237 7820 7820 202d 2037 7820 7820   227x x  - 7x x 
│ │ │ │ +0003ec00: 202d 2031 3432 3378 2078 2020 2d20 3134   - 1423x x  - 14
│ │ │ │ +0003ec10: 3436 7820 7820 202d 2031 3537 3878 2078  46x x  - 1578x x
│ │ │ │ +0003ec20: 207c 0a7c 2020 2020 3720 2020 2020 2020   |.|    7       
│ │ │ │ +0003ec30: 2030 2038 2020 2020 2020 2031 2038 2020   0 8       1 8  
│ │ │ │ +0003ec40: 2020 2020 2032 2038 2020 2020 2033 2038       2 8     3 8
│ │ │ │ +0003ec50: 2020 2020 2020 2020 3420 3820 2020 2020          4 8     
│ │ │ │ +0003ec60: 2020 2035 2038 2020 2020 2020 2020 3620     5 8        6 
│ │ │ │ +0003ec70: 387c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  8|.|------------
│ │ │ │  0003ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003ecc0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0003ecd0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -0003ece0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ -0003ecf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ed00: 2020 2020 2020 2032 2020 207c 0a7c 2b20         2   |.|+ 
│ │ │ │ -0003ed10: 3135 3131 7820 7820 202b 2039 3137 7820  1511x x  + 917x 
│ │ │ │ -0003ed20: 2c20 3132 3730 7820 202d 2033 3931 7820  , 1270x  - 391x 
│ │ │ │ -0003ed30: 7820 202d 2031 3131 3678 2020 2d20 3238  x  - 1116x  - 28
│ │ │ │ -0003ed40: 3778 2078 2020 2b20 3635 3378 2078 2020  7x x  + 653x x  
│ │ │ │ -0003ed50: 2b20 3136 3433 7820 202b 207c 0a7c 2020  + 1643x  + |.|  
│ │ │ │ -0003ed60: 2020 2020 2037 2038 2020 2020 2020 2038       7 8       8
│ │ │ │ -0003ed70: 2020 2020 2020 2030 2020 2020 2020 2030         0       0
│ │ │ │ -0003ed80: 2031 2020 2020 2020 2020 3120 2020 2020   1        1     
│ │ │ │ -0003ed90: 2020 3020 3220 2020 2020 2020 3120 3220    0 2       1 2 
│ │ │ │ -0003eda0: 2020 2020 2020 2032 2020 207c 0a7c 2d2d         2   |.|--
│ │ │ │ -0003edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ecc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003ecd0: 2020 2020 2032 2020 2020 2020 2032 2020       2       2  
│ │ │ │ +0003ece0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ecf0: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +0003ed00: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +0003ed10: 207c 0a7c 2b20 3135 3131 7820 7820 202b   |.|+ 1511x x  +
│ │ │ │ +0003ed20: 2039 3137 7820 2c20 3132 3730 7820 202d   917x , 1270x  -
│ │ │ │ +0003ed30: 2033 3931 7820 7820 202d 2031 3131 3678   391x x  - 1116x
│ │ │ │ +0003ed40: 2020 2d20 3238 3778 2078 2020 2b20 3635    - 287x x  + 65
│ │ │ │ +0003ed50: 3378 2078 2020 2b20 3136 3433 7820 202b  3x x  + 1643x  +
│ │ │ │ +0003ed60: 207c 0a7c 2020 2020 2020 2037 2038 2020   |.|       7 8  
│ │ │ │ +0003ed70: 2020 2020 2038 2020 2020 2020 2030 2020       8       0  
│ │ │ │ +0003ed80: 2020 2020 2030 2031 2020 2020 2020 2020       0 1        
│ │ │ │ +0003ed90: 3120 2020 2020 2020 3020 3220 2020 2020  1       0 2     
│ │ │ │ +0003eda0: 2020 3120 3220 2020 2020 2020 2032 2020    1 2        2  
│ │ │ │ +0003edb0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003ee00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee10: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0003ee20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ee00: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003ee10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ee20: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  0003ee30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ee40: 2020 2020 2020 2020 2020 207c 0a7c 3136             |.|16
│ │ │ │ -0003ee50: 3233 7820 7820 202b 2035 3134 7820 7820  23x x  + 514x x 
│ │ │ │ -0003ee60: 202d 2031 3478 2078 2020 2d20 3930 7820   - 14x x  - 90x 
│ │ │ │ -0003ee70: 202b 2031 3233 3278 2078 2020 2d20 3134   + 1232x x  - 14
│ │ │ │ -0003ee80: 3334 7820 7820 202b 2031 3239 3678 2078  34x x  + 1296x x
│ │ │ │ -0003ee90: 2020 2b20 2020 2020 2020 207c 0a7c 2020    +        |.|  
│ │ │ │ -0003eea0: 2020 2030 2033 2020 2020 2020 2031 2033     0 3       1 3
│ │ │ │ -0003eeb0: 2020 2020 2020 3220 3320 2020 2020 2033        2 3      3
│ │ │ │ -0003eec0: 2020 2020 2020 2020 3020 3420 2020 2020          0 4     
│ │ │ │ -0003eed0: 2020 2031 2034 2020 2020 2020 2020 3220     1 4        2 
│ │ │ │ -0003eee0: 3420 2020 2020 2020 2020 207c 0a7c 2d2d  4          |.|--
│ │ │ │ -0003eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ee40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ee50: 207c 0a7c 3136 3233 7820 7820 202b 2035   |.|1623x x  + 5
│ │ │ │ +0003ee60: 3134 7820 7820 202d 2031 3478 2078 2020  14x x  - 14x x  
│ │ │ │ +0003ee70: 2d20 3930 7820 202b 2031 3233 3278 2078  - 90x  + 1232x x
│ │ │ │ +0003ee80: 2020 2d20 3134 3334 7820 7820 202b 2031    - 1434x x  + 1
│ │ │ │ +0003ee90: 3239 3678 2078 2020 2b20 2020 2020 2020  296x x  +       
│ │ │ │ +0003eea0: 207c 0a7c 2020 2020 2030 2033 2020 2020   |.|     0 3    
│ │ │ │ +0003eeb0: 2020 2031 2033 2020 2020 2020 3220 3320     1 3      2 3 
│ │ │ │ +0003eec0: 2020 2020 2033 2020 2020 2020 2020 3020       3        0 
│ │ │ │ +0003eed0: 3420 2020 2020 2020 2031 2034 2020 2020  4        1 4    
│ │ │ │ +0003eee0: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │ +0003eef0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003ef20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003ef40: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003ef50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ef40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003ef50: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  0003ef60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ef70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ef80: 2020 2020 2020 3220 2020 207c 0a7c 3135        2    |.|15
│ │ │ │ -0003ef90: 3232 7820 7820 202b 2031 3336 7820 202d  22x x  + 136x  -
│ │ │ │ -0003efa0: 2036 3233 7820 7820 202d 2036 3037 7820   623x x  - 607x 
│ │ │ │ -0003efb0: 7820 202b 2031 3878 2078 2020 2b20 3839  x  + 18x x  + 89
│ │ │ │ -0003efc0: 3678 2078 2020 2d20 3239 7820 7820 202b  6x x  - 29x x  +
│ │ │ │ -0003efd0: 2031 3035 3978 2020 2d20 207c 0a7c 2020   1059x  -  |.|  
│ │ │ │ -0003efe0: 2020 2033 2034 2020 2020 2020 2034 2020     3 4       4  
│ │ │ │ -0003eff0: 2020 2020 2030 2035 2020 2020 2020 2031       0 5       1
│ │ │ │ -0003f000: 2035 2020 2020 2020 3220 3520 2020 2020   5      2 5     
│ │ │ │ -0003f010: 2020 3320 3520 2020 2020 2034 2035 2020    3 5      4 5  
│ │ │ │ -0003f020: 2020 2020 2020 3520 2020 207c 0a7c 2d2d        5    |.|--
│ │ │ │ -0003f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003ef80: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0003ef90: 207c 0a7c 3135 3232 7820 7820 202b 2031   |.|1522x x  + 1
│ │ │ │ +0003efa0: 3336 7820 202d 2036 3233 7820 7820 202d  36x  - 623x x  -
│ │ │ │ +0003efb0: 2036 3037 7820 7820 202b 2031 3878 2078   607x x  + 18x x
│ │ │ │ +0003efc0: 2020 2b20 3839 3678 2078 2020 2d20 3239    + 896x x  - 29
│ │ │ │ +0003efd0: 7820 7820 202b 2031 3035 3978 2020 2d20  x x  + 1059x  - 
│ │ │ │ +0003efe0: 207c 0a7c 2020 2020 2033 2034 2020 2020   |.|     3 4    
│ │ │ │ +0003eff0: 2020 2034 2020 2020 2020 2030 2035 2020     4       0 5  
│ │ │ │ +0003f000: 2020 2020 2031 2035 2020 2020 2020 3220       1 5      2 
│ │ │ │ +0003f010: 3520 2020 2020 2020 3320 3520 2020 2020  5       3 5     
│ │ │ │ +0003f020: 2034 2035 2020 2020 2020 2020 3520 2020   4 5        5   
│ │ │ │ +0003f030: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003f080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f080: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003f090: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f0a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f0b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f0c0: 2020 2020 3220 2020 2020 207c 0a7c 3130      2      |.|10
│ │ │ │ -0003f0d0: 3533 7820 7820 202b 2031 3634 3378 2078  53x x  + 1643x x
│ │ │ │ -0003f0e0: 2020 2b20 3136 3532 7820 7820 202d 2031    + 1652x x  - 1
│ │ │ │ -0003f0f0: 3139 3078 2078 2020 2d20 3130 3733 7820  190x x  - 1073x 
│ │ │ │ -0003f100: 7820 202b 2031 3437 3078 2078 2020 2d20  x  + 1470x x  - 
│ │ │ │ -0003f110: 3934 3478 2020 2d20 2020 207c 0a7c 2020  944x  -    |.|  
│ │ │ │ -0003f120: 2020 2030 2036 2020 2020 2020 2020 3120     0 6        1 
│ │ │ │ -0003f130: 3620 2020 2020 2020 2032 2036 2020 2020  6        2 6    
│ │ │ │ -0003f140: 2020 2020 3320 3620 2020 2020 2020 2034      3 6        4
│ │ │ │ -0003f150: 2036 2020 2020 2020 2020 3520 3620 2020   6        5 6   
│ │ │ │ -0003f160: 2020 2020 3620 2020 2020 207c 0a7c 2d2d      6      |.|--
│ │ │ │ -0003f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f0c0: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0003f0d0: 207c 0a7c 3130 3533 7820 7820 202b 2031   |.|1053x x  + 1
│ │ │ │ +0003f0e0: 3634 3378 2078 2020 2b20 3136 3532 7820  643x x  + 1652x 
│ │ │ │ +0003f0f0: 7820 202d 2031 3139 3078 2078 2020 2d20  x  - 1190x x  - 
│ │ │ │ +0003f100: 3130 3733 7820 7820 202b 2031 3437 3078  1073x x  + 1470x
│ │ │ │ +0003f110: 2078 2020 2d20 3934 3478 2020 2d20 2020   x  - 944x  -   
│ │ │ │ +0003f120: 207c 0a7c 2020 2020 2030 2036 2020 2020   |.|     0 6    
│ │ │ │ +0003f130: 2020 2020 3120 3620 2020 2020 2020 2032      1 6        2
│ │ │ │ +0003f140: 2036 2020 2020 2020 2020 3320 3620 2020   6        3 6   
│ │ │ │ +0003f150: 2020 2020 2034 2036 2020 2020 2020 2020       4 6        
│ │ │ │ +0003f160: 3520 3620 2020 2020 2020 3620 2020 2020  5 6       6     
│ │ │ │ +0003f170: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003f1c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f1c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003f1d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f1e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f1f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f200: 2020 2020 2020 2020 2020 327c 0a7c 3933            2|.|93
│ │ │ │ -0003f210: 7820 7820 202d 2031 3837 7820 7820 202d  x x  - 187x x  -
│ │ │ │ -0003f220: 2039 3934 7820 7820 202d 2031 3431 3578   994x x  - 1415x
│ │ │ │ -0003f230: 2078 2020 2d20 3232 3978 2078 2020 2d20   x  - 229x x  - 
│ │ │ │ -0003f240: 3739 3678 2078 2020 2b20 3136 3432 7820  796x x  + 1642x 
│ │ │ │ -0003f250: 7820 202b 2031 3630 3078 207c 0a7c 2020  x  + 1600x |.|  
│ │ │ │ -0003f260: 2030 2037 2020 2020 2020 2031 2037 2020   0 7       1 7  
│ │ │ │ -0003f270: 2020 2020 2032 2037 2020 2020 2020 2020       2 7        
│ │ │ │ -0003f280: 3320 3720 2020 2020 2020 3420 3720 2020  3 7       4 7   
│ │ │ │ -0003f290: 2020 2020 3520 3720 2020 2020 2020 2036      5 7        6
│ │ │ │ -0003f2a0: 2037 2020 2020 2020 2020 377c 0a7c 2d2d   7        7|.|--
│ │ │ │ -0003f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f200: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f210: 327c 0a7c 3933 7820 7820 202d 2031 3837  2|.|93x x  - 187
│ │ │ │ +0003f220: 7820 7820 202d 2039 3934 7820 7820 202d  x x  - 994x x  -
│ │ │ │ +0003f230: 2031 3431 3578 2078 2020 2d20 3232 3978   1415x x  - 229x
│ │ │ │ +0003f240: 2078 2020 2d20 3739 3678 2078 2020 2b20   x  - 796x x  + 
│ │ │ │ +0003f250: 3136 3432 7820 7820 202b 2031 3630 3078  1642x x  + 1600x
│ │ │ │ +0003f260: 207c 0a7c 2020 2030 2037 2020 2020 2020   |.|   0 7      
│ │ │ │ +0003f270: 2031 2037 2020 2020 2020 2032 2037 2020   1 7       2 7  
│ │ │ │ +0003f280: 2020 2020 2020 3320 3720 2020 2020 2020        3 7       
│ │ │ │ +0003f290: 3420 3720 2020 2020 2020 3520 3720 2020  4 7       5 7   
│ │ │ │ +0003f2a0: 2020 2020 2036 2037 2020 2020 2020 2020       6 7        
│ │ │ │ +0003f2b0: 377c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  7|.|------------
│ │ │ │  0003f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003f300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f300: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003f310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f340: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -0003f350: 3334 3478 2078 2020 2b20 3930 3578 2078  344x x  + 905x x
│ │ │ │ -0003f360: 2020 2b20 3130 3332 7820 7820 202d 2035    + 1032x x  - 5
│ │ │ │ -0003f370: 3338 7820 7820 202d 2038 3931 7820 7820  38x x  - 891x x 
│ │ │ │ -0003f380: 202b 2031 3234 3378 2078 2020 2b20 3132   + 1243x x  + 12
│ │ │ │ -0003f390: 3930 7820 7820 202b 2020 207c 0a7c 2020  90x x  +   |.|  
│ │ │ │ -0003f3a0: 2020 2020 3020 3820 2020 2020 2020 3120      0 8       1 
│ │ │ │ -0003f3b0: 3820 2020 2020 2020 2032 2038 2020 2020  8        2 8    
│ │ │ │ -0003f3c0: 2020 2033 2038 2020 2020 2020 2034 2038     3 8       4 8
│ │ │ │ -0003f3d0: 2020 2020 2020 2020 3520 3820 2020 2020          5 8     
│ │ │ │ -0003f3e0: 2020 2036 2038 2020 2020 207c 0a7c 2d2d     6 8     |.|--
│ │ │ │ -0003f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f350: 207c 0a7c 2d20 3334 3478 2078 2020 2b20   |.|- 344x x  + 
│ │ │ │ +0003f360: 3930 3578 2078 2020 2b20 3130 3332 7820  905x x  + 1032x 
│ │ │ │ +0003f370: 7820 202d 2035 3338 7820 7820 202d 2038  x  - 538x x  - 8
│ │ │ │ +0003f380: 3931 7820 7820 202b 2031 3234 3378 2078  91x x  + 1243x x
│ │ │ │ +0003f390: 2020 2b20 3132 3930 7820 7820 202b 2020    + 1290x x  +  
│ │ │ │ +0003f3a0: 207c 0a7c 2020 2020 2020 3020 3820 2020   |.|      0 8   
│ │ │ │ +0003f3b0: 2020 2020 3120 3820 2020 2020 2020 2032      1 8        2
│ │ │ │ +0003f3c0: 2038 2020 2020 2020 2033 2038 2020 2020   8       3 8    
│ │ │ │ +0003f3d0: 2020 2034 2038 2020 2020 2020 2020 3520     4 8        5 
│ │ │ │ +0003f3e0: 3820 2020 2020 2020 2036 2038 2020 2020  8        6 8    
│ │ │ │ +0003f3f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003f440: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -0003f450: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -0003f460: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +0003f430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f440: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003f450: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │ +0003f460: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  0003f470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f480: 2020 2020 2020 3220 2020 207c 0a7c 3439        2    |.|49
│ │ │ │ -0003f490: 3078 2078 2020 2d20 3131 3438 7820 2c20  0x x  - 1148x , 
│ │ │ │ -0003f4a0: 3136 3133 7820 202b 2031 3735 7820 7820  1613x  + 175x x 
│ │ │ │ -0003f4b0: 202d 2031 3334 3678 2020 2d20 3130 3030   - 1346x  - 1000
│ │ │ │ -0003f4c0: 7820 7820 202d 2031 3231 3778 2078 2020  x x  - 1217x x  
│ │ │ │ -0003f4d0: 2d20 3732 3978 2020 2d20 207c 0a7c 2020  - 729x  -  |.|  
│ │ │ │ -0003f4e0: 2020 3720 3820 2020 2020 2020 2038 2020    7 8        8  
│ │ │ │ -0003f4f0: 2020 2020 2030 2020 2020 2020 2030 2031       0       0 1
│ │ │ │ -0003f500: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -0003f510: 2030 2032 2020 2020 2020 2020 3120 3220   0 2        1 2 
│ │ │ │ -0003f520: 2020 2020 2020 3220 2020 207c 0a7c 2d2d        2    |.|--
│ │ │ │ -0003f530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f480: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +0003f490: 207c 0a7c 3439 3078 2078 2020 2d20 3131   |.|490x x  - 11
│ │ │ │ +0003f4a0: 3438 7820 2c20 3136 3133 7820 202b 2031  48x , 1613x  + 1
│ │ │ │ +0003f4b0: 3735 7820 7820 202d 2031 3334 3678 2020  75x x  - 1346x  
│ │ │ │ +0003f4c0: 2d20 3130 3030 7820 7820 202d 2031 3231  - 1000x x  - 121
│ │ │ │ +0003f4d0: 3778 2078 2020 2d20 3732 3978 2020 2d20  7x x  - 729x  - 
│ │ │ │ +0003f4e0: 207c 0a7c 2020 2020 3720 3820 2020 2020   |.|    7 8     
│ │ │ │ +0003f4f0: 2020 2038 2020 2020 2020 2030 2020 2020     8       0    
│ │ │ │ +0003f500: 2020 2030 2031 2020 2020 2020 2020 3120     0 1        1 
│ │ │ │ +0003f510: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ +0003f520: 2020 3120 3220 2020 2020 2020 3220 2020    1 2       2   
│ │ │ │ +0003f530: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003f540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003f580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f580: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003f590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f5a0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +0003f5a0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  0003f5b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f5c0: 2020 2020 2020 2020 2020 207c 0a7c 3132             |.|12
│ │ │ │ -0003f5d0: 3936 7820 7820 202b 2031 3435 3678 2078  96x x  + 1456x x
│ │ │ │ -0003f5e0: 2020 2b20 3734 3578 2078 2020 2b20 3533    + 745x x  + 53
│ │ │ │ -0003f5f0: 3978 2020 2b20 3532 3578 2078 2020 2d20  9x  + 525x x  - 
│ │ │ │ -0003f600: 3831 3178 2078 2020 2b20 3735 3378 2078  811x x  + 753x x
│ │ │ │ -0003f610: 2020 2b20 2020 2020 2020 207c 0a7c 2020    +        |.|  
│ │ │ │ -0003f620: 2020 2030 2033 2020 2020 2020 2020 3120     0 3        1 
│ │ │ │ -0003f630: 3320 2020 2020 2020 3220 3320 2020 2020  3       2 3     
│ │ │ │ -0003f640: 2020 3320 2020 2020 2020 3020 3420 2020    3       0 4   
│ │ │ │ -0003f650: 2020 2020 3120 3420 2020 2020 2020 3220      1 4       2 
│ │ │ │ -0003f660: 3420 2020 2020 2020 2020 207c 0a7c 2d2d  4          |.|--
│ │ │ │ -0003f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f5c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f5d0: 207c 0a7c 3132 3936 7820 7820 202b 2031   |.|1296x x  + 1
│ │ │ │ +0003f5e0: 3435 3678 2078 2020 2b20 3734 3578 2078  456x x  + 745x x
│ │ │ │ +0003f5f0: 2020 2b20 3533 3978 2020 2b20 3532 3578    + 539x  + 525x
│ │ │ │ +0003f600: 2078 2020 2d20 3831 3178 2078 2020 2b20   x  - 811x x  + 
│ │ │ │ +0003f610: 3735 3378 2078 2020 2b20 2020 2020 2020  753x x  +       
│ │ │ │ +0003f620: 207c 0a7c 2020 2020 2030 2033 2020 2020   |.|     0 3    
│ │ │ │ +0003f630: 2020 2020 3120 3320 2020 2020 2020 3220      1 3       2 
│ │ │ │ +0003f640: 3320 2020 2020 2020 3320 2020 2020 2020  3       3       
│ │ │ │ +0003f650: 3020 3420 2020 2020 2020 3120 3420 2020  0 4       1 4   
│ │ │ │ +0003f660: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │ +0003f670: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003f6c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -0003f6d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f6c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003f6d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  0003f6e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f6f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f700: 2020 2020 2020 2020 2032 207c 0a7c 3133           2 |.|13
│ │ │ │ -0003f710: 3632 7820 7820 202b 2031 3632 3978 2020  62x x  + 1629x  
│ │ │ │ -0003f720: 2d20 3834 3078 2078 2020 2b20 3531 3378  - 840x x  + 513x
│ │ │ │ -0003f730: 2078 2020 2b20 3432 3978 2078 2020 2b20   x  + 429x x  + 
│ │ │ │ -0003f740: 3834 3278 2078 2020 2b20 3134 3134 7820  842x x  + 1414x 
│ │ │ │ -0003f750: 7820 202d 2033 3038 7820 207c 0a7c 2020  x  - 308x  |.|  
│ │ │ │ -0003f760: 2020 2033 2034 2020 2020 2020 2020 3420     3 4        4 
│ │ │ │ -0003f770: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ -0003f780: 3120 3520 2020 2020 2020 3220 3520 2020  1 5       2 5   
│ │ │ │ -0003f790: 2020 2020 3320 3520 2020 2020 2020 2034      3 5        4
│ │ │ │ -0003f7a0: 2035 2020 2020 2020 2035 207c 0a7c 2d2d   5       5 |.|--
│ │ │ │ -0003f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f700: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +0003f710: 207c 0a7c 3133 3632 7820 7820 202b 2031   |.|1362x x  + 1
│ │ │ │ +0003f720: 3632 3978 2020 2d20 3834 3078 2078 2020  629x  - 840x x  
│ │ │ │ +0003f730: 2b20 3531 3378 2078 2020 2b20 3432 3978  + 513x x  + 429x
│ │ │ │ +0003f740: 2078 2020 2b20 3834 3278 2078 2020 2b20   x  + 842x x  + 
│ │ │ │ +0003f750: 3134 3134 7820 7820 202d 2033 3038 7820  1414x x  - 308x 
│ │ │ │ +0003f760: 207c 0a7c 2020 2020 2033 2034 2020 2020   |.|     3 4    
│ │ │ │ +0003f770: 2020 2020 3420 2020 2020 2020 3020 3520      4       0 5 
│ │ │ │ +0003f780: 2020 2020 2020 3120 3520 2020 2020 2020        1 5       
│ │ │ │ +0003f790: 3220 3520 2020 2020 2020 3320 3520 2020  2 5       3 5   
│ │ │ │ +0003f7a0: 2020 2020 2034 2035 2020 2020 2020 2035       4 5       5
│ │ │ │ +0003f7b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003f800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f800: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003f810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f840: 2020 2020 3220 2020 2020 207c 0a7c 2b20      2      |.|+ 
│ │ │ │ -0003f850: 3134 3135 7820 7820 202d 2031 3436 3178  1415x x  - 1461x
│ │ │ │ -0003f860: 2078 2020 2d20 3131 3335 7820 7820 202b   x  - 1135x x  +
│ │ │ │ -0003f870: 2037 3031 7820 7820 202b 2037 3636 7820   701x x  + 766x 
│ │ │ │ -0003f880: 7820 202b 2037 3835 7820 7820 202b 2031  x  + 785x x  + 1
│ │ │ │ -0003f890: 3530 3378 2020 2b20 2020 207c 0a7c 2020  503x  +    |.|  
│ │ │ │ -0003f8a0: 2020 2020 2030 2036 2020 2020 2020 2020       0 6        
│ │ │ │ -0003f8b0: 3120 3620 2020 2020 2020 2032 2036 2020  1 6        2 6  
│ │ │ │ -0003f8c0: 2020 2020 2033 2036 2020 2020 2020 2034       3 6       4
│ │ │ │ -0003f8d0: 2036 2020 2020 2020 2035 2036 2020 2020   6       5 6    
│ │ │ │ -0003f8e0: 2020 2020 3620 2020 2020 207c 0a7c 2d2d      6      |.|--
│ │ │ │ -0003f8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f840: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +0003f850: 207c 0a7c 2b20 3134 3135 7820 7820 202d   |.|+ 1415x x  -
│ │ │ │ +0003f860: 2031 3436 3178 2078 2020 2d20 3131 3335   1461x x  - 1135
│ │ │ │ +0003f870: 7820 7820 202b 2037 3031 7820 7820 202b  x x  + 701x x  +
│ │ │ │ +0003f880: 2037 3636 7820 7820 202b 2037 3835 7820   766x x  + 785x 
│ │ │ │ +0003f890: 7820 202b 2031 3530 3378 2020 2b20 2020  x  + 1503x  +   
│ │ │ │ +0003f8a0: 207c 0a7c 2020 2020 2020 2030 2036 2020   |.|       0 6  
│ │ │ │ +0003f8b0: 2020 2020 2020 3120 3620 2020 2020 2020        1 6       
│ │ │ │ +0003f8c0: 2032 2036 2020 2020 2020 2033 2036 2020   2 6       3 6  
│ │ │ │ +0003f8d0: 2020 2020 2034 2036 2020 2020 2020 2035       4 6       5
│ │ │ │ +0003f8e0: 2036 2020 2020 2020 2020 3620 2020 2020   6        6     
│ │ │ │ +0003f8f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003f940: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f940: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003f950: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003f970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003f980: 2020 2020 2020 2020 2020 327c 0a7c 3134            2|.|14
│ │ │ │ -0003f990: 3778 2078 2020 2b20 3932 3978 2078 2020  7x x  + 929x x  
│ │ │ │ -0003f9a0: 2d20 3132 3230 7820 7820 202d 2038 3533  - 1220x x  - 853
│ │ │ │ -0003f9b0: 7820 7820 202b 2034 3933 7820 7820 202b  x x  + 493x x  +
│ │ │ │ -0003f9c0: 2032 3236 7820 7820 202b 2031 3431 3678   226x x  + 1416x
│ │ │ │ -0003f9d0: 2078 2020 2b20 3238 3078 207c 0a7c 2020   x  + 280x |.|  
│ │ │ │ -0003f9e0: 2020 3020 3720 2020 2020 2020 3120 3720    0 7       1 7 
│ │ │ │ -0003f9f0: 2020 2020 2020 2032 2037 2020 2020 2020         2 7      
│ │ │ │ -0003fa00: 2033 2037 2020 2020 2020 2034 2037 2020   3 7       4 7  
│ │ │ │ -0003fa10: 2020 2020 2035 2037 2020 2020 2020 2020       5 7        
│ │ │ │ -0003fa20: 3620 3720 2020 2020 2020 377c 0a7c 2d2d  6 7       7|.|--
│ │ │ │ -0003fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003f980: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003f990: 327c 0a7c 3134 3778 2078 2020 2b20 3932  2|.|147x x  + 92
│ │ │ │ +0003f9a0: 3978 2078 2020 2d20 3132 3230 7820 7820  9x x  - 1220x x 
│ │ │ │ +0003f9b0: 202d 2038 3533 7820 7820 202b 2034 3933   - 853x x  + 493
│ │ │ │ +0003f9c0: 7820 7820 202b 2032 3236 7820 7820 202b  x x  + 226x x  +
│ │ │ │ +0003f9d0: 2031 3431 3678 2078 2020 2b20 3238 3078   1416x x  + 280x
│ │ │ │ +0003f9e0: 207c 0a7c 2020 2020 3020 3720 2020 2020   |.|    0 7     
│ │ │ │ +0003f9f0: 2020 3120 3720 2020 2020 2020 2032 2037    1 7        2 7
│ │ │ │ +0003fa00: 2020 2020 2020 2033 2037 2020 2020 2020         3 7      
│ │ │ │ +0003fa10: 2034 2037 2020 2020 2020 2035 2037 2020   4 7       5 7  
│ │ │ │ +0003fa20: 2020 2020 2020 3620 3720 2020 2020 2020        6 7       
│ │ │ │ +0003fa30: 377c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  7|.|------------
│ │ │ │  0003fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003fa80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003fa80: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  0003fa90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003faa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fac0: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -0003fad0: 3778 2078 2020 2b20 3136 3332 7820 7820  7x x  + 1632x x 
│ │ │ │ -0003fae0: 202b 2035 3230 7820 7820 202b 2031 3235   + 520x x  + 125
│ │ │ │ -0003faf0: 3978 2078 2020 2b20 3135 3778 2078 2020  9x x  + 157x x  
│ │ │ │ -0003fb00: 2b20 3135 3936 7820 7820 202b 2036 3535  + 1596x x  + 655
│ │ │ │ -0003fb10: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -0003fb20: 2020 3020 3820 2020 2020 2020 2031 2038    0 8        1 8
│ │ │ │ -0003fb30: 2020 2020 2020 2032 2038 2020 2020 2020         2 8      
│ │ │ │ -0003fb40: 2020 3320 3820 2020 2020 2020 3420 3820    3 8       4 8 
│ │ │ │ -0003fb50: 2020 2020 2020 2035 2038 2020 2020 2020         5 8      
│ │ │ │ -0003fb60: 2036 2038 2020 2020 2020 207c 0a7c 2d2d   6 8       |.|--
│ │ │ │ -0003fb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003fac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fad0: 207c 0a7c 2d20 3778 2078 2020 2b20 3136   |.|- 7x x  + 16
│ │ │ │ +0003fae0: 3332 7820 7820 202b 2035 3230 7820 7820  32x x  + 520x x 
│ │ │ │ +0003faf0: 202b 2031 3235 3978 2078 2020 2b20 3135   + 1259x x  + 15
│ │ │ │ +0003fb00: 3778 2078 2020 2b20 3135 3936 7820 7820  7x x  + 1596x x 
│ │ │ │ +0003fb10: 202b 2036 3535 7820 7820 202d 2020 2020   + 655x x  -    
│ │ │ │ +0003fb20: 207c 0a7c 2020 2020 3020 3820 2020 2020   |.|    0 8     
│ │ │ │ +0003fb30: 2020 2031 2038 2020 2020 2020 2032 2038     1 8       2 8
│ │ │ │ +0003fb40: 2020 2020 2020 2020 3320 3820 2020 2020          3 8     
│ │ │ │ +0003fb50: 2020 3420 3820 2020 2020 2020 2035 2038    4 8        5 8
│ │ │ │ +0003fb60: 2020 2020 2020 2036 2038 2020 2020 2020         6 8      
│ │ │ │ +0003fb70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  0003fb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -0003fbc0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -0003fbd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003fbc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +0003fbd0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │  0003fbe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fbf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fc00: 2020 2020 2020 2020 2020 207c 0a7c 3432             |.|42
│ │ │ │ -0003fc10: 7820 7820 202d 2035 3836 7820 7d29 2020  x x  - 586x })  
│ │ │ │ -0003fc20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fc00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fc10: 207c 0a7c 3432 7820 7820 202d 2035 3836   |.|42x x  - 586
│ │ │ │ +0003fc20: 7820 7d29 2020 2020 2020 2020 2020 2020  x })            
│ │ │ │  0003fc30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fc40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fc50: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003fc60: 2037 2038 2020 2020 2020 2038 2020 2020   7 8       8    
│ │ │ │ -0003fc70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fc50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fc60: 207c 0a7c 2020 2037 2038 2020 2020 2020   |.|   7 8      
│ │ │ │ +0003fc70: 2038 2020 2020 2020 2020 2020 2020 2020   8              
│ │ │ │  0003fc80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fc90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fca0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0003fcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003fca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fcb0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935  -----------+.|i5
│ │ │ │ -0003fd00: 203a 2074 696d 6520 6465 6772 6565 4d61   : time degreeMa
│ │ │ │ -0003fd10: 7020 7068 6920 2020 2020 2020 2020 2020  p phi           
│ │ │ │ +0003fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003fd00: 2d2b 0a7c 6935 203a 2074 696d 6520 6465  -+.|i5 : time de
│ │ │ │ +0003fd10: 6772 6565 4d61 7020 7068 6920 2020 2020  greeMap phi     
│ │ │ │  0003fd20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fd30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fd40: 2020 2020 2020 2020 2020 207c 0a7c 202d             |.| -
│ │ │ │ -0003fd50: 2d20 7573 6564 2030 2e30 3439 3732 3438  - used 0.0497248
│ │ │ │ -0003fd60: 7320 2863 7075 293b 2030 2e30 3439 3732  s (cpu); 0.04972
│ │ │ │ -0003fd70: 3538 7320 2874 6872 6561 6429 3b20 3073  58s (thread); 0s
│ │ │ │ -0003fd80: 2028 6763 2920 2020 2020 2020 2020 2020   (gc)           
│ │ │ │ -0003fd90: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003fda0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fd40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fd50: 207c 0a7c 202d 2d20 7573 6564 2030 2e30   |.| -- used 0.0
│ │ │ │ +0003fd60: 3538 3433 3134 7320 2863 7075 293b 2030  584314s (cpu); 0
│ │ │ │ +0003fd70: 2e30 3538 3433 3136 7320 2874 6872 6561  .0584316s (threa
│ │ │ │ +0003fd80: 6429 3b20 3073 2028 6763 2920 2020 2020  d); 0s (gc)     
│ │ │ │ +0003fd90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fda0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003fdb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fdc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fdd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fde0: 2020 2020 2020 2020 2020 207c 0a7c 6f35             |.|o5
│ │ │ │ -0003fdf0: 203d 2031 2020 2020 2020 2020 2020 2020   = 1            
│ │ │ │ +0003fde0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fdf0: 207c 0a7c 6f35 203d 2031 2020 2020 2020   |.|o5 = 1      
│ │ │ │  0003fe00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fe10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003fe20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003fe30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d             |.+--
│ │ │ │ -0003fe40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003fe30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fe40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.+------------
│ │ │ │  0003fe50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fe60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  0003fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -0003fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936  -----------+.|i6
│ │ │ │ -0003fe90: 203a 202d 2d20 436f 6d70 6f73 6520 7068   : -- Compose ph
│ │ │ │ -0003fea0: 693a 505e 382d 2d2d 3e50 5e31 3420 7769  i:P^8--->P^14 wi
│ │ │ │ -0003feb0: 7468 2061 206c 696e 6561 7220 7072 6f6a  th a linear proj
│ │ │ │ -0003fec0: 6563 7469 6f6e 2050 5e31 342d 2d2d 3e50  ection P^14--->P
│ │ │ │ -0003fed0: 5e38 2066 726f 6d20 2020 207c 0a7c 2020  ^8 from    |.|  
│ │ │ │ -0003fee0: 2020 202d 2d20 6f66 2064 696d 656e 7369     -- of dimensi
│ │ │ │ -0003fef0: 6f6e 2035 2028 736f 2074 6861 7420 7468  on 5 (so that th
│ │ │ │ -0003ff00: 6520 636f 6d70 6f73 6974 696f 6e20 7068  e composition ph
│ │ │ │ -0003ff10: 6927 3a50 5e38 2d2d 2d3e 505e 3820 6d75  i':P^8--->P^8 mu
│ │ │ │ -0003ff20: 7374 2068 6176 6520 2020 207c 0a7c 2020  st have    |.|  
│ │ │ │ -0003ff30: 2020 2070 6869 273d 7068 692a 6d61 7028     phi'=phi*map(
│ │ │ │ -0003ff40: 7269 6e67 5031 342c 7269 6e67 5038 2c66  ringP14,ringP8,f
│ │ │ │ -0003ff50: 6f72 2069 2074 6f20 3820 6c69 7374 2072  or i to 8 list r
│ │ │ │ -0003ff60: 616e 646f 6d28 312c 7269 6e67 5031 3429  andom(1,ringP14)
│ │ │ │ -0003ff70: 2920 2020 2020 2020 2020 207c 0a7c 2020  )          |.|  
│ │ │ │ -0003ff80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +0003fe90: 2d2b 0a7c 6936 203a 202d 2d20 436f 6d70  -+.|i6 : -- Comp
│ │ │ │ +0003fea0: 6f73 6520 7068 693a 505e 382d 2d2d 3e50  ose phi:P^8--->P
│ │ │ │ +0003feb0: 5e31 3420 7769 7468 2061 206c 696e 6561  ^14 with a linea
│ │ │ │ +0003fec0: 7220 7072 6f6a 6563 7469 6f6e 2050 5e31  r projection P^1
│ │ │ │ +0003fed0: 342d 2d2d 3e50 5e38 2066 726f 6d20 2020  4--->P^8 from   
│ │ │ │ +0003fee0: 207c 0a7c 2020 2020 202d 2d20 6f66 2064   |.|     -- of d
│ │ │ │ +0003fef0: 696d 656e 7369 6f6e 2035 2028 736f 2074  imension 5 (so t
│ │ │ │ +0003ff00: 6861 7420 7468 6520 636f 6d70 6f73 6974  hat the composit
│ │ │ │ +0003ff10: 696f 6e20 7068 6927 3a50 5e38 2d2d 2d3e  ion phi':P^8--->
│ │ │ │ +0003ff20: 505e 3820 6d75 7374 2068 6176 6520 2020  P^8 must have   
│ │ │ │ +0003ff30: 207c 0a7c 2020 2020 2070 6869 273d 7068   |.|     phi'=ph
│ │ │ │ +0003ff40: 692a 6d61 7028 7269 6e67 5031 342c 7269  i*map(ringP14,ri
│ │ │ │ +0003ff50: 6e67 5038 2c66 6f72 2069 2074 6f20 3820  ngP8,for i to 8 
│ │ │ │ +0003ff60: 6c69 7374 2072 616e 646f 6d28 312c 7269  list random(1,ri
│ │ │ │ +0003ff70: 6e67 5031 3429 2920 2020 2020 2020 2020  ngP14))         
│ │ │ │ +0003ff80: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  0003ff90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ffa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  0003ffb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ffc0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -0003ffd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -0003ffe0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -0003fff0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040000: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ -00040010: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -00040020: 203d 206d 6170 2028 7269 6e67 5038 2c20   = map (ringP8, 
│ │ │ │ -00040030: 7269 6e67 5038 2c20 7b2d 2037 3830 7820  ringP8, {- 780x 
│ │ │ │ -00040040: 202d 2035 3036 7820 7820 202b 2031 3533   - 506x x  + 153
│ │ │ │ -00040050: 3778 2020 2d20 3133 3278 2078 2020 2d20  7x  - 132x x  - 
│ │ │ │ -00040060: 3932 3878 2078 2020 2020 207c 0a7c 2020  928x x     |.|  
│ │ │ │ -00040070: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040080: 2020 2020 2020 2020 2020 2020 2020 2030                 0
│ │ │ │ -00040090: 2020 2020 2020 2030 2031 2020 2020 2020         0 1      
│ │ │ │ -000400a0: 2020 3120 2020 2020 2020 3020 3220 2020    1       0 2   
│ │ │ │ -000400b0: 2020 2020 3120 3220 2020 207c 0a7c 2020      1 2    |.|  
│ │ │ │ -000400c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ffc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003ffd0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +0003ffe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +0003fff0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00040000: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00040010: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040020: 207c 0a7c 6f36 203d 206d 6170 2028 7269   |.|o6 = map (ri
│ │ │ │ +00040030: 6e67 5038 2c20 7269 6e67 5038 2c20 7b2d  ngP8, ringP8, {-
│ │ │ │ +00040040: 2037 3830 7820 202d 2035 3036 7820 7820   780x  - 506x x 
│ │ │ │ +00040050: 202b 2031 3533 3778 2020 2d20 3133 3278   + 1537x  - 132x
│ │ │ │ +00040060: 2078 2020 2d20 3932 3878 2078 2020 2020   x  - 928x x    
│ │ │ │ +00040070: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │ +00040080: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040090: 2020 2020 2030 2020 2020 2020 2030 2031       0       0 1
│ │ │ │ +000400a0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ +000400b0: 3020 3220 2020 2020 2020 3120 3220 2020  0 2       1 2   
│ │ │ │ +000400c0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000400d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000400e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000400f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040100: 2020 2020 2020 2020 2020 207c 0a7c 6f36             |.|o6
│ │ │ │ -00040110: 203a 2052 696e 674d 6170 2072 696e 6750   : RingMap ringP
│ │ │ │ -00040120: 3820 3c2d 2d20 7269 6e67 5038 2020 2020  8 <-- ringP8    
│ │ │ │ -00040130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040110: 207c 0a7c 6f36 203a 2052 696e 674d 6170   |.|o6 : RingMap
│ │ │ │ +00040120: 2072 696e 6750 3820 3c2d 2d20 7269 6e67   ringP8 <-- ring
│ │ │ │ +00040130: 5038 2020 2020 2020 2020 2020 2020 2020  P8              
│ │ │ │  00040140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040150: 2020 2020 2020 2020 2020 207c 0a7c 2d2d             |.|--
│ │ │ │ -00040160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040160: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00040170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000401a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6120  -----------|.|a 
│ │ │ │ -000401b0: 6765 6e65 7261 6c20 7375 6273 7061 6365  general subspace
│ │ │ │ -000401c0: 206f 6620 505e 3134 2020 2020 2020 2020   of P^14        
│ │ │ │ +000401a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000401b0: 2d7c 0a7c 6120 6765 6e65 7261 6c20 7375  -|.|a general su
│ │ │ │ +000401c0: 6273 7061 6365 206f 6620 505e 3134 2020  bspace of P^14  
│ │ │ │  000401d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000401e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000401f0: 2020 2020 2020 2020 2020 207c 0a7c 6465             |.|de
│ │ │ │ -00040200: 6772 6565 2065 7175 616c 2074 6f20 6465  gree equal to de
│ │ │ │ -00040210: 6728 4728 312c 3529 293d 3134 2920 2020  g(G(1,5))=14)   
│ │ │ │ -00040220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000401f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040200: 207c 0a7c 6465 6772 6565 2065 7175 616c   |.|degree equal
│ │ │ │ +00040210: 2074 6f20 6465 6728 4728 312c 3529 293d   to deg(G(1,5))=
│ │ │ │ +00040220: 3134 2920 2020 2020 2020 2020 2020 2020  14)             
│ │ │ │  00040230: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040240: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -00040250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040250: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  00040260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040290: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000402a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000402a0: 207c 0a7c 2020 2020 2020 2020 2020 2020   |.|            
│ │ │ │  000402b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000402c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000402d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000402e0: 2020 2020 2020 2020 2020 207c 0a7c 2020             |.|  
│ │ │ │ -000402f0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +000402e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000402f0: 207c 0a7c 2020 2020 2020 3220 2020 2020   |.|      2     
│ │ │ │  00040300: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040310: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00040320: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040330: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -00040340: 3338 3678 2020 2d20 3130 3278 2078 2020  386x  - 102x x  
│ │ │ │ -00040350: 2b20 3432 3278 2078 2020 2b20 3732 3578  + 422x x  + 725x
│ │ │ │ -00040360: 2078 2020 2d20 3130 3733 7820 202d 2039   x  - 1073x  - 9
│ │ │ │ -00040370: 3035 7820 7820 202d 2038 3330 7820 7820  05x x  - 830x x 
│ │ │ │ -00040380: 202b 2031 3530 3078 2078 207c 0a7c 2020   + 1500x x |.|  
│ │ │ │ -00040390: 2020 2020 3220 2020 2020 2020 3020 3320      2       0 3 
│ │ │ │ -000403a0: 2020 2020 2020 3120 3320 2020 2020 2020        1 3       
│ │ │ │ -000403b0: 3220 3320 2020 2020 2020 2033 2020 2020  2 3        3    
│ │ │ │ -000403c0: 2020 2030 2034 2020 2020 2020 2031 2034     0 4       1 4
│ │ │ │ -000403d0: 2020 2020 2020 2020 3220 347c 0a7c 2d2d          2 4|.|--
│ │ │ │ -000403e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040310: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040320: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00040330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040340: 207c 0a7c 2b20 3338 3678 2020 2d20 3130   |.|+ 386x  - 10
│ │ │ │ +00040350: 3278 2078 2020 2b20 3432 3278 2078 2020  2x x  + 422x x  
│ │ │ │ +00040360: 2b20 3732 3578 2078 2020 2d20 3130 3733  + 725x x  - 1073
│ │ │ │ +00040370: 7820 202d 2039 3035 7820 7820 202d 2038  x  - 905x x  - 8
│ │ │ │ +00040380: 3330 7820 7820 202b 2031 3530 3078 2078  30x x  + 1500x x
│ │ │ │ +00040390: 207c 0a7c 2020 2020 2020 3220 2020 2020   |.|      2     
│ │ │ │ +000403a0: 2020 3020 3320 2020 2020 2020 3120 3320    0 3       1 3 
│ │ │ │ +000403b0: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ +000403c0: 2033 2020 2020 2020 2030 2034 2020 2020   3       0 4    
│ │ │ │ +000403d0: 2020 2031 2034 2020 2020 2020 2020 3220     1 4        2 
│ │ │ │ +000403e0: 347c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  4|.|------------
│ │ │ │  000403f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00040420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00040430: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00040440: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040430: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00040440: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │  00040450: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040460: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040470: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -00040480: 3237 3678 2078 2020 2b20 3135 3333 7820  276x x  + 1533x 
│ │ │ │ -00040490: 202d 2036 3533 7820 7820 202b 2031 3535   - 653x x  + 155
│ │ │ │ -000404a0: 3878 2078 2020 2b20 3933 3978 2078 2020  8x x  + 939x x  
│ │ │ │ -000404b0: 2d20 3134 3332 7820 7820 202b 2034 3632  - 1432x x  + 462
│ │ │ │ -000404c0: 7820 7820 202d 2020 2020 207c 0a7c 2020  x x  -     |.|  
│ │ │ │ -000404d0: 2020 2020 3320 3420 2020 2020 2020 2034      3 4        4
│ │ │ │ -000404e0: 2020 2020 2020 2030 2035 2020 2020 2020         0 5      
│ │ │ │ -000404f0: 2020 3120 3520 2020 2020 2020 3220 3520    1 5       2 5 
│ │ │ │ -00040500: 2020 2020 2020 2033 2035 2020 2020 2020         3 5      
│ │ │ │ -00040510: 2034 2035 2020 2020 2020 207c 0a7c 2d2d   4 5       |.|--
│ │ │ │ -00040520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040480: 207c 0a7c 2b20 3237 3678 2078 2020 2b20   |.|+ 276x x  + 
│ │ │ │ +00040490: 3135 3333 7820 202d 2036 3533 7820 7820  1533x  - 653x x 
│ │ │ │ +000404a0: 202b 2031 3535 3878 2078 2020 2b20 3933   + 1558x x  + 93
│ │ │ │ +000404b0: 3978 2078 2020 2d20 3134 3332 7820 7820  9x x  - 1432x x 
│ │ │ │ +000404c0: 202b 2034 3632 7820 7820 202d 2020 2020   + 462x x  -    
│ │ │ │ +000404d0: 207c 0a7c 2020 2020 2020 3320 3420 2020   |.|      3 4   
│ │ │ │ +000404e0: 2020 2020 2034 2020 2020 2020 2030 2035       4       0 5
│ │ │ │ +000404f0: 2020 2020 2020 2020 3120 3520 2020 2020          1 5     
│ │ │ │ +00040500: 2020 3220 3520 2020 2020 2020 2033 2035    2 5        3 5
│ │ │ │ +00040510: 2020 2020 2020 2034 2035 2020 2020 2020         4 5      
│ │ │ │ +00040520: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00040530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00040560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00040570: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00040560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040570: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  00040580: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040590: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000405a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000405b0: 2020 2020 2020 3220 2020 207c 0a7c 3332        2    |.|32
│ │ │ │ -000405c0: 3978 2020 2d20 3932 7820 7820 202b 2036  9x  - 92x x  + 6
│ │ │ │ -000405d0: 3631 7820 7820 202d 2031 3239 3878 2078  61x x  - 1298x x
│ │ │ │ -000405e0: 2020 2d20 3638 3478 2078 2020 2b20 3730    - 684x x  + 70
│ │ │ │ -000405f0: 7820 7820 202d 2037 3135 7820 7820 202b  x x  - 715x x  +
│ │ │ │ -00040600: 2031 3039 3378 2020 2b20 207c 0a7c 2020   1093x  +  |.|  
│ │ │ │ -00040610: 2020 3520 2020 2020 2030 2036 2020 2020    5      0 6    
│ │ │ │ -00040620: 2020 2031 2036 2020 2020 2020 2020 3220     1 6        2 
│ │ │ │ -00040630: 3620 2020 2020 2020 3320 3620 2020 2020  6       3 6     
│ │ │ │ -00040640: 2034 2036 2020 2020 2020 2035 2036 2020   4 6       5 6  
│ │ │ │ -00040650: 2020 2020 2020 3620 2020 207c 0a7c 2d2d        6    |.|--
│ │ │ │ -00040660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000405b0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +000405c0: 207c 0a7c 3332 3978 2020 2d20 3932 7820   |.|329x  - 92x 
│ │ │ │ +000405d0: 7820 202b 2036 3631 7820 7820 202d 2031  x  + 661x x  - 1
│ │ │ │ +000405e0: 3239 3878 2078 2020 2d20 3638 3478 2078  298x x  - 684x x
│ │ │ │ +000405f0: 2020 2b20 3730 7820 7820 202d 2037 3135    + 70x x  - 715
│ │ │ │ +00040600: 7820 7820 202b 2031 3039 3378 2020 2b20  x x  + 1093x  + 
│ │ │ │ +00040610: 207c 0a7c 2020 2020 3520 2020 2020 2030   |.|    5      0
│ │ │ │ +00040620: 2036 2020 2020 2020 2031 2036 2020 2020   6       1 6    
│ │ │ │ +00040630: 2020 2020 3220 3620 2020 2020 2020 3320      2 6       3 
│ │ │ │ +00040640: 3620 2020 2020 2034 2036 2020 2020 2020  6      4 6      
│ │ │ │ +00040650: 2035 2036 2020 2020 2020 2020 3620 2020   5 6        6   
│ │ │ │ +00040660: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00040670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000406a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000406b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000406a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000406b0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  000406c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000406d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000406e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000406f0: 2020 2020 2020 2020 2032 207c 0a7c 3538           2 |.|58
│ │ │ │ -00040700: 3178 2078 2020 2b20 3332 3978 2078 2020  1x x  + 329x x  
│ │ │ │ -00040710: 2b20 3435 3478 2078 2020 2d20 3931 3178  + 454x x  - 911x
│ │ │ │ -00040720: 2078 2020 2d20 3834 7820 7820 202d 2031   x  - 84x x  - 1
│ │ │ │ -00040730: 3435 3278 2078 2020 2d20 3830 3978 2078  452x x  - 809x x
│ │ │ │ -00040740: 2020 2b20 3132 3032 7820 207c 0a7c 2020    + 1202x  |.|  
│ │ │ │ -00040750: 2020 3020 3720 2020 2020 2020 3120 3720    0 7       1 7 
│ │ │ │ -00040760: 2020 2020 2020 3220 3720 2020 2020 2020        2 7       
│ │ │ │ -00040770: 3320 3720 2020 2020 2034 2037 2020 2020  3 7      4 7    
│ │ │ │ -00040780: 2020 2020 3520 3720 2020 2020 2020 3620      5 7       6 
│ │ │ │ -00040790: 3720 2020 2020 2020 2037 207c 0a7c 2d2d  7        7 |.|--
│ │ │ │ -000407a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000406f0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00040700: 207c 0a7c 3538 3178 2078 2020 2b20 3332   |.|581x x  + 32
│ │ │ │ +00040710: 3978 2078 2020 2b20 3435 3478 2078 2020  9x x  + 454x x  
│ │ │ │ +00040720: 2d20 3931 3178 2078 2020 2d20 3834 7820  - 911x x  - 84x 
│ │ │ │ +00040730: 7820 202d 2031 3435 3278 2078 2020 2d20  x  - 1452x x  - 
│ │ │ │ +00040740: 3830 3978 2078 2020 2b20 3132 3032 7820  809x x  + 1202x 
│ │ │ │ +00040750: 207c 0a7c 2020 2020 3020 3720 2020 2020   |.|    0 7     
│ │ │ │ +00040760: 2020 3120 3720 2020 2020 2020 3220 3720    1 7       2 7 
│ │ │ │ +00040770: 2020 2020 2020 3320 3720 2020 2020 2034        3 7      4
│ │ │ │ +00040780: 2037 2020 2020 2020 2020 3520 3720 2020   7        5 7   
│ │ │ │ +00040790: 2020 2020 3620 3720 2020 2020 2020 2037      6 7        7
│ │ │ │ +000407a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000407b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000407c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000407d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000407e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000407f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000407e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000407f0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00040800: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040810: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040820: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040830: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -00040840: 3133 3533 7820 7820 202b 2031 3530 3378  1353x x  + 1503x
│ │ │ │ -00040850: 2078 2020 2b20 3438 3278 2078 2020 2b20   x  + 482x x  + 
│ │ │ │ -00040860: 3839 3378 2078 2020 2d20 3634 3378 2078  893x x  - 643x x
│ │ │ │ -00040870: 2020 2b20 3539 3878 2078 2020 2b20 3131    + 598x x  + 11
│ │ │ │ -00040880: 3078 2078 2020 2b20 2020 207c 0a7c 2020  0x x  +    |.|  
│ │ │ │ -00040890: 2020 2020 2030 2038 2020 2020 2020 2020       0 8        
│ │ │ │ -000408a0: 3120 3820 2020 2020 2020 3220 3820 2020  1 8       2 8   
│ │ │ │ -000408b0: 2020 2020 3320 3820 2020 2020 2020 3420      3 8       4 
│ │ │ │ -000408c0: 3820 2020 2020 2020 3520 3820 2020 2020  8       5 8     
│ │ │ │ -000408d0: 2020 3620 3820 2020 2020 207c 0a7c 2d2d    6 8      |.|--
│ │ │ │ -000408e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040830: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040840: 207c 0a7c 2b20 3133 3533 7820 7820 202b   |.|+ 1353x x  +
│ │ │ │ +00040850: 2031 3530 3378 2078 2020 2b20 3438 3278   1503x x  + 482x
│ │ │ │ +00040860: 2078 2020 2b20 3839 3378 2078 2020 2d20   x  + 893x x  - 
│ │ │ │ +00040870: 3634 3378 2078 2020 2b20 3539 3878 2078  643x x  + 598x x
│ │ │ │ +00040880: 2020 2b20 3131 3078 2078 2020 2b20 2020    + 110x x  +   
│ │ │ │ +00040890: 207c 0a7c 2020 2020 2020 2030 2038 2020   |.|       0 8  
│ │ │ │ +000408a0: 2020 2020 2020 3120 3820 2020 2020 2020        1 8       
│ │ │ │ +000408b0: 3220 3820 2020 2020 2020 3320 3820 2020  2 8       3 8   
│ │ │ │ +000408c0: 2020 2020 3420 3820 2020 2020 2020 3520      4 8       5 
│ │ │ │ +000408d0: 3820 2020 2020 2020 3620 3820 2020 2020  8       6 8     
│ │ │ │ +000408e0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000408f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00040920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00040930: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00040940: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ -00040950: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00040920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040930: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00040940: 2020 2032 2020 2020 2020 2020 3220 2020     2        2   
│ │ │ │ +00040950: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00040960: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040970: 2020 2020 2020 2020 3220 207c 0a7c 3130          2  |.|10
│ │ │ │ -00040980: 3634 7820 7820 202d 2034 3732 7820 2c20  64x x  - 472x , 
│ │ │ │ -00040990: 2d20 3532 3278 2020 2d20 3538 3378 2078  - 522x  - 583x x
│ │ │ │ -000409a0: 2020 2b20 3133 3339 7820 202b 2031 3533    + 1339x  + 153
│ │ │ │ -000409b0: 3578 2078 2020 2d20 3133 3137 7820 7820  5x x  - 1317x x 
│ │ │ │ -000409c0: 202b 2031 3131 3378 2020 2d7c 0a7c 2020   + 1113x  -|.|  
│ │ │ │ -000409d0: 2020 2037 2038 2020 2020 2020 2038 2020     7 8       8  
│ │ │ │ -000409e0: 2020 2020 2020 3020 2020 2020 2020 3020        0       0 
│ │ │ │ -000409f0: 3120 2020 2020 2020 2031 2020 2020 2020  1        1      
│ │ │ │ -00040a00: 2020 3020 3220 2020 2020 2020 2031 2032    0 2        1 2
│ │ │ │ -00040a10: 2020 2020 2020 2020 3220 207c 0a7c 2d2d          2  |.|--
│ │ │ │ -00040a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040970: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00040980: 207c 0a7c 3130 3634 7820 7820 202d 2034   |.|1064x x  - 4
│ │ │ │ +00040990: 3732 7820 2c20 2d20 3532 3278 2020 2d20  72x , - 522x  - 
│ │ │ │ +000409a0: 3538 3378 2078 2020 2b20 3133 3339 7820  583x x  + 1339x 
│ │ │ │ +000409b0: 202b 2031 3533 3578 2078 2020 2d20 3133   + 1535x x  - 13
│ │ │ │ +000409c0: 3137 7820 7820 202b 2031 3131 3378 2020  17x x  + 1113x  
│ │ │ │ +000409d0: 2d7c 0a7c 2020 2020 2037 2038 2020 2020  -|.|     7 8    
│ │ │ │ +000409e0: 2020 2038 2020 2020 2020 2020 3020 2020     8        0   
│ │ │ │ +000409f0: 2020 2020 3020 3120 2020 2020 2020 2031      0 1        1
│ │ │ │ +00040a00: 2020 2020 2020 2020 3020 3220 2020 2020          0 2     
│ │ │ │ +00040a10: 2020 2031 2032 2020 2020 2020 2020 3220     1 2        2 
│ │ │ │ +00040a20: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00040a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00040a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00040a70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040a70: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00040a80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040a90: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00040a90: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │  00040aa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040ab0: 2020 2020 2020 2020 2020 207c 0a7c 3136             |.|16
│ │ │ │ -00040ac0: 3978 2078 2020 2b20 3134 3430 7820 7820  9x x  + 1440x x 
│ │ │ │ -00040ad0: 202d 2031 3635 3778 2078 2020 2b20 3732   - 1657x x  + 72
│ │ │ │ -00040ae0: 3178 2020 2b20 3430 7820 7820 202d 2031  1x  + 40x x  - 1
│ │ │ │ -00040af0: 3537 3678 2078 2020 2d20 3336 3778 2078  576x x  - 367x x
│ │ │ │ -00040b00: 2020 2b20 3235 3778 2078 207c 0a7c 2020    + 257x x |.|  
│ │ │ │ -00040b10: 2020 3020 3320 2020 2020 2020 2031 2033    0 3        1 3
│ │ │ │ -00040b20: 2020 2020 2020 2020 3220 3320 2020 2020          2 3     
│ │ │ │ -00040b30: 2020 3320 2020 2020 2030 2034 2020 2020    3      0 4    
│ │ │ │ -00040b40: 2020 2020 3120 3420 2020 2020 2020 3220      1 4       2 
│ │ │ │ -00040b50: 3420 2020 2020 2020 3320 347c 0a7c 2d2d  4       3 4|.|--
│ │ │ │ -00040b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040ab0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040ac0: 207c 0a7c 3136 3978 2078 2020 2b20 3134   |.|169x x  + 14
│ │ │ │ +00040ad0: 3430 7820 7820 202d 2031 3635 3778 2078  40x x  - 1657x x
│ │ │ │ +00040ae0: 2020 2b20 3732 3178 2020 2b20 3430 7820    + 721x  + 40x 
│ │ │ │ +00040af0: 7820 202d 2031 3537 3678 2078 2020 2d20  x  - 1576x x  - 
│ │ │ │ +00040b00: 3336 3778 2078 2020 2b20 3235 3778 2078  367x x  + 257x x
│ │ │ │ +00040b10: 207c 0a7c 2020 2020 3020 3320 2020 2020   |.|    0 3     
│ │ │ │ +00040b20: 2020 2031 2033 2020 2020 2020 2020 3220     1 3        2 
│ │ │ │ +00040b30: 3320 2020 2020 2020 3320 2020 2020 2030  3       3      0
│ │ │ │ +00040b40: 2034 2020 2020 2020 2020 3120 3420 2020   4        1 4   
│ │ │ │ +00040b50: 2020 2020 3220 3420 2020 2020 2020 3320      2 4       3 
│ │ │ │ +00040b60: 347c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  4|.|------------
│ │ │ │  00040b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00040ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00040bb0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00040ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040bb0: 2d7c 0a7c 2020 2020 2020 2032 2020 2020  -|.|       2    
│ │ │ │  00040bc0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040bd0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040be0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040bf0: 2020 2032 2020 2020 2020 207c 0a7c 2d20     2       |.|- 
│ │ │ │ -00040c00: 3134 3534 7820 202b 2031 3631 3278 2078  1454x  + 1612x x
│ │ │ │ -00040c10: 2020 2b20 3135 3239 7820 7820 202d 2031    + 1529x x  - 1
│ │ │ │ -00040c20: 3036 3878 2078 2020 2b20 3536 3078 2078  068x x  + 560x x
│ │ │ │ -00040c30: 2020 2d20 3134 3431 7820 7820 202b 2036    - 1441x x  + 6
│ │ │ │ -00040c40: 3038 7820 202d 2020 2020 207c 0a7c 2020  08x  -     |.|  
│ │ │ │ -00040c50: 2020 2020 2034 2020 2020 2020 2020 3020       4        0 
│ │ │ │ -00040c60: 3520 2020 2020 2020 2031 2035 2020 2020  5        1 5    
│ │ │ │ -00040c70: 2020 2020 3220 3520 2020 2020 2020 3320      2 5       3 
│ │ │ │ -00040c80: 3520 2020 2020 2020 2034 2035 2020 2020  5        4 5    
│ │ │ │ -00040c90: 2020 2035 2020 2020 2020 207c 0a7c 2d2d     5       |.|--
│ │ │ │ -00040ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040bf0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00040c00: 207c 0a7c 2d20 3134 3534 7820 202b 2031   |.|- 1454x  + 1
│ │ │ │ +00040c10: 3631 3278 2078 2020 2b20 3135 3239 7820  612x x  + 1529x 
│ │ │ │ +00040c20: 7820 202d 2031 3036 3878 2078 2020 2b20  x  - 1068x x  + 
│ │ │ │ +00040c30: 3536 3078 2078 2020 2d20 3134 3431 7820  560x x  - 1441x 
│ │ │ │ +00040c40: 7820 202b 2036 3038 7820 202d 2020 2020  x  + 608x  -    
│ │ │ │ +00040c50: 207c 0a7c 2020 2020 2020 2034 2020 2020   |.|       4    
│ │ │ │ +00040c60: 2020 2020 3020 3520 2020 2020 2020 2031      0 5        1
│ │ │ │ +00040c70: 2035 2020 2020 2020 2020 3220 3520 2020   5        2 5   
│ │ │ │ +00040c80: 2020 2020 3320 3520 2020 2020 2020 2034      3 5        4
│ │ │ │ +00040c90: 2035 2020 2020 2020 2035 2020 2020 2020   5       5      
│ │ │ │ +00040ca0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00040cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00040ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00040cf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040cf0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00040d00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040d10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040d20: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00040d30: 2020 2020 2020 2020 2020 207c 0a7c 3932             |.|92
│ │ │ │ -00040d40: 7820 7820 202d 2031 3030 3678 2078 2020  x x  - 1006x x  
│ │ │ │ -00040d50: 2b20 3238 3578 2078 2020 2b20 3130 3278  + 285x x  + 102x
│ │ │ │ -00040d60: 2078 2020 2d20 3339 3778 2078 2020 2b20   x  - 397x x  + 
│ │ │ │ -00040d70: 3636 7820 7820 202d 2036 3433 7820 202d  66x x  - 643x  -
│ │ │ │ -00040d80: 2033 3878 2078 2020 2b20 207c 0a7c 2020   38x x  +  |.|  
│ │ │ │ -00040d90: 2030 2036 2020 2020 2020 2020 3120 3620   0 6        1 6 
│ │ │ │ -00040da0: 2020 2020 2020 3220 3620 2020 2020 2020        2 6       
│ │ │ │ -00040db0: 3320 3620 2020 2020 2020 3420 3620 2020  3 6       4 6   
│ │ │ │ -00040dc0: 2020 2035 2036 2020 2020 2020 2036 2020     5 6       6  
│ │ │ │ -00040dd0: 2020 2020 3020 3720 2020 207c 0a7c 2d2d      0 7    |.|--
│ │ │ │ -00040de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040d20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040d30: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00040d40: 207c 0a7c 3932 7820 7820 202d 2031 3030   |.|92x x  - 100
│ │ │ │ +00040d50: 3678 2078 2020 2b20 3238 3578 2078 2020  6x x  + 285x x  
│ │ │ │ +00040d60: 2b20 3130 3278 2078 2020 2d20 3339 3778  + 102x x  - 397x
│ │ │ │ +00040d70: 2078 2020 2b20 3636 7820 7820 202d 2036   x  + 66x x  - 6
│ │ │ │ +00040d80: 3433 7820 202d 2033 3878 2078 2020 2b20  43x  - 38x x  + 
│ │ │ │ +00040d90: 207c 0a7c 2020 2030 2036 2020 2020 2020   |.|   0 6      
│ │ │ │ +00040da0: 2020 3120 3620 2020 2020 2020 3220 3620    1 6       2 6 
│ │ │ │ +00040db0: 2020 2020 2020 3320 3620 2020 2020 2020        3 6       
│ │ │ │ +00040dc0: 3420 3620 2020 2020 2035 2036 2020 2020  4 6      5 6    
│ │ │ │ +00040dd0: 2020 2036 2020 2020 2020 3020 3720 2020     6      0 7   
│ │ │ │ +00040de0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00040df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00040e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00040e30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040e30: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00040e40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040e50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040e60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040e70: 3220 2020 2020 2020 2020 207c 0a7c 3133  2          |.|13
│ │ │ │ -00040e80: 3830 7820 7820 202b 2031 3036 3978 2078  80x x  + 1069x x
│ │ │ │ -00040e90: 2020 2d20 3432 3678 2078 2020 2b20 3131    - 426x x  + 11
│ │ │ │ -00040ea0: 3437 7820 7820 202b 2039 3832 7820 7820  47x x  + 982x x 
│ │ │ │ -00040eb0: 202b 2031 3078 2078 2020 2d20 3636 3278   + 10x x  - 662x
│ │ │ │ -00040ec0: 2020 2b20 3136 7820 7820 207c 0a7c 2020    + 16x x  |.|  
│ │ │ │ -00040ed0: 2020 2031 2037 2020 2020 2020 2020 3220     1 7        2 
│ │ │ │ -00040ee0: 3720 2020 2020 2020 3320 3720 2020 2020  7       3 7     
│ │ │ │ -00040ef0: 2020 2034 2037 2020 2020 2020 2035 2037     4 7       5 7
│ │ │ │ -00040f00: 2020 2020 2020 3620 3720 2020 2020 2020        6 7       
│ │ │ │ -00040f10: 3720 2020 2020 2030 2038 207c 0a7c 2d2d  7      0 8 |.|--
│ │ │ │ -00040f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040e70: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │ +00040e80: 207c 0a7c 3133 3830 7820 7820 202b 2031   |.|1380x x  + 1
│ │ │ │ +00040e90: 3036 3978 2078 2020 2d20 3432 3678 2078  069x x  - 426x x
│ │ │ │ +00040ea0: 2020 2b20 3131 3437 7820 7820 202b 2039    + 1147x x  + 9
│ │ │ │ +00040eb0: 3832 7820 7820 202b 2031 3078 2078 2020  82x x  + 10x x  
│ │ │ │ +00040ec0: 2d20 3636 3278 2020 2b20 3136 7820 7820  - 662x  + 16x x 
│ │ │ │ +00040ed0: 207c 0a7c 2020 2020 2031 2037 2020 2020   |.|     1 7    
│ │ │ │ +00040ee0: 2020 2020 3220 3720 2020 2020 2020 3320      2 7       3 
│ │ │ │ +00040ef0: 3720 2020 2020 2020 2034 2037 2020 2020  7        4 7    
│ │ │ │ +00040f00: 2020 2035 2037 2020 2020 2020 3620 3720     5 7      6 7 
│ │ │ │ +00040f10: 2020 2020 2020 3720 2020 2020 2030 2038        7      0 8
│ │ │ │ +00040f20: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00040f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00040f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00040f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00040f70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040f70: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00040f80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040f90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00040fa0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00040fb0: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -00040fc0: 3135 3631 7820 7820 202b 2031 3539 3778  1561x x  + 1597x
│ │ │ │ -00040fd0: 2078 2020 2b20 3531 3278 2078 2020 2b20   x  + 512x x  + 
│ │ │ │ -00040fe0: 3132 3838 7820 7820 202d 2031 3235 3378  1288x x  - 1253x
│ │ │ │ -00040ff0: 2078 2020 2b20 3133 3137 7820 7820 202b   x  + 1317x x  +
│ │ │ │ -00041000: 2031 3438 3178 2078 2020 2d7c 0a7c 2020   1481x x  -|.|  
│ │ │ │ -00041010: 2020 2020 2031 2038 2020 2020 2020 2020       1 8        
│ │ │ │ -00041020: 3220 3820 2020 2020 2020 3320 3820 2020  2 8       3 8   
│ │ │ │ -00041030: 2020 2020 2034 2038 2020 2020 2020 2020       4 8        
│ │ │ │ -00041040: 3520 3820 2020 2020 2020 2036 2038 2020  5 8        6 8  
│ │ │ │ -00041050: 2020 2020 2020 3720 3820 207c 0a7c 2d2d        7 8  |.|--
│ │ │ │ -00041060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00040fb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00040fc0: 207c 0a7c 2b20 3135 3631 7820 7820 202b   |.|+ 1561x x  +
│ │ │ │ +00040fd0: 2031 3539 3778 2078 2020 2b20 3531 3278   1597x x  + 512x
│ │ │ │ +00040fe0: 2078 2020 2b20 3132 3838 7820 7820 202d   x  + 1288x x  -
│ │ │ │ +00040ff0: 2031 3235 3378 2078 2020 2b20 3133 3137   1253x x  + 1317
│ │ │ │ +00041000: 7820 7820 202b 2031 3438 3178 2078 2020  x x  + 1481x x  
│ │ │ │ +00041010: 2d7c 0a7c 2020 2020 2020 2031 2038 2020  -|.|       1 8  
│ │ │ │ +00041020: 2020 2020 2020 3220 3820 2020 2020 2020        2 8       
│ │ │ │ +00041030: 3320 3820 2020 2020 2020 2034 2038 2020  3 8        4 8  
│ │ │ │ +00041040: 2020 2020 2020 3520 3820 2020 2020 2020        5 8       
│ │ │ │ +00041050: 2036 2038 2020 2020 2020 2020 3720 3820   6 8        7 8 
│ │ │ │ +00041060: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000410a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000410b0: 2020 3220 2020 2020 2020 2032 2020 2020    2        2    
│ │ │ │ -000410c0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000410d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000410e0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ -000410f0: 2020 2020 2020 2020 2020 207c 0a7c 3335             |.|35
│ │ │ │ -00041100: 3478 202c 202d 2036 3430 7820 202d 2031  4x , - 640x  - 1
│ │ │ │ -00041110: 3535 3178 2078 2020 2b20 3436 3978 2020  551x x  + 469x  
│ │ │ │ -00041120: 2b20 3134 3832 7820 7820 202d 2031 3539  + 1482x x  - 159
│ │ │ │ -00041130: 3378 2078 2020 2d20 3938 3678 2020 2b20  3x x  - 986x  + 
│ │ │ │ -00041140: 3437 3178 2078 2020 2b20 207c 0a7c 2020  471x x  +  |.|  
│ │ │ │ -00041150: 2020 3820 2020 2020 2020 2030 2020 2020    8        0    
│ │ │ │ -00041160: 2020 2020 3020 3120 2020 2020 2020 3120      0 1       1 
│ │ │ │ -00041170: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ -00041180: 2020 3120 3220 2020 2020 2020 3220 2020    1 2       2   
│ │ │ │ -00041190: 2020 2020 3020 3320 2020 207c 0a7c 2d2d      0 3    |.|--
│ │ │ │ -000411a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000410a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000410b0: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │ +000410c0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +000410d0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +000410e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000410f0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00041100: 207c 0a7c 3335 3478 202c 202d 2036 3430   |.|354x , - 640
│ │ │ │ +00041110: 7820 202d 2031 3535 3178 2078 2020 2b20  x  - 1551x x  + 
│ │ │ │ +00041120: 3436 3978 2020 2b20 3134 3832 7820 7820  469x  + 1482x x 
│ │ │ │ +00041130: 202d 2031 3539 3378 2078 2020 2d20 3938   - 1593x x  - 98
│ │ │ │ +00041140: 3678 2020 2b20 3437 3178 2078 2020 2b20  6x  + 471x x  + 
│ │ │ │ +00041150: 207c 0a7c 2020 2020 3820 2020 2020 2020   |.|    8       
│ │ │ │ +00041160: 2030 2020 2020 2020 2020 3020 3120 2020   0        0 1   
│ │ │ │ +00041170: 2020 2020 3120 2020 2020 2020 2030 2032      1        0 2
│ │ │ │ +00041180: 2020 2020 2020 2020 3120 3220 2020 2020          1 2     
│ │ │ │ +00041190: 2020 3220 2020 2020 2020 3020 3320 2020    2       0 3   
│ │ │ │ +000411a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000411b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000411c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000411d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000411e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000411f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041200: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +000411e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000411f0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00041200: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00041210: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041220: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041230: 2020 2020 2020 2020 2032 207c 0a7c 3631           2 |.|61
│ │ │ │ -00041240: 3278 2078 2020 2b20 3132 3238 7820 7820  2x x  + 1228x x 
│ │ │ │ -00041250: 202b 2031 3135 3678 2020 2d20 3733 3178   + 1156x  - 731x
│ │ │ │ -00041260: 2078 2020 2b20 3135 3033 7820 7820 202d   x  + 1503x x  -
│ │ │ │ -00041270: 2036 3238 7820 7820 202b 2036 3734 7820   628x x  + 674x 
│ │ │ │ -00041280: 7820 202d 2037 3939 7820 207c 0a7c 2020  x  - 799x  |.|  
│ │ │ │ -00041290: 2020 3120 3320 2020 2020 2020 2032 2033    1 3        2 3
│ │ │ │ -000412a0: 2020 2020 2020 2020 3320 2020 2020 2020          3       
│ │ │ │ -000412b0: 3020 3420 2020 2020 2020 2031 2034 2020  0 4        1 4  
│ │ │ │ -000412c0: 2020 2020 2032 2034 2020 2020 2020 2033       2 4       3
│ │ │ │ -000412d0: 2034 2020 2020 2020 2034 207c 0a7c 2d2d   4       4 |.|--
│ │ │ │ -000412e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041230: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00041240: 207c 0a7c 3631 3278 2078 2020 2b20 3132   |.|612x x  + 12
│ │ │ │ +00041250: 3238 7820 7820 202b 2031 3135 3678 2020  28x x  + 1156x  
│ │ │ │ +00041260: 2d20 3733 3178 2078 2020 2b20 3135 3033  - 731x x  + 1503
│ │ │ │ +00041270: 7820 7820 202d 2036 3238 7820 7820 202b  x x  - 628x x  +
│ │ │ │ +00041280: 2036 3734 7820 7820 202d 2037 3939 7820   674x x  - 799x 
│ │ │ │ +00041290: 207c 0a7c 2020 2020 3120 3320 2020 2020   |.|    1 3     
│ │ │ │ +000412a0: 2020 2032 2033 2020 2020 2020 2020 3320     2 3        3 
│ │ │ │ +000412b0: 2020 2020 2020 3020 3420 2020 2020 2020        0 4       
│ │ │ │ +000412c0: 2031 2034 2020 2020 2020 2032 2034 2020   1 4       2 4  
│ │ │ │ +000412d0: 2020 2020 2033 2034 2020 2020 2020 2034       3 4       4
│ │ │ │ +000412e0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000412f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041330: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041330: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00041340: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041350: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041360: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ -00041370: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -00041380: 3131 3337 7820 7820 202b 2038 3434 7820  1137x x  + 844x 
│ │ │ │ -00041390: 7820 202b 2035 3839 7820 7820 202d 2036  x  + 589x x  - 6
│ │ │ │ -000413a0: 3636 7820 7820 202b 2038 3239 7820 7820  66x x  + 829x x 
│ │ │ │ -000413b0: 202d 2031 3032 3478 2020 2d20 3137 3078   - 1024x  - 170x
│ │ │ │ -000413c0: 2078 2020 2b20 2020 2020 207c 0a7c 2020   x  +      |.|  
│ │ │ │ -000413d0: 2020 2020 2030 2035 2020 2020 2020 2031       0 5       1
│ │ │ │ -000413e0: 2035 2020 2020 2020 2032 2035 2020 2020   5       2 5    
│ │ │ │ -000413f0: 2020 2033 2035 2020 2020 2020 2034 2035     3 5       4 5
│ │ │ │ -00041400: 2020 2020 2020 2020 3520 2020 2020 2020          5       
│ │ │ │ -00041410: 3020 3620 2020 2020 2020 207c 0a7c 2d2d  0 6        |.|--
│ │ │ │ -00041420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041360: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ +00041370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041380: 207c 0a7c 2b20 3131 3337 7820 7820 202b   |.|+ 1137x x  +
│ │ │ │ +00041390: 2038 3434 7820 7820 202b 2035 3839 7820   844x x  + 589x 
│ │ │ │ +000413a0: 7820 202d 2036 3636 7820 7820 202b 2038  x  - 666x x  + 8
│ │ │ │ +000413b0: 3239 7820 7820 202d 2031 3032 3478 2020  29x x  - 1024x  
│ │ │ │ +000413c0: 2d20 3137 3078 2078 2020 2b20 2020 2020  - 170x x  +     
│ │ │ │ +000413d0: 207c 0a7c 2020 2020 2020 2030 2035 2020   |.|       0 5  
│ │ │ │ +000413e0: 2020 2020 2031 2035 2020 2020 2020 2032       1 5       2
│ │ │ │ +000413f0: 2035 2020 2020 2020 2033 2035 2020 2020   5       3 5    
│ │ │ │ +00041400: 2020 2034 2035 2020 2020 2020 2020 3520     4 5        5 
│ │ │ │ +00041410: 2020 2020 2020 3020 3620 2020 2020 2020        0 6       
│ │ │ │ +00041420: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041470: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041470: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00041480: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041490: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000414a0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -000414b0: 2020 2020 2020 2020 2020 207c 0a7c 3435             |.|45
│ │ │ │ -000414c0: 3078 2078 2020 2b20 3134 3937 7820 7820  0x x  + 1497x x 
│ │ │ │ -000414d0: 202b 2031 3230 3478 2078 2020 2d20 3930   + 1204x x  - 90
│ │ │ │ -000414e0: 3778 2078 2020 2b20 3136 3231 7820 7820  7x x  + 1621x x 
│ │ │ │ -000414f0: 202d 2034 3137 7820 202b 2031 3239 3778   - 417x  + 1297x
│ │ │ │ -00041500: 2078 2020 2b20 2020 2020 207c 0a7c 2020   x  +      |.|  
│ │ │ │ -00041510: 2020 3120 3620 2020 2020 2020 2032 2036    1 6        2 6
│ │ │ │ -00041520: 2020 2020 2020 2020 3320 3620 2020 2020          3 6     
│ │ │ │ -00041530: 2020 3420 3620 2020 2020 2020 2035 2036    4 6        5 6
│ │ │ │ -00041540: 2020 2020 2020 2036 2020 2020 2020 2020         6        
│ │ │ │ -00041550: 3020 3720 2020 2020 2020 207c 0a7c 2d2d  0 7        |.|--
│ │ │ │ -00041560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000414a0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +000414b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000414c0: 207c 0a7c 3435 3078 2078 2020 2b20 3134   |.|450x x  + 14
│ │ │ │ +000414d0: 3937 7820 7820 202b 2031 3230 3478 2078  97x x  + 1204x x
│ │ │ │ +000414e0: 2020 2d20 3930 3778 2078 2020 2b20 3136    - 907x x  + 16
│ │ │ │ +000414f0: 3231 7820 7820 202d 2034 3137 7820 202b  21x x  - 417x  +
│ │ │ │ +00041500: 2031 3239 3778 2078 2020 2b20 2020 2020   1297x x  +     
│ │ │ │ +00041510: 207c 0a7c 2020 2020 3120 3620 2020 2020   |.|    1 6     
│ │ │ │ +00041520: 2020 2032 2036 2020 2020 2020 2020 3320     2 6        3 
│ │ │ │ +00041530: 3620 2020 2020 2020 3420 3620 2020 2020  6       4 6     
│ │ │ │ +00041540: 2020 2035 2036 2020 2020 2020 2036 2020     5 6       6  
│ │ │ │ +00041550: 2020 2020 2020 3020 3720 2020 2020 2020        0 7       
│ │ │ │ +00041560: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000415a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000415b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000415a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000415b0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  000415c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000415d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000415e0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000415f0: 2020 2020 2020 2020 2020 207c 0a7c 3134             |.|14
│ │ │ │ -00041600: 3434 7820 7820 202b 2034 7820 7820 202b  44x x  + 4x x  +
│ │ │ │ -00041610: 2033 3938 7820 7820 202b 2039 3936 7820   398x x  + 996x 
│ │ │ │ -00041620: 7820 202d 2031 3033 3178 2078 2020 2b20  x  - 1031x x  + 
│ │ │ │ -00041630: 3233 3978 2078 2020 2b20 3330 3378 2020  239x x  + 303x  
│ │ │ │ -00041640: 2b20 3132 3135 7820 7820 207c 0a7c 2020  + 1215x x  |.|  
│ │ │ │ -00041650: 2020 2031 2037 2020 2020 2032 2037 2020     1 7     2 7  
│ │ │ │ -00041660: 2020 2020 2033 2037 2020 2020 2020 2034       3 7       4
│ │ │ │ -00041670: 2037 2020 2020 2020 2020 3520 3720 2020   7        5 7   
│ │ │ │ -00041680: 2020 2020 3620 3720 2020 2020 2020 3720      6 7       7 
│ │ │ │ -00041690: 2020 2020 2020 2030 2038 207c 0a7c 2d2d         0 8 |.|--
│ │ │ │ -000416a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000415e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000415f0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00041600: 207c 0a7c 3134 3434 7820 7820 202b 2034   |.|1444x x  + 4
│ │ │ │ +00041610: 7820 7820 202b 2033 3938 7820 7820 202b  x x  + 398x x  +
│ │ │ │ +00041620: 2039 3936 7820 7820 202d 2031 3033 3178   996x x  - 1031x
│ │ │ │ +00041630: 2078 2020 2b20 3233 3978 2078 2020 2b20   x  + 239x x  + 
│ │ │ │ +00041640: 3330 3378 2020 2b20 3132 3135 7820 7820  303x  + 1215x x 
│ │ │ │ +00041650: 207c 0a7c 2020 2020 2031 2037 2020 2020   |.|     1 7    
│ │ │ │ +00041660: 2032 2037 2020 2020 2020 2033 2037 2020   2 7       3 7  
│ │ │ │ +00041670: 2020 2020 2034 2037 2020 2020 2020 2020       4 7        
│ │ │ │ +00041680: 3520 3720 2020 2020 2020 3620 3720 2020  5 7       6 7   
│ │ │ │ +00041690: 2020 2020 3720 2020 2020 2020 2030 2038      7        0 8
│ │ │ │ +000416a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000416b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000416c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000416d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000416e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000416f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000416e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000416f0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00041700: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041710: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041720: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041730: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -00041740: 3833 7820 7820 202b 2031 3537 3178 2078  83x x  + 1571x x
│ │ │ │ -00041750: 2020 2d20 3135 3433 7820 7820 202d 2039    - 1543x x  - 9
│ │ │ │ -00041760: 3235 7820 7820 202d 2036 3934 7820 7820  25x x  - 694x x 
│ │ │ │ -00041770: 202b 2031 3531 7820 7820 202d 2035 3230   + 151x x  - 520
│ │ │ │ -00041780: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -00041790: 2020 2031 2038 2020 2020 2020 2020 3220     1 8        2 
│ │ │ │ -000417a0: 3820 2020 2020 2020 2033 2038 2020 2020  8        3 8    
│ │ │ │ -000417b0: 2020 2034 2038 2020 2020 2020 2035 2038     4 8       5 8
│ │ │ │ -000417c0: 2020 2020 2020 2036 2038 2020 2020 2020         6 8      
│ │ │ │ -000417d0: 2037 2038 2020 2020 2020 207c 0a7c 2d2d   7 8       |.|--
│ │ │ │ -000417e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041730: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041740: 207c 0a7c 2d20 3833 7820 7820 202b 2031   |.|- 83x x  + 1
│ │ │ │ +00041750: 3537 3178 2078 2020 2d20 3135 3433 7820  571x x  - 1543x 
│ │ │ │ +00041760: 7820 202d 2039 3235 7820 7820 202d 2036  x  - 925x x  - 6
│ │ │ │ +00041770: 3934 7820 7820 202b 2031 3531 7820 7820  94x x  + 151x x 
│ │ │ │ +00041780: 202d 2035 3230 7820 7820 202b 2020 2020   - 520x x  +    
│ │ │ │ +00041790: 207c 0a7c 2020 2020 2031 2038 2020 2020   |.|     1 8    
│ │ │ │ +000417a0: 2020 2020 3220 3820 2020 2020 2020 2033      2 8        3
│ │ │ │ +000417b0: 2038 2020 2020 2020 2034 2038 2020 2020   8       4 8    
│ │ │ │ +000417c0: 2020 2035 2038 2020 2020 2020 2036 2038     5 8       6 8
│ │ │ │ +000417d0: 2020 2020 2020 2037 2038 2020 2020 2020         7 8      
│ │ │ │ +000417e0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000417f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041830: 2020 3220 2020 2020 2020 2020 3220 2020    2         2   
│ │ │ │ -00041840: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00041850: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041860: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ -00041870: 2020 2020 2020 2020 2020 207c 0a7c 3838             |.|88
│ │ │ │ -00041880: 3078 202c 202d 2031 3231 3078 2020 2d20  0x , - 1210x  - 
│ │ │ │ -00041890: 3232 3278 2078 2020 2b20 3138 3578 2020  222x x  + 185x  
│ │ │ │ -000418a0: 2b20 3234 3578 2078 2020 2b20 3130 3539  + 245x x  + 1059
│ │ │ │ -000418b0: 7820 7820 202d 2033 3232 7820 202b 2032  x x  - 322x  + 2
│ │ │ │ -000418c0: 3338 7820 7820 202b 2020 207c 0a7c 2020  38x x  +   |.|  
│ │ │ │ -000418d0: 2020 3820 2020 2020 2020 2020 3020 2020    8         0   
│ │ │ │ -000418e0: 2020 2020 3020 3120 2020 2020 2020 3120      0 1       1 
│ │ │ │ -000418f0: 2020 2020 2020 3020 3220 2020 2020 2020        0 2       
│ │ │ │ -00041900: 2031 2032 2020 2020 2020 2032 2020 2020   1 2       2    
│ │ │ │ -00041910: 2020 2030 2033 2020 2020 207c 0a7c 2d2d     0 3     |.|--
│ │ │ │ -00041920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041830: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │ +00041840: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00041850: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00041860: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041870: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00041880: 207c 0a7c 3838 3078 202c 202d 2031 3231   |.|880x , - 121
│ │ │ │ +00041890: 3078 2020 2d20 3232 3278 2078 2020 2b20  0x  - 222x x  + 
│ │ │ │ +000418a0: 3138 3578 2020 2b20 3234 3578 2078 2020  185x  + 245x x  
│ │ │ │ +000418b0: 2b20 3130 3539 7820 7820 202d 2033 3232  + 1059x x  - 322
│ │ │ │ +000418c0: 7820 202b 2032 3338 7820 7820 202b 2020  x  + 238x x  +  
│ │ │ │ +000418d0: 207c 0a7c 2020 2020 3820 2020 2020 2020   |.|    8       
│ │ │ │ +000418e0: 2020 3020 2020 2020 2020 3020 3120 2020    0       0 1   
│ │ │ │ +000418f0: 2020 2020 3120 2020 2020 2020 3020 3220      1       0 2 
│ │ │ │ +00041900: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +00041910: 2032 2020 2020 2020 2030 2033 2020 2020   2       0 3    
│ │ │ │ +00041920: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041970: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041980: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00041960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041970: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00041980: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00041990: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000419a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000419b0: 2020 2020 2020 2020 2020 207c 0a7c 3936             |.|96
│ │ │ │ -000419c0: 3278 2078 2020 2b20 3132 3630 7820 7820  2x x  + 1260x x 
│ │ │ │ -000419d0: 202d 2031 3538 3178 2020 2b20 3530 7820   - 1581x  + 50x 
│ │ │ │ -000419e0: 7820 202b 2031 3335 3278 2078 2020 2d20  x  + 1352x x  - 
│ │ │ │ -000419f0: 3134 3635 7820 7820 202b 2031 3535 3578  1465x x  + 1555x
│ │ │ │ -00041a00: 2078 2020 2b20 2020 2020 207c 0a7c 2020   x  +      |.|  
│ │ │ │ -00041a10: 2020 3120 3320 2020 2020 2020 2032 2033    1 3        2 3
│ │ │ │ -00041a20: 2020 2020 2020 2020 3320 2020 2020 2030          3      0
│ │ │ │ -00041a30: 2034 2020 2020 2020 2020 3120 3420 2020   4        1 4   
│ │ │ │ -00041a40: 2020 2020 2032 2034 2020 2020 2020 2020       2 4        
│ │ │ │ -00041a50: 3320 3420 2020 2020 2020 207c 0a7c 2d2d  3 4        |.|--
│ │ │ │ -00041a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000419b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000419c0: 207c 0a7c 3936 3278 2078 2020 2b20 3132   |.|962x x  + 12
│ │ │ │ +000419d0: 3630 7820 7820 202d 2031 3538 3178 2020  60x x  - 1581x  
│ │ │ │ +000419e0: 2b20 3530 7820 7820 202b 2031 3335 3278  + 50x x  + 1352x
│ │ │ │ +000419f0: 2078 2020 2d20 3134 3635 7820 7820 202b   x  - 1465x x  +
│ │ │ │ +00041a00: 2031 3535 3578 2078 2020 2b20 2020 2020   1555x x  +     
│ │ │ │ +00041a10: 207c 0a7c 2020 2020 3120 3320 2020 2020   |.|    1 3     
│ │ │ │ +00041a20: 2020 2032 2033 2020 2020 2020 2020 3320     2 3        3 
│ │ │ │ +00041a30: 2020 2020 2030 2034 2020 2020 2020 2020       0 4        
│ │ │ │ +00041a40: 3120 3420 2020 2020 2020 2032 2034 2020  1 4        2 4  
│ │ │ │ +00041a50: 2020 2020 2020 3320 3420 2020 2020 2020        3 4       
│ │ │ │ +00041a60: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041ab0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00041aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041ab0: 2d7c 0a7c 2020 2020 2032 2020 2020 2020  -|.|     2      
│ │ │ │  00041ac0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ad0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ae0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041af0: 2032 2020 2020 2020 2020 207c 0a7c 3133   2         |.|13
│ │ │ │ -00041b00: 3333 7820 202b 2031 3336 3278 2078 2020  33x  + 1362x x  
│ │ │ │ -00041b10: 2b20 3133 3635 7820 7820 202b 2031 3136  + 1365x x  + 116
│ │ │ │ -00041b20: 3878 2078 2020 2d20 3134 3031 7820 7820  8x x  - 1401x x 
│ │ │ │ -00041b30: 202b 2031 3439 7820 7820 202d 2036 3532   + 149x x  - 652
│ │ │ │ -00041b40: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -00041b50: 2020 2034 2020 2020 2020 2020 3020 3520     4        0 5 
│ │ │ │ -00041b60: 2020 2020 2020 2031 2035 2020 2020 2020         1 5      
│ │ │ │ -00041b70: 2020 3220 3520 2020 2020 2020 2033 2035    2 5        3 5
│ │ │ │ -00041b80: 2020 2020 2020 2034 2035 2020 2020 2020         4 5      
│ │ │ │ -00041b90: 2035 2020 2020 2020 2020 207c 0a7c 2d2d   5         |.|--
│ │ │ │ -00041ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041af0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +00041b00: 207c 0a7c 3133 3333 7820 202b 2031 3336   |.|1333x  + 136
│ │ │ │ +00041b10: 3278 2078 2020 2b20 3133 3635 7820 7820  2x x  + 1365x x 
│ │ │ │ +00041b20: 202b 2031 3136 3878 2078 2020 2d20 3134   + 1168x x  - 14
│ │ │ │ +00041b30: 3031 7820 7820 202b 2031 3439 7820 7820  01x x  + 149x x 
│ │ │ │ +00041b40: 202d 2036 3532 7820 202b 2020 2020 2020   - 652x  +      
│ │ │ │ +00041b50: 207c 0a7c 2020 2020 2034 2020 2020 2020   |.|     4      
│ │ │ │ +00041b60: 2020 3020 3520 2020 2020 2020 2031 2035    0 5        1 5
│ │ │ │ +00041b70: 2020 2020 2020 2020 3220 3520 2020 2020          2 5     
│ │ │ │ +00041b80: 2020 2033 2035 2020 2020 2020 2034 2035     3 5       4 5
│ │ │ │ +00041b90: 2020 2020 2020 2035 2020 2020 2020 2020         5        
│ │ │ │ +00041ba0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041bf0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041bf0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00041c00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041c10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041c20: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00041c30: 2020 2020 2020 2020 2020 207c 0a7c 3133             |.|13
│ │ │ │ -00041c40: 3738 7820 7820 202d 2035 3537 7820 7820  78x x  - 557x x 
│ │ │ │ -00041c50: 202d 2031 3132 7820 7820 202b 2032 3678   - 112x x  + 26x
│ │ │ │ -00041c60: 2078 2020 2b20 3331 3578 2078 2020 2b20   x  + 315x x  + 
│ │ │ │ -00041c70: 3131 3178 2078 2020 2b20 3135 3932 7820  111x x  + 1592x 
│ │ │ │ -00041c80: 202d 2032 3833 7820 7820 207c 0a7c 2020   - 283x x  |.|  
│ │ │ │ -00041c90: 2020 2030 2036 2020 2020 2020 2031 2036     0 6       1 6
│ │ │ │ -00041ca0: 2020 2020 2020 2032 2036 2020 2020 2020         2 6      
│ │ │ │ -00041cb0: 3320 3620 2020 2020 2020 3420 3620 2020  3 6       4 6   
│ │ │ │ -00041cc0: 2020 2020 3520 3620 2020 2020 2020 2036      5 6        6
│ │ │ │ -00041cd0: 2020 2020 2020 2030 2037 207c 0a7c 2d2d         0 7 |.|--
│ │ │ │ -00041ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041c20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041c30: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00041c40: 207c 0a7c 3133 3738 7820 7820 202d 2035   |.|1378x x  - 5
│ │ │ │ +00041c50: 3537 7820 7820 202d 2031 3132 7820 7820  57x x  - 112x x 
│ │ │ │ +00041c60: 202b 2032 3678 2078 2020 2b20 3331 3578   + 26x x  + 315x
│ │ │ │ +00041c70: 2078 2020 2b20 3131 3178 2078 2020 2b20   x  + 111x x  + 
│ │ │ │ +00041c80: 3135 3932 7820 202d 2032 3833 7820 7820  1592x  - 283x x 
│ │ │ │ +00041c90: 207c 0a7c 2020 2020 2030 2036 2020 2020   |.|     0 6    
│ │ │ │ +00041ca0: 2020 2031 2036 2020 2020 2020 2032 2036     1 6       2 6
│ │ │ │ +00041cb0: 2020 2020 2020 3320 3620 2020 2020 2020        3 6       
│ │ │ │ +00041cc0: 3420 3620 2020 2020 2020 3520 3620 2020  4 6       5 6   
│ │ │ │ +00041cd0: 2020 2020 2036 2020 2020 2020 2030 2037       6       0 7
│ │ │ │ +00041ce0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041d30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041d30: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00041d40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041d60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041d70: 2020 2032 2020 2020 2020 207c 0a7c 2d20     2       |.|- 
│ │ │ │ -00041d80: 3134 3534 7820 7820 202b 2039 3037 7820  1454x x  + 907x 
│ │ │ │ -00041d90: 7820 202b 2032 3132 7820 7820 202b 2034  x  + 212x x  + 4
│ │ │ │ -00041da0: 3030 7820 7820 202b 2031 3034 3978 2078  00x x  + 1049x x
│ │ │ │ -00041db0: 2020 2d20 3838 3278 2078 2020 2d20 3134    - 882x x  - 14
│ │ │ │ -00041dc0: 3239 7820 202d 2020 2020 207c 0a7c 2020  29x  -     |.|  
│ │ │ │ -00041dd0: 2020 2020 2031 2037 2020 2020 2020 2032       1 7       2
│ │ │ │ -00041de0: 2037 2020 2020 2020 2033 2037 2020 2020   7       3 7    
│ │ │ │ -00041df0: 2020 2034 2037 2020 2020 2020 2020 3520     4 7        5 
│ │ │ │ -00041e00: 3720 2020 2020 2020 3620 3720 2020 2020  7       6 7     
│ │ │ │ -00041e10: 2020 2037 2020 2020 2020 207c 0a7c 2d2d     7       |.|--
│ │ │ │ -00041e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041d70: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00041d80: 207c 0a7c 2d20 3134 3534 7820 7820 202b   |.|- 1454x x  +
│ │ │ │ +00041d90: 2039 3037 7820 7820 202b 2032 3132 7820   907x x  + 212x 
│ │ │ │ +00041da0: 7820 202b 2034 3030 7820 7820 202b 2031  x  + 400x x  + 1
│ │ │ │ +00041db0: 3034 3978 2078 2020 2d20 3838 3278 2078  049x x  - 882x x
│ │ │ │ +00041dc0: 2020 2d20 3134 3239 7820 202d 2020 2020    - 1429x  -    
│ │ │ │ +00041dd0: 207c 0a7c 2020 2020 2020 2031 2037 2020   |.|       1 7  
│ │ │ │ +00041de0: 2020 2020 2032 2037 2020 2020 2020 2033       2 7       3
│ │ │ │ +00041df0: 2037 2020 2020 2020 2034 2037 2020 2020   7       4 7    
│ │ │ │ +00041e00: 2020 2020 3520 3720 2020 2020 2020 3620      5 7       6 
│ │ │ │ +00041e10: 3720 2020 2020 2020 2037 2020 2020 2020  7        7      
│ │ │ │ +00041e20: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041e70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041e70: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00041e80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041e90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00041ea0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041eb0: 2020 2020 2020 2020 2020 207c 0a7c 3138             |.|18
│ │ │ │ -00041ec0: 3378 2078 2020 2b20 3135 3731 7820 7820  3x x  + 1571x x 
│ │ │ │ -00041ed0: 202d 2031 3238 3678 2078 2020 2d20 3131   - 1286x x  - 11
│ │ │ │ -00041ee0: 3739 7820 7820 202b 2031 3331 3978 2078  79x x  + 1319x x
│ │ │ │ -00041ef0: 2020 2b20 3234 3078 2078 2020 2d20 3131    + 240x x  - 11
│ │ │ │ -00041f00: 3030 7820 7820 202b 2020 207c 0a7c 2020  00x x  +   |.|  
│ │ │ │ -00041f10: 2020 3020 3820 2020 2020 2020 2031 2038    0 8        1 8
│ │ │ │ -00041f20: 2020 2020 2020 2020 3220 3820 2020 2020          2 8     
│ │ │ │ -00041f30: 2020 2033 2038 2020 2020 2020 2020 3420     3 8        4 
│ │ │ │ -00041f40: 3820 2020 2020 2020 3520 3820 2020 2020  8       5 8     
│ │ │ │ -00041f50: 2020 2036 2038 2020 2020 207c 0a7c 2d2d     6 8     |.|--
│ │ │ │ -00041f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00041ec0: 207c 0a7c 3138 3378 2078 2020 2b20 3135   |.|183x x  + 15
│ │ │ │ +00041ed0: 3731 7820 7820 202d 2031 3238 3678 2078  71x x  - 1286x x
│ │ │ │ +00041ee0: 2020 2d20 3131 3739 7820 7820 202b 2031    - 1179x x  + 1
│ │ │ │ +00041ef0: 3331 3978 2078 2020 2b20 3234 3078 2078  319x x  + 240x x
│ │ │ │ +00041f00: 2020 2d20 3131 3030 7820 7820 202b 2020    - 1100x x  +  
│ │ │ │ +00041f10: 207c 0a7c 2020 2020 3020 3820 2020 2020   |.|    0 8     
│ │ │ │ +00041f20: 2020 2031 2038 2020 2020 2020 2020 3220     1 8        2 
│ │ │ │ +00041f30: 3820 2020 2020 2020 2033 2038 2020 2020  8        3 8    
│ │ │ │ +00041f40: 2020 2020 3420 3820 2020 2020 2020 3520      4 8       5 
│ │ │ │ +00041f50: 3820 2020 2020 2020 2036 2038 2020 2020  8        6 8    
│ │ │ │ +00041f60: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00041f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00041f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00041fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00041fb0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00041fc0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00041fd0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00041fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041fb0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00041fc0: 2020 2032 2020 2020 2020 2032 2020 2020     2       2    
│ │ │ │ +00041fd0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │  00041fe0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00041ff0: 2020 2020 2020 3220 2020 207c 0a7c 3135        2    |.|15
│ │ │ │ -00042000: 3030 7820 7820 202d 2033 3438 7820 2c20  00x x  - 348x , 
│ │ │ │ -00042010: 3130 3531 7820 202d 2031 3332 3578 2078  1051x  - 1325x x
│ │ │ │ -00042020: 2020 2b20 3133 3534 7820 202d 2033 3436    + 1354x  - 346
│ │ │ │ -00042030: 7820 7820 202d 2031 3533 3278 2078 2020  x x  - 1532x x  
│ │ │ │ -00042040: 2d20 3436 3678 2020 2b20 207c 0a7c 2020  - 466x  +  |.|  
│ │ │ │ -00042050: 2020 2037 2038 2020 2020 2020 2038 2020     7 8       8  
│ │ │ │ -00042060: 2020 2020 2030 2020 2020 2020 2020 3020       0        0 
│ │ │ │ -00042070: 3120 2020 2020 2020 2031 2020 2020 2020  1        1      
│ │ │ │ -00042080: 2030 2032 2020 2020 2020 2020 3120 3220   0 2        1 2 
│ │ │ │ -00042090: 2020 2020 2020 3220 2020 207c 0a7c 2d2d        2    |.|--
│ │ │ │ -000420a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00041ff0: 2020 2020 2020 2020 2020 2020 3220 2020              2   
│ │ │ │ +00042000: 207c 0a7c 3135 3030 7820 7820 202d 2033   |.|1500x x  - 3
│ │ │ │ +00042010: 3438 7820 2c20 3130 3531 7820 202d 2031  48x , 1051x  - 1
│ │ │ │ +00042020: 3332 3578 2078 2020 2b20 3133 3534 7820  325x x  + 1354x 
│ │ │ │ +00042030: 202d 2033 3436 7820 7820 202d 2031 3533   - 346x x  - 153
│ │ │ │ +00042040: 3278 2078 2020 2d20 3436 3678 2020 2b20  2x x  - 466x  + 
│ │ │ │ +00042050: 207c 0a7c 2020 2020 2037 2038 2020 2020   |.|     7 8    
│ │ │ │ +00042060: 2020 2038 2020 2020 2020 2030 2020 2020     8       0    
│ │ │ │ +00042070: 2020 2020 3020 3120 2020 2020 2020 2031      0 1        1
│ │ │ │ +00042080: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ +00042090: 2020 3120 3220 2020 2020 2020 3220 2020    1 2       2   
│ │ │ │ +000420a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000420b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000420c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000420d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000420e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000420f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000420e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000420f0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00042100: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042110: 3220 2020 2020 2020 2020 2020 2020 2020  2               
│ │ │ │ +00042110: 2020 2020 2020 3220 2020 2020 2020 2020        2         
│ │ │ │  00042120: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042130: 2020 2020 2020 2020 2020 207c 0a7c 3136             |.|16
│ │ │ │ -00042140: 3378 2078 2020 2d20 3635 3978 2078 2020  3x x  - 659x x  
│ │ │ │ -00042150: 2d20 3239 3178 2078 2020 2b20 3936 3678  - 291x x  + 966x
│ │ │ │ -00042160: 2020 2b20 3738 3978 2078 2020 2b20 3339    + 789x x  + 39
│ │ │ │ -00042170: 3378 2078 2020 2b20 3430 3378 2078 2020  3x x  + 403x x  
│ │ │ │ -00042180: 2d20 3131 3939 7820 7820 207c 0a7c 2020  - 1199x x  |.|  
│ │ │ │ -00042190: 2020 3020 3320 2020 2020 2020 3120 3320    0 3       1 3 
│ │ │ │ -000421a0: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ -000421b0: 3320 2020 2020 2020 3020 3420 2020 2020  3       0 4     
│ │ │ │ -000421c0: 2020 3120 3420 2020 2020 2020 3220 3420    1 4       2 4 
│ │ │ │ -000421d0: 2020 2020 2020 2033 2034 207c 0a7c 2d2d         3 4 |.|--
│ │ │ │ -000421e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042130: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042140: 207c 0a7c 3136 3378 2078 2020 2d20 3635   |.|163x x  - 65
│ │ │ │ +00042150: 3978 2078 2020 2d20 3239 3178 2078 2020  9x x  - 291x x  
│ │ │ │ +00042160: 2b20 3936 3678 2020 2b20 3738 3978 2078  + 966x  + 789x x
│ │ │ │ +00042170: 2020 2b20 3339 3378 2078 2020 2b20 3430    + 393x x  + 40
│ │ │ │ +00042180: 3378 2078 2020 2d20 3131 3939 7820 7820  3x x  - 1199x x 
│ │ │ │ +00042190: 207c 0a7c 2020 2020 3020 3320 2020 2020   |.|    0 3     
│ │ │ │ +000421a0: 2020 3120 3320 2020 2020 2020 3220 3320    1 3       2 3 
│ │ │ │ +000421b0: 2020 2020 2020 3320 2020 2020 2020 3020        3       0 
│ │ │ │ +000421c0: 3420 2020 2020 2020 3120 3420 2020 2020  4       1 4     
│ │ │ │ +000421d0: 2020 3220 3420 2020 2020 2020 2033 2034    2 4        3 4
│ │ │ │ +000421e0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000421f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042230: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │ +00042220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042230: 2d7c 0a7c 2020 2020 2020 3220 2020 2020  -|.|      2     
│ │ │ │  00042240: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042250: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042260: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00042270: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -00042280: 3537 3078 2020 2d20 3933 7820 7820 202d  570x  - 93x x  -
│ │ │ │ -00042290: 2034 3932 7820 7820 202d 2034 3138 7820   492x x  - 418x 
│ │ │ │ -000422a0: 7820 202b 2037 3133 7820 7820 202d 2031  x  + 713x x  - 1
│ │ │ │ -000422b0: 3332 3378 2078 2020 2d20 3133 3834 7820  323x x  - 1384x 
│ │ │ │ -000422c0: 202d 2038 3330 7820 7820 207c 0a7c 2020   - 830x x  |.|  
│ │ │ │ -000422d0: 2020 2020 3420 2020 2020 2030 2035 2020      4      0 5  
│ │ │ │ -000422e0: 2020 2020 2031 2035 2020 2020 2020 2032       1 5       2
│ │ │ │ -000422f0: 2035 2020 2020 2020 2033 2035 2020 2020   5       3 5    
│ │ │ │ -00042300: 2020 2020 3420 3520 2020 2020 2020 2035      4 5        5
│ │ │ │ -00042310: 2020 2020 2020 2030 2036 207c 0a7c 2d2d         0 6 |.|--
│ │ │ │ -00042320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042260: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042270: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00042280: 207c 0a7c 2d20 3537 3078 2020 2d20 3933   |.|- 570x  - 93
│ │ │ │ +00042290: 7820 7820 202d 2034 3932 7820 7820 202d  x x  - 492x x  -
│ │ │ │ +000422a0: 2034 3138 7820 7820 202b 2037 3133 7820   418x x  + 713x 
│ │ │ │ +000422b0: 7820 202d 2031 3332 3378 2078 2020 2d20  x  - 1323x x  - 
│ │ │ │ +000422c0: 3133 3834 7820 202d 2038 3330 7820 7820  1384x  - 830x x 
│ │ │ │ +000422d0: 207c 0a7c 2020 2020 2020 3420 2020 2020   |.|      4     
│ │ │ │ +000422e0: 2030 2035 2020 2020 2020 2031 2035 2020   0 5       1 5  
│ │ │ │ +000422f0: 2020 2020 2032 2035 2020 2020 2020 2033       2 5       3
│ │ │ │ +00042300: 2035 2020 2020 2020 2020 3420 3520 2020   5        4 5   
│ │ │ │ +00042310: 2020 2020 2035 2020 2020 2020 2030 2036       5       0 6
│ │ │ │ +00042320: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00042330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042370: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042370: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00042380: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042390: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000423a0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -000423b0: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -000423c0: 3534 7820 7820 202d 2033 3036 7820 7820  54x x  - 306x x 
│ │ │ │ -000423d0: 202b 2037 3039 7820 7820 202b 2034 3231   + 709x x  + 421
│ │ │ │ -000423e0: 7820 7820 202d 2039 3534 7820 7820 202d  x x  - 954x x  -
│ │ │ │ -000423f0: 2032 3939 7820 202b 2031 3035 3378 2078   299x  + 1053x x
│ │ │ │ -00042400: 2020 2d20 2020 2020 2020 207c 0a7c 2020    -        |.|  
│ │ │ │ -00042410: 2020 2031 2036 2020 2020 2020 2032 2036     1 6       2 6
│ │ │ │ -00042420: 2020 2020 2020 2033 2036 2020 2020 2020         3 6      
│ │ │ │ -00042430: 2034 2036 2020 2020 2020 2035 2036 2020   4 6       5 6  
│ │ │ │ -00042440: 2020 2020 2036 2020 2020 2020 2020 3020       6        0 
│ │ │ │ -00042450: 3720 2020 2020 2020 2020 207c 0a7c 2d2d  7          |.|--
│ │ │ │ -00042460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000423a0: 2020 2020 2020 2020 2020 2032 2020 2020             2    
│ │ │ │ +000423b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000423c0: 207c 0a7c 2d20 3534 7820 7820 202d 2033   |.|- 54x x  - 3
│ │ │ │ +000423d0: 3036 7820 7820 202b 2037 3039 7820 7820  06x x  + 709x x 
│ │ │ │ +000423e0: 202b 2034 3231 7820 7820 202d 2039 3534   + 421x x  - 954
│ │ │ │ +000423f0: 7820 7820 202d 2032 3939 7820 202b 2031  x x  - 299x  + 1
│ │ │ │ +00042400: 3035 3378 2078 2020 2d20 2020 2020 2020  053x x  -       
│ │ │ │ +00042410: 207c 0a7c 2020 2020 2031 2036 2020 2020   |.|     1 6    
│ │ │ │ +00042420: 2020 2032 2036 2020 2020 2020 2033 2036     2 6       3 6
│ │ │ │ +00042430: 2020 2020 2020 2034 2036 2020 2020 2020         4 6      
│ │ │ │ +00042440: 2035 2036 2020 2020 2020 2036 2020 2020   5 6       6    
│ │ │ │ +00042450: 2020 2020 3020 3720 2020 2020 2020 2020      0 7         
│ │ │ │ +00042460: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00042470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000424a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000424b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000424a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000424b0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  000424c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000424d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000424e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000424f0: 2020 2032 2020 2020 2020 207c 0a7c 3130     2       |.|10
│ │ │ │ -00042500: 3830 7820 7820 202b 2036 3836 7820 7820  80x x  + 686x x 
│ │ │ │ -00042510: 202b 2031 3730 7820 7820 202d 2031 3237   + 170x x  - 127
│ │ │ │ -00042520: 3278 2078 2020 2d20 3136 3631 7820 7820  2x x  - 1661x x 
│ │ │ │ -00042530: 202b 2031 3233 3578 2078 2020 2b20 3135   + 1235x x  + 15
│ │ │ │ -00042540: 3533 7820 202d 2020 2020 207c 0a7c 2020  53x  -     |.|  
│ │ │ │ -00042550: 2020 2031 2037 2020 2020 2020 2032 2037     1 7       2 7
│ │ │ │ -00042560: 2020 2020 2020 2033 2037 2020 2020 2020         3 7      
│ │ │ │ -00042570: 2020 3420 3720 2020 2020 2020 2035 2037    4 7        5 7
│ │ │ │ -00042580: 2020 2020 2020 2020 3620 3720 2020 2020          6 7     
│ │ │ │ -00042590: 2020 2037 2020 2020 2020 207c 0a7c 2d2d     7       |.|--
│ │ │ │ -000425a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000424f0: 2020 2020 2020 2020 2032 2020 2020 2020           2      
│ │ │ │ +00042500: 207c 0a7c 3130 3830 7820 7820 202b 2036   |.|1080x x  + 6
│ │ │ │ +00042510: 3836 7820 7820 202b 2031 3730 7820 7820  86x x  + 170x x 
│ │ │ │ +00042520: 202d 2031 3237 3278 2078 2020 2d20 3136   - 1272x x  - 16
│ │ │ │ +00042530: 3631 7820 7820 202b 2031 3233 3578 2078  61x x  + 1235x x
│ │ │ │ +00042540: 2020 2b20 3135 3533 7820 202d 2020 2020    + 1553x  -    
│ │ │ │ +00042550: 207c 0a7c 2020 2020 2031 2037 2020 2020   |.|     1 7    
│ │ │ │ +00042560: 2020 2032 2037 2020 2020 2020 2033 2037     2 7       3 7
│ │ │ │ +00042570: 2020 2020 2020 2020 3420 3720 2020 2020          4 7     
│ │ │ │ +00042580: 2020 2035 2037 2020 2020 2020 2020 3620     5 7        6 
│ │ │ │ +00042590: 3720 2020 2020 2020 2037 2020 2020 2020  7        7      
│ │ │ │ +000425a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000425b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000425c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000425d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000425e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000425f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000425e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000425f0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00042600: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042610: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042620: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042630: 2020 2020 2020 2020 2020 207c 0a7c 3134             |.|14
│ │ │ │ -00042640: 3534 7820 7820 202d 2031 3431 3178 2078  54x x  - 1411x x
│ │ │ │ -00042650: 2020 2d20 3131 3935 7820 7820 202d 2039    - 1195x x  - 9
│ │ │ │ -00042660: 3632 7820 7820 202b 2037 3337 7820 7820  62x x  + 737x x 
│ │ │ │ -00042670: 202d 2033 3930 7820 7820 202b 2039 3537   - 390x x  + 957
│ │ │ │ -00042680: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -00042690: 2020 2030 2038 2020 2020 2020 2020 3120     0 8        1 
│ │ │ │ -000426a0: 3820 2020 2020 2020 2032 2038 2020 2020  8        2 8    
│ │ │ │ -000426b0: 2020 2033 2038 2020 2020 2020 2034 2038     3 8       4 8
│ │ │ │ -000426c0: 2020 2020 2020 2035 2038 2020 2020 2020         5 8      
│ │ │ │ -000426d0: 2036 2038 2020 2020 2020 207c 0a7c 2d2d   6 8       |.|--
│ │ │ │ -000426e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042630: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042640: 207c 0a7c 3134 3534 7820 7820 202d 2031   |.|1454x x  - 1
│ │ │ │ +00042650: 3431 3178 2078 2020 2d20 3131 3935 7820  411x x  - 1195x 
│ │ │ │ +00042660: 7820 202d 2039 3632 7820 7820 202b 2037  x  - 962x x  + 7
│ │ │ │ +00042670: 3337 7820 7820 202d 2033 3930 7820 7820  37x x  - 390x x 
│ │ │ │ +00042680: 202b 2039 3537 7820 7820 202b 2020 2020   + 957x x  +    
│ │ │ │ +00042690: 207c 0a7c 2020 2020 2030 2038 2020 2020   |.|     0 8    
│ │ │ │ +000426a0: 2020 2020 3120 3820 2020 2020 2020 2032      1 8        2
│ │ │ │ +000426b0: 2038 2020 2020 2020 2033 2038 2020 2020   8       3 8    
│ │ │ │ +000426c0: 2020 2034 2038 2020 2020 2020 2035 2038     4 8       5 8
│ │ │ │ +000426d0: 2020 2020 2020 2036 2038 2020 2020 2020         6 8      
│ │ │ │ +000426e0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000426f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042730: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00042740: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ -00042750: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00042720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042730: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00042740: 2020 2020 3220 2020 2020 2020 2032 2020      2        2  
│ │ │ │ +00042750: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00042760: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042770: 2020 2020 3220 2020 2020 207c 0a7c 3135      2      |.|15
│ │ │ │ -00042780: 3338 7820 7820 202b 2031 3233 3478 202c  38x x  + 1234x ,
│ │ │ │ -00042790: 202d 2035 3039 7820 202b 2039 7820 7820   - 509x  + 9x x 
│ │ │ │ -000427a0: 202d 2031 3536 3378 2020 2d20 3731 3078   - 1563x  - 710x
│ │ │ │ -000427b0: 2078 2020 2d20 3634 3278 2078 2020 2b20   x  - 642x x  + 
│ │ │ │ -000427c0: 3534 3178 2020 2b20 2020 207c 0a7c 2020  541x  +    |.|  
│ │ │ │ -000427d0: 2020 2037 2038 2020 2020 2020 2020 3820     7 8        8 
│ │ │ │ -000427e0: 2020 2020 2020 2030 2020 2020 2030 2031         0     0 1
│ │ │ │ -000427f0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00042800: 3020 3220 2020 2020 2020 3120 3220 2020  0 2       1 2   
│ │ │ │ -00042810: 2020 2020 3220 2020 2020 207c 0a7c 2d2d      2      |.|--
│ │ │ │ -00042820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042770: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00042780: 207c 0a7c 3135 3338 7820 7820 202b 2031   |.|1538x x  + 1
│ │ │ │ +00042790: 3233 3478 202c 202d 2035 3039 7820 202b  234x , - 509x  +
│ │ │ │ +000427a0: 2039 7820 7820 202d 2031 3536 3378 2020   9x x  - 1563x  
│ │ │ │ +000427b0: 2d20 3731 3078 2078 2020 2d20 3634 3278  - 710x x  - 642x
│ │ │ │ +000427c0: 2078 2020 2b20 3534 3178 2020 2b20 2020   x  + 541x  +   
│ │ │ │ +000427d0: 207c 0a7c 2020 2020 2037 2038 2020 2020   |.|     7 8    
│ │ │ │ +000427e0: 2020 2020 3820 2020 2020 2020 2030 2020      8        0  
│ │ │ │ +000427f0: 2020 2030 2031 2020 2020 2020 2020 3120     0 1        1 
│ │ │ │ +00042800: 2020 2020 2020 3020 3220 2020 2020 2020        0 2       
│ │ │ │ +00042810: 3120 3220 2020 2020 2020 3220 2020 2020  1 2       2     
│ │ │ │ +00042820: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00042830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042870: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042870: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00042880: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042890: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00042890: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  000428a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000428b0: 2020 2020 2020 2020 2020 207c 0a7c 3232             |.|22
│ │ │ │ -000428c0: 3078 2078 2020 2d20 3132 3134 7820 7820  0x x  - 1214x x 
│ │ │ │ -000428d0: 202d 2031 3678 2078 2020 2b20 3130 3038   - 16x x  + 1008
│ │ │ │ -000428e0: 7820 202d 2031 3038 3878 2078 2020 2b20  x  - 1088x x  + 
│ │ │ │ -000428f0: 3735 3578 2078 2020 2d20 3838 3678 2078  755x x  - 886x x
│ │ │ │ -00042900: 2020 2d20 2020 2020 2020 207c 0a7c 2020    -        |.|  
│ │ │ │ -00042910: 2020 3020 3320 2020 2020 2020 2031 2033    0 3        1 3
│ │ │ │ -00042920: 2020 2020 2020 3220 3320 2020 2020 2020        2 3       
│ │ │ │ -00042930: 2033 2020 2020 2020 2020 3020 3420 2020   3        0 4   
│ │ │ │ -00042940: 2020 2020 3120 3420 2020 2020 2020 3220      1 4       2 
│ │ │ │ -00042950: 3420 2020 2020 2020 2020 207c 0a7c 2d2d  4          |.|--
│ │ │ │ -00042960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000428b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000428c0: 207c 0a7c 3232 3078 2078 2020 2d20 3132   |.|220x x  - 12
│ │ │ │ +000428d0: 3134 7820 7820 202d 2031 3678 2078 2020  14x x  - 16x x  
│ │ │ │ +000428e0: 2b20 3130 3038 7820 202d 2031 3038 3878  + 1008x  - 1088x
│ │ │ │ +000428f0: 2078 2020 2b20 3735 3578 2078 2020 2d20   x  + 755x x  - 
│ │ │ │ +00042900: 3838 3678 2078 2020 2d20 2020 2020 2020  886x x  -       
│ │ │ │ +00042910: 207c 0a7c 2020 2020 3020 3320 2020 2020   |.|    0 3     
│ │ │ │ +00042920: 2020 2031 2033 2020 2020 2020 3220 3320     1 3      2 3 
│ │ │ │ +00042930: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00042940: 3020 3420 2020 2020 2020 3120 3420 2020  0 4       1 4   
│ │ │ │ +00042950: 2020 2020 3220 3420 2020 2020 2020 2020      2 4         
│ │ │ │ +00042960: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00042970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000429a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000429b0: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -000429c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000429a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000429b0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +000429c0: 2020 2020 3220 2020 2020 2020 2020 2020      2           
│ │ │ │  000429d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000429e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000429f0: 2020 2020 2020 2020 2020 207c 0a7c 3134             |.|14
│ │ │ │ -00042a00: 3333 7820 7820 202b 2031 3135 3478 2020  33x x  + 1154x  
│ │ │ │ -00042a10: 2b20 3136 3237 7820 7820 202d 2031 3534  + 1627x x  - 154
│ │ │ │ -00042a20: 3778 2078 2020 2d20 3935 3178 2078 2020  7x x  - 951x x  
│ │ │ │ -00042a30: 2b20 3836 3678 2078 2020 2b20 3136 3378  + 866x x  + 163x
│ │ │ │ -00042a40: 2078 2020 2d20 2020 2020 207c 0a7c 2020   x  -      |.|  
│ │ │ │ -00042a50: 2020 2033 2034 2020 2020 2020 2020 3420     3 4        4 
│ │ │ │ -00042a60: 2020 2020 2020 2030 2035 2020 2020 2020         0 5      
│ │ │ │ -00042a70: 2020 3120 3520 2020 2020 2020 3220 3520    1 5       2 5 
│ │ │ │ -00042a80: 2020 2020 2020 3320 3520 2020 2020 2020        3 5       
│ │ │ │ -00042a90: 3420 3520 2020 2020 2020 207c 0a7c 2d2d  4 5        |.|--
│ │ │ │ -00042aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000429f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042a00: 207c 0a7c 3134 3333 7820 7820 202b 2031   |.|1433x x  + 1
│ │ │ │ +00042a10: 3135 3478 2020 2b20 3136 3237 7820 7820  154x  + 1627x x 
│ │ │ │ +00042a20: 202d 2031 3534 3778 2078 2020 2d20 3935   - 1547x x  - 95
│ │ │ │ +00042a30: 3178 2078 2020 2b20 3836 3678 2078 2020  1x x  + 866x x  
│ │ │ │ +00042a40: 2b20 3136 3378 2078 2020 2d20 2020 2020  + 163x x  -     
│ │ │ │ +00042a50: 207c 0a7c 2020 2020 2033 2034 2020 2020   |.|     3 4    
│ │ │ │ +00042a60: 2020 2020 3420 2020 2020 2020 2030 2035      4        0 5
│ │ │ │ +00042a70: 2020 2020 2020 2020 3120 3520 2020 2020          1 5     
│ │ │ │ +00042a80: 2020 3220 3520 2020 2020 2020 3320 3520    2 5       3 5 
│ │ │ │ +00042a90: 2020 2020 2020 3420 3520 2020 2020 2020        4 5       
│ │ │ │ +00042aa0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00042ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042af0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00042ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042af0: 2d7c 0a7c 2020 2020 2032 2020 2020 2020  -|.|     2      
│ │ │ │  00042b00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042b10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042b20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042b30: 2020 2020 2020 2020 2020 327c 0a7c 3131            2|.|11
│ │ │ │ -00042b40: 3432 7820 202d 2036 3638 7820 7820 202b  42x  - 668x x  +
│ │ │ │ -00042b50: 2031 3336 3178 2078 2020 2b20 3133 3234   1361x x  + 1324
│ │ │ │ -00042b60: 7820 7820 202d 2034 3930 7820 7820 202b  x x  - 490x x  +
│ │ │ │ -00042b70: 2032 3832 7820 7820 202d 2031 3133 3378   282x x  - 1133x
│ │ │ │ -00042b80: 2078 2020 2d20 3631 3278 207c 0a7c 2020   x  - 612x |.|  
│ │ │ │ -00042b90: 2020 2035 2020 2020 2020 2030 2036 2020     5       0 6  
│ │ │ │ -00042ba0: 2020 2020 2020 3120 3620 2020 2020 2020        1 6       
│ │ │ │ -00042bb0: 2032 2036 2020 2020 2020 2033 2036 2020   2 6       3 6  
│ │ │ │ -00042bc0: 2020 2020 2034 2036 2020 2020 2020 2020       4 6        
│ │ │ │ -00042bd0: 3520 3620 2020 2020 2020 367c 0a7c 2d2d  5 6       6|.|--
│ │ │ │ -00042be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042b40: 327c 0a7c 3131 3432 7820 202d 2036 3638  2|.|1142x  - 668
│ │ │ │ +00042b50: 7820 7820 202b 2031 3336 3178 2078 2020  x x  + 1361x x  
│ │ │ │ +00042b60: 2b20 3133 3234 7820 7820 202d 2034 3930  + 1324x x  - 490
│ │ │ │ +00042b70: 7820 7820 202b 2032 3832 7820 7820 202d  x x  + 282x x  -
│ │ │ │ +00042b80: 2031 3133 3378 2078 2020 2d20 3631 3278   1133x x  - 612x
│ │ │ │ +00042b90: 207c 0a7c 2020 2020 2035 2020 2020 2020   |.|     5      
│ │ │ │ +00042ba0: 2030 2036 2020 2020 2020 2020 3120 3620   0 6        1 6 
│ │ │ │ +00042bb0: 2020 2020 2020 2032 2036 2020 2020 2020         2 6      
│ │ │ │ +00042bc0: 2033 2036 2020 2020 2020 2034 2036 2020   3 6       4 6  
│ │ │ │ +00042bd0: 2020 2020 2020 3520 3620 2020 2020 2020        5 6       
│ │ │ │ +00042be0: 367c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  6|.|------------
│ │ │ │  00042bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042c30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042c30: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00042c40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042c50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042c60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042c70: 2020 2020 2020 2020 2020 207c 0a7c 2b20             |.|+ 
│ │ │ │ -00042c80: 3830 3578 2078 2020 2d20 3132 3678 2078  805x x  - 126x x
│ │ │ │ -00042c90: 2020 2b20 3132 3936 7820 7820 202d 2039    + 1296x x  - 9
│ │ │ │ -00042ca0: 3733 7820 7820 202b 2031 3237 3178 2078  73x x  + 1271x x
│ │ │ │ -00042cb0: 2020 2d20 3136 3436 7820 7820 202b 2038    - 1646x x  + 8
│ │ │ │ -00042cc0: 3434 7820 7820 202b 2020 207c 0a7c 2020  44x x  +   |.|  
│ │ │ │ -00042cd0: 2020 2020 3020 3720 2020 2020 2020 3120      0 7       1 
│ │ │ │ -00042ce0: 3720 2020 2020 2020 2032 2037 2020 2020  7        2 7    
│ │ │ │ -00042cf0: 2020 2033 2037 2020 2020 2020 2020 3420     3 7        4 
│ │ │ │ -00042d00: 3720 2020 2020 2020 2035 2037 2020 2020  7        5 7    
│ │ │ │ -00042d10: 2020 2036 2037 2020 2020 207c 0a7c 2d2d     6 7     |.|--
│ │ │ │ -00042d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042c70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042c80: 207c 0a7c 2b20 3830 3578 2078 2020 2d20   |.|+ 805x x  - 
│ │ │ │ +00042c90: 3132 3678 2078 2020 2b20 3132 3936 7820  126x x  + 1296x 
│ │ │ │ +00042ca0: 7820 202d 2039 3733 7820 7820 202b 2031  x  - 973x x  + 1
│ │ │ │ +00042cb0: 3237 3178 2078 2020 2d20 3136 3436 7820  271x x  - 1646x 
│ │ │ │ +00042cc0: 7820 202b 2038 3434 7820 7820 202b 2020  x  + 844x x  +  
│ │ │ │ +00042cd0: 207c 0a7c 2020 2020 2020 3020 3720 2020   |.|      0 7   
│ │ │ │ +00042ce0: 2020 2020 3120 3720 2020 2020 2020 2032      1 7        2
│ │ │ │ +00042cf0: 2037 2020 2020 2020 2033 2037 2020 2020   7       3 7    
│ │ │ │ +00042d00: 2020 2020 3420 3720 2020 2020 2020 2035      4 7        5
│ │ │ │ +00042d10: 2037 2020 2020 2020 2036 2037 2020 2020   7       6 7    
│ │ │ │ +00042d20: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00042d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042d70: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │ +00042d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042d70: 2d7c 0a7c 2020 2020 2032 2020 2020 2020  -|.|     2      
│ │ │ │  00042d80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042d90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00042da0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042db0: 2020 2020 2020 2020 2020 207c 0a7c 3130             |.|10
│ │ │ │ -00042dc0: 3733 7820 202d 2031 3435 3278 2078 2020  73x  - 1452x x  
│ │ │ │ -00042dd0: 2d20 3131 3132 7820 7820 202d 2031 3431  - 1112x x  - 141
│ │ │ │ -00042de0: 7820 7820 202b 2031 3736 7820 7820 202d  x x  + 176x x  -
│ │ │ │ -00042df0: 2031 3537 3978 2078 2020 2d20 3738 7820   1579x x  - 78x 
│ │ │ │ -00042e00: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -00042e10: 2020 2037 2020 2020 2020 2020 3020 3820     7        0 8 
│ │ │ │ -00042e20: 2020 2020 2020 2031 2038 2020 2020 2020         1 8      
│ │ │ │ -00042e30: 2032 2038 2020 2020 2020 2033 2038 2020   2 8       3 8  
│ │ │ │ -00042e40: 2020 2020 2020 3420 3820 2020 2020 2035        4 8      5
│ │ │ │ -00042e50: 2038 2020 2020 2020 2020 207c 0a7c 2d2d   8         |.|--
│ │ │ │ -00042e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042db0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042dc0: 207c 0a7c 3130 3733 7820 202d 2031 3435   |.|1073x  - 145
│ │ │ │ +00042dd0: 3278 2078 2020 2d20 3131 3132 7820 7820  2x x  - 1112x x 
│ │ │ │ +00042de0: 202d 2031 3431 7820 7820 202b 2031 3736   - 141x x  + 176
│ │ │ │ +00042df0: 7820 7820 202d 2031 3537 3978 2078 2020  x x  - 1579x x  
│ │ │ │ +00042e00: 2d20 3738 7820 7820 202b 2020 2020 2020  - 78x x  +      
│ │ │ │ +00042e10: 207c 0a7c 2020 2020 2037 2020 2020 2020   |.|     7      
│ │ │ │ +00042e20: 2020 3020 3820 2020 2020 2020 2031 2038    0 8        1 8
│ │ │ │ +00042e30: 2020 2020 2020 2032 2038 2020 2020 2020         2 8      
│ │ │ │ +00042e40: 2033 2038 2020 2020 2020 2020 3420 3820   3 8        4 8 
│ │ │ │ +00042e50: 2020 2020 2035 2038 2020 2020 2020 2020       5 8        
│ │ │ │ +00042e60: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00042e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042eb0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ec0: 2020 2020 2020 2032 2020 2032 2020 2020         2   2    
│ │ │ │ -00042ed0: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ -00042ee0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00042ef0: 2020 2020 2020 2020 2020 207c 0a7c 3834             |.|84
│ │ │ │ -00042f00: 3878 2078 2020 2d20 3133 3635 7820 7820  8x x  - 1365x x 
│ │ │ │ -00042f10: 202b 2037 3131 7820 2c20 7820 202b 2031   + 711x , x  + 1
│ │ │ │ -00042f20: 3534 3378 2078 2020 2d20 3130 3736 7820  543x x  - 1076x 
│ │ │ │ -00042f30: 202b 2034 3933 7820 7820 202d 2035 3236   + 493x x  - 526
│ │ │ │ -00042f40: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -00042f50: 2020 3620 3820 2020 2020 2020 2037 2038    6 8        7 8
│ │ │ │ -00042f60: 2020 2020 2020 2038 2020 2030 2020 2020         8   0    
│ │ │ │ -00042f70: 2020 2020 3020 3120 2020 2020 2020 2031      0 1        1
│ │ │ │ -00042f80: 2020 2020 2020 2030 2032 2020 2020 2020         0 2      
│ │ │ │ -00042f90: 2031 2032 2020 2020 2020 207c 0a7c 2d2d   1 2       |.|--
│ │ │ │ -00042fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042eb0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00042ec0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ +00042ed0: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00042ee0: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ +00042ef0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00042f00: 207c 0a7c 3834 3878 2078 2020 2d20 3133   |.|848x x  - 13
│ │ │ │ +00042f10: 3635 7820 7820 202b 2037 3131 7820 2c20  65x x  + 711x , 
│ │ │ │ +00042f20: 7820 202b 2031 3534 3378 2078 2020 2d20  x  + 1543x x  - 
│ │ │ │ +00042f30: 3130 3736 7820 202b 2034 3933 7820 7820  1076x  + 493x x 
│ │ │ │ +00042f40: 202d 2035 3236 7820 7820 202b 2020 2020   - 526x x  +    
│ │ │ │ +00042f50: 207c 0a7c 2020 2020 3620 3820 2020 2020   |.|    6 8     
│ │ │ │ +00042f60: 2020 2037 2038 2020 2020 2020 2038 2020     7 8       8  
│ │ │ │ +00042f70: 2030 2020 2020 2020 2020 3020 3120 2020   0        0 1   
│ │ │ │ +00042f80: 2020 2020 2031 2020 2020 2020 2030 2032       1       0 2
│ │ │ │ +00042f90: 2020 2020 2020 2031 2032 2020 2020 2020         1 2      
│ │ │ │ +00042fa0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00042fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00042fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00042fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00042ff0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00042fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00042ff0: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  00043000: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043010: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00043010: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00043020: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043030: 2020 2020 2020 2020 2020 207c 0a7c 3836             |.|86
│ │ │ │ -00043040: 3878 2020 2d20 3538 3278 2078 2020 2d20  8x  - 582x x  - 
│ │ │ │ -00043050: 3939 3678 2078 2020 2b20 3230 3678 2078  996x x  + 206x x
│ │ │ │ -00043060: 2020 2d20 3431 3978 2020 2b20 3132 3538    - 419x  + 1258
│ │ │ │ -00043070: 7820 7820 202d 2033 3931 7820 7820 202b  x x  - 391x x  +
│ │ │ │ -00043080: 2031 3030 3278 2078 2020 2d7c 0a7c 2020   1002x x  -|.|  
│ │ │ │ -00043090: 2020 3220 2020 2020 2020 3020 3320 2020    2       0 3   
│ │ │ │ -000430a0: 2020 2020 3120 3320 2020 2020 2020 3220      1 3       2 
│ │ │ │ -000430b0: 3320 2020 2020 2020 3320 2020 2020 2020  3       3       
│ │ │ │ -000430c0: 2030 2034 2020 2020 2020 2031 2034 2020   0 4       1 4  
│ │ │ │ -000430d0: 2020 2020 2020 3220 3420 207c 0a7c 2d2d        2 4  |.|--
│ │ │ │ -000430e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043030: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043040: 207c 0a7c 3836 3878 2020 2d20 3538 3278   |.|868x  - 582x
│ │ │ │ +00043050: 2078 2020 2d20 3939 3678 2078 2020 2b20   x  - 996x x  + 
│ │ │ │ +00043060: 3230 3678 2078 2020 2d20 3431 3978 2020  206x x  - 419x  
│ │ │ │ +00043070: 2b20 3132 3538 7820 7820 202d 2033 3931  + 1258x x  - 391
│ │ │ │ +00043080: 7820 7820 202b 2031 3030 3278 2078 2020  x x  + 1002x x  
│ │ │ │ +00043090: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │ +000430a0: 3020 3320 2020 2020 2020 3120 3320 2020  0 3       1 3   
│ │ │ │ +000430b0: 2020 2020 3220 3320 2020 2020 2020 3320      2 3       3 
│ │ │ │ +000430c0: 2020 2020 2020 2030 2034 2020 2020 2020         0 4      
│ │ │ │ +000430d0: 2031 2034 2020 2020 2020 2020 3220 3420   1 4        2 4 
│ │ │ │ +000430e0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000430f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00043130: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -00043140: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043130: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00043140: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  00043150: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043160: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043170: 2020 2020 2020 2020 2032 207c 0a7c 3135           2 |.|15
│ │ │ │ -00043180: 3339 7820 7820 202b 2039 3331 7820 202d  39x x  + 931x  -
│ │ │ │ -00043190: 2031 3530 3478 2078 2020 2b20 3831 3078   1504x x  + 810x
│ │ │ │ -000431a0: 2078 2020 2b20 3332 3478 2078 2020 2b20   x  + 324x x  + 
│ │ │ │ -000431b0: 3133 3536 7820 7820 202b 2033 3133 7820  1356x x  + 313x 
│ │ │ │ -000431c0: 7820 202b 2037 3732 7820 207c 0a7c 2020  x  + 772x  |.|  
│ │ │ │ -000431d0: 2020 2033 2034 2020 2020 2020 2034 2020     3 4       4  
│ │ │ │ -000431e0: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ -000431f0: 3120 3520 2020 2020 2020 3220 3520 2020  1 5       2 5   
│ │ │ │ -00043200: 2020 2020 2033 2035 2020 2020 2020 2034       3 5       4
│ │ │ │ -00043210: 2035 2020 2020 2020 2035 207c 0a7c 2d2d   5       5 |.|--
│ │ │ │ -00043220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043170: 2020 2020 2020 2020 2020 2020 2020 2032                 2
│ │ │ │ +00043180: 207c 0a7c 3135 3339 7820 7820 202b 2039   |.|1539x x  + 9
│ │ │ │ +00043190: 3331 7820 202d 2031 3530 3478 2078 2020  31x  - 1504x x  
│ │ │ │ +000431a0: 2b20 3831 3078 2078 2020 2b20 3332 3478  + 810x x  + 324x
│ │ │ │ +000431b0: 2078 2020 2b20 3133 3536 7820 7820 202b   x  + 1356x x  +
│ │ │ │ +000431c0: 2033 3133 7820 7820 202b 2037 3732 7820   313x x  + 772x 
│ │ │ │ +000431d0: 207c 0a7c 2020 2020 2033 2034 2020 2020   |.|     3 4    
│ │ │ │ +000431e0: 2020 2034 2020 2020 2020 2020 3020 3520     4        0 5 
│ │ │ │ +000431f0: 2020 2020 2020 3120 3520 2020 2020 2020        1 5       
│ │ │ │ +00043200: 3220 3520 2020 2020 2020 2033 2035 2020  2 5        3 5  
│ │ │ │ +00043210: 2020 2020 2034 2035 2020 2020 2020 2035       4 5       5
│ │ │ │ +00043220: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00043230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00043270: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043270: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00043280: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043290: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000432a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000432b0: 2032 2020 2020 2020 2020 207c 0a7c 2b20   2         |.|+ 
│ │ │ │ -000432c0: 3239 3978 2078 2020 2b20 3131 3836 7820  299x x  + 1186x 
│ │ │ │ -000432d0: 7820 202b 2037 3138 7820 7820 202b 2034  x  + 718x x  + 4
│ │ │ │ -000432e0: 3037 7820 7820 202d 2036 3478 2078 2020  07x x  - 64x x  
│ │ │ │ -000432f0: 2d20 3832 3878 2078 2020 2d20 3133 3933  - 828x x  - 1393
│ │ │ │ -00043300: 7820 202b 2039 3478 2078 207c 0a7c 2020  x  + 94x x |.|  
│ │ │ │ -00043310: 2020 2020 3020 3620 2020 2020 2020 2031      0 6        1
│ │ │ │ -00043320: 2036 2020 2020 2020 2032 2036 2020 2020   6       2 6    
│ │ │ │ -00043330: 2020 2033 2036 2020 2020 2020 3420 3620     3 6      4 6 
│ │ │ │ -00043340: 2020 2020 2020 3520 3620 2020 2020 2020        5 6       
│ │ │ │ -00043350: 2036 2020 2020 2020 3020 377c 0a7c 2d2d   6      0 7|.|--
│ │ │ │ -00043360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000432b0: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │ +000432c0: 207c 0a7c 2b20 3239 3978 2078 2020 2b20   |.|+ 299x x  + 
│ │ │ │ +000432d0: 3131 3836 7820 7820 202b 2037 3138 7820  1186x x  + 718x 
│ │ │ │ +000432e0: 7820 202b 2034 3037 7820 7820 202d 2036  x  + 407x x  - 6
│ │ │ │ +000432f0: 3478 2078 2020 2d20 3832 3878 2078 2020  4x x  - 828x x  
│ │ │ │ +00043300: 2d20 3133 3933 7820 202b 2039 3478 2078  - 1393x  + 94x x
│ │ │ │ +00043310: 207c 0a7c 2020 2020 2020 3020 3620 2020   |.|      0 6   
│ │ │ │ +00043320: 2020 2020 2031 2036 2020 2020 2020 2032       1 6       2
│ │ │ │ +00043330: 2036 2020 2020 2020 2033 2036 2020 2020   6       3 6    
│ │ │ │ +00043340: 2020 3420 3620 2020 2020 2020 3520 3620    4 6       5 6 
│ │ │ │ +00043350: 2020 2020 2020 2036 2020 2020 2020 3020         6      0 
│ │ │ │ +00043360: 377c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  7|.|------------
│ │ │ │  00043370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000433a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000433b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000433a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000433b0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  000433c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000433d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000433e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000433f0: 2020 3220 2020 2020 2020 207c 0a7c 2d20    2        |.|- 
│ │ │ │ -00043400: 3239 3078 2078 2020 2d20 3736 3678 2078  290x x  - 766x x
│ │ │ │ -00043410: 2020 2b20 3935 3078 2078 2020 2d20 3634    + 950x x  - 64
│ │ │ │ -00043420: 3078 2078 2020 2b20 3236 3578 2078 2020  0x x  + 265x x  
│ │ │ │ -00043430: 2d20 3136 3430 7820 7820 202d 2031 3430  - 1640x x  - 140
│ │ │ │ -00043440: 3378 2020 2d20 2020 2020 207c 0a7c 2020  3x  -      |.|  
│ │ │ │ -00043450: 2020 2020 3120 3720 2020 2020 2020 3220      1 7       2 
│ │ │ │ -00043460: 3720 2020 2020 2020 3320 3720 2020 2020  7       3 7     
│ │ │ │ -00043470: 2020 3420 3720 2020 2020 2020 3520 3720    4 7       5 7 
│ │ │ │ -00043480: 2020 2020 2020 2036 2037 2020 2020 2020         6 7      
│ │ │ │ -00043490: 2020 3720 2020 2020 2020 207c 0a7c 2d2d    7        |.|--
│ │ │ │ -000434a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000433f0: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00043400: 207c 0a7c 2d20 3239 3078 2078 2020 2d20   |.|- 290x x  - 
│ │ │ │ +00043410: 3736 3678 2078 2020 2b20 3935 3078 2078  766x x  + 950x x
│ │ │ │ +00043420: 2020 2d20 3634 3078 2078 2020 2b20 3236    - 640x x  + 26
│ │ │ │ +00043430: 3578 2078 2020 2d20 3136 3430 7820 7820  5x x  - 1640x x 
│ │ │ │ +00043440: 202d 2031 3430 3378 2020 2d20 2020 2020   - 1403x  -     
│ │ │ │ +00043450: 207c 0a7c 2020 2020 2020 3120 3720 2020   |.|      1 7   
│ │ │ │ +00043460: 2020 2020 3220 3720 2020 2020 2020 3320      2 7       3 
│ │ │ │ +00043470: 3720 2020 2020 2020 3420 3720 2020 2020  7       4 7     
│ │ │ │ +00043480: 2020 3520 3720 2020 2020 2020 2036 2037    5 7        6 7
│ │ │ │ +00043490: 2020 2020 2020 2020 3720 2020 2020 2020          7       
│ │ │ │ +000434a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000434b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000434c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000434d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000434e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000434f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000434e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000434f0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00043500: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043510: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043520: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043530: 2020 2020 2020 2020 2020 207c 0a7c 3132             |.|12
│ │ │ │ -00043540: 3678 2078 2020 2b20 3839 3178 2078 2020  6x x  + 891x x  
│ │ │ │ -00043550: 2d20 3135 3139 7820 7820 202d 2039 3237  - 1519x x  - 927
│ │ │ │ -00043560: 7820 7820 202d 2031 3333 3578 2078 2020  x x  - 1335x x  
│ │ │ │ -00043570: 2d20 3134 3438 7820 7820 202d 2078 2078  - 1448x x  - x x
│ │ │ │ -00043580: 2020 2d20 2020 2020 2020 207c 0a7c 2020    -        |.|  
│ │ │ │ -00043590: 2020 3020 3820 2020 2020 2020 3120 3820    0 8       1 8 
│ │ │ │ -000435a0: 2020 2020 2020 2032 2038 2020 2020 2020         2 8      
│ │ │ │ -000435b0: 2033 2038 2020 2020 2020 2020 3420 3820   3 8        4 8 
│ │ │ │ -000435c0: 2020 2020 2020 2035 2038 2020 2020 3620         5 8    6 
│ │ │ │ -000435d0: 3820 2020 2020 2020 2020 207c 0a7c 2d2d  8          |.|--
│ │ │ │ -000435e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043530: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043540: 207c 0a7c 3132 3678 2078 2020 2b20 3839   |.|126x x  + 89
│ │ │ │ +00043550: 3178 2078 2020 2d20 3135 3139 7820 7820  1x x  - 1519x x 
│ │ │ │ +00043560: 202d 2039 3237 7820 7820 202d 2031 3333   - 927x x  - 133
│ │ │ │ +00043570: 3578 2078 2020 2d20 3134 3438 7820 7820  5x x  - 1448x x 
│ │ │ │ +00043580: 202d 2078 2078 2020 2d20 2020 2020 2020   - x x  -       
│ │ │ │ +00043590: 207c 0a7c 2020 2020 3020 3820 2020 2020   |.|    0 8     
│ │ │ │ +000435a0: 2020 3120 3820 2020 2020 2020 2032 2038    1 8        2 8
│ │ │ │ +000435b0: 2020 2020 2020 2033 2038 2020 2020 2020         3 8      
│ │ │ │ +000435c0: 2020 3420 3820 2020 2020 2020 2035 2038    4 8        5 8
│ │ │ │ +000435d0: 2020 2020 3620 3820 2020 2020 2020 2020      6 8         
│ │ │ │ +000435e0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000435f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00043630: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │ -00043640: 2020 2020 2032 2020 2020 2020 2020 2020       2          
│ │ │ │ -00043650: 2020 2020 2020 2020 3220 2020 2020 2020          2       
│ │ │ │ +00043620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043630: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +00043640: 2020 2020 3220 2020 2020 2032 2020 2020      2      2    
│ │ │ │ +00043650: 2020 2020 2020 2020 2020 2020 2020 3220                2 
│ │ │ │  00043660: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043670: 2020 2020 3220 2020 2020 207c 0a7c 3131      2      |.|11
│ │ │ │ -00043680: 3033 7820 7820 202d 2031 3135 3278 202c  03x x  - 1152x ,
│ │ │ │ -00043690: 2038 3231 7820 202b 2035 3538 7820 7820   821x  + 558x x 
│ │ │ │ -000436a0: 202d 2031 3137 3478 2020 2d20 3136 3878   - 1174x  - 168x
│ │ │ │ -000436b0: 2078 2020 2b20 3938 3678 2078 2020 2b20   x  + 986x x  + 
│ │ │ │ -000436c0: 3739 3078 2020 2b20 2020 207c 0a7c 2020  790x  +    |.|  
│ │ │ │ -000436d0: 2020 2037 2038 2020 2020 2020 2020 3820     7 8        8 
│ │ │ │ -000436e0: 2020 2020 2030 2020 2020 2020 2030 2031       0       0 1
│ │ │ │ -000436f0: 2020 2020 2020 2020 3120 2020 2020 2020          1       
│ │ │ │ -00043700: 3020 3220 2020 2020 2020 3120 3220 2020  0 2       1 2   
│ │ │ │ -00043710: 2020 2020 3220 2020 2020 207c 0a7c 2d2d      2      |.|--
│ │ │ │ -00043720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043670: 2020 2020 2020 2020 2020 3220 2020 2020            2     
│ │ │ │ +00043680: 207c 0a7c 3131 3033 7820 7820 202d 2031   |.|1103x x  - 1
│ │ │ │ +00043690: 3135 3278 202c 2038 3231 7820 202b 2035  152x , 821x  + 5
│ │ │ │ +000436a0: 3538 7820 7820 202d 2031 3137 3478 2020  58x x  - 1174x  
│ │ │ │ +000436b0: 2d20 3136 3878 2078 2020 2b20 3938 3678  - 168x x  + 986x
│ │ │ │ +000436c0: 2078 2020 2b20 3739 3078 2020 2b20 2020   x  + 790x  +   
│ │ │ │ +000436d0: 207c 0a7c 2020 2020 2037 2038 2020 2020   |.|     7 8    
│ │ │ │ +000436e0: 2020 2020 3820 2020 2020 2030 2020 2020      8      0    
│ │ │ │ +000436f0: 2020 2030 2031 2020 2020 2020 2020 3120     0 1        1 
│ │ │ │ +00043700: 2020 2020 2020 3020 3220 2020 2020 2020        0 2       
│ │ │ │ +00043710: 3120 3220 2020 2020 2020 3220 2020 2020  1 2       2     
│ │ │ │ +00043720: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00043730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00043770: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043770: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00043780: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043790: 2032 2020 2020 2020 2020 2020 2020 2020   2              
│ │ │ │ +00043790: 2020 2020 2020 2032 2020 2020 2020 2020         2        
│ │ │ │  000437a0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000437b0: 2020 2020 2020 2020 2020 207c 0a7c 3534             |.|54
│ │ │ │ -000437c0: 3978 2078 2020 2b20 3831 3778 2078 2020  9x x  + 817x x  
│ │ │ │ -000437d0: 2b20 3133 3936 7820 7820 202b 2036 3935  + 1396x x  + 695
│ │ │ │ -000437e0: 7820 202b 2031 3231 3178 2078 2020 2b20  x  + 1211x x  + 
│ │ │ │ -000437f0: 3837 3878 2078 2020 2d20 3130 3631 7820  878x x  - 1061x 
│ │ │ │ -00043800: 7820 202d 2020 2020 2020 207c 0a7c 2020  x  -       |.|  
│ │ │ │ -00043810: 2020 3020 3320 2020 2020 2020 3120 3320    0 3       1 3 
│ │ │ │ -00043820: 2020 2020 2020 2032 2033 2020 2020 2020         2 3      
│ │ │ │ -00043830: 2033 2020 2020 2020 2020 3020 3420 2020   3        0 4   
│ │ │ │ -00043840: 2020 2020 3120 3420 2020 2020 2020 2032      1 4        2
│ │ │ │ -00043850: 2034 2020 2020 2020 2020 207c 0a7c 2d2d   4         |.|--
│ │ │ │ -00043860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000437b0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000437c0: 207c 0a7c 3534 3978 2078 2020 2b20 3831   |.|549x x  + 81
│ │ │ │ +000437d0: 3778 2078 2020 2b20 3133 3936 7820 7820  7x x  + 1396x x 
│ │ │ │ +000437e0: 202b 2036 3935 7820 202b 2031 3231 3178   + 695x  + 1211x
│ │ │ │ +000437f0: 2078 2020 2b20 3837 3878 2078 2020 2d20   x  + 878x x  - 
│ │ │ │ +00043800: 3130 3631 7820 7820 202d 2020 2020 2020  1061x x  -      
│ │ │ │ +00043810: 207c 0a7c 2020 2020 3020 3320 2020 2020   |.|    0 3     
│ │ │ │ +00043820: 2020 3120 3320 2020 2020 2020 2032 2033    1 3        2 3
│ │ │ │ +00043830: 2020 2020 2020 2033 2020 2020 2020 2020         3        
│ │ │ │ +00043840: 3020 3420 2020 2020 2020 3120 3420 2020  0 4       1 4   
│ │ │ │ +00043850: 2020 2020 2032 2034 2020 2020 2020 2020       2 4        
│ │ │ │ +00043860: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00043870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000438a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000438b0: 2020 2020 2020 2020 2020 2020 2032 2020               2  
│ │ │ │ -000438c0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +000438a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000438b0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │ +000438c0: 2020 2032 2020 2020 2020 2020 2020 2020     2            
│ │ │ │  000438d0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  000438e0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -000438f0: 2020 2020 2020 2020 2020 207c 0a7c 3132             |.|12
│ │ │ │ -00043900: 3434 7820 7820 202d 2038 3830 7820 202b  44x x  - 880x  +
│ │ │ │ -00043910: 2031 3430 3978 2078 2020 2d20 3536 3778   1409x x  - 567x
│ │ │ │ -00043920: 2078 2020 2b20 3132 3430 7820 7820 202b   x  + 1240x x  +
│ │ │ │ -00043930: 2031 3132 3678 2078 2020 2d20 3132 3632   1126x x  - 1262
│ │ │ │ -00043940: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -00043950: 2020 2033 2034 2020 2020 2020 2034 2020     3 4       4  
│ │ │ │ -00043960: 2020 2020 2020 3020 3520 2020 2020 2020        0 5       
│ │ │ │ -00043970: 3120 3520 2020 2020 2020 2032 2035 2020  1 5        2 5  
│ │ │ │ -00043980: 2020 2020 2020 3320 3520 2020 2020 2020        3 5       
│ │ │ │ -00043990: 2034 2035 2020 2020 2020 207c 0a7c 2d2d   4 5       |.|--
│ │ │ │ -000439a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000438f0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043900: 207c 0a7c 3132 3434 7820 7820 202d 2038   |.|1244x x  - 8
│ │ │ │ +00043910: 3830 7820 202b 2031 3430 3978 2078 2020  80x  + 1409x x  
│ │ │ │ +00043920: 2d20 3536 3778 2078 2020 2b20 3132 3430  - 567x x  + 1240
│ │ │ │ +00043930: 7820 7820 202b 2031 3132 3678 2078 2020  x x  + 1126x x  
│ │ │ │ +00043940: 2d20 3132 3632 7820 7820 202b 2020 2020  - 1262x x  +    
│ │ │ │ +00043950: 207c 0a7c 2020 2020 2033 2034 2020 2020   |.|     3 4    
│ │ │ │ +00043960: 2020 2034 2020 2020 2020 2020 3020 3520     4        0 5 
│ │ │ │ +00043970: 2020 2020 2020 3120 3520 2020 2020 2020        1 5       
│ │ │ │ +00043980: 2032 2035 2020 2020 2020 2020 3320 3520   2 5        3 5 
│ │ │ │ +00043990: 2020 2020 2020 2034 2035 2020 2020 2020         4 5      
│ │ │ │ +000439a0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  000439b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000439c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  000439d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -000439e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -000439f0: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +000439e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +000439f0: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  00043a00: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043a10: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043a20: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043a30: 2020 2020 2020 2020 2020 327c 0a7c 3439            2|.|49
│ │ │ │ -00043a40: 3078 2020 2b20 3135 3533 7820 7820 202b  0x  + 1553x x  +
│ │ │ │ -00043a50: 2031 3237 3678 2078 2020 2b20 3830 3578   1276x x  + 805x
│ │ │ │ -00043a60: 2078 2020 2b20 3537 3678 2078 2020 2d20   x  + 576x x  - 
│ │ │ │ -00043a70: 3130 3736 7820 7820 202b 2031 3631 3778  1076x x  + 1617x
│ │ │ │ -00043a80: 2078 2020 2d20 3439 3578 207c 0a7c 2020   x  - 495x |.|  
│ │ │ │ -00043a90: 2020 3520 2020 2020 2020 2030 2036 2020    5        0 6  
│ │ │ │ -00043aa0: 2020 2020 2020 3120 3620 2020 2020 2020        1 6       
│ │ │ │ -00043ab0: 3220 3620 2020 2020 2020 3320 3620 2020  2 6       3 6   
│ │ │ │ -00043ac0: 2020 2020 2034 2036 2020 2020 2020 2020       4 6        
│ │ │ │ -00043ad0: 3520 3620 2020 2020 2020 367c 0a7c 2d2d  5 6       6|.|--
│ │ │ │ -00043ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043a30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043a40: 327c 0a7c 3439 3078 2020 2b20 3135 3533  2|.|490x  + 1553
│ │ │ │ +00043a50: 7820 7820 202b 2031 3237 3678 2078 2020  x x  + 1276x x  
│ │ │ │ +00043a60: 2b20 3830 3578 2078 2020 2b20 3537 3678  + 805x x  + 576x
│ │ │ │ +00043a70: 2078 2020 2d20 3130 3736 7820 7820 202b   x  - 1076x x  +
│ │ │ │ +00043a80: 2031 3631 3778 2078 2020 2d20 3439 3578   1617x x  - 495x
│ │ │ │ +00043a90: 207c 0a7c 2020 2020 3520 2020 2020 2020   |.|    5       
│ │ │ │ +00043aa0: 2030 2036 2020 2020 2020 2020 3120 3620   0 6        1 6 
│ │ │ │ +00043ab0: 2020 2020 2020 3220 3620 2020 2020 2020        2 6       
│ │ │ │ +00043ac0: 3320 3620 2020 2020 2020 2034 2036 2020  3 6        4 6  
│ │ │ │ +00043ad0: 2020 2020 2020 3520 3620 2020 2020 2020        5 6       
│ │ │ │ +00043ae0: 367c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  6|.|------------
│ │ │ │  00043af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00043b30: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043b30: 2d7c 0a7c 2020 2020 2020 2020 2020 2020  -|.|            
│ │ │ │  00043b40: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043b50: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043b60: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043b70: 2020 2020 2020 2020 2020 207c 0a7c 2d20             |.|- 
│ │ │ │ -00043b80: 3735 3078 2078 2020 2d20 3237 3778 2078  750x x  - 277x x
│ │ │ │ -00043b90: 2020 2b20 3534 3478 2078 2020 2b20 3134    + 544x x  + 14
│ │ │ │ -00043ba0: 3739 7820 7820 202d 2037 3834 7820 7820  79x x  - 784x x 
│ │ │ │ -00043bb0: 202d 2036 3478 2078 2020 2d20 3132 3033   - 64x x  - 1203
│ │ │ │ -00043bc0: 7820 7820 202b 2020 2020 207c 0a7c 2020  x x  +     |.|  
│ │ │ │ -00043bd0: 2020 2020 3020 3720 2020 2020 2020 3120      0 7       1 
│ │ │ │ -00043be0: 3720 2020 2020 2020 3220 3720 2020 2020  7       2 7     
│ │ │ │ -00043bf0: 2020 2033 2037 2020 2020 2020 2034 2037     3 7       4 7
│ │ │ │ -00043c00: 2020 2020 2020 3520 3720 2020 2020 2020        5 7       
│ │ │ │ -00043c10: 2036 2037 2020 2020 2020 207c 0a7c 2d2d   6 7       |.|--
│ │ │ │ -00043c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043b70: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ +00043b80: 207c 0a7c 2d20 3735 3078 2078 2020 2d20   |.|- 750x x  - 
│ │ │ │ +00043b90: 3237 3778 2078 2020 2b20 3534 3478 2078  277x x  + 544x x
│ │ │ │ +00043ba0: 2020 2b20 3134 3739 7820 7820 202d 2037    + 1479x x  - 7
│ │ │ │ +00043bb0: 3834 7820 7820 202d 2036 3478 2078 2020  84x x  - 64x x  
│ │ │ │ +00043bc0: 2d20 3132 3033 7820 7820 202b 2020 2020  - 1203x x  +    
│ │ │ │ +00043bd0: 207c 0a7c 2020 2020 2020 3020 3720 2020   |.|      0 7   
│ │ │ │ +00043be0: 2020 2020 3120 3720 2020 2020 2020 3220      1 7       2 
│ │ │ │ +00043bf0: 3720 2020 2020 2020 2033 2037 2020 2020  7        3 7    
│ │ │ │ +00043c00: 2020 2034 2037 2020 2020 2020 3520 3720     4 7      5 7 
│ │ │ │ +00043c10: 2020 2020 2020 2036 2037 2020 2020 2020         6 7      
│ │ │ │ +00043c20: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d   |.|------------
│ │ │ │  00043c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │  00043c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ -00043c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020  -----------|.|  
│ │ │ │ -00043c70: 2020 3220 2020 2020 2020 2020 2020 2020    2             
│ │ │ │ +00043c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d  ----------------
│ │ │ │ +00043c70: 2d7c 0a7c 2020 2020 3220 2020 2020 2020  -|.|    2       
│ │ │ │  00043c80: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043c90: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │  00043ca0: 2020 2020 2020 2020 2020 2020 2020 2020                  
│ │ │ │ -00043cb0: 2020 2020 2020 2020 2020 207c 0a7c 3430             |.|40
│ │ │ │ -00043cc0: 3578 2020 2b20 3130 3133 7820 7820 202b  5x  + 1013x x  +
│ │ │ │ -00043cd0: 2036 3034 7820 7820 202b 2031 3330 3178   604x x  + 1301x
│ │ │ │ -00043ce0: 2078 2020 2b20 3130 3033 7820 7820 202b   x  + 1003x x  +
│ │ │ │ -00043cf0: 2032 3335 7820 7820 202b 2036 3936 7820   235x x  + 696x 
│ │ │ │ -00043d00: 7820 202b 2020 2020 2020 207c 0a7c 2020  x  +       |.|  
│ │ │ │ -00043d10: 2020 3720 2020 2020 2020 2030 2038 2020    7        0 8  
│ │ │ │ -00043d20: 2020 2020 2031 2038 2020 2020 2020 2020       1 8      

TRUNCATED DUE TO SIZE LIMIT: 10485760 bytes